Ultimate periodicity problem for linear numeration systems

Abstract

We address the following decision problem. Given a numeration system [Formula: see text] and a [Formula: see text]-recognizable set [Formula: see text], i.e. the set of its greedy [Formula: see text]-representations is recognized by a finite automaton, decide whether or not [Formula: see text] is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on [Formula: see text]-adic methods, the DFA given as input provides a bound on the admissible periods to test.

Automatic sequences of rank two

“…In fact, later results showed that this is actually efficiently decidable; see Leroux [25] and Marsault and Sakarovitch [28]. For other related work, see [3,6,11,15,24,26,31].…”