= Vinogradov mean value
{c}
{title2=$J_{k,r}(Z)=\int_{[0,1]^r}|\sum_{z\le Z}e(\sum_{j=1}^r\theta_jz^j)|^{2k}\,d\boldsymbol\theta$}
By <orthogonality of integer Fourier modes>, this integral counts pairs of $k$-tuples with the same first $r$ power sums. Diagonal pairs give $J_{k,r}(Z)\gg Z^k$. There are $O_{k,r}(Z^{r(r+1)/2})$ possible moment vectors; the <Cauchy-Schwarz inequality> gives $J_{k,r}(Z)\gg Z^{2k-r(r+1)/2}$. Upper bounds measure how much arithmetic coincidence remains beyond these necessary contributions, and enter the <Vinogradov mean-value method for a bilinear exponential sum>.
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