# Character sums in complex half-planes

Sergei V. Konyagin^{[1]}; Vsevolod F. Lev^{[2]}

- [1] Department of Mechanics and Mathematics Moscow State University Moscow, Russia
- [2] Department of Mathematics Haifa University at Oranim Tivon 36006, Israel

Journal de Théorie des Nombres de Bordeaux (2004)

- Volume: 16, Issue: 3, page 587-606
- ISSN: 1246-7405

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topKonyagin, Sergei V., and Lev, Vsevolod F.. "Character sums in complex half-planes." Journal de Théorie des Nombres de Bordeaux 16.3 (2004): 587-606. <http://eudml.org/doc/249259>.

@article{Konyagin2004,

abstract = {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 $\chi \in \{\widehat\{G\}\}$ such that $\sum _\{a\in A\} \chi (a)\in P$.},

affiliation = {Department of Mechanics and Mathematics Moscow State University Moscow, Russia; Department of Mathematics Haifa University at Oranim Tivon 36006, Israel},

author = {Konyagin, Sergei V., Lev, Vsevolod F.},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {character group; Fourier coefficients; character sums},

language = {eng},

number = {3},

pages = {587-606},

publisher = {Université Bordeaux 1},

title = {Character sums in complex half-planes},

url = {http://eudml.org/doc/249259},

volume = {16},

year = {2004},

}

TY - JOUR

AU - Konyagin, Sergei V.

AU - Lev, Vsevolod F.

TI - Character sums in complex half-planes

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2004

PB - Université Bordeaux 1

VL - 16

IS - 3

SP - 587

EP - 606

AB - 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 $\chi \in {\widehat{G}}$ such that $\sum _{a\in A} \chi (a)\in P$.

LA - eng

KW - character group; Fourier coefficients; character sums

UR - http://eudml.org/doc/249259

ER -

## References

top- J. Bourgain, Sur le minimum d’une somme de cosinus, [On the minimum of a sum of cosines]. Acta Arithmetica 45 (4) (1986), 381–389. Zbl0615.42001MR847298
- A.S. Belov, S.V. Konyagin, On the conjecture of Littlewood and minima of even trigonometric polynomials. Harmonic analysis from the Pichorides viewpoint (Anogia, 1995), 1–11, Publ. Math. Orsay, 96-01, Univ. Paris XI, Orsay, 1996. Zbl0862.42001MR1426370
- S.V. Konyagin, On the Littlewood problem Izv. Akad. Nauk SSSR Ser. Mat. 45 (2) (1981), 243–265. (English translation: Mathematics of the USSR - Izvestiya 45 (2) (1982), 205–225.) Zbl0493.42004MR616222
- S.V. Konyagin, V. Lev, On the distribution of exponential sums. Integers 0 (2000), #A1 (electronic). Zbl0968.11031MR1759419
- O.C. McGehee, L. Pigno, B. Smith, Hardy’s inequality and the Littlewood conjecture. Bull. Amer. Math. Soc. (N.S.) 5 (1) (1981), 71–72. Zbl0485.42001MR614316
- O.C. McGehee, L. Pigno, B. Smith, Hardy’s inequality and the ${L}^{1}$ norm of exponential sums. Annals of Mathematics (2) 113 (3) (1981), 613–618. Zbl0473.42001MR621019

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