Comptes Rendus
Number Theory/Mathematical Analysis
New results on expanders
Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 717-721.

Based on purely analytical methods, we exhibit new families of expanders in SL2(p) (p prime) and SU(2), contributing to conjectures of A. Lubotzky and P. Sarnak.

En utilisant d'une approche purement analytique, nous obtenons de nouvelle familles d'expanseurs dans des groupes SL2(p) (p primier) et SU(2). Nos résultats contribuent à des conjectures de A. Lubotzky et P. Sarnak.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.02.032

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
@article{CRMATH_2006__342_10_717_0,
     author = {Jean Bourgain and Alex Gamburd},
     title = {New results on expanders},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {717--721},
     publisher = {Elsevier},
     volume = {342},
     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.02.032},
     language = {en},
}
TY  - JOUR
AU  - Jean Bourgain
AU  - Alex Gamburd
TI  - New results on expanders
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 717
EP  - 721
VL  - 342
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2006.02.032
LA  - en
ID  - CRMATH_2006__342_10_717_0
ER  - 
%0 Journal Article
%A Jean Bourgain
%A Alex Gamburd
%T New results on expanders
%J Comptes Rendus. Mathématique
%D 2006
%P 717-721
%V 342
%N 10
%I Elsevier
%R 10.1016/j.crma.2006.02.032
%G en
%F CRMATH_2006__342_10_717_0
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/

[1] J. Bourgain, On the Erdös–Volkmann and Katz–Tao ring conjectures GAFA, 13 (2003) 334–365

[2] J. Bourgain, A. Gamburd, Uniform expansion bounds for Cayley graphs of SL2(Fp), 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 SL2(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 SL2(Z/pZ), 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

Cited by Sources:

Comments - Policy