Towards a characterization of the star-free sets of integers

Let U be a numeration system, a set X of integers is U-star-free if the set made up of the U-representations of the elements in X is a star-free regular language. Answering a question of A. de Luca and A. Restivo, we obtain a complete logical characterization of the U-star-free sets of integers for suitable numeration systems related to a Pisot number and in particular for integer base systems. For these latter systems, we study as well the problem of the base dependence. Finally, the case of k-adic systems is also investigated.