Arithmetic Study Group: 'Topological games, Cantor sets and Diophantine approximation: Some applications.'
21 April 2015 14:00 in CM 103
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 briefly outline 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 within the field of Diophantine apprximation. In particular, I will describe how the method outlined above can be applied to problems surrounding the mixed Littlewood conjecture.
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