A general mass transference principle

Abstract

The Mass Transference Principle proved by Beresnevich and Velani in 2006 is a celebrated and highly influential result which allows us to infer Hausdorff measure statements for lim sup sets of balls in 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 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 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 lim sup sets.2000 Mathematics Subject Classification: Primary 11J83, 28A78; Secondary 11K60.

Mass transference principle from rectangles to rectangles in Diophantine approximation

“…Theorem 2.2 is also generalized to the case when the resonant sets {R α : α ∈ J} are planes in R d by Allen & Beresnevich [2] and general space with the intersection property similar to that for affine space by Allen & Baker [1].…”

A mass transference principle and sets with large intersections

“…If this is not the case, we can just replace every ball B(x j ; r j ) by a ball B(x j ; 2r j ) with twice as large diameter, which does not inuence any of the assumptions in the theorem. 1 Since E has full Lebesgue measure, for every n there exists a number m n such that the set E n = mn j=n B(x j ; r j ) has Lebesgue measure (E n ) > 1 1=n.…”

“…Allen and V. Beresnevich [2] proved a mass transference principle for shrinking neighbourhoods of l-dimensional subspaces. This was further developed by D. Allen and S. Baker [1], who proved a mass transference principle for shrinking neighbourhoods of sets of much more general form.…”

A New Hausdorff Content Bound for Limsup Sets

“…MathSciNet knew, in 2021, of 97 references to their work; see e.g. [1,2,10,22,25,32,33,34]. Many of these are various generalizations.…”

“…We will assume that ξ is a doubling gauge function: there exists a constant D so that for any two open balls B, B with B ⊂ B and rad(B) ≤ rad(B ) ≤ 2rad(B), we have D −1 ξ(B) ≤ ξ(B ) ≤ Dξ(B). 1 If we wish to emphasize the constant, we will call such a function a D−doubling gauge function.…”