Journal de Th´
eorie des Nombres
de Bordeaux
16
(2004), 587–606
Character sums in complex half-planes
par
Sergei V. KONYAGIN
et
Vsevolod F. LEV
R´esum´e. SoitAun sous-ensemble fini d’un groupe ab´elien Get P un demi-plan ferm´e du plan complexe contenant z´ero. Nous montrons qu’il existe un coefficient de Fourier non-trivial de la fonction indicatrice de A qui appartient `a P (si A ne poss`ede pas une structure sp´eciale explicite). Autrement dit, il existe un caract`ere non-trivialχ∈Gb tel quePa∈Aχ(a)∈P.
Abstract. LetA be a finite subset of an abelian group Gand letP be a closed half-plane of the complex plane, containing zero. We show that (unlessA possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the in-dicator function ofA which belongs toP. In other words, there exists a non-trivial characterχ∈Gbsuch that Pa
∈Aχ(a)∈P.
Sergei V.Konyagin
Department of Mechanics and Mathematics Moscow State University
Moscow, Russia
E-mail:[email protected]
Vsevolod F.Lev
Department of Mathematics Haifa University at Oranim Tivon 36006, Israel
E-mail:[email protected]
