An improved upper bound on the covering radius of the logarithmic lattice of $\mathbb{Q}({\zeta }_n)$

Abstract

Let ${\mathbb{R}}^m$ be endowed with the Euclidean metric. The covering radius of a lattice $\Lambda \subset {\mathbb{R}}^m$ is the least distance $r$ such that, given any point of ${\mathbb{R}}^m$, the distance from that point to $\Lambda$ 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 ${\mathbb{K}}^{\ast }\to {\mathbb{R}}^m$. In this paper, we examine those lattices which arise from the cyclotomic number field $\mathbb{Q}({\zeta }_n)$, for a given positive integer $n\ge 5$ such that $n\not\equiv 2(\mathrm{mod}4)$. 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.

8 pages