Large sets of integers and hierarchy of mixing properties of measure preserving systems

Abstract

We consider a hierarchy of notions of largeness for subsets of Z (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βZ to establish the connections between various notions of largeness and apply the obtained results to the study of sets of “fat intersections” R2 A,B = {n ∈ Z : μ(A ∩ T nB) > μ(A)μ(B) − 2}. Among other things we show that the sets R2 A,B allow one to distinguish between various notions of mixing and introduce an interesting class of weakly but not mildly mixing systems. Some of our results on fat intersections are established in a more general context of unitary Z-actions.