On a Special Metric in Cyclotomic Fields
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909371812806656 |
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| author | Saettone, Katerina Zaharescu, Alexandru Zhang, Zhuo |
| author_facet | Saettone, Katerina Zaharescu, Alexandru Zhang, Zhuo |
| contents | Let $p$ be an odd prime, and let $ω$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(ω)$. We prove that this metric has several remarkable properties, such as invariance under the action of the Galois group. Furthermore, we show that points in the ring of integers $\mathcal{O}_K$ behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in $\mathcal{O}_K$ centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other, when $p$ and $N$ are large enough. When suitably normalized, this distance is exactly $1/\sqrt{6}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22687 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Special Metric in Cyclotomic Fields Saettone, Katerina Zaharescu, Alexandru Zhang, Zhuo Number Theory Let $p$ be an odd prime, and let $ω$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(ω)$. We prove that this metric has several remarkable properties, such as invariance under the action of the Galois group. Furthermore, we show that points in the ring of integers $\mathcal{O}_K$ behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in $\mathcal{O}_K$ centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other, when $p$ and $N$ are large enough. When suitably normalized, this distance is exactly $1/\sqrt{6}$. |
| title | On a Special Metric in Cyclotomic Fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.22687 |