Weighted twisted inhomogeneous Diophantine approximation

Abstract

We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let $A$ be matrix of real numbers, $\Psi$ an $n$-tuple of monotonic decreasing functions, and let $W_{A}(\Psi)$ be the set of points that infinitely often lie in a $\Psi(q)$-neighbourhood of the sequence $\{Aq\}_{q\in\mathbb{N}}$. We prove that the set $ W_{A}(\Psi)$ has zero-full Lebesgue measure under convergent-divergent sum conditions with some mild assumptions on $A$ and the approximating functions $\Psi$. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (Adv. Math. 2023), and Wang & Wu (Math. Ann. 2021) respectively.