  Consider a non-standard numeration system like the one built over the
Fibonacci sequence where nonnegative integers are represented by words over
$\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the
language of their greedy representations in this system is accepted by a finite
automaton, we consider the problem of deciding whether or not $X$ is a finite
union of arithmetic progressions. We obtain a decision procedure for this
problem, under some hypothesis about the considered numeration system. In a
second part, we obtain an analogous decision result for a particular class of
abstract numeration systems built on an infinite regular language.
