Farey-subgraphs and Continued Fractions

TL;DR

This paper investigates subgraphs of the Farey graph, characterizes their connectivity, introduces a new class of continued fractions related to these subgraphs, and explores their properties and connections to paths within the graphs.

Contribution

It establishes the connectivity criteria for Farey subgraphs and introduces $ ext{F}_N$-continued fractions, linking them to graph paths and analyzing their existence and uniqueness.

Findings

$ ext{F}_N$ is connected iff $N$ is 1 or a prime power

Defined $ ext{F}_N$-continued fractions and related them to graph paths

Analyzed existence and uniqueness of $ ext{F}_N$-continued fraction expansions

Abstract

In this note, we study a family of subgraphs of the Farey graph, denoted as $\mathcal{F}_N$ for every $N\in\mathbb{N}.$ We show that $\mathcal{F}_N$ is connected if and only if $N$ is either equal to one or a prime power. We introduce a class of continued fractions referred to as $\mathcal{F}_N$-continued fractions for each $N>1.$ We establish a relation between $\mathcal{F}_N$-continued fractions and certain paths from infinity in the graph $\mathcal{F}_N.$ We discuss existence and uniqueness of $\mathcal{F}_N$-continued fraction expansions of real numbers.