subsequences of sin(n)

Question

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 $\sin(n^\alpha)$ ?

Answer 1

Some answers, I eventually found.

In this book, https://web.maths.unsw.edu.au/%7Ejosefdick/preprints/KuipersNied_book.pdf we have a nice Theorem 3.2 stating that for any polynomial with real coefficients and at least one irrational coefficient, p(n) is unif. distributed mod 1. In my setting, $\sin(n^2) =\sin(\pi n^2/\pi)$ the polynomial p(n) = n^2/\pi$ satisfies the criterion.

Another situation is $\sin(n^n)$ that seems to be treated here https://www.jstor.org/stable/43678733.