On Falconer's distance set problem in the plane

TL;DR

This paper proves that in the plane, a compact set with Hausdorff dimension greater than 1.25 guarantees the existence of a point from which the set of distances to other points has positive measure.

Contribution

It advances Falconer's distance set problem by establishing a new threshold for the Hausdorff dimension in the plane.

Findings

Sets with Hausdorff dimension > 5/4 have a point with a distance set of positive measure.

The result improves previous bounds for the distance set problem in the plane.

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.