Arithmetic properties of arguments of algebraic numbers on the unit circle

TL;DR

This paper investigates the Diophantine and transcendence properties of arguments of algebraic numbers on the unit circle, establishing new results about their irrationality, transcendence, and Diophantine nature using bounds for linear forms in logarithms.

Contribution

It proves that the argument of any algebraic number on the unit circle not a root of unity is Diophantine, and that certain angles satisfying a tangent relation are both transcendental and Diophantine.

Findings

Arguments of algebraic numbers on the unit circle are Diophantine.

Angles satisfying specific tangent relations are transcendental and Diophantine.

Algebraic angles are either rational multiples of pi or transcendental.

Abstract

An irrational number $\theta$ is called Diophantine if there exist $c>0$ and $\tau < \infty$ such that $\left| \theta - \frac{p}{q} \right| \ge \frac{c}{q^\tau}$ holds for every $(p,q) \in \mathbb{Z} \times \mathbb{N}$. In this paper, we study Diophantine and transcendence properties of some real numbers. Using lower bounds for linear forms in logarithms, we show that if $\beta \in \mathbb{C}$ is an algebraic number with $|\beta|=1$ that is not a root of unity, then $\frac{\operatorname{Arg}(\beta)}{2\pi}$ is Diophantine. We also prove that if $\beta = e^{i\alpha}$ is algebraic, then $\frac{\alpha}{\pi}$ is either rational or transcendental. As a consequence, we obtain that if $n \ge 2$ is an integer and $\alpha \in \left(0,\frac{\pi}{2}\right)$ satisfies $n \tan \alpha = \tan(n \alpha)$, then $\frac{\alpha}{2\pi}$ is both Diophantine and transcendental, and $\alpha$ is transcendental. This extends a result of [V. Cyr, A number theoretic question arising in the geometry of plane curves and in billiard dynamics, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3035--3040], which establishes that $\frac{\alpha}{2\pi}$ is irrational.