Mann pairs

Abstract:

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.