On sets of pointwise recurrence and dynamically thick sets

A set A ⊆ N A ⊆ N upper A subset of or equal to double struck upper N is a set of pointwise recurrence if for all minimal dynamical systems ( X , T ) $(X, T)$ left parenthesis upper X comma upper T right parenthesis , all x ∈ X x ∈ X x element of upper X , and all open neighborhoods U ⊆ X U ⊆ X upper U subset of or equal to upper X of x , there exists a time n ∈ A n ∈ A n element of upper A such that T n x ∈ U Tⁿ x ∈ U upper T Superscript n Baseline x element of upper U . The set A is dynamically thick if the same holds for all non-empty, open sets U ⊆ X U ⊆ X upper U subset of or equal to upper 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, and Maass [Variations on topological recurrence. Monatsh. Math. 179 (1) (2016), 57–89] and Glasner, Tsankov, Weiss, and Zucker [Bernoulli disjointness. Duke Math. J . 170 (4) (2021), 615–651]. 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 Glasner and Weiss [Interpolation sets for subalgebras of <jats:altern