Towards a characterization of the star-free sets of integers

TL;DR

This paper characterizes the star-free sets of integers in various numeration systems, providing logical descriptions and exploring base dependence, especially for systems related to Pisot numbers and k-adic systems.

Contribution

It offers a complete logical characterization of U-star-free sets for certain numeration systems, answering a previously open question.

Findings

Logical characterization for systems related to Pisot numbers

Analysis of base dependence in integer representations

Investigation of k-adic systems

Abstract

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.