Explicit Salem sets and applications to metrical Diophantine approximation

Abstract

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $\Psi :\mathbb{Z}\to [0,\infty )$ be positive on $Q$, and let $\theta \in \mathbb{R}$. Define $$E(Q,\Psi ,\theta )=\{x\in \mathbb{R}:\Vert qx-\theta \Vert \le \Psi (q)\text{ for infinitely many q∈Q}\}.$$ 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.