Hamed Mousavi

Pointwise ergodic theorem along the primes

This talk focuses on recent progress on pointwise ergodic theorems along the set of prime numbers, a field of study initiated by Bourgain in the late 80s. In the course of strengthening the work of Bourgain we use hard-analytic methods to derive delicate structure theorems for the set of prime numbers that are inaccessible using combinatorial methods. We will then touch on generalizations of the ergodic theorems to other number fields.

We conclude by discussing current research on bilinear ergodic theorems, following recent breakthrough work of Krause-Mirek-Tao. As a corollary of our work, we present a quantitative version of the Green-Tao-Ziegler Theorem: for instance, whenever E\subset {1,...,N} avoids the progressions {x,x-p,x-p^2} with p prime, then necessarily |E|<< N/(\log\log N)^c