Weighted refined decoupling estimates and application to Falconer distance set problem

TL;DR

This paper develops weighted refined decoupling estimates and applies them to improve results on Falconer's distance set problem, showing that sets with sufficiently large Hausdorff dimension have points with positive measure distance sets.

Contribution

It introduces weighted refined decoupling estimates and provides an alternative proof for a Falconer distance set result, extending previous bounds on Hausdorff dimension.

Findings

Sets with Hausdorff dimension > d/2 + 1/4 - 1/(8d+4) have points with positive measure distance sets.

Weighted refined decoupling estimates are established and may be of independent interest.

Alternative proof of a Falconer distance set result for d ≥ 4.

Abstract

We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: 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 $d\geq 4$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive Lebesgue measure. Aside from this application, the weighted refined decoupling estimates may be of independent interest.