New improvement to Falconer distance set problem in higher dimensions

TL;DR

This paper advances the Falconer distance set problem in higher dimensions by establishing new bounds on the Hausdorff dimension of sets ensuring positive measure of pinned distance sets, improving previous results.

Contribution

It provides improved dimension thresholds for the Falconer distance problem and bounds for pinned distance sets in dimensions three and higher.

Findings

Established new dimension bounds for positive measure of pinned distance sets.

Improved previous bounds by Du-Zhang and Du-Iosevich-Ou-Wang-Zhang.

Derived lower bounds for Hausdorff dimension of pinned distance sets in specific ranges.

Abstract

We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive Lebesgue measure. This improves upon bounds of Du-Zhang and Du-Iosevich-Ou-Wang-Zhang in all dimensions $d \ge 3$. We also prove lower bounds for Hausdorff dimension of pinned distance sets when $\dim_H (E) \in (\frac{d}{2} - \frac{1}{4} - \frac{3}{8d+4}, \frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4})$, which improves upon bounds of Harris and Wang-Zheng in dimensions $d \ge 3$.