Leírás
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, UK), 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, UK), we have extended these results to a more general class of sets which include smooth manifolds and certain fractal sets.