New bounds on the dimensions of planar distance sets

Abstract

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset {\mathbb{R}}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff dimension at least $37/54\approx 0.685$. Moreover, if $s\in (1,3/2]$, then outside of a set of exceptional $y$ of Hausdorff dimension at most $1$, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff dimension $\ge \frac{2}{3}s$ and packing dimension at least $\frac{1}{4}(1+s+\sqrt{3s(2-s)})\ge 0.933$. These estimates improve upon the existing ones by Bourgain, Wolff, Peres-Schlag and Iosevich-Liu for sets of Hausdorff dimension $>1$. 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.

60 pages, 2 figures. Incorporates referee comments. To appear in GAFA

References in corpus (5)

• On the dimension and smoothness of radial projections
• An $L^2$-identity and pinned distance problem
• On the Hausdorff dimension of pinned distance sets
• Pinned distance problem, slicing measures and local smoothing estimates
• On distance sets, box-counting and Ahlfors-regular sets

Cited by in corpus (12)

• On Falconer's distance set problem in the plane
• An $L^2$-identity and pinned distance problem
• Microlocal decoupling inequalities and the distance problem on Riemannian manifolds
• The Assouad dimension of Kakeya sets in ${\mathbb{R}}^3$
• On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem
• Hausdorff dimension of pinned distance sets and the $L^2$-method
• Intersection between pencils of tubes, discretized sum-product, and radial projections
• Embedding distance graphs in finite field vector spaces
• An improved dimensional threshold for the angle problem
• Inverse theorems for discretized sums and $L^q$ norms of convolutions in ${\mathbb{R}}^d$
• On Hausdorff dimension of radial projections
• Incidence estimates for well spaced tubes