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Sparse mean value estimates, algebraic number solution counting, and non-Archimedean Fourier analysis

Ben Johnsrude

Abstract

We demonstrate two applications of Fourier decoupling theorems over non-Archimedean local fields to real-variable problems. These include short mean value estimates for exponential sums, canonical-scale mean value estimates for exponential sums arising from phase functions with coefficients arising from the traces of powers of algebraic numbers, and solution counting bounds for Vinogradov systems whose indeterminates are families of algebraic numbers. We also record an example where real and $\mathfrak p$-adic decoupling estimates differ.

Sparse mean value estimates, algebraic number solution counting, and non-Archimedean Fourier analysis

Abstract

We demonstrate two applications of Fourier decoupling theorems over non-Archimedean local fields to real-variable problems. These include short mean value estimates for exponential sums, canonical-scale mean value estimates for exponential sums arising from phase functions with coefficients arising from the traces of powers of algebraic numbers, and solution counting bounds for Vinogradov systems whose indeterminates are families of algebraic numbers. We also record an example where real and -adic decoupling estimates differ.

Paper Structure

This paper contains 4 sections, 5 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mean values on sparse subsets of [0,1]k
  3. Integrals arising from decoupling over completions of algebraic fields
  4. An obstruction for transference arguments

Key Result

Theorem 2.5

Let $\mathfrak p$, $\mathbb P$, $N$, $r$, and $\Omega$ be as above. Suppose $\sigma_j\leq|e_j|$ for all $j$. Then we have the bound If $\epsilon_j=\max(1,\max_{\bf n\in\Omega}|\sum_\ell c_j^\ell(\bf n/N)^{e_j^\ell}|)^{-1}$, then

Figures (3)

  • Figure 1: Domains for discrete restriction for the parabola in $\mathbb{R}^2$.
  • Figure 2: Domains for discrete restriction for the moment curve in $\mathbb{R}^3$.
  • Figure 3: Domains for discrete restriction for the paraboloid in $\mathbb{R}^3$.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: Transference of mean value estimates
  • proof
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • ...and 14 more