Representation of numbers in nonclassical numeration systems

Abstract

Numeration systems, the bases of which are defined by a linear recurrence with integer coefficients, are considered. Conditions on the recurrence are given under which the function of normalization which transforms any representation of an integer into the normal one-obtained by the usual algorithm-can be realized by a finite automaton. Addition is a particular case of normalization. The same questions are discussed for the representation of real numbers in basis theta , where theta is a real number >1. In particular it is shown that, if theta is a Pisot number, then the normalization and the addition in basis theta are computable by a finite automaton.<>