# A Mass Transference Principle for systems of linear forms and its applications

## TL;DR

This paper extends the Mass Transference Principle to systems of linear forms, enabling the transfer of measure results in Diophantine approximation to Hausdorff measures, with broad applications including inhomogeneous and constrained settings.

## Contribution

It introduces a general form of the Mass Transference Principle for linear systems, expanding its applicability to various Diophantine approximation problems and measure transference.

## Key findings

- Established Hausdorff measure counterparts of Khintchine-Groshev theorems.
- Extended measure transference to inhomogeneous and constrained cases.
- Applied results to recent theorems with primitivity constraints.

## Abstract

In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine-Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira [8].