Some consequences of the Polynomial Freiman–Ruzsa Conjecture

Mei-Chu Chang

doi:10.1016/j.crma.2009.04.006

Combinatorics/Number Theory

Some consequences of the Polynomial Freiman–Ruzsa Conjecture
[Quelques conséquences de la conjecture polynomiale de Freiman–Ruzsa]

Présenté par : Jean Bourgain

Mei-Chu Chang ¹

¹ Department of Mathematics, University of California, Riverside, CA 92521, USA

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 583-588.

Résumé
Abstract

En supposant la conjecture polynomiale faible de Freiman–Ruzsa, on en déduit certaines conséquences sur les ensembles sommes-produits ainsi que sur la croissance de sous-ensembles de ${SL}_{3} (C)$ .

Assuming the Weak Polynomial Freiman–Ruzsa Conjecture, we derive some consequences on sum-products and the growth of subsets of ${SL}_{3} (C)$ .

Reçu le : 2008-12-25
Accepté le : 2009-04-02
Publié le : 2009-05-05

DOI : 10.1016/j.crma.2009.04.006

Affiliations des auteurs :

Mei-Chu Chang ¹

¹ Department of Mathematics, University of California, Riverside, CA 92521, USA

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     author = {Mei-Chu Chang},
     title = {Some consequences of the {Polynomial} {Freiman{\textendash}Ruzsa} {Conjecture}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {583--588},
     publisher = {Elsevier},
     volume = {347},
     number = {11-12},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.006},
     language = {en},
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Mei-Chu Chang. Some consequences of the Polynomial Freiman–Ruzsa Conjecture. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 583-588. doi : 10.1016/j.crma.2009.04.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.006/

Bibliographie
Cité par

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[2] J. Bourgain; M.-C. Chang On the size of k-fold sum and product sets of integers, JAMS, Volume 17 (2004) no. 2, pp. 473-497

[3] J. Bourgain, M.-C. Chang, Sum-product theorems in algebraic number fields, Journal d'Analyse Mathematique, in press

[4] M.-C. Chang A polynomial bound in Freiman's Theorem, Duke Math. J., Volume 113 (2002) no. 3, pp. 399-419

[5] M.-C. Chang Product theorems in SL2 and SL3, J. Math. Jussieu, Volume 7 (2008) no. 1, pp. 1-25

[6] M.-C. Chang, On product sets in ${SL}_{2}$ and ${SL}_{3}$ , preprint

[7] J.-H. Evertse; H. Schlickewei; W. Schmidt Linear equations in variables which lie in a multiplicative group, Ann. Math., Volume 155 (2002), pp. 807-836

[8] H. Helfgott, Growth and generation in ${SL}_{3} (Z / Z_{p})$ , preprint, 2008

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

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