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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     title = {$\beta $-shift, syst\`emes de num\'eration et automates},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {473--498},
     publisher = {Universit\'e Bordeaux I},
     volume = {7},
     number = {2},
     year = {1995},
     mrnumber = {1378592},
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     url = {http://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. http://www.numdam.org/item/JTNB_1995__7_2_473_0/

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