Does the sequence $\{\sin(en)\}$ converge or diverge? Is it known if $\{\sin(en)\}$ converges or diverges?

Also, I have a more general question.  For almost every rational $r$, I think we can say that $\{\sin(rn)\}$ diverges.  Does that statement hold if we only assume $r$ is a real number?
The sequence $(\sin rn)_{n\in\mathbb N}$ converges if and only if $r$ is a multiple of $\pi$ (in which case it is the constant sequence $0,0,0,\ldots$).

If $r$ is any rational-but-not-integral multiple of $\pi$, then the sequence is periodic but not constant, and therefore doesn't converge.

If $r/\pi$ is irrational, then values in the sequence are dense in $(-1,1)$ -- in particular there are infinitely many elements in $(-1,-\frac12)$ and also infinitely many elements in $(\frac12,1)$, so it cannot converge in this case either.
It does not convergence for all $e$ and therfore diverges. We can easily see this if we pick $e = \frac{\pi}{2}$. For $n \in \mathbb{N}$ we now see that $\sin(\frac{\pi}{2} n) =  0$  if $n$ is even. While if $n$ is odd, this is either $-1$ or $1$. This is not convergent and thus divergent.