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.

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.

DOI : https://doi.org/10.1051/ita/2014013
Classification : 11K16,  68R15
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://www.numdam.org/articles/10.1051/ita/2014013/}
}
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://www.numdam.org/articles/10.1051/ita/2014013/

[1] B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 307 (2003) 47-75. | MR 2014730 | Zbl 1059.68083

[2] P. Ambrož, D. Dombek, Z. Masáková and E. Pelantová, Numbers with integer expansion in the numeration system with negative base. Funct. Approx. Comment. Math. 47 (2012) 241-266. | MR 3051451 | Zbl 1271.11009

[3] L. Balková, E. Pelantová and Š. Starosta, Sturmian jungle (or garden?) on multilateral alphabets. RAIRO: ITA 44 (2010) 443-470. | Numdam | MR 2775406 | Zbl 1211.68295

[4] L. Balková, E. Pelantová and O. Turek, Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers. RAIRO: ITA 41 (2007) 307-328. | Numdam | MR 2354360 | Zbl 1144.11009

[5] F. Bassino, 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 1966122 | Zbl 1152.11342

[6] Č. Burdík, Ch. Frougny, J.P. Gazeau and R. Krejcar, Beta-Integers as Natural Counting Systems for Quasicrystals. J. Phys. A: Math. Gen. 31 (1998) 6449-6472. | MR 1644115 | Zbl 0941.52019

[7] A. Elkharrat, Ch. Frougny, J.P. Gazeau and J.-L. Verger-Gaugry, Symmetry groups for beta-lattices. Theoret. Comput. Sci. 319 (2004) 281-305. | MR 2074957 | Zbl 1068.52028

[8] S. Fabre, Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR 1311222 | Zbl 0872.11017

[9] P. Fogg, 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 1970385 | Zbl 1014.11015

[10] P. Góra, Invariant densities for generalized β-maps. Ergodic Theory Dyn. Systems 27 (2007) 1583-1598. | MR 2358979 | Zbl 1123.37015

[11] L.S. Guimond, Z. Masáková and E. Pelantová, Combinatorial properties of infinite words associated with cut-and-project sequences, J. Théor. Nombres Bordeaux 15 (2003) 697-725. | Numdam | MR 2142232 | Zbl 1076.68055

[12] S. Ito and T. Sadahiro, (− β)-expansions of real numbers. Integers 9 (2009) 239-259. | MR 2534912 | Zbl 1191.11005

[13] C. Kalle, Isomorphisms between positive and negative beta-transformations, Ergodic Theory Dyn. Systems 32 (2014) 153-170. | MR 3163028 | Zbl 1287.37024

[14] L. Liao and W. Steiner, Dynamical properties of the negative beta-transformation. Ergodic Theory Dyn. Systems 32 (2012) 1673-1690. | MR 2974214 | Zbl 1266.37017

[15] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002). | MR 1905123 | Zbl 1221.68183

[16] M. Morse and G. Hedlund, Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 61 (1940) 1-42. | JFM 66.0188.03 | MR 745 | Zbl 0022.34003

[17] Z. Masáková and E. Pelantová, Purely periodic expansions in systems with negative base. Acta Math. Hungar 139 (2013) 208-227. | Zbl 1286.11118

[18] Z. Masáková, E. Pelantová and T. Vávra, Arithmetics in number systems with negative base. Theoret. Comput. Sci. 412 (2011) 835-845. | Zbl 1226.11015

[19] Z. Masáková and T. Vávra, Numeration systems with negative base β for quadratic Pisot numbers. Kybernetika 47 (2011) 74-92. | Zbl 1227.11033

[20] R.V. Moody, 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] W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401-416. | MR 142719 | Zbl 0099.28103

[22] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957) 477-493. | MR 97374 | Zbl 0079.08901

[23] M. Rigo, P. Salimov and E. Vandomme, Some properties of abelian return words. J. Integer Sequences 16 (2013) 13.2.5. | MR 3032388 | Zbl 1297.68195

[24] W. Steiner, On the structure of (− β)-integers. RAIRO: ITA 46 (2012) 181-200. | MR 2904969

[25] W.P. Thurston, Groups, tilings, and finite state automata. AMS Colloquium Lecture Notes. American Mathematical Society, Boulder (1989).

[26] O. Turek, Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO: ITA 41 (2007) 123-135. | Numdam | MR 2350639 | Zbl 1146.68410