Asymptotic dynamics on amenable groups and van der Corput sets
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| Format: | Preprint |
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2024
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| _version_ | 1866908688844849152 |
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| author | Farhangi, Sohail Tucker-Drob, Robin |
| author_facet | Farhangi, Sohail Tucker-Drob, Robin |
| contents | We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the Følner sequence used to define it. This result has been discovered independently by Saúl Rodríguez Martín. Both ours and Rodríguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve Følner sequences remain equivalent for arbitrary countably infinite groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00806 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic dynamics on amenable groups and van der Corput sets Farhangi, Sohail Tucker-Drob, Robin Dynamical Systems Functional Analysis Group Theory 43A07, 22D40 We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the Følner sequence used to define it. This result has been discovered independently by Saúl Rodríguez Martín. Both ours and Rodríguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve Følner sequences remain equivalent for arbitrary countably infinite groups. |
| title | Asymptotic dynamics on amenable groups and van der Corput sets |
| topic | Dynamical Systems Functional Analysis Group Theory 43A07, 22D40 |
| url | https://arxiv.org/abs/2409.00806 |