Multiple tilings associated to d-Bonacci beta-expansions

Abstract

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 $$d-1$$ .