Fractional parts of powers of negative rationals
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917359047933952 |
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| author | Lu, Qing Zheng, Weizhe |
| author_facet | Lu, Qing Zheng, Weizhe |
| contents | We prove that for any real number $ξ\neq 0$ and any coprime integers $p>q\ge1$ such that $ξ$ is irrational or $q>1$, the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-p/q)^n)_{n\ge 0}$ is not contained in any interval of length less than $(1+q/p-q^2/p^2)/p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16794 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional parts of powers of negative rationals Lu, Qing Zheng, Weizhe Number Theory 11J54 (Primary) 11J71, 68R15 (Secondary) We prove that for any real number $ξ\neq 0$ and any coprime integers $p>q\ge1$ such that $ξ$ is irrational or $q>1$, the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(ξ(-p/q)^n)_{n\ge 0}$ is not contained in any interval of length less than $(1+q/p-q^2/p^2)/p$. |
| title | Fractional parts of powers of negative rationals |
| topic | Number Theory 11J54 (Primary) 11J71, 68R15 (Secondary) |
| url | https://arxiv.org/abs/2603.16794 |