Explicit Salem sets and applications to metrical Diophantine approximation

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 \begin{equation*} E(Q,\Psi ,\theta ) = \{ x \in {\mathbb {R}} : \| q x - \theta \| \leq \Psi (q) \text { for infinitely many $q \in Q$} \}. \end{equation*} 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 by determining the Hausdorff dimension of $E(Q,\Psi ,\theta )$ in various cases. We also prove a multidimensional analogue of our result.