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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
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Abstract

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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.

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

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