Alon–Balogh–Morris–Samotij conjecture on sets with small doubling in [n]

Let $[n]=\{1,\ldots,n\}$, and let $s$ and $K$ be parameters. A set $J\subset[n]$ has doubling constant at most $K$ when $|J+J|\leq K|J|$. Alon–Balogh–Morris–Samotij conjecture. For every $\delta>0$, there exists $C>0$ such that, whenever $s\geq C\log n$ and $K\leq s/C$, the number of sets $J\subset[n]$ satisfying $|J|=s$ and $|J+J|\leq K|J|$ is at most

$$2^{\delta s}\binom{\frac{1}{2}Ks}{s}.$$

This conjecture gives the sharp expected upper bound, up to the factor $2^{\delta s}$, for counting $s$-element subsets of $[n]$ with bounded doubling constant. The paper proves the conjecture in the relevant growing-parameter regime, improving earlier results that handled fixed $K$; consequently, the conjecture is solved.