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

Let ξ ≠ = 0 and α > 1 be reals. We prove that the fractional parts {ξ αⁿ}, n = 1, 2, 3, ..., take every value only finitely many times except for the case when α is the root of an integer: α = q ^1/d, where q ≥ 2 and d ≥ 1 are integers and ξ is a rational factor of a nonnegative integer power of α.