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This paper deals with the distribution of $\alpha \zeta^{n} \bmod 1$, where$\alpha\neq 0,\zeta>1$ are fixed real numbers and $n$ runs through the positiveintegers. Denote by $\Vert.\Vert$ the distance to the nearest integer. Weinvestigate the case of $\alpha\zeta^{n}$ all lying in prescribed smallintervals modulo $1$ for all large $n$, with focus on the case $\Vert\alpha\zeta^{n}\Vert \leq \epsilon$ for small $\epsilon>0$. We are particularlyinterested in what we call cardinality gap phenomena. For example for fixed$\zeta>1$ and small $\epsilon>0$ there are at most countably many values of$\alpha$ such that $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ for all large$n$, whereas larger $\epsilon$ induces an uncountable set. We investigate thevalue of $\epsilon$ at which the gap occurs. We will pay particular attentionto the case of algebraic and, more specific, rational $\zeta>1$. Resultsconcerning Pisot and Salem numbers such as some contribution to Mahler's$3/2$-problem are implicitly deduced. We study similar questions for fixed$\alpha\neq 0$ as well.

Integral powers of numbers in small intervals modulo $1$: The cardinality gap phenomenon