Vol. 15, No. 1, 2022

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Discordant sets and ergodic Ramsey theory

Vitaly Bergelson, Jake Huryn and Rushil Raghavan

Vol. 15 (2022), No. 1, 89–130
Abstract

We explore the properties of nonpiecewise 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 Szemerédi’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.

Keywords
Ramsey theory, ergodic theory, topological dynamics, upper density, piecewise syndetic
Mathematical Subject Classification
Primary: 05D10
Secondary: 37A44
Milestones
Received: 18 February 2021
Revised: 24 June 2021
Accepted: 27 June 2021
Published: 14 March 2022

Communicated by Steven J. Miller
Authors
Vitaly Bergelson
Department of Mathematics
The Ohio State University
Columbus, OH
United States
Jake Huryn
Department of Mathematics
The Ohio State University
Columbus, OH
United States
Rushil Raghavan
Department of Mathematics
The Ohio State University
Columbus, OH
United States