Limit points of fractional parts of geometric sequences

Let $\alpha > 1$ be an algebraic number and $\xi$ a nonzero real number. In this paper, we compute the range of the fractional parts ${\xi \alpha^n} (n=0,1, \ldots)$. In particular, we estimate the maximal and minimal limit points. Our results show, for example, that if $\theta (=24.97 \ldots)$ is the unique zero of the polynomial $2X^2-50X+1$ with $X > 1$, then there exists a nonzero $\xi^*$ satisfying $\limsup_{n \to \infty} {\xi^*\theta^n} \leq 0.02127 \ldots$. On the other hand, we also prove for any nonzero $\xi$ that \limsup_{n\to \infty}{\xi \theta^n} \geq 0.02003 \ldots .