Dimension of Diophantine approximation and some applications in harmonic analysis

TL;DR

This paper constructs new sets based on Diophantine approximation in Euclidean space and explores their applications in harmonic analysis, including Hausdorff and Fourier dimensions, with implications for sum-product problems and Fourier restriction.

Contribution

It introduces a novel family of sets with prescribed Hausdorff and Fourier dimensions, and provides the first construction of measures capturing both dimensions simultaneously.

Findings

Sharpness of Ren and Wang's sum-product result confirmed.

Constructed sets with prescribed Hausdorff and Fourier dimensions.

Provided new insights into Fourier restriction phenomena.

Abstract

In this paper we construct a new family of sets based on Diophantine approximation in the Euclidean space, and consider their applications in several problems in harmonic analysis. Our first application is on the Hausdorff dimension of our sets. We show a recent result of Ren and Wang on the ABC sum-product problem is sharp. Higher dimensional cases and the relation to orthogonal projections are also discussed. Some conjectures are proposed. In addition to Hausdorff dimension, we also consider Fourier dimension. For every $0\leq t\leq s\leq 1$, we are able to construct a subset of $\mathbb{R}$ that has Hausdorff dimension $s$ and Fourier dimension $t$, together with a measure $\mu$ that captures both dimensions, i.e., $$\mu(B(x,r))\lesssim_\epsilon r^{s-\epsilon} \ \text{and} \ |\hat{\mu}(\xi)|\lesssim_\epsilon |\xi|^{-t/2 +\epsilon}, \ \forall\,\epsilon>0.$$ It is fundamental but the very first such result in the literature. Our last result is to provide a viewpoint of the sharpness of Fourier restriction over general measures from dimensions of sets and measures.