On the number of sets with small sumset

Abstract: We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and $|A + A| \leq m$ is at most [2^{o(s)}\binom{\frac{m+\beta}{2}}{s},] where $\beta$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ 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 $A \subseteq {1,\ldots,n}$ with $|A| = s$ and $|A + A| \leq m$. Under the same assumption that $m \ll s^2/(\log n)^2$, we show that with high probability there exists an arithmetic progression $P \subseteq \mathbb{Z}$ 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 results is a more efficient container-type theorem developed for sets with small sumset, which gives an essentially optimal collection of containers. The proof of this combines an adapted hypergraph container lemma, that caters to the asymmetric setup as well, with a novel ``preprocessing'' graph container lemma, which allows the hypergraph container lemma to be called upon significantly less times than was necessary before.