A unified approach to mass transference principle and large intersection property

Abstract

The mass transference principle, discovered by Beresnevich and Velani (2006) [5], is a landmark result in metric Diophantine approximation that allows us to obtain the Hausdorff measure theory of $\lim \sup$ sets. Another important tool is the notion of a large intersection property, introduced and systematically studied by Falconer (1994) [9]. The former mainly focuses on passing between full (Lebesgue) measure and full Hausdorff measure statements, while the latter transfers full Hausdorff content statement to Hausdorff dimension. From this perspective, the proofs of the two results are similar but often treated in different ways.

In this paper, we establish a general mass transference principle from the viewpoint of Hausdorff content, aiming to provide a unified proof for the aforementioned results. More precisely, this principle enables us to transfer the Hausdorff content bounds of a sequence of open sets $E_n$ to the full Hausdorff measure statement and the large intersection property for $\lim \sup E_n$. One of the advantages of our approach is that the verification of the Hausdorff content bound does not require the construction of Cantor-like subset, resulting in a much simpler proof. As an application, we provide simpler proofs for several mass transference principles.