SOLVED
Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$. Is it true that
\[\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{(\log \lvert A\rvert)^C}\]
for some constant $C>0$?
This was proved by Solymosi
[So09d], in the strong form
\[\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.\]
See also [52].
