A Short Proof of Bernoulli Disjointness via the Local Lemma

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.