# 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.

## Key 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.