Sets of transfer times with small densities

Abstract: 

We will consider the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-theoretical extensions of some classical sumset theorems in compact abelian groups due to Kneser. Based on a joint work with Michael Bjorklund (Chalmers) and Ilya Shkredov (Steklov Institute, Moscow).