We consider numeration systems with base β and - β, for quadratic Pisot numbers β and focus on comparing the combinatorial structure of the sets Zβ and Z- β of numbers with integer expansion in base β, resp. - β. Our main result is the comparison of languages of infinite words uβ and u- β coding the ordering of distances between consecutive β- and (- β)-integers. It turns out that for a class of roots β of x2 - mx - m, the languages coincide, while for other quadratic Pisot numbers the language of uβ can be identified only with the language of a morphic image of u- β. We also study the group structure of (- β)-integers.
Mots-clés : quadratic Pisot numbers, beta-integers, negative base
@article{ITA_2014__48_3_341_0,
author = {Mas\'akov\'a, Z. and V\'avra, T.},
title = {Integers in number systems with positive and negative quadratic {Pisot} base},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {341--367},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
doi = {10.1051/ita/2014013},
mrnumber = {3302492},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ita/2014013/}
}
TY - JOUR AU - Masáková, Z. AU - Vávra, T. TI - Integers in number systems with positive and negative quadratic Pisot base JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2014 SP - 341 EP - 367 VL - 48 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ita/2014013/ DO - 10.1051/ita/2014013 LA - en ID - ITA_2014__48_3_341_0 ER -
%0 Journal Article %A Masáková, Z. %A Vávra, T. %T Integers in number systems with positive and negative quadratic Pisot base %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2014 %P 341-367 %V 48 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ita/2014013/ %R 10.1051/ita/2014013 %G en %F ITA_2014__48_3_341_0
Masáková, Z.; Vávra, T. Integers in number systems with positive and negative quadratic Pisot base. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014) no. 3, pp. 341-367. doi : 10.1051/ita/2014013. http://archive.numdam.org/articles/10.1051/ita/2014013/
[1] , Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 307 (2003) 47-75. | MR | Zbl
[2] , , and , Numbers with integer expansion in the numeration system with negative base. Funct. Approx. Comment. Math. 47 (2012) 241-266. | MR | Zbl
[3] , and , Sturmian jungle (or garden?) on multilateral alphabets. RAIRO: ITA 44 (2010) 443-470. | Numdam | MR | Zbl
[4] , and , Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers. RAIRO: ITA 41 (2007) 307-328. | Numdam | MR | Zbl
[5] , β-expansions for cubic Pisot numbers, in Proc. of 5th Latin American Theoretical Informatics Symposium, LATIN'02. Vol. 2286 Lect. Note Comput. Sci. Springer-Verlag (2002) 141-152. | MR | Zbl
[6] , , and , Beta-Integers as Natural Counting Systems for Quasicrystals. J. Phys. A: Math. Gen. 31 (1998) 6449-6472. | MR | Zbl
[7] , , and , Symmetry groups for beta-lattices. Theoret. Comput. Sci. 319 (2004) 281-305. | MR | Zbl
[8] , Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR | Zbl
[9] , Substitutions in Dynamics, Arithmetics and Combinatorics, edited by V. Berthé, S. Ferenczi, C. Mauduit, A. Siegel, vol. 1794 of Lect. Note Math. Ser. Springer (2002). | MR | Zbl
[10] , Invariant densities for generalized β-maps. Ergodic Theory Dyn. Systems 27 (2007) 1583-1598. | MR | Zbl
[11] , and , Combinatorial properties of infinite words associated with cut-and-project sequences, J. Théor. Nombres Bordeaux 15 (2003) 697-725. | Numdam | MR | Zbl
[12] S. Ito and T. Sadahiro, (− β)-expansions of real numbers. Integers 9 (2009) 239-259. | MR | Zbl
[13] , Isomorphisms between positive and negative beta-transformations, Ergodic Theory Dyn. Systems 32 (2014) 153-170. | MR | Zbl
[14] and , Dynamical properties of the negative beta-transformation. Ergodic Theory Dyn. Systems 32 (2012) 1673-1690. | MR | Zbl
[15] , Algebraic combinatorics on words. Cambridge University Press (2002). | MR | Zbl
[16] and , Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 61 (1940) 1-42. | JFM | MR | Zbl
[17] and , Purely periodic expansions in systems with negative base. Acta Math. Hungar 139 (2013) 208-227. | Zbl
[18] , and , Arithmetics in number systems with negative base. Theoret. Comput. Sci. 412 (2011) 835-845. | Zbl
[19] and , Numeration systems with negative base β for quadratic Pisot numbers. Kybernetika 47 (2011) 74-92. | Zbl
[20] , Model sets: A Survey, in From Quasicrystals to More Complex Systems (Les Houches) edited by F. Axel, F. Denoyer, J.-P. Gazeau. Springer (2000).
[21] , On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401-416. | MR | Zbl
[22] , Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957) 477-493. | MR | Zbl
[23] , and , Some properties of abelian return words. J. Integer Sequences 16 (2013) 13.2.5. | MR | Zbl
[24] , On the structure of (− β)-integers. RAIRO: ITA 46 (2012) 181-200. | MR
[25] , Groups, tilings, and finite state automata. AMS Colloquium Lecture Notes. American Mathematical Society, Boulder (1989).
[26] , Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO: ITA 41 (2007) 123-135. | Numdam | MR | Zbl
Cité par Sources :
