Question

The well-known Liouville theorem asserts that an irrational algebraic number $\alpha$ cannot have too good rational approximations, namely $|\alpha-p/q|\ge C(\alpha)/q^k$ where $k$ is the degree of $\alpha$. I wonder whether a similar result holds for the argument of an algebraic number that happens to lie on the unit circle.

For example, consider $\theta=\arccos(1/3)$. It is the argument of a root $\alpha$ of the polynomial $x^2-\frac23x+1$. Is there a similar lower bound for $|\theta/\pi-p/q|$, or equivalently, for $|\alpha^q-1|$? (Note that $|\alpha^q-1|\approx q\cdot |\theta/2\pi-p/q|$).

More generally, let $\alpha\in\mathbb C$ be an algebraic number, $|\alpha|=1$ and $\alpha$ is not a root of unity. Is it always true that $|\alpha^q-1|\ge C(\alpha)/q^k$ where $k$ depends only on the degree of $\alpha$ (or perhaps equals the degree minus one)?

Answer 1

The answer is affirmative by a result of Fel'dman: An improvement of the estimate of a linear form in the logarithms of algebraic numbers (Russian), Mat. Sb. (N.S.) 77 (119) 1968, 423–436, MR0232736.

See also Theorem 3.1 in Baker's Transcendental Number Theory.