Explicit Salem sets and applications to metrical Diophantine approximation

TL;DR

This paper establishes lower bounds on the Fourier dimension of certain sets defined by Diophantine approximation conditions, providing explicit Salem sets and advancing understanding of their dimensional properties in number theory.

Contribution

It introduces new explicit Salem sets via Fourier dimension bounds and extends results to higher dimensions in the context of Diophantine approximation.

Findings

Lower bounds on Fourier dimension of approximation sets

Construction of explicit Salem sets

Determination of Hausdorff dimension in new cases

Abstract

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $\Psi: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $\theta \in \mathbb{R}$. Define $$ E(Q,\Psi,\theta) = \{ x \in \mathbb{R} : \| q x - \theta \| \leq \Psi(q) \text{ for infinitely many $q \in Q$} \}. $$ We prove a lower bound on the Fourier dimension of $E(Q,\Psi,\theta)$. This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimension of $E(Q,\Psi,\theta)$ in new cases. We also prove a higher-dimensional analog of our result.