On sets of numbers rationally represented in a rational base number system

TL;DR

This paper proves that certain sets of numbers represented in rational base systems, which are closed under addition and form rational languages, are not finitely generated additive monoids, using a novel combinatorial property.

Contribution

It introduces the bounded left iteration property and demonstrates its relevance to rational base numeration system languages, providing new insights into their algebraic structure.

Findings

Sets closed under addition in rational base systems are not finitely generated monoids

The bounded left iteration property characterizes these languages

Rational base representations have unique combinatorial properties

Abstract

In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition of a strong combinatorial property on languages : the bounded left iteration property. It is both an unnatural property in usual formal language theory (as it contradicts any kind of pumping lemma) and an ideal fit to the languages defined through rational base number systems.