no. 2
β-shift, systèmes de numération et automates
Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 473-498

In this note we prove that the language of a numeration system is the language of a β -shift under some assumptions on the basis. We deduce from this result a partial answer to the question when the language of a numeration system is regular. Moreover, we give a characterization of the arithmetico-geometric sequences and the mixed radix sequences that are basis of a numeration system for which the language is regular. Finally, we study the Ostrowski systems of numeration and give another proof of the result of J. Shallit : the Ostrowski systems having a regular langage are exactly the ones associated to a quadratic number.

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     author = {Loraud, Nathalie},
     title = {$\beta $-shift, syst\`emes de num\'eration et automates},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {473--498},
     year = {1995},
     publisher = {Universit\'e Bordeaux I},
     volume = {7},
     number = {2},
     mrnumber = {1378592},
     zbl = {0843.11013},
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     url = {https://www.numdam.org/item/JTNB_1995__7_2_473_0/}
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Loraud, Nathalie. $\beta $-shift, systèmes de numération et automates. Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 473-498. https://www.numdam.org/item/JTNB_1995__7_2_473_0/

[Be 1]: A. Bertrand, Comment écrire les nombres entiers dans une base qui n'est pas entière. à paraître dans Acta. Math. Acad. Sci. Hungar. | Zbl

[Be 2]: A. Bertrand, Questions diverses relatives aux systèmes codés: applications au θ-shift. Preprint

[Be 3]: A. Bertrand, Nombres de Perron et problèmes de rationnalité. S. M. F. (1991), 198-200.

[Be 4]: A. Bertrand, Le θ-shift sans peine. en préparation.

[Be 5]: A. Bertrand, Développement en base θ, répartition modulo un de la suite (xθn) n≽0, langages codés et θ-shift. Bull. Soc. math. France 114 (1986), 271-323. | Zbl | Numdam | EuDML

[Bl]: F. Blanchard, β-expansions and symbolic dynamics. Theoret. Comput. Sci. 65 (1989), 131-141. | Zbl

[Br]: A. Brauer, On algebraic equations with all but one root in the interior of the unit circle. Math. Nachr. 4 (1951), 250-257. | Zbl | MR

[Co]: A. Cobham, Uniform tag sequences. Math. Syst. Theory 6 (1972), 164-192. | Zbl | MR

[Fr 1]: A.S. Fraenkel, Systems of numeration. Amer. Math. Monthly 92 (1985), 105-114. | Zbl | MR

[Fr 2]: A.S. Fraenkel, The use and usefulness of numeration systems. Inform. and Comput. 81 (1989), 46-61. | Zbl | MR

[Fro 1]: C. Frougny, Representations of numbers and finite automata. Math. Syst. Theory 25 (1992), 37-60. | Zbl | MR

[Fro 2]: C. Frougny, Linear Numeration Systems of Order Two. Inform. & Comput. 77 (1988), 233-259. | Zbl | MR

[Fro 3]: C. Frougny, Systèmes de numération linéaires et θ-représentations. Theoret. Comput. Sci. 94 (1992), 223-236. | Zbl

[F-So]: C. Frougny, B. Solomyak, Finite β-expansions. Ergod. Th. & Dynam. Sys. 12 (1992), 713-723. | Zbl

[G-L-T]: P.J. Grabner, P. Liardet et R.F. Tichy, Odometers and systems of numerations. Acta Arith. to appear. | Zbl | MR

[G-T 1]: P.J. Grabner, R.F. Tichy, Contributions to Digit Expansions with Respect to Linear Recurrences. J. Number Th. 36 (1990), 160-169. | Zbl | MR

[G-T 2]: P.J. Grabner, R.F. Tichy, α-expansions, Linear recurrences and the Sum-of-Digits Function. Manuscripta Math. 70 (1991), 311-324. | Zbl

[I-T]: S. Ito and Y. Takahashi, Markov subshifts and realization of β-expansions. J. Math. Soc. Japan 26 1 (1974), 33-55. | Zbl

[Li]: D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4 (1984), 283-300. | Zbl | MR

[L-S]: J. Shallit, H.W. Lenstra, Continued fractions and linear recurrences. Math. Comp. 61 (1993), 351-354. | Zbl | MR

[Os]: A. Ostrowski, Bemerkungen zur Theorie der Diophantishen Approximationene. Abh. Math. Sem. Hamburg 1 (1922), 77-98. | JFM

[Pa]: W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. | Zbl

[P-T]: A. Pethö, R.F. Tichy, On Digit Expansions with Respect to Linear Recurrences. J. Number Th. 33 (1989), 243-256. | Zbl | MR

[Re]: A. Reyi, Representations for real numbers and their ergodic properties. Acta Math. Ac. Sci. Hungar. 8 (1957), 477-493. | Zbl | MR

[Sc]: K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269-278. | Zbl | MR

[Sh 1]: J. Shallit, A generalization of automatic sequences. Theoret. Comput. Sci. 61 (1988), 1-16. | Zbl | MR

[Sh 2]: J. Shallit, Numeration Systems, Linear Recurrences, and Regular Sets. Inform. and comput. to appear | Zbl | MR