Ultimate periodicity problem for linear numeration systems

We address the following decision problem. Given a numeration system U and a U-recognizable set of non-negative integers X, 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.