Quotient rings of integers from a metric point of view

TL;DR

This paper explores the limits of quotient rings of integers under Gromov-Hausdorff convergence, revealing their relation to real numbers and their properties as metric fields.

Contribution

It introduces the concept of Gromov-Hausdorff limits of quotient rings of integers and analyzes their structure and relation to real and rational fields.

Findings

Limit rings exist as Gromov-Hausdorff limits of quotient rings.

These limit rings are dense in the real numbers but not identical to them.

Limit rings can be equipped with an order relation.

Abstract

The theory of Gromov-Hausdorff convergence is applied to sequences of quotient rings of integers. It is shown the existence of limit rings (fields) as the Gromov-Hausdorff limits of sequences of metric quotient rings. The relation of these constructions with the field of the reals $\mathbb{R}$ is discussed, showing that they are dense in $\mathbb{R}$ but that they cannot be identified with the real field or with the rational field $\mathbb{Q}$, at least when $\mathbb{R}$ and $\mathbb{Q}$ are endowed with the usual metric structures. It is also shown that the limit rings can be endowed with an order relation.