  We study the number of $s$-element subsets $J$ of a given abelian group $G$,
such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and
Samotij, and improving a result of Green and Morris, who proved the conjecture
for $K$ fixed, we provide an upper bound on the number of such sets which is
tight up to a factor of $2^{o(s)},$ when $G=\mathbb{Z}$ and $K=o(s/(\log
n)^3)$. We also provide a generalization of this result to arbitrary abelian
groups which is tight up to a factor of $2^{o(s)}$ in many cases. The main tool
used in the proof is the asymmetric container lemma, introduced recently by
Morris, Samotij and Saxton.
