Hostname: page-component-76c49bb84f-vcmr7 Total loading time: 0 Render date: 2025-07-03T19:37:05.474Z Has data issue: false hasContentIssue false
  • English
  • Français

The Three Gap Theorem (Steinhaus Conjecture)

Published online by Cambridge University Press:  09 April 2009

Tony van Ravenstein
Tony van Ravenstein
Affiliation:
Department of Mathematics, The University of Wollongong, Wollongong, N.S.W. 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 is concerned with the distribution of N points placed consecutively around the circle by an angle of α. We offer a new proof of the Steinhaus Conjecture which states that, for all irrational α and all N, the points partition the circle into arcs or gaps of at least two, and at most three, different lengths. We then investigate the partitioning of a gap as more points are included on the circle. The analysis leads to an interesting geometrical interpretation of the simple continued fraction expansion of α.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 3 , December 1988 , pp. 360 - 370
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Chung, F. R. K. and Graham, R. L., ‘On the set of distances determined by the union of arithmetic progressions’, Ars Combinatoria I (1976), 57–76.Google Scholar
[2]Halton, J. H., ‘The distribution of the sequence {n §} (n = 0, 1, 2, …)’, Proc. Cambridge Phil. Soc. 71 (1965), 665–670.CrossRefGoogle Scholar
[3]Hartman, S., ‘Über die Abstände von Punkten n§ auf der Kreisperipherie,’ Ann. Soc. Polon. Math. 25 (1952), 110–114.Google Scholar
[4]Khintchine, A. Y. (translated by Wynn, P.), Continued fractions (P. Noordhoff Ltd., Groningen, 1963.)Google Scholar
[5]Knuth, D. E., The art of computer programming, Vol. 3, Ex. 6.4.10 (Addison-Wesley, Reading, Mass., 1968.)Google Scholar
[6]Liang, F. M., ‘A short proof of the 3d distance theorem,’ Discrete Math. 28 (1979), 325–326.CrossRefGoogle Scholar
[7]Slater, N. B., ‘The distribution of the integers N for which {θN} < φ,’ Proc. Cambridge. Philos. Soc. 46 (1950), 525–543.CrossRefGoogle Scholar
[8]Slater, N. B., ‘Gaps and steps for the sequence nθ mod 1,’ Proc. Cambridge Philos. Soc. 73 (1967), 1115–1122.CrossRefGoogle Scholar
[9]Sós, V. T., ‘On the theory of diophantine approximations. I,’ Acta Math. Acad. Sci. Hungar. 8 (1957), 461–472.CrossRefGoogle Scholar
[10]Sós, V. T., ‘On the distribution mod 1 of the sequence nα,’ Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1 (1958), 127–134.Google Scholar
[11]Świerckowski, S., ‘On successive settings of an arc on the circumference of a circle,’ Fund. Math. 46 (1958), 187–189.CrossRefGoogle Scholar
[12]Surányi, J., ‘Über dei Anordnung der Vielfachen einer reelen Zahl mod 1,’ Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1 (1958), 107–111.Google Scholar