Cellular automata and powers of p/q

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 ≥ 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 ξ(p∕q)ⁿ, (n = 0, 1, 2, …) for some ξ > 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 ξ(p∕q)ⁿ for any ξ > 0.