Balanced Stick Breaking
Fran\c{c}ois Cl\'ement, Stefan Steinerberger

TL;DR
This paper investigates the ratios of lengths in infinite sequences of points on the circle, improving bounds on how balanced these segments can be, and confirms a conjecture using discrepancy estimates for the van der Corput sequence.
Contribution
It establishes a new bound on the ratio of maximum to minimum lengths of r consecutive intervals, refining previous results and confirming a conjecture.
Findings
The ratio can be as small as 1 + c log r / r.
Refined discrepancy estimates for the van der Corput sequence are used.
The work confirms a conjecture of Brethouwer.
Abstract
Consider an infinite sequence on the unit circle . We may interpret the first elements as places where the `circular stick' is broken into a total of pieces. It is clear that they cannot all be the same length all the time. de Bruijn and Erd\H{o}s (1949) show that the ratio of the largest to the smallest has to be arbitrarily close to 2 infinitely many times which is sharp. They also consider the problem of balancing the length of consecutive intervals and prove We prove that this ratio can be as small as . This is done by means of refined discrepancy estimates for the van der Corput sequence over very short intervals and proves a conjecture of…
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Taxonomy
TopicsMathematical Approximation and Integration · Analytic Number Theory Research · Markov Chains and Monte Carlo Methods
