Vol. 14, No. 7, 2020

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Burgess bounds for short character sums evaluated at forms

Lillian B. Pierce and Junyan Xu

Vol. 14 (2020), No. 7, 1911–1951
DOI: 10.2140/ant.2020.14.1911
Abstract

We establish a Burgess bound for short multiplicative character sums in arbitrary dimensions, in which the character is evaluated at a homogeneous form that belongs to a very general class of “admissible” forms. This n-dimensional Burgess bound is nontrivial for sums over boxes of sidelength at least qβ, with β > 12 1(2(n + 1)). This is the first Burgess bound that applies in all dimensions to generic forms of arbitrary degree. Our approach capitalizes on a recent stratification result for complete multiplicative character sums evaluated at rational functions, due to the second author.

Keywords
character sums
Mathematical Subject Classification 2010
Primary: 11L40
Milestones
Received: 17 July 2019
Revised: 14 December 2019
Accepted: 6 February 2020
Published: 18 August 2020
Authors
Lillian B. Pierce
Duke University
Durham, NC
United States
Junyan Xu
Indiana University
Bloomington, IN
United States