On a Special Metric in Cyclotomic Fields

Abstract

Let $p$ be an odd prime, and let $\ddot{I}‰$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(\ddot{I}‰)$. We prove that this metric has several remarkable properties, such as invariance under the action of the Galois group. Furthermore, we show that points in the ring of integers ${\mathcal{O}}_K$ behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in ${\mathcal{O}}_K$ centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other, when $p$ and $N$ are large enough. When suitably normalized, this distance is exactly $1/\sqrt{6}$.

13 pages