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

TL;DR

This paper establishes bounds on character sums over multidimensional parallelepipeds in finite fields of order p^2 and p^3, extending previous results to higher dimensions with explicit error terms.

Contribution

It provides new bounds for multiplicative character sums over specific geometric regions in finite fields of order p^2 and p^3, with conditions on the character's restriction to the prime field.

Findings

Bounds are valid for parallelepipeds with size at least p^{n(1/4+ε)}.

Results distinguish cases based on whether the character is trivial on the prime field.

Explicit decay rate of the character sum in terms of the size of the parallelepiped.

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.