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.
@article{ITA_2014__48_3_341_0, author = {Mas\'akov\'a, Z. and V\'avra, T.}, title = {Integers in number systems with positive and negative quadratic Pisot base}, journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications}, volume = {48}, year = {2014}, pages = {341-367}, doi = {10.1051/ita/2014013}, mrnumber = {3302492}, language = {en}, url = {http://dml.mathdoc.fr/item/ITA_2014__48_3_341_0} }
Masáková, Z.; Vávra, T. Integers in number systems with positive and negative quadratic Pisot base. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 48 (2014) pp. 341-367. doi : 10.1051/ita/2014013. http://gdmltest.u-ga.fr/item/ITA_2014__48_3_341_0/
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