Burgess bounds for multi-dimensional short mixed character sums

Published

Journal Article

© 2015 Elsevier Inc. 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.

Full Text

Duke Authors

Cited Authors

  • Pierce, LB

Published Date

  • June 1, 2016

Published In

Volume / Issue

  • 163 /

Start / End Page

  • 172 - 210

International Standard Serial Number (ISSN)

  • 0022-314X

Digital Object Identifier (DOI)

  • 10.1016/j.jnt.2015.08.022

Citation Source

  • Scopus