Quantitative expansivity for ergodic $\mathbb{Z}^d$ actions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929517127270400 |
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| author | Fish, Alexander Skinner, Sean |
| author_facet | Fish, Alexander Skinner, Sean |
| contents | We study expansiveness properties of positive measure subsets of ergodic $\mathbb{Z}^d$-actions along two different types of structured subsets of $\mathbb{Z}^d$, namely, cyclic subgroups and images of integer polynomials. We prove quantitative expansiveness properties in both cases and strengthen combinatorial results obtained by Björklund and Fish in arXiv:2401.03724, and Bulinski and Fish in arXiv:2102.05862. Our methods unify and strengthen earlier approaches used in arXiv:2401.03724 and arXiv:2102.05862 and to our surprise, also yield a counterexample to a certain pinned variant of the polynomial Bogolyubov theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative expansivity for ergodic $\mathbb{Z}^d$ actions Fish, Alexander Skinner, Sean Dynamical Systems We study expansiveness properties of positive measure subsets of ergodic $\mathbb{Z}^d$-actions along two different types of structured subsets of $\mathbb{Z}^d$, namely, cyclic subgroups and images of integer polynomials. We prove quantitative expansiveness properties in both cases and strengthen combinatorial results obtained by Björklund and Fish in arXiv:2401.03724, and Bulinski and Fish in arXiv:2102.05862. Our methods unify and strengthen earlier approaches used in arXiv:2401.03724 and arXiv:2102.05862 and to our surprise, also yield a counterexample to a certain pinned variant of the polynomial Bogolyubov theorem. |
| title | Quantitative expansivity for ergodic $\mathbb{Z}^d$ actions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.18363 |