## Abstract

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 N and along cosets of multiplicative subsemigroups of 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.

Original language | English (US) |
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Pages (from-to) | 5891-5921 |

Number of pages | 31 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 39 |

Issue number | 10 |

DOIs | |

State | Published - 2019 |

## Keywords

- Geometric progressions
- Minimal topological dynamical systems
- Multiplicative upper Banach density
- Multiplicatively large sets
- Return time sets
- Syndetic sets
- sets sets.

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics