  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-\v{C}ech compactification $\beta G$ contains the
smallest ideal $K(\beta G)$ of $\beta G$ 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).
