An analogue of a theorem of Kurzweil

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 $\Vert q\theta -s\Vert <\psi (q)$, 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).

Cited by in corpus (2)

• A characterization of bad approximability
• Geometric and Measure-Theoretic Shrinking Targets in Dynamical Systems