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 $1\le s\le k^2-1$. In this way, we prove the main conjecture for such even moments when $1\le s\le \frac{1}{4}(k+1{)}^2$, 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.

arXiv commentv3. 2013-12-18. We corrected the relation in the second display following (3.5) in the proof of Lemma 3.3, and corrected the argument leading from (5.1) to the 2nd display preceding (5.2) in the proof of Lemma 5.1. We rearranged the proof of Lemma 7.1 a bit, and corrected a number of very small typos. arXiv admin note: text overlap with arXiv:1112.0358, arXiv:1310.8447