Logic Seminar

Given a discrete group G, two G-flows X and Y are said to be disjoint if their only joining is the trivial joining 2^X. Earlier this year, it was shown by Glasner, Tsankov, Weiss and Zucker that the Bernoulli shift 2^G is disjoint from every minimal G-flow. Their proof is by cases depending on the group, and relies heavily on difficult machinery developed for ICC groups due to Frisch, Tamuz and Vahidi Ferdowsi. Recently, Bernshteyn has found a much shorter proof eliminating all casework, which reduces the problem to an application of the Lovász Local Lemma from combinatorics. We will present this proof, which proceeds via showing a result interesting in its own right, namely that if U is a nonempty open set in 2^G, then then there is some n such that any union of n translates of U always contains an orbit.