this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund.

. --- To each number fi ? 1 correspond abelian groups in R d , of the form fi = P d i=1 Z fi e i , which obey fi fi ae fi . The set Z fi of beta-integers is a countable set of numbers : it is precisely the set of real numbers which are polynomial in fi when they are written in "basis fi", and Z fi = Zwhen fi 2 N. We prove here a list of arithmetic properties of Z fi : addition, multiplication, relation with integers, when fi is a quadratic Pisot-Vijayaraghavan unit (quasicrystallographic inflation factors are particular examples). We also consider the case of a cubic Pisot-Vijayaraghavan unit associated with the seven-fold cyclotomic ring. At the end, we show how the point sets fi are vertices of d-dimensional tilings. R'esum'e. --- ` A chaque nombre fi ? 1 correspondent des groupes ab'eliens dans R d , de la forme fi = P d i=1 Z fi e i , et qui satisfont fi fi ae fi . L'ensemble Z fi des betaentiers est un ensemble d'enombrable de nombres, qui est form'e de t...

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this paper. The fi-integers are defined via a numeration system with base fi, see below. In

Let ` ? 1 be a non-integral real number such that the `-expansion of every positive integer is finite. If the set of `-expansions of all the positive integers is a context-free language, then ` must be a quadratic Pisot unit. R'esum'e Soit ` ? 1 un nombre r'eel non entier tel que le `-d'eveloppement de tout entier positif soit fini. Si l'on suppose que l'ensemble des `-d'eveloppements des entiers positifs forme un langage alg'ebrique, alors ` doit etre un nombre de Pisot quadratique unitaire. In [1], to which this note is an addendum, it was proved (Theorem 2) that, if ` is a quadratic Pisot unit, then there exists a letter-to-letter finite two-tape automaton that maps the representation of any integer in a linear numeration system canonically associated with ` onto the "folded" `-expansion of that integer (we refer to [1] for definitions and references). As an immediate consequence ([1, Corollary 4]), it follows that, if ` is a quadratic Pisot unit, then the set of `-expansions of all...

We present a construction of symmetry plane-groups for quasiperiodic point-sets in the plane, named beta-lattices. The algebraic framework is issued from counting systems called beta-integers, determined by Pisot-Vijayaraghavan (PV) algebraic integers beta > 1. The beta-integer sets can be equipped with abelian group structures and internal multiplicative laws. These arithmetic structures lead to freely generated symmetry plane-groups for beta-lattices, based on repetitions of discrete "adapted rotations and translations" in the plane. Hence beta-lattices, endowed with these adapted rotations and translations, can be viewed like lattices. Moreover, beta-lattices tend to behave asymptotically like lattices.

Abstract. — We study real numbers β with the curious property that the β-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let γ(β) denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less than c have a purely periodic β-expansion. We prove that γ(β) is irrational for a class of cubic Pisot units that contains the smallest Pisot number η. This result is motivated by the observation of Akiyama and Scheicher that γ(η) = 0.666 666 666 086 · · · is surprisingly close to 2/3. 1.