Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910681115131904 |
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| author | Frantzikinakis, Nikos Host, Bernard Kra, Bryna |
| author_facet | Frantzikinakis, Nikos Host, Bernard Kra, Bryna |
| contents | A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$ is a real polynomial with at least one non-constant irrational coefficient, then the set $\{P(2^m3^n)\colon m,n\in \mathbb{N}\}$ is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$ Frantzikinakis, Nikos Host, Bernard Kra, Bryna Dynamical Systems Combinatorics Primary: 37B20, Secondary: 37A44, 37C85, 11J71, 05B10 A subset $R$ of integers is a set of Bohr recurrence if every rotation on $\mathbb{T}^d$ returns arbitrarily close to zero under some non-zero multiple of $R$. We show that the set $\{k!\, 2^m3^n\colon k,m,n\in \mathbb{N}\}$ is a set of Bohr recurrence. This is a particular case of a more general statement about images of such sets under any integer polynomial with zero constant term. We also show that if $P$ is a real polynomial with at least one non-constant irrational coefficient, then the set $\{P(2^m3^n)\colon m,n\in \mathbb{N}\}$ is dense in $\mathbb{T}$, thus providing a joint generalization of two well-known results, one of Furstenberg and one of Weyl. |
| title | Bohr recurrence and density of non-lacunary semigroups of $\mathbb{N}$ |
| topic | Dynamical Systems Combinatorics Primary: 37B20, Secondary: 37A44, 37C85, 11J71, 05B10 |
| url | https://arxiv.org/abs/2406.01353 |