A Short Character Sum in $\mathbb{F}_{p^3}$

TL;DR

The paper establishes a new nontrivial, uniform bound for short multiplicative character sums over two-dimensional grids in $\mathbb{F}_{p^3}$, namely $|\sum_{x\in I}\sum_{y\in J}\chi(x+\omega y)|\le p^{-\delta}|I||J|$ for intervals $|I|,|J|\ge p^{3/8+\varepsilon}$ and $\omega\notin\mathbb{F}_p$. The approach extends Burgess-type methods to higher field extensions by developing multiplicative-energy estimates for intervals in $\mathbb{F}_{p^d}$, coupled with Hölder-type decompositions and Weil bounds; these yield nontrivial bounds uniformly in the parameter $\omega$. The results generalize to codimension-one sublattices in $\mathbb{F}_{p^d}$ and are applied to obtain uniform short-sum estimates for binary cubic forms over $\mathbb{F}_p$. In particular, for $d=3$ the work recovers a bound with threshold $\rho_3=3/8$ and provides an actionable framework for higher-degree homogeneous forms, advancing Burgess-type bounds in broader finite-field settings.

Abstract

We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in $\mathbb{F}_{p^3}$ and higher-dimensional lattices in $\mathbb{F}_{p^d}$, extending an earlier work of Mei-Chu Chang on Burgess inequality in $\mathbb{F}_{p^2}$. In particular, we show that for intervals of size $p^{3/8+\varepsilon}$, the sum $\sum_{x, y} χ(x + ωy)$, with $ω\in \mathbb{F}_{p^3} \setminus \mathbb{F}_p$, exhibits nontrivial cancellation uniformly in $ω$. This is further generalized to codimension-one sublattices in $\mathbb{F}_{p^d}$, and applied to obtain an alternative estimate for character sums on binary cubic forms.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Corollary 5
• Lemma 6
• proof
• proof
• Theorem