Arithmetic properties of arguments of algebraic numbers on the unit circle

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.