| Title:
|
Remarks on Steinhaus’ property and ratio sets of sets of positive integers (English) |
| Author:
|
Šalát, Tibor |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
50 |
| Issue:
|
1 |
| Year:
|
2000 |
| Pages:
|
175-183 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| MSC:
|
11A41 |
| MSC:
|
11B05 |
| MSC:
|
11B83 |
| MSC:
|
11K55 |
| idZBL:
|
Zbl 1034.11010 |
| idMR:
|
MR1745470 |
| . |
| Date available:
|
2009-09-24T10:31:25Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127559 |
| . |
| Reference:
|
[1] T. M. Apostol: Introduction to Analytic Number Theory.Springer-Verlag, New York-Heidelberg-Berlin, 1976. Zbl 0335.10001, MR 0434929 |
| Reference:
|
[2] T. C. Brown, A. R. Freedman: Arithmetic progressions in lacunary sets.Rocky Mountain J. Math. 17 (1987), 587–596. MR 0908265, 10.1216/RMJ-1987-17-3-587 |
| Reference:
|
[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. MR 1043085 |
| Reference:
|
[4] J. Bukor, M. Kmeťová, J. Tóth: Notes on ratio sets of sets of natural numbers.Acta Math. (Nitra) 2 (1995), 35–40. |
| Reference:
|
[5] D. Hobby, D. M. Silberger: Quotients of primes.Amer. Math. Monthly 100 (1993), 50–52. MR 1197643, 10.2307/2324814 |
| Reference:
|
[6] J. Nagata: Modern General Topology.North-Holland Publ. Comp. Amsterdam-London-Groningen-New York, 1974. MR 0474164 |
| Reference:
|
[7] W. Narkiewicz, T. Šalát: A theorem of H. Steinhaus and $(R)$-dense sets of positive integers.Czechoslovak Math. J. 34(109) (1984), 355–361. MR 0761418 |
| Reference:
|
[8] H. H. Ostmann: Additive Zahlentheorie I.Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956. Zbl 0072.03101, MR 0098721 |
| Reference:
|
[9] T. Šalát: Cantorsche Entwicklungen der reellen Zahlen und das Husdorffsche.Mass. Publ. Math. Inst. Hung. Acad. Sci. 6 (1961), 15–41. MR 0147465 |
| Reference:
|
[10] T. Šalát: On Hausdorff measure of linear sets (Russian).Czechoslovak Math. J. 11(86) (1961), 24–56. MR 0153802 |
| Reference:
|
[11] T. Šalát: Über die Cantorsche Reihen.Czechoslovak Math. J. 18(93) (1968), 25–56. |
| Reference:
|
[12] T. Šalát: On ratio sets of sets of natural numbers.Acta Arith. 15 (1969), 273–278. MR 0242756, 10.4064/aa-15-3-273-278 |
| Reference:
|
[13] T. Šalát: Quotientbasen und $(R)$-dichte Mengen.Acta Arithm. 19 (1971), 63–78. Zbl 0218.10071, MR 0292788, 10.4064/aa-19-1-63-78 |
| Reference:
|
[14] P. Starni: Answers to two questions concerning quotients of primes.Amer. Math. Monthly 102 (1995), 347–349. Zbl 0828.11004, MR 1328019, 10.2307/2974957 |
| Reference:
|
[15] W. Sierpiński: Elementary Theory of Numbers.PWN, Warszawa, 1964. MR 0175840 |
| . |
