Balanced Stick Breaking

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 $(x_k)_{k=1}^{\infty}$ on the unit circle $\mathbb{S}^1$. We may interpret the first $n$ elements $(x_k)_{k=1}^{n}$ as places where the `circular stick' $\mathbb{S}^1$ is broken into a total of $n+1$ 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 $r$ consecutive intervals and prove $$ \frac{\max \mbox{length of}~r~\mbox{consecutive intervals}}{\min \mbox{length of}~r~\mbox{consecutive intervals}} \geq 1 + \frac{1}{r}.$$ We prove that this ratio can be as small as $1 + c \log{r}/ r$. This is done by means of refined discrepancy estimates for the van der Corput sequence over very short intervals and proves a conjecture of Brethouwer.