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On the radii of Voronoi cells of rings of integers
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On the radii of Voronoi cells of rings of integers

Frauke M Bleher, Ted Chinburg, Xuxi Ding, Nadia Heninger and Daniele Micciancio
ArXiv.org
Cornell University
12/22/2025
DOI: 10.48550/arxiv.2512.19518
url
https://doi.org/10.48550/arxiv.2512.19518View
Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

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.
 Mathematics - Number Theory

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