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 recurrent is equivalent to distality, which partially answers a question of Auslander and Furstenberg (Trans. Amer. Math. Soc. 343, 1994, 221–232).

💡 Research Summary

The paper investigates the interplay between return‑time sets and product recurrence for actions of countable infinite discrete groups. Its two main achievements are (i) a precise combinatorial characterization of the return‑time sets of piecewise‑syndetic recurrent points in terms of quasi‑central sets, and (ii) a structural result showing that, for any closed subsemigroup S of the Stone–Čech compactification βG that contains the smallest ideal K(βG), S‑product recurrence coincides with distality. Both results answer, at least partially, a question posed by Auslander and Furstenberg in 1994.

Background and definitions.
For a G‑system (X,G) (X compact Hausdorff, G acting continuously), the return‑time set of a point x to a neighbourhood U is N(x,U)={g∈G : g·x∈U}. A point is recurrent if N(x,U) is infinite for every neighbourhood U; it is piecewise‑syndetic recurrent if each N(x,U) is a piecewise‑syndetic subset of G (i.e., the intersection of a thick set and a syndetic set). In the classical ℕ‑case, Furstenberg showed that return‑time sets of recurrent points are IP‑sets, and central sets were later introduced to capture richer combinatorial structure. Hindman et al. defined quasi‑central sets as those belonging to an idempotent p in the closure of the smallest ideal K(βℕ).

Theorem 1.2 – Return‑time sets = quasi‑central sets.
The authors prove that a subset F⊂G contains the return‑time set of some piecewise‑syndetic recurrent point if and only if F is quasi‑central. The forward direction uses the fact that for a piecewise‑syndetic recurrent point x, the pair (x,x) has a return‑time set N((x,x),V×V)=N(x,V) that is piecewise‑syndetic for every neighbourhood V; this set is shown to be quasi‑central via the algebraic structure of βG. The converse constructs, from a given quasi‑central set F, a dynamical system (Σ₂,σ) (the full two‑symbol shift) and a point z whose neighbourhood