# Lower bounds for $|A+A|$ if $A$ contains only perfect squares

Let $A$ a set with $|A|=n$ that contains only perfect squares of integers. What lower bounds can we give for $|A+A|$?

I think the lower bound $\gg \frac{n^2}{\sqrt{log \,n}}$ holds (this would be the best bound possible, with "equality" for $A=\{1^2,2^2,...n^2\}$). However, this estimate seems really hard.

I couldn't even prove the bound $|A+A|>Cn$ for all constant $C$ (for big $n$). I think proving this bound would already be a good start. Any idea or technique related to the problem is appreciated.

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The current record is $|A+A| \geq \log(|A|)^{c\log\log(|A|)}|A|$ due to Schoen (in 2011), using his near optimal form of Freiman's theorem. Note this is just shy of a power gain.