%0 Journal Article
%T Mahler's $Z$-number and $3/2$ number systems
%A Shigeki Akiyama
%J Uniform Distribution Theory
%D 2008
%I Mathematical Institute of the Slovak Academy of Sciences
%X 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.
%K Irregularity of distribution
%K number system
%K Z-number
%U http://www.boku.ac.at/MATH/udt/vol03/no2/Akiyama08-2.pdf