Comptes Rendus
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
Jean Bourgain 1

1 Institute for Advanced Study, Princeton, NJ 08540, USA
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Jean Bourgain. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.019/

[1] J. Bourgain Mordell's exponential sum estimate revisited, JAMS, Volume 18 (2005) no. 2, pp. 477-499

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

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

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

[7] 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., Volume 73 (2006), pp. 380-398

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

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

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