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

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 $\underset{‾}{\sigma }(\alpha ,\zeta ),\bar{\sigma }(\alpha ,\zeta )$ 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 $\zeta$.