|Publication ahead of print|
RAIRO-Theor. Inf. Appl.
|Published online||31 January 2018|
Cellular automata and powers of p∕q☆
Department of Mathematics and Statistics, University of Turku,
Turun yliopisto, Finland
* Corresponding author: firstname.lastname@example.org
Accepted: 1 December 2017
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.
Mathematics Subject Classification: 11J71 / 37A25 / 68Q80
Key words: Distribution modulo 1 / Z-numbers / cellular automata / ergodicity / strongly mixing
© 2018, EDP Sciences
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