Vinogradov’s mean value theorem via efficient congruencing, II

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We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order 2s, with 1⩽s⩽k2−1. Thereby, we show that quasi-diagonal behavior holds when s=o(k2), and we obtain near-optimal estimates for 1⩽s⩽14k2+k and optimal estimates for s⩾k2−1. In this way we come halfway to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring’s problem is established for sums of s kth powers of natural numbers whenever s⩾2k2−2k−8 (k⩾6).