Numeration systems without a dominant root and regularity

TL;DR

This paper explores the regularity of representations in positional numeration systems without a dominant root, establishing a broader connection to alternate base systems and providing a comprehensive characterization of regular languages.

Contribution

It introduces a general link between positional and alternate base numeration systems, extending previous results and characterizing systems that produce regular languages.

Findings

Established a broader connection between numeration systems and alternate base systems.

Provided a full characterization of systems generating regular languages.

Discussed the effectiveness of the new characterization method.

Abstract

Positional numeration systems are a large family of numeration systems used to represent natural numbers. Whether the set of all representations forms a regular language or not is one of the most important questions that can be asked of such a system. This question was investigated in a 1998 article by Hollander. Central to his analysis is a property linking positional numeration systems and R\'{e}nyi numeration systems, which use a real base to represent real numbers. However, this link only exists when the initial numeration system has a dominant root, which is not a necessary condition for regularity. In this article, we show a more general link between positional numeration systems and alternate base numeration systems, a family generalizing R\'{e}nyi systems. We then take advantage of this link to provide a full characterization of those numeration systems that generate a regular language. We also discuss the effectiveness of our method, and comment Hollander's results and conjecture in the light of ours.