  In 1996, Michaux and Villemaire considered integer relations $R$ which are
not definable in Presburger Arithmetic. That is, not definable in first-order
logic over integers with the addition function and the order relation
(FO[N,+,<]-definable relations). They proved that, for each such $R$, there
exists a FO[N,+,<,$R$]-formula $\nu_{R}(x)$ which defines a set of integers
which is not ultimately periodic, i.e. not FO[N,+,<]-definable.
  It is proven in this paper that the formula $\nu(x)$ can be chosen such that
it does not depend on the interpretation of $R$. It is furthermore proven that
$\nu(x)$ can be chosen such that it defines an expanding set. That is, an
infinite set of integers such that the distance between two successive elements
is not bounded.
