New results on expanders

Jean Bourgain; Alex Gamburd

doi:10.1016/j.crma.2006.02.032

Number Theory/Mathematical Analysis

New results on expanders
[Nouveaux résultats sur les expanseurs]

Présenté par : Jean Bourgain

Jean Bourgain ¹ ; Alex Gamburd ^1,²

¹ Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
² Department of Mathematics, University of California, Santa Cruz, CA 95064, USA

Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 717-721.

Résumé
Abstract

En utilisant d'une approche purement analytique, nous obtenons de nouvelle familles d'expanseurs dans des groupes ${SL}_{2} (p)$ (p primier) et $SU (2)$ . Nos résultats contribuent à des conjectures de A. Lubotzky et P. Sarnak.

Based on purely analytical methods, we exhibit new families of expanders in ${SL}_{2} (p)$ (p prime) and $SU (2)$ , contributing to conjectures of A. Lubotzky and P. Sarnak.

Reçu le : 2006-02-10
Accepté le : 2006-02-21
Publié le : 2006-04-11

DOI : 10.1016/j.crma.2006.02.032

Affiliations des auteurs :

Jean Bourgain ¹ ; Alex Gamburd ^{1, 2}

¹ Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
² Department of Mathematics, University of California, Santa Cruz, CA 95064, USA

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     title = {New results on expanders},
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     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.02.032},
     language = {en},
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Jean Bourgain; Alex Gamburd. New results on expanders. Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 717-721. doi : 10.1016/j.crma.2006.02.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.032/

Bibliographie
Cité par

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[2] J. Bourgain, A. Gamburd, Uniform expansion bounds for Cayley graphs of ${SL}_{2} (F_{p})$ , preprint

[3] J. Bourgain, A. Gamburd, On the spectral gap for finitely-generated subgroups of $SU (2)$ , preprint

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

[5] J. Bourgain, A. Glibichuk, S. Konyagin, Estimate for the number of sums and product and for exponential sums in fields of prime order, Proc. LMS, in press

[6] J. Conway; C. Radin Quaquaversal tilings and rotations, Invent. Math., Volume 132 (1998), pp. 179-188

[7] C.M. Dawson; M.A. Nielsen The Solovay–Kitaev algorithm, Quantum Inf. Comput., Volume 6 (2006), pp. 81-95

[8] B. Draco; L. Sadun; D. Van Wieren Growth rates in the quaquaversal tiling, Comput. Geom., Volume 23 (2000) no. 3, pp. 419-435

[9] A. Gamburd Spectral gap for infinite index “congruence” subgroups of ${SL}_{2} (Z)$ , Israel J. Math., Volume 127 (2002), pp. 157-200

[10] A. Gamburd; D. Jakobson; P. Sarnak Spectra of elements in the group ring of $SU (2)$ , J. Eur. Math. Soc., Volume 1 (1999), pp. 51-85

[11] H. Helfgott, Growth and generation in ${SL}_{2} (Z / p Z)$ , preprint, 2005

[12] N. Katz; T. Tao Some connections between Falconer's distance set conjecture and sets of Furstenberg type, New York J. Math., Volume 7 (2001), pp. 149-187

[13] A. Lubotzky Discrete Groups, Expanding Graphs and Invariant Measures, Progr. Math., vol. 195, Birkhäuser, 1994

[14] A. Lubotzky Cayley graphs: eigenvalues, expanders and random walk (P. Rowbinson, ed.), Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., vol. 218, Cambridge Univ. Press, 1995, pp. 155-189

[15] A. Lubotzky; B. Weiss Groups and expanders (J. Friedaman, ed.), DIMACS Ser. Discrete Math. and Theoret. Comput. Sci., vol. 10, Amer. Math. Soc., 1993, pp. 95-109

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

[17] A. Selberg On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., vol. VII, Amer. Math. Soc., 1965, pp. 1-15

[18] T. Tao, Non-commutative sum set estimates, preprint

[19] T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press, 2006, in press

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