I am looking for good diophantine approximations for a specific class of irrational numbers.

Let $e^{2 \pi i \theta}$ be a complex algebraic number. I would like a result to the effect that $\theta$ can be approximated well; more specifically, for any constant $k$, I would like for the inequality

$|n \theta -m| < \frac{1}{k n}$

to have infinitely many integer solutions in $n$ and $m$.

What I know is that Hurwitz's theorem guarantees a value of $k$ of at least $\sqrt{5}$, and that Khinchin's theorem asserts that, for any given $k$, the inequality $|n \alpha -m| < \frac{1}{k n}$ will have infinitely many solutions for $\textit{almost}$ $\textit{all}\ $ irrational numbers $\alpha$.

Are there any other relevant results I can use here? And is it plausible to conjecture that irrational numbers of the form $\theta$ are somehow mysteriously guaranteed to have good approximations (i.e., with any value of $k$) as given above?