The main conjecture in Vinogradov’s Mean Value Theorem is true if s>=k(k+1)

You need to know: Just basic arithmetic for the Theorem. You may like to learn what is Vinogradov’s Mean Value Theorem and Waring’s problem to better understand the context.

Background: For positive integers s,k, and X, let $J_{s,k}(X)$ be the number of integral solutions of the system of equations $x_1^j+\dots+x_s^j = y_1^j+\dots+y_s^j$ , $1 \leq j \leq k$ , such that $1\leq x_i,y_i \leq X$ for $1\leq i \leq s$ .

The Theorem: On 3rd December 2010, Trevor Wooley submitted to the Annals of Mathematics a paper in which he proved that for any natural numbers $k\geq 2$ and $s \geq k(k + 1)$ , and any real $\epsilon>0$ , there exist a constant C such that $J_{s,k}(X) \leq C X^{2s-\frac{1}{2}k(k+1)+\epsilon}$ .

Short context: A famous conjecture, known as the main conjecture in Vinogradov’s Mean Value Theorem, predicts that $J_{s,k}(X) \leq C X^\epsilon(X^s+X^{2s-\frac{1}{2}k(k+1)})$ for any $s\geq 1$ and $k\geq 2$ . This estimate, if true, would be optimal up to constant and $X^\epsilon$ factors. The Theorem implies that it holds if $s \geq k(k + 1)$ . In the same paper, Wooley demonstrated several applications of the Theorem, for example, to Waring’s problem. In a later work, Bourgain, Demeter, and Guth proved the conjecture in general.