The Metrical Theory of Simultaneously Small Linear Forms

TL;DR

This paper develops a comprehensive metrical theory for Diophantine approximation involving linear forms that are simultaneously small, establishing key theorems that generalize classical results to Hausdorff measures and dimensions.

Contribution

It provides a complete Khintchine--Groshev type theorem and its Hausdorff measure generalization for simultaneous small linear forms, advancing the metric number theory.

Findings

Established a Khintchine--Groshev type theorem for small linear forms.

Generalized results to Hausdorff measures and dimensions.

Provided a complete Hausdorff dimension theory for the problem.

Abstract

In this paper we investigate the metrical theory of Diophantine approximation associated with linear forms that are simultaneously small for infinitely many integer vectors; i.e. forms which are close to the origin. A complete Khintchine--Groshev type theorem is established, as well as its Hausdorff measure generalization. The latter implies the complete Hausdorff dimension theory.