Arithmetic properties of arguments of algebraic numbers on the unit circle

Abstract

An irrational number $θ$ is called Diophantine if there exist $c>0$ and $τ< \infty$ such that $\left| θ- \frac{p}{q} \right| \ge \frac{c}{q^τ}$ 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 $β\in \mathbb{C}$ is an algebraic number with $|β|=1$ that is not a root of unity, then $\frac{\operatorname{Arg}(β)}{2π}$ is Diophantine. We also prove that if $β= e^{iα}$ is algebraic, then $\fracαπ$ is either rational or transcendental. As a consequence, we obtain that if $n \ge 2$ is an integer and $α\in \left(0,\fracπ{2}\right)$ satisfies $n \tan α= \tan(n α)$, then $\fracα{2π}$ is both Diophantine and transcendental, and $α$ 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 $\fracα{2π}$ is irrational.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Example 2.2
• Theorem 2.3: Baker-Wüstholz BakerWuestholz1993
• Theorem 2.4: Gelfond-Schneider Niven1956
• Corollary 2.5
• proof
• Theorem 2.6: Hermite-Lindemann Niven1956
• Proposition 4.1