An uncountable ergodic Roth theorem and applications

TL;DR

This paper proves an uncountable ergodic Roth theorem for amenable groups, extending previous results to non-countable and non-separable spaces, with applications in combinatorics and recurrence theory.

Contribution

It introduces the first uncountable amenable ergodic Roth theorem, broadening the scope of ergodic theory and combinatorial applications beyond countable groups.

Findings

Extended Roth theorem to uncountable amenable groups

Applied to triangular patterns in arbitrary amenable groups

Established a new uniformity aspect in double recurrence

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