Counting sets with given doubling via dimension

We determine, up to a factor of $2^{o(k)}$, the number of $k$-sets $A\subset \{1,\dots ,n\}$ such that $\mathrm{\mid }A+A\mathrm{\mid }\le m$, where $k=\mathrm{\Theta }(\log n)$ and $m\le k^{1+\alpha }$, for small $\alpha >0$, answering a question of Green and Morris.