A note on zero-one laws in metrical Diophantine approximation

TL;DR

This paper extends zero-one laws in metrical Diophantine approximation for systems of linear forms, generalizing previous one-dimensional results and discussing potential future research directions.

Contribution

It introduces a generalized zero-one law for multivariate Diophantine approximation, expanding upon classical one-dimensional theorems by Cassels and Gallagher.

Findings

Established a zero-one law for systems of linear forms

Extended classical results to higher dimensions

Discussed open problems and potential generalizations

Abstract

In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a discussion on possible generalisations including a selection of various open problems.