Ultimate periodicity problem for linear numeration systems

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.

39 pages, 2 figures. This is an improved version of the original submission. It clarifies some arguments taking into account several comments from reviews