Geodesic
Extremal properties of product sets
Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 239-245 
K. Ford. Extremal properties of product sets. Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 239-245. http://geodesic.mathdoc.fr/item/TRSPY_2018_303_a16/
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     author = {K. Ford},
     title = {Extremal properties of product sets},
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     year = {2018},
     volume = {303},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2018_303_a16/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Candela P., Rué J., Serra O., “Memorial to Javier Cilleruelo: A problem list”, Integers, 18 (2018), A28 | MR

[2] J. Cilleruelo, D. S. Ramana, O. Ramaré, “Quotient and product sets of thin subsets of the positive integers”, Proc. Steklov Inst. Math., 296 (2017), 52–64 | DOI | MR | MR | Zbl

[3] Ford K., “Generalized Smirnov statistics and the distribution of prime factors”, Funct. approx. Comment. math., 37:1 (2007), 119–129 | DOI | MR | Zbl

[4] Ford K., “The distribution of integers with a divisor in a given interval”, Ann. Math. Ser. 2, 168:2 (2008), 367–433 | DOI | MR | Zbl

[5] Ford K., “Integers with a divisor in $(y,2y]$”, Anatomy of integers, CRM Proc. Lect. Notes, 46, Amer. Math. Soc., Providence, RI, 2008, 65–80 | DOI | MR | Zbl

[6] Hall R. R., Tenenbaum G., Divisors, Cambridge Tracts Math., 90, Cambridge Univ. Press, Cambridge, 1988 | MR | Zbl