A general mass transference principle

Demi Allen*, Simon Baker

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
98 Downloads (Pure)

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 lim sup sets of balls in Rn 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 lim sup 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 Rn. 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 lim sup sets.

Original languageEnglish
Number of pages38
JournalSelecta Mathematica
Volume25
Issue number39
Early online date7 Jun 2019
DOIs
Publication statusPublished - 1 Aug 2019

Keywords

  • Diophantine Approximation
  • Hausdorff measures
  • lim sup sets
  • Mass Transference Principle

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