On sets of pointwise recurrence and dynamically thick sets

Abstract A set <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> $A \subseteq \mathbb {N}$ upper A subset of or equal to double struck upper N is a set of pointwise recurrence if for all minimal dynamical systems <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $(X, T)$ left parenthesis upper X comma upper T right parenthesis , all <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:math> $x \in X$ x element of upper X , and all open neighborhoods <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>U</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>X</mml:mi> </mml:math> $U \subseteq X$ upper U subset of or equal to upper X of x , there exists a time <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:math> $n \in A$ n element of upper A such that <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mi>T</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>U</mml:mi> </mml:math> $T^n x \in U$ 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>U</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>X</mml:mi> </mml:math> $U \subseteq X$ 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