2024-11-0920221078-094710.3934/dcds.20221112-s2.0-85136134185https://hdl.handle.net/20.500.14288/2689We 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.pdfMathematics, appliedMathematicsJournal Article1553-5231https://doi.org/10.3934/dcds.2022111835551000001Q2NOIR03822