Burgess bounds for multi-dimensional short mixed character sums

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 f, 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 k≥1 dimensions that recapture the Burgess bound in dimension 1. Moreover, we show that by embedding any given polynomial f into an advantageously chosen translation–dilation invariant system constructed in terms of f, we may in many cases significantly improve the bound for the associated character sum, due to a novel phenomenon that occurs only in dimensions k≥2.