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Integers in number systems with positive and negative quadratic Pisot base

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

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.

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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RAIRO - Theoretical Informatics and Applications
  • ISSN: 0988-3754
  • EISSN: 1290-385X
  • URL: /core/journals/rairo-theoretical-informatics-and-applications
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