About one question about the set of rational numbers determined by the quotients of two subsets
https://doi.org/10.22405/2226-8383-2024-25-4-154-157
Abstract
In this paper, we aim to derive a quantitative version of a problem on the size of a set of fractions 𝐴/𝐴, in the case where 𝐴 is a given finite set of natural numbers lying in the interval [1, 𝑛], having positive asymptotic density 𝛼 > 0 as 𝑛 → ∞.
About the Author
Yurii Nikolaevich ShteinikovRussian Federation
References
1. Cilleruelo, J., Ramana, D.S., Ramare, O. “Quotients and product sets of thin subsets of the positive integers”, Proceedings of the Steklov Institute of Mathematics, vol. 296, no. 1, pp. 52–64.
2. Shteinikov, Yu.N. 2023, “Sets with Extremal Product Property and Its Variations”, Mathematical Notes, vol. 114, no. 6, pp. 1357-–1364.
3. Erd˝os, Paul. “An asymptotic inequality in the theory of numbers”„ Vestnik Leningrad. Univ. vol.15, no. 13, pp. 41—49.
4. Ford K. “The distribution of integers with a divisor in a given interval”„ Annals of Mathematics. Second Series. vol.168, no. 2, pp. 367-–433.
5. Hall R.R., Tenenbaum G. 1988, “Divisors”, Cambridge Tracts Math., 90, Cambridge Univ. Press, Cambridge.
6. Shteinikov Yu. 2017, “On the product sets of rational numbers”, Proceedings of the Steklov Institute of Mathematics, vol. 296, no. 1, pp. 243–250.
7. Cilleruelo J. 2016, “A note on product sets of rationals”„ International Journal of Number Theory, Vol. 12, no. 05, pp. 1415–1420.
Review
For citations:
Shteinikov Yu.N. About one question about the set of rational numbers determined by the quotients of two subsets. Chebyshevskii Sbornik. 2024;25(4):154-157. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-4-154-157