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

On the rate of accumulation of $\alpha\zeta^{n}$ mod 1 to 0