Uniform definition of sets using relations and complement of Presburger   Arithmetic
  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. 
Arthur Milchior
math.LO