---
abstract: 'If $E \subset {\mathbb{R}}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x \in E$ so that the set of distances $\{ |x-y| \}_{y \in E}$ has positive Lebesgue measure.'
author:
- 'Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang'
title: 'On Falconer’s distance set problem in the plane'
---

introduction
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.125in

For a set $E \subset {\mathbb{R}}^d$, define the distance set

$$\Delta(E)=\{|p-p'|: p,p' \in E\}.$$

Falconer’s distance problem ([@Falc86]) is about the connection between the Hausdorff dimension of a set $E$ and the size of $\Delta(E)$. Given a compact set $E$ in ${\Bbb R}^d$, $d \ge 2$, Falconer asked how large the Hausdorff dimension of $E$ needs to be to ensure that the Lebesgue measure of $\Delta(E)$ is positive. He proved that if $dim_{{\mathcal H}}(E)>\frac{d+1}{2}$, then ${\mathcal L}(\Delta(E))>0$. Using an example based on the integer lattice, he showed for every $s \leq \frac{d}{2}$ there exist sets of Hausdorff dimension $s$ for which ${\mathcal L}(\Delta(E))=0$. This led him to conjecture that if $dim_{{\mathcal H}}(E)>\frac{d}{2}$, then the Lebesgue measure of the distance set is positive. This is known as the Falconer Distance Conjecture.

In [@W99], Wolff proved that 