On the cubic Weyl sum

TL;DR

This paper improves bounds on cubic Weyl sums for short summation ranges, advancing understanding of exponential sums involving cubic polynomials and providing sharper estimates within specific parameter ranges.

Contribution

It introduces a novel estimate for cubic Weyl sums that surpasses previous bounds from Weyl differencing for certain short ranges of summation.

Findings

Established a new bound for cubic Weyl sums with improved decay rate.

Extended the valid range of summation lengths where the estimate applies.

Built on ideas of Enflo to achieve these sharper bounds.

Abstract

We obtain an estimate for the cubic Weyl sum which improves the bound obtained from Weyl differencing for short ranges of summation. In particular, we show that for any $\varepsilon>0$ there exists some $\delta>0$ such that for any coprime integers $a,q$ and real number $\gamma$ we have \begin{align*} \sum_{1\le n \le N}e\left(\frac{an^3}{q}+\gamma n\right)\ll (qN)^{1/4} q^{-\delta}, \end{align*} provided $q^{1/3+\varepsilon}\le N \le q^{1/2-\varepsilon}$. Our argument builds on some ideas of Enflo.