Balanced Stick Breaking

TL;DR

This work studies how to balance $r$ consecutive intervals on the circle by refining de Bruijn–Erdős bounds. It proves a universal bound of $1 + c\log r/r$ for the ratio of the largest to smallest among any $r$ consecutive gaps and shows this rate is tight up to the logarithm, via two classical sequences. The authors then establish tight small-scale discrepancy results for the van der Corput sequence in base 2 and the Golden Ratio Kronecker sequence, using new tools such as an Ordering Lemma and a Two Gap Lemma rooted in binary and Fibonacci structure. The findings advance understanding of irregularity of distribution at small scales and have potential implications for quasi-MMC methods and related pair-correlation phenomena, while confirming the Brethouwer conjecture up to a logarithmic factor.

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 Erdős (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.

Figures (3)

• Figure 1: Iteratively breaking a stick.
• Figure 2: Setting $x_0 = 0$ and taking the first 660 elements of the van der Corput sequence, the plot shows $x_{i+53} - x_i$.
• Figure :

Theorems & Definitions (15)

• Theorem 1: de Bruijn and Erdős deb, 1949
• Theorem 2
• Theorem 3
• Lemma 1: Main Lemma
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof : Proof of Lemma \ref{['lem:induc']}