Advertisement

Finitely additive measures on groups and rings

  • Sophie Frisch
  • Milan Paštéka
  • Robert F. Tichy
  • Reinhard Winkler
Article

Abstract

On topological groups a natural finitely additive measure can be defined via compactifications. It is closely related to Hartman's concept of uniform distribution on non-compact groups (cf. [3]). Applications to several situations are possible. Some results of M. Paštéka and other authors on uniform distribution with respect to translation invariant finitely additive probability measures on Dedekind domains are transferred to more general situations. Furthermore it is shown that the range of a polynomial of degree ≥2 on a ring of algebraic integers has measure 0.

Keywords

Haar Measure Finite Index Ideal Measure Algebraic Integer Dedekind Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Buck R. C.,The measure theoretic approach to density, Amer. J. Math.68 (1946), 560–580.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Dikranjan D. N., Prodanov I. R., Stoyanov L. N.,Topological Groups, Marcel Dekker Inc., New York, (1990).MATHGoogle Scholar
  3. [3]
    Hartman S.,Remarks on equidistribution on non-compact groups, Comp. Math.16 (1964), 66–71.MATHMathSciNetGoogle Scholar
  4. [4]
    Hewitt E., Ross K. A.,Abstract harmonic analysis, 2nd edition, Springer, New York, 1979.MATHGoogle Scholar
  5. [5]
    Kuipers L., Niederreiter H.,Uniform distribution of sequences, J. Wiley and Sons, New York, (1974).MATHGoogle Scholar
  6. [6]
    Lo S.K., Niederreiter H.,Banach-Buck measure, density, and uniform distribution in rings of algebraic integers, Pac. J. Math.61, No. 1 (1975), 191–208.MATHMathSciNetGoogle Scholar
  7. [7]
    Niederreiter H., Lo S. K.,Permutation Polynomials over Rings of Algebraic Integers, Abh. Math. Sem. Hamburg49 (1979) 126–139.MATHMathSciNetGoogle Scholar
  8. [8]
    Niven I.,Uniform distribution of sequences of integers, Comp. Math.16 (1964), 158–160.MATHMathSciNetGoogle Scholar
  9. [9]
    Okutsu K.,On the measure on the set of positive integers, Proc. Japan Acad.,69 A, 173–174 (1993).MathSciNetGoogle Scholar
  10. [10]
    Paštéka M.,The measure density on Dedekind domains, Ricerche di matematica,45, to appear.Google Scholar
  11. [11]
    Paštéka M.,Submeasures and uniform distribution on Dedekind rings, Tatra Mountain Math. Publ., to appear.Google Scholar
  12. [12]
    Paštéka M., Tichy R. F.,Distribution problems in Dedekind domains and submeasures, Ann. Univ. Ferrara,40 (1994), 191–206.Google Scholar
  13. [13]
    Tichy R. F., Turnwald G.,Uniform distribution of recurrences in Dedekind domains, Acta Arith.,46 (1985), 81–89.MATHMathSciNetGoogle Scholar
  14. [14]
    Tichy R., Turnwald G.,Weak uniform distribution of u n+1=aun+b in Dedekind domains, Manuscripta Math.,61 (1988), 11–22.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Turnwald G.,Einige Bemerkungen zur diskreten Gleichverteilung, Wiener Seminarberichte 1984–1986, inZahlentheoretische Analysis II, E. Hlawka (ed.) Lecture Notes in Mathematics 1262, Springer, Berlin, 1987, pp. 144–149.CrossRefGoogle Scholar
  16. [16]
    van der Waerden B. L.,Algebra I+II, Springer, Berlin, 1967.Google Scholar
  17. [17]
    Warner S.,Topological Rings, Mathematics Studies178, North Holland, Amsterdam, 1993.MATHGoogle Scholar
  18. [18]
    Wan D.,A p-adic Lifting Lemma and Its Applications to Permutation Polynomials, in “Finite Fields, Coding Theory, and Advances in Computing” (Proc. of Conf. in Las Vegas, Nevada, 1991), G. L. Mullen and P. J.-S. Shiue eds., Dekker, 1993.Google Scholar

Copyright information

© Springer 1999

Authors and Affiliations

  • Sophie Frisch
    • 1
  • Milan Paštéka
    • 2
  • Robert F. Tichy
    • 1
  • Reinhard Winkler
    • 3
    • 4
  1. 1.Institut für MathematikTU GrazGrazAustria
  2. 2.Faculty of SciencesUniversity of OstravaOstravaCzech Republic
  3. 3.Institut für Algebra und ComputermathematikTU WienWienAustria
  4. 4.Institut für Diskrete MathematikAustrian Academy of SciencesWienAustria

Personalised recommendations