Fractional Parts of Dense Additive Subgroups of Real Numbers

TL;DR

This paper studies the structure of dense additive subgroups of real numbers containing integers, focusing on their fractional parts, and provides a model-theoretic analysis showing the theory is model-complete and decomposes into standard and infinitesimal parts.

Contribution

It axiomatizes the theory of fractional parts of dense additive subgroups of reals and proves model-completeness using a Feferman-Vaught type argument.

Findings

The theory of these fractional parts is model-complete.

Any sufficiently saturated model decomposes into a standard part and ordered semigroups.

The structure includes infinitely small and large elements.

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.