Abstract
The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in \({F_{p^2}}\) and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.A. Burgess, On character sums and primitive roots, Proc. London Math. Soc (3) 12 (1962), 179–192
D.A. Burgess, Character sums and primitive roots in finite fields, Proc. London Math. Soc (3) 37 (1967), 11–35.
Burgess D.A.: A note on character sums of binary quadratic forms. JLMS 43, 271–274 (1968)
Chang M.-C.: Factorization in generalized arithmetic progressions and applications to the Erdős–Szemerédi sum-product problems. Geom. Funct. Anal. 13, 720–736 (2003)
Chang M.-C.: On a question of Davenport and Lewis and new character sum bounds in finite fields. Duke Math. J. 145(3), 409–442 (2008)
H. Davenport, D. Lewis, Character sums and primitive roots in finite fields, Rend. Circ. Matem. Palermo-Serie II-Tomo XII-Anno (1963), 129–136.
Friedlander J., Iwaniec H.: Estimates for character sums. Proc. Amer. Math. Soc. 119(2), 265–372 (1993)
Karacuba A.A.: Estimates of character sums. Math. USSR-Izvestija 4(1), 19–29 (1970)
T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press, 2006.
Acknowledgments
The author would like to thank the referees for helpful comments.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Chang, MC. Burgess Inequality In \({\mathbb {F}_{p^2}}\) . Geom. Funct. Anal. 19, 1001–1016 (2009). https://doi.org/10.1007/s00039-009-0031-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-009-0031-5