Approximating the main conjecture in Vinogradov’s mean value theorem, submitted

Abstract. We apply multigrade efficient congruencing to estimate Vino-gradov’s integral of degree k for moments of order 2s, establishing strongly diagonal behaviour for 1 6 s 6 12k(k + 1) − 13k + o(k). In particular, as k → ∞, we confirm the main conjecture in Vinogradov’s mean value theorem for 100 % of the critical interval 1 6 s 6 12k(k + 1). 1