Arithmetic Meyer sets and finite automata

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 .