Multiple tilings associated to d-Bonacci beta-expansions

Let $\beta\in(1,2)$ be a Pisot unit and consider the symmetric $\beta$-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For $\beta$ a $d$-Bonacci number, i.e., Pisot root of $x^d-x^{d-1}-\dots-x-1$ we show that the Rauzy fractals form a multiple tiling with covering degree $d-1$.