In Diophantine Approximation we are often interested in the Lebesgue and Hausdorff measures of certain lim sup 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 lim sup sets arising from sequences of balls in Rk. Subsequently, they extended this Mass Transference Principle to the more general situation in which the lim sup sets arise from sequences of neighbourhoods of “approximating” planes.
In this talk, Demi Allen aims to discuss two recent strengthenings and generalisations of this latter result. Firstly, in a joint work with Victor Beresnevich (York), they have removed some potentially restrictive conditions from the statement given by Beresnevich and Velani. The improvement they obtain yields a number of interesting applications in Diophantine Approximation. Secondly, in a joint work with Simon Baker (Warwick), they 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 pure mathematics colloquium at the University of St Andrews. Check out upcoming talks.