Divergence of $\sum_{n=0}^\infty |\sin \omega n|$

Question

I'm looking for a simple argument to show that $$ \sum_{n=0}^\infty |\sin \omega n| $$ does not converge for $\omega \neq k \pi$, $k \in \mathbb{Z}$.

If $\omega = \frac{1}{b}\pi$ with $b \in \mathbb{Z}$ and $b > 1$, then the proof is quite easy because the sequence is periodic with period $2b$.

Now, in the general case (where $\omega$ and $\pi$ are possibly incommensurable), is there a simple argument to establish this proof?