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arXiv · 2501.12325

Estimates for short character sums evaluated at homogeneous polynomials

Rena Chu
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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.

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Rena Chu. 2025-01-21. Estimates for short character sums evaluated at homogeneous polynomials. https://arxiv.org/abs/2501.12325

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