Logic Seminar
Anush Tserunyan
UIUC
A general van der Corput lemma and underlying Ramsey theory
Abstract: A major theme in ergodic Ramsey theory and multiplicative combinatorics is proving multiple recurrence results for certian doubly recurrent (mixing) actions of semigroups. The amplification of double to multiple recurrence is usually done using a so-called van der Corput difference (ratio) lemma for a suitable filter on the semigroup. Particular instances of this lemma (for concrete filters) have been known proven before (by Furstenberg, Bergelson-McCutcheon, and others), with a different proof for each filter. We define a general class of filters on semigroups, which includes all of the filters for which the van der Corput lemma was known. For the filters in this class (call them Delta-filters), we prove a Ramsey theorem related to labeling edges between the semigroup elements with their ratios. An application of this theorem yields a van der Corput lemma for Delta-filters, generalizing all its previous instances.
Tuesday March 4, 2014 at 4:00 PM in SEO 427