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On a theorem of van der Corput on uniform distribution

Published online by Cambridge University Press:  24 October 2008

W. J. Coles
W. J. Coles
Affiliation:
King's CollegeCambridge University of UtahSalt Lake City Utah, U.S.A.
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Extract

Van der Corput has shown (2), using a general criterion of Weyl (1), that a necessary and sufficient condition, that a sequence of points Pn = (αn, βn) (n = 1, 2,…) in two-dimensional space be uniformly distributed modulo 1, is that for all pairs of integers (u, v) other than u = v = 0 the one-dimensional sequence (uαn + vβn) (n = 1, 2,…) is uniformly distributed modulo 1. The object of this paper is to give a quantitative form to the sufficiency part of this qualitative criterion.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 4 , October 1957 , pp. 781 - 789
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

(1)Weyl, H.Math. Ann. (1916), 313.CrossRefGoogle Scholar
(2)Van Der Corput, J. C.Acta Math., Stockh. 56 (1931), 373 and 59 (1932), 209.CrossRefGoogle Scholar
(3)Erdős, P. and Turán, P.Proc. Acad. Sci. Amst. 51 (1948), 1146, 1262 [ = Indag. Math. 10 (1948), 370, 406].Google Scholar