  In this paper, we study the linear structure of sets $A \subset
\mathbb{F}_2^n$ with doubling constant $\sigma(A)<2$, where
$\sigma(A):=\frac{|A+A|}{|A|}$. In particular, we show that $A$ is contained in
a small affine subspace. We also show that $A$ can be covered by at most four
shifts of some subspace $V$ with $|V|\leq |A|$. Finally, we classify all binary
sets with small doubling constant.
