On the number of sets with a given doubling constant

Abstract

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 2o(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 2ο(s) in many cases. The main tool used in the proof is the asymmetric container lemma, introduced recently by Morris, Samotij and Saxton.