Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

Abstract.

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type \(\sum_{x_{1} \in A_{1},\ldots, x_{r} \in A_{r}} e_{p}(x_{1},\ldots x_{r})\) under nearly optimal conditions on \(|A_{1}| \cdots |A_{r}|\). It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of \({\mathbb{F}_{p}^{*}}\), obtaining a nontrivial estimate provided \(log |H| > C {\frac{log p}{log log p}}\). This is a further improvement on [BGK].

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jean Bourgain.

Additional information

Received: May 2007, Revision: October 2007, Accepted: October 2007

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bourgain, J. Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources. GAFA Geom. funct. anal. 18, 1477–1502 (2009). https://doi.org/10.1007/s00039-008-0691-6

Download citation

Keywords and phrases:

  • Prime field
  • exponential sum

AMS Mathematics Subject Classification:

  • 11L07
  • 11L05
  • 11T23