2017/10/16 by Kari, Jarkko, Kopra, Johan
#11J71 #37A25 #68Q80 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1710.05737
We consider one-dimensional cellular automata Fp,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 Fp,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 Fp,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.