Number Theory
Sum–product theorems and exponential sum bounds in residue classes for general modulus
Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, pp. 349-352.

The purpose of this Note is to extend (in the appropriate formulation) the sum–product theorem in Zq=Z/qZ (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 Zq, 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.

Dans cette Note, nous généralisons (avec un énoncé approprié) le théorème somme–produit dans Zq=Z/qZ, 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 Zq, 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).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.01.019
Bourgain, Jean 1

1 Institute for Advanced Study, Princeton, NJ 08540, USA
@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, Volume 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] Bourgain, J. Mordell's exponential sum estimate revisited, JAMS, Volume 18 (2005) no. 2, pp. 477-499

[2] Bourgain, J. Exponential sum estimates over subgroups of Zq, q arbitrary, J. Analyse, Volume 97 (2005), pp. 317-356

[3] Bourgain, J. Estimates on exponential sums related to the Diffie–Hellman distributions, GAFA, Volume 15 (2005) no. 1, pp. 1-34

[4] Bourgain, J.; Chang, M. Exponential sum estimates over subgroups and almost subgroups of Zq, 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 SL2(Fp), Ann. of Math., in press

[6] Bourgain, J.; Gamburd, A.; Sarnak, P. Sieving and expanders, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006) no. 3, pp. 155-159

[7] Bourgain, J.; Glibichuk, A.; Konyagin, S. 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] Bourgain, J.; Katz, N.; Tao, T. A sum–product estimate in finite fields and applications, GAFA, Volume 14 (2004), pp. 27-57

[9] Tao, T.; Vu, V. Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105, Cambridge University Press, 2006

Cited by Sources: