Fractional parts of dense additive subgroups of real numbers

Abstract

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.