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On a metrical theorem of Weyl

Published online by Cambridge University Press:  26 February 2010

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham Hill. Egham, TW20 OEX, Surrey.
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Extract

The following theorem appears in Weyl's famous memoir [3] of 1916.

THEOREM A. Let λ1 ≤ λ2 ≤ … ≤ λn ≤ … be an increasing sequence of positive integers. Suppose that, of the numbers λ1, … λn, the first h1 are equal to each other, then the following h2 and so on, and finally that the last hm coincide. Let hj. If

the sequenceis uniformly distributed (mod 1)for almost all x, in the Lebesgue sense.

Type
Research Article
Information
Mathematika , Volume 22 , Issue 1 , June 1975 , pp. 29 - 33
Copyright
Copyright © University College London 1975

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References

1. Chung, K. L.. A course in probability theory (Harcourt, Brace and World. New York, 1968).Google Scholar
2. Davenport, H., Erdös, P. and LeVeque, W. J.. “On Weyl's criterion for uniform distribution” Michigan Math. J., 10 (1963), 311–314.CrossRefGoogle Scholar
3. Weyl, H.. “Über die Gleichverteilung von Zahlen mod Eins”, Math. Ann., 77 (1916), 313–352.CrossRefGoogle Scholar
4. Marstrand, J. M.. “On Khinchin's conjecture about strong uniform distribution”, Proc. Lond. Math. Soc, 21 (1970), 540–56.CrossRefGoogle Scholar