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

Abstract

We establish Burgess-type bounds for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$ under a purely volumetric condition. We show that for a box $B \subset \mathbb{F}_{p^n}$, nontrivial cancellation occurs whenever $|B| \ge p^{n(1/4+\varepsilon)}$, without imposing lower bounds on the individual side lengths. This removes the coordinate-wise restrictions present in earlier results and extends work of Gabdullin for $n=2,3$ to arbitrary dimension. The proof combines methods from the geometry of numbers with multiplicative energy estimates and bounds for character sums due to Katz.