A decision problem for ultimately periodic sets in non-standard numeration systems

Given a linear numeration system U and a set X (include in N) such that repU(X) is recognized by a (deterministic) finite automaton. Is it decidable whether or not X is ultimately periodic, i.e., whether or not X is a finite union of arithmetic progressions? Honkala showed that this problem turns out to be decidable for the usual b-ary numeration system (b greater than 2) defined by U_n = bU_{n-1} for n greater than 1 and U_0 = 1. In this work, we give a decision procedure for this problem for a large class of linear numeration systems