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arXiv · 2002.03435

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

Lillian B. Pierce
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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 $q^β$ with $β> 1/2 - 1/(2(n+1))$ 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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Lillian B. Pierce. 2021-05-26. Burgess bounds for short character sums evaluated at forms II: the mixed case. https://arxiv.org/abs/2002.03435

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