= Hua's lemma
{c}
{title2=$\int_0^1|\sum_{x\le n}e(\theta x^k)|^{2^j}\,d\theta\ll_{k,j,\eta}n^{2^j-j+\eta}$}
{wiki}
For $1\le j\le k$ and every $\eta>0$, the displayed estimate bounds a mean value of a polynomial <exponential sum>. For cubes its eighth-moment bound is $O_\eta(n^{5+\eta})$. It complements pointwise <Weyl differencing> estimates: a ninth moment over a region is bounded by its supremum there times this eighth moment. https://pub.math.leidenuniv.nl/~evertsejh/ant20-8.pdf[Leiden lecture notes, Section 8.1] give the cubic formulation.
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