On a Special Metric in Cyclotomic Fields

TL;DR

This paper introduces a novel metric on cyclotomic fields that exhibits invariance under Galois actions and reveals uniform distance properties among algebraic integers, with implications for understanding their geometric structure.

Contribution

The paper defines a new metric on cyclotomic fields and proves its invariance and uniformity properties, providing insights into the geometric behavior of algebraic integers.

Findings

The metric is invariant under Galois group actions.

Points in the ring of integers show near-uniform distances in large hypercubes.

Normalized distances approach 1/√6 for large parameters.

Abstract

Let $p$ be an odd prime, and let $\omega$ be a primitive $p$th root of unity. In this paper, we introduce a metric on the cyclotomic field $K=\mathbb{Q}(\omega)$. 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 $1/\sqrt{6}$.