On the radii of Voronoi cells of rings of integers

TL;DR

This paper investigates the covering radius of the ring of integers in number fields, exploring bounds related to the degree of the field and comparing L^2 and L^ norms, contributing to understanding geometric measures in algebraic number theory.

Contribution

It introduces bounds on the covering radius (K) of rings of integers in number fields, relating it to the degree and comparing L^2 and L^ norms, advancing geometric analysis in number theory.

Findings

Bounded (K) by explicit powers of the degree n(K) for certain families

Compared L^2 and L^ norms in the context of Voronoi cells

Provided insights into the limitations of lower bounds for (K)

Abstract

Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $\Delta(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another measure of the size of $K$ given by the covering radius $\mu(K)$ of the ring of integers $O_K$ of $K$. Here $\mu(K)$ is the $L^2$ radius $||V_2(K)||_2$ of the $L^2$ Voronoi cell $V_2(K)$ of $O_K$, where $V_2(K)$ is the set of points in $\mathbb{R} \otimes_{\mathbb{Q}} K$ that are at least as close to the origin as they are to any non-zero element of $O_K$. To put a limit on what lower bounds one can prove for $\mu(K)$ in terms of $n(K)$, we study infinite families of $K$ of increasing degree for which $\mu(K)$ can be bounded above by an explicit power of $n(K)$. We also study analogous questions when the $L^2$ norm is replaced by the $L^\infty$ norm.