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 ${\phi }_j\in \mathbb{Z}[t]$ $(1\le j\le k)$ 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 \[ \int_{[0,1)^k} \left| \sum_{|n|\le X} {\mathfrak a}_n e(α_1φ_1(n)+\ldots +α_kφ_k(n)) \right|^{2s} {\rm d}{\boldsymbol α} \ll X^ε\left( \sum_{|n|\le X} |{\mathfrak a}_n|^2\right)^s. \] 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.

84pp; minor corrections, enhanced explanation on page 44

References in corpus (12)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three
• Translation invariance, exponential sums, and Waring's problem
• Perturbations of Weyl sums
• Mean value estimates for odd cubic Weyl sums
• Rational solutions of pairs of diagonal equations, one cubic and one quadratic
• Discrete Fourier restriction via efficient congruencing: basic principles
• Improved ${\ell }^p$-Boundedness for Integral $k$-Spherical Maximal Functions
• Approximating the main conjecture in Vinogradov's mean value theorem
• The Cubic Case of Vinogradov's Mean Value Theorem --- A Simplified Approach to Wooley's "Efficient Congruencing"
• Effective Vinogradov's mean value theorem via efficient boxing
• On a binary system of Prendiville: The cubic case

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• On integer solutions of Parsell-Vinogradov systems
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• Reversing a philosophy: from counting to square functions and decoupling
• A short proof of ${\ell }^2$ decoupling for the moment curve
• Efficient congruencing in ellipsephic sets: the quadratic case
• ${\ell }^2$ Decoupling in ${\mathbb{R}}^2$ for curves with Vanishing Curvature
• Efficient congruencing in ellipsephic sets: the general case
• Discrete restriction for $(x,x^3)$ and related topics
• Effective Vinogradov's mean value theorem via efficient boxing
• Hybrid bounds on two-parametric family Weyl sums along smooth curves
• Weak approximation and the Hilbert property for Campana points
• A bilinear proof of decoupling for the cubic moment curve
• An $L^4$ maximal estimate for quadratic Weyl sums
• Restricted mean value theorems and metric theory of restricted Weyl sums
• Squarefree Integers in Arithmetic Progressions to Smooth Moduli
• Paucity problems and some relatives of Vinogradov's mean value theorem
• Integral solutions to systems of diagonal equations
• On a Hybrid Version of the Vinogradov Mean Value Theorem
• Improved discrete restriction for the parabola
• Large sieve inequalities with power moduli and Waring's problem
• Burgess bounds for short character sums evaluated at forms II: the mixed case
• Arithmetic Ramsey theory over the primes
• Hausdorff dimension of the large values of Weyl sums