Geodesic
On sum sets of sets having small product set
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 304-316

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We improve the sum–product result of Solymosi in $\mathbb R$; namely, we prove that $\max \{|A+A|,|AA|\}\gg |A|^{4/3+c}$, where $c>0$ is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets $A\subset \mathbb R$ with $|AA| \le |A|^{4/3}$. 
@article{TM_2015_290_a24,
     author = {S. V. Konyagin and I. D. Shkredov},
     title = {On sum sets of sets having small product set},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {304--316},
     publisher = {mathdoc},
     volume = {290},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2015_290_a24/}
}
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S. V. Konyagin; I. D. Shkredov. On sum sets of sets having small product set. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 304-316. http://geodesic.mathdoc.fr/item/TM_2015_290_a24/