The Prime times of twisted Diophantine approximation

The seminal work of Kurzweil (1955) provides for any fixed badly approximable $\alpha$ and monotonically decreasing $\psi$ a Khintchine-type statement on the set of the inhomogeneous real parameters $\gamma$ for which $\lVert n \alpha + \gamma\rVert \leq \psi(n)$ has infinitely many integer solutions, and further shows that the assumption of $\alpha$ being badly approximable is necessary. In this article, we generalize Kurzweil's statement to restricting $n \in \mathcal{A}$, where $\mathcal{A} \subseteq \mathbb{N}$ is a set with some multiplicative structure. We show that for badly approximable $\alpha$, the result of Kurzweil extends to a general class of sets $\mathcal{A}$, which allows us to establish the Kurzweil-type result in particular along the primes and along the sums of two squares. Furthermore, we construct non-trivial sets $\mathcal{A}$ where the assumption of $\alpha$ being badly approximable is necessary. In particular, this criterion applies to $\mathcal{A}$ being the set of square-free numbers, providing a novel characterization of the badly approximable numbers. These statements in particular allow for improving the best known bounds for $\lVert n \alpha + \gamma\rVert \leq \psi(n)$ for infinitely many $n \in \mathcal{A}$ for fixed badly approximable $\alpha$ and for various sets $\mathcal{A}$ of number-theoretic interest when accepting an exceptional set for $\gamma$ of Lebesgue measure $0$.