A general mass transference principle

• Demi Allen ^[1] ; Simon Baker ^[2]
  1. [1] University of Bristol

     University of Bristol

     Reino Unido

     
  2. [2] University of Warwick

     University of Warwick

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 3, 2019
• Idioma: inglés
• DOI: 10.1007/s00029-019-0484-9
• Enlaces
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• Resumen

  • 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.