Analysis Seminar

Given a set $E$ of Hausdorff dimension $s > d/2$ in $\mathbb{R}^d$ , Falconer conjectured that its distance set $\Delta(E)=\{ |x-y|: x, y \in E\}$ should have positive Lebesgue measure. When $d$ is even, we show that $\dim_H E>d/2+1/4$ implies $|\Delta(E)| > 0$. This improves upon the work of Wolff, Erdogan, Du-Zhang, etc. Our tools include Orponen's radial projection theorem and refined decoupling estimates.

This is joint work with Guth, Iosevich, and Ou and with Du, Iosevich, Ou, and Zhang.