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

TL;DR

This paper proves Burgess-type bounds for short multiplicative character sums over finite fields, extending previous results to arbitrary dimensions without coordinate restrictions.

Contribution

It introduces a volumetric condition for nontrivial cancellation in character sums over $_{p^n}$, removing coordinate-wise restrictions and generalizing earlier work.

Findings

Nontrivial cancellation occurs when $|B| \,\ge\, p^{n(1/4+\varepsilon)}$

Extends results of Gabdullin to arbitrary dimension

Combines geometry of numbers, energy estimates, and Katz's bounds

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.