Comptes Rendus
no. 3
Combinatorics
New results on the Erdös–Szemerédi sum-product problems
[Nouveaux résultats sur les problèmes sommes-produits d'Erdös et Szemerédi]

Présenté par : Jean Bourgain

Mei-Chu Chang1
1 University of California at Riverside, Department of Mathematics, Riverside, CA 92521, USA
Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 201-205.

Dans cette Note, nous présentons diverses contributions (ou solutions) à des questions concernant la taille d'ensembles somme et produit d'ensembles finis d'entiers (ou de nombres complexes), posées dans [8]. Nous introduisons également quelques méthodes nouvelles dans ce domaine de recherche.

In this Note, we present several contributions (or solutions) to problems related to the sizes of sum sets and product sets of integers (or complex numbers), considered in [8]. We also introduce some new methods in this area of research.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00018-9
Mei-Chu Chang 1

1 University of California at Riverside, Department of Mathematics, Riverside, CA 92521, USA 
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Mei-Chu Chang. New results on the Erdös–Szemerédi sum-product problems. Comptes Rendus. Mathématique, Volume 336 (2003) no. 3, pp. 201-205. doi : 10.1016/S1631-073X(03)00018-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00018-9/

[1] L.I. Borevich; I.R. Shafarevich Number Theory, Academic Press, 1966

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

[3] M. Chang, Erdös–Szemerédi problem on sum set and product set, Ann. of Math., submitted

[4] M. Chang, Factorization in generalized arithmetic progressions and applications to the Erdös–Szemerédi sum-product problems, Preprint, 2002

[5] G. Elekes On the number of sums and products, Acta Arith., Volume 81 (1997) no. 4, pp. 365-367

[6] G. Elekes, M. Nathanson, I. Ruzsa, Convexity and sumsets, J. Number Theory, to appear

[7] G. Elekes, J. Ruzsa, Few sums, many products, Preprint

[8] P. Erdös, E. Szemerédi, On sums and products of integers, in: P. Erdös, L. Alpàr, G. Haláz (Eds.), Stud. Pure Math., pp. 215–218

[9] Laczkovich, I. Ruzsa, Preprint

[10] M. Nathanson Additive Number Theory, Springer, 1996

[11] W. Rudin Trigonometric series with gaps, J. Math. Mech., Volume 9 (1960), pp. 203-227

[12] E. Szemerédi; W. Trotter Extremal problems in discrete geometry, Combinatorics, Volume 3 (1983) no. 3–4, pp. 387-392

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