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The three gap theorem (Steinhaus conjecture). (English) Zbl 0663.10039
Author’s abstract: “This paper is concerned with the distribution of N points placed consecutively around the circle by an angle of $$\alpha$$. We offer a new proof of the Steinhaus Conjecture which states that, for all irrational $$\alpha$$ 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 $$\alpha$$.”
Reviewer: P.Kiss

##### MSC:
 11J71 Distribution modulo one 11J04 Homogeneous approximation to one number 11B75 Other combinatorial number theory