  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.
