On the cubic Weyl sum

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.