This work concerns the study of arithmetical and statistical properties of infinite words and sequences of integers generated by a substitution on an infinite denumerable alphabet or by a deterministic automata with an infinite denumerable set of states. In particular, we prove that if is a sequence of integers generated by an automaton whose associated graph represents a random walk with zero average on a -dimensional lattice, then the sequence is uniformly distributed modulo 1 if and only if .
L’objet de ce travail est d’étudier les propriétés arithmétiques et statistiques des mots infinis et des suites de nombres entiers engendrés par des substitutions sur un alphabet infini ou par des automates déterministes ayant un nombre infini dénombrable d’états. En particulier, nous montrons que si est une suite de nombres entiers engendrée par un automate dont le graphe étiqueté associé représente une marche aléatoire de moyenne nulle sur un réseau de ( entier positif), alors la suite est équirépartie modulo 1 si et seulement si .
Mot clés : mots infinis, substitutions, automates, équirépartition modulo 1
Keywords: Infinite words, substitutions, automata, uniform distribution modulo 1
@article{AIF_2006__56_7_2525_0, author = {Mauduit, Christian}, title = {Propri\'et\'es arithm\'etiques des substitutions et automates infinis}, journal = {Annales de l'Institut Fourier}, pages = {2525--2549}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {7}, year = {2006}, doi = {10.5802/aif.2248}, zbl = {1147.11016}, mrnumber = {2290789}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/aif.2248/} }
TY - JOUR AU - Mauduit, Christian TI - Propriétés arithmétiques des substitutions et automates infinis JO - Annales de l'Institut Fourier PY - 2006 SP - 2525 EP - 2549 VL - 56 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2248/ DO - 10.5802/aif.2248 LA - fr ID - AIF_2006__56_7_2525_0 ER -
%0 Journal Article %A Mauduit, Christian %T Propriétés arithmétiques des substitutions et automates infinis %J Annales de l'Institut Fourier %D 2006 %P 2525-2549 %V 56 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2248/ %R 10.5802/aif.2248 %G fr %F AIF_2006__56_7_2525_0
Mauduit, Christian. Propriétés arithmétiques des substitutions et automates infinis. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2525-2549. doi : 10.5802/aif.2248. http://www.numdam.org/articles/10.5802/aif.2248/
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