Lower bounds for Z-numbers

Let $p/q$ be a rational noninteger number with $p>q\geq2$. A real number $\lambda>0$ is a $Z_{p/q}$-number if $\{\lambda(p/q)^n\}<1/q$ for every nonnegative integer $n$, where $\{x\}$ denotes the fractional part of $x$. We develop several algorithms to search for $Z_{p/q}$-numbers, and use them to determine lower bounds on such numbers for several $p$ and $q$. It is shown, for instance, that if there is a $Z_{3/2}$-number, then it is greater than $2^{57}$. We also explore some connections between these problems and some questions regarding iterated maps on integers.