On the Delone property of (-β)-integers

These sets were introduced in the domain of quasicrystallography, see e.g. [2]. It is not difficult to see that Z−β =Z when β ∈Z, and that Z−β = {0} when β < 1+ √ 5 2 . For β ≥ 1+ √ 5 2 , Ambrož et al. [1] showed that Z−β can be described by the fixed point of an anti-morphism on a possibly infinite alphabet. They also calculated explicitely the set of distances between consecutive (−β )-integers when T n −β ( −β β+1 ) ≤ 0 and T 2n−1 −β ( −β β+1 ) ≥ 1−⌊β⌋ β for all n ≥ 1. It seems to be difficult to extend their methods to the general case. For the case when β is an Yrrap number, i.e., when { T n −β ( −β β+1 ) | n ≥ 0 }