Ukrainian Mathematical Journal

, Volume 65, Issue 9, pp 1384–1393 | Cite as

Thin Subsets of Groups

  • I. V. Protasov
  • S. Slobodyanyuk
Article
  • 49 Downloads

For a group G and a natural number m; a subset A of G is called m-thin if, for each finite subset F of G; there exists a finite subset K of G such that |FgA| ≤ m for all gG \ K: We show that each m-thin subset of an Abelian group G of cardinality ℵn; n = 0, 1,… can be split into ≤ mn+1 1-thin subsets. On the other hand, we construct a group G of cardinality ℵω and select a 2-thin subset of G which cannot be split into finitely many 1-thin subsets.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ie. Lutsenko and I. V. Protasov, “Sparse, thin and other subsets of groups,” Int. J. Algebra Comput., 11, 491–510 (2009).Google Scholar
  2. 2.
    Ie. Lutsenko and I. V. Protasov, “Relatively thin and sparse subsets of groups,” Ukr. Math. J., 63, No. 2, 216–225 (2011).Google Scholar
  3. 3.
    Ie. Lutsenko and I. Protasov, “Thin subsets of balleans,” Appl. Gen. Topology, 11, No. 2, 89–93 (2010).Google Scholar
  4. 4.
    I. V. Protasov, “Selective survey on subset combinatorics of groups,” Ukr. Math. Bull., 7, 220–257 (2010).Google Scholar
  5. 5.
    I. Protasov, “Partitions of groups into thin subsets,” Algebra Discrete Math., 11, 88–92 (2011).MathSciNetGoogle Scholar
  6. 6.
    T. Banakh and N. Lyaskovska, “On thin complete ideals of subsets of groups,” Ukr. Math. J., 63, No. 6, 741–754 (2011).CrossRefMathSciNetGoogle Scholar
  7. 7.
    O. Petrenko and I. V. Protasov, “Thin ultrafilters,” Notre Dame J. Formal Logic., 53, 79–88 (2012).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    I. Protasov and M. Zarichnyi, “General asymptology,” Math. Stud. Monogr. Ser., VNTL Publ., Lviv, 12 (2007).Google Scholar
  9. 9.
    J. Roe, “Lectures on coarse geometry,” AMS Univ. Lect. Ser., RI, Providence, 31 (2003).Google Scholar
  10. 10.
    R. Engelking, General Topology, PWN, Warszawa (1985).Google Scholar
  11. 11.
    J. C. Simms, “Another characterization of Alephs: decompositions of hyperspace,” Notre Dame J. Formal Logic., 38, 19–36 (1997).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    R. O. Davies, “The power of the continuum on some propositions of the plane geometry,” Fundam. Math., 52, 277–281 (1963).MATHGoogle Scholar
  13. 13.
    R. O. Davies, “On a problem of Erd¨os concerning decomposition of the plane,” Proc. Cambridge Phil. Soc., 59, 33–36 (1963).CrossRefMATHGoogle Scholar
  14. 14.
    N. Hindman and D. Strauss, Algebra in the Stone–Cěch Compactification: Theory and Applications, Walter de Gruyter, Berlin; New York (1998).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • I. V. Protasov
    • 1
  • S. Slobodyanyuk
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations