Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-12-01T00:20:12.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Exceptional Sets in Uniform Distribution

Published online by Cambridge University Press:  20 January 2009

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let B be a measurable set of real numbers in (0,1) of Lebesgue measure |B| and let x1, …, xn be real. Then

denotes the number of j (1 ≦j≦n) for which the fractional part {xj}∈B. The discrepancy of x1, …, xn is

where the supremum is taken over all intervals I in [0,1].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 22 , Issue 2 , June 1979 , pp. 145 - 160
Copyright
Copyright © Edinburgh Mathematical Society 1979

References

REFERENCES

(1) Baker, A. and Schmidt, W. M., Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc. (3) 21 (1970), 1–11.CrossRefGoogle Scholar
(2) Baker, R. C., Slowly growing sequences and discrepancy modulo one, Ada Arith. 23 (1973), 279–293.Google Scholar
(3) Baker, R. C., Khinchin's conjecture and Marstrand's theorem, Mathematika 21 (1974), 248–260.CrossRefGoogle Scholar
(4) Baker, R. C., Dyadic methods in the measure theory of numbers, Trans. Amer. Math. Soc. 221 (1976), 419–432.CrossRefGoogle Scholar
(5) Behnke, H., Zur Theorie der diophantischen Approximationen, Abh. Math. Sent. Univ. Hamburg 3 (1924), 261–318.CrossRefGoogle Scholar
(6) Kahane, J-P. and Salem, R., Ensembles parfaits et series trigonometriques (Hermann, Paris, 1963).Google Scholar
(7) Kuipers, L. and Niederreiter, H., Uniform distribution of sequences (Wiley, New York, 1974).Google Scholar
(8) LeVeque, W. J., On the frequency of small fractional parts in certain real sequences III, J.fiir reine und angew. Math. 202 (1959), 215–220.Google Scholar
(9) Rogers, C. A., Hausdorff measures (Cambridge University Press, 1970).Google Scholar