  In this paper we study the distribution of the sequence $(\alpha
\zeta^{n})_{n\geq 1}$ mod $1$, where $\alpha,\zeta$ are fixed positive real
numbers, with special focus on the accumulation point $0$. For this purpose we
introduce approximation constants
$\underline{\sigma}(\alpha,\zeta),\overline{\sigma}(\alpha, \zeta)$ and study
their properties in dependence of $\alpha,\zeta$, distinguishing in particular
the cases of Pisot numbers, algebraic non Pisot numbers and transcendental
values of $\alpha$ as well as $\zeta$.
