On the number of sets with a given doubling constant

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 = ο ( 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 ο ( s ) in many cases. The main tool used in the proof is the asymmetric container lemma, introduced recently by Morris, Samotij and Saxton.