On sumsets in ${\mathbb{F}}_2^n$

Abstract

Let ${\mathbb{F}}_2$ be the finite field of two elements, ${\mathbb{F}}_2^n$ be the vector space of dimension $n$ over ${\mathbb{F}}_2$. For sets $A,B\subseteq {\mathbb{F}}_2^n$, their sumset is defined as the set of all pairwise sums $a+b$ with $a\in A,b\in B$. Ben Green and Terence Tao proved that, let $K\ge 1$, if$A,B\subseteq {\mathbb{F}}_2^n$ and $|A+B|\le K|A{|}^{\frac{1}{2}}|B{|}^{\frac{1}{2}}$, then there exists a subspace $H\subseteq {\mathbb{F}}_2^n$ with $$|H|\gg \exp (-O(\sqrt{K}\log K))|A|$$ and $x,y\in {\mathbb{F}}_2^n$ such that $$|A\cap (x+H){|}^{\frac{1}{2}}|B\cap (y+H){|}^{\frac{1}{2}}\ge \frac{1}{2K}|H|.$$ In this note, we shall use the method of Green and Tao with some modification to prove that if $$|H|\gg \exp (-O(\sqrt{K}))|A|,$$ then the above conclusion still holds true.