Enhancements of van der Corput's Difference Theorem and connections to the ergodic hierarchy of mixing.

Abstract: We will examine three commonly used variants of van der Corput's Difference Theorem (vdCDT) in Hilbert spaces and show that they are associated with the notions of weak mixing, strong mixing, and Bernoullicity. We will then use this association to derive 2 new vdCDTs corresponding to ergodicity and mild mixing. We remark that our methods naturally yield vdCDTs for a class of unbounded sequences of vectors. We will then obtain an application to recurrence in measure preserving systems by giving a partial answer to a question of Frantzikinakis. If time permits, we will also discuss analogues of these vdCDTs in the context of uniform distribution and the classes of "mixing distributions" that they produce.

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University