manuscripta mathematica

, Volume 1, Issue 3, pp 259–288 | Cite as

Diskrepanz in kompakten abelschen Gruppen I

  • Edmund Hlawka
  • Harald Niederreiter


It is the aim of this paper to generalize the notion of discrepancy of a sequence to compact abelian groups satisfying the second axiom of countability. This concept of discrepancy which depends heavily on the study of certain generating subsets of the character group includes the hitherto known case of the n-dimensional unit cube. The definition is subsequently justified by transferring well-known theorems of the classical theory to the general case. This first part mainly deals with the algebraic point of view which doesn't possess an analogy in the theory of uniform distribution mod 1. The proofs involve a lot of group theoretic argument. Theorems concerning distribution and approximation will be presented in the second part.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    CIGLER, J. - HELMBERG, G.: Neuere Entwicklungen der Theorie der Gleichverteilung. J.-ber.Deutsch. Math.-Verein. 64, 1–50 (1961).Google Scholar
  2. [2]
    ECKMANN, B.: Über monothetische Gruppen. Commentarii math. Helvet. 16, 249–263 (1943/44).Google Scholar
  3. [3]
    HELMBERG, G.: Abstract theory of uniform distribution. Compositio math. 16, 72–82 (1964).Google Scholar
  4. [4]
    HEWITT, E. - ROSS, K.A.: Abstract harmonic analysis I. Grundlehren Bd. 115, Berlin-Heidelberg-New York: Springer 1963.Google Scholar
  5. [5]
    HLAWKA, E.: Discrepancy and uniform distribution of sequences. Compositio math. 16, 83–91 (1964).Google Scholar
  6. [6]
    PONTRJAGIN, L.S.: Topologische Gruppen II. Leipzig: Teubner 1957.Google Scholar
  7. [7]
    SCHMEIDLER, W.: Bemerkungen zur Theorie der abzählbaren Abelschen Gruppen. Math. Z. 6, 274–280 (1920).Google Scholar
  8. [8]
    SCOTT, W.R.: Group Theory. Englewood Cliffs, N.J.: Prentice Hall, Inc. 1964.Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • Edmund Hlawka
    • 1
  • Harald Niederreiter
    • 1
  1. 1.Mathematisches Institut der Universität WienWien IX

Personalised recommendations