On two classes of dense 2-generator subgroups in $\mathbb C$

TL;DR

This paper investigates the behavior of dense two-generator subgroups in the complex plane, revealing that the set of argument limit values at any point is either finite or the entire interval [-π,π].

Contribution

It characterizes the possible argument limit sets of powers of generators in dense two-generator subgroups of , providing a clear dichotomy.

Findings

The set of argument limit values is either finite or the entire interval [-π,π].

The result applies to all points in .

It advances understanding of the structure of dense multiplicative subgroups in .

Abstract

We consider dense 2-generator multiplicative subgroups in $\mathbb C$ and show that for each point $z\in \mathbb C$ the set of limit values for the arguments of the powers of each generator at the point $z$ is either finite or is $[-\pi,\pi]$