The Mass Transference Principle: Ten years on

Demi Allen, Sascha Troscheit

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

Abstract

In this article we discuss the Mass Transference Principle due to Beresnevich and Velani and survey several generalisations and variants, both deterministic and random. Using a Hausdorff measure analogue of the inhomogeneous Khintchine–Groshev Theorem, proved recently via an extension of the Mass Transference Principle to systems of linear forms, we give an alternative proof of (most cases of) a general inhomogeneous Jarnik–Besicovitch Theorem which was originally proved by Levesley. We additionally show that without monotonicity Levesley’s theorem no longer holds in general. Thereafter, we discuss recent advances by Wang, Wu and Xu towards mass transference principles where one transitions from limsup sets defined by balls to limsup sets defined by rectangles (rather than from “balls to balls” as is the case in the original Mass Transference Principle). Furthermore, we consider mass transference principles for transitioning from rectangles to rectangles and extend known results using a slicing technique. We end this article with a brief survey of random analogues of the Mass Transference Principle.
Original languageEnglish
Title of host publicationHorizons of Fractal Geometry and Complex Dimensions
EditorsErin Pearse, John Rock, Tony Samuel, Robert Niemeyer
PublisherAmerican Mathematical Society
Pages1-33
Number of pages33
Publication statusPublished - 2019

Publication series

NameAMS Contemporary Mathematics
Volume731

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