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Department of Mathematical Sciences

# Seminar Archives

## Pure Maths Colloquium: Topological games, Cantor sets and Diophantine approximation: An introduction.

Presented by Stephen Harrap , Durham University

20 April 2015 16:00 in CM221

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 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 and will be briefly explained if time permits. The talk will be aimed at a broad audience.