Group Theory/Number Theory
Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order
Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 75-80.

Our first result is a ‘sum–product’ theorem for subsets A of the finite field ${𝔽}_{p}$, p prime, providing a lower bound on max(|A+A|,|A·A|). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group ${𝔽}_{p}^{*}$.

Notre premier résultat est un théorème « sommes–produits » pour des sous-ensembles A d'un corps fini ${𝔽}_{p}$, p un nombre premier, donnant une minoration du max(|A+A|,|A·A|). Comme corollaire et résultat principal, on en déduit de nouvelles bornes sur les sommes exponentielles associées à des sous-groupes du groupe multiplicatif ${𝔽}_{p}^{*}$.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00281-4

Jean Bourgain 1, 2; S.V. Konyagin 3

1 School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
2 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
3 Department of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia
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Jean Bourgain; S.V. Konyagin. Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 75-80. doi : 10.1016/S1631-073X(03)00281-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00281-4/

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