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

Trevor D. Wooley

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 $ε>0$, one has \[ \int_{[0,1)^k} \left| \sum_{|n|\le X} {\mathfrak a}_n e(α_1\varphi_1(n)+\ldots +α_k\varphi_k(n)) \right|^{2s} {\rm d}{\boldsymbol α} \ll X^ε\left( \sum_{|n|\le X} |{\mathfrak a}_n|^2\right)^s. \] 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.

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

Abstract

We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when is a system of polynomials with non-vanishing Wronskian, and , then for all complex sequences , and for each , one has As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents , recovering the recent conclusions of the author (for ) and Bourgain, Demeter and Guth (for ). In contrast with the -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.

Paper Structure

This paper contains 17 sections, 43 theorems, 590 equations.

Table of Contents

  1. Introduction
  2. A crude outline of nested efficient congruencing
  3. The infrastructure for nested efficient congruencing
  4. Translation-dilation invariant families
  5. The base of the induction: the trivial case $k=1$
  6. The initial conditioning process
  7. Harnessing approximate translation-dilation invariance
  8. The iterative step
  9. Distilling multigrade combinations into monograde processes
  10. The proof of Theorem \ref{['theorem3.1']}
  11. Solutions of congruences in short intervals
  12. Mean values of exponential sums
  13. A remark on Tarry's problem
  14. Analogues of Hua's lemma, and Waring's problem
  15. Vinogradov's mean value theorem in number fields
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose that $\varphi_j\in {\mathbb Z}[t]$$(1\leqslant j\leqslant k)$ is a system of polynomials with $W(t;{\boldsymbol \varphi})\ne 0$. Let $s$ be a positive real number with $s\leqslant k(k+1)/2$. Also, suppose that $({\mathfrak a}_n)_{n\in {\mathbb Z}}$ is a sequence of complex numbers. Then for In particular, under these conditions, one has

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • Lemma 6.1
  • ...and 33 more