On a multilinear character sum of Burgess

Bourgain, Jean; Chang, Mei-Chu

doi:10.1016/j.crma.2009.12.013

Number Theory

On a multilinear character sum of Burgess
[Sur les sommes de caractères multi-linéaires de Burgess]

Présenté par : Jean Bourgain

Bourgain, Jean ¹ ; Chang, Mei-Chu ²

¹ School of Mathematics, Institute for Advanced Study, Olden lane, Princeton, NJ 08540, USA
² Department of Mathematics, University of California, Riverside, 900 University Avenue, Riverside, CA 92521, USA

Comptes Rendus. Mathématique, Tome 348 (2010) no. 3-4, pp. 115-120.

Résumé
Abstract

Soit p un nombre premier suffisamment grand et ${(L_{i})}_{1 ⩽ i ⩽ n}$ un système non-dégénéré de formes linéaires sur $F_{p}$ en n variables. Nous obtenons une estimée non-triviale de la somme incomplète

\sum_{x \in \prod_{i = 1}^{n} [a_{i}, a_{i} + H]} X (\prod_{j = 1}^{n} L_{j} (x)),

où

X \neq 1

est un caractère multiplicatif

(\mod p)

H > p^{\frac{1}{4} + ε}

Let p be a sufficiently large prime and ${(L_{i})}_{1 ⩽ i ⩽ n}$ a nondegenerate system of linear forms in n variables over $F_{p}$ . We establish a nontrivial estimate on the incomplete character sum

\sum_{x \in \prod_{i = 1}^{n} [a_{i}, a_{i} + H]} X (\prod_{j = 1}^{n} L_{j} (x)),

provided

H > p^{\frac{1}{4} + ε}

Reçu le : 2009-11-29
Accepté le : 2009-12-10
Publié le : 2010-01-20

DOI : 10.1016/j.crma.2009.12.013

Affiliations des auteurs :

Bourgain, Jean ¹ ; Chang, Mei-Chu ²

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Bourgain, Jean; Chang, Mei-Chu. On a multilinear character sum of Burgess. Comptes Rendus. Mathématique, Tome 348 (2010) no. 3-4, pp. 115-120. doi : 10.1016/j.crma.2009.12.013. http://www.numdam.org/articles/10.1016/j.crma.2009.12.013/

Bibliographie
Cité par

[1] Burgess, D.A. The distribution of quadratic residues and non-residues, Mathematica, Volume 4 (1957), pp. 106-112

[2] Burgess, D.A. A note on character sums for binary quadratic forms, J. London Math. Soc., Volume 43 (1968), pp. 271-274

[3] Chang, M.-C. On a question of Davenport and Lewis and new character sum bounds in finite fields, Duke Math. J., Volume 145 (2008) no. 3, pp. 409-442

[4] M.-C. Chang, Character sums in $F_{p^{2}}$ , GAFA, in press

[5] S.V. Konyagin, Estimates of character sums in finite fields, Matematicheskie Zametki (in Russian), in press

[6] C. Lekkerkerker, Geometry of Numbers, North-Holland Mathematical Library, vol. 37, 1987

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