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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.14531 |
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Table of Contents:
- We improve the standard Weyl estimate for quartic exponential sums in which the argument is a quadratic irrational. Specifically we show that \[\sum_{n\le N} e(αn^4)\ll_{\ep,α}N^{5/6+\ep}\] for any $\ep>0$ and any quadratic irrational $α\in\R-\Q$. Classically one would have had the exponent $7/8+\ep$ for such $α$. In contrast to the author's earlier work \cite{cubweyl} on cubic Weyl sums (which was conditional on the $abc$-conjecture), we show that the van der Corput $AB$-steps are sufficient for the quartic case, rather than the $BAAB$-process needed for the cubic sum.