beta-expansions

Non-standard number representation has proved to be useful in the speed-up of some algorithms, and in the modelization of solids called quasicrystals. Using tools from automata theory we study the set Zβ of β-integers, 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

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.

Entropy quotients and correct digits in

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 β-integers – one dimensional models of quasicrystals. We are interested in β-integers 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 β-integers. We will solve a particular case for β being a quadratic non-simple Parry number. 1

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