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 $\{\omega_1,\ldots\omega_n\}$ be a basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$. Let, further, $N_i,H_i$ be integers such that $1\leq H_i\leq p$, $\,\,i=1,\ldots,n$. Define $n$-dimensional parallelepiped $B\subseteq\mathbb{F}_{p^n}$ as follows: $$B=\left\{\sum_{i=1}^nx_i\omega_i \,:\, N_i+1\leq x_i\leq N_i+H_i, \,\,\, 1\leq i\leq n\right\}. $$ Let $n\in\{2,3\}$, $\chi$ be a nontrivial multiplicative character of $\mathbb{F}_{p^n}$ and $|B|\geq p^{n(1/4+\varepsilon)}$, and let us assume that $H_1\leq\ldots\leq H_n$. Then we prove that $$\left|\sum_{x\in B}\chi(x)\right|\ll_{\varepsilon} |B|p^{-\varepsilon^2/12}, $$ if $\chi|_{\mathbb{F}_p}$ is not identical, and $$\left|\sum_{x\in B}\chi(x)\right|\ll_{\varepsilon} |B|p^{-\varepsilon^2/12}+|B\cap \omega_n\mathbb{F}_p| $$ otherwise.