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An uncountable ergodic Roth theorem and applications

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Schmid, Polona durcik
Greenfeld, Rachel
Iseli, Annina
Jamneshan
Madrid, Jose

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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 Gamma-systems for uniformly amenable groups Gamma. As a special case, we obtain this uniformity over all Z-systems, and our result seems to be novel already in this case. Our uncountable Roth theorem is crucial in the proof of both of these results.

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American Institute of Mathematical Sciences

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Mathematics, applied, Mathematics

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Discrete and Continuous Dynamical Systems

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10.3934/dcds.2022111

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