Simplices in large sets and directional expansion in ergodic actions
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916515893215232 |
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| author | Björklund, Michael Fish, Alexander |
| author_facet | Björklund, Michael Fish, Alexander |
| contents | In this paper we study ergodic $\mathbb{Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all $r$-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplices in large sets and directional expansion in ergodic actions Björklund, Michael Fish, Alexander Dynamical Systems Combinatorics In this paper we study ergodic $\mathbb{Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all $r$-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham. |
| title | Simplices in large sets and directional expansion in ergodic actions |
| topic | Dynamical Systems Combinatorics |
| url | https://arxiv.org/abs/2401.03724 |