  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.
