[Estimes sommes–produits et sur les sommes exponentielles associées à des sous-groupes d'un corps d'ordre premier]
Notre premier résultat est un théorème « sommes–produits » pour des sous-ensembles A d'un corps fini , p un nombre premier, donnant une minoration du max(|A+A|,|A·A|). Comme corollaire et résultat principal, on en déduit de nouvelles bornes sur les sommes exponentielles associées à des sous-groupes du groupe multiplicatif .
Our first result is a ‘sum–product’ theorem for subsets A of the finite field , p prime, providing a lower bound on max(|A+A|,|A·A|). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group .
Accepté le :
Publié le :
@article{CRMATH_2003__337_2_75_0,
author = {Jean Bourgain and S.V. Konyagin},
title = {Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order},
journal = {Comptes Rendus. Math\'ematique},
pages = {75--80},
publisher = {Elsevier},
volume = {337},
number = {2},
year = {2003},
doi = {10.1016/S1631-073X(03)00281-4},
language = {en},
}
TY - JOUR AU - Jean Bourgain AU - S.V. Konyagin TI - Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order JO - Comptes Rendus. Mathématique PY - 2003 SP - 75 EP - 80 VL - 337 IS - 2 PB - Elsevier DO - 10.1016/S1631-073X(03)00281-4 LA - en ID - CRMATH_2003__337_2_75_0 ER -
%0 Journal Article %A Jean Bourgain %A S.V. Konyagin %T Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order %J Comptes Rendus. Mathématique %D 2003 %P 75-80 %V 337 %N 2 %I Elsevier %R 10.1016/S1631-073X(03)00281-4 %G en %F CRMATH_2003__337_2_75_0
Jean Bourgain; S.V. Konyagin. Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 75-80. doi : 10.1016/S1631-073X(03)00281-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00281-4/
[1] J. Bourgain, N. Katz, T. Tao, A sum–product estimate in finite fields and their applications, ArXiv: v1, January 29, 2003, to appear in GAFA | arXiv
[2] Borel subrings of the reals, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 1121-1129
[3] On the umber of sums and products, Acta Arith., Volume 81 (1997), pp. 1121-1129
[4] On sums and the products of integers (P. Erdős; L. Alpár; G. Halász, eds.), Studies in Pure Mathematics, Akadémiai Kiadó–Birkhäuser, Budapest–Basel, 1983, pp. 213-218 (To the memory of Paul Turan)
[5] Sums and products from a finite set of real numbers, Ramanujan J., Volume 2 (1998), pp. 59-66
[6] New bounds for Gauss sums derived from k-th powers, and for Heilbronn's exponential sums, Quart. J. Math., Volume 51 (2000), pp. 221-235
[7] Estimates of trigonometric sums over subgroups and Gaussian sums, IV International Conference “Modern Problems of Number Theory and its Applications” dedicated to 180th anniversary of P.L. Chebyshev and 110th anniversary of I.M. Vinogradov, Topical Problems, Part 3, Department of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow, 2002, pp. 86-114 ([in Russian])
[8] Exponential Sums and their Applications, Kluwer Academic, Dordrecht, 1992
[9] Character Sums with Exponential Functions and their Applications, Cambridge Tracts in Math., 136, Cambridge University Press, Cambridge, 1999
[10] On sums and products of integers, Proc. Amer. Math. Soc., Volume 125 (1997), pp. 9-16
[11] Estimates for Gauss sums, Math. Notes, Volume 50 (1991), pp. 140-146
[12] J. Solymosi, On a question of Erdős and Szemerédi, Preprint, 2003
Cité par Sources :
Commentaires - Politique
New bounds on exponential sums related to the Diffie–Hellman distributions
Jean Bourgain
C. R. Math (2004)
Sum–product theorems and exponential sum bounds in residue classes for general modulus
Jean Bourgain
C. R. Math (2007)