On the radii of Voronoi cells of rings of integers

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.