Optimal Farey sequence for the Congruence subgroup $\Gamma_0(2^{n})$

TL;DR

This paper establishes an optimal upper bound for the Farey sequence associated with the congruence subgroup (2^n), linking it to fundamental domains and generators of the subgroup.

Contribution

It proves the existence and uniqueness of a Farey sequence with a specific upper bound for (2^n) and relates sequence elements to fundamental domain vertices and generators.

Findings

The Farey sequence for (2^n) has an upper bound of 2^{n-1}.

There is a unique sequence element equal to 2^{n-1}.

Each sequence element corresponds to a fundamental domain vertex and a generator.

Abstract

We prove that $\Gamma_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain of $\Gamma_0(2^n)$ whose side-pairings give a set of independent generators of $\Gamma_0(2^n)$.