We construct finite families of 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 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 , 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”.
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Jean Bourgain 1
@article{CRMATH_2009__347_7-8_357_0, author = {Jean Bourgain}, 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}, }
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.009/
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