Character sums in complex half-planes
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, p. 587-606

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 χG ^ such that aA χ(a)P.

Soit A un sous-ensemble fini d’un groupe abélien G et P un demi-plan fermé du plan complexe contenant zéro. Nous montrons qu’il existe un coefficient de Fourier non-trivial de la fonction indicatrice de A qui appartient à P (si A ne possède pas une structure spéciale explicite). Autrement dit, il existe un caractère non-trivial χG ^ tel que aA χ(a)P.

@article{JTNB_2004__16_3_587_0,
     author = {Konyagin, Sergei V. and Lev, Vsevolod F.},
     title = {Character sums in complex half-planes},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {3},
     year = {2004},
     pages = {587-606},
     doi = {10.5802/jtnb.463},
     mrnumber = {2144960},
     zbl = {1068.43004},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2004__16_3_587_0}
}
Konyagin, Sergei V.; Lev, Vsevolod F. Character sums in complex half-planes. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 3, pp. 587-606. doi : 10.5802/jtnb.463. http://www.numdam.org/item/JTNB_2004__16_3_587_0/

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