Vinogradov's mean-value theorem

By considering the X^s solutions where :x_i=y_i, (1\le i\le s) one can see that J_{s,k}(X)\gg X^s. A more careful analysis (see Vaughan equation 7.4) provides the lower bound :J_{s,k}\gg X^s+X^{2s-\frac12k(k+1)}. ==Proof of the Main conjecture==

The main conjecture of Vinogradov's mean value theorem was that the upper bound is close to this lower bound. More specifically that for any \epsilon>0 we have :J_{s,k}(X)\ll X^{s+\epsilon}+X^{2s-\frac12k(k+1)+\epsilon}. This was proved by Jean Bourgain, Ciprian Demeter, and Larry Guth and by a different method by Trevor Wooley. If :s\ge \frac12k(k+1) this is equivalent to the bound :J_{s,k}(X)\ll X^{2s-\frac12k(k+1)+\epsilon}. Similarly if s\le \frac12k(k+1) the conjectural form is equivalent to the bound :J_{s,k}(X)\ll X^{s+\epsilon}. Stronger forms of the theorem lead to an asymptotic expression for J_{s,k}, in particular for large s relative to k the expression :J_{s,k}\sim \mathcal C(s,k)X^{2s-\frac12k(k+1)}, where \mathcal C(s,k) is a fixed positive number depending on at most s and k, holds, see Theorem 1.2 in. == History ==

Vinogradov's original theorem of 1935 showed that for fixed s,k with :s\ge k^2\log (k^2+k)+\frac14k^2+\frac54 k+1 there exists a positive constant D(s,k) such that :J_{s,k}(X)\le D(s,k)(\log X)^{2s}X^{2s-\frac12k(k+1)+\frac12}. Although this was a ground-breaking result, it falls short of the full conjectured form. Instead it demonstrates the conjectured form when \epsilon>\frac12. Vinogradov's approach was improved upon by Karatsuba and Stechkin who showed that for s\ge k there exists a positive constant D(s,k) such that :J_{s,k}(X)\le D(s,k)X^{2s-\frac12k(k+1)+\eta_{s,k}}, where :\eta_{s,k}=\frac12 k^2\left(1-\frac1k\right)^{\left[\frac sk\right]}\le k^2e^{-s/k^2}. Noting that for :s>k^2(2\log k-\log\epsilon) we have :\eta_{s,k}, this proves that the conjectural form holds for s of this size. The method can be sharpened further to prove the asymptotic estimate :J_{s,k}\sim \mathcal C(s,k)X^{2s-\frac12k(k+1)}, for large s in terms of k. In 2012 Wooley improved the range of s for which the conjectural form holds. He proved that for :k\ge 2 and s\ge k(k+1) and for any \epsilon>0 we have :J_{s,k}(X)\ll X^{2s-\frac12k(k+1)+\epsilon}. Ford and Wooley have shown that the conjectural form is established for small s in terms of k. Specifically they show that for :k\ge 4 and :1\le s\le \frac14(k+1)^2 for any \epsilon>0 we have :J_{s,k}(X)\ll X^{s+\epsilon}. == References ==