Estimates for Character Sums in Finite Fields, $\mathbb{F}_{p^n}$

Abstract

We improve upon our previous result \cite{Ch} on short character sums over finite fields $\mathbb{F}_{p^n}$. Specifically, we show that for intervals $I_1,I_2,\ldots,I_k$ of lengths of at least $p^{n/4k+\varepsilon}$ and $k\leq n$, the sum \[ \sum_{x_0\in I_0,\; x_1\in I_1,\;\ldots,\; x_{k}\in I_{k}} χ\!\bigl(x_1ω_1 + x_2ω_2 + \cdots + x_{k}ω_{k}\bigr) \] exhibits nontrivial cancellation, where $\{ω_1,\ldots,ω_n\}$ is any basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$. More generally, let $|I_1|\le |I_2|\le \cdots \le |I_n| \le \sqrt{p/2}$ and fix $\varepsilon>0$ such that $ |I_1|\cdots|I_n| \ge p^{\,n(1/4+\varepsilon)}. $ Then, for any nontrivial multiplicative character $χ$ of $\mathbb{F}_{p^n}$, we have \[ \Biggl|\sum_{x_1\in I_1,\;\ldots,\;x_n\in I_n} χ\!\Bigl(\sum_{i=1}^{n} x_i ω_i\Bigr)\Biggr| \;\ll\; |I_1|\cdots|I_n|\, p^{-δ(\varepsilon)}, \] where \[ δ(\varepsilon) = \varepsilon^2 \frac{1-\frac{1}{2n}}{\left(1+\frac{1}{4n}\right)\left(2-\frac{1}{2n}\right)}. \] The proof relies on Minkowski's second theorem, which is applied to estimate the multiplicative energy of sets arising from products of intervals.

Theorems & Definitions (22)

• Theorem 1.1: Konyagin Kon
• Theorem 1.2: Gabdullin GB
• Theorem 1.3: Main Theorem
• Remark 1
• Theorem 1.4: Chattopadhyay Ch
• Corollary 1
• Definition 2: Multiplicative energy
• Definition 3: Successive minima
• Proposition 4: Minkowski’s second theorem
• proof