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We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with \({1\leqslant s\leqslant k^{2}-1}\). In this way, we prove the main conjecture for such even moments when \({1\leqslant s\leqslant \tfrac{1}{4}(k+1)^{2}}\), showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of s, these finding application to the asymptotic formula in Waring’s problem.

On Vinogradov’s mean value theorem: strongly diagonal behaviour via efficient congruencing