AN UNCOUNTABLE ERGODIC ROTH THEOREM AND APPLICATIONS

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

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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 languageEnglish
Pages (from-to)5509-5540
Number of pages32
JournalDiscrete and Continuous Dynamical Systems
Volume42
Issue number11
DOIs
StatePublished - Nov 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 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.

FundersFunder number
Deutsche ForschungsgemeinschaftJA 2512/3-1
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung181898

    Keywords

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

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