An example concerning set addition in F_2^n

Abstract

We construct sets $A,B$ in a vector space over ${\mathbb{F}}_2$ with the property that $A$ is "statistically" almost closed under addition by $B$ in the sense that $a+b$ almost always lies in $A$ when $a\in A,b\in B$, but which is extremely far from being "combinatorially" almost closed under addition by $B$: if $A^{\prime }\subset A$, $B^{\prime }\subset B$ and $A^{\prime }+B^{\prime }$ is comparable in size to $A^{\prime }$ then $|B^{\prime }|⪅|B{|}^{1/2}$.

5 pages, to appear in "Harmonic analysis, approximation theory and number theory", special volume of Proc Steklov Math. Inst. in honour of the 60th birthday of Sergei Konyagin