Convergence of below series

Question

For what $k \geq1 $, will the series $\sum_{n=1}^{\infty} 1/(n^k|sin(n)|)$ converge?

Answer 1

The problem is that large values of the summand correspond to numerators of good rational approximations to $\pi$, but we don't have a very good handle on irrationality measures for $\pi$. See this paper of Alekseyev.