Geodesic
Quotient and product sets of thin subsets of the positive integers
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and combinatorial number theory, Tome 296 (2017), pp. 58-71

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We study the cardinalities of $A/A$ and $AA$ for thin subsets $A$ of the set of the first $n$ positive integers. In particular, we consider the typical size of these quantities for random sets $A$ of zero density and compare them with the sizes of $A/A$ and $AA$ for subsets of the shifted primes and the set of sums of two integral squares. 
@article{TM_2017_296_a4,
     author = {J. Cilleruelo and D. S. Ramana and O. Ramar\'e},
     title = {Quotient and product sets of thin subsets of the positive integers},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {58--71},
     publisher = {mathdoc},
     volume = {296},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_296_a4/}
}
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J. Cilleruelo; D. S. Ramana; O. Ramaré. Quotient and product sets of thin subsets of the positive integers. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic and combinatorial number theory, Tome 296 (2017), pp. 58-71. http://geodesic.mathdoc.fr/item/TM_2017_296_a4/