质量转移原理与大交集性质的统一方法
A unified approach to mass transference principle and large intersection property
AI总结：
本文从Hausdorff容度视角建立一般质量转移原理，统一证明满Hausdorff测度陈述与大交集性质，避免构造类Cantor子集，从而简化多个质量转移原理的证明。
摘要：
质量转移原理由Beresnevich和Velani发现[Ann Math (2), 2006]，是丢番图逼近中一个具有里程碑意义的结果，它使我们能够获得$\limsup$集合的Hausdorff测度理论。另一个重要工具是大交集性质的概念，由Falconer引入并系统研究[J. Lond. Math. Soc. (2), 1994]。前者主要关注在满（Lebesgue）测度与满Hausdorff测度陈述之间进行传递，而后者则将满Hausdorff容度陈述转移到Hausdorff维数。从这个角度来看，这两个结果的证明非常相似，但通常以不同的方式处理。在本文中，我们从Hausdorff容度的视角建立了一个一般的质量转移原理，旨在为上述结果提供统一的证明。更准确地说，该原理使我们能够将一列开集$E_n$的Hausdorff容度界转移到$\limsup E_n$的满Hausdorff测度陈述和大交集性质。我们方法的一个优点是，Hausdorff容度界的验证不需要构造类Cantor子集，从而使证明大为简化。作为一个应用，我们为若干质量转移原理提供了更简单的证明。
英文摘要：
The mass transference principle, discovered by Beresnevich and Velani [Ann Math (2), 2006], is a landmark result in Diophantine approximation that allows us to obtain the Hausdorff measure theory of $\limsup$ set. Another important tool is the notion of large intersection property, introduced and systematically studied by Falconer [J. Lond. Math. Soc. (2), 1994]. 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 quite 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 allows us to transfer the Hausdorff content bounds of a sequence of open sets $E_n$ to the full Hausdorff measure statement and large intersection property for $\limsup E_n$. 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.
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