### Computation of L_{⊕} for several cubic Pisot numbers

*Julien Bernat*

#### Abstract

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.Full Text: PostScript PDF