Geodesic
Extremal properties of product sets
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 239-245

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We find the nearly optimal size of a set $A\subset [N] := \{1,\dots ,N\}$ so that the product set $AA$ satisfies either (i) $|AA| \sim |A|^2/2$ or (ii) $|AA| \sim |[N][N]|$. This settles problems recently posed in a paper of J. Cilleruelo, D. S. Ramana and O. Ramaré. 
@article{TM_2018_303_a16,
     author = {K. Ford},
     title = {Extremal properties of product sets},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {239--245},
     publisher = {mathdoc},
     volume = {303},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_303_a16/}
}
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K. Ford. Extremal properties of product sets. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 239-245. http://geodesic.mathdoc.fr/item/TM_2018_303_a16/