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A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture

James Aaronson

Abstract

Let $p$ be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p$ can be stated as follows. If $φ: \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ is a function such that $φ(x+y) - φ(x) - φ(y)$ takes values in some set $S$, then there is a linear map $\tildeφ : \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ with the property that $φ- \tildeφ$ takes at most $|S|^{O(1)}$ values. A strong variant of this conjecture states that, in fact, there is a linear map $\tildeφ$ such that $φ- \tildeφ$ takes values in $tS$ for some constant $t$. In this note, we discuss a counterexample to this conjecture.

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture

Abstract

Let be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over can be stated as follows. If is a function such that takes values in some set , then there is a linear map with the property that takes at most values. A strong variant of this conjecture states that, in fact, there is a linear map such that takes values in for some constant . In this note, we discuss a counterexample to this conjecture.

Paper Structure

This paper contains 2 sections, 2 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. The construction

Key Result

Theorem 2.1

Given $t$, suppose that $n \geqslant 12t + 7$. View $\mathbb{F}_p^N$ as the space of all (not necessarily linear) maps from $\mathbb{F}_p^n$ to $\mathbb{F}_p$ (in particular, $N = p^{p^n}$). Let $\phi : \mathbb{F}_p^n \rightarrow \mathbb{F}_p^N$ be defined as follows. If $v$ is an element of $\mathb where $f : \mathbb{F}_p^n \rightarrow \mathbb{F}_p$. Let $S \subseteq \mathbb{F}_p^N$ denote the se

Theorems & Definitions (6)

  • Conjecture 1.1
  • Conjecture 1.2
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof