On a question of Davenport and Lewis and new character sum bounds in finite fields

Abstract

Let $\chi$ be a nontrivial multiplicative character of ${\mathbb{F}}_{p^n}$. We obtain the following results.

(1) Let $ϵ>0$ be given. If $B=\{\sum _{j=1}^n{x}_j{\omega }_j :x_j\in [N_j+1,N_j+H_j]\cap \mathbb{Z},j=1,\dots ,n\}$ is a box satisfying ${\Pi }_{j=1}^n{H}_j>p^{(2/5+ϵ)n},$ then for $p>p(ϵ)$ we have, denoting $\chi$ a nontrivial multiplicative character, ${\displaystyle |\sum _{x\in B}\chi (x)|{\ll }_n{p}^{-{ϵ}^2/4}\left|B\right|}$ unless $n$ is even, $\chi$ is principal on a subfield $F_2$ of size $p^{n/2}$, and ${\max }_{\xi }|B\cap \xi F_2|>p^{-ϵ}\left|B\right|$.

(2) Assume that $A,B\subset {\mathbb{F}}_p$ so that ${\displaystyle \left|A\right|>p^{(4/9)+ϵ},\left|B\right|>p^{(4/9)+ϵ},|B+B|<K\left|B\right|.}$ Then ${\displaystyle |\sum _{x\in A,y\in B}\chi (x+y)|<p^{-\tau }\left|A\right| \left|B\right|.}$

(3) Let $I\subset {\mathbb{F}}_p$ be an interval with $\left|I\right|=p^{\beta }$, and let $D\subset {\mathbb{F}}_p$ be a $p^{\beta }$-spaced set with $\left|D\right|=p^{\sigma }$. Assume that $2\beta +\sigma -\beta \sigma /(1-\beta )>1/2+\delta$. Then for a nonprincipal multiplicative character $\chi$, ${\displaystyle |\sum _{x\in I,y\in D}\chi (x+y)|<p^{-{\delta }^2/12}\left|I\right|\left|D\right|.}$ We also slightly improve a result of Karacuba [K3]