Topological games, Cantor sets and Diophantine approximation

Abstract: 

When attacking various difficult problems in the field of Diophantine approximation the application of certain topological games has proven extremely fruitful in recent times due to the amenable properties of the associated 'winning' sets. Other problems in Diophantine approximation have recently been solved via the method of constructing certain tree-like structures inside the Diophantine set of interest. In this talk I will discuss how one broad method of  tree-like construction, namely the class of 'generalised Cantor sets', can be formalized for use in a wide variety of problems.  By introducing a further class of so-called 'Cantor-winning' sets we may then provide a criterion for arbitrary sets in a metric space to satisfy the desirable properties usually attributed to winning sets, and so in some sense unify the two above approaches.  Applications of this new framework include new answers to questions relating to the mixed Littlewood conjecture and the $\times2, \times3$  problem.