Abstract

In this paper we establish a general form of the mass transference principle for systems of linear forms conjectured in 2009. 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.

Highlights

  • The main goal of this paper is to settle a problem posed in [BBDV09] regarding the mass transference principle, a technique in geometric measure theory that was originally discovered in [BV06a] having primarily been motivated by applications in metric number theory

  • Recall that the sets of interest in metric number theory often arise as the upper limit of a sequence of ‘elementary’ sets, such as balls, and satisfy elegant zero-one laws

  • Recall that if (Ei)i∈N is a sequence of sets the upper limit or lim sup of this sequence is defined as lim sup Ei : = {x : x ∈ Ei for infinitely many i ∈ N}

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Summary

Introduction

The main goal of this paper is to settle a problem posed in [BBDV09] regarding the mass transference principle, a technique in geometric measure theory that was originally discovered in [BV06a] having primarily been motivated by applications in metric number theory. The original mass transference principle [BV06a] stated above is a result regarding lim sup sets which arise from sequences of balls. For the sake of completeness, we remark here that recently some progress has been made towards extending the mass transference principle to deal with lim sup sets defined by sequences of rectangles [WWX15]. The mass transference principle result of [BV06b] carries some technical conditions which arise as a consequence of the ‘slicing’ technique that was used for the proof. These conditions were conjectured to be unnecessary and verifying that this is the case is the main purpose of this paper.

Some applications of Theorem 1
Preliminaries to the Proof of Theorem 1
Proof of Theorem 1
The desired properties of Kη
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