Bohr and Measure Recurrent Sets

Given a probability preserving system (X, \mu, T) and a set U of positive measure contained in X we denote by N(U,U) the set of integers n such that the measure of U intersected with T^n(U) is positive. These sets are called return-time sets and are of very special nature. For instance, Poincaré recurrence theorem tells us that the set must have bounded gaps while Sarkozy-Furstenberg theorem tells us that it must have a square. The subject of this talk is a very old question (going back to Følner-1954 if not earlier) whether they give rise to the same family of the sets as when we restrict ourselves to compact group rotations. This was answered negatively by Kříž in 1987 and recently it was proved by Griesmer that a return-time set need not contain any translate of a return-time set arising from compact group rotations. In this talk, I will try to sketch some of these proofs and give a flavour of results and questions in this direction.