J Austral Math Soc Ser A 45 pp360--370, 1988.

The Three Gap Theorem (Steinhaus Conjecture)

Tony van Ravenstein

(Received 18 November 1986; revised 6 August 1987)

Abstract

This paper is concerned with the distribution of N points places consecutively around the circle by an angle of a. We offer a new proof of the Steinhaus Conjecture which states that, for all irrational a 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. Tha analysis leads to an interesting geometrical interpretation of the simple continued fraction expansion of a.

1980 AMS Subject Classification (1985 Revision): 10F40, 10F05

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Authors

Tony van Ravenstein: Department of Mathematics, The University of Wollongong, Wollongong, N.S.W. 2500, Australia.

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Last Modified: Wed Feb 19 10:27:48 2003