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A general mass transference principle

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If you have a question about this talk, please contact Diogo Oliveira E Silva.

In Diophantine Approximation we are often interested in the Lebesgue and Hausdorff measures of certain limsup sets. In 2006, motivated by such considerations, Beresnevich and Velani proved a remarkable result – the Mass Transference Principle – which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for limsup sets arising from sequences of balls in R^k. Subsequently, they extended this Mass Transference Principle to the more general situation in which the limsup sets arise from sequences of neighbourhoods of “approximating” planes. In this talk, I aim to discuss two recent strengthenings and generalisations of this latter result. Firstly, in a joint work with Victor Beresnevich (York), we have removed some potentially restrictive conditions from the statement given by Beresnevich and Velani. The improvement we obtain yields a number of interesting applications in Diophantine Approximation. Secondly, in a joint work with Simon Baker (Warwick), we have extended these results to a more general class of sets which include smooth manifolds and certain fractal sets.

This talk is part of the Analysis Seminar series.

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