Mann pairs

Mann proved in the 1960s that for any n ≥ 1 there is a finite set E of n-tuples (n₁,⋯,n_n) of complex roots of unity with the following property: if a₁,⋯,a_n are any rational numbers and ζ₁,⋯,ζ_n are any complex roots of unity such that ∑_i=1ⁿ a_iζ _{i= 1} and ∑_i∈I a_iζ_i,=0 for all nonempty I ⊆{1,⋯,n},then (Ζ₁,⋯, Ζ_n) ∈ E. Taking an arbitrary field k instead of ℚ and any multiplicative group in an extension field of k instead of the group of roots of unity, this property defines what we call a Mann pair (k, Γ). We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.