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-{C}ech compactification β G contains the smallest ideal K(β G) of β G then S-product recurrent is equivalent to distality, which partially answers a question of Auslander and Furstenberg (Trans. Amer. Math. Soc. 343, 1994, 221--232).