New improvement to Falconer distance set problem in higher dimensions

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 $d\ge 3$, then there is a point $x\in E$ such that the pinned distance set $\hat{I}{”}_x(E)$ 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 $d\ge 3$.

33 pages; slightly changed the introduction