We consider one-dimensional cellular automata which multiply numbers by in base for relatively prime integers and . By studying the structure of traces with respect to we show that for (and then as a simple corollary for ) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence , for some . To the other direction, by studying the measure theoretical properties of , , we show that for there are finite unions of intervals approximating the unit interval arbitrarily well wich don't contain the fractional parts of the whole sequence for any .
Mots-clés : Distribution modulo 1, Z-numbers, cellular automata, ergodicity, strongly mixing
@article{ITA_2017__51_4_191_0, author = {Kari, Jarkko and Kopra, Johan}, title = {Cellular automata and powers of p\ensuremath{/}q}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, pages = {191--204}, publisher = {EDP-Sciences}, volume = {51}, number = {4}, year = {2017}, doi = {10.1051/ita/2017014}, mrnumber = {3782820}, zbl = {1432.11081}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ita/2017014/} }
TY - JOUR AU - Kari, Jarkko AU - Kopra, Johan TI - Cellular automata and powers of p∕q JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2017 SP - 191 EP - 204 VL - 51 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ita/2017014/ DO - 10.1051/ita/2017014 LA - en ID - ITA_2017__51_4_191_0 ER -
%0 Journal Article %A Kari, Jarkko %A Kopra, Johan %T Cellular automata and powers of p∕q %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2017 %P 191-204 %V 51 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ita/2017014/ %R 10.1051/ita/2017014 %G en %F ITA_2017__51_4_191_0
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://archive.numdam.org/articles/10.1051/ita/2017014/
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