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


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