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

TL;DR

This paper improves the upper bound on the covering radius of the logarithmic lattice derived from cyclotomic number fields, providing tighter estimates and asymptotic optimality for these bounds.

Contribution

It offers a refined upper bound on the covering radius of lattices from cyclotomic fields, advancing previous results and establishing asymptotic optimality.

Findings

Improved upper bound on the covering radius for cyclotomic lattice

Asymptotic optimality of the new bound

Bound expressed in terms of n and prime factors

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.