For which $\alpha \in \mathbb R$ can one say that $\forall \epsilon > 0$ there $\exists N \geq 1$ such that $\forall i \in \mathbb N$ one has that some $n\in \{1, \dots, N\}$ is a solution to
$$ \min_{m\in \mathbb Z} |\alpha^{in} - m| \leq \epsilon. $$
Can one find such a bound $N$ for all real numbers $\alpha$? Can someone see any sufficient conditions on $\alpha$ such that this works out? What about $\alpha = e$, that is $\alpha^{in} = \exp(in)$?
Thanks
It's true for any algebraic integer $\alpha$ having exactly one conjugate outside the unit circle, since for such $\alpha$ the distance from $\alpha^n$ to the nearest integer goes to zero as $n$ goes to infinity. I suspect it's a very hard question for other real $\alpha$ --- it's a notoriously hard problem to say anything much about small values of the fractional part of $(3/2)^n$, for example, a question of Mahler related to Waring's problem (Mahler's paper is here).