Mann pairs

Mann proved in the 1960s that for any $n\ge 1$ there is a finite set $E$ of $n$-tuples $(\eta_1,\dots, \eta_n)$ of complex roots of unity with the following property: if $a_1,\dots,a_n$ are any rational numbers and $\zeta_1,\dots,\zeta_n$ are any complex roots of unity such that $\sum_{i=1}^n a_i\zeta_i=1$ and $\sum_{i\in I} a_i \zeta_i\ne 0$ for all nonempty $I\subseteq \{1,\dots,n\}$, then $(\zeta_1,\dots,\zeta_n)\in E$. Taking an arbitrary field $\mathbf{k}$ instead of $\mathbb{Q}$ and any multiplicative group in an extension field of $\mathbf{k}$ instead of the group of roots of unity, this property defines what we call a Mann pair $(\mathbf{k}, \Gamma)$. We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.