Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T04:37:32.393Z Has data issue: false hasContentIssue false
  • English
  • Français

On the discrepancy of the sequence formed from multiples of an irrational number

Published online by Cambridge University Press:  17 April 2009

Tony van Ravenstein
Tony van Ravenstein
Affiliation:
Department of Mathematics, University of Wollongong, Wollongon, New South Wales 2500, Australia.
Rights & Permissions [Opens in a new window]

Abstract

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.

This paper demonstrates a connection between two measures of discrepancy of sequences which arise in the theory of uniform distribution modulo one. The sequence formed from the non-negative integer multiples of an irrational number ξ is investigated and, by an application of the “Steinhaus Conjecture”, some values of the two discrepancies are obtained using continued fractions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 3 , June 1985 , pp. 329 - 338
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Halton, J.H., “The distribution of the sequence {nξ} (n = 0, 1, 2, …)“, Proc. Cambridge Philos. Soc. 61 (1965), 665–670.CrossRefGoogle Scholar
[2]Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers, 4th ed. (Clarendon Press, Oxford, 1960; reprinted 1971).Google Scholar
[3]Khintchine, A.Y., Continued fractions (translated by Wynn, P.. Noordhoff, Groningen, 1963).Google Scholar
[4]Kuipers, L. and Neiderreiter, H., Uniform distribution of sequences (John Wiley and Sons, New York, London, 1974).Google Scholar
[5]Révuz, A., “Sur la repartition des points eνiθ”, C.R. Acad. Sci. 228 (1949), 1966–1967.Google Scholar
[6]Slater, N.B., “Gaps and steps for the sequence nθ mod 1”, Proc. Cambridge Philos. Soc. 63 (1967), 1115–1122.CrossRefGoogle Scholar
[7]Sós, V.T., “On the theory of Diophantine approximations, I”, Acta Math. Acad. Sci. Hungar. 8 (1957), 461–472.CrossRefGoogle Scholar
[8]Sós, V.T., “On the distribution mod 1 of the sequence nα” Ann. Univ. Sci. Pudapest. Eötvös Sect. Math. 1 (1958), 127–134.Google Scholar
[9]Świerckowski, S., “On successive settings of an arc on the circumference of a circle”, Fund. Math. 46 (1958), 187–189.CrossRefGoogle Scholar