Definable Sets in Mann Pairs

Consider structures (Ω, k, Γ) where Ω is an algebraically closed field of characteristic zero, k is a subfield, and Γ is a subgroup of the multiplicative group of Ω. Certain pairs (k, Γ) have been singled out as Mann pairs in [4 van den Dries , L. , Günayd?n , A. ( 2010 ). Mann pairs . Trans. Amer. Math. Soc. 362 : 2393 – 2414 . [Google Scholar]]. We give new examples of such Mann pairs, we axiomatize for each Mann pair (k, Γ) the first-order theory of (Ω, k, Γ) in a cleaner way than in [4 van den Dries , L. , Günayd?n , A. ( 2010 ). Mann pairs . Trans. Amer. Math. Soc. 362 : 2393 – 2414 . [Google Scholar]], and, as the main result of the article, we characterize the subsets of Ω ⁿ that are definable in (Ω, k, Γ).