  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.
