Arithmetic Meyer sets and finite automata

  • Authors:

  • Shigeki Akiyama;Frédérique Bassino;Christiane Frougny

  • Affiliations:

  • Department of Mathematics, Niigata University, Japan;Institut Gaspard Monge, Université de Marne-la-Vallée, France;LIAFA, CNRS & Universitéé Paris and Université Paris, France

  • Venue:
  • Information and Computation
  • Year:
  • 2005

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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 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.