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 Z(beta) of beta-integers, that is, the set of real numbers which have a zero fractional part when expanded in a real base 8, for a given beta > 1. In particular, when beta is a Pisot number-like the golden mean-, the set Z(beta) is a Meyer set, which implies that there exists a finite set F (which depends only on beta) such that Z(beta) - Z(beta) subset of Z(beta) + 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 Z(beta) - Z(beta) as a sum belonging to Z(beta) + F. (c) 2005 Elsevier Inc. All rights reserved.