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arXiv · 1302.4655

Integers in number systems with positive and negative quadratic Pisot base

Zuzana Masáková·Tomáš Vávra
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Abstract

We consider numeration systems with base $\beta$ and $-\beta$, for quadratic Pisot numbers $\beta$ and focus on comparing the combinatorial structure of the sets $\Z_\beta$ and $\Z_{-\beta}$ of numbers with integer expansion in base $\beta$, resp. $-\beta$. Our main result is the comparison of languages of infinite words $u_\beta$ and $u_{-\beta}$ coding the ordering of distances between consecutive $\beta$- and $(-\beta)$-integers. It turns out that for a class of roots $\beta$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_\beta$ can be identified only with the language of a morphic image of $u_{-\beta}$. We also study the group structure of $(-\beta)$-integers.

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Zuzana Masáková, Tomáš Vávra. 2013-02-19. Integers in number systems with positive and negative quadratic Pisot base. https://doi.org/10.1051/ita/2014013

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