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Explicit Salem sets and applications to metrical Diophantine approximation

Let $Q$ be an infinite subset of $\mathbb{Z}$, let $Ψ: \mathbb{Z} \rightarrow [0,\infty)$ be positive on $Q$, and let $θ\in \mathbb{R}$. Define $$ E(Q,Ψ,θ) = \{ x \in \mathbb{R} : \| q x - θ\| \leq Ψ(q) \text{ for infinitely many $q \in Q$} \}. $$ We prove a lower bound on the Fourier dimension of $E(Q,Ψ,θ)$. 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 $E(Q,Ψ,θ)$ in new cases. We also prove a higher-dimensional analog of our result.