Ultimate periodicity problem for linear numeration systems

TL;DR

This paper proves the decidability of the ultimate periodicity problem for sets recognized by finite automata in a broad class of linear numeration systems, using arithmetical and p-adic methods.

Contribution

It establishes the decidability of the ultimate periodicity problem for $U$-recognizable sets in linear recurrent numeration systems, extending previous results.

Findings

Decidability is achieved for a large class of linear numeration systems.

A bound on the periods is derived from the automaton and arithmetical properties.

p-adic methods are employed to analyze the recurrence equations.

Abstract

We address the following decision problem. Given a numeration system $U$ and a $U$-recognizable set $X\subseteq\mathbb{N}$, i.e. the set of its greedy $U$-representations is recognized by a finite automaton, decide whether or not $X$ is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linearly recurrent sequences. Based on arithmetical considerations about the recurrence equation and on $p$-adic methods, the DFA given as input provides a bound on the admissible periods to test.