Abstract:Recently, Glasner, Tsankov, Weiss, and Zucker showed that if $\Gamma$ is an infinite discrete group, then every minimal $\Gamma$-flow is disjoint from the Bernoulli shift $2^\Gamma$. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lovász Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems.
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- Anton Bernshteyn
- Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- MR Author ID: 1104079
- Email: email@example.com
- Received by editor(s): July 22, 2019
- Received by editor(s) in revised form: April 27, 2020
- Published electronically: August 5, 2020
- Additional Notes: This research was partially supported by the NSF grant DMS-1954014.
- Communicated by: Nimish Shah
- © Copyright 2020 Copyright is retained by the author.
- Journal: Proc. Amer. Math. Soc. 148 (2020), 5235-5240
- MSC (2010): Primary 37B05, 37B10; Secondary 05D40
- DOI: https://doi.org/10.1090/proc/15151
- MathSciNet review: 4163835