Erdős Problem #818

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 $C>0$?

#818: [Er91]

additive combinatorics

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].

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