On the cubic Weyl sum

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.