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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 onthe Fourier dimension of $E(Q,\Psi,\theta)$. This generalizes theorems ofKaufman and Bluhm and yields new explicit examples of Salem sets. We giveapplications to metrical Diophantine approximation, including determining theHausdorff dimension of $E(Q,\Psi,\theta)$ in new cases. We also prove ahigher-dimensional analog of our result.

Explicit Salem sets and applications to metrical Diophantine approximation