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On a Special Metric in Cyclotomic Fields

Katerina Saettone, Alexandru Zaharescu, Zhuo Zhang

Abstract

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

On a Special Metric in Cyclotomic Fields

Abstract

Let be an odd prime, and let be a primitive th root of unity. In this paper, we introduce a metric on the cyclotomic field . 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 behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other, when and are large enough. When suitably normalized, this distance is exactly .

Paper Structure

This paper contains 14 sections, 14 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Notations and definition of the metric
  3. Notations and setup
  4. Definition of the metric
  5. Properties of the metric
  6. Invariance under the Galois group action
  7. An analogue of Krasner's lemma
  8. Computing the metric in coordinates
  9. The normalized distance
  10. Almost all points in $B(p,N)$ are almost equi-distanced
  11. Computation of the second moment
  12. Computation of the fourth moment
  13. Computation of the second moment about the mean
  14. Proof of \ref{['Main theorem']}

Key Result

Theorem 1.1

For any $\varepsilon>0$, there exists an absolute and effectively computable constant $A(\varepsilon)$ such that if $N,p>A(\varepsilon)$, then

Theorems & Definitions (32)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2: Krasner's lemma
  • proof
  • Theorem 3.3
  • ...and 22 more