  Let $q > 1$ be a real number and let $m=m(q)$ be the largest integer smaller
than $q$. It is well known that each number $x \in J_q:=[0, \sum_{i=1}^{\infty}
m q^{-i}]$ can be written as $x=\sum_{i=1}^{\infty}{c_i}q^{-i}$ with integer
coefficients $0 \le c_i < q$. If $q$ is a non-integer, then almost every $x \in
J_q$ has continuum many expansions of this form. In this note we consider some
properties of the set $\mathcal{U}_q$ consisting of numbers $x \in J_q$ having
a unique representation of this form. More specifically, we compare the size of
the sets $\mathcal{U}_q$ and $\mathcal{U}_r$ for values $q$ and $r$ satisfying
$1< q < r$ and $m(q)=m(r)$.
