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Discordant sets
and ergodic Ramsey theory
Vitaly Bergelson, Jake Huryn and Rushil Raghavan
Vol. 15 (2022), No. 1, 89–130
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Abstract |
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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. |
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Keywords
Ramsey theory, ergodic theory, topological dynamics, upper
density, piecewise syndetic
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Mathematical Subject Classification
Primary: 05D10
Secondary: 37A44
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Milestones
Received: 18 February 2021
Revised: 24 June 2021
Accepted: 27 June 2021
Published: 14 March 2022
Communicated by Steven J. Miller
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Authors |
| Vitaly Bergelson | |
| Department of Mathematics The Ohio State University Columbus, OH United States |
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| Jake Huryn | |
| Department of Mathematics The Ohio State University Columbus, OH United States |
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| Rushil Raghavan | |
| Department of Mathematics The Ohio State University Columbus, OH United States |
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