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17
Factor complexity of infinite words associated with nonsimple Parry
, 812
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Arithmetic Meyer sets and finite automata
 Information and Computation
, 2005
"... Nonstandard number representation has proved to be useful in the speedup of some algorithms, and in the modelization of solids called quasicrystals. Using tools from automata theory we study the set Zβ of βintegers, that is, the set of real numbers which have a zero fractional part when expanded ..."
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Nonstandard number representation has proved to be useful in the speedup of some algorithms, and in the modelization of solids called quasicrystals. Using tools from automata theory we study the set Zβ of βintegers, that is, the set of real numbers which have a zero fractional part when expanded in a real base β, for a given β> 1. In particular, when β is a Pisot number — like the golden mean —, the set Zβ is a Meyer set, which implies that there exists a finite set F (which depends only on β) such that Zβ − Zβ ⊂ Zβ + F. Such a finite set F, even of minimal size, is not uniquely determined. In this paper we give a method to construct the sets F and an algorithm, whose complexity is exponential in time and space, to minimize their size. We also give a finite transducer that performs the decomposition of the elements of Zβ − Zβ as a sum belonging to Zβ + F. 1
TILINGS FOR PISOT BETA NUMERATION
"... Abstract. For a (nonunit) Pisot number β, several collections of tiles are associated with βnumeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the βtransformation and a Euclidean one made of integral betatiles. We ..."
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Abstract. For a (nonunit) Pisot number β, several collections of tiles are associated with βnumeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the βtransformation and a Euclidean one made of integral betatiles. We show that all these collections (except possibly the periodic translation of the central tile) are tilings if one of them is a tiling or, equivalently, the weak finiteness property (W) holds. We also obtain new results on rational numbers with purely periodic βexpansions; in particular, we calculate γ(β) for all quadratic β with β 2 = aβ +b, gcd(a,b) = 1. hal00869984, version 1 4 Oct 2013 1.
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 equipp ..."
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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.
Repetitions in betaintegers
, 812
"... Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this Letter, we study the maximal possible repetition of the same motif occurring in βintegers – one dimensional models of quasicrystals. We are interested in βintegers realizing only a fi ..."
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Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this Letter, we study the maximal possible repetition of the same motif occurring in βintegers – one dimensional models of quasicrystals. We are interested in βintegers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding βintegers. We will solve a particular case for β being a quadratic nonsimple Parry number. 1
On gaps in Rényi βexpansions of unity for . . .
, 2006
"... Let β> 1 be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi βexpansion dβ(1) of unity which controls the set Zβ of βintegers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in dβ(1) are shown t ..."
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Let β> 1 be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi βexpansion dβ(1) of unity which controls the set Zβ of βintegers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in dβ(1) are shown to exhibit a “gappiness” asymptotically bounded above by log(M(β)) / log(β), where M(β) is the Mahler measure of β. The proof of this result provides in a natural way a new classification of algebraic numbers> 1 with classes called Q (j) i which we compare to BertrandMathis’s classification with classes C1 to C5 (reported in an article by Blanchard). This new classification relies on the maximal asymptotic “quotient of the gap ” value of the “gappy ” power series associated with dβ(1). As a corollary, all Salem numbers are in the class C1 ∪ Q (1) 0 ∪ Q(2) 0 ∪ Q(3) 0; this result is also directly proved using a recent generalization of the ThueSiegelRoth Theorem given by Corvaja.