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

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 positive integers. Denote by $\Vert.\Vert$ the distance to the nearest integer. We investigate the case of $\alpha\zeta^{n}$ all lying in prescribed small intervals modulo $1$ for all large $n$, with focus on the case $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ for small $\epsilon>0$. We are particularly interested 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 the value of $\epsilon$ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational $\zeta>1$. Results concerning 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.