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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
Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 12000 Praha 2, Czech Republic..;
T. Vávra
Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 12000 Praha 2, Czech Republic..;
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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.

Research Article
© EDP Sciences 2014

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