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).

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

1 Institute for Advanced Study, Princeton, NJ 08540, USA
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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/

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[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

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