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A short proof of Bernoulli disjointness via the local lemma


Author: Anton Bernshteyn
Journal: Proc. Amer. Math. Soc. 148 (2020), 5235-5240
MSC (2010): Primary 37B05, 37B10; Secondary 05D40
DOI: https://doi.org/10.1090/proc/15151
Published electronically: August 5, 2020
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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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Additional Information

Anton Bernshteyn
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
MR Author ID: 1104079
Email: abernsht@math.cmu.edu

DOI: https://doi.org/10.1090/proc/15151
Keywords: Disjointness, minimal flows, Bernoulli flow, Lov\'asz Local Lemma
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
Article copyright: © Copyright 2020 Copyright is retained by the author.