If $f\in \mathbb Q[x]$, and $f$ is not a constant function, will $\{\sin f(n)\}^{+\infty}_{n=1}$ always be dense in $[-1,1]$? I think the key part is that whether $f(n)\alpha-[f(n)\alpha]$ is dense in $[0,1]$, here $f\in \mathbb Q[x]$, $\alpha \in \mathbb Q^c$, $[f(n)\alpha]$ means the biggest integer smaller than $f(n)\alpha$.
All the literature I can found is about the case of $\{\sin n\}^{+\infty}_{n=1}$, it is said that $n\alpha-[n\alpha]$ is distributed uniformly on $[0,1]$ for any given irrational number $\alpha$. I guess $f(n)\alpha-[f(n)\alpha]$ will be no longer distributed uniformly on $[0,1]$, but it seems that $f(n)\alpha-[f(n)\alpha]$ is dense in $[0,1]$.
I also found some books about continued fraction, but I am not sure whether they are related to my question. I would also be appreciated if anyone can tell me the name of related research. Right now I even don't know the keyword I should use in google scholar.