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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.20544 |
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| _version_ | 1866908470043738112 |
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| author | Punch, James |
| author_facet | Punch, James |
| contents | Let $\mathbb{R}^m$ be endowed with the Euclidean metric. The covering radius of a lattice $Λ\subset \mathbb{R}^m$ is the least distance $r$ such that, given any point of $\mathbb{R}^m$, the distance from that point to $Λ$ is not more than $r$. Lattices can occur via the unit group of the ring of integers in an algebraic number field $\mathbb{K}$, by applying a logarithmic embedding $\mathbb{K}^*\rightarrow \mathbb{R}^m$. In this paper, we examine those lattices which arise from the cyclotomic number field $\mathbb{Q}(ζ_n)$, for a given positive integer $n\geq5$ such that $n\not \equiv 2\pmod{4}$. We then provide improvements to an upper bound in (de Araujo, 2024), and conclude with an upper bound on the covering radius for this lattice in terms of $n$ and the number of its distinct prime factors. In particular, we improve Lemma 2 of (de Araujo, 2024) and show that, asymptotically, it can be improved no further. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An improved upper bound on the covering radius of the logarithmic lattice of $\mathbb{Q}(ζ_n)$ Punch, James Number Theory 11H31 Let $\mathbb{R}^m$ be endowed with the Euclidean metric. The covering radius of a lattice $Λ\subset \mathbb{R}^m$ is the least distance $r$ such that, given any point of $\mathbb{R}^m$, the distance from that point to $Λ$ is not more than $r$. Lattices can occur via the unit group of the ring of integers in an algebraic number field $\mathbb{K}$, by applying a logarithmic embedding $\mathbb{K}^*\rightarrow \mathbb{R}^m$. In this paper, we examine those lattices which arise from the cyclotomic number field $\mathbb{Q}(ζ_n)$, for a given positive integer $n\geq5$ such that $n\not \equiv 2\pmod{4}$. We then provide improvements to an upper bound in (de Araujo, 2024), and conclude with an upper bound on the covering radius for this lattice in terms of $n$ and the number of its distinct prime factors. In particular, we improve Lemma 2 of (de Araujo, 2024) and show that, asymptotically, it can be improved no further. |
| title | An improved upper bound on the covering radius of the logarithmic lattice of $\mathbb{Q}(ζ_n)$ |
| topic | Number Theory 11H31 |
| url | https://arxiv.org/abs/2507.20544 |