Pure Maths Colloquium: Demi Allen

This talk is part of the Pure Maths Colloquium at the University of St Andrews. Check out our upcoming talks at https://theran.lt/pure-colloquium/.

Where: Lecture Theatre D
When: Nov 15 2018 @ 16.00
Video: Not recorded
Speaker: Demi Allen University of Manchester
Title: A general mass transference principle

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.