Integers in number systems with positive and negative quadratic Pisot base

We consider numeration systems with base $\beta$ and $-\beta$, for quadratic Pisot numbers $\beta$ and focus on comparing the combinatorial structure of the sets ${\mathbb{Z}}_{\beta }$ and ${\mathbb{Z}}_{-\beta }$ 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.