A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems

TL;DR

This paper presents a decision procedure to determine whether sets of integers, represented in non-standard numeration systems like Fibonacci-based systems, are finite unions of arithmetic progressions, extending to certain abstract numeration systems.

Contribution

It introduces a decision method for analyzing sets in non-standard numeration systems and extends results to specific classes of abstract numeration systems built on infinite regular languages.

Findings

Decidability of whether a set is a finite union of arithmetic progressions

Applicable to numeration systems based on Fibonacci and similar sequences

Extends to certain abstract numeration systems with regular languages

Abstract

Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over $\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not $X$ is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language.