New bounds on the dimensions of planar distance sets

TL;DR

This paper establishes improved lower bounds on the Hausdorff and packing dimensions of distance sets and pinned distance sets for planar sets with Hausdorff dimension greater than one, advancing previous results.

Contribution

It introduces a novel multi-scale measure decomposition technique with constrained scale choices, leading to sharper bounds on distance set dimensions.

Findings

Distance set of a set with Hausdorff dimension >1 has dimension at least 37/54.

Pinned distance sets have Hausdorff dimension at least 2/3 times the original set's dimension.

Pinned distance sets have packing dimension at least 0.933 under certain conditions.

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 $\ge \tfrac{2}{3}s$ and packing dimension at least $ \tfrac{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.