  We consider in this paper 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. Our proofs
are based on a correspondence principle for action sets which was developed
previously by the first two authors.
