Mahler's $Z$-number and $3/2$ number systems

We improve the results in \cite{Akiyama-Frougny-Sakarovitch:06} on the characterization of multiple points inrational based number system, in connection with Mahler's $Z$-number problem. As a by-product, we show that when $p > q^2$, there exists a positive $x$ such that the fractional part of $x (p/q)^n (n=0,1, \dots)$ stays in a Cantor set (Theorem \ref{Largep2}). Hausdorff dimension of the set is positive but tends to zero as $p \rightarrow \infty$ when $q$ is fixed.