On a Kurzweil type theorem via ubiquity

TL;DR

This paper extends Kurzweil's zero-one law in inhomogeneous Diophantine approximation by relaxing the badly approximable condition, using ubiquitous systems to prove divergence results and exploring related theorems.

Contribution

It introduces a new approach to Kurzweil's theorem by removing the badly approximable assumption through the construction of ubiquitous systems.

Findings

Proved the divergent part of a Kurzweil type theorem without the badly approximable assumption.

Developed a method using ubiquitous systems for inhomogeneous Diophantine approximation.

Discussed related results and potential extensions of Kurzweil's theorem.

Abstract

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.