Return time sets and product recurrence

Let G be a countable infinite discrete group. We show that a subset F of G contains a return time set of some piecewise syndetic recurrent point x in a compact Hausdorff space X with a G-action if and only if F is a quasi-central set. As an application, we show that if a nonempty closed subsemigroup S of the Stone–Čech compactification βG contains the smallest ideal K(βG) of βG then S-product recurrence is equivalent to distality, which partially answers a question of Auslander and Furstenberg [Trans. Amer. Math. Soc. 343 (1994), 221–232].