Asymptotic Behavior of Beta-Integers

Abstract

Beta-integers ("β-integers") are those numbers which are the counterparts of integers when real numbers are expressed in an irrational base β > 1. In quasicrystalline studies, β-integers supersede the "crystallographic" ordinary integers. When the number β is a Parry number, the corresponding β-integers realize only a finite number of distances between consecutive elements and are in this sense the most comparable to ordinary integers. In this paper, we point out the similarity of β-integers and ordinary integers in the asymptotic sense, in particular for a subclass of Parry numbers - Pisot numbers for which their Parry and minimal polynomial coincide

Arithmetics in number systems with a negative base

“…There are also other aspects of numeration in non-standard systems that deserve to be explored. Properties, known for Rényi numeration with positive base, likely to hold also for the case of negative base, are for example the description of the distribution of (−β)-integers, given for positive base in [2], or the connection of (−β)-numeration to substitution dynamical systems established for β-integers in [6]. Many such properties are studied in [13] for number systems with positive base which are generalizations of the Rényi numeration.…”

Repetitions in Beta-Integers

“…As it holds for coefficients of the continued fraction of β that a 2n−1 = 1 and a 2n = p − 1, it suffices to consider even n in (1). We obtain then finally…”

“…As we have already said, the reader may find the notions of the Rényi expansion of unity, βintegers, distances between neighbors in Z β etc. in our precedent Letter [1]. However, in order to follow the ideas in the sequel, it is sufficient to know that u β is the unique fixed point of a morphism canonically associated with parameters p, q characterizing non-simple quadratic Parry numbers β.…”

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

“…We have thus proved existence of constants so that (5) is valid. Let now v (1) , v (2) be factors of the infinite word ψ(u) with |v (1)…”

“…A class of one-dimensional sets studied for the BDL property is the family of the sets Z β for β > 1, where Z β is formed by the so-called β -integers, as defined in [6]. In [2], Gazeau et al have shown that Z β is BDL if β is a Pisot number. (Recall that α is a Pisot number if it is an algebraic integer greater than 1 with all algebraic conjugates in the interior of the unit disc.…”