Fractional Parts of Dense Additive Subgroups of Real Numbers

Given a dense additive subgroup $G$ of $\mathbb R$ containing $\mathbb Z$, we consider its intersection $\mathbb G$ with the interval $[0,1[$ with the induced order and the group structure given by addition modulo $1$. We axiomatize the theory of $\mathbb G$ and show it is model-complete, using a Feferman-Vaught type argument. We show that any sufficiently saturated model decomposes into a product of a "standard" part and two ordered semigroups of infinitely small and infinitely large elements.