Mann pairs

Abstract

Mann proved in the 1960s that for any n ≥ 1 n\ge 1 there is a finite set E E of n n -tuples ( η 1 , … , η n ) (\eta _1,\dots , \eta _n) of complex roots of unity with the following property: if a 1 , … , a n a_1,\dots ,a_n are any rational numbers and ζ 1 , … , ζ n \zeta _1,\dots ,\zeta _n are any complex roots of unity such that ∑ i = 1 n a i ζ i = 1 \sum _{i=1}^n a_i\zeta _i=1 and ∑ i ∈ I a i ζ i ≠ 0 \sum _{i\in I} a_i \zeta _ie 0 for all nonempty I ⊆ { 1 , … , n } I\subseteq \{1,\dots ,n\} , then ( ζ 1 , … , ζ n ) ∈ E (\zeta _1,\dots ,\zeta _n)\in E . Taking an arbitrary field k \mathbf {k} instead of Q \mathbb {Q} and any multiplicative group in an extension field of k \mathbf {k} instead of the group of roots of unity, this property defines what we call a Mann pair ( k , Γ ) (\mathbf {k}, \Gamma ) . We show that Mann pairs are robust in certain ways, construct various kinds of Mann pairs, and characterize them model-theoretically.