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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.30959 |
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Table of Contents:
- Let $(x_n)_{n\geq 1}$ be a sequence of distinct points on the unit circle. After the first $n$ points are inserted, the circle is divided into $n$ intervals. For a fixed integer $r\geq 1$, let $M_n^{(r)}$ and $m_n^{(r)}$ denote respectively the largest and smallest total lengths of $r$ consecutive intervals. A theorem of de Bruijn and Erdős gives \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}}\geq 1+\frac1r . \] The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\geq 2$ remain much less understood. We prove the improved lower bound \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{r}{r^2-1} \qquad (r\geq 2). \] In particular, for two consecutive intervals the lower bound becomes $5/3$, improving the de Bruijn--Erdős bound $3/2$.