An analogue of a theorem of Kurzweil

TL;DR

This paper explores a modified version of Kurzweil's inhomogeneous Diophantine approximation theorem, establishing a new condition on continued fractions that characterizes when the theorem's equivalence holds under the added hypothesis.

Contribution

It introduces a new condition on continued fraction expansions that characterizes the validity of a modified Kurzweil theorem, expanding recent research.

Findings

Identifies a specific condition on continued fractions equivalent to the modified theorem

Shows the original theorem fails under the added hypothesis

Provides a new criterion linking continued fractions and Diophantine approximation

Abstract

A theorem of Kurzweil ('55) on inhomogeneous Diophantine approximation states that if $\theta$ is an irrational number, then the following are equivalent: (A) for every decreasing positive function $\psi$ such that $\sum_{q = 1}^\infty \psi(q) = \infty$, and for almost every $s\in\mathbb R$, there exist infinitely many $q\in\mathbb N$ such that $\|q\theta - s\| < \psi(q)$, and (B) $\theta$ is badly approximable. This theorem is not true if one adds to condition (A) the hypothesis that the function $q\mapsto q\psi(q)$ is decreasing. In this paper we find a condition on the continued fraction expansion of $\theta$ which is equivalent to the modified version of condition (A). This expands on a recent paper of D. H. Kim ('14).