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/(log n)², then the number of subsets A ⊆ Y with $|A| = s$ and |A + A| ≤ m is at most \[2ᵒ⁽ˢ⁾{m+β/2}{s},\] where β 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/(log 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.