Document Zbl 1337.28015

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Math. (2)</a> <a href="/?q=in%3A336262" title="Articles in this Issue">180, No. 2, 773-822 (2014)</a>. </div> <div class="abstract">Let $\Phi :=\{\varphi_i : i\in \Lambda\}$ be a finite collection of linear contractions of the form $\varphi_i (x) := r_i x + a_i$, where $|r_i| < 1$ and $a_i\in \mathbb R$. Let $X\neq\emptyset$ be the attractor of the iterated function system $\Phi$ and let the self-similar measure associated with $\Phi$ and a probability vector $(p_i)_{i\in \Lambda}$ be denoted by $\mu$. The author investigates the Hausdorff dimension $\dim \mu$ of $\mu$ in case the images $\varphi_i X$ have significant overlap. The main result is the following: If $\dim\mu < \min\{1, s\}$, where $s$ is the similarity dimension of $X$, then there exist superexponentially close cylinders at small enough scales. As a consequence of this result, the Furstenberg conjecture on the projections of the one-dimensional Sierpiński gasket is proven and some progress on the Bernoulli convolutions problem is achieved.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A13393">Peter Massopust (München)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6346461">Cited in <strong>9</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6346461">Cited in <strong>158</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A37L40" title="MSC2020">37L40</a> </td> <td class="space"> Invariant measures for infinite-dimensional dissipative dynamical systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37C35" title="MSC2020">37C35</a> </td> <td class="space"> Orbit growth in dynamical systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A28D20" title="MSC2020">28D20</a> </td> <td class="space"> Entropy and other invariants </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aadditive+combinatorics">additive combinatorics</a>; <a href="/?q=ut%3ABernoulli+convolutions">Bernoulli convolutions</a>; <a href="/?q=ut%3Aconvolution">convolution</a>; <a href="/?q=ut%3Aentropy">entropy</a>; <a href="/?q=ut%3AHausdorff+dimension">Hausdorff dimension</a>; <a href="/?q=ut%3AMarstrand%27s+projection+theorem">Marstrand’s projection theorem</a>; <a href="/?q=ut%3Aself-similarity">self-similarity</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 1337.28015" data-ciurl="/ci/06346461" data-biburl="/bibtex/06346461.bib" data-amsurl="/amsrefs/06346461.bib" data-xmlurl="/xml/06346461.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06346461.pdf" title="Zbl 1337.28015 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/10.4007/annals.2014.180.2.7" aria-label="DOI for “On self-similar sets with overlaps and inverse theorems for entropy”" title="10.4007/annals.2014.180.2.7">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1212.1873"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">D. 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