Comptes Rendus
Number Theory
Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors
Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 463-466.

In this Note, we extend the results of Bourgain, Konyagin and Glibichuk to certain composite moduli q involving few ‘large’ primes. First a ‘sum-product’ theorem for subsets A of Zq is obtained, ensuring that |A+A|+|A.A|>c|A|1+ɛ provided |A|<q1δ and A does not have a ‘large’ intersection with a translate of a subring. Next, exponential sum estimates are established. In particular nontrivial bounds are obtained for the exponential sums associated to a multiplicative subgroup H<Zq*, with applications to Heilbronn-type sums.

Nous présentons dans cette Note une extension des résultats obtenus par Bourgain, Konyagin et Glibichuk pour les modules composés q dont la factorization ne comporte qu'un nombre borné de nombres premiers ‘grands’. D'abord nous démontrons un théorème « somme-produit » pour les sous-ensembles A de Zq, affirmant que |A+A|+|A.A|>c|A|1+ɛ si |A|<q1δ et n'a pas de « grosse » intersection avec une translatée d'un sous-anneau de Zq. Ensuite on obtient des estimées sur des sommes exponentielles, en particulier associées à des sous-groupes multiplicatifs H<Zq*. Ils s'appliquent aux sommes de type Heilbronn pour lesquelles on établit des estimées non-trivials.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.08.007
Jean Bourgain 1; Mei-Chu Chang 2

1 School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
2 Department of Mathematics, University of California, Riverside, California, USA
@article{CRMATH_2004__339_7_463_0,
     author = {Jean Bourgain and Mei-Chu Chang},
     title = {Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {463--466},
     publisher = {Elsevier},
     volume = {339},
     number = {7},
     year = {2004},
     doi = {10.1016/j.crma.2004.08.007},
     language = {en},
}
TY  - JOUR
AU  - Jean Bourgain
AU  - Mei-Chu Chang
TI  - Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 463
EP  - 466
VL  - 339
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2004.08.007
LA  - en
ID  - CRMATH_2004__339_7_463_0
ER  - 
%0 Journal Article
%A Jean Bourgain
%A Mei-Chu Chang
%T Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors
%J Comptes Rendus. Mathématique
%D 2004
%P 463-466
%V 339
%N 7
%I Elsevier
%R 10.1016/j.crma.2004.08.007
%G en
%F CRMATH_2004__339_7_463_0
Jean Bourgain; Mei-Chu Chang. Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors. Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 463-466. doi : 10.1016/j.crma.2004.08.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.007/

[1] J. Bourgain, Mordell's exponential sum estimate revisited, preprint, 2004; JAMS, submitted for publication

[2] J. Bourgain; S. Konyagin Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003) no. 2, pp. 75-80

[3] 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., submitted for publication

[4] J. Bourgain; N. Katz; T. Tao A sum-product estimate in finite fields and their applications, Geom. Funct. Anal. (GAFA), Volume 14 (2004) no. 1, pp. 27-57

[5] R. Heath-Brown An estimate for Heilbronn's exponential sum, Proc. Conf. in honor of H. Halberstam, Birkhäuser, Boston, MA (1996), pp. 451-463

[6] R. Heath-Brown; S. Konyagin New bounds for Gauss sums derived from kth powers, and for Heilbronn's exponential sums, Quart. J. Math., Volume 51 (2000), pp. 221-235

[7] S. Konyagin; I. Shparlinski Character Sums with Exponential Functions and their Applications, Cambridge Tracts in Mathematics, vol. 136, Cambridge University Press, Cambridge, 1999

[8] R. Odoni Trigonometric sums of Heilbronn's type, Math. Proc. Comb. Philos. Soc., Volume 98 (1985), pp. 389-396

Cited by Sources:

Comments - Policy


Articles of potential interest

Sum–product theorems and exponential sum bounds in residue classes for general modulus

Jean Bourgain

C. R. Math (2007)


Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order

Jean Bourgain; S.V. Konyagin

C. R. Math (2003)


A Gauss sum estimate in arbitrary finite fields

Jean Bourgain; Mei-Chu Chang

C. R. Math (2006)