**Sanju Velani**

University of York, UK

January 21, 2016

**Metric Diophantine approximation: the Lebesgue and Hausdorff theories**:
There are two fundamental results in the classical theory of metric
Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We discuss these theorems and show that Lebesgue statement implies the general Hausdorff statement. The key is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for limsup sets to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of limsup sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.