Sanju Velani: 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.

This is an analysis colloquium style talk.