Character sums in complex half - planes par

Let A be a finite subset of an abelian group G and let P be a closed half-plane of the complex plane, containing zero. We show that (unless A possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of A which belongs to P . In other words, there exists a non-trivial character χ ∈ Ĝ such that ∑ a∈A χ(a) ∈ P . 1. Summary of results Let G be an abelian group and let Ĝ denote the dual group. For a finite subset A ⊆ G and a character χ ∈ Ĝ we write SA(χ) := ∑