Abstract
The Mass Transference Principle proved by Beresnevich and Velani (Ann. Math. (2) 164(3):971–992, 2006) is a celebrated and highly influential result which allows us to infer Hausdorff measure statements for limsup sets of balls in mathbb {R}^n from a priori weaker Lebesgue measure statements. The Mass Transference Principle and subsequent generalisations have had a profound impact on several areas of mathematics, especially Diophantine Approximation. In the present paper, we prove a considerably more general form of the Mass Transference Principle which extends known results of this type in several distinct directions. In particular, we establish a Mass Transference Principle for limsup sets defined via neighbourhoods of sets satisfying a certain local scaling property. Such sets include self-similar sets satisfying the open set condition and smooth compact manifolds embedded in mathbb {R}^n. Furthermore, our main result is applicable in locally compact metric spaces and allows one to transfer Hausdorff g-measure statements to Hausdorff f-measure statements. We conclude the paper with an application of our mass transference principle to a general class of random limsup sets.
Highlights
1.1 Background In Diophantine Approximation, Dynamical Systems, and Probability Theory, many sets of interest can be characterised as lim sup sets
When the sequence (E j ) j∈N is a collection of balls, a powerful tool in determining the metric properties of lim sup j→∞ E j is the Mass Transference Principle [5], which allows us to infer Hausdorff measure statements from seemingly less general Lebesgue measure statements
Compared with the Mass Transference Principle, this general theorem applies to lim sup sets of balls in more general metric spaces and deals with a larger class of (Hausdorff) measures
Summary
In Diophantine Approximation, Dynamical Systems, and Probability Theory, many sets of interest can be characterised as lim sup sets. When the sequence (E j ) j∈N is a collection of balls, a powerful tool in determining the metric properties of lim sup j→∞ E j is the Mass Transference Principle [5], which allows us to infer Hausdorff measure statements from seemingly less general Lebesgue measure statements. Compared with the Mass Transference Principle, this general theorem applies to lim sup sets of balls in more general metric spaces and deals with a larger class of (Hausdorff) measures. The Mass Transference Principle was originally motivated by a desire to establish a Hausdorff measure analogue of the famous Duffin–Schaeffer Conjecture in Metric Number Theory. Since their initial announcement, the Mass Transference Principle and Theorem MTP* have been shown to have applications in many distinct areas of mathematics. For further applications of the Mass Transference Principle and Theorem MTP* see [2,4,5,16,24]
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