# Character sums in complex half-planes

• [1] Department of Mechanics and Mathematics Moscow State University Moscow, Russia
• [2] Department of Mathematics Haifa University at Oranim Tivon 36006, Israel
• Volume: 16, Issue: 3, page 587-606
• ISSN: 1246-7405

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## Abstract

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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 $\chi \in \stackrel{^}{G}$ such that ${\sum }_{a\in A}\chi \left(a\right)\in P$.

## How to cite

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Konyagin, 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

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1. 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
2. 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
3. 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
4. S.V. Konyagin, V. Lev, On the distribution of exponential sums. Integers 0 (2000), #A1 (electronic). Zbl0968.11031MR1759419
5. 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
6. 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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