On the fractional parts of the natural powers of a fixed number

Let $\xi\ne0$ and $\alpha>1$ be reals. We prove that the fractional parts $\{\xi\alpha^n\}$, $n=12,3,\dots$, take every value only finitely many times except for the case when $\alpha$ is the root of an integer: $\alpha=q^{1/d}$, where $q\geqslant2$ and $d\geqslant1$ are integers and $\xi$ is a rational factor of a nonnegative integer power of $\alpha$.