Joint IAS/PU Groups and Dynamics Seminar

A common heuristic in Diophantine approximation asserts that different arithmetic expansions (such as digit and continued-fraction expansions) should be independent of one another. In some cases, these elementary heuristics have led to surprisingly deep questions in dynamics, with Furstenberg’s x2,x3-conjectures being notably famous examples, and with recent developments in other cases drawing on Benoist-Quint’s work on measure rigidity, Bourgain-Gamburd’s expansion machinery, strong bounds towards Ramanujan-Petersson conjectures, among others. 

In this talk, I will survey some of these developments and open questions, and report on recent joint work with Manuel Luethi and Barak Weiss. A simple-to-state special case of our results is that natural measures on products of Cantor’s middle thirds set give 0 mass to all weighted, exceptional sets in Diophantine approximation. A new feature of the proof is a measure rigidity statement for certain random walks on the space of lattices admitting stationary, non-invariant measures.