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Remarks on Steinhaus' property and ratio sets of sets of positive integers

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Abstract

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.

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References

  1. T. M. Apostol: Introduction to Analytic Number Theory. Springer-Verlag, New York-Heidelberg-Berlin, 1976.

    Google Scholar 

  2. T. C. Brown, A. R. Freedman: Arithmetic progressions in lacunary sets. Rocky Mountain J. Math. 17 (1987), 587–596.

    Google Scholar 

  3. T. C. Brown, A. R. Freedman: The uniform density of sets of integers and Fermat's last theorem. C. R. Math. Rep. Acad. Sci. Canada XII (1990), 1–6.

    Google Scholar 

  4. J. Bukor, M. Kmeťová, J. Tóth: Notes on ratio sets of sets of natural numbers. Acta Math. (Nitra) 2 (1995), 35–40.

    Google Scholar 

  5. D. Hobby, D. M. Silberger: Quotients of primes. Amer. Math. Monthly 100 (1993), 50–52.

    Google Scholar 

  6. J. Nagata: Modern General Topology. North-Holland Publ. Comp. Amsterdam-London-Groningen-New York, 1974.

    Google Scholar 

  7. W. Narkiewicz, T. Šalát: A theorem of H. Steinhaus and (R)-dense sets of positive integer Czechoslovak Math. J. 34(109) (1984), 355–361.

    Google Scholar 

  8. H. H. Ostmann: Additive Zahlentheorie I. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956.

    Google Scholar 

  9. T. Šalát: Cantorsche Entwicklungen der reellen Zahlen und das Husdorffsche. Mass. Publ. Math. Inst. Hung. Acad. Sci. 6 (1961), 15–41.

    Google Scholar 

  10. T. Šalát: On Hausdorff measure of linear sets (Russian). Czechoslovak Math. J. 11(86) (1961), 24–56.

    Google Scholar 

  11. T. Šalát: Über die Cantorsche Reihen. Czechoslovak Math. J. 18(93) (1968), 25–56.

    Google Scholar 

  12. T. Šalát: On ratio sets of sets of natural numbers. Acta Arith. 15 (1969), 273–278; Corrigendum: Acta Arith. 16 (1969).

    Google Scholar 

  13. T. Šalát: Quotientbasen und (R)-dichte Mengen. Acta Arithm. 19 (1971), 63–78.

    Google Scholar 

  14. P. Starni: Answers to two questions concerning quotients of primes. Amer. Math. Monthly 102 (1995), 347–349.

    Google Scholar 

  15. W. Sierpiński: Elementary Theory of Numbers. PWN, Warszawa, 1964.

    Google Scholar 

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Šalát, T. Remarks on Steinhaus' property and ratio sets of sets of positive integers. Czechoslovak Mathematical Journal 50, 175–183 (2000). https://doi.org/10.1023/A:1022457724187

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