Fields with a dense-codense linearly independent multiplicative subgroup

We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \({\mathbb {S}}(R)\), satisfying a density/codensity condition (in the sense of geometric theories). Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and real closed field cases, the resulting theory is near model complete and the expansion preserves many nice model theoretic conditions related to the complexity of definable sets such as stability and NIP.