Burgess bounds for multi-dimensional short mixed character sums

TL;DR

This paper extends Burgess bounds to multi-dimensional short mixed character sums involving multivariate polynomials and Dirichlet characters, using advanced methods to improve bounds especially in higher dimensions.

Contribution

It introduces a multi-dimensional Burgess method combined with Vinogradov Mean Value Theorems, and demonstrates improved bounds through polynomial embedding in higher dimensions.

Findings

Burgess bounds are established for multi-dimensional sums.

Embedding polynomials into invariant systems can significantly improve bounds.

The results generalize the classical Burgess bound to higher dimensions.

Abstract

This paper proves Burgess bounds for short mixed character sums in multi-dimensional settings. The mixed character sums we consider involve both an exponential evaluated at a real-valued multivariate polynomial, and a product of multiplicative Dirichlet characters. We combine a multi-dimensional Burgess method with recent results on multi-dimensional Vinogradov Mean Value Theorems for translation-dilation invariant systems in order to prove character sum bounds in any dimension that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding the given polynomial into an advantageously chosen translation-dilation invariant system constructed in terms of that polynomial, we may in many cases significantly improve the bound for the associated character sum, due to a novel phenomenon that occurs only in dimensions two and higher.