Arithmetic properties of arguments of algebraic numbers on the unit circle

An irrational number $\theta$ is called Diophantine if there exist $c>0$ and $\tau <\mathrm{\infty }$ such that $\mid \theta -\frac{p}{q}\mid \ge \frac{c}{q^{\tau }}$ 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 $\mathrm{\mid }\beta \mathrm{\mid }=1$ that is not a root of unity, then $\frac{\mathrm{Arg}(\beta )}{2\pi }$ 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 $\alpha \in (0,\frac{\pi }{2})$ 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.