Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

Frederick Manners

Abstract

We provide a new proof of the inverse theorem for the Gowers $U^{s+1}$-norm over groups $H=\mathbb Z/N\mathbb Z$ for $N$ prime. This proof gives reasonable quantitative bounds (the worst parameters are double-exponential), and in particular does not make use of regularity or non-standard analysis, both of which are new for $s \ge 3$ in this setting.

Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

Abstract

We provide a new proof of the inverse theorem for the Gowers -norm over groups for prime. This proof gives reasonable quantitative bounds (the worst parameters are double-exponential), and in particular does not make use of regularity or non-standard analysis, both of which are new for in this setting.

Paper Structure

This paper contains 38 sections, 55 theorems, 439 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Applications
  4. Approximate polynomials
  5. Limitations and bounds
  6. Structure of the paper
  7. Notation and definitions
  8. Acknowledgements
  9. From 1% polynomials to global structure
  10. Outline of the argument
  11. Systems of cubes
  12. Connectivity and dividing into non-interacting pieces
  13. Extending to a global function
  14. Iterating global structure and polynomial hierarchies
  15. Improving the global structure
  16. ...and 23 more sections

Key Result

Theorem 1.1.2

Fix an integer $s \ge 1$. Let $N$ be a (large) prime and write $H = \mathbb{Z}/N\mathbb{Z}$. If $f \colon H \to \mathbb{C}$ is a function such that $\|f\|_\infty \le 1$ and $\|f\|_{U^{s+1}} \ge \delta$, then there exists a one-bounded nilsequence $\psi \colon H \to \mathbb{C}$ with degree $s$, dimen where if $s \le 3$ then and if $s \ge 4$ then

Figures (2)

  • Figure 1: Glueing cubes together with $k=2$, $i=1$
  • Figure 2: A tricube configuration

Theorems & Definitions (206)

  • Definition 1.1.1
  • Theorem 1.1.2
  • Definition 1.3.1
  • Definition 1.3.2
  • Theorem 1.3.3
  • Definition 2.2.1
  • Remark 2.2.2
  • Lemma 2.2.3
  • proof
  • Corollary 2.2.4
  • ...and 196 more