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We study the number of s-element subsets J of a given abelian group G, such that |J + J| ≤ 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 = 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. 2 δs 1 2 Ks s sets J ⊂ [n] with |J| = s and |J + J| ≤ K|J|.The conjecture was later confirmed for K constant by Green and Morris [7]; in fact they proved a slightly more general result: for each fixed K and as s → ∞, the number Research partially supported by CNPq. 1 of sets J ⊂ [n] with |J| = s and |J + J| ≤ K|J| is at most 2 o(s) 1 2 Ks s n ⌊K+o(1)⌋ .The authors of [7] used this result to bound the size of the largest clique in a random Cayley graph and recently the result was also applied by Balogh, Liu, Sharifzadeh and Treglown [2] to determine the number of maximal sum-free sets in [n]. Our main theorem confirms Conjecture 1.1 for all K = o(s/(log n) 3 ). Theorem 1.2. Let s, n be integers and 2 ≤ K ≤ o s (log n) 3 . The number of sets J ⊂ [n] with |J| = s such that |J + J| ≤ K|J| is at most 2 o(s) 1 2 Ks s .We will in fact prove stronger bounds on the error term than those stated above, see Theorem 4.1. Nevertheless, we are unable to prove the conjecture in the range K = Ω(s/(log n) 3 ), and actually the conjecture is false for a certain range of values of s and K ≫ s/ log n. More precisely, for any integers n, s, and any positive numbers K, ǫ with min{s, n 1/2−ǫ } ≥ K ≥ 4 log(24C)s ǫ log n , there are at least n 2 K 4 Ks 8 s − K 4 ≥ CKs s sets J ⊂ [n] with |J| = s and |J + J| ≤ Ks. The construction 1 is very simple: let P be an arithmetic progression of size Ks/8 and set J = J 0 ∪ J 1 , where J 0 is any subset of P of size s − K/4, and J 1 is any subset of [n] \ P of size K/4. For convenience we provide the details in the appendix. Our methods also allow us to characterize the typical structure of an s-set with doubling constant K, and obtain the following result. Theorem 1.3. Let s, n be integers and 2 ≤ K ≤ o s (log n) 3 . For almost all sets J ⊂ [n]with |J| = s such that |J + J| ≤ K|J|, there is a set T ⊂ J such that J \ T is contained in an arithmetic progression of size 1+o(1) 2 Ks and |T | = o(s). In the case s = Ω(n) (and hence K = O(1)), this result was proved by Mazur [10]. We will provide better bounds for the error terms in Theorem 5.1, below.

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $\lambda> 2$ and every $k \geqslant (\log n)^4$: if $\omega \rightarrow \infty $ as $n \rightarrow \infty $ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \leqslant \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega $.

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