Vinogradov's mean value theorem via efficient congruencing, II

TL;DR

This paper advances the understanding of Vinogradov's mean value theorem by applying efficient congruencing to obtain near-optimal estimates for moments, with implications for Waring's problem and related areas.

Contribution

It introduces improved bounds for Vinogradov's integral using efficient congruencing, moving closer to the main conjecture and extending applications in additive number theory.

Findings

Established near-optimal estimates for moments 1<=s<= (1/4)k^2 + k

Proved optimal estimates for s >= k^2 - 1

Confirmed asymptotic formulas in Waring's problem for s >= 2k^2 - 2k - 8

Abstract

We apply the efficient congruencing method to estimate Vinogradov's integral for moments of order 2s, with 1<=s<=k^2-1. Thereby, we show that quasi-diagonal behaviour holds when s=o(k^2), we obtain near-optimal estimates for 1<=s<=(1/4)k^2+k, and optimal estimates for s>=k^2-1. In this way we come half way to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type, and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s>=2k^2-2k-8 (k>=6).