Geodesic
On the size of the quotient of two subsets of positive integers
Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 279-287.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a nontrivial lower bound for the size of the set $A/B$, where $A$ and $B$ are subsets of the interval $[1,Q]$.
Keywords: integers, divisibility, energy of sets. 
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Yu. N. Shteinikov. On the size of the quotient of two subsets of positive integers. Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 279-287. http://geodesic.mathdoc.fr/item/TRSPY_2018_303_a19/

[1] Bourgain J., Konyagin S. V., Shparlinski I. E., “Product sets of rationals, multiplicative translates of subgroups in residue rings, and fixed points of the discrete logarithm”, Int. Math. Res. Not., 2008 (2008), rnn090 | MR | Zbl

[2] Cilleruelo J., “On product sets of rationals”, Int. J. Number Theory, 12:5 (2016), 1415–1420 | DOI | MR | Zbl

[3] Cilleruelo J., Garaev M. Z., “Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications”, Math. Proc. Cambridge Philos. Soc., 160:3 (2016), 477–494 | DOI | MR | Zbl

[4] 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

[5] Hildebrand A., Tenenbaum G., “Integers without large prime factors”, J. théor. nombres Bord., 5:2 (1993), 411–484 | DOI | MR | Zbl

[6] S. V. Konyagin, I. D. Shkredov, “On sum sets of sets having small product set”, Proc. Steklov Inst. Math., 290 (2015), 288–299 | DOI | MR | Zbl

[7] S. V. Konyagin, I. D. Shkredov, “New results on sums and products in $\mathbb R$”, Proc. Steklov Inst. Math., 294 (2016), 78–88 | DOI | MR | Zbl

[8] Prachar K., Primzahlverteilung, Springer, Berlin, 1957 | MR | Zbl

[9] Schnirelmann L., “Über additive Eigenschaften von Zahlen”, Math. Ann., 107 (1933), 649–690 | DOI | MR

[10] Yu. N. Shteinikov, “On the product sets of rational numbers”, Proc. Steklov Inst. Math., 296 (2017), 243–250 | DOI | MR | MR | Zbl

[11] Yu. N. Shteinikov, “Addendum to J. Cilleruelo, D. S. Ramana, and O. Ramaré's paper ‘Quotient and product sets of thin subsets of the positive integers’”, Proc. Steklov Inst. Math., 296 (2017), 251–255 | DOI | MR | Zbl

[12] Tao T., Vu V. H., Additive combinatorics, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl