AN UNCOUNTABLE ERGODIC ROTH THEOREM AND APPLICATIONS

Polona Durcik; Rachel Greenfeld; Annina Iseli; Asgar Jamneshan; José Madrid

doi:10.3934/dcds.2022111

AN UNCOUNTABLE ERGODIC ROTH THEOREM AND APPLICATIONS

Polona Durcik, Rachel Greenfeld, Annina Iseli, Asgar Jamneshan, José Madrid

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for -systems for uniformly amenable groups ⋯ Our uncountable Roth theorem is crucial in the proof of both of these results.

Original language	English
Pages (from-to)	5509-5540
Number of pages	32
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	42
Issue number	11
DOIs	https://doi.org/10.3934/dcds.2022111
State	Published - Nov 2022
Externally published	Yes

Bibliographical note

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© 2022 American Institute of Mathematical Sciences. All rights reserved.

Funding

2020 Mathematics Subject Classification. Primary: 37A15, 37A30; Secondary: 05D10. Key words and phrases. Uncountable ergodic theory, ergodic Ramsey theory, ergodic Roth theorem, amenable groups, syndetic sets, uniformity in recurrence, Furstenberg correspondence principle. R. Greenfeld was partially supported by the Eric and Wendy Schmidt Postdoctoral Award. A. Iseli was partially supported by the Swiss National Science Foundation,Project no. 181898). A. Jamneshan was supported by DFG-research fellowship JA 2512/3-1. ∗Corresponding author: Polona Durcik.

Funders	Funder number
Deutsche Forschungsgemeinschaft	JA 2512/3-1
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung	181898

Keywords

Furstenberg correspondence principle
Uncountable ergodic theory
amenable groups
ergodic Ramsey theory
ergodic Roth theorem
syndetic sets
uniformity in recurrence

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AB - We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for -systems for uniformly amenable groups ⋯ Our uncountable Roth theorem is crucial in the proof of both of these results.

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KW - amenable groups

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AN UNCOUNTABLE ERGODIC ROTH THEOREM AND APPLICATIONS

Abstract

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