Eine Ungleichung von van der Corput und Kemperman

An inequality of van der Corput and Kemperman

Abstract

The tool of van der Corput's difference theorem in the theory of uniform distribution is his so-called fundamental inequality.Kemperman showed that even the non-constructive proofs of the difference theorem byBass, Bertrandias andCigler implicitly use a more general form of van der Corput's fundamental inequality. In this article, the inequality which constitutes the basis of the difference theorem will be proved under a very general setting, applications will be demonstrated in connection with the uniform distribution of products of linear forms and a quantitative version of the difference theorem, i. e. an estimation of discrepancies, will be derived.

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Literatur

  1. [1]

    Bass, J.: Suites uniformement denses, moyennes trigonométriques, fonctions pseudo-aléatoires. Bull. Soc. Math. France87, 1–64 (1959).

    Google Scholar 

  2. [2]

    Bass, J., Bertrandias, J.-P.: Moyennes de sommes trigonométriques et fonctions d'autocorrelation. C. R. Acad. Sci. Paris245, 2457–2459 (1957).

    Google Scholar 

  3. [3]

    Bertrandias, J.-P.: Suites pseudo-aléatoires et critères d'équirépartition modulo un. Compositio Math.16, 23–28 (1964).

    Google Scholar 

  4. [4]

    Cassels, J. W. S.: A new inequality with application to the theory of diophantine approximation. Math. Ann.126, 108–118 (1953).

    Google Scholar 

  5. [5]

    Cigler, J.: Asymptotische Verteilung reeller Zahlen mod 1. Monatsh. Math.64, 201–225 (1960).

    Google Scholar 

  6. [6]

    Cigler, J.: The fundamental theorem of van der Corput on uniform distribution and its generalizations. Compositio Math.16, 29–34 (1964).

    Google Scholar 

  7. [7]

    Cigler, J.: Über eine Verallgemeinerung des Hauptsatzes der Theorie der Gleichverteilung. J. reine angew. Math.210, 141–147 (1962).

    Google Scholar 

  8. [8]

    van der Corput, J. G.: Diophantische Ungleichungen I. Zur Gleichverteilung modulo Eins. Acta Math.56, 373–456 (1931).

    Google Scholar 

  9. [9]

    van der Corput, J. G.: Neue zahlentheoretische Abschätzungen. II. Math. Z.29, 397–426 (1929).

    Google Scholar 

  10. [10]

    van der Corput, J. G., Pisot, C.: Sur la discrépance modulo un. Indag. Math.1, 143–153, 184–195, 260–269 (1939).

    Google Scholar 

  11. [11]

    Erdös, P.: On a problem in the theory of uniform distribution. I, II. Indag. Math.10, 370–378, 406–413 (1948).

    Google Scholar 

  12. [12]

    Hiergeist, F. X.: Some generalizations and extensions of uniformly distributed sequences. Thesis, Univ. Pittsburgh. 1964.

  13. [13]

    Hlawka, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung. Ann. Mat. Pura Appl. (IV)54, 325–333 (1961).

    Google Scholar 

  14. [14]

    Hlawka, E.: Theorie der Gleichverteilung. Mannheim: Bibliographisches Institut. 1979.

    Google Scholar 

  15. [15]

    Hlawka, E.: ÜberC-Gleichverteilung. Ann. Mat. Pura Appl. (IV)49, 311–326 (1960).

    Google Scholar 

  16. [16]

    Hlawka, E.: Über die Diskrepanz mehrdimensionaler Folgen modulo 1. Math. Z.77, 273–284 (1961).

    Google Scholar 

  17. [17]

    Hlawka, E.: Zur formalen Theorie der Gleichverteilung in kompakten Gruppen. Rend. Circ. Mat. Palermo (2)4, 33–47 (1955).

    Google Scholar 

  18. [18]

    Kemperman, J. H. B.: Probability methods in the theory of distribution modulo one. Compositio Math.16, 106–137 (1964).

    Google Scholar 

  19. [19]

    Kemperman, J. H. B.: On the distribution of a sequence in a compact group. Compositio Math.16, 138–157 (1964).

    Google Scholar 

  20. [20]

    Koksma, J. F.: Ein algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1. Mathematika B (Zutphen)11, 7–11 (1942/43).

    Google Scholar 

  21. [21]

    Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. New York: J. Wiley & Sons. 1974.

    Google Scholar 

  22. [22]

    Niederreiter, H., Philipp, W.: On a theorem of Erdös and Turán on uniform distribution. Proc. Number Theory Conference (Boulder, Colo. 1972), pp. 180–182. Boulder: Univ. Colorado. 1972.

    Google Scholar 

  23. [23]

    Taschner, R. J.: Der Differenzensatz von van der Corput und gleichverteilte Funktionen. J. reine angew. Math.307/308, 325–340 (1979).

    Google Scholar 

  24. [24]

    Weyl, H.: Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann.77, 313–352 (1916).

    Google Scholar 

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Taschner, R.J. Eine Ungleichung von van der Corput und Kemperman. Monatshefte für Mathematik 91, 139–152 (1981). https://doi.org/10.1007/BF01295144

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