For which pairs of real numbers $(\alpha,x)$ with $\alpha>1$ does the sequence $\sin(\alpha^n\pi x)$ converge?
As pointed out in a comment on this question, given an integer value of $\alpha$, the sequence converges for all integer values of $x$. In the short time since that question was posted, I have learned that the premise of the question - that $\sin(\alpha^n\pi x)$ almost never converges to $0$ - is probably wrong.
This makes me very sad.
It also makes me want to know when $\sin(\alpha^n\pi x)$ does converge - to $0$ or any other real number.
