Comptes Rendus
Number Theory
On a multilinear character sum of Burgess
Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 115-120.

Let p be a sufficiently large prime and (Li)1in a nondegenerate system of linear forms in n variables over Fp. We establish a nontrivial estimate on the incomplete character sum

xi=1n[ai,ai+H]X(j=1nLj(x)),
provided H>p14+ε.

Soit p un nombre premier suffisamment grand et (Li)1in un système non-dégénéré de formes linéaires sur Fp en n variables. Nous obtenons une estimée non-triviale de la somme incomplète

xi=1n[ai,ai+H]X(j=1nLj(x)),
X1 est un caractère multiplicatif (modp) et H>p14+ε.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.12.013
Jean Bourgain 1; Mei-Chu Chang 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
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Jean Bourgain; Mei-Chu Chang. On a multilinear character sum of Burgess. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 115-120. doi : 10.1016/j.crma.2009.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.12.013/

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

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

[3] M.-C. Chang 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 Fp2, 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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