Fibonacci numbers and a metric on coprime pairs

TL;DR

This work defines a metric on the set of coprime natural-number pairs and studies its large-scale geometry via quasi-isometries. It constructs a concrete $(1,1)$-quasi-isometric embedding of $(\mathbb{N},|\cdot|)$ into $(\mathcal{X}, d_\varphi)$ by sending $n$ to the consecutive Fibonacci pair $\{F_n,F_{n+1}\}$, with the distance governed by Diophantine representations of Fibonacci relations. The key technical step derives sharp lower and upper bounds for $q_{\{F_n,F_{n+1}\}}(F_m)$ from $F_m = xF_n + yF_{n+1}$, yielding the coarse linear control needed for the embedding. The approach extends to generalizations using $k$-Fibonacci numbers and to $\ell$-coprime collections, establishing analogous $(1,1)$-embeddings with corresponding metallic ratios and extended distance functions. Collectively, the results reveal how Fibonacci-type sequences encode the coarse geometry of spaces of coprime-number tuples.

Abstract

In this paper, we introduce a metric on the set of pairs of coprime natural numbers. We explicitly construct a quasi-isometric embedding from the set of natural numbers into this metric space via Fibonacci numbers.

Theorems & Definitions (24)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3: Honsberger's identity
• proof
• Lemma 2.4
• proof
• Lemma 3.1: Lower bound