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Integers in number systems with positive and negative quadratic Pisot base

Published online by Cambridge University Press:  13 June 2014

Z. Masáková  and
T. Vávra
Z. Masáková
Affiliation:
Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 12000 Praha 2, Czech Republic.. zuzana.masakova@fjfi.cvut.cz; t.vavra@seznam.cz
T. Vávra
Affiliation:
Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 12000 Praha 2, Czech Republic.. zuzana.masakova@fjfi.cvut.cz; t.vavra@seznam.cz
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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 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.

Keywords

Quadratic Pisot numbersbeta-integersnegative base
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 48 , Issue 3: Special issue in the honor of the 14th “Journées montoises d’informatique théorique”. I. , July 2014 , pp. 341 - 367
Copyright
© EDP Sciences 2014

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