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 $\mathbb{K}^*\rightarrow \mathbb{R}^m$. In this paper, we examine those lattices which arise from the cyclotomic number field $\mathbb{Q}(\zeta_n)$, for a given positive integer $n\geq5$ 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.