New Bounds on the Dimensions of Planar Distance Sets

By Tamás Keleti, Pablo Shmerkin. Geometric and Functional Analysis, 2018

48 citations recorded by the source corpus.

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\subset {\mathbb {R}}^2$$\end{document} is a Borel set of Hausdorff dimension s>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>1$$\end{document}, then its distance set has Hausdorff dimension at least 37/54≈0.685\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$37/54\approx 0.685$$\end{document}. Moreover, if s∈(1,3/2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (1,3/2]$$\end{document}, then outside of a set of exceptional y of Hausdorff dimension at most 1, the pinned distance set {|x-y|:x∈A}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ |x-y|:x\in A\}$$\end{document} has Hausdorff dimension ≥23s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge \tfrac{2}{3}s$$\end{document} and packing dimension at least 14(1+s+3s(2-s))≥0.933\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \tfrac{1}{4}(1+s+\sqrt{3s(2-s)}) \ge 0.933$$\end{document}. These estimates improve upon the existing ones by Bourgain, Wolff, Peres–Schlag and Iosevich–Liu for sets of Hausdorff dimension >1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$>1$$\end{document}. Our proof uses a multi-scale decomposition of measures in which, unlike previous works, we are able to choose the scales subject to certain constrains. This leads to a combinatorial problem, which is a key new ingredient of our approach, and which we solve completely by optimizing certain variation of Lipschitz functions.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/cff9d1530d82dd3dcf64cb8db63d31d951eed755

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Citations in the corpus

Weighted restriction estimates and application to Falconer distance set problem (2018, background): “We also point the interested reader to [12, 19, 21, 24] and the references therein for more results along this line of research.”
Incidence Estimates for Well Spaced Tubes (2019, methodology): “. Theorem 1.4 proves this conjecture up a factor of δǫ for sets E that are as widely spaced as possible. In the other direction, there has been some remarkable work by Orponen [16] and KeletiShmerkin [14] on the case when Eis tightly spaced. We say that Eis an Ahlfors-David regular set of δ-balls if”
Assouad Dimension and Fractal Geometry (2020, background): “That said, since 2010 the problem has been receiving a lot of attention in the literature, and significant progress is being made, see for example, [252, 253, 225, 167, 124].”
On Falconer’s distance set problem in the plane (2018, methodology): “ that the Hausdor dimension of a compact set E•R2 is equal to s¡1 and Kis as in Theorem 1.3. Then there exists xPEsuch that the upper Minkowski dimension of 2 x;KpEqis ¥ 4s 3 3 . Keleti and Shmerkin ([25]) obtained the lower bound 1 4 p1 s a 3sp2 sqqin the case of the Euclidean metric. Their estimat”
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane (2021, background): “The results of [19, 28] concern projections, and the information from different scales is put together by means of entropy.”