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Publication - Dr Demi Allen

    A General Mass Transference Principle

    Citation

    Allen, D & Baker, S, 2019, ‘A General Mass Transference Principle’. Selecta Mathematica, vol 25.

    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 $\limsup$ sets of balls in $\R^n$ from \emph{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.

    Full details in the University publications repository