Decomposition of decidable first-order logics over integers and reals

We tackle the issue of representing infinite sets of real-valued vectors. This paper introduces an operator for combining integer and real sets. Using this operator we decompose three well-known logics extending Presburger with reals. Our decomposition splits a logic into two parts : one integer and one decimal (i.e. on the interval [0, 1[). We also give a basis for an implementation of our representation.