Estimates for short character sums evaluated at homogeneous polynomials

TL;DR

This paper establishes nontrivial bounds for short Dirichlet character sums evaluated at homogeneous polynomials across various dimensions, using finite field techniques and energy bounds, applicable to sums over small boxes.

Contribution

It introduces new bounds for short character sums at homogeneous polynomials in multiple dimensions, leveraging finite field character relationships and energy estimates.

Findings

Bounds are nontrivial for sums over boxes as short as p^{1/4 + κ}

Methods connect characters mod p with characters over finite field extensions

Utilizes bounds on multiplicative energy in finite fields

Abstract

Let $p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as $p^{1/4 + \kappa}$ for any $\kappa>0$. Our methods capitalize on the relationship between characters mod $p$ and characters over finite field extensions as well as bounds on the multiplicative energy of sets in products of finite fields.