An Estimate of Incomplete Mixed Character Sums

Abstract

Dedicated to Endre Szemerédi for his 70th birthday. In this note we consider incomplete mixed character sums over a finite field $\mathbb{F}_{p^n }$ of the form $\sum\nolimits_{x \in B_H } {\psi \left( {f\left( x \right)} \right)\chi \left( x \right)}$ where is an additive character, $f\left( x \right) \in \mathbb{F}_{p^n }$ a polynomial, x a non-trivial multiplicative character and B _H a ‘box’ of the form $B_H = \left\{ {\sum\nolimits_{j = 1}^n {xj\omega j:x_j \left[ {1,H} \right]} } \right\}$. (Here $\left\{ {\omega _i } \right\}_{i = 1}^n$ is an arbitrary basis of F_pn over F_p.)