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

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 $\mathrm{\mid }B\mathrm{\mid }\ge p^{n(1\mathrm{/}4+\epsilon )}$, 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.