A Ramsey theorem on semigroups and a general van der Corput lemma

TL;DR

This paper generalizes the van der Corput lemma for semigroup actions by introducing a new class of filters called -filters, supported by a Ramsey theorem on semigroup-labeled graphs, advancing multiple recurrence results in arithmetic combinatorics.

Contribution

It defines -filters that respect a new notion of differentiation on semigroups and proves a generalized van der Corput lemma for these filters, unifying previous special cases.

Findings

Introduces -filters that respect differentiation on semigroups

Proves a generalized van der Corput lemma for -filters

Establishes a Ramsey theorem for graphs on semigroups with labeled edges

Abstract

A major theme in arithmetic combinatorics is proving multiple recurrence results on semigroups (such as Szemer\'edi's theorem) and this can often be done using methods of ergodic Ramsey theory. What usually lies at the heart of such proofs is that, for actions of semigroups, a certain kind of one recurrence (mixing along a filter) amplifies itself to multiple recurrence. This amplification is proved using a so-called van der Corput difference lemma for a suitable filter on the semigroup. Particular instances of this lemma (for concrete 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 the 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 all its previous instances. This is done via proving a Ramsey theorem for graphs on the semigroup with edges between the semigroup elements labeled by their ratios.