Arithmetic Meyer sets and finite automata |
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Authors: | Shigeki Akiyama, Fr d rique Bassino,Christiane Frougny |
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Affiliation: | aDepartment of Mathematics, Niigata University, Japan;bInstitut Gaspard Monge, Université de Marne-la-Vallée, France;cLIAFA, CNRS & Université Paris 7, and Université Paris 8, France |
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Abstract: | 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 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 is a Meyer set, which implies that there exists a finite set F (which depends only on β) such that . 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 as a sum belonging to . |
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