Multigrade efficient congruencing and Vinogradov's mean value theorem

Abstract

We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\frac{5}{8}{k}^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.

48pp; modest revisions in light of referee comments

References in corpus (1)

• Vinogradov's mean value theorem via efficient congruencing, II

Cited by in corpus (9)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• Translation invariance, exponential sums, and Waring's problem
• Perturbations of Weyl sums
• Additive representation in short intervals, II: sums of two like powers
• Discrete Fourier restriction via efficient congruencing: basic principles
• Approximating the main conjecture in Vinogradov's mean value theorem
• The asymptotic formula in Waring's problem: higher order expansions
• On the Waring--Goldbach problem for eighth and higher powers