Return time sets and product recurrence
Volume 273 / 2026
Fundamenta Mathematicae 273 (2026), 101-138
MSC: Primary 37B20; Secondary 37B05
DOI: 10.4064/fm241006-2-1
Published online: 27 May 2026
Abstract
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 $\beta G$ contains the smallest ideal $K(\beta G)$ of $\beta G$ 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].
