Fractional parts of dense additive subgroups of real numbers

Given a dense additive subgroup G of R containing Z, we consider its intersection G with the interval inverted right perpendicular 0, 1 right perpendicular with the induced order and the group structure given by addition modulo 1. We axiomatize the theory of 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.