## Abstract

Mann proved in the 1960s that for any n ≥ 1 there is a finite set E of n-tuples (n_{1},⋯,n_{n}) of complex roots of unity with the following property: if a_{1},⋯,a_{n} are any rational numbers and ζ_{1},⋯,ζ_{n} are any complex roots of unity such that ∑_{i=1}^{n} a_{i}ζ _{i= 1} and ∑_{i∈I} a_{i}ζ_{i},=0 for all nonempty I ⊆{1,⋯,n},then (Ζ_{1},⋯, 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.

Original language | English (US) |
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Pages (from-to) | 2393-2414 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 362 |

Issue number | 5 |

DOIs | |

State | Published - May 2010 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics