Comptes Rendus
Number Theory/Mathematical Analysis
A Gauss sum estimate in arbitrary finite fields
Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 643-646.

We establish bounds on exponential sums xFqψ(xn) where q=pm, p prime, and ψ an additive character on Fq. They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order (m2). More precisely, a non-trivial estimate is obtained provided n satisfies gcd(n,q1pν1)<pνq1ε for all 1ν<m, ν|m, where ε>0 is arbitrary.

On etabli des bornes sur les sommes d'exponentielles xFqψ(xn)q=pm, p est premier et ψ est un caractère additif de Fq. Il s'agit d'une extension des résultats de Bourgain, Glibichuk, et Konyagin pour un corps qui n'est pas d'ordre premier, c'est-à-dire m2. On obtient une estimée non-triviale pour tout n satisfaisant la condition pgcd(n,q1pν1)<pνq1ε pour tout 1ν<m,ν|m et où ε>0 est arbitraire.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.01.022
Jean Bourgain 1; Mei-Chu Chang 2

1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
2 Mathematics Department, University of California, Riverside, CA 92521, USA
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Jean Bourgain; Mei-Chu Chang. A Gauss sum estimate in arbitrary finite fields. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 643-646. doi : 10.1016/j.crma.2006.01.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.022/

[1] J. Bourgain, M.-C. Chang, Exponential sum estimates over subgroups and almost subgroups of Zq, where q is composite with few factors, GAFA, in press

[2] J. Bourgain, A. Glibichuk, S. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc., in press

[3] J. Bourgain; N. Katz; T. Tao A sum-product estimate in finite fields and their applications, GAFA, Volume 14 (2004) no. 1, pp. 27-57

[4] S. Konyagin; I. Shparlinski Character Sums with Exponential Functions and their Applications, Cambridge Univ. Press, Cambridge, 1999

[5] I. Shparlinski Bounds on Gauss sums in finite fields, Proc. Amer. Math. Soc., Volume 132 (2006) no. 10, pp. 2817-2824

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