Estimates for character sums in finite fields of order $p^2$ and $p^3$

Abstract

Let $p$ be a prime number, ${\mathbb{F}}_{p^n}$ be the finite field of order $p^n$, and $\{{\omega }_1,\dots {\omega }_n\}$ be a basis of ${\mathbb{F}}_{p^n}$ over ${\mathbb{F}}_p$. Let, further, $N_i,H_i$ be integers such that $1\le H_i\le p$, $i=1,\dots ,n$. Define $n$-dimensional parallelepiped $B\subseteq {\mathbb{F}}_{p^n}$ as follows: $$B=\left\{\sum _{i=1}^n{x}_i{\omega }_i:N_i+1\le x_i\le N_i+H_i,1\le i\le n\right\}.$$ Let $n\in \{2,3\}$, $\chi$ be a nontrivial multiplicative character of ${\mathbb{F}}_{p^n}$ and $|B|\ge p^{n(1/4+\epsilon )}$, and let us assume that $H_1\le \dots \le H_n$. Then we prove that $$\left|\sum _{x\in B}\chi (x)\right|{\ll }_{\epsilon }|B|p^{-{\epsilon }^2/12},$$ if $\chi {|}_{{\mathbb{F}}_p}$ is not identical, and $$\left|\sum _{x\in B}\chi (x)\right|{\ll }_{\epsilon }|B|p^{-{\epsilon }^2/12}+|B\cap {\omega }_n{\mathbb{F}}_p|$$ otherwise.