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

TL;DR

This paper advances the efficient congruencing method to estimate Vinogradov's integral, proving the main conjecture for certain moments and revealing strongly diagonal behavior, with implications for Waring's problem.

Contribution

It introduces improvements to the efficient congruencing technique, establishing the main conjecture for specific moments of Vinogradov's integral and analyzing their diagonal behavior.

Findings

Proves the main conjecture for moments with 1 ≤ s ≤ (k+1)^2/4.

Shows moments exhibit strongly diagonal behavior in this range.

Provides applications to the asymptotic formula in Waring's problem.

Abstract

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