Real numbers having ultimately periodic representations in abstract numeration systems

TL;DR

This paper characterizes numbers with ultimately periodic representations in generalized abstract numeration systems based on regular languages, linking classical theta-expansions with these new representations, especially for Pisot numbers.

Contribution

It introduces a characterization of ultimately periodic representations in generalized systems and explores their syntactical properties, connecting them to classical theta-expansions.

Findings

Characterization of numbers with ultimately periodic representations in generalized systems

Analysis of syntactical properties of these representations

Equivalence established between classical theta-expansions and generalized representations for Pisot numbers

Abstract

Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical "theta"-expansions with our generalized representations in some special case related to a Pisot number "theta".