Discordant sets and ergodic Ramsey theory

TL;DR

This paper investigates discordant sets with positive upper density in infinite amenable groups, revealing their role in Ramsey theory and connecting diverse mathematical areas to generalize and unify existing results.

Contribution

It introduces a unified framework for discordant sets, generalizes previous constructions, and presents new results linking ergodic theory, number theory, and dynamics.

Findings

Discordant sets are involved in key Ramsey theory questions.

New constructions and results about discordant sets are provided.

The work unifies approaches across multiple mathematical disciplines.

Abstract

We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemer\'{e}di's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.