Return time sets and product recurrence

TL;DR

This paper characterizes subsets of countable groups that contain return time sets of recurrent points, linking quasi-central sets to recurrence and exploring the relationship between product recurrence and distality within the Stone-ech compactification.

Contribution

It establishes a characterization of return time sets via quasi-central sets and connects product recurrence with distality in the context of the Stone-ech compactification.

Findings

A subset contains a return time set iff it is quasi-central.

S-product recurrence is equivalent to distality if S contains the smallest ideal of ech G.

Partially answers a question of Auslander and Furstenberg.

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-\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).