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

TL;DR

This paper presents a counterexample to a strong version of the Polynomial Freiman-Ruzsa conjecture over finite fields, challenging assumptions about the structure of approximate homomorphisms.

Contribution

The paper constructs a specific counterexample demonstrating that the strong variant of the conjecture does not hold universally.

Findings

Counterexample disproves the strong conjecture

Challenges assumptions about the structure of approximate homomorphisms

Implications for additive combinatorics and related fields

Abstract

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