A Gauss sum estimate in arbitrary finite fields

Jean Bourgain; Mei-Chu Chang

doi:10.1016/j.crma.2006.01.022

Number Theory/Mathematical Analysis

A Gauss sum estimate in arbitrary finite fields

Presented by: Jean Bourgain

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

Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 643-646

Abstract (Original language)
Résumé

We establish bounds on exponential sums $\sum_{x \in F_{q}} ψ (x^{n})$ where $q = p^{m}$ , p prime, and ψ an additive character on $F_{q}$ . They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order $(m ⩾ 2)$ . More precisely, a non-trivial estimate is obtained provided n satisfies $\gcd (n, \frac{q - 1}{p^{ν} - 1}) < p^{- ν} q^{1 - ε}$ for all $1 ⩽ ν < m$ , $ν | m$ , where $ε > 0$ is arbitrary.

On etabli des bornes sur les sommes d'exponentielles $\sum_{x \in F_{q}} ψ (x^{n})$ où $q = p^{m}$ , p est premier et ψ est un caractère additif de $F_{q}$ . 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 $m ⩾ 2$ . On obtient une estimée non-triviale pour tout n satisfaisant la condition $pgcd (n, \frac{q - 1}{p^{ν} - 1}) < p^{- ν} q^{1 - ε}$ pour tout $1 ⩽ ν < m, ν | m$ et où $ε > 0$ est arbitraire.

Received: 2005-09-07
Accepted: 2006-01-23
Published online: 2006-03-29

DOI: 10.1016/j.crma.2006.01.022

Author's affiliations:

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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     author = {Jean Bourgain and Mei-Chu Chang},
     title = {A {Gauss} sum estimate in arbitrary finite fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {643--646},
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     publisher = {Elsevier},
     volume = {342},
     number = {9},
     doi = {10.1016/j.crma.2006.01.022},
     language = {en},
}

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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

References
Cited by

[1] J. Bourgain, M.-C. Chang, Exponential sum estimates over subgroups and almost subgroups of $Z_{q}^{*}$ , 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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