Pure Maths Colloquium: Demi Allen

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 \(\mathbb{R}^k\). 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.