Kevin Hughes

Towards an ergodic Hasse principle

We will study singular variants of higher dimensional versions of Brikhoff's ergodic theorem; in particular the discrete spherical averages of Magyar--Stein--Wainger over lacunary sequences. This program began with Bourgain's study of averages along the squares which notably used the Calderon transference principle to reduce the problem to averages on the integers, the circle method to study these integral averages and transference principles to relate these to operators in harmonic analysis. Pushing forward this program we connect this to the Hasse principle in number theory, strong approximation and the Seeger--Wainger--Wright theorem for (Euclidean) spherical maximal functions.