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 $\mathrm{span}(B)\le K^{O_L(1)}\cdot |A|$.

10 pages

References in corpus (1)

• On the Bogolyubov-Ruzsa lemma