Let $k$ be a natural number. Apparently it is possible to show that there is a choice of $0<n_1\leq n_2\leq \cdots \leq n_k$ with $n_i$ integers such that as $k\rightarrow \infty,$
$$ \max_{\theta \in (0,2\pi]} \left|\sum_{i=1}^k \sin n_i \theta\right|\leq ck^{2/3}$$
for some constant $c>0,$ but I cannot find a reference.
Edit: Regarding the comment, I am happy with even a sequence $(n_i)$ which changes with $k$, to start with.
For background see this MO question here