[Théorèmes sommes–produits et sommes exponentielles dans les classes de résidus pour module arbitraire]
Dans cette Note, nous généralisons (avec un énoncé approprié) le théorème somme–produit dans , où q est arbitraire (pour q premier, un tel résultat fût obtenu dans [J. Bourgain, N. Katz, T. Tao, A sum–product estimate in finite fields and applications, GAFA 14 (2004) 27–57 ; J. Bourgain, A. Glibichuk, S. Konyagin, Estimate for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. 73 (2006) 380–398], pour q un nombre composé avec un nombre de facteurs premiers borné dans [J. Bourgain, M. Chang, Exponential sum estimates over subgroups and almost subgroups of , where q is composite with few factors, GAFA 16 (2) (2006) 327–366], et pour q un produit simple dans [J. Bourgain, A. Gamburd, P. Sarnak, Sieving and expanders, C. R. Acad. Sci. Paris, Ser. I 343 (3) (2006) 155–159]. Nous en déduisons également des estimées sur certaines sommes exponentielles (mod q).
The purpose of this Note is to extend (in the appropriate formulation) the sum–product theorem in (established in [J. Bourgain, N. Katz, T. Tao, A sum–product estimate in finite fields and applications, GAFA 14 (2004) 27–57; J. Bourgain, A. Glibichuk, S. Konyagin, Estimate for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc. 73 (2006) 380–398] for q prime, in [J. Bourgain, M. Chang, Exponential sum estimates over subgroups and almost subgroups of , where q is composite with few factors, GAFA 16 (2) (2006) 327–366] for q composite with few factors and in [J. Bourgain, A. Gamburd, P. Sarnak, Sieving and expanders, C. R. Acad. Sci. Paris, Ser. I 343 (3) (2006) 155–159] for q square free) to the case of arbitrary modulus. Consequences to exponential sum bounds (mod q) are given.
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@article{CRMATH_2007__344_6_349_0,
author = {Bourgain, Jean},
title = {Sum{\textendash}product theorems and exponential sum bounds in residue classes for general modulus},
journal = {Comptes Rendus. Math\'ematique},
pages = {349--352},
publisher = {Elsevier},
volume = {344},
number = {6},
year = {2007},
doi = {10.1016/j.crma.2007.01.019},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.01.019/}
}
TY - JOUR AU - Bourgain, Jean TI - Sum–product theorems and exponential sum bounds in residue classes for general modulus JO - Comptes Rendus. Mathématique PY - 2007 SP - 349 EP - 352 VL - 344 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.01.019/ DO - 10.1016/j.crma.2007.01.019 LA - en ID - CRMATH_2007__344_6_349_0 ER -
%0 Journal Article %A Bourgain, Jean %T Sum–product theorems and exponential sum bounds in residue classes for general modulus %J Comptes Rendus. Mathématique %D 2007 %P 349-352 %V 344 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.01.019/ %R 10.1016/j.crma.2007.01.019 %G en %F CRMATH_2007__344_6_349_0
Bourgain, Jean. Sum–product theorems and exponential sum bounds in residue classes for general modulus. Comptes Rendus. Mathématique, Tome 344 (2007) no. 6, pp. 349-352. doi : 10.1016/j.crma.2007.01.019. http://www.numdam.org/articles/10.1016/j.crma.2007.01.019/
[1] Mordell's exponential sum estimate revisited, JAMS, Volume 18 (2005) no. 2, pp. 477-499
[2] Exponential sum estimates over subgroups of , q arbitrary, J. Analyse, Volume 97 (2005), pp. 317-356
[3] Estimates on exponential sums related to the Diffie–Hellman distributions, GAFA, Volume 15 (2005) no. 1, pp. 1-34
[4] Exponential sum estimates over subgroups and almost subgroups of , where q is composite with few factors, GAFA, Volume 16 (2006) no. 2, pp. 327-366
[5] J. Bourgain, A. Gamburd, Uniform expansion bounds for Cayley graphs of , Ann. of Math., in press
[6] Sieving and expanders, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006) no. 3, pp. 155-159
[7] Estimate for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc., Volume 73 (2006), pp. 380-398
[8] A sum–product estimate in finite fields and applications, GAFA, Volume 14 (2004), pp. 27-57
[9] Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, 2006
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