On Kurzweil's 0-1 Law in Inhomogeneous Diophantine Approximation

TL;DR

This paper establishes a precise criterion for inhomogeneous Diophantine approximation that generalizes and improves upon Kurzweil's classical results, with implications for both real numbers and formal Laurent series.

Contribution

It provides a necessary and sufficient condition for inhomogeneous approximation that encompasses several previous theorems and extends to formal Laurent series.

Findings

Established a full criterion for inhomogeneous approximation for almost all real numbers.

Unified and extended previous results by Kurzweil and Tseng.

Presented an analogue of the main result in the setting of formal Laurent series.

Abstract

We give a sufficient and necessary condition such that for almost all $s\in{\mathbb R}$ \[ \|n\theta-s\|<\psi(n)\qquad\text{for infinitely many}\ n\in{\mathbb N}, \] where $\theta$ is fixed and $\psi(n)$ is a positive, non-increasing sequence. This improves upon an old result of Kurzweil and contains several previous results as special cases: two theorems of Kurzweil, a theorem of Tseng and a recent result of the second author. Moreover, we also discuss an analogue of our result in the field of formal Laurent series which has similar consequences.