Alternating geometric progressions modulo one and Sturmian words
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917379853778944 |
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| author | Lu, Qing Zheng, Weizhe |
| author_facet | Lu, Qing Zheng, Weizhe |
| contents | Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $ξ$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-b)^n)_{n\ge 0}$ is contained in an interval of length $b^{-1}+b^{-2}-b^{-3}$. In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22045 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Alternating geometric progressions modulo one and Sturmian words Lu, Qing Zheng, Weizhe Number Theory Combinatorics 11J54 (Primary) 11A63, 11J71, 68R15 (Secondary) Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $ξ$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-b)^n)_{n\ge 0}$ is contained in an interval of length $b^{-1}+b^{-2}-b^{-3}$. In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval. |
| title | Alternating geometric progressions modulo one and Sturmian words |
| topic | Number Theory Combinatorics 11J54 (Primary) 11A63, 11J71, 68R15 (Secondary) |
| url | https://arxiv.org/abs/2603.22045 |