Geodesic
New results on sums and products in~$\mathbb R$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 87-98

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We improve previous sum-product estimates in $\mathbb R$; namely, we prove the inequality $\max\{|A+A|,|AA|\}\gg|A|^{4/3+c}$, where $c$ is any number less than $5/9813$. New lower bounds for sums of sets with small product set are found. We also obtain results on the additive and multiplicative energies; in particular, we improve a result of Balog and Wooley. 
@article{TM_2016_294_a4,
     author = {S. V. Konyagin and I. D. Shkredov},
     title = {New results on sums and products in~$\mathbb R$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {87--98},
     publisher = {mathdoc},
     volume = {294},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2016_294_a4/}
}
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S. V. Konyagin; I. D. Shkredov. New results on sums and products in~$\mathbb R$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 87-98. http://geodesic.mathdoc.fr/item/TM_2016_294_a4/