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Fourier growth of degree $2$ polynomials

Lars Becker, Joseph Slote, Alexander Volberg, Haonan Zhang

Abstract

We prove bounds for the absolute sum of all level-$k$ Fourier coefficients for $(-1)^{p(x)}$, where polynomial $p:\mathbf{F}_2^n \to \mathbf{F}_2$ is of degree $1$ or degree $2$.

Fourier growth of degree $2$ polynomials

Abstract

We prove bounds for the absolute sum of all level- Fourier coefficients for , where polynomial is of degree or degree .

Paper Structure

This paper contains 9 sections, 11 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm deg 2']} by induction
  3. Another approach via the structure theorem of Dickson
  4. Vectors of boolean weight $k$ in affine subspace of $\mathbf{F}_2^n$.
  5. Estimating the Fourier weight
  6. Sharpness
  7. Degree 3 case
  8. Other arguments
  9. Why the induction argument for $k > 2$ does not work

Key Result

Theorem 1

For any polynomial $p: \mathbf{F}^n_2 \to \mathbf{F}_2$ of degree at most $2$, the $k$-th Fourier weight of $f(x)=(-1)^{p(x)}$ is at most $(1 + \sqrt{2})^k$, that is,

Theorems & Definitions (17)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Theorem 4: McWS and Lemma 2.4 BGGM
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 7 more