A modular Szemerédi–Trotter theorem for hyperbolas

Jean Bourgain

doi:10.1016/j.crma.2012.09.011

Logic

A modular Szemerédi–Trotter theorem for hyperbolas
[Un théorème de type Szemerédi–Trotter modulaire pour hyperboles]

Présenté par : Jean Bourgain

Jean Bourgain ¹

¹ School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 793-796.

Résumé
Abstract

Nous démontrons une version du théorème de Szemerédi–Trotter pour des familles dʼhyperboles dans $F_{p} \times F_{p}$ .

We establish a Szemerédi–Trotter type result for hyperbolas in $F_{p} \times F_{p}$ .

Reçu le : 2012-08-20
Accepté le : 2012-09-18
Publié le : 2012-09-01

DOI : 10.1016/j.crma.2012.09.011

Affiliations des auteurs :

Jean Bourgain ¹

¹ School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

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     title = {A modular {Szemer\'edi{\textendash}Trotter} theorem for hyperbolas},
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Jean Bourgain. A modular Szemerédi–Trotter theorem for hyperbolas. Comptes Rendus. Mathématique, Volume 350 (2012) no. 17-18, pp. 793-796. doi : 10.1016/j.crma.2012.09.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.011/

Bibliographie
Cité par

[1] J. Bourgain More on the sum-product phenomenon in prime fields and its applications, Int. J. Number Theory, Volume 1 (2005) no. 1, pp. 1-32

[2] J. Bourgain; A. Gamburd Uniform expansion bounds for Cayley graphs of ${SL}_{2} (F_{p})$ , Annals of Math., Volume 167 (2008), pp. 625-642

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

[4] H. Helfgott Growth and generation in ${SL}_{2} (Z / p Z)$ , Annals of Math., Volume 167 (2008) no. 2, pp. 601-623

[5] H. Helfgott; M. Rudnev An explicit incidence theorem in $F_{p}$ | arXiv

[6] P. Sarnak; X. Xue Bounds for multiplicities of automorphic representations, Duke Math. J., Volume 64 (1991), pp. 207-227

Cité par Sources :

^☆ The research was partially supported by NSF grants DMS-0808042 and DMS-0835373.

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