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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 uniformity aspect in the double recurrence theorem for Γ-systems for arbitrary uniformly amenable groups Γ. Our uncountable Roth theorem is crucial in the proof of both of these results.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems, volume 42, issue 11, in 2022 following peer review. This article may not exactly replicate the final published version. The definitive publisher-authenticated version is available online at

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American Institute of Mathematical Sciences (AIMS)



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