Nested efficient congruencing and relatives of Vinogradov's mean value theorem

TL;DR

This paper introduces a nested efficient congruencing method to estimate Vinogradov mean values, confirming the main conjecture for all exponents and extending applicability to algebraic number fields and function fields.

Contribution

The paper develops a nested variant of efficient congruencing that proves the main conjecture in Vinogradov's mean value theorem for all exponents, without relying on multilinear Kakeya estimates.

Findings

Confirmed the main conjecture for all exponents k

Extended methods to algebraic number fields and function fields

Provided a flexible approach avoiding multilinear Kakeya estimates

Abstract

We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $\varphi_j\in \mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing Wronskian, and $s\le k(k+1)/2$, then for all complex sequences $(\mathfrak a_n)$, and for each $\epsilon>0$, one has \[ \int_{[0,1)^k} \left| \sum_{|n|\le X} {\mathfrak a}_n e(\alpha_1\varphi_1(n)+\ldots +\alpha_k\varphi_k(n)) \right|^{2s} {\rm d}{\boldsymbol \alpha} \ll X^\epsilon \left( \sum_{|n|\le X} |{\mathfrak a}_n|^2\right)^s. \] As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents $k$, recovering the recent conclusions of the author (for $k=3$) and Bourgain, Demeter and Guth (for $k\ge 4$). In contrast with the $l^2$-decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.