Details

Lillian B. Pierce ^[a]

Burgess bounds for short character sums evaluated at forms II: the mixed case

Pages: 151-179
Received: 10 February 2020
Accepted in revised form: 8 September 2020
Mathematics Subject Classification (2010): 11L40.
Keywords: Character sums, Vinogradov Mean Value Theorem.
Author address:
[a]: Department of Mathematics, Duke University, 120 Science Drive, Durham NC 27708 USA

Pierce is partially supported by NSF CAREER grant DMS-1652173, a Sloan Research Fellowship, and the AMS Joan and Joseph Birman Fellowship.

Abstract: This work proves a Burgess bound for short mixed character sums in \(n\) dimensions. The non-principal multiplicative character of prime conductor \(q\) may be evaluated at any ''admissible'' form, and the additive character may be evaluated at any real-valued polynomial. The resulting upper bound for the mixed character sum is nontrivial when the length of the sum is at least \(q^{\beta}\) with \(\beta> 1/2 - 1/(2(n+1))\) in each coordinate. This work capitalizes on the recent stratification of multiplicative character sums due to Xu, and the resolution of the Vinogradov Mean Value Theorem in arbitrary dimensions.

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