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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 Shteinikov
Scientific Research Institute of System Analysis
Russian 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

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ISSN 2226-8383 (Print)