An uncountable ergodic Roth theorem and applications

Polona Durcik; Rachel Greenfeld; Annina Iseli; Asgar Jamneshan; José Madrid

doi:10.3934/dcds.2022111

Article Contents

2022, Volume 42, Issue 11: 5509-5540. Doi: 10.3934/dcds.2022111

This issue Previous Article Energy conservation and regularity for the 3D magneto-hydrodynamics equations Next Article Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

An uncountable ergodic Roth theorem and applications

1.
Schmid College of Science and Technology, Chapman University, Orange, CA, USA
2.
Department of Mathematics, University of California, Los Angeles, CA, USA
3.
Department of Mathematics, University of Fribourg, Fribourg, Switzerland
4.
Department of Mathematics, Koç University, Istanbul, Turkey

^*Corresponding author: Polona Durcik
^*Corresponding author: Polona Durcik

Received: April 2022

Early access: August 2022

Published: November 2022

R. Greenfeld was partially supported by the Eric and Wendy Schmidt Postdoctoral Award. A. Iseli was partially supported by the Swiss National Science Foundation,Project no. 181898). A. Jamneshan was supported by DFG-research fellowship JA 2512/3-1..

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

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Mathematics Subject Classification: Primary: 37A15, 37A30; Secondary: 05D10.

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