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

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.