Twisted approximation with restricted denominators

In a twisted approximation problem, one fixes a real number $\alpha$ and studies the set of points $\gamma\in [0,1]$ that can be approximated by the fractional parts $n\alpha\, (\bmod\, 1)$, with an error prescribed by a given function $\psi(n)$. I will discuss recent work with Manuel Hauke on twisted approximation under the additional constraint that $n$ must lie in some given integer sequence. We prove Jarnik-type theorems that hold for almost every fixed $\alpha$, where the "almost every" is with respect to a measure of positive Fourier dimension. These results answer questions of Kristensen and Persson.