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Miller","url":"https://williams.academia.edu/StevenJMiller","image":"https://0.academia-photos.com/13733/4607/4518/s200_steven_j..miller.jpg","sameAs":["http://www.williams.edu/go/math/sjmiller/public_html/index.htm"]},"dateCreated":"2008-10-30T03:37:58-07:00","dateModified":"2025-03-22T16:02:46-07:00","name":"Steven J. Miller","description":"My main research interests are in number theory (especially random matrix theory, elliptic curves and additive and computational number theory), probability and statistics (especially Benford's law, linear programming and sabermetrics). I'm married with two kids, and I enjoyed sailing, reading and tennis when I had the time. \n\nI have written or refereed articles in the following fields (in addition to mathematics): accounting, biology, economics, geology, marketing, sabermetrics and statistics, and am always looking for interesting projects.\n\nI have written a book (with Ramin Takloo-Bighash), \"An Invitation to Modern Number Theory\" (Princeton University Press), whose purpose is to introduce undergraduates and beginning graduate students to modern number theory (either as a standard textbook, or through suggested research problems). 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Miller" border="0" onerror="if (this.src != '//a.academia-assets.com/images/s200_no_pic.png') this.src = '//a.academia-assets.com/images/s200_no_pic.png';" width="200" height="200" src="https://0.academia-photos.com/13733/4607/4518/s200_steven_j..miller.jpg" /></div><div class="title-container"><h1 class="ds2-5-heading-sans-serif-sm">Steven J. Miller</h1><div class="affiliations-container fake-truncate js-profile-affiliations"><div><a class="u-tcGrayDarker" href="https://williams.academia.edu/">Williams College</a>, <a class="u-tcGrayDarker" href="https://williams.academia.edu/Departments/Mathematics_and_Statistics/Documents">Mathematics and Statistics</a>, <span class="u-tcGrayDarker">Faculty Member</span></div></div></div></div><div class="sidebar-cta-container"><button class="ds2-5-button hidden profile-cta-button grow js-profile-follow-button" data-broccoli-component="user-info.follow-button" data-click-track="profile-user-info-follow-button" data-follow-user-fname="Steven J." data-follow-user-id="13733" data-follow-user-source="profile_button" data-has-google="false"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">add</span>Follow</button><button class="ds2-5-button hidden profile-cta-button grow js-profile-unfollow-button" data-broccoli-component="user-info.unfollow-button" data-click-track="profile-user-info-unfollow-button" data-unfollow-user-id="13733"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">done</span>Following</button></div></div><div class="user-stats-container"><a><div class="stat-container js-profile-followers"><p class="label">Followers</p><p class="data">65,647</p></div></a><a><div class="stat-container js-profile-followees" data-broccoli-component="user-info.followees-count" data-click-track="profile-expand-user-info-following"><p class="label">Following</p><p class="data">0</p></div></a><span><div class="stat-container"><p class="label"><span class="js-profile-total-view-text">Public Views</span></p><p class="data"><span class="js-profile-view-count"></span></p></div></span></div><div class="user-bio-container"><div class="profile-bio fake-truncate js-profile-about" style="margin: 0px;">My main research interests are in number theory (especially random matrix theory, elliptic curves and additive and computational number theory), probability and statistics (especially Benford's law, linear programming and sabermetrics). I'm married with two kids, and I enjoyed sailing, reading and tennis when I had the time. <br /><br />I have written or refereed articles in the following fields (in addition to mathematics): accounting, biology, economics, geology, marketing, sabermetrics and statistics, and am always looking for interesting projects.<br /><br />I have written a book (with Ramin Takloo-Bighash), "An Invitation to Modern Number Theory" (Princeton University Press), whose purpose is to introduce undergraduates and beginning graduate students to modern number theory (either as a standard textbook, or through suggested research problems). The book's homepage is: http://www.williams.edu/go/math/sjmiller/public_html/book/index.html<br /><span class="u-fw700">Supervisors: </span>Sarnak, Iwaniec<br /><span class="u-fw700">Phone: </span>413-597-3293<br /><b>Address: </b>Bronfman Science Center<br />Williams College<br />Williamstown, MA 01267<br /><div class="js-profile-less-about u-linkUnstyled u-tcGrayDarker u-textDecorationUnderline u-displayNone">less</div></div></div><div class="suggested-academics-container"><div class="suggested-academics--header"><h3 class="ds2-5-heading-sans-serif-xs">Related Authors</h3></div><ul class="suggested-user-card-list" data-nosnippet="true"><div class="suggested-user-card"><div class="suggested-user-card__avatar social-profile-avatar-container"><a data-nosnippet="" href="https://oxford.academia.edu/BenHambly"><img class="profile-avatar u-positionAbsolute" alt="Ben Hambly related author profile picture" border="0" onerror="if (this.src != '//a.academia-assets.com/images/s200_no_pic.png') 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id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://williams.academia.edu/StevenJMiller","location":"/StevenJMiller","scheme":"https","host":"williams.academia.edu","port":null,"pathname":"/StevenJMiller","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Mathematics"]}" data-trace="false" data-dom-id="Pill-react-component-1f934ac0-c7a4-4687-9552-0b742652ebd1"></div> <div id="Pill-react-component-1f934ac0-c7a4-4687-9552-0b742652ebd1"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="13733" href="https://www.academia.edu/Documents/in/Number_Theory"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Number Theory"]}" 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id="social-redesign-work-container"><div class="upload-header"><h2 class="ds2-5-heading-sans-serif-xs">Uploads</h2></div><div class="documents-container backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Steven J. Miller</h3></div><div class="js-work-strip profile--work_container" data-work-id="126812071"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812071/On_the_spectral_distribution_of_large_weighted_random_regular_graphs"><img alt="Research paper thumbnail of On the spectral distribution of large weighted random regular graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/120633519/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812071/On_the_spectral_distribution_of_large_weighted_random_regular_graphs">On the spectral distribution of large weighted random regular graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 28, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vert...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. In this paper we explore the case of weighted graphs. More precisely, given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique 'eigendistribution', i.e., a weight distribution W such that the associated limiting spectral distribution is a rescaling of W. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d 2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8df736694155275ae384bddd91676411" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633519,"asset_id":126812071,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633519/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812071"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812071"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812071; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812071]").text(description); $(".js-view-count[data-work-id=126812071]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812071; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812071']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8df736694155275ae384bddd91676411" } } $('.js-work-strip[data-work-id=126812071]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812071,"title":"On the spectral distribution of large weighted random regular graphs","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. In this paper we explore the case of weighted graphs. More precisely, given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique 'eigendistribution', i.e., a weight distribution W such that the associated limiting spectral distribution is a rescaling of W. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d 2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.","publication_date":{"day":28,"month":6,"year":2013,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633519},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812071/On_the_spectral_distribution_of_large_weighted_random_regular_graphs","translated_internal_url":"","created_at":"2025-01-05T02:41:59.995-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633519,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633519/thumbnails/1.jpg","file_name":"1306.pdf","download_url":"https://www.academia.edu/attachments/120633519/download_file","bulk_download_file_name":"On_the_spectral_distribution_of_large_we.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633519/1306-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_spectral_distribution_of_large_we.pdf\u0026Expires=1742779652\u0026Signature=dnKfIwMhMqn4wCreenC4wcAC1L3xvcBAShXK1dA~-wg6yOAjZ~zc94uFjFB0xELiesiPB~ODNAMOSb-WgbFTyKtrnAF4chVocTUD9ZKsL1VhcLxYUJwsrp95Lk3dbw7Ccq2jzlGiWagvluFF3RrnqvbeM~wYJENDkI59XTd6Avenkh28e7HfyHy0lnK1NitlwY9r494-jSoOaBl3u-SBywkbdoA8Q~z0APFhGlQBZ-qDM0O-IFMlQPcPyQs2F6dxUTh4CUHjdHzVGDc4-JQVIg86KStQRuvQQxex5IBfKfU059Yoj5IwsNhuUU-cIWoum8u5Bg~D5USfiJmpqzg2Xw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_spectral_distribution_of_large_weighted_random_regular_graphs","translated_slug":"","page_count":19,"language":"en","content_type":"Work","summary":"McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. In this paper we explore the case of weighted graphs. More precisely, given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique 'eigendistribution', i.e., a weight distribution W such that the associated limiting spectral distribution is a rescaling of W. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d 2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Martin and...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A more sums than differences (MSTD) set A is a subset of Z for which |A + A| > |A -A|. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of {1, . . . , n} are MSTD as n → ∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of Z. In particular we show that every finite subset of Z can be transformed into an element of I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in I consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of Z from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0eb37c874403add77abba933eba3763f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633518,"asset_id":126812070,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633518/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812070"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812070"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812070; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812070]").text(description); $(".js-view-count[data-work-id=126812070]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812070; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812070']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0eb37c874403add77abba933eba3763f" } } $('.js-work-strip[data-work-id=126812070]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812070,"title":"A geometric perspective on the MSTD question","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A more sums than differences (MSTD) set A is a subset of Z for which |A + A| \u003e |A -A|. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of {1, . . . , n} are MSTD as n → ∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of Z. In particular we show that every finite subset of Z can be transformed into an element of I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in I consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of Z from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.","publication_date":{"day":2,"month":9,"year":2017,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633518},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812070/A_geometric_perspective_on_the_MSTD_question","translated_internal_url":"","created_at":"2025-01-05T02:41:59.572-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633518,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633518/thumbnails/1.jpg","file_name":"1709.pdf","download_url":"https://www.academia.edu/attachments/120633518/download_file","bulk_download_file_name":"A_geometric_perspective_on_the_MSTD_ques.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633518/1709-libre.pdf?1736077063=\u0026response-content-disposition=attachment%3B+filename%3DA_geometric_perspective_on_the_MSTD_ques.pdf\u0026Expires=1742779652\u0026Signature=hAcb28cRJqWQ~x3ZEanWfcArnfXT9gCAGYu1XGSdZuABIlNZ3E9Ke9Z6OgX2n3Xc-5FO1HqRghPTeYQUUKExlZV9aZSlr~8N3rnJKwuNZrtXxINQC~lzLN6rv4a2p~NARSIXe1zwimTDn-hT17zBKgaapLUjnyTy~xKQ2Ik4GE7pCkMEFFKRLuNvpWPKgqWQe56G-Gk4W-WUOTFt3dfh~zjEz~kbruTaFAA3xnn5wERh7D4nTqwLF2Gz3S3n0akEGqEzf-aHQ9qdemoP5wjxd8QNyUlbGAZjYwgWYtCMIcvCenLq63-WzWHLLJOtb-qWWEYvThAVWS8N8s2Ijiw2Hw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_geometric_perspective_on_the_MSTD_question","translated_slug":"","page_count":22,"language":"en","content_type":"Work","summary":"A more sums than differences (MSTD) set A is a subset of Z for which |A + A| \u003e |A -A|. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of {1, . . . , n} are MSTD as n → ∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of Z. In particular we show that every finite subset of Z can be transformed into an element of I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in I consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of Z from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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We examine related problems from placing n rooks. We prove that as n → ∞, the probability rapidly tends to 1 that the fraction of safe squares from a random placement converges to 1/e 2 . Our interest in the problem is showing how to view the involved algebra to obtain the simple, closed form limiting fraction. In particular, we see the power of many of the key concepts in probability: binary indicator variables, linearity of expectation, variances and covariances, Chebyshev's inequality, and Stirling's formula.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9a4bf32c094fefc07346ff34d50d8dd2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633517,"asset_id":126812069,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633517/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812069"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812069"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812069; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812069]").text(description); $(".js-view-count[data-work-id=126812069]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812069; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812069']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9a4bf32c094fefc07346ff34d50d8dd2" } } $('.js-work-strip[data-work-id=126812069]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812069,"title":"When Rooks Miss: Probability through Chess","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A famous (and hard) chess problem asks what is the maximum number of safe squares possible in placing n queens on an n × n board. 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When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call the M&M Game: Given k people, each simultaneously flips a fair coin, with each eating an M&M on a head and not eating on a tail. The process then continues until all M&M'S are consumed, and two people are deemed to die at the same time if they run out of M&M'S together 1 . This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. There are many ways to determine the probability of a tie, which allow us in this article to use this problem as a springboard to a lot of great mathematics, including memoryless process, combinatorics, statistical inference, graph theory, and hypergeometric functions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="321214489d737e404219ba57977de082" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633522,"asset_id":126812068,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633522/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812068"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812068"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812068; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812068]").text(description); $(".js-view-count[data-work-id=126812068]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812068; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812068']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "321214489d737e404219ba57977de082" } } $('.js-work-strip[data-work-id=126812068]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812068,"title":"The M\u0026M Game: From Morsels to Modern Mathematics","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"The M\u0026M Game: Teaching Probability through Play","grobid_abstract":"To an adult, it's obvious that the day of someone's death is not precisely determined by the day of birth, but it's a very different story for a child. When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call the M\u0026M Game: Given k people, each simultaneously flips a fair coin, with each eating an M\u0026M on a head and not eating on a tail. The process then continues until all M\u0026M'S are consumed, and two people are deemed to die at the same time if they run out of M\u0026M'S together 1 . This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. 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When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call the M\u0026M Game: Given k people, each simultaneously flips a fair coin, with each eating an M\u0026M on a head and not eating on a tail. The process then continues until all M\u0026M'S are consumed, and two people are deemed to die at the same time if they run out of M\u0026M'S together 1 . This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. 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The Allies needed accura...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">During World War II the German army used tanks to devastating advantage. The Allies needed accurate estimates of their tank production and deployment. They used two approaches to find these values: spies, and statistics. This note describes the statistical approach. Assuming the tanks are labeled consecutively starting at 1, if we observe k serial numbers from an unknown number N of tanks, with the maximum observed value m, then the best estimate for N is m(1+1/k)-1. This is now known as the German Tank Problem, and is a terrific example of the applicability of mathematics and statistics in the real world. The first part of the paper reproduces known results, specifically deriving this estimate and comparing its effectiveness to that of the spies. The second part presents a result we have not found in print elsewhere, the generalization to the case where the smallest value is not necessarily 1. We emphasize in detail why we are able to obtain such clean, closed-form expressions for the estimates, and conclude with an appendix highlighting how to use this problem to teach regression and how statistics can help us find functional relationships.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ff49102743a773dd4af33619ed6374fc" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633521,"asset_id":126812067,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633521/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812067"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812067"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812067; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812067]").text(description); $(".js-view-count[data-work-id=126812067]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812067; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812067']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "ff49102743a773dd4af33619ed6374fc" } } $('.js-work-strip[data-work-id=126812067]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812067,"title":"Lessons from the German Tank Problem","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"During World War II the German army used tanks to devastating advantage. 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The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, [3] studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input N → ∞, the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input N involving the Catalan numbers. The works [3] and Cuzensa et al. determined how to achieve a shortest and longest possible Zeckendorf game on a given input N , respectively: we establish that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞, both players win with probability 1/2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞. We conclude the work with many open problems.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="45888380c9299a10a9fcb1ec1fc874fd" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633516,"asset_id":126812064,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633516/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812064"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812064"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812064; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812064]").text(description); $(".js-view-count[data-work-id=126812064]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812064; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812064']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "45888380c9299a10a9fcb1ec1fc874fd" } } $('.js-work-strip[data-work-id=126812064]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812064,"title":"Towards the Gaussianity of Random Zeckendorf Games","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Zeckendorf proved that any positive integer has a unique decomposition as a sum of non-consecutive Fibonacci numbers, indexed by Baird, Epstein, Flint, and Miller [3] defined the two-player Zeckendorf game, where two players take turns acting on a multiset of Fibonacci numbers that always sums to N . The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, [3] studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input N → ∞, the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input N involving the Catalan numbers. The works [3] and Cuzensa et al. determined how to achieve a shortest and longest possible Zeckendorf game on a given input N , respectively: we establish that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞, both players win with probability 1/2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞. 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We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞, both players win with probability 1/2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞. We conclude the work with many open problems.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i th and j th squares are adjacent if and only if |i-j| ∈ {1, 3, 4} or {i, j} = {1, 3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an) ∞ n=1 is defined by letting an to be the smallest positive integer that cannot be written as ℓ∈L a ℓ for some set L ⊂ [n -1] with |i -j| / ∈ S for all i, j ∈ L. We discuss our results governing the growth of S-LID sequences, as well as results proving that many families of sets S yield S-LID sequences which follow simple recurrence relations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f869dadee457f8ca2fa182fbf8423e05" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633509,"asset_id":126812060,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633509/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812060"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812060"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812060; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812060]").text(description); $(".js-view-count[data-work-id=126812060]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812060; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812060']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f869dadee457f8ca2fa182fbf8423e05" } } $('.js-work-strip[data-work-id=126812060]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812060,"title":"Recurrence Relations for $S$-Legal Index Difference Sequences","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"Recurrence Relations in S-Legal Index Sequences","grobid_abstract":"Zeckendorf's Theorem implies that the Fibonacci number Fn is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i th and j th squares are adjacent if and only if |i-j| ∈ {1, 3, 4} or {i, j} = {1, 3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an) ∞ n=1 is defined by letting an to be the smallest positive integer that cannot be written as ℓ∈L a ℓ for some set L ⊂ [n -1] with |i -j| / ∈ S for all i, j ∈ L. 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Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i th and j th squares are adjacent if and only if |i-j| ∈ {1, 3, 4} or {i, j} = {1, 3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an) ∞ n=1 is defined by letting an to be the smallest positive integer that cannot be written as ℓ∈L a ℓ for some set L ⊂ [n -1] with |i -j| / ∈ S for all i, j ∈ L. We discuss our results governing the growth of S-LID sequences, as well as results proving that many families of sets S yield S-LID sequences which follow simple recurrence relations.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Faulhaber conjectured (later proved by Jacobi) that for k odd, S k (n) can be written as a polynomial of S1(n), and for k even, S k (n) can be written as ). We give a proof of a variant of this result, namely that for any k there is a polynomial g k (x, y) such that S k (n) = g(S1(n), S2(n)). The novel proof yields a recursive formula to evaluate S k (n) as a polynomial of n that has roughly half the number of terms as the classical recursive formula that uses Pascal's identity.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d65ce77ed60bf81328a8cca4b7d443fa" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633507,"asset_id":126812058,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633507/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812058"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812058"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812058; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812058]").text(description); $(".js-view-count[data-work-id=126812058]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812058; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812058']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d65ce77ed60bf81328a8cca4b7d443fa" } } $('.js-work-strip[data-work-id=126812058]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812058,"title":"On the sum of $k$-th powers in terms of earlier sums","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"For k a positive integer let S k (n) = 1 k + 2 k + • • • + n k , i.e., S k (n) is the sum of the first k-th powers. Faulhaber conjectured (later proved by Jacobi) that for k odd, S k (n) can be written as a polynomial of S1(n), and for k even, S k (n) can be written as ). We give a proof of a variant of this result, namely that for any k there is a polynomial g k (x, y) such that S k (n) = g(S1(n), S2(n)). The novel proof yields a recursive formula to evaluate S k (n) as a polynomial of n that has roughly half the number of terms as the classical recursive formula that uses Pascal's identity.","publication_date":{"day":15,"month":12,"year":2019,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633507},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812058/On_the_sum_of_k_th_powers_in_terms_of_earlier_sums","translated_internal_url":"","created_at":"2025-01-05T02:41:54.198-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633507,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633507/thumbnails/1.jpg","file_name":"1912.pdf","download_url":"https://www.academia.edu/attachments/120633507/download_file","bulk_download_file_name":"On_the_sum_of_k_th_powers_in_terms_of_ea.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633507/1912-libre.pdf?1736077055=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_sum_of_k_th_powers_in_terms_of_ea.pdf\u0026Expires=1742723721\u0026Signature=IS-uDepmBKuD44ixfNk0tLCZqM5Q3L7NdywfyVMZvYy~o736HLstFkyOyuaBNyvLspA3SYZuOQV--wkJzH~WHkh-LlG6-WBeONGqjrB2ymcJ6H44PtsPuT9SjueJUJh7PVzV5Nm4qxM~IQmEi23OZFUapZIntjMYUG-fygzMGEwY7T1wxnE-jeri5THtl3tktAwam4d0xpEDtfoXM~AbiHtAAjGDc5gpkq1Buh3O2A7WIms-18Q8DnzkGLOt0DUzZpTw2JfQFcRWeemx5P3~sY3T3DBI3nbu6FN5tpW6onoE4af9mwjnS1MpyL~tuGxbb6R8hlP~Kf-p4bEAs4xNpA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_sum_of_k_th_powers_in_terms_of_earlier_sums","translated_slug":"","page_count":5,"language":"en","content_type":"Work","summary":"For k a positive integer let S k (n) = 1 k + 2 k + • • • + n k , i.e., S k (n) is the sum of the first k-th powers. Faulhaber conjectured (later proved by Jacobi) that for k odd, S k (n) can be written as a polynomial of S1(n), and for k even, S k (n) can be written as ). We give a proof of a variant of this result, namely that for any k there is a polynomial g k (x, y) such that S k (n) = g(S1(n), S2(n)). The novel proof yields a recursive formula to evaluate S k (n) as a polynomial of n that has roughly half the number of terms as the classical recursive formula that uses Pascal's identity.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller","email":"bkFGOHBvdmplZVk0WmU3NHJTVFlzVWpNbGpGQ0ZXZ0tIK2N5NEx2a2xyMzFTZkFCMDByQUJNcVRXSXhLWjNpZi0tbDBvYTdyQktMemRSOVRScFF2SVpqUT09--2c7a05e1a51448d8b9c59776c733f86e0e02f0d8"},"attachments":[{"id":120633507,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633507/thumbnails/1.jpg","file_name":"1912.pdf","download_url":"https://www.academia.edu/attachments/120633507/download_file","bulk_download_file_name":"On_the_sum_of_k_th_powers_in_terms_of_ea.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633507/1912-libre.pdf?1736077055=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_sum_of_k_th_powers_in_terms_of_ea.pdf\u0026Expires=1742723721\u0026Signature=IS-uDepmBKuD44ixfNk0tLCZqM5Q3L7NdywfyVMZvYy~o736HLstFkyOyuaBNyvLspA3SYZuOQV--wkJzH~WHkh-LlG6-WBeONGqjrB2ymcJ6H44PtsPuT9SjueJUJh7PVzV5Nm4qxM~IQmEi23OZFUapZIntjMYUG-fygzMGEwY7T1wxnE-jeri5THtl3tktAwam4d0xpEDtfoXM~AbiHtAAjGDc5gpkq1Buh3O2A7WIms-18Q8DnzkGLOt0DUzZpTw2JfQFcRWeemx5P3~sY3T3DBI3nbu6FN5tpW6onoE4af9mwjnS1MpyL~tuGxbb6R8hlP~Kf-p4bEAs4xNpA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":2254318,"name":"Sums of Powers","url":"https://www.academia.edu/Documents/in/Sums_of_Powers"},{"id":3777178,"name":"Polynomial","url":"https://www.academia.edu/Documents/in/Polynomial"}],"urls":[{"id":46362528,"url":"http://arxiv.org/pdf/1912.07171"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812056"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812056/Infinite_Families_of_Partitions_into_MSTD_Subsets"><img alt="Research paper thumbnail of Infinite Families of Partitions into MSTD Subsets" class="work-thumbnail" src="https://attachments.academia-assets.com/120633506/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812056/Infinite_Families_of_Partitions_into_MSTD_Subsets">Infinite Families of Partitions into MSTD Subsets</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 16, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A set A is MSTD (more-sum-than-difference) if |A+A| > |A-A|. Though MSTD sets are rare, Martin an...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A set A is MSTD (more-sum-than-difference) if |A+A| > |A-A|. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r → ∞. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r → ∞. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition {1, 2, . . . , r} (for r sufficiently large) into k ≥ 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r ≥ R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r → ∞.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="76e9c566b50ae50f5dfe17c9ba00bd29" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633506,"asset_id":126812056,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633506/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812056"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812056"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812056; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812056]").text(description); $(".js-view-count[data-work-id=126812056]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812056; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812056']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "76e9c566b50ae50f5dfe17c9ba00bd29" } } $('.js-work-strip[data-work-id=126812056]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812056,"title":"Infinite Families of Partitions into MSTD Subsets","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A set A is MSTD (more-sum-than-difference) if |A+A| \u003e |A-A|. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r → ∞. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r → ∞. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition {1, 2, . . . , r} (for r sufficiently large) into k ≥ 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r ≥ R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r → ∞.","publication_date":{"day":16,"month":8,"year":2018,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633506},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812056/Infinite_Families_of_Partitions_into_MSTD_Subsets","translated_internal_url":"","created_at":"2025-01-05T02:41:53.861-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633506,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633506/thumbnails/1.jpg","file_name":"1808.05460.pdf","download_url":"https://www.academia.edu/attachments/120633506/download_file","bulk_download_file_name":"Infinite_Families_of_Partitions_into_MST.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633506/1808.05460-libre.pdf?1736077068=\u0026response-content-disposition=attachment%3B+filename%3DInfinite_Families_of_Partitions_into_MST.pdf\u0026Expires=1742696126\u0026Signature=Di~F3NUvmUNECeZqUbMZrai2Clr30AtlAlnBJxheQUQbdNAZk7k1pn3355GP5WOlXNse0423EcCNn5TqNtsIUBbqMfCfK~DJgmEn9DTMX4ymWgE6PdoCXtXCwJJ4Tm7-DjUEsi~hDH5rsvdDrXfk2mdoSYW6QaR9e9hX3hN1XsPAG4g5X2kyMU1V92WEm-1Wt6wc7eWRbbMQhlXj2Fa-d2rHPB97-gYRLH4T7Aunvm1DJCJFVCZCeDdyGqZZtrAi5vK7m6iHb7dTOoiE9pjaDhjXmbT35~oM8pFaVbHok7kpfM-Rpteo45lLeJk17pI71l8VZwTz5cqv6Jx6o6YHUg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Infinite_Families_of_Partitions_into_MSTD_Subsets","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"A set A is MSTD (more-sum-than-difference) if |A+A| \u003e |A-A|. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r → ∞. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r → ∞. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition {1, 2, . . . , r} (for r sufficiently large) into k ≥ 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r ≥ R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r → ∞.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633506,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633506/thumbnails/1.jpg","file_name":"1808.05460.pdf","download_url":"https://www.academia.edu/attachments/120633506/download_file","bulk_download_file_name":"Infinite_Families_of_Partitions_into_MST.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633506/1808.05460-libre.pdf?1736077068=\u0026response-content-disposition=attachment%3B+filename%3DInfinite_Families_of_Partitions_into_MST.pdf\u0026Expires=1742696126\u0026Signature=Di~F3NUvmUNECeZqUbMZrai2Clr30AtlAlnBJxheQUQbdNAZk7k1pn3355GP5WOlXNse0423EcCNn5TqNtsIUBbqMfCfK~DJgmEn9DTMX4ymWgE6PdoCXtXCwJJ4Tm7-DjUEsi~hDH5rsvdDrXfk2mdoSYW6QaR9e9hX3hN1XsPAG4g5X2kyMU1V92WEm-1Wt6wc7eWRbbMQhlXj2Fa-d2rHPB97-gYRLH4T7Aunvm1DJCJFVCZCeDdyGqZZtrAi5vK7m6iHb7dTOoiE9pjaDhjXmbT35~oM8pFaVbHok7kpfM-Rpteo45lLeJk17pI71l8VZwTz5cqv6Jx6o6YHUg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":524840,"name":"Integers","url":"https://www.academia.edu/Documents/in/Integers"}],"urls":[{"id":46362526,"url":"https://arxiv.org/pdf/1808.05460.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812055"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812055/A_Quick_Introduction_to_Benford%CA%BCs_Law"><img alt="Research paper thumbnail of A Quick Introduction to Benfordʼs Law" class="work-thumbnail" src="https://attachments.academia-assets.com/120633505/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812055/A_Quick_Introduction_to_Benford%CA%BCs_Law">A Quick Introduction to Benfordʼs Law</a></div><div class="wp-workCard_item"><span>Princeton University Press eBooks</span><span>, May 26, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The history of Benford's Law is a fascinating and unexpected story of the interplay between theor...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The history of Benford's Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford's Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford's Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford's Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford's Law on a solid foundation. There we explore several different categorizations of Benford's Law, and rigorously prove that certain systems satisfy these conditions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="73a18c257fa50e33d7e6227b926ac13d" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633505,"asset_id":126812055,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633505/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812055"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812055"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812055; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812055]").text(description); $(".js-view-count[data-work-id=126812055]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812055; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812055']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "73a18c257fa50e33d7e6227b926ac13d" } } $('.js-work-strip[data-work-id=126812055]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812055,"title":"A Quick Introduction to Benfordʼs Law","translated_title":"","metadata":{"publisher":"Princeton University Press","grobid_abstract":"The history of Benford's Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford's Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford's Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford's Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford's Law on a solid foundation. There we explore several different categorizations of Benford's Law, and rigorously prove that certain systems satisfy these conditions.","publication_date":{"day":26,"month":5,"year":2015,"errors":{}},"publication_name":"Princeton University Press eBooks","grobid_abstract_attachment_id":120633505},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812055/A_Quick_Introduction_to_Benford%CA%BCs_Law","translated_internal_url":"","created_at":"2025-01-05T02:41:53.534-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633505,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633505/thumbnails/1.jpg","file_name":"s10527.pdf","download_url":"https://www.academia.edu/attachments/120633505/download_file","bulk_download_file_name":"A_Quick_Introduction_to_Benfords_Law.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633505/s10527-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DA_Quick_Introduction_to_Benfords_Law.pdf\u0026Expires=1742779652\u0026Signature=DIPmJPUNLgh2vNUGgAQe1Xe21ZK0BZVcaZ6ZRwYAqMCyn9qU7KcFJUjTlZp3eRyQAklptFelv~eZ9L0LeNT9kecG~CPEDleKug6moc0OB6TXefGJsY6WYVEEfQ2tT-WsmjKBcZkE-D0stN6QfERqIQcmX7WJvBuL5avcSm8O2fOA3GkKqKB3CM7oi-pYN8AtJw6gIPjyq5hGb2GOP8tpCzD6RDyALCHImcSFZFUnJwff1skBUxkqC7tSSpywca5XJu0gfAfjvKyEQIZveeZpqc2H3801rBYjEqrNQnio~kTLaMmLB6opA7KgNPWRU5JyMBtVgs7TK9pAXd9KC3LyEg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_Quick_Introduction_to_Benfordʼs_Law","translated_slug":"","page_count":16,"language":"en","content_type":"Work","summary":"The history of Benford's Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford's Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford's Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford's Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford's Law on a solid foundation. There we explore several different categorizations of Benford's Law, and rigorously prove that certain systems satisfy these conditions.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633505,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633505/thumbnails/1.jpg","file_name":"s10527.pdf","download_url":"https://www.academia.edu/attachments/120633505/download_file","bulk_download_file_name":"A_Quick_Introduction_to_Benfords_Law.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633505/s10527-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DA_Quick_Introduction_to_Benfords_Law.pdf\u0026Expires=1742779652\u0026Signature=DIPmJPUNLgh2vNUGgAQe1Xe21ZK0BZVcaZ6ZRwYAqMCyn9qU7KcFJUjTlZp3eRyQAklptFelv~eZ9L0LeNT9kecG~CPEDleKug6moc0OB6TXefGJsY6WYVEEfQ2tT-WsmjKBcZkE-D0stN6QfERqIQcmX7WJvBuL5avcSm8O2fOA3GkKqKB3CM7oi-pYN8AtJw6gIPjyq5hGb2GOP8tpCzD6RDyALCHImcSFZFUnJwff1skBUxkqC7tSSpywca5XJu0gfAfjvKyEQIZveeZpqc2H3801rBYjEqrNQnio~kTLaMmLB6opA7KgNPWRU5JyMBtVgs7TK9pAXd9KC3LyEg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":724,"name":"Economics","url":"https://www.academia.edu/Documents/in/Economics"}],"urls":[{"id":46362525,"url":"http://assets.press.princeton.edu/chapters/s10527.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812054"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812054/Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences"><img alt="Research paper thumbnail of Legal Decompositions Arising from Non-positive Linear Recurrences" class="work-thumbnail" src="https://attachments.academia-assets.com/120633504/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812054/Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences">Legal Decompositions Arising from Non-positive Linear Recurrences</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 30, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adj...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s, b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="944b47ffb00cb3be0a08b923615cec4a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633504,"asset_id":126812054,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633504/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812054"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812054"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812054; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812054]").text(description); $(".js-view-count[data-work-id=126812054]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812054; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812054']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "944b47ffb00cb3be0a08b923615cec4a" } } $('.js-work-strip[data-work-id=126812054]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812054,"title":"Legal Decompositions Arising from Non-positive Linear Recurrences","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s, b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.","publication_date":{"day":30,"month":6,"year":2016,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633504},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812054/Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences","translated_internal_url":"","created_at":"2025-01-05T02:41:53.209-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633504,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633504/thumbnails/1.jpg","file_name":"1606.09312.pdf","download_url":"https://www.academia.edu/attachments/120633504/download_file","bulk_download_file_name":"Legal_Decompositions_Arising_from_Non_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633504/1606.09312-libre.pdf?1736077064=\u0026response-content-disposition=attachment%3B+filename%3DLegal_Decompositions_Arising_from_Non_po.pdf\u0026Expires=1742779652\u0026Signature=aGIrPGlqzNkWN3Keovq0yX46Fs7ZUO5fTEiPQMoUoHYyiRkLVqI~y44YksJis194j0-YKI~-1Ysx1grnlDC6Rdb4AY-eqMBceJZizwnqG3y8VIYh-Ux50O~J0Ux-jfCuNDjYcFhw6LdlSs0oRP8rYeBCMRM9arUEfwfBbdGZW4Yu6nHgOQN27c2rnPohwNaYiEXJEiQFNlS5moszb~O9d0uHfbgNmI~jtcxSCEYciUIfz1r5PrY180tphoTh4qmSRQOJCzVr6JNyzA8rCDDJ2fOGTLUSIC6RpQXAM0cPQFXhRoQfejIypiGU6Uy5JnFj8G~YcV3J6MwnEUtPLuNJMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences","translated_slug":"","page_count":21,"language":"en","content_type":"Work","summary":"Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s, b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633504,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633504/thumbnails/1.jpg","file_name":"1606.09312.pdf","download_url":"https://www.academia.edu/attachments/120633504/download_file","bulk_download_file_name":"Legal_Decompositions_Arising_from_Non_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633504/1606.09312-libre.pdf?1736077064=\u0026response-content-disposition=attachment%3B+filename%3DLegal_Decompositions_Arising_from_Non_po.pdf\u0026Expires=1742779652\u0026Signature=aGIrPGlqzNkWN3Keovq0yX46Fs7ZUO5fTEiPQMoUoHYyiRkLVqI~y44YksJis194j0-YKI~-1Ysx1grnlDC6Rdb4AY-eqMBceJZizwnqG3y8VIYh-Ux50O~J0Ux-jfCuNDjYcFhw6LdlSs0oRP8rYeBCMRM9arUEfwfBbdGZW4Yu6nHgOQN27c2rnPohwNaYiEXJEiQFNlS5moszb~O9d0uHfbgNmI~jtcxSCEYciUIfz1r5PrY180tphoTh4qmSRQOJCzVr6JNyzA8rCDDJ2fOGTLUSIC6RpQXAM0cPQFXhRoQfejIypiGU6Uy5JnFj8G~YcV3J6MwnEUtPLuNJMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":550512,"name":"Fibonacci Number","url":"https://www.academia.edu/Documents/in/Fibonacci_Number"},{"id":1317768,"name":"Quilt","url":"https://www.academia.edu/Documents/in/Quilt"}],"urls":[{"id":46362524,"url":"https://arxiv.org/pdf/1606.09312.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812053"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812053/Geometric_progression_free_sets_over_quadratic_number_fields"><img alt="Research paper thumbnail of Geometric-progression-free sets over quadratic number fields" class="work-thumbnail" src="https://attachments.academia-assets.com/120633501/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812053/Geometric_progression_free_sets_over_quadratic_number_fields">Geometric-progression-free sets over quadratic number fields</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 2, 2014</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a par...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a7e9150069b79c8a4f3ef8bae4c62de5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633501,"asset_id":126812053,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633501/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812053"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812053"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812053; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812053]").text(description); $(".js-view-count[data-work-id=126812053]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812053; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812053']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a7e9150069b79c8a4f3ef8bae4c62de5" } } $('.js-work-strip[data-work-id=126812053]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812053,"title":"Geometric-progression-free sets over quadratic number fields","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets \"greedily,\" a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.","publication_date":{"day":2,"month":12,"year":2014,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633501},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812053/Geometric_progression_free_sets_over_quadratic_number_fields","translated_internal_url":"","created_at":"2025-01-05T02:41:52.756-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633501,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633501/thumbnails/1.jpg","file_name":"1412.pdf","download_url":"https://www.academia.edu/attachments/120633501/download_file","bulk_download_file_name":"Geometric_progression_free_sets_over_qua.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633501/1412-libre.pdf?1736077067=\u0026response-content-disposition=attachment%3B+filename%3DGeometric_progression_free_sets_over_qua.pdf\u0026Expires=1742779652\u0026Signature=KXOFW-S3vMqfipT-OFp7aCTFNZzZ2VCio9Q2k71UCh0mPaLrAKapEwodS7iYkoyuUNirceI2vq09GxRDTmnzL6luW8NGDaIS2~lJIs-qeuzp7QDfSp-32yVWInA947zYXC7tIl1ARN0flK-pqTek-awg7ZztPBJakbuK9Ip4S6ahUDe6bfGf2zpgqdorDCQojKvrGxOfB3CnYVZkcakiHGVo0C7b~cZf3DZU2ViwtnaOsONv6vwdJzuf-0MRx40HdsYCT-I6kPLJcyfLzMQ1W7IOXQ8frRthufJbWyobWgvNvrkU2yFPHow7iR37DtCHNbufhNHorJx5~a88V1ozRw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Geometric_progression_free_sets_over_quadratic_number_fields","translated_slug":"","page_count":19,"language":"en","content_type":"Work","summary":"In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets \"greedily,\" a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Many of the mo...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset S + S = {x + y : x, y ∈ S} of a set of integers S. A finite set of integers A is sumdominant if |A + A| > |A -A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately 4.5 • 10 -4 ). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that |A + A| = |A -A| tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of {0, . . ., n} have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6e2bf624b1cb5fb9f06c544c8f9444fe" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633502,"asset_id":126812051,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633502/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812051"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812051"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812051; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812051]").text(description); $(".js-view-count[data-work-id=126812051]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812051; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812051']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "6e2bf624b1cb5fb9f06c544c8f9444fe" } } $('.js-work-strip[data-work-id=126812051]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812051,"title":"Most Subsets are Balanced in Finite Groups","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset S + S = {x + y : x, y ∈ S} of a set of integers S. A finite set of integers A is sumdominant if |A + A| \u003e |A -A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately 4.5 • 10 -4 ). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that |A + A| = |A -A| tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of {0, . . ., n} have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.","publication_date":{"day":10,"month":8,"year":2013,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633502},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812051/Most_Subsets_are_Balanced_in_Finite_Groups","translated_internal_url":"","created_at":"2025-01-05T02:41:52.105-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633502,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633502/thumbnails/1.jpg","file_name":"1308.pdf","download_url":"https://www.academia.edu/attachments/120633502/download_file","bulk_download_file_name":"Most_Subsets_are_Balanced_in_Finite_Grou.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633502/1308-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DMost_Subsets_are_Balanced_in_Finite_Grou.pdf\u0026Expires=1742723722\u0026Signature=UIg9WSnK~EwcYa8uz-sXVfiEHg2Kh4hPenVza3tr0QXHFy769TCMWT61MD-QyZkzHUPjW07ziX2UpqD4vyDPgGMPo1NkAP2MCDJCOQqoE8Zn9x6VRUNsUC0w1WkY3rAQAvoF~C3k6evhUkpQ1Ymq8KMSbx3qnE4dTmqfn8duyW7bLZ8kYg2wK8jowV59lHWHTUFHqFazsWPZ2b8nPvJOdFNbOc3tGKnP-LIeMMcPwkDjrIgt6T~A9LhegI2q-4wXt7VmV1xFiQ2xD69GqZwuugOa3XSDvdlm5gp0L1CpRQCrZtjAELKia7l3ZrpvNPRpX~ULvGexw0DaVdRq3kau0Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Most_Subsets_are_Balanced_in_Finite_Groups","translated_slug":"","page_count":11,"language":"en","content_type":"Work","summary":"The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset S + S = {x + y : x, y ∈ S} of a set of integers S. A finite set of integers A is sumdominant if |A + A| \u003e |A -A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately 4.5 • 10 -4 ). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that |A + A| = |A -A| tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of {0, . . ., n} have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller","email":"NUNDaU10MklWNmlDNnB0UmFmQzdMd1pnbFdKL2dqaWtzRGZFYTNnRnVwL0FaMkJtc1hNWWtBNDhnY2ZwditwcC0tKzhFWGM2Q3VQNktUWDZpeVF1RkRudz09--bdb86de1b2654cf51a36f2f6f904c711004a2100"},"attachments":[{"id":120633502,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633502/thumbnails/1.jpg","file_name":"1308.pdf","download_url":"https://www.academia.edu/attachments/120633502/download_file","bulk_download_file_name":"Most_Subsets_are_Balanced_in_Finite_Grou.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633502/1308-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DMost_Subsets_are_Balanced_in_Finite_Grou.pdf\u0026Expires=1742723722\u0026Signature=UIg9WSnK~EwcYa8uz-sXVfiEHg2Kh4hPenVza3tr0QXHFy769TCMWT61MD-QyZkzHUPjW07ziX2UpqD4vyDPgGMPo1NkAP2MCDJCOQqoE8Zn9x6VRUNsUC0w1WkY3rAQAvoF~C3k6evhUkpQ1Ymq8KMSbx3qnE4dTmqfn8duyW7bLZ8kYg2wK8jowV59lHWHTUFHqFazsWPZ2b8nPvJOdFNbOc3tGKnP-LIeMMcPwkDjrIgt6T~A9LhegI2q-4wXt7VmV1xFiQ2xD69GqZwuugOa3XSDvdlm5gp0L1CpRQCrZtjAELKia7l3ZrpvNPRpX~ULvGexw0DaVdRq3kau0Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":37345,"name":"Discrete Mathematics","url":"https://www.academia.edu/Documents/in/Discrete_Mathematics"},{"id":1463317,"name":"Contents","url":"https://www.academia.edu/Documents/in/Contents"},{"id":2239835,"name":"Finite abelian groups","url":"https://www.academia.edu/Documents/in/Finite_abelian_groups"},{"id":2570814,"name":"conjecture","url":"https://www.academia.edu/Documents/in/conjecture"}],"urls":[{"id":46362522,"url":"http://arxiv.org/pdf/1308.2344"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812050"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812050/Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes"><img alt="Research paper thumbnail of Sets characterized by missing sums and differences in dilating polytopes" class="work-thumbnail" src="https://attachments.academia-assets.com/120633533/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812050/Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes">Sets characterized by missing sums and differences in dilating polytopes</a></div><div class="wp-workCard_item"><span>Journal of Number Theory</span><span>, Dec 1, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A sum-dominant set is a finite set A of integers such that |A + A| > |A -A|. As a typical pair of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A sum-dominant set is a finite set A of integers such that |A + A| > |A -A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sumdominant subsets of {0, . . . , n} is bounded below by a positive constant as n → ∞. Hegarty then extended their work and showed that for any prescribed s, d ∈ N 0 , the proportion ρ s,d n of subsets of {0, . . . , n} that are missing exactly s sums in {0, . . . , 2n} and exactly 2d differences in {-n, . . . , n} also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) -L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9ebb998152265fbdb7ccc8b7ff9e7895" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633533,"asset_id":126812050,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633533/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812050"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812050"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812050; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812050]").text(description); $(".js-view-count[data-work-id=126812050]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812050; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812050']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9ebb998152265fbdb7ccc8b7ff9e7895" } } $('.js-work-strip[data-work-id=126812050]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812050,"title":"Sets characterized by missing sums and differences in dilating polytopes","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"A sum-dominant set is a finite set A of integers such that |A + A| \u003e |A -A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sumdominant subsets of {0, . . . , n} is bounded below by a positive constant as n → ∞. Hegarty then extended their work and showed that for any prescribed s, d ∈ N 0 , the proportion ρ s,d n of subsets of {0, . . . , n} that are missing exactly s sums in {0, . . . , 2n} and exactly 2d differences in {-n, . . . , n} also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) -L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit.","publication_date":{"day":1,"month":12,"year":2015,"errors":{}},"publication_name":"Journal of Number Theory","grobid_abstract_attachment_id":120633533},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812050/Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes","translated_internal_url":"","created_at":"2025-01-05T02:41:51.322-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633533,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633533/thumbnails/1.jpg","file_name":"MSTD_ddim52.pdf","download_url":"https://www.academia.edu/attachments/120633533/download_file","bulk_download_file_name":"Sets_characterized_by_missing_sums_and_d.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633533/MSTD_ddim52-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DSets_characterized_by_missing_sums_and_d.pdf\u0026Expires=1742779652\u0026Signature=ZQ-8fncJLShFLjQuVOWL-l21JPfA1zIAn2MWzW9ze5rNrQnWz3u18fW~T9qQ7Zz5ynpPp3WGMjUdM3bJVoO5srFQwweRutrs03RWEUcHGzeCqparh7mwgXsGDs6Vg-fANx0-7gDvUbsInGuOMBov5K9rffl3hRDELexxr5nVWFecMWkf6RoI-2h~rlz358WslTXYWcTn1Xo22ARMupT1XyCfMMDhfz97aLQObTm9KEzMDlSTiHS4O-IOlwIA8zQOeglhju9WWXWQ2-QKp6KWEUk2~d53koHFQ52i1IjEmfzTs941bM1PrPCGBZPtLNq8E7aJXwoy4Bg1PsVZ8qsACA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes","translated_slug":"","page_count":23,"language":"en","content_type":"Work","summary":"A sum-dominant set is a finite set A of integers such that |A + A| \u003e |A -A|. 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Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) -L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633533,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633533/thumbnails/1.jpg","file_name":"MSTD_ddim52.pdf","download_url":"https://www.academia.edu/attachments/120633533/download_file","bulk_download_file_name":"Sets_characterized_by_missing_sums_and_d.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633533/MSTD_ddim52-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DSets_characterized_by_missing_sums_and_d.pdf\u0026Expires=1742779652\u0026Signature=ZQ-8fncJLShFLjQuVOWL-l21JPfA1zIAn2MWzW9ze5rNrQnWz3u18fW~T9qQ7Zz5ynpPp3WGMjUdM3bJVoO5srFQwweRutrs03RWEUcHGzeCqparh7mwgXsGDs6Vg-fANx0-7gDvUbsInGuOMBov5K9rffl3hRDELexxr5nVWFecMWkf6RoI-2h~rlz358WslTXYWcTn1Xo22ARMupT1XyCfMMDhfz97aLQObTm9KEzMDlSTiHS4O-IOlwIA8zQOeglhju9WWXWQ2-QKp6KWEUk2~d53koHFQ52i1IjEmfzTs941bM1PrPCGBZPtLNq8E7aJXwoy4Bg1PsVZ8qsACA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":1463317,"name":"Contents","url":"https://www.academia.edu/Documents/in/Contents"},{"id":2138291,"name":"Polytope","url":"https://www.academia.edu/Documents/in/Polytope"}],"urls":[{"id":46362521,"url":"https://doi.org/10.1016/j.jnt.2015.04.027"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812038"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812038/Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions"><img alt="Research paper thumbnail of Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions" class="work-thumbnail" src="https://attachments.academia-assets.com/120633491/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812038/Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions">Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 8, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="38b6e030b30b9765d8a2b10cd75b8bce" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633491,"asset_id":126812038,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633491/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812038"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812038"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812038; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812038]").text(description); $(".js-view-count[data-work-id=126812038]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812038; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812038']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "38b6e030b30b9765d8a2b10cd75b8bce" } } $('.js-work-strip[data-work-id=126812038]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812038,"title":"Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.","publication_date":{"day":8,"month":8,"year":2005,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633491},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812038/Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions","translated_internal_url":"","created_at":"2025-01-05T02:40:30.404-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633491,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633491/thumbnails/1.jpg","file_name":"0508150.pdf","download_url":"https://www.academia.edu/attachments/120633491/download_file","bulk_download_file_name":"Investigations_of_Zeros_Near_the_Central.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633491/0508150-libre.pdf?1736077076=\u0026response-content-disposition=attachment%3B+filename%3DInvestigations_of_Zeros_Near_the_Central.pdf\u0026Expires=1742723722\u0026Signature=JbzWFpMXDVPYLgaRvq7ynvnC8koxxQlxO38LR8ywtefXhS5bSLzrW0~GwbWDTuMzAEwJMNCxmQgmuDsyUfxkDWehnrmopSELmL9EKYrG5P0iTu3LI2YMKTX-bL4iBOWbDIlmtRFtYVopaIwVT4wU4HWwtyL89jerPtlzq2hZlicozv-gozwXEBuJH9tKA5sIj2~94zyrnB36BJmde1voP4iaGygibMcppqg3zlj0iqJHCs3Az~QD2JU2s1ZUtD~UBICWU1sLUkdguuf5undPHhT6zE1flcDMCjOOyNd8fWCO4mDP~1~fl33eVOL3s2MoJdJ1BIa8~8wK8NsgXTZkvQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions","translated_slug":"","page_count":37,"language":"en","content_type":"Work","summary":"We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125376742"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125376742/One_Level_density_for_holomorphic_cusp_forms_of_arbitrary_level"><img alt="Research paper thumbnail of One-Level density for holomorphic cusp forms of arbitrary level" class="work-thumbnail" src="https://attachments.academia-assets.com/119432040/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125376742/One_Level_density_for_holomorphic_cusp_forms_of_arbitrary_level">One-Level density for holomorphic cusp forms of arbitrary level</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Apr 11, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomor...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="12bf6a7f9cc7a1e513875000846cc0df" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119432040,"asset_id":125376742,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119432040/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125376742"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125376742"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125376742; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125376742]").text(description); $(".js-view-count[data-work-id=125376742]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125376742; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125376742']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "12bf6a7f9cc7a1e513875000846cc0df" } } $('.js-work-strip[data-work-id=125376742]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125376742,"title":"One-Level density for holomorphic cusp forms of arbitrary level","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125376741"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125376741/Maine_Qu_ebec_Number_Theory_Conference_A_Shifted_Twisted_Second_Moment_and_Gaps_Between_Zeros_for_L_functions_Associated_to_Holomorphic_Cusp_Forms"><img alt="Research paper thumbnail of Maine-Qu ebec Number Theory Conference: A Shifted Twisted Second Moment and Gaps Between Zeros for L-functions Associated to Holomorphic Cusp Forms" class="work-thumbnail" src="https://attachments.academia-assets.com/119432026/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125376741/Maine_Qu_ebec_Number_Theory_Conference_A_Shifted_Twisted_Second_Moment_and_Gaps_Between_Zeros_for_L_functions_Associated_to_Holomorphic_Cusp_Forms">Maine-Qu ebec Number Theory Conference: A Shifted Twisted Second Moment and Gaps Between Zeros for L-functions Associated to Holomorphic Cusp Forms</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let f be a modular cusp form of weight κ and level q, f has a (normalized) Fourier series at infi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let f be a modular cusp form of weight κ and level q, f has a (normalized) Fourier series at infinity: We are interested in the distribution of the zeros of our automorphic L-function along the critical line.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="041751bd9721c1f5f763928cbad8f50c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119432026,"asset_id":125376741,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119432026/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125376741"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125376741"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125376741; 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The simplest cases are base-b expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d6a6c94d6316108647f4b1d4bcf3f4f8" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119432024,"asset_id":125376740,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119432024/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125376740"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125376740"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125376740; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125376740]").text(description); $(".js-view-count[data-work-id=125376740]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125376740; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125376740']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d6a6c94d6316108647f4b1d4bcf3f4f8" } } $('.js-work-strip[data-work-id=125376740]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125376740,"title":"A Probabilistic Approach to Generalized Zeckendorf Decompositions","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-b expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.","publication_date":{"day":9,"month":5,"year":2014,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":119432024},"translated_abstract":null,"internal_url":"https://www.academia.edu/125376740/A_Probabilistic_Approach_to_Generalized_Zeckendorf_Decompositions","translated_internal_url":"","created_at":"2024-11-08T04:25:18.956-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":119432024,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/119432024/thumbnails/1.jpg","file_name":"1405.pdf","download_url":"https://www.academia.edu/attachments/119432024/download_file","bulk_download_file_name":"A_Probabilistic_Approach_to_Generalized.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/119432024/1405-libre.pdf?1731070905=\u0026response-content-disposition=attachment%3B+filename%3DA_Probabilistic_Approach_to_Generalized.pdf\u0026Expires=1742779652\u0026Signature=ZUQS9xD84j2L-chsKnseEvreXJyOzJNzR4s-dszWYUJP0i40ud34vS8EpnJ7cAzAMHnAgdbJyzojoDCSohSkN1h0MMSN02lI311FjZE4U~MZzF3g~RBRZVMUJfheyHvfYCVWMI9qr05hEdwekAkbbbgU3G1wdPUyUXi--Ygb4E3CYZKK3GmNCMf0o-JR7-0lpM5bgiypiPOcIalku3BOWz4akCqPkVjO~aB0C0sPAzNUMnWp9Ftgyir-~UYQk4I6eAo9m8rIPAzcjTrPqxffF4-loOpiClHmTMW1UKjtk8BF6QouhWRwc9Ogj~fVXf7Ms7fxdocCuxW7trO-ZRWhUw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_Probabilistic_Approach_to_Generalized_Zeckendorf_Decompositions","translated_slug":"","page_count":41,"language":"en","content_type":"Work","summary":"Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-b expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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In this paper we explore the case of weighted graphs. More precisely, given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique 'eigendistribution', i.e., a weight distribution W such that the associated limiting spectral distribution is a rescaling of W. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d 2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8df736694155275ae384bddd91676411" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633519,"asset_id":126812071,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633519/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812071"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812071"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812071; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812071]").text(description); $(".js-view-count[data-work-id=126812071]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812071; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812071']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8df736694155275ae384bddd91676411" } } $('.js-work-strip[data-work-id=126812071]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812071,"title":"On the spectral distribution of large weighted random regular graphs","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. In this paper we explore the case of weighted graphs. More precisely, given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique 'eigendistribution', i.e., a weight distribution W such that the associated limiting spectral distribution is a rescaling of W. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d 2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.","publication_date":{"day":28,"month":6,"year":2013,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633519},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812071/On_the_spectral_distribution_of_large_weighted_random_regular_graphs","translated_internal_url":"","created_at":"2025-01-05T02:41:59.995-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633519,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633519/thumbnails/1.jpg","file_name":"1306.pdf","download_url":"https://www.academia.edu/attachments/120633519/download_file","bulk_download_file_name":"On_the_spectral_distribution_of_large_we.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633519/1306-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_spectral_distribution_of_large_we.pdf\u0026Expires=1742779652\u0026Signature=dnKfIwMhMqn4wCreenC4wcAC1L3xvcBAShXK1dA~-wg6yOAjZ~zc94uFjFB0xELiesiPB~ODNAMOSb-WgbFTyKtrnAF4chVocTUD9ZKsL1VhcLxYUJwsrp95Lk3dbw7Ccq2jzlGiWagvluFF3RrnqvbeM~wYJENDkI59XTd6Avenkh28e7HfyHy0lnK1NitlwY9r494-jSoOaBl3u-SBywkbdoA8Q~z0APFhGlQBZ-qDM0O-IFMlQPcPyQs2F6dxUTh4CUHjdHzVGDc4-JQVIg86KStQRuvQQxex5IBfKfU059Yoj5IwsNhuUU-cIWoum8u5Bg~D5USfiJmpqzg2Xw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_spectral_distribution_of_large_weighted_random_regular_graphs","translated_slug":"","page_count":19,"language":"en","content_type":"Work","summary":"McKay proved that the limiting spectral measures of the ensembles of d-regular graphs with N vertices converge to Kesten's measure as N → ∞. In this paper we explore the case of weighted graphs. More precisely, given a large d-regular graph we assign random weights, drawn from some distribution W, to its edges. We study the relationship between W and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other results, we establish the existence of a unique 'eigendistribution', i.e., a weight distribution W such that the associated limiting spectral distribution is a rescaling of W. Initial investigations suggested that the eigendistribution was the semi-circle distribution, which by Wigner's Law is the limiting spectral measure for real symmetric matrices. We prove this is not the case, though the deviation between the eigendistribution and the semi-circular density is small (the first seven moments agree, and the difference in each higher moment is O(1/d 2 )). Our analysis uses combinatorial results about closed acyclic walks in large trees, which may be of independent interest.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Martin and...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A more sums than differences (MSTD) set A is a subset of Z for which |A + A| > |A -A|. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of {1, . . . , n} are MSTD as n → ∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of Z. In particular we show that every finite subset of Z can be transformed into an element of I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in I consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of Z from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0eb37c874403add77abba933eba3763f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633518,"asset_id":126812070,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633518/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812070"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812070"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812070; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812070]").text(description); $(".js-view-count[data-work-id=126812070]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812070; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812070']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0eb37c874403add77abba933eba3763f" } } $('.js-work-strip[data-work-id=126812070]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812070,"title":"A geometric perspective on the MSTD question","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A more sums than differences (MSTD) set A is a subset of Z for which |A + A| \u003e |A -A|. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of {1, . . . , n} are MSTD as n → ∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of Z. In particular we show that every finite subset of Z can be transformed into an element of I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in I consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of Z from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.","publication_date":{"day":2,"month":9,"year":2017,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633518},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812070/A_geometric_perspective_on_the_MSTD_question","translated_internal_url":"","created_at":"2025-01-05T02:41:59.572-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633518,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633518/thumbnails/1.jpg","file_name":"1709.pdf","download_url":"https://www.academia.edu/attachments/120633518/download_file","bulk_download_file_name":"A_geometric_perspective_on_the_MSTD_ques.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633518/1709-libre.pdf?1736077063=\u0026response-content-disposition=attachment%3B+filename%3DA_geometric_perspective_on_the_MSTD_ques.pdf\u0026Expires=1742779652\u0026Signature=hAcb28cRJqWQ~x3ZEanWfcArnfXT9gCAGYu1XGSdZuABIlNZ3E9Ke9Z6OgX2n3Xc-5FO1HqRghPTeYQUUKExlZV9aZSlr~8N3rnJKwuNZrtXxINQC~lzLN6rv4a2p~NARSIXe1zwimTDn-hT17zBKgaapLUjnyTy~xKQ2Ik4GE7pCkMEFFKRLuNvpWPKgqWQe56G-Gk4W-WUOTFt3dfh~zjEz~kbruTaFAA3xnn5wERh7D4nTqwLF2Gz3S3n0akEGqEzf-aHQ9qdemoP5wjxd8QNyUlbGAZjYwgWYtCMIcvCenLq63-WzWHLLJOtb-qWWEYvThAVWS8N8s2Ijiw2Hw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_geometric_perspective_on_the_MSTD_question","translated_slug":"","page_count":22,"language":"en","content_type":"Work","summary":"A more sums than differences (MSTD) set A is a subset of Z for which |A + A| \u003e |A -A|. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of {1, . . . , n} are MSTD as n → ∞. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of Z. In particular we show that every finite subset of Z can be transformed into an element of I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in I consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of Z from a single such set A; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to A.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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We examine related problems from placing n rooks. We prove that as n → ∞, the probability rapidly tends to 1 that the fraction of safe squares from a random placement converges to 1/e 2 . Our interest in the problem is showing how to view the involved algebra to obtain the simple, closed form limiting fraction. In particular, we see the power of many of the key concepts in probability: binary indicator variables, linearity of expectation, variances and covariances, Chebyshev's inequality, and Stirling's formula.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9a4bf32c094fefc07346ff34d50d8dd2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633517,"asset_id":126812069,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633517/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812069"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812069"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812069; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812069]").text(description); $(".js-view-count[data-work-id=126812069]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812069; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812069']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9a4bf32c094fefc07346ff34d50d8dd2" } } $('.js-work-strip[data-work-id=126812069]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812069,"title":"When Rooks Miss: Probability through Chess","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A famous (and hard) chess problem asks what is the maximum number of safe squares possible in placing n queens on an n × n board. 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When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call the M&M Game: Given k people, each simultaneously flips a fair coin, with each eating an M&M on a head and not eating on a tail. The process then continues until all M&M'S are consumed, and two people are deemed to die at the same time if they run out of M&M'S together 1 . This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. There are many ways to determine the probability of a tie, which allow us in this article to use this problem as a springboard to a lot of great mathematics, including memoryless process, combinatorics, statistical inference, graph theory, and hypergeometric functions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="321214489d737e404219ba57977de082" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633522,"asset_id":126812068,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633522/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812068"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812068"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812068; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812068]").text(description); $(".js-view-count[data-work-id=126812068]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812068; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812068']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "321214489d737e404219ba57977de082" } } $('.js-work-strip[data-work-id=126812068]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812068,"title":"The M\u0026M Game: From Morsels to Modern Mathematics","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"The M\u0026M Game: Teaching Probability through Play","grobid_abstract":"To an adult, it's obvious that the day of someone's death is not precisely determined by the day of birth, but it's a very different story for a child. When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call the M\u0026M Game: Given k people, each simultaneously flips a fair coin, with each eating an M\u0026M on a head and not eating on a tail. The process then continues until all M\u0026M'S are consumed, and two people are deemed to die at the same time if they run out of M\u0026M'S together 1 . This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. 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When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call the M\u0026M Game: Given k people, each simultaneously flips a fair coin, with each eating an M\u0026M on a head and not eating on a tail. The process then continues until all M\u0026M'S are consumed, and two people are deemed to die at the same time if they run out of M\u0026M'S together 1 . This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. 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The Allies needed accura...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">During World War II the German army used tanks to devastating advantage. The Allies needed accurate estimates of their tank production and deployment. They used two approaches to find these values: spies, and statistics. This note describes the statistical approach. Assuming the tanks are labeled consecutively starting at 1, if we observe k serial numbers from an unknown number N of tanks, with the maximum observed value m, then the best estimate for N is m(1+1/k)-1. This is now known as the German Tank Problem, and is a terrific example of the applicability of mathematics and statistics in the real world. The first part of the paper reproduces known results, specifically deriving this estimate and comparing its effectiveness to that of the spies. The second part presents a result we have not found in print elsewhere, the generalization to the case where the smallest value is not necessarily 1. We emphasize in detail why we are able to obtain such clean, closed-form expressions for the estimates, and conclude with an appendix highlighting how to use this problem to teach regression and how statistics can help us find functional relationships.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ff49102743a773dd4af33619ed6374fc" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633521,"asset_id":126812067,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633521/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812067"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812067"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812067; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812067]").text(description); $(".js-view-count[data-work-id=126812067]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812067; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812067']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "ff49102743a773dd4af33619ed6374fc" } } $('.js-work-strip[data-work-id=126812067]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812067,"title":"Lessons from the German Tank Problem","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"During World War II the German army used tanks to devastating advantage. 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The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, [3] studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input N → ∞, the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input N involving the Catalan numbers. The works [3] and Cuzensa et al. determined how to achieve a shortest and longest possible Zeckendorf game on a given input N , respectively: we establish that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞, both players win with probability 1/2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞. We conclude the work with many open problems.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="45888380c9299a10a9fcb1ec1fc874fd" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633516,"asset_id":126812064,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633516/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812064"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812064"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812064; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812064]").text(description); $(".js-view-count[data-work-id=126812064]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812064; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812064']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "45888380c9299a10a9fcb1ec1fc874fd" } } $('.js-work-strip[data-work-id=126812064]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812064,"title":"Towards the Gaussianity of Random Zeckendorf Games","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Zeckendorf proved that any positive integer has a unique decomposition as a sum of non-consecutive Fibonacci numbers, indexed by Baird, Epstein, Flint, and Miller [3] defined the two-player Zeckendorf game, where two players take turns acting on a multiset of Fibonacci numbers that always sums to N . The game terminates when no possible moves remain, and the final player to perform a move wins. Notably, [3] studied the setting of random games: the game proceeds by choosing an available move uniformly at random, and they conjecture that as the input N → ∞, the distribution of random game lengths converges to a Gaussian. We prove that certain sums of move counts is constant, and find a lower bound on the number of shortest games on input N involving the Catalan numbers. The works [3] and Cuzensa et al. determined how to achieve a shortest and longest possible Zeckendorf game on a given input N , respectively: we establish that for any input N , the range of possible game lengths constitutes an interval of natural numbers: every game length between the shortest and longest game lengths can be achieved. We further the study of probabilistic aspects of random Zeckendorf games. We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞, both players win with probability 1/2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞. 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We study two probability measures on the space of all Zeckendorf games on input N : the uniform measure, and the measure induced by choosing moves uniformly at random at any given position. Under both measures that in the limit N → ∞, both players win with probability 1/2. We also find natural partitions of the collection of all Zeckendorf games of a fixed input N , on which we observe weak convergence to a Gaussian in the limit N → ∞. We conclude the work with many open problems.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i th and j th squares are adjacent if and only if |i-j| ∈ {1, 3, 4} or {i, j} = {1, 3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an) ∞ n=1 is defined by letting an to be the smallest positive integer that cannot be written as ℓ∈L a ℓ for some set L ⊂ [n -1] with |i -j| / ∈ S for all i, j ∈ L. We discuss our results governing the growth of S-LID sequences, as well as results proving that many families of sets S yield S-LID sequences which follow simple recurrence relations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f869dadee457f8ca2fa182fbf8423e05" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633509,"asset_id":126812060,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633509/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812060"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812060"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812060; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812060]").text(description); $(".js-view-count[data-work-id=126812060]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812060; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812060']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "f869dadee457f8ca2fa182fbf8423e05" } } $('.js-work-strip[data-work-id=126812060]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812060,"title":"Recurrence Relations for $S$-Legal Index Difference Sequences","translated_title":"","metadata":{"publisher":"Cornell University","ai_title_tag":"Recurrence Relations in S-Legal Index Sequences","grobid_abstract":"Zeckendorf's Theorem implies that the Fibonacci number Fn is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i th and j th squares are adjacent if and only if |i-j| ∈ {1, 3, 4} or {i, j} = {1, 3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an) ∞ n=1 is defined by letting an to be the smallest positive integer that cannot be written as ℓ∈L a ℓ for some set L ⊂ [n -1] with |i -j| / ∈ S for all i, j ∈ L. 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Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i th and j th squares are adjacent if and only if |i-j| ∈ {1, 3, 4} or {i, j} = {1, 3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an) ∞ n=1 is defined by letting an to be the smallest positive integer that cannot be written as ℓ∈L a ℓ for some set L ⊂ [n -1] with |i -j| / ∈ S for all i, j ∈ L. We discuss our results governing the growth of S-LID sequences, as well as results proving that many families of sets S yield S-LID sequences which follow simple recurrence relations.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Faulhaber conjectured (later proved by Jacobi) that for k odd, S k (n) can be written as a polynomial of S1(n), and for k even, S k (n) can be written as ). We give a proof of a variant of this result, namely that for any k there is a polynomial g k (x, y) such that S k (n) = g(S1(n), S2(n)). The novel proof yields a recursive formula to evaluate S k (n) as a polynomial of n that has roughly half the number of terms as the classical recursive formula that uses Pascal's identity.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d65ce77ed60bf81328a8cca4b7d443fa" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633507,"asset_id":126812058,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633507/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812058"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812058"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812058; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812058]").text(description); $(".js-view-count[data-work-id=126812058]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812058; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812058']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d65ce77ed60bf81328a8cca4b7d443fa" } } $('.js-work-strip[data-work-id=126812058]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812058,"title":"On the sum of $k$-th powers in terms of earlier sums","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"For k a positive integer let S k (n) = 1 k + 2 k + • • • + n k , i.e., S k (n) is the sum of the first k-th powers. Faulhaber conjectured (later proved by Jacobi) that for k odd, S k (n) can be written as a polynomial of S1(n), and for k even, S k (n) can be written as ). We give a proof of a variant of this result, namely that for any k there is a polynomial g k (x, y) such that S k (n) = g(S1(n), S2(n)). The novel proof yields a recursive formula to evaluate S k (n) as a polynomial of n that has roughly half the number of terms as the classical recursive formula that uses Pascal's identity.","publication_date":{"day":15,"month":12,"year":2019,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633507},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812058/On_the_sum_of_k_th_powers_in_terms_of_earlier_sums","translated_internal_url":"","created_at":"2025-01-05T02:41:54.198-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633507,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633507/thumbnails/1.jpg","file_name":"1912.pdf","download_url":"https://www.academia.edu/attachments/120633507/download_file","bulk_download_file_name":"On_the_sum_of_k_th_powers_in_terms_of_ea.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633507/1912-libre.pdf?1736077055=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_sum_of_k_th_powers_in_terms_of_ea.pdf\u0026Expires=1742723721\u0026Signature=IS-uDepmBKuD44ixfNk0tLCZqM5Q3L7NdywfyVMZvYy~o736HLstFkyOyuaBNyvLspA3SYZuOQV--wkJzH~WHkh-LlG6-WBeONGqjrB2ymcJ6H44PtsPuT9SjueJUJh7PVzV5Nm4qxM~IQmEi23OZFUapZIntjMYUG-fygzMGEwY7T1wxnE-jeri5THtl3tktAwam4d0xpEDtfoXM~AbiHtAAjGDc5gpkq1Buh3O2A7WIms-18Q8DnzkGLOt0DUzZpTw2JfQFcRWeemx5P3~sY3T3DBI3nbu6FN5tpW6onoE4af9mwjnS1MpyL~tuGxbb6R8hlP~Kf-p4bEAs4xNpA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"On_the_sum_of_k_th_powers_in_terms_of_earlier_sums","translated_slug":"","page_count":5,"language":"en","content_type":"Work","summary":"For k a positive integer let S k (n) = 1 k + 2 k + • • • + n k , i.e., S k (n) is the sum of the first k-th powers. Faulhaber conjectured (later proved by Jacobi) that for k odd, S k (n) can be written as a polynomial of S1(n), and for k even, S k (n) can be written as ). We give a proof of a variant of this result, namely that for any k there is a polynomial g k (x, y) such that S k (n) = g(S1(n), S2(n)). The novel proof yields a recursive formula to evaluate S k (n) as a polynomial of n that has roughly half the number of terms as the classical recursive formula that uses Pascal's identity.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller","email":"bkFGOHBvdmplZVk0WmU3NHJTVFlzVWpNbGpGQ0ZXZ0tIK2N5NEx2a2xyMzFTZkFCMDByQUJNcVRXSXhLWjNpZi0tbDBvYTdyQktMemRSOVRScFF2SVpqUT09--2c7a05e1a51448d8b9c59776c733f86e0e02f0d8"},"attachments":[{"id":120633507,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633507/thumbnails/1.jpg","file_name":"1912.pdf","download_url":"https://www.academia.edu/attachments/120633507/download_file","bulk_download_file_name":"On_the_sum_of_k_th_powers_in_terms_of_ea.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633507/1912-libre.pdf?1736077055=\u0026response-content-disposition=attachment%3B+filename%3DOn_the_sum_of_k_th_powers_in_terms_of_ea.pdf\u0026Expires=1742723721\u0026Signature=IS-uDepmBKuD44ixfNk0tLCZqM5Q3L7NdywfyVMZvYy~o736HLstFkyOyuaBNyvLspA3SYZuOQV--wkJzH~WHkh-LlG6-WBeONGqjrB2ymcJ6H44PtsPuT9SjueJUJh7PVzV5Nm4qxM~IQmEi23OZFUapZIntjMYUG-fygzMGEwY7T1wxnE-jeri5THtl3tktAwam4d0xpEDtfoXM~AbiHtAAjGDc5gpkq1Buh3O2A7WIms-18Q8DnzkGLOt0DUzZpTw2JfQFcRWeemx5P3~sY3T3DBI3nbu6FN5tpW6onoE4af9mwjnS1MpyL~tuGxbb6R8hlP~Kf-p4bEAs4xNpA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":2254318,"name":"Sums of Powers","url":"https://www.academia.edu/Documents/in/Sums_of_Powers"},{"id":3777178,"name":"Polynomial","url":"https://www.academia.edu/Documents/in/Polynomial"}],"urls":[{"id":46362528,"url":"http://arxiv.org/pdf/1912.07171"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812056"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812056/Infinite_Families_of_Partitions_into_MSTD_Subsets"><img alt="Research paper thumbnail of Infinite Families of Partitions into MSTD Subsets" class="work-thumbnail" src="https://attachments.academia-assets.com/120633506/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812056/Infinite_Families_of_Partitions_into_MSTD_Subsets">Infinite Families of Partitions into MSTD Subsets</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 16, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A set A is MSTD (more-sum-than-difference) if |A+A| > |A-A|. Though MSTD sets are rare, Martin an...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A set A is MSTD (more-sum-than-difference) if |A+A| > |A-A|. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r → ∞. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r → ∞. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition {1, 2, . . . , r} (for r sufficiently large) into k ≥ 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r ≥ R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r → ∞.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="76e9c566b50ae50f5dfe17c9ba00bd29" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633506,"asset_id":126812056,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633506/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812056"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812056"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812056; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812056]").text(description); $(".js-view-count[data-work-id=126812056]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812056; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812056']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "76e9c566b50ae50f5dfe17c9ba00bd29" } } $('.js-work-strip[data-work-id=126812056]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812056,"title":"Infinite Families of Partitions into MSTD Subsets","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A set A is MSTD (more-sum-than-difference) if |A+A| \u003e |A-A|. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r → ∞. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r → ∞. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition {1, 2, . . . , r} (for r sufficiently large) into k ≥ 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r ≥ R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r → ∞.","publication_date":{"day":16,"month":8,"year":2018,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633506},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812056/Infinite_Families_of_Partitions_into_MSTD_Subsets","translated_internal_url":"","created_at":"2025-01-05T02:41:53.861-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633506,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633506/thumbnails/1.jpg","file_name":"1808.05460.pdf","download_url":"https://www.academia.edu/attachments/120633506/download_file","bulk_download_file_name":"Infinite_Families_of_Partitions_into_MST.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633506/1808.05460-libre.pdf?1736077068=\u0026response-content-disposition=attachment%3B+filename%3DInfinite_Families_of_Partitions_into_MST.pdf\u0026Expires=1742696126\u0026Signature=Di~F3NUvmUNECeZqUbMZrai2Clr30AtlAlnBJxheQUQbdNAZk7k1pn3355GP5WOlXNse0423EcCNn5TqNtsIUBbqMfCfK~DJgmEn9DTMX4ymWgE6PdoCXtXCwJJ4Tm7-DjUEsi~hDH5rsvdDrXfk2mdoSYW6QaR9e9hX3hN1XsPAG4g5X2kyMU1V92WEm-1Wt6wc7eWRbbMQhlXj2Fa-d2rHPB97-gYRLH4T7Aunvm1DJCJFVCZCeDdyGqZZtrAi5vK7m6iHb7dTOoiE9pjaDhjXmbT35~oM8pFaVbHok7kpfM-Rpteo45lLeJk17pI71l8VZwTz5cqv6Jx6o6YHUg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Infinite_Families_of_Partitions_into_MSTD_Subsets","translated_slug":"","page_count":20,"language":"en","content_type":"Work","summary":"A set A is MSTD (more-sum-than-difference) if |A+A| \u003e |A-A|. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of {1, 2, . . . , r} as r → ∞. Later, Asada et al. showed that there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into two MSTD subsets as r → ∞. However, the method is probabilistic and does not give explicit decompositions. Continuing this work, we provide an efficient method to partition {1, 2, . . . , r} (for r sufficiently large) into k ≥ 2 MSTD subsets, positively answering a question raised by Asada et al. as to whether this is possible for all such k. Next, let R(k) be the smallest integer such that for all r ≥ R(k), {1, 2, . . . , r} can be k-decomposed into MSTD subsets. We establish rough lower and upper bounds for R(k). Lastly, we provide a sufficient condition on when there exists a positive constant lower bound for the proportion of decompositions of {1, 2, . . . , r} into k MSTD subsets as r → ∞.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633506,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633506/thumbnails/1.jpg","file_name":"1808.05460.pdf","download_url":"https://www.academia.edu/attachments/120633506/download_file","bulk_download_file_name":"Infinite_Families_of_Partitions_into_MST.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633506/1808.05460-libre.pdf?1736077068=\u0026response-content-disposition=attachment%3B+filename%3DInfinite_Families_of_Partitions_into_MST.pdf\u0026Expires=1742696126\u0026Signature=Di~F3NUvmUNECeZqUbMZrai2Clr30AtlAlnBJxheQUQbdNAZk7k1pn3355GP5WOlXNse0423EcCNn5TqNtsIUBbqMfCfK~DJgmEn9DTMX4ymWgE6PdoCXtXCwJJ4Tm7-DjUEsi~hDH5rsvdDrXfk2mdoSYW6QaR9e9hX3hN1XsPAG4g5X2kyMU1V92WEm-1Wt6wc7eWRbbMQhlXj2Fa-d2rHPB97-gYRLH4T7Aunvm1DJCJFVCZCeDdyGqZZtrAi5vK7m6iHb7dTOoiE9pjaDhjXmbT35~oM8pFaVbHok7kpfM-Rpteo45lLeJk17pI71l8VZwTz5cqv6Jx6o6YHUg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":524840,"name":"Integers","url":"https://www.academia.edu/Documents/in/Integers"}],"urls":[{"id":46362526,"url":"https://arxiv.org/pdf/1808.05460.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812055"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812055/A_Quick_Introduction_to_Benford%CA%BCs_Law"><img alt="Research paper thumbnail of A Quick Introduction to Benfordʼs Law" class="work-thumbnail" src="https://attachments.academia-assets.com/120633505/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812055/A_Quick_Introduction_to_Benford%CA%BCs_Law">A Quick Introduction to Benfordʼs Law</a></div><div class="wp-workCard_item"><span>Princeton University Press eBooks</span><span>, May 26, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The history of Benford's Law is a fascinating and unexpected story of the interplay between theor...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The history of Benford's Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford's Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford's Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford's Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford's Law on a solid foundation. There we explore several different categorizations of Benford's Law, and rigorously prove that certain systems satisfy these conditions.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="73a18c257fa50e33d7e6227b926ac13d" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633505,"asset_id":126812055,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633505/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812055"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812055"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812055; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812055]").text(description); $(".js-view-count[data-work-id=126812055]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812055; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812055']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "73a18c257fa50e33d7e6227b926ac13d" } } $('.js-work-strip[data-work-id=126812055]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812055,"title":"A Quick Introduction to Benfordʼs Law","translated_title":"","metadata":{"publisher":"Princeton University Press","grobid_abstract":"The history of Benford's Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford's Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford's Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford's Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford's Law on a solid foundation. There we explore several different categorizations of Benford's Law, and rigorously prove that certain systems satisfy these conditions.","publication_date":{"day":26,"month":5,"year":2015,"errors":{}},"publication_name":"Princeton University Press eBooks","grobid_abstract_attachment_id":120633505},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812055/A_Quick_Introduction_to_Benford%CA%BCs_Law","translated_internal_url":"","created_at":"2025-01-05T02:41:53.534-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633505,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633505/thumbnails/1.jpg","file_name":"s10527.pdf","download_url":"https://www.academia.edu/attachments/120633505/download_file","bulk_download_file_name":"A_Quick_Introduction_to_Benfords_Law.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633505/s10527-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DA_Quick_Introduction_to_Benfords_Law.pdf\u0026Expires=1742779652\u0026Signature=DIPmJPUNLgh2vNUGgAQe1Xe21ZK0BZVcaZ6ZRwYAqMCyn9qU7KcFJUjTlZp3eRyQAklptFelv~eZ9L0LeNT9kecG~CPEDleKug6moc0OB6TXefGJsY6WYVEEfQ2tT-WsmjKBcZkE-D0stN6QfERqIQcmX7WJvBuL5avcSm8O2fOA3GkKqKB3CM7oi-pYN8AtJw6gIPjyq5hGb2GOP8tpCzD6RDyALCHImcSFZFUnJwff1skBUxkqC7tSSpywca5XJu0gfAfjvKyEQIZveeZpqc2H3801rBYjEqrNQnio~kTLaMmLB6opA7KgNPWRU5JyMBtVgs7TK9pAXd9KC3LyEg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_Quick_Introduction_to_Benfordʼs_Law","translated_slug":"","page_count":16,"language":"en","content_type":"Work","summary":"The history of Benford's Law is a fascinating and unexpected story of the interplay between theory and applications. From its beginnings in understanding the distribution of digits in tables of logarithms, the subject has grown enormously. Currently hundreds of papers are being written by accountants, computer scientists, engineers, mathematicians, statisticians and many others. In this chapter we start by stating Benford's Law of digit bias and describing its history. We discuss its origins and give numerous examples of data sets that follow this law, as well as some that do not. From these examples we extract several explanations as to the prevalence of Benford's Law, which are described in greater detail later in the book. We end by quickly summarizing many of the diverse situations in which Benford's Law holds, and why an observation that began in looking at the wear and tear in tables of logarithms has become a major tool in subjects as diverse as detecting tax fraud and building efficient computers. We then continue in the next chapters with rigorous derivations, and then launch into a survey of some of the many applications. In particular, in the next chapter we put Benford's Law on a solid foundation. There we explore several different categorizations of Benford's Law, and rigorously prove that certain systems satisfy these conditions.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633505,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633505/thumbnails/1.jpg","file_name":"s10527.pdf","download_url":"https://www.academia.edu/attachments/120633505/download_file","bulk_download_file_name":"A_Quick_Introduction_to_Benfords_Law.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633505/s10527-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DA_Quick_Introduction_to_Benfords_Law.pdf\u0026Expires=1742779652\u0026Signature=DIPmJPUNLgh2vNUGgAQe1Xe21ZK0BZVcaZ6ZRwYAqMCyn9qU7KcFJUjTlZp3eRyQAklptFelv~eZ9L0LeNT9kecG~CPEDleKug6moc0OB6TXefGJsY6WYVEEfQ2tT-WsmjKBcZkE-D0stN6QfERqIQcmX7WJvBuL5avcSm8O2fOA3GkKqKB3CM7oi-pYN8AtJw6gIPjyq5hGb2GOP8tpCzD6RDyALCHImcSFZFUnJwff1skBUxkqC7tSSpywca5XJu0gfAfjvKyEQIZveeZpqc2H3801rBYjEqrNQnio~kTLaMmLB6opA7KgNPWRU5JyMBtVgs7TK9pAXd9KC3LyEg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":724,"name":"Economics","url":"https://www.academia.edu/Documents/in/Economics"}],"urls":[{"id":46362525,"url":"http://assets.press.princeton.edu/chapters/s10527.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812054"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812054/Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences"><img alt="Research paper thumbnail of Legal Decompositions Arising from Non-positive Linear Recurrences" class="work-thumbnail" src="https://attachments.academia-assets.com/120633504/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812054/Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences">Legal Decompositions Arising from Non-positive Linear Recurrences</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jun 30, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adj...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s, b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="944b47ffb00cb3be0a08b923615cec4a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633504,"asset_id":126812054,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633504/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812054"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812054"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812054; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812054]").text(description); $(".js-view-count[data-work-id=126812054]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812054; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812054']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "944b47ffb00cb3be0a08b923615cec4a" } } $('.js-work-strip[data-work-id=126812054]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812054,"title":"Legal Decompositions Arising from Non-positive Linear Recurrences","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s, b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.","publication_date":{"day":30,"month":6,"year":2016,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633504},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812054/Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences","translated_internal_url":"","created_at":"2025-01-05T02:41:53.209-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633504,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633504/thumbnails/1.jpg","file_name":"1606.09312.pdf","download_url":"https://www.academia.edu/attachments/120633504/download_file","bulk_download_file_name":"Legal_Decompositions_Arising_from_Non_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633504/1606.09312-libre.pdf?1736077064=\u0026response-content-disposition=attachment%3B+filename%3DLegal_Decompositions_Arising_from_Non_po.pdf\u0026Expires=1742779652\u0026Signature=aGIrPGlqzNkWN3Keovq0yX46Fs7ZUO5fTEiPQMoUoHYyiRkLVqI~y44YksJis194j0-YKI~-1Ysx1grnlDC6Rdb4AY-eqMBceJZizwnqG3y8VIYh-Ux50O~J0Ux-jfCuNDjYcFhw6LdlSs0oRP8rYeBCMRM9arUEfwfBbdGZW4Yu6nHgOQN27c2rnPohwNaYiEXJEiQFNlS5moszb~O9d0uHfbgNmI~jtcxSCEYciUIfz1r5PrY180tphoTh4qmSRQOJCzVr6JNyzA8rCDDJ2fOGTLUSIC6RpQXAM0cPQFXhRoQfejIypiGU6Uy5JnFj8G~YcV3J6MwnEUtPLuNJMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Legal_Decompositions_Arising_from_Non_positive_Linear_Recurrences","translated_slug":"","page_count":21,"language":"en","content_type":"Work","summary":"Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with leading term positive. We investigate legal decompositions arising from two new sequences: the (s, b)-Generacci sequence and the Fibonacci Quilt sequence. Both satisfy recurrence relations with leading term zero, and thus previous results and techniques do not apply. These sequences exhibit drastically different behavior. We show that the (s, b)-Generacci sequence leads to unique legal decompositions, whereas not only do we have non-unique legal decompositions with the Fibonacci Quilt sequence, we also have that in this case the average number of legal decompositions grows exponentially. Another interesting difference is that while in the (s, b)-Generacci case the greedy algorithm always leads to a legal decomposition, in the Fibonacci Quilt setting the greedy algorithm leads to a legal decomposition (approximately) 93% of the time. In the (s, b)-Generacci case, we again have Gaussian behavior in the number of summands as well as for the Fibonacci Quilt sequence when we restrict to decompositions resulting from a modified greedy algorithm.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633504,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633504/thumbnails/1.jpg","file_name":"1606.09312.pdf","download_url":"https://www.academia.edu/attachments/120633504/download_file","bulk_download_file_name":"Legal_Decompositions_Arising_from_Non_po.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633504/1606.09312-libre.pdf?1736077064=\u0026response-content-disposition=attachment%3B+filename%3DLegal_Decompositions_Arising_from_Non_po.pdf\u0026Expires=1742779652\u0026Signature=aGIrPGlqzNkWN3Keovq0yX46Fs7ZUO5fTEiPQMoUoHYyiRkLVqI~y44YksJis194j0-YKI~-1Ysx1grnlDC6Rdb4AY-eqMBceJZizwnqG3y8VIYh-Ux50O~J0Ux-jfCuNDjYcFhw6LdlSs0oRP8rYeBCMRM9arUEfwfBbdGZW4Yu6nHgOQN27c2rnPohwNaYiEXJEiQFNlS5moszb~O9d0uHfbgNmI~jtcxSCEYciUIfz1r5PrY180tphoTh4qmSRQOJCzVr6JNyzA8rCDDJ2fOGTLUSIC6RpQXAM0cPQFXhRoQfejIypiGU6Uy5JnFj8G~YcV3J6MwnEUtPLuNJMw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":550512,"name":"Fibonacci Number","url":"https://www.academia.edu/Documents/in/Fibonacci_Number"},{"id":1317768,"name":"Quilt","url":"https://www.academia.edu/Documents/in/Quilt"}],"urls":[{"id":46362524,"url":"https://arxiv.org/pdf/1606.09312.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812053"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812053/Geometric_progression_free_sets_over_quadratic_number_fields"><img alt="Research paper thumbnail of Geometric-progression-free sets over quadratic number fields" class="work-thumbnail" src="https://attachments.academia-assets.com/120633501/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812053/Geometric_progression_free_sets_over_quadratic_number_fields">Geometric-progression-free sets over quadratic number fields</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Dec 2, 2014</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a par...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets "greedily," a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a7e9150069b79c8a4f3ef8bae4c62de5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633501,"asset_id":126812053,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633501/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812053"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812053"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812053; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812053]").text(description); $(".js-view-count[data-work-id=126812053]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812053; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812053']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a7e9150069b79c8a4f3ef8bae4c62de5" } } $('.js-work-strip[data-work-id=126812053]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812053,"title":"Geometric-progression-free sets over quadratic number fields","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets \"greedily,\" a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.","publication_date":{"day":2,"month":12,"year":2014,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633501},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812053/Geometric_progression_free_sets_over_quadratic_number_fields","translated_internal_url":"","created_at":"2025-01-05T02:41:52.756-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633501,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633501/thumbnails/1.jpg","file_name":"1412.pdf","download_url":"https://www.academia.edu/attachments/120633501/download_file","bulk_download_file_name":"Geometric_progression_free_sets_over_qua.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633501/1412-libre.pdf?1736077067=\u0026response-content-disposition=attachment%3B+filename%3DGeometric_progression_free_sets_over_qua.pdf\u0026Expires=1742779652\u0026Signature=KXOFW-S3vMqfipT-OFp7aCTFNZzZ2VCio9Q2k71UCh0mPaLrAKapEwodS7iYkoyuUNirceI2vq09GxRDTmnzL6luW8NGDaIS2~lJIs-qeuzp7QDfSp-32yVWInA947zYXC7tIl1ARN0flK-pqTek-awg7ZztPBJakbuK9Ip4S6ahUDe6bfGf2zpgqdorDCQojKvrGxOfB3CnYVZkcakiHGVo0C7b~cZf3DZU2ViwtnaOsONv6vwdJzuf-0MRx40HdsYCT-I6kPLJcyfLzMQ1W7IOXQ8frRthufJbWyobWgvNvrkU2yFPHow7iR37DtCHNbufhNHorJx5~a88V1ozRw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Geometric_progression_free_sets_over_quadratic_number_fields","translated_slug":"","page_count":19,"language":"en","content_type":"Work","summary":"In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding 3-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid 3-term geometric progressions. When unique factorization fails or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets \"greedily,\" a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometricprogression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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Many of the mo...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset S + S = {x + y : x, y ∈ S} of a set of integers S. A finite set of integers A is sumdominant if |A + A| > |A -A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately 4.5 • 10 -4 ). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that |A + A| = |A -A| tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of {0, . . ., n} have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6e2bf624b1cb5fb9f06c544c8f9444fe" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633502,"asset_id":126812051,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633502/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812051"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812051"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812051; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812051]").text(description); $(".js-view-count[data-work-id=126812051]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812051; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812051']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "6e2bf624b1cb5fb9f06c544c8f9444fe" } } $('.js-work-strip[data-work-id=126812051]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812051,"title":"Most Subsets are Balanced in Finite Groups","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset S + S = {x + y : x, y ∈ S} of a set of integers S. A finite set of integers A is sumdominant if |A + A| \u003e |A -A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately 4.5 • 10 -4 ). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that |A + A| = |A -A| tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of {0, . . ., n} have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.","publication_date":{"day":10,"month":8,"year":2013,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633502},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812051/Most_Subsets_are_Balanced_in_Finite_Groups","translated_internal_url":"","created_at":"2025-01-05T02:41:52.105-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633502,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633502/thumbnails/1.jpg","file_name":"1308.pdf","download_url":"https://www.academia.edu/attachments/120633502/download_file","bulk_download_file_name":"Most_Subsets_are_Balanced_in_Finite_Grou.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633502/1308-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DMost_Subsets_are_Balanced_in_Finite_Grou.pdf\u0026Expires=1742723722\u0026Signature=UIg9WSnK~EwcYa8uz-sXVfiEHg2Kh4hPenVza3tr0QXHFy769TCMWT61MD-QyZkzHUPjW07ziX2UpqD4vyDPgGMPo1NkAP2MCDJCOQqoE8Zn9x6VRUNsUC0w1WkY3rAQAvoF~C3k6evhUkpQ1Ymq8KMSbx3qnE4dTmqfn8duyW7bLZ8kYg2wK8jowV59lHWHTUFHqFazsWPZ2b8nPvJOdFNbOc3tGKnP-LIeMMcPwkDjrIgt6T~A9LhegI2q-4wXt7VmV1xFiQ2xD69GqZwuugOa3XSDvdlm5gp0L1CpRQCrZtjAELKia7l3ZrpvNPRpX~ULvGexw0DaVdRq3kau0Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Most_Subsets_are_Balanced_in_Finite_Groups","translated_slug":"","page_count":11,"language":"en","content_type":"Work","summary":"The sumset is one of the most basic and central objects in additive number theory. Many of the most important problems (such as Goldbach's conjecture and Fermat's Last theorem) can be formulated in terms of the sumset S + S = {x + y : x, y ∈ S} of a set of integers S. A finite set of integers A is sumdominant if |A + A| \u003e |A -A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O'Bryant proved a very small positive percentage are sum-dominant if the sets are chosen uniformly at random (through work of Zhao we know this percentage is approximately 4.5 • 10 -4 ). While most sets are difference-dominant in the integer case, this is not the case when we take subsets of many finite groups. We show that if we take subsets of larger and larger finite groups uniformly at random, then not only does the probability of a set being sum-dominant tend to zero but the probability that |A + A| = |A -A| tends to one, and hence a typical set is balanced in this case. The cause of this marked difference in behavior is that subsets of {0, . . ., n} have a fringe, whereas finite groups do not. We end with a detailed analysis of dihedral groups, where the results are in striking contrast to what occurs for subsets of integers.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller","email":"NUNDaU10MklWNmlDNnB0UmFmQzdMd1pnbFdKL2dqaWtzRGZFYTNnRnVwL0FaMkJtc1hNWWtBNDhnY2ZwditwcC0tKzhFWGM2Q3VQNktUWDZpeVF1RkRudz09--bdb86de1b2654cf51a36f2f6f904c711004a2100"},"attachments":[{"id":120633502,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633502/thumbnails/1.jpg","file_name":"1308.pdf","download_url":"https://www.academia.edu/attachments/120633502/download_file","bulk_download_file_name":"Most_Subsets_are_Balanced_in_Finite_Grou.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633502/1308-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DMost_Subsets_are_Balanced_in_Finite_Grou.pdf\u0026Expires=1742723722\u0026Signature=UIg9WSnK~EwcYa8uz-sXVfiEHg2Kh4hPenVza3tr0QXHFy769TCMWT61MD-QyZkzHUPjW07ziX2UpqD4vyDPgGMPo1NkAP2MCDJCOQqoE8Zn9x6VRUNsUC0w1WkY3rAQAvoF~C3k6evhUkpQ1Ymq8KMSbx3qnE4dTmqfn8duyW7bLZ8kYg2wK8jowV59lHWHTUFHqFazsWPZ2b8nPvJOdFNbOc3tGKnP-LIeMMcPwkDjrIgt6T~A9LhegI2q-4wXt7VmV1xFiQ2xD69GqZwuugOa3XSDvdlm5gp0L1CpRQCrZtjAELKia7l3ZrpvNPRpX~ULvGexw0DaVdRq3kau0Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":37345,"name":"Discrete Mathematics","url":"https://www.academia.edu/Documents/in/Discrete_Mathematics"},{"id":1463317,"name":"Contents","url":"https://www.academia.edu/Documents/in/Contents"},{"id":2239835,"name":"Finite abelian groups","url":"https://www.academia.edu/Documents/in/Finite_abelian_groups"},{"id":2570814,"name":"conjecture","url":"https://www.academia.edu/Documents/in/conjecture"}],"urls":[{"id":46362522,"url":"http://arxiv.org/pdf/1308.2344"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812050"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812050/Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes"><img alt="Research paper thumbnail of Sets characterized by missing sums and differences in dilating polytopes" class="work-thumbnail" src="https://attachments.academia-assets.com/120633533/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812050/Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes">Sets characterized by missing sums and differences in dilating polytopes</a></div><div class="wp-workCard_item"><span>Journal of Number Theory</span><span>, Dec 1, 2015</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A sum-dominant set is a finite set A of integers such that |A + A| > |A -A|. As a typical pair of...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A sum-dominant set is a finite set A of integers such that |A + A| > |A -A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sumdominant subsets of {0, . . . , n} is bounded below by a positive constant as n → ∞. Hegarty then extended their work and showed that for any prescribed s, d ∈ N 0 , the proportion ρ s,d n of subsets of {0, . . . , n} that are missing exactly s sums in {0, . . . , 2n} and exactly 2d differences in {-n, . . . , n} also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) -L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9ebb998152265fbdb7ccc8b7ff9e7895" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633533,"asset_id":126812050,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633533/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812050"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812050"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812050; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812050]").text(description); $(".js-view-count[data-work-id=126812050]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812050; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812050']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9ebb998152265fbdb7ccc8b7ff9e7895" } } $('.js-work-strip[data-work-id=126812050]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812050,"title":"Sets characterized by missing sums and differences in dilating polytopes","translated_title":"","metadata":{"publisher":"Elsevier BV","grobid_abstract":"A sum-dominant set is a finite set A of integers such that |A + A| \u003e |A -A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sumdominant subsets of {0, . . . , n} is bounded below by a positive constant as n → ∞. Hegarty then extended their work and showed that for any prescribed s, d ∈ N 0 , the proportion ρ s,d n of subsets of {0, . . . , n} that are missing exactly s sums in {0, . . . , 2n} and exactly 2d differences in {-n, . . . , n} also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) -L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit.","publication_date":{"day":1,"month":12,"year":2015,"errors":{}},"publication_name":"Journal of Number Theory","grobid_abstract_attachment_id":120633533},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812050/Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes","translated_internal_url":"","created_at":"2025-01-05T02:41:51.322-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633533,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633533/thumbnails/1.jpg","file_name":"MSTD_ddim52.pdf","download_url":"https://www.academia.edu/attachments/120633533/download_file","bulk_download_file_name":"Sets_characterized_by_missing_sums_and_d.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633533/MSTD_ddim52-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DSets_characterized_by_missing_sums_and_d.pdf\u0026Expires=1742779652\u0026Signature=ZQ-8fncJLShFLjQuVOWL-l21JPfA1zIAn2MWzW9ze5rNrQnWz3u18fW~T9qQ7Zz5ynpPp3WGMjUdM3bJVoO5srFQwweRutrs03RWEUcHGzeCqparh7mwgXsGDs6Vg-fANx0-7gDvUbsInGuOMBov5K9rffl3hRDELexxr5nVWFecMWkf6RoI-2h~rlz358WslTXYWcTn1Xo22ARMupT1XyCfMMDhfz97aLQObTm9KEzMDlSTiHS4O-IOlwIA8zQOeglhju9WWXWQ2-QKp6KWEUk2~d53koHFQ52i1IjEmfzTs941bM1PrPCGBZPtLNq8E7aJXwoy4Bg1PsVZ8qsACA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Sets_characterized_by_missing_sums_and_differences_in_dilating_polytopes","translated_slug":"","page_count":23,"language":"en","content_type":"Work","summary":"A sum-dominant set is a finite set A of integers such that |A + A| \u003e |A -A|. 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Specifically, let P be a polytope in R D with vertices in Z D , and let ρ s,d n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) -L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in L(nP ) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P . A direct corollary of these results is that if P is additionally point symmetric, the proportion of sum-dominant subsets of L(nP ) also remains positive in the limit.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. Miller","url":"https://williams.academia.edu/StevenJMiller"},"attachments":[{"id":120633533,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633533/thumbnails/1.jpg","file_name":"MSTD_ddim52.pdf","download_url":"https://www.academia.edu/attachments/120633533/download_file","bulk_download_file_name":"Sets_characterized_by_missing_sums_and_d.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633533/MSTD_ddim52-libre.pdf?1736077061=\u0026response-content-disposition=attachment%3B+filename%3DSets_characterized_by_missing_sums_and_d.pdf\u0026Expires=1742779652\u0026Signature=ZQ-8fncJLShFLjQuVOWL-l21JPfA1zIAn2MWzW9ze5rNrQnWz3u18fW~T9qQ7Zz5ynpPp3WGMjUdM3bJVoO5srFQwweRutrs03RWEUcHGzeCqparh7mwgXsGDs6Vg-fANx0-7gDvUbsInGuOMBov5K9rffl3hRDELexxr5nVWFecMWkf6RoI-2h~rlz358WslTXYWcTn1Xo22ARMupT1XyCfMMDhfz97aLQObTm9KEzMDlSTiHS4O-IOlwIA8zQOeglhju9WWXWQ2-QKp6KWEUk2~d53koHFQ52i1IjEmfzTs941bM1PrPCGBZPtLNq8E7aJXwoy4Bg1PsVZ8qsACA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":1463317,"name":"Contents","url":"https://www.academia.edu/Documents/in/Contents"},{"id":2138291,"name":"Polytope","url":"https://www.academia.edu/Documents/in/Polytope"}],"urls":[{"id":46362521,"url":"https://doi.org/10.1016/j.jnt.2015.04.027"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="126812038"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/126812038/Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions"><img alt="Research paper thumbnail of Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions" class="work-thumbnail" src="https://attachments.academia-assets.com/120633491/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/126812038/Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions">Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Aug 8, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="38b6e030b30b9765d8a2b10cd75b8bce" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":120633491,"asset_id":126812038,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/120633491/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="126812038"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="126812038"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 126812038; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=126812038]").text(description); $(".js-view-count[data-work-id=126812038]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 126812038; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='126812038']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "38b6e030b30b9765d8a2b10cd75b8bce" } } $('.js-work-strip[data-work-id=126812038]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":126812038,"title":"Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.","publication_date":{"day":8,"month":8,"year":2005,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":120633491},"translated_abstract":null,"internal_url":"https://www.academia.edu/126812038/Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions","translated_internal_url":"","created_at":"2025-01-05T02:40:30.404-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":13733,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":120633491,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/120633491/thumbnails/1.jpg","file_name":"0508150.pdf","download_url":"https://www.academia.edu/attachments/120633491/download_file","bulk_download_file_name":"Investigations_of_Zeros_Near_the_Central.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/120633491/0508150-libre.pdf?1736077076=\u0026response-content-disposition=attachment%3B+filename%3DInvestigations_of_Zeros_Near_the_Central.pdf\u0026Expires=1742723722\u0026Signature=JbzWFpMXDVPYLgaRvq7ynvnC8koxxQlxO38LR8ywtefXhS5bSLzrW0~GwbWDTuMzAEwJMNCxmQgmuDsyUfxkDWehnrmopSELmL9EKYrG5P0iTu3LI2YMKTX-bL4iBOWbDIlmtRFtYVopaIwVT4wU4HWwtyL89jerPtlzq2hZlicozv-gozwXEBuJH9tKA5sIj2~94zyrnB36BJmde1voP4iaGygibMcppqg3zlj0iqJHCs3Az~QD2JU2s1ZUtD~UBICWU1sLUkdguuf5undPHhT6zE1flcDMCjOOyNd8fWCO4mDP~1~fl33eVOL3s2MoJdJ1BIa8~8wK8NsgXTZkvQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Investigations_of_Zeros_Near_the_Central_Point_of_Elliptic_Curve_L_Functions","translated_slug":"","page_count":37,"language":"en","content_type":"Work","summary":"We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve Et in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater repulsion in the subset of curves of rank r + 2 than in the subset of curves of rank r in a rank r family. For curves with comparable conductors, the behavior of rank 2 curves in a rank 0 one-parameter family over Q is statistically different from that of rank 2 curves from a rank 2 family. Unlike excess rank calculations, the repulsion decreases markedly as the conductors increase, and we conjecture that the r family zeros do not repel in the limit. Finally, the differences between adjacent normalized zeros near the central point are statistically independent of the repulsion, family rank and rank of the curves in the subset. Specifically, the differences between adjacent normalized zeros are statistically equal for all curves investigated with rank 0, 2 or 4 and comparable conductors from one-parameter families of rank 0 or 2 over Q. (1.4) 3 We normalize all L-functions to have functional equation s → 1s, and thus central point is at s = 1 2 . 4 The group of rational function solutions (x(T ), y(T )) ∈ Q(T ) 2 to y 2 = x 3 + A(T )x + B(T ) is isomorphic to Z r ⊕ T, where T is the torsion part and r is the rank.","owner":{"id":13733,"first_name":"Steven J.","middle_initials":null,"last_name":"Miller","page_name":"StevenJMiller","domain_name":"williams","created_at":"2008-10-30T03:37:58.032-07:00","display_name":"Steven J. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125376742"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125376742/One_Level_density_for_holomorphic_cusp_forms_of_arbitrary_level"><img alt="Research paper thumbnail of One-Level density for holomorphic cusp forms of arbitrary level" class="work-thumbnail" src="https://attachments.academia-assets.com/119432040/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125376742/One_Level_density_for_holomorphic_cusp_forms_of_arbitrary_level">One-Level density for holomorphic cusp forms of arbitrary level</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Apr 11, 2016</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomor...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milićević, which is of use for other problems as well.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="12bf6a7f9cc7a1e513875000846cc0df" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119432040,"asset_id":125376742,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119432040/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125376742"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125376742"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125376742; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125376742]").text(description); $(".js-view-count[data-work-id=125376742]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125376742; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125376742']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "12bf6a7f9cc7a1e513875000846cc0df" } } $('.js-work-strip[data-work-id=125376742]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125376742,"title":"One-Level density for holomorphic cusp forms of arbitrary level","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="125376741"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/125376741/Maine_Qu_ebec_Number_Theory_Conference_A_Shifted_Twisted_Second_Moment_and_Gaps_Between_Zeros_for_L_functions_Associated_to_Holomorphic_Cusp_Forms"><img alt="Research paper thumbnail of Maine-Qu ebec Number Theory Conference: A Shifted Twisted Second Moment and Gaps Between Zeros for L-functions Associated to Holomorphic Cusp Forms" class="work-thumbnail" src="https://attachments.academia-assets.com/119432026/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/125376741/Maine_Qu_ebec_Number_Theory_Conference_A_Shifted_Twisted_Second_Moment_and_Gaps_Between_Zeros_for_L_functions_Associated_to_Holomorphic_Cusp_Forms">Maine-Qu ebec Number Theory Conference: A Shifted Twisted Second Moment and Gaps Between Zeros for L-functions Associated to Holomorphic Cusp Forms</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Let f be a modular cusp form of weight κ and level q, f has a (normalized) Fourier series at infi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Let f be a modular cusp form of weight κ and level q, f has a (normalized) Fourier series at infinity: We are interested in the distribution of the zeros of our automorphic L-function along the critical line.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="041751bd9721c1f5f763928cbad8f50c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119432026,"asset_id":125376741,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119432026/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125376741"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125376741"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125376741; 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The simplest cases are base-b expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="d6a6c94d6316108647f4b1d4bcf3f4f8" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":119432024,"asset_id":125376740,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/119432024/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="125376740"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="125376740"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 125376740; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=125376740]").text(description); $(".js-view-count[data-work-id=125376740]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 125376740; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='125376740']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "d6a6c94d6316108647f4b1d4bcf3f4f8" } } $('.js-work-strip[data-work-id=125376740]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":125376740,"title":"A Probabilistic Approach to Generalized Zeckendorf Decompositions","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. 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