Sets with Extremal Product Property and Its Variations
Matematičeskie zametki, Tome 114 (2023) no. 6, pp. 922-930.

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In the present paper, we refine the lower bound for the size of the set $A$ of finite intervals of positive integers such that the size of the set $AA$ is asymptotically equal to $|A|^2/2$. Arguing by analogy with the previous work by Ford (2018) with minor optimizations, we refine the previous estimate. In this paper, we borrow the problems, approaches, and arguments of reasoning proposed by Ford.
Keywords: product, set, divisor
Mots-clés : prime.
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Yu. N. Shteinikov. Sets with Extremal Product Property and Its Variations. Matematičeskie zametki, Tome 114 (2023) no. 6, pp. 922-930. http://geodesic.mathdoc.fr/item/MZM_2023_114_6_a9/

[1] K. Ford, “Ekstremalnye svoistva proizvedenii mnozhestv”, Garmonicheskii analiz, teoriya priblizhenii i teoriya chisel, Sbornik statei. K 60-letiyu so dnya rozhdeniya akademika S. V. Konyagina, Trudy MIAN, 303, Nauka, M., 2018, 239–245 | DOI | MR

[2] P. Erdös, “An asymptotic inequality in the theory of numbers”, Vestnik Leningrad. Univ., 15 (1960), 41–49 | MR

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

[4] Kh. Silleruelo, D. S. Ramana, O. Ramare, “Chastnye i proizvedeniya podmnozhestv nulevoi plotnosti mnozhestva naturalnykh chisel”, Analiticheskaya i kombinatornaya teoriya chisel, Sbornik statei. K 125-letiyu so dnya rozhdeniya akademika I. M. Vinogradova, Trudy MIAN, 296, Nauka, M., 2017, 58–71 | DOI | MR

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

[6] A. Selberg, “Note on a paper by L. G. Sathe”, J. Indian Math. Soc., 18 (1954), 83–87 | MR

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

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