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Integers in number systems with positive and negative quadraticPisot 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 comparingthe combinatorial structure of the sets Zβ and Z− β of numberswith integer expansion in base β, resp. − β. Our main result is the comparison of languagesof infinite words uβ andu−β coding the ordering of distances betweenconsecutive β- and (−β)-integers. It turns out that for a class of rootsβ ofx2 −mx − m, the languages coincide,while for other quadratic Pisot numbers the language of uβ can be identified onlywith the language of a morphic image of u− β. We also study thegroup structure of (−β)-integers.

Keywords

Quadratic Pisot numbersbeta-integersnegative base

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