Integers in number systems with positive and negative quadratic Pisot base
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 48 (2014) no. 3, pp. 341-367.

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.

DOI: 10.1051/ita/2014013
Classification: 11K16,  68R15
Keywords: quadratic Pisot numbers, beta-integers, negative base
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Masáková, Z.; Vávra, T. Integers in number systems with positive and negative quadratic Pisot base. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 48 (2014) no. 3, pp. 341-367. doi : 10.1051/ita/2014013. http://www.numdam.org/articles/10.1051/ita/2014013/

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