Comptes Rendus
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 * .

Received:
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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[5] K. Ford Sums and products from a finite set of real numbers, Ramanujan J., Volume 2 (1998), pp. 59-66

[6] D.R. Heath-Brown; S.V. Konyagin New bounds for Gauss sums derived from k-th powers, and for Heilbronn's exponential sums, Quart. J. Math., Volume 51 (2000), pp. 221-235

[7] S.V. Konyagin Estimates of trigonometric sums over subgroups and Gaussian sums, IV International Conference “Modern Problems of Number Theory and its Applications” dedicated to 180th anniversary of P.L. Chebyshev and 110th anniversary of I.M. Vinogradov, Topical Problems, Part 3, Department of Mechanics and Mathematics, Moscow Lomonosov State University, Moscow, 2002, pp. 86-114 ([in Russian])

[8] N.M. Korobov Exponential Sums and their Applications, Kluwer Academic, Dordrecht, 1992

[9] S.V. Konyagin; I.E. Shparlinski Character Sums with Exponential Functions and their Applications, Cambridge Tracts in Math., 136, Cambridge University Press, Cambridge, 1999

[10] M. Nathanson On sums and products of integers, Proc. Amer. Math. Soc., Volume 125 (1997), pp. 9-16

[11] I.E. Shparlinski Estimates for Gauss sums, Math. Notes, Volume 50 (1991), pp. 140-146

[12] J. Solymosi, On a question of Erdős and Szemerédi, Preprint, 2003

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