The bad and rough rotation is Poissonian
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909632908230656 |
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| author | Hauke, Manuel |
| author_facet | Hauke, Manuel |
| contents | Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_nα)_{n \in \mathbb{N}} \mod 1$ with $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{N} $ and $α$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \in \mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $α$, $(a_nα)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence $(a_nα)_{n \in \mathbb{N}} \mod 1$ with these properties. Further, we show that this result is false for Lebesgue almost every $α$, thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod $d$ in diophantine Bohr sets which might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The bad and rough rotation is Poissonian Hauke, Manuel Number Theory 11J71 11N25 11N80 11K45 11K60 11J83 Motivated by the Berry-Tabor Conjecture and the seminal work of Rudnick-Sarnak, the fine-scale properties of sequences $(a_nα)_{n \in \mathbb{N}} \mod 1$ with $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{N} $ and $α$ irrational have been extensively studied in the last decades. In this article, we prove that for $(a_n)_{n \in \mathbb{N}}$ arising from the set of rough numbers with explicit roughness parameters and any badly approximable $α$, $(a_nα)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders, and consequently, Poissonian gaps. This is the first known explicit sequence $(a_nα)_{n \in \mathbb{N}} \mod 1$ with these properties. Further, we show that this result is false for Lebesgue almost every $α$, thereby disproving a conjecture of Larcher and Stockinger [Math. Proc. Camb. Phil. Soc. 2020]. The method of proof makes use of an equidistribution result mod $d$ in diophantine Bohr sets which might be of independent interest. |
| title | The bad and rough rotation is Poissonian |
| topic | Number Theory 11J71 11N25 11N80 11K45 11K60 11J83 |
| url | https://arxiv.org/abs/2506.01736 |