Computation of L ⊕ for several cubic Pisot numbers

In this article, we are dealing with β-numeration, which is a generalization of numeration in a non-integer base. We consider the class of simple Parry numbers such that d β (1) = 0.k 1 d-1  k d with d ∈ ℕ, d ≥ 2 and k 1  ≥ k d  ≥ 1. We prove that these elements define Rauzy fractals that are stable under a central symmetry. We use this result to compute, for several cases of cubic Pisot units, the maximal length among the lengths of the finite β-fractional parts of sums of two β-integers, denoted by L ⊕. In particular, we prove that L ⊕  = 5 in the Tribonacci case