Cubic Weyl inequality ($|S(\theta )|{\ll }_{\eta }{n}^{1+\eta }(q^{-1}+n^{-1}+q/n^3{)}^{1/4}$)

If $S(\theta )={\sum }_{x\le n}e(\theta x^3)$ and $|\theta -a/q|\le q^{-2}$ with $(a,q)=1$, the displayed estimate holds. Twice applying Weyl differencing bounds $|S{|}^4$ by $O(n^3+n{\sum }_{0<|h|,|l|<n}\min (n,\Vert 6\theta hl{\Vert }^{-1}))$. Group by $6hl$ and use the subpower bound for the divisor function. The reciprocal fractional-part sum near a rational bounds the resulting sum by $O(\log (2q)(n^3/q+n^2+n+q))$. Take fourth roots, absorbing logarithms into $n^{\eta }$ when $q\le n^3$; for larger $q$ the trivial bound suffices. Leiden lecture notes, Section 7.2 develop this cubic differencing estimate.