An uncountable ergodic Roth theorem and applications

2021/01/03 by Durcik, Polona, Greenfeld, Rachel, Iseli, Annina +2
#05D10 #37A15 #37A30 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.00685

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

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