On sets of pointwise recurrence and dynamically thick sets

A set $A \subseteq \mathbb{N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X, T)$, all $x \in X$, and all open neighborhoods $U \subseteq X$ of $x$, there exists a time $n \in A$ such that $T^n x \in U$. The set $A$ is dynamically thick if the same holds for all non-empty, open sets $U \subseteq X$. Our main results give combinatorial characterizations of sets of pointwise recurrence and dynamically thick sets that allow us to answer questions of Host, Kra, Maass and Glasner, Tsankov, Weiss, and Zucker. We also introduce and study a local version of dynamical thickness called dynamical piecewise syndeticity. We show that dynamically piecewise syndetic sets are piecewise syndetic, generalizing results of Dong, Glasner, Huang, Shao, Weiss, and Ye. The proofs involve the algebra of families of large sets, dynamics on the space of ultrafilters, and our recent characterization of dynamically syndetic sets.

💡 Research Summary

This paper investigates two families of subsets of the natural numbers that arise from recurrence phenomena in minimal topological dynamical systems: sets of pointwise recurrence (denoted dc T) and sets of dynamically thick (denoted d T). A set A ⊂ ℕ is a set of pointwise recurrence if for every minimal system (X,T), every point x∈X and every open neighbourhood U of x there exists n∈A with Tⁿx∈U. A set A is dynamically thick if the same condition holds for every non‑empty open set U⊂X, without requiring x∈U. While set recurrence (intersection of U with its image) has been extensively studied, the pointwise version and the “thick” version have received far less attention. The authors provide purely combinatorial characterizations of both families, answer several open questions, and introduce a new notion—dynamically piecewise syndetic sets—that bridges these families with classical piecewise syndetic sets.

Main results.
Theorem A gives three equivalent descriptions of a set of pointwise recurrence: (i) the original definition; (ii) a strengthening of the classical combinatorial criterion for topological recurrence—namely, for every syndetic set S there exists a finite subset F⊂A such that for every syndetic S′⊂S we have F∩(S−S′)=∅; (iii) a “finite‑blocking” condition stating that for any superset B⊇A one can remove a finite set from the complement of B to make B∪(B−F) thick. This shows that pointwise recurrence can be detected by a finite obstruction against all syndetic differences.

Theorem B characterizes dynamically thick sets analogously: (i) the definition; (ii) for every piecewise syndetic set S there is a finite F⊂A such that S−F is thick; (iii) for any superset B⊇A one can delete a finite set from the complement of B to obtain a thick set B−F. Thus dynamic thickness is equivalent to the ability to “thin out” any piecewise syndetic set by a finite amount while preserving thickness.

Theorem C provides a negative answer to a question of Host, Kra and Maass: not every set of pointwise recurrence for distal systems is an IP‑set. The authors construct a family of sets of the form
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