Cantor-winning sets in Diophantine approximation

Dzmitry Badziahin (Sydney)

Abstract

Winning sets were invented by W. Schmidt in 1960's. On one hand they share several remarkable properties:
* They have full Hausdorff dimension (even though they can have zero Lebesgue measure).
* Countable intersection of winning sets is again winning and therefore has full Hausdorff dimension.
* The property of being winning is invariant under bi-Lipschitz homeomorphisms.
On the other hand there are natural examples of Schmidt winning sets in the area of Diophantine approximation. For example the set of badly approximable real numbers and more generally the set of badly approximable points in R^n is winning. This immediately gives a lot of information about the structure of these sets.

Later many other notions of sets which share the same "winning properties" appeared. In the talk we will introduce one of those: Cantor-winning sets. They appear to be quite useful, because many interesting set arising in Diophantine approximation can be shown to be Cantor-winning. If time permits I will provide several results of this kind.