Ergodic characterization of van der Corput sets

Abstract

We prove an analogue to the well-known equivalence of intersective sets and Poincaré recurrent sets, in a stronger setting. We show that a set D is van der Corput, if and only if for each Hilbert space H, unitary operator U, and x ∈ H such that the projection of x to the kernel of (U − I) is nonvanishing, there exists d ∈ D such that (U d x, x) = 0. We also characterize the smallest such d.Mathematics Subject Classification. 37A45, 11P99, 37B20.