Comptes Rendus
Mathematical Analysis
Expanders and dimensional expansion
Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 357-362.

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).

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”.

Published online:
DOI: 10.1016/j.crma.2009.02.009
Jean Bourgain 1

1 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
     author = {Jean Bourgain},
     title = {Expanders and dimensional expansion},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {357--362},
     publisher = {Elsevier},
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     number = {7-8},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.009},
     language = {en},
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DO  - 10.1016/j.crma.2009.02.009
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%T Expanders and dimensional expansion
%J Comptes Rendus. Mathématique
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Jean Bourgain. Expanders and dimensional expansion. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 357-362. doi : 10.1016/j.crma.2009.02.009.

[1] J. Bourgain; A. Gamburd 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] P. Sarnak; X. Xue 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] T. Tao; V. Vu Additive Combinatorics, Cambridge University Press, 2006

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

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