The main conjecture in Vinogradov’s Mean Value Theorem is true

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

Background: For positive integers s,k, and N, let $J_{s,k}(N)$ 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 $1\leq x_i,y_i \leq N$ for $1\leq i \leq s$ .

The Theorem: On 4th December 2015, Jean Bourgain, Ciprian Demeter, and Larry Guth submitted to arxiv and the Annals of Mathematics a paper in which they proved that for any natural numbers $s\geq 1$ and $k\geq 2$ , and any real $\epsilon>0$ , there exist a constant $C=C(s,k,\epsilon)$ such that $J_{s,k}(N) \leq C N^\epsilon(N^s+N^{2s-\frac{1}{2}k(k+1)})$ for all $N\geq 2$ .

Short context: The Theorem confirms a famous conjecture, known as the main conjecture in Vinogradov’s Mean Value Theorem. The estimate is optimal up to constant and $N^\epsilon$ factors. Before 2015, the conjecture was proved if $s \geq k(k + 1)$ , and, more recently, for $k\leq 3$ . The Theorem confirms the conjecture in general.