no. 7-8
Mathematical Analysis
Expanders and dimensional expansion
[Expanseurs et expansion dimensionnelle]

Présenté par : Jean Bourgain

Bourgain, Jean1
1 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 357-362.

On construit une famille finie d'éléments de SL2(R), arbitrairement proches de l'identité, telle que l'opérateur de Hecke associé agissant par transformation de Moebius ait un trou spectral uniforme (en un sense restreint approprié).

Cela donne des systèmes finis de transformations monotones de l'intervalle ayant la propriété d'expansion. Ensuite, par l'approche de Dvir et Shpilka (2008), on obtient une solution au problème de Wigderson (2004) sur “l'expansion dimensionnelle”.

We construct finite families of SL2(R) elements that are arbitrary close to identity and such that the corresponding Hecke operator, acting by Moebius transformation, has a uniform spectral gap (in a suitably restricted sense). This provides finite systems of monotone transformations of the interval [0,1] with the expansion property. Combined with the approach from Dvir and Shpilka (2008), we obtain a solution to the “dimension expander” problem from Wigderson (2004).

Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.02.009
Bourgain, Jean 1

1 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA 
@article{CRMATH_2009__347_7-8_357_0,
     author = {Bourgain, Jean},
     title = {Expanders and dimensional expansion},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {357--362},
     publisher = {Elsevier},
     volume = {347},
     number = {7-8},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.009},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2009.02.009/}
}
TY  - JOUR
AU  - Bourgain, Jean
TI  - Expanders and dimensional expansion
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 357
EP  - 362
VL  - 347
IS  - 7-8
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2009.02.009/
DO  - 10.1016/j.crma.2009.02.009
LA  - en
ID  - CRMATH_2009__347_7-8_357_0
ER  - 
%0 Journal Article
%A Bourgain, Jean
%T Expanders and dimensional expansion
%J Comptes Rendus. Mathématique
%D 2009
%P 357-362
%V 347
%N 7-8
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2009.02.009/
%R 10.1016/j.crma.2009.02.009
%G en
%F CRMATH_2009__347_7-8_357_0
Bourgain, Jean. Expanders and dimensional expansion. Comptes Rendus. Mathématique, Tome 347 (2009) no. 7-8, pp. 357-362. doi : 10.1016/j.crma.2009.02.009. http://archive.numdam.org/articles/10.1016/j.crma.2009.02.009/

[1] Bourgain, J.; Gamburd, A. On the spectral gap for finitely-generated subgroups of SU(2), Invent. Math., Volume 171 (2008) no. 1, pp. 83-121

[2] E. Breuillard, A strong Tits alternative, preprint 08

[3] Z. Dvir, A. Shpilka, Towards dimension expanders over finite fields, preprint 08

[4] A. Lubotzky, Y. Zelmanov, Dimension expanders, preprint 08

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

[6] T. Tao, Product sets estimates for non-commutative groups, Combinatorica, in press

[7] Tao, T.; Vu, V. Additive Combinatorics, Cambridge University Press, 2006

[8] A. Wigderson, A lecture at IPAM, UCLA, February 2004

Cité par Sources :