Burgess bounds for short character sums evaluated at forms II: the mixed case
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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2020
- DOI:
- 10.48550/arXiv.2002.03435
- arXiv:
- arXiv:2002.03435
- Bibcode:
- 2020arXiv200203435P
- Keywords:
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- Mathematics - Number Theory
- E-Print:
- 19 pages