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Recurrence, rigidity, and popular differences

Published online by Cambridge University Press:  28 September 2017

JOHN T. GRIESMER
JOHN T. GRIESMER*
Affiliation:
Department of Applied Mathematics & Statistics, Colorado School of Mines, 1500 Illinois Street, Golden, CO 80401, USA email jtgriesmer@gmail.com
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Abstract

We construct a set of integers $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. Here ‘set of rigidity’ means that enumerating $S$ as $(s_{n})_{n\in \mathbb{N}}$ produces a rigidity sequence. This construction generalizes or strengthens results of Katznelson, Saeki (on equidistribution and the Bohr topology), Forrest (on sets of recurrence and strong recurrence), and Fayad and Kanigowski (on rigidity sequences). The construction also provides a density analogue of Julia Wolf’s results on popular differences in finite abelian groups.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1299 - 1316
Copyright
© Cambridge University Press, 2017 

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