Let $G$ ($G=\mathbb{Z}^n_2$ for my case) be a additive group and $A$ be a subset of $G$. For any set $S\subseteq G$ define its doubling as $$\sigma (S)=\dfrac{S+S}{S}$$ Suppose $A$ has small doubling. i.e. $\sigma (A)\leq c$ for some real number $c\geq 1$. What can we say about $\sigma (A+A)$? I know in general it is not true that $\sigma (A+A)\leq\sigma (A)$. But does there exist sets $A$ satisfying $$\sigma (2^k A)\leq\sigma(2^{k1}A)\ \ \ \ \forall k\geq 0$$ and can we characterize them?
