Mass Transference Principles and Applications in Diophantine Approximation

Abstract

This thesis is concerned with the Mass Transference Principle and its applications in Diophantine approximation. The Mass Transference Principle, proved by Beresnevich and Velani in 2006, is a powerful result allowing for the transference of Lebesgue measure statements for $\limsup$ sets arising from sequences of balls in $\mathbb{R}^k$ to Hausdorff measure statements. The significance of this result is especially prominent in Diophantine approximation, where many sets of interest arise naturally as $\limsup$ sets. We establish a general form of the Mass Transference Principle for systems of linear forms conjectured by Beresnevich, Bernik, Dodson and Velani in 2009. This improves upon an earlier result in this direction due to Beresnevich and Velani from 2006. In addition, we present a number of applications of this ``new'' mass transference principle for linear forms to problems in Diophantine approximation, some of which were previously out of reach when using the result of Beresnevich and Velani. 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. Using a Hausdorff measure analogue of the inhomogeneous Khintchine--Groshev Theorem (established via the mass transference principle for linear forms), we give an alternative proof of most cases of a general inhomogeneous Jarn\'{\i}k--Besicovitch Theorem which was originally proved by Levesley in 1998. We additionally show that without monotonicity Levesley's theorem no longer holds in general. We conclude this thesis by discussing the concept of a mass transference principle for rectangles. In particular, we demonstrate how some known results may be extended using a slicing technique