A structure theorem for polynomial return-time sets in minimal systems
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916005990629376 |
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| author | Glasscock, Daniel Koutsogiannis, Andreas Le, Anh N. Moreira, Joel Richter, Florian K. Robertson, Donald |
| author_facet | Glasscock, Daniel Koutsogiannis, Andreas Le, Anh N. Moreira, Joel Richter, Florian K. Robertson, Donald |
| contents | We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A structure theorem for polynomial return-time sets in minimal systems Glasscock, Daniel Koutsogiannis, Andreas Le, Anh N. Moreira, Joel Richter, Florian K. Robertson, Donald Dynamical Systems 37B20 (Primary) 37B05 (Secondary) We investigate the structure of return-time sets determined by orbits along polynomial tuples in minimal topological dynamical systems. Building on the topological characteristic factor theory of Glasner, Huang, Shao, Weiss, and Ye, we prove a structure theorem showing that, in a minimal system, return-time sets coincide -- up to a non-piecewise syndetic set -- with those in its maximal infinite-step pronilfactor. As applications, we establish three new multiple recurrence theorems concerning linear recurrence along dynamically defined syndetic sets and polynomial recurrence along arithmetic progressions in minimal and totally minimal systems. We also show how our main theorem can be used to prove that two previously separate conjectures -- one due to Glasner, Huang, Shao, Weiss, and Ye and the other due to Leibman -- are equivalent. |
| title | A structure theorem for polynomial return-time sets in minimal systems |
| topic | Dynamical Systems 37B20 (Primary) 37B05 (Secondary) |
| url | https://arxiv.org/abs/2511.02080 |