It is well known that $( \sin(n) )_{n\ge 1}$ is dense in $[-1,1]$. Often one falls in exercise calculation on subsequences of this one: like $\sin(n^2)$ for example; let us generalize right away to $\sin(n^\alpha)$ for $\alpha>0$. These sequences should be dense as well, but that is not clear right away.
For $\sin(n^2)$ for example, Weyl's argument would require to show that $\sum_{k=1}^n \exp( i k^2 m) = o(n)$ which is not clear to me, since I cannot calculate explicitly this sum. Is there some nice trick to handle the density of $\sin(n^\alpha)$ ?