Forests of Gaussian numbers

TL;DR

This paper extends the concept of the Calkin-Wilf tree to positive complex numbers using subsemigroups of the modular group, creating forests that partition the complex domain and analyzing their fundamental properties.

Contribution

It introduces a novel construction of trees in the complex plane based on modular semigroups, generalizing rational trees to complex numbers.

Findings

Partition of positive complex numbers into forests

Explicit description of cusps and fundamental domains

Generalization of rational trees to complex domain

Abstract

The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. In this paper, certain subsemigroups of the modular group are used to construct similar trees in the set $\mathcal{D}_0$ of positive complex numbers. Associated to each semigroup is a forest of trees that partitions $\mathcal{D}_0$. The set of cusps and the fundamental domain of the semigroup are defined and computed.