Kurt Gödel Research Center

2016 seminar talk: Differentiation of subsets of semigroups, a Ramsey theorem, and a van der Corput lemma

Talk held by Anush Tserunyan (University of Illinois at Urbana-Champaign, USA) at the KGRC seminar on 2016-05-12.

Abstract

A major theme in ergodic Ramsey theory is proving multiple recurrence results for measure-preserving actions of semigroups. What often lies at the heart of these results is that mixing ($\approx$ "chaotic") along a suitable filter on the semigroup amplifies itself to multiple mixing ($\approx$ even more "chaotic") along the same filter. This amplification is usually proved using a so-called van der Corput difference lemma. Instances of this lemma for specific filters have been proven before by Furstenberg, Bergelson–McCutcheon, and others, with a somewhat different proof in each case. We define a notion of differentiation for subsets of semigroups and isolate a class of filters that respect this notion. The filters in this class (call them $\partial$-filters) include all those, for which the van der Corput lemma was known, and our main result is a van der Corput lemma for $\partial$-filters, which thus generalizes its previous instances. This is done via proving a Ramsey theorem for graphs on the semigroup.