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<p>We establish the dimension version of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions <i>d</i>=2 or 3, we obtain the first explicit improvements over the classical 1/2 bound for the dimensions of distance sets of general Borel sets of dimension <i>d</i>/2. For example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension 1 has Hausdorff dimension at least <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2024_696_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="TEX">$(\sqrt{5}-1)/2\approx 0.618$</EquationSource> </InlineEquation>. In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension <i>d</i>/2. These results rely on new estimates for the dimensions of radial projections that may have independent interest.</p>

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On the Distance Sets Spanned by Sets of Dimension d/2 in \(\mathbb{R}^{d}\)

  • 期刊论文

  • 发表日期：

    2025年1月9日

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  • Pablo Shmerkin,
  • Hong Wang

摘要

We establish the dimension version of Falconer’s distance set conjecture for sets of equal Hausdorff and packing dimension (in particular, for Ahlfors-regular sets) in all ambient dimensions. In dimensions d=2 or 3, we obtain the first explicit improvements over the classical 1/2 bound for the dimensions of distance sets of general Borel sets of dimension d/2. For example, we show that the set of distances spanned by a planar Borel set of Hausdorff dimension 1 has Hausdorff dimension at least $(\sqrt{5}-1)/2\approx 0.618$ . In higher dimensions we obtain explicit estimates for the lower Minkowski dimension of the distance sets of sets of dimension d/2. These results rely on new estimates for the dimensions of radial projections that may have independent interest.