Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A unified approach to mass transference principle and large intersection property

Yubin He

TL;DR

The paper develops a unified mass transference principle (MTP) based on Hausdorff content that implies both full Hausdorff measure statements and the large intersection property for limsup sets. By avoiding Cantor-type constructions and employing a weighted, Frostman-type framework with net measures, it provides simpler proofs for classical MTP results (ball-to-open-set and rectangle-to-rectangle) and extends to dynamical settings via local ubiquity concepts. It also shows how local scaling and dynamical Diophantine approximation fit naturally into the content-based transfer, offering a cohesive approach that broadens the scope and applicability of mass transference and large intersection theory. The results yield new corollaries under weaker ubiquity assumptions and provide streamlined proofs for both static and dynamical limsup sets, with potential impact on metric Diophantine approximation and related dynamical systems analysis.

Abstract

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.

A unified approach to mass transference principle and large intersection property

TL;DR

The paper develops a unified mass transference principle (MTP) based on Hausdorff content that implies both full Hausdorff measure statements and the large intersection property for limsup sets. By avoiding Cantor-type constructions and employing a weighted, Frostman-type framework with net measures, it provides simpler proofs for classical MTP results (ball-to-open-set and rectangle-to-rectangle) and extends to dynamical settings via local ubiquity concepts. It also shows how local scaling and dynamical Diophantine approximation fit naturally into the content-based transfer, offering a cohesive approach that broadens the scope and applicability of mass transference and large intersection theory. The results yield new corollaries under weaker ubiquity assumptions and provide streamlined proofs for both static and dynamical limsup sets, with potential impact on metric Diophantine approximation and related dynamical systems analysis.

Abstract

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 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 to the full Hausdorff measure statement and large intersection property for . 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.
Paper Structure (24 sections, 37 theorems, 262 equations)

This paper contains 24 sections, 37 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. MTP from balls to open sets
  3. MTP from rectangles to rectangles
  4. Dynamical dimension transference principle
  5. Main results
  6. Large intersection property
  7. Statement of results
  8. Properties of the class $\mathscr G^f(X)$
  9. Net measure and large intersection property
  10. Proof of Theorem \ref{['t:LIPfull']}
  11. Proof: MTP from balls to open sets
  12. Weighted Hausdorff measure and Frostman-type lemma
  13. Construction of the subset of $F_n\cap B$
  14. A measure $\mu$ supported on $K$
  15. Proof: Weaker condition for the class $\mathscr G^f(X)$
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $f$ be a dimension function such that $f\preceq g$. Assume that $\{B(x_n,r_n)\}_{n\ge 1}$ is a sequence of balls in $X$ with $r_n\to 0$ as $n\to\infty$. If then,

Theorems & Definitions (67)

  • Definition 1.1: $g$-Ahlfors regular
  • Remark 1
  • Theorem 1.2: BV06
  • Definition 1.3: $\kappa$-scaling property
  • Remark 2
  • Theorem 1.4: AB19
  • Theorem 1.5: KR21 and Zh21
  • Remark 3
  • Definition 1.6: Local ubiquity system for rectangles
  • Definition 1.7: Uniform local ubiquity system for rectangles
  • ...and 57 more