An Example Concerning Set Addition in $$\mathbb{F}_2^n$$

We construct sets A and B in a vector space over \(\mathbb{F}\) 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 ∈ A and b ∈ B, but which is extremely far from being “combinatorially” almost closed under addition by B: if A′⊂ A, B′⊂ B and A′ + B′ is comparable in size to A′, then |B′| ⪅ |B|1/2.