On the number of sets with small sumset

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group G and an n-element subset Y G we show that if m s²/ (n) ², then the number of subsets A Y with |A| = s and |A + A| m is at most \2^o (s) m+{2}s, \ where is the size of the largest subgroup of G of size at most (1+o (1) ) m. This bound is sharp for Z and many other groups. Our result improves the one of Campos and nearly bridges the remaining gap in a conjecture of Alon, Balogh, Morris, and Samotij. We also explore the behaviour of uniformly chosen random sets A \1, , n\ with |A| = s and |A + A| m. Under the same assumption that m s²/ (n) ², we show that with high probability there exists an arithmetic progression P Z of size at most m/2 + o (m) containing all but o (s) elements of A. Analogous results are obtained for asymmetric sumsets, improving results by Campos, Coulson, Serra, and W\"otzel. The main tool behind our proofs is a graph container theorem combined with a variant of an asymmetric hypergraph container theorem.