Geodesic
Some remarks on sets with small quotient set
Sbornik. Mathematics, Tome 208 (2017) no. 12, pp. 1854-1868

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We prove that for any finite set $A\subset \mathbb{R}$ with $|A/A| \ll |A|$ we have $|A-A| \gg |A|^{5/3 - o(1)}$. We also show that for such sets $|A+A+A| \gg |A|^{2-o(1)}$. Bibliography: 22 titles.
Keywords: additive combinatorics, sum-product phenomenon, Erdős-Szemerédi conjecture. 
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     author = {I. D. Shkredov},
     title = {Some remarks on sets with small quotient set},
     journal = {Sbornik. Mathematics},
     pages = {1854--1868},
     publisher = {mathdoc},
     volume = {208},
     number = {12},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_12_a6/}
}
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I. D. Shkredov. Some remarks on sets with small quotient set. Sbornik. Mathematics, Tome 208 (2017) no. 12, pp. 1854-1868. http://geodesic.mathdoc.fr/item/SM_2017_208_12_a6/