A modular Szemerédi–Trotter theorem for hyperbolas

Bourgain, Jean

doi:10.1016/j.crma.2012.09.011

Logic

A modular Szemerédi–Trotter theorem for hyperbolas

Presented by: Jean Bourgain

Bourgain, Jean ¹

¹School 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

Abstract (Original language)
Résumé

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

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

Received: 2012-08-20
Accepted: 2012-09-18
Published online: 2012-09-01

DOI: 10.1016/j.crma.2012.09.011

Author's affiliations:

Bourgain, Jean ¹

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

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Bourgain, Jean. 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

References
Cited by

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

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

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

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

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

Cited by Sources:

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