The PFR Conjecture Holds for Two Opposing Special Cases

Abstract: Let $A \subseteq F_2^n$ be a set with $|2A| = K|A|$. We prove that if (1) for at least a fraction $1-K^{-9}$ of all $s \in 2A$, the set $(A+s) \cap A$ has size at most $L\cdot|A|/K$, or (2) for at least a fraction $K^{-L}$ of all $s \in 2A$, the set $(A+s) \cap A$ has size at least $|A|\cdot(1- K^{-1/L})$, then there is a subset $B \subseteq A$ of size $|A|/K^{O_L(1)}$ such that $\mathrm{span}(B) \leq K^{O_L(1)}\cdot|A|$.