# A Short Proof of Bernoulli Disjointness via the Local Lemma

## TL;DR

This paper provides a concise, combinatorial proof that all minimal group actions are disjoint from Bernoulli shifts, utilizing the Lovász Local Lemma to unify the argument across all groups.

## Contribution

It introduces a short, self-contained combinatorial proof of Bernoulli disjointness for all groups using the Lovász Local Lemma, simplifying previous complex proofs.

## Key findings

- Unified proof applicable to all groups
- Utilizes Lovász Local Lemma in dynamical systems
- Simplifies previous complex arguments

## 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 Lov\'{a}sz Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems.