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Sep 10, 2015 at 23:49 comment added Brendan Murphy Theorem 1.2 of Li and Roche-Newton's paper arxiv.org/pdf/1111.5159v1.pdf gives a slightly better lower bound of $|A|^{24/19}$. This result also gives the best lower bound on $|A+A|$ when $|AA|$ or $|S+S|$ is very small.
Sep 6, 2015 at 20:41 comment added Terry Tao This appears to be a special case of Corollary 3.2 of the Elekes-Nathanson-Ruzsa result linked in Boris's comment: ams.org/mathscinet-getitem?mr=1772612
Sep 6, 2015 at 15:33 history edited Mark Lewko CC BY-SA 3.0
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Sep 6, 2015 at 15:28 history answered Mark Lewko CC BY-SA 3.0