Key points are not available for this paper at this time.
Let p be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over Fₚ can be stated as follows. If ϕ: Fₚⁿ FₚN is a function such that ϕ (x+y) - ϕ (x) - ϕ (y) takes values in some set S, then there is a linear map ϕ: Fₚⁿ FₚN with the property that ϕ- ϕ takes at most |S|^O (1) values. A strong variant of this conjecture states that, in fact, there is a linear map ϕ such that ϕ- ϕ takes values in tS for some constant t. In this note, we discuss a counterexample to this conjecture.
James Aaronson (Tue,) studied this question.