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 $\mathbb{F}<em>{p<sup>n}$. Specifically, we show that for intervals $I_1,I_2,\ldots,I_k$ of lengths of at least $p<sup>{n/4k+\varepsilon}$ 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_1ω1 + x_2ω_2 + \cdots + x{k}ω{k}\bigr) ] exhibits nontrivial cancellation, where ${ω_1,\ldots,ω_n}$ is any basis of $\mathbb{F}</em>{p<sup>n}$ over $\mathbb{F}<em>p$. More generally, let $|I_1|\le |I_2|\le \cdots \le |I_n| \le \sqrt{p/2}$ and fix $\varepsilon&gt;0$ such that $$|I_1|\cdots|I_n| \ge p^{{\,n(1/4+\varepsilon)}.}$$ Then, for any nontrivial multiplicative character $χ$ of $\mathbb{F}</em>{p<sup>n}$, 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.