Multiplicative combinatorial properties of return time sets in minimal dynamical systems

Daniel Glasscock; Andreas Koutsogiannis; Florian Karl Richter

doi:10.3934/dcds.2019258

Article Contents

2019, Volume 39, Issue 10: 5891-5921. Doi: 10.3934/dcds.2019258

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Multiplicative combinatorial properties of return time sets in minimal dynamical systems

1.
Department of Mathematics, Northeastern University, Boston, MA, USA
2.
Department of Mathematics, The Ohio State University, Columbus, OH, USA
3.
Department of Mathematics, Northwestern University, Evanston, IL, USA

Received: October 2018

Revised: March 2019

Published: July 2019

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We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual set of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $ \mathbb{N} $ and along cosets of multiplicative subsemigroups of $ \mathbb{N} $, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.

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Mathematics Subject Classification: Primary: 37B05; Secondary: 05D01.

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