Symmetrized β -integers

This article deals with β-numeration systems, which are numeration systems with a non-integral base β>1. In this framework, there exist elements which naturally play the role of integers, which are called β-integers. The set of non-negative β-integers, denoted by , has various equivalent definitions which arise from different points of view. Nevertheless, these definitions may be generalized on negative real numbers in a non-unique way, depending on the chosen framework.We focus in this article on confluent Parry units. They are the positive roots of the polynomials , where the integers d and k satisfy d≥2 and k≥1. For any of these numbers, we prove that there exists a discrete subset of R, that we denote by , which is locally isomorphic to , and such that . Moreover, is a model set and satisfies the inflation property: there exists λ>1 such that (λ is then called the inflation factor for ). Finally, we compute inflation factors for of the form βi, i being a positive integer.