  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 $A' \subset A$, $B' \subset B$ and $A' + B'$ is comparable in size to
$A'$ then $|B'| \lessapprox |B|^{1/2}$.
