= Cubic Weyl inequality
{title2=$|S(\theta)|\ll_\eta n^{1+\eta}(q^{-1}+n^{-1}+q/n^3)^{1/4}$}
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 $O(n^3+n\sum_{0<|h|,|l|<n}\min(n,\|6\theta hl\|^{-1}))$. 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. https://pub.math.leidenuniv.nl/~evertsejh/ant20-7.pdf[Leiden lecture notes, Section 7.2] develop this cubic differencing estimate.
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