On Fourier decay and the distance set problem

TL;DR

This paper advances the understanding of the Falconer distance set problem by establishing new Fourier analytic conditions that ensure the distance set has full Hausdorff dimension, surpassing previous thresholds in higher dimensions.

Contribution

It introduces improved Fourier dimension and spectrum criteria that guarantee full Hausdorff dimension of distance sets, surpassing the classical $d/2$ threshold in dimensions $d \, \geq \, 5$.

Findings

Sets with Fourier dimension at least 2 have full Hausdorff dimension distance sets.

Sets with Fourier spectrum at least $d/4+1$ at $ heta=1/2$ have full Hausdorff dimension distance sets.

Constructed examples demonstrate the near sharpness of the results.

Abstract

We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild assumptions, we are able to beat the $d/2$ dimension threshold in dimensions $d \geq 5$. For example, we show that (in any ambient spatial dimension $d$) a Borel set with Fourier dimension at least $2$ has a distance set of full Hausdorff dimension. We also show that (in any ambient spatial dimension $d$) a Borel set with Fourier spectrum at least $d/4+1$ at $\theta=1/2$ has a distance set of full Hausdorff dimension. In particular, this can hold for sets with Fourier dimension zero (provided $d \geq 4$). We also consider pinned variants of these problems and construct examples that demonstrate the sharpness (or near sharpness) of our results.