Twisted approximation with restricted denominators

Given an increasing integer sequence $(a_n)$, a real number $\alpha$, and a sequence $\psi(n)$, we study the set $W$ of real numbers $\gamma$ for which $a_n\alpha - \gamma$ is a distance less than $\psi(n)$ away from an integer. This is often referred to as twisted Diophantine approximation, in this case with denominators restricted to the given sequence $(a_n)$. Our main results are about the size of $W$, and they hold for almost every $\alpha$, with respect to a measure of positive Fourier dimension, for example Lebesgue measure. Our results extend recent work of Kristensen and Persson, and answer questions that they posed.