p-Adic Diophantine approximation with respect to fractal measures

Abstract: I will give an introduction to Diophantine approximation problems starting with the famous Sprindzuk Conjecture (which is now a theorem by Kleinbock and Margulis, who solved this using homogeneous dynamics). Next, I will talk about p-adic Diophantine approximation and how it is different than the real case. In a very recent work with Anish Ghosh and Victor Beresnevich we solved a conjecture of Kleinbock and Tomanov, which shows pushforward of a fractal measure by ‘nice’ functions exhibits ‘nice’ Diophantine properties. In particular, we prove p-adic analogue of a result by Kleinbock, Lindenstrauss and Weiss on friendly measures. I will talk about how lack of the mean value theorem makes life difficult in the p-adic fields. (No prior knowledge on this subject will be assumed!)

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar