Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete Fourier restriction via efficient congruencing: basic principles

Trevor D. Wooley

Abstract

We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(α_1n+\ldots +α_kn^k) \right|^{2s}\,{\rm d}{\mathbf α}\ll N^{s-k(k+1)/2}\biggl( \sum_{|n|\le N}|\mathfrak a_n|^2\biggr)^s.$$ Bounds for the constant in the associated periodic Strichartz inequality from $L^{2s}$ to $l^2$ of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from $l^2$ to $L^{s'}$, where $s'=2s/(2s-1)$. These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject.

Discrete Fourier restriction via efficient congruencing: basic principles

Abstract

We show that whenever , then for any complex sequence , one has Bounds for the constant in the associated periodic Strichartz inequality from to of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from to , where . These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject.

Paper Structure

This paper contains 9 sections, 19 theorems, 266 equations.

Table of Contents

  1. Introduction
  2. The infrastructure for efficient congruencing
  3. Some consequences of latent translation-dilation invariance
  4. The conditioning process
  5. The efficient congruencing step
  6. The iterative process
  7. The Keil-Zhao device
  8. Some consequences of Theorem \ref{['theorem1.1']}
  9. Wider applications of weighted efficient congruencing

Key Result

Theorem 1.1

Suppose that $k\geqslant 2$ and $s\geqslant k(k+1)$. Then for any $\varepsilon>0$, and any complex sequence $({\mathfrak a}_n)_{n\in {\mathbb Z}}$, one hasWe employ the convention that whenever $G:[0,1)^k\rightarrow {\mathbb C}$ is integrable, then $\oint G({\boldsymbol \alpha}){\,{\rm d}}{\boldsymb Moreover, when $s>k(k+1)$, one may take $\varepsilon=0$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Conjecture 1.4: Main Conjecture
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 27 more