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An inverse theorem for the Gowers U^{s+1}[N]-norm

Ben Green, Terence Tao, Tamar Ziegler

Abstract

We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} > δthen there is a bounded-complexity s-step nilsequence F(g(n)Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. Erratum (added April 2024): a 6-page erratum is available as a separate PDF.

An inverse theorem for the Gowers U^{s+1}[N]-norm

Abstract

We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} > δthen there is a bounded-complexity s-step nilsequence F(g(n)Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. Erratum (added April 2024): a 6-page erratum is available as a separate PDF.

Paper Structure

This paper contains 19 sections, 69 theorems, 492 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Basic notation
  4. The polynomial formulation of ${\operatorname{GI}}(s)$
  5. Taking ultralimits
  6. Nilcharacters and symbols in one and several variables
  7. A more detailed outline of the argument
  8. A variant of Gowers's Cauchy-Schwarz argument
  9. Frequencies and representations
  10. Linear independence and the sunflower lemma
  11. Obtaining bracket-linear behaviour
  12. Building a nilobject
  13. The symmetry argument
  14. Basic theory of ultralimits
  15. Polynomial algebra
  16. ...and 4 more sections

Key Result

Theorem 1.3

For any $s \geqslant 3$, the inverse conjecture for the $U^{s+1}[N]$-norm, ${\operatorname{GI}}(s)$, is true.

Theorems & Definitions (182)

  • Conjecture 1.2: ${\operatorname{GI}}(s)$
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Definition 4.1: Polynomial nilsequence
  • Example 4.2: Linear nilsequences are polynomial nilsequences
  • Example 4.3: Polynomial phases are polynomial nilsequences
  • Example 4.4: Combinations of monomials are polynomials
  • Conjecture 4.5: ${\operatorname{GI}}(s)$, polynomial formulation
  • Definition 5.1: Lipschitz functions
  • ...and 172 more