Burgess-Type volume centric Bounds for Character Sums over $\mathbb{F}_{p^n}$

We establish a Burgess-type bound for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$. Define box $B$ by $$ B=\left\{ \sum_{i=1}^{n} x_i\omega_i : N_i+1 \leq x_i \leq N_i+H_i,\; 1 \leq i \leq n \right\} \subseteq \mathbb{F}_{p^{n}},$$ where $N_i$ and $H_i$ are integers that satisfy $1 \leq H_i \leq p \text{ for all } 1 \leq i \leq n$ and $H_1\leq H_2\leq \cdots\leq H_n$. We show that for the box $B \subset \mathbb{F}_{p^n}$ with the first $(n-2)$ sides of length at least of some size with respect to the last two side lengths, a nontrivial cancellation occurs whenever $|B| \ge p^{n(1/4+\varepsilon)}$.
This extends earlier work of Gabdullin in dimensions $n=2,3$ to arbitrary dimension. The proof combines methods from the geometry of numbers, multiplicative energy estimates, and bounds for character sums due to Katz.