Substitutions, abstract number systems and the space filling property
Annales de l'Institut Fourier, Volume 56 (2006) no. 7, p. 2345-2389

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo 1 and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

Dans cet article nous étudions des mots multidimensionnels engendrés par des points fixes de substitutions, et obtenus en projetant les points entiers sur la demi-droite brisée correspondante. Nous montrons que pour une grande classe de substitutions le mot correspondant est la restriction d’une fonction linéaire modulo 1 et qu’il est possible de décider si le mot résultant remplit l’espace. La preuve utilise des réseaux et le système de numération abstrait associé à la substitution.

DOI : https://doi.org/10.5802/aif.2243
Classification:  11A63,  68R15,  37B10,  05A05,  05B25,  52C07,  52C22,  68Q45,  11K16
Keywords: Substitutions, limit word, discretisation of the hyperplane, lattices, automata, abstract number systems
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     author = {Fuchs, Clemens and Tijdeman, Robert},
     title = {Substitutions, abstract number systems and the space filling property},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {7},
     year = {2006},
     pages = {2345-2389},
     doi = {10.5802/aif.2243},
     mrnumber = {2290784},
     zbl = {pre05176572},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_7_2345_0}
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Fuchs, Clemens; Tijdeman, Robert. Substitutions, abstract number systems and the space filling property. Annales de l'Institut Fourier, Volume 56 (2006) no. 7, pp. 2345-2389. doi : 10.5802/aif.2243. http://www.numdam.org/item/AIF_2006__56_7_2345_0/

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