Optimal balancing ratio for consecutive intervals

Determine the optimal asymptotic lower bound for the limsup ratio between the largest and smallest lengths of blocks of r consecutive intervals generated by an arbitrary infinite sequence of points on the unit circle, for each integer r≥2.

Background

The paper studies sequences of points on the unit circle as successive break points of a circular stick. After n points have been placed, the points determine n+1 intervals, and the central balancing question concerns how comparable the lengths of consecutive blocks of r intervals can remain over time.

de Bruijn and Erdős established the universal lower bound 1+1/r for the limsup ratio of the largest to smallest r-block lengths. The paper constructs sequences for which the ratio is at most 1+c log(r)/r for sufficiently large n, thereby narrowing the possible asymptotic behavior but not determining the optimal bound for every r≥2.