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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