 
Summary: Uniform Distribution Theory 3 (2008), no.2, 9199
uniform
distribution
theory
MAHLER'S ZNUMBER AND 3/2 NUMBER
SYSTEMS
Shigeki Akiyama
ABSTRACT. We improve the results in [1] on the characterization of multiple
points in rational based number system, in connection with Mahler's Znumber
problem. As a byproduct, we show that when p > q2, there exists a positive
x such that the fractional part of x(p/q)n (n = 0, 1, . . . ) stays in a Cantor set
(Theorem 2.5). Hausdorff dimension of the set is positive but tends to zero as
p when q is fixed.
Communicated by Yann Bugeaud
1. Representations in a rational base
Let us review the result in [1]. Let p, q be coprime integers with p > q > 1
and consider a digit set A = {0, 1, . . . , p  1}. Every positive integer u has a
unique representation:
u = u0
1
