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Symbolic Dynamics and Finite Automata
, 1999
"... 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. ..."
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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.
Additive And Multiplicative Properties Of Point Sets Based On BetaIntegers
"... .  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 betaintegers 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", ..."
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Cited by 12 (0 self)
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.  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 betaintegers 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 PisotVijayaraghavan unit (quasicrystallographic inflation factors are particular examples). We also consider the case of a cubic PisotVijayaraghavan unit associated with the sevenfold cyclotomic ring. At the end, we show how the point sets fi are vertices of ddimensional 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...
BetaIntegers As A Group
, 1999
"... this paper. The fiintegers are defined via a numeration system with base fi, see below. In ..."
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Cited by 4 (0 self)
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this paper. The fiintegers are defined via a numeration system with base fi, see below. In
RATIONAL NUMBERS WITH PURELY PERIODIC βEXPANSION
"... We study real numbers β with the curious property that the βexpansion 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 c ..."
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Cited by 3 (3 self)
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We study real numbers β with the curious property that the βexpansion 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 βexpansion. 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.
On the ContextFreeness of the ThetaExpansions of the Integers
"... Let ` ? 1 be a nonintegral real number such that the `expansion of every positive integer is finite. If the set of `expansions of all the positive integers is a contextfree language, then ` must be a quadratic Pisot unit. R'esum'e Soit ` ? 1 un nombre r'eel non entier tel que le `d'eveloppement ..."
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Let ` ? 1 be a nonintegral real number such that the `expansion of every positive integer is finite. If the set of `expansions of all the positive integers is a contextfree 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 lettertoletter finite twotape 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...
Symmetry Groups for BetaLattices
, 2003
"... We present a construction of symmetry planegroups for quasiperiodic pointsets in the plane, named betalattices. The algebraic framework is issued from counting systems called betaintegers, determined by PisotVijayaraghavan (PV) algebraic integers beta > 1. The betainteger sets can be equipped ..."
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We present a construction of symmetry planegroups for quasiperiodic pointsets in the plane, named betalattices. The algebraic framework is issued from counting systems called betaintegers, determined by PisotVijayaraghavan (PV) algebraic integers beta > 1. The betainteger sets can be equipped with abelian group structures and internal multiplicative laws. These arithmetic structures lead to freely generated symmetry planegroups for betalattices, based on repetitions of discrete "adapted rotations and translations" in the plane. Hence betalattices, endowed with these adapted rotations and translations, can be viewed like lattices. Moreover, betalattices tend to behave asymptotically like lattices.
Fully Subtractive Algorithm, Tribonacci numeration and connectedness of discrete planes By
"... We investigate connections between a well known multidimensional continued fraction algorithm, the socalled fully subtractive algorithm, the finiteness property for βnumeration, and the connectedness of arithmetic discrete hyperplanes. A discrete hyperplane is said to be critical if its thickness ..."
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We investigate connections between a well known multidimensional continued fraction algorithm, the socalled fully subtractive algorithm, the finiteness property for βnumeration, and the connectedness of arithmetic discrete hyperplanes. A discrete hyperplane is said to be critical if its thickness is equal to the infimum of the set of thicknesses for which discrete hyperplanes with the same normal vector are connected. We focus on particular planes the parameters of which belong to the cubic extension generated by the Tribonacci number, we prove connectedness in the critical case, and we exhibit an intriguing tree structure rooted at the origin.