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We consider in this paper the set of transfer times between two measurablesubsets of positive measures in an ergodic probability measure-preservingsystem of a countable abelian group. If the lower asymptotic density of thetransfer times is small, then we prove this set must be either periodic orSturmian. Our results can be viewed as ergodic-theoretical extensions of someclassical sumset theorems in compact abelian groups due to Kneser. Our proofsare based on a correspondence principle for action sets which was developedpreviously by the first two authors.

Sets of transfer times with small densities