Vinogradov mean value ($J_{k,r}(Z)={\int }_{[0,1{]}^r}|{\sum }_{z\le Z}e({\sum }_{j=1}^r{\theta }_j{z}^j){|}^{2k}d\mathbf{\theta }$)

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.

Expand moments of the inner sum and group equal power-sum differences. Their multiplicities are bounded by $J_{k,r}(Z)$ through the Cauchy-Schwarz inequality. Two applications of the Holder inequality yield a bound for $|U{|}^{4k^2}$ containing $Z^{8k^2-4k}{J}_{k,r}(Z{)}^2$ and a product of short geometric-sum bounds over the moment differences. Thus a sharp mean-value estimate combines with rational approximation or spacing of the coefficients to prove cancellation. When all coefficients are integers, $U=\lfloor Z{\rfloor }^2$, so mean-value estimates alone cannot force cancellation.