  We consider one-dimensional cellular automata $F_{p,q}$ which multiply
numbers by $p/q$ in base $pq$ for relatively prime integers $p$ and $q$. By
studying the structure of traces with respect to $F_{p,q}$ we show that for
$p\geq 2q-1$ (and then as a simple corollary for $p>q>1$) there are arbitrarily
small finite unions of intervals which contain the fractional parts of the
sequence $\xi(p/q)^n$, ($n=0,1,2,\dots$) for some $\xi>0$. To the other
direction, by studying the measure theoretical properties of $F_{p,q}$, we show
that for $p>q>1$ there are finite unions of intervals approximating the unit
interval arbitrarily well which don't contain the fractional parts of the whole
sequence $\xi(p/q)^n$ for any $\xi>0$.
