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Integers in number systems with positive and negative quadratic Pisot base

  • Z. Masáková (a1) and T. Vávra (a1)

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 Zβ of numbers with integer expansion in base β, resp. − β. Our main result is the comparison of languages of infinite words uβ and uβ coding the ordering of distances between consecutive β- and (− β)-integers. It turns out that for a class of roots β of x2mxm, 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 uβ. We also study the group structure of (− β)-integers.

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[1] Adamczewski, B., Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 307 (2003) 4775.
[2] Ambrož, P., Dombek, D., Masáková, Z. and Pelantová, E., Numbers with integer expansion in the numeration system with negative base. Funct. Approx. Comment. Math. 47 (2012) 241266.
[3] Balková, L., Pelantová, E. and Starosta, Š., Sturmian jungle (or garden?) on multilateral alphabets. RAIRO: ITA 44 (2010) 443470.
[4] Balková, L., Pelantová, E. and Turek, O., Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers. RAIRO: ITA 41 (2007) 307328.
[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.
[6] Burdík, Č., Frougny, Ch., Gazeau, J.P. and Krejcar, R., Beta-Integers as Natural Counting Systems for Quasicrystals. J. Phys. A: Math. Gen. 31 (1998) 64496472.
[7] Elkharrat, A., Frougny, Ch., Gazeau, J.P. and Verger-Gaugry, J.-L., Symmetry groups for beta-lattices. Theoret. Comput. Sci. 319 (2004) 281305.
[8] Fabre, S., Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219236.
[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).
[10] Góra, P., Invariant densities for generalized β-maps. Ergodic Theory Dyn. Systems 27 (2007) 15831598.
[11] Guimond, L.S., Masáková, Z. and Pelantová, E., Combinatorial properties of infinite words associated with cut-and-project sequences, J. Théor. Nombres Bordeaux 15 (2003) 697725.
[12] Ito, S. and Sadahiro, T., (− β)-expansions of real numbers. Integers 9 (2009) 239259.
[13] Kalle, C., Isomorphisms between positive and negative beta-transformations, Ergodic Theory Dyn. Systems 32 (2014) 153170.
[14] Liao, L. and Steiner, W., Dynamical properties of the negative beta-transformation. Ergodic Theory Dyn. Systems 32 (2012) 16731690.
[15] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002).
[16] Morse, M. and Hedlund, G., Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 61 (1940) 142.
[17] Masáková, Z. and Pelantová, E., Purely periodic expansions in systems with negative base. Acta Math. Hungar 139 (2013) 208227.
[18] Masáková, Z., Pelantová, E. and Vávra, T., Arithmetics in number systems with negative base. Theoret. Comput. Sci. 412 (2011) 835845.
[19] Masáková, Z. and Vávra, T., Numeration systems with negative base β for quadratic Pisot numbers. Kybernetika 47 (2011) 7492.
[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] Parry, W., On the β-expansions of real numbers. Acta Math. Acad. Sci. Hung. 11 (1960) 401416.
[22] Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8 (1957) 477493.
[23] Rigo, M., Salimov, P. and Vandomme, E., Some properties of abelian return words. J. Integer Sequences 16 (2013) 13.2.5.
[24] Steiner, W., On the structure of (− β)-integers. RAIRO: ITA 46 (2012) 181200.
[25] W.P. Thurston, Groups, tilings, and finite state automata. AMS Colloquium Lecture Notes. American Mathematical Society, Boulder (1989).
[26] Turek, O., Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO: ITA 41 (2007) 123135.
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RAIRO - Theoretical Informatics and Applications
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