Document Zbl 0893.52015

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II: Lattice tilings.</strong> <i>(English)</i> <a class="label nowrap" href="/0893.52015">Zbl 0893.52015</a> </h2> <div class="source"> <a href="/serials/2184" title="Journal Profile">J. Fourier Anal. Appl.</a> <a href="/?q=in%3A37294" title="Articles in this Issue">3, No. 1, 83-102 (1997)</a>. </div> <div class="abstract">Summary: Let $A$ be an expanding $n\times n$ integer matrix with $|\text{det} (A) |=m$. A standard digit set ${\mathcal D}$ for $A$ is any complete set of coset representatives for $\mathbb{Z}^n/A (\mathbb{Z}^n)$. Associated to a given ${\mathcal D}$ is a set $T(A, {\mathcal D})$, which is the attractor of an affine iterated function system, satisfying $T=\cup_{d\in {\mathcal D}} (T+d)$. It is known that $T(A, {\mathcal D})$ tiles $\mathbb{R}^n$ by some subset of $\mathbb{Z}^n$.<br class="zbmathjax-paragraph">This paper proves that every standard digit set ${\mathcal D}$ gives a set $T(A, {\mathcal D})$ that tiles $\mathbb{R}^n$ with a lattice tiling.<br class="zbmathjax-paragraph">[For part I see the authors, J. Lond. Math. Soc., II. Ser. 54, No. 1, 161-179 (1996; see the paper above)].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A987898">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A987898">Cited in <strong>95</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A52C22" title="MSC2020">52C22</a> </td> <td class="space"> Tilings in $n$ dimensions (aspects of discrete geometry) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A28A80" title="MSC2020">28A80</a> </td> <td class="space"> Fractals </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A42C15" title="MSC2020">42C15</a> </td> <td class="space"> General harmonic expansions, frames </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52C07" title="MSC2020">52C07</a> </td> <td class="space"> Lattices and convex bodies in $n$ dimensions (aspects of discrete geometry) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adigit+set">digit set</a>; <a href="/?q=ut%3Aself-affine+tile">self-affine tile</a>; <a href="/?q=ut%3Alattice+tiling">lattice tiling</a>; <a href="/?q=ut%3Aquasi-product+form">quasi-product form</a>; <a href="/?q=ut%3Awavelet">wavelet</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0893.52014">Zbl 0893.52014</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0893.52015" data-ciurl="/ci/00987898" data-biburl="/bibtex/00987898.bib" data-amsurl="/amsrefs/00987898.bib" data-xmlurl="/xml/00987898.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00987898.pdf" title="Zbl 0893.52015 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF02647948" aria-label="DOI for “Integral self-affine tiles in $\mathbb{R}^n$. II: Lattice tilings”" title="10.1007/BF02647948">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://eudml.org/doc/59503" title="Open Access at EuDML">EuDML</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bandt, C. (1991). Self-similar sets 5. Integer matrices and fractal tilings ofR n .Proc. Amer. Math Soc. 112, 549–562. · <a href="/0743.58027" class="nowrap">Zbl 0743.58027</a></td> </tr><tr> <td>[2]</td> <td class="space">Berger, M. A., and Wang, Y. (1992). Multidimensional two-scale dilation equations. InWavelets. A Tutorial in Theory and Applications C. K. Chui, ed., Academic Press, New York, 295–323. · <a href="/0767.65002" class="nowrap">Zbl 0767.65002</a></td> </tr><tr> <td>[3]</td> <td class="space">Cassels, J. W. S. (1957). 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