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Cancellation of Abelian groups of finite rank modulo elementary equivalence.

Francis Oger (1990)

Mathematica Scandinavica

Carriers of torsion-free groups

R. S. Pierce, C. I. Vinsonhaler (1990)

Rendiconti del Seminario Matematico della Università di Padova

Character sums in complex half-planes

Sergei V. Konyagin, Vsevolod F. Lev (2004)

Journal de Théorie des Nombres de Bordeaux

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 $χ \in \hat{G}$ such that $\sum_{a \in A} χ (a) \in P$ .