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Solution of a problem of Knuth on complete uniform distribution of sequences

  • Harald Niederreiter (a1) and Robert F. Tichy (a2)

Abstract

It is proved that the sequence is completely uniformly distributed modulo 1 for almost all real numbers x with |x|> 1, if (an) is an arbitrary sequence of distinct positive integers.

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References

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1.Franklin, J. N.. Deterministic simulation of random processes. Math. Comp., 17 (1963), 2859.
2.Hlawka, E.. Theorie der Gleichverteilung (Bibliographisches Inst., Mannheim-Wien-Zürich, 1979).
3.Knuth, D. E.. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms (Addison-Wesley, Reading, Mass., 1969, 2nd ed., 1981).
4.Koksma, J. F.. Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins. Compositio Math., 2 (1935), 250258.
5.Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (John Wiley and Sons, New York, 1974).
6.Natanson, I. P.. Konstruktive Funktionentheorie (Akademie-Verlag, Berlin, 1955).
7.Weyl, H.. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77 (1916), 313352.
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