Cellular automata and powers of p∕q
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 16th "Journées Montoises d’Informatique Théorique", Tome 51 (2017) no. 4, pp. 191-204.

We consider one-dimensional cellular automata F p , q which multiply numbers by p ∕ q in base p q for relatively prime integers p and q . By studying the structure of traces with respect to F p , q we show that for p ≥ 2 q - 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 wich don't contain the fractional parts of the whole sequence ξ ( p / q ) n for any ξ > 0 .

DOI : 10.1051/ita/2017014
Classification : 11J71, 37A25, 68Q80
Mots-clés : Distribution modulo 1, Z-numbers, cellular automata, ergodicity, strongly mixing
Kari, Jarkko 1 ; Kopra, Johan 1

1 
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     title = {Cellular automata and powers of p\ensuremath{/}q},
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Kari, Jarkko; Kopra, Johan. Cellular automata and powers of p∕q. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 16th "Journées Montoises d’Informatique Théorique", Tome 51 (2017) no. 4, pp. 191-204. doi : 10.1051/ita/2017014. http://www.numdam.org/articles/10.1051/ita/2017014/

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