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

IF 2.4 1区 数学 Q1 MATHEMATICS
Pablo Shmerkin, Hong Wang
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引用次数: 0

Abstract

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.

$\mathbb{R}^{d}$中d/2维集张成的距离集
我们建立了Falconer距离集猜想的维度版本,适用于所有环境维中相等的Hausdorff和包装维数的集合(特别是对于ahlfors -正则集)。在维数d=2或3的情况下,我们首次获得了维数d/2的一般Borel集的距离集维数在经典1/2界上的显式改进。例如,我们证明了一个Hausdorff维数为1的平面Borel集所张成的距离集的Hausdorff维数至少为\((\sqrt{5}-1)/2\approx 0.618\)。在高维中,我们得到了维数为d/2的集合的距离集的下闵可夫斯基维的显式估计。这些结果依赖于对可能具有独立意义的径向投影尺寸的新估计。
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来源期刊
CiteScore
3.70
自引率
4.50%
发文量
34
审稿时长
6-12 weeks
期刊介绍: Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.
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