Disjointness for measurably distal group actions and applications

We generalize Berg’s notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure-preserving system. As a corollary, we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener–Wintner-type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence