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 numbers, beta-integers, negative base
@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},
pages = {341--367},
year = {2014},
publisher = {EDP Sciences},
volume = {48},
number = {3},
doi = {10.1051/ita/2014013},
mrnumber = {3302492},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ita/2014013/}
}
TY - JOUR AU - Masáková, Z. AU - Vávra, T. TI - Integers in number systems with positive and negative quadratic Pisot base JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2014 SP - 341 EP - 367 VL - 48 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ita/2014013/ DO - 10.1051/ita/2014013 LA - en ID - ITA_2014__48_3_341_0 ER -
%0 Journal Article %A Masáková, Z. %A Vávra, T. %T Integers in number systems with positive and negative quadratic Pisot base %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2014 %P 341-367 %V 48 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ita/2014013/ %R 10.1051/ita/2014013 %G en %F 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) no. 3, pp. 341-367. doi: 10.1051/ita/2014013
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