To each number $\beta>1$ correspond abelian groups in $\R^d$, of the form $\Lambda_\b=\sum_{i=1}^{d} \Z_\b {\bf e}_i$, which obey $\b \Lambda_\b \subset \Lambda_\b$. The set $\Z_{\beta}$ of beta-integers is a countable set of numbers : it is precisely the set of real numbers which are polynomial in $\beta$ when they are written in ''basis $\beta$'', and $\Z_{\beta} =\Z$ when $\beta \in \N $. We prove here a list of arithmetic properties of $\Z_{\beta}$: addition, multiplication, relation with integers, when $\beta$ is a quadratic Pisot-Vijayaraghavan unit (quasicrystallographic inflation factors are particular examples). We also consider the case of a cubic Pisot-Vijayaraghavan unit associated with the seven-fold cyclotomic ring. At the end, we show how the point sets $\Lambda_\beta$ are vertices of $d$-dimensional tilings.