Vinogradov’s mean value theorem via efficient congruencing

Abstract

We obtain estimates for Vinogradov’s integral that for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of $s$ $k$th powers of natural numbers whenever $s\ge 2k^2+2k-3$.