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

Abstract

We enhance the efficient congruencing method for estimating Vinogradov’s integral for moments of order 2s, with $1⩽s⩽k^2-1$. In this way, we prove the main conjecture for such even moments when $1⩽s⩽\frac{1}{4}{(k+1)}^2$, 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.