Sur les ensembles d'entiers reconnaissables

TL;DR

This paper proves that any subset of positive integers recognizable in two multiplicatively independent Bertrand numeration systems must be a finite union of arithmetic progressions, revealing a structural limitation of recognizability across such systems.

Contribution

It establishes a new structural characterization of recognizable sets in two multiplicatively independent Bertrand systems, showing they are finite unions of arithmetic progressions.

Findings

Recognizable sets in two independent Bertrand systems are finite unions of arithmetic progressions.

The result links recognizability with classical number theory structures.

It extends understanding of the limitations of numeration system recognizability.

Abstract

Let U and V be two Bertrand numeration systems, and, a and b the two Parry numbers there are naturally associated with. Suppose they are multiplicatively independent. We prove that, if E is a subset of positive integers which is both U and V recognizable, then E is a finite union of arithmetical progressions.