On the radii of Voronoi cells of rings of integers
Frauke M. Bleher, Ted Chinburg, Xuxi Ding, Nadia Heninger, Daniele Micciancio
TL;DR
The paper investigates the L^2 and L^∞ radii of Voronoi cells for rings of integers in number fields as a size measure beyond discriminants. It introduces the exponent ν_2, capturing how small the L^2 radius can be relative to degree n(K), and proves 1/2 ≤ ν_2 ≤ 0.6609..., while showing ν_∞ ≤ 0.5849... via parallel L^∞ analysis. Central to the approach is constructing infinite families of fields N arising from two-step towers of elementary abelian 2-extensions with unramified tops (via BC’s method), enabling explicit, controlled fundamental domains by an ε-parameter refinement that yields concrete bounds on the radii. The paper also derives lower bounds tying ||V_2(K)||_2 to the root discriminant δ(K) and demonstrates that cyclotomic fields do not produce sublinear growth in μ(K), reinforcing the nontriviality of the ν_2 bounds and highlighting the relationship between discriminants, class groups, and Voronoi geometry in number fields.
Abstract
Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $Δ(K)$ 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 $μ(K)$ of the ring of integers $O_K$ of $K$. Here $μ(K)$ 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 $μ(K)$ in terms of $n(K)$, we study infinite families of $K$ of increasing degree for which $μ(K)$ 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.