Various types of Winning sets and relations between them

Abstract:

Winning sets were initially introduced by W. Schmidt. He used them to solve several problems in Diophantine approximation about the structure of the so called badly approximable sets. Schmidt winning sets are defined in complete metric spaces and satisfy several remarkable properties, which make them interesting on their own:

 (W1) They have full Hausdorff dimension;

 (W2) The intersection of countably many winning sets is winning;

 (W3) The image of a winning set under any bi-Lipshitz map is winning.

Later, many other versions of winning sets were introduced: strongly winning, absolute winning, hyperplane absolute winning, etc. All of them satisfy similar properties to (W1) - (W3). 

In my talk I will give a brief introduction to Schmidt games, winning sets and some of their variations. Then I will explain how they are applied to problems in Diophantine approximation. At the end of the talk I will introduce the most recent Cantor winning and potential winning sets and will discuss their relations with more classical collections of winning sets.