# A Dirichlet approximation theorem for group actions

## TL;DR

This paper generalizes Dirichlet's Diophantine approximation theorem to the setting of compact group actions on metric spaces, including noncommutative groups like the unitary group, with applications to complex spheres.

## Contribution

It introduces a noncommutative Dirichlet approximation theorem for compact group actions, extending classical results to new noncommutative contexts.

## Key findings

- Established a noncommutative Dirichlet theorem for compact groups of isometries.
- Applied the theorem to the unitary group acting on the complex sphere.
- Provided new approximation results in noncommutative geometric settings.

## Abstract

If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting.