On expansions of the real field by complex subgroups

TL;DR

This paper constructs specific complex subgroups that, when used to expand the real field, exhibit well-behaved model-theoretic properties, challenging previous classification results.

Contribution

It introduces finite rank multiplicative subgroups of complex numbers that expand the real field in a controlled, well-understood manner, extending prior classifications.

Findings

Expansion by these subgroups is model-theoretically well-behaved

Classification by Hieronymi does not extend to two-generator subgroups

Provides new examples of tame expansions of the real field

Abstract

We construct a class of finite rank multiplicative subgroups of the complex numbers such that the expansion of the real field by such a group is model-theoretically well-behaved. As an application we show that a classification of expansions of the real field by cyclic multiplicative subgroups of the complex numbers due to Hieronymi does not even extend to expansions by subgroups with two generators.