Weighted twisted inhomogeneous Diophantine approximation

TL;DR

This paper extends classical inhomogeneous Diophantine approximation results to a multidimensional weighted setting, establishing measure and dimension properties of approximation sets using advanced geometric and measure-theoretic tools.

Contribution

It provides a weighted multidimensional analogue of Kurzweil's theorem, incorporating recent developments in weighted ubiquity and mass transference principles.

Findings

The set of points approximable infinitely often has zero or full measure depending on sum conditions.

Hausdorff dimension results are established for the approximation sets.

The work generalizes classical theorems to a weighted, multidimensional context.

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.