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Quasipolynomial inverse theorem for the $\mathsf{U}^4(\mathbb{F}_p^n)$ norm

Luka Milićević

Abstract

The inverse theory for Gowers uniformity norms is one of the central topics in additive combinatorics and one of the most important aspects of the theory is the question of bounds. In this paper, we prove a quasipolynomial inverse theorem for the $\mathsf{U}^4$ norm in finite vector spaces. The proof follows a different strategy compared to the existing quantitative inverse theorems. In particular, the argument relies on a novel argument, which we call the abstract Balog-Szemerédi-Gowers theorem, and combines several other ingredients such as algebraic regularity method, bilinear Bogolyubov argument and algebraic dependent random choice.

Quasipolynomial inverse theorem for the $\mathsf{U}^4(\mathbb{F}_p^n)$ norm

Abstract

The inverse theory for Gowers uniformity norms is one of the central topics in additive combinatorics and one of the most important aspects of the theory is the question of bounds. In this paper, we prove a quasipolynomial inverse theorem for the norm in finite vector spaces. The proof follows a different strategy compared to the existing quantitative inverse theorems. In particular, the argument relies on a novel argument, which we call the abstract Balog-Szemerédi-Gowers theorem, and combines several other ingredients such as algebraic regularity method, bilinear Bogolyubov argument and algebraic dependent random choice.

Paper Structure

This paper contains 13 sections, 21 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic regularity method
  4. Detailed proof overview
  5. Freiman bihomomorphisms and small rank condition
  6. Getting all additive 16-tuples with small rank combinations
  7. Full system of linear maps
  8. Changing the category---system of partially defined maps
  9. Making domain a bilinear variety
  10. Bilinear variety with many respected additive quadruples
  11. Extending bihomomorphisms from a dense set of columns
  12. Structure theorem for Freiman bihomomorphisms
  13. Inverse theorem for $\mathsf{U}^4$ norm

Key Result

Theorem 2

Suppose that $f : \mathbb{F}_p^n \to \mathbb{D}$ is a function such that $\|f\|_{\mathsf{U}^4} \geq c$. Then there exists a non-classical cubic polynomial $q : \mathbb{F}_p^n \to \mathbb{T}$ such that

Theorems & Definitions (60)

  • Definition 1
  • Theorem 2: Inverse theorem for $\mathsf{U}^4$ norm in $\mathbb{F}_p^n$
  • Theorem 4: Structure theorem for Freiman bihomomorphisms
  • Lemma 5: Gowers GowU4, dependent random choice (also Sudakov, Szémeredi, Vu SudSzeVu)
  • Lemma 6
  • proof
  • Theorem 7: Robust Bogolyubov-Ruzsa lemma SchSis
  • Lemma 8
  • Theorem 9: Structure theorem for approximate homomorphisms
  • Theorem 10: Algebraic regularity lemma
  • ...and 50 more