AI 中文总结
本文解决了 de Bruijn–Erdős 连续间隙问题，通过比较区间计数和有限前缀差异界，证明了最大与最小 r-跨度之比的下界，匹配已知上界并证实猜想。
AI 中文摘要
设 $(x_n)_{n\geq1}$ 为单位圆上一列互不相同的点。一个 $r$-跨度是由插入点确定的 $r$ 个连续间隙的总长度。记 $M_n^{(r)}$ 和 $m_n^{(r)}$ 分别为前 $n$ 次插入后最大和最小的 $r$-跨度。我们证明，对于每一个充分大的 $r$，\\[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{\log r}{100r}. \\] 这一结果与 Clément 和 Steinerberger 的上界在绝对常数意义下匹配，证明了 de Bruijn 和 Erdős 在 1949 年提出的一个猜想，并回答了 Brethouwer 的一个问题。证明方法比较不同时刻的区间计数，并对短区间内的点应用一个有限前缀差异界。
英文摘要
Let $(x_n)_{n\geq1}$ be a sequence of distinct points on the unit circle. An $r$-span is the total length of $r$ consecutive gaps determined by the inserted points. Write $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest $r$-spans after the first $n$ insertions. We prove that there is an absolute constant $c>0$ such that, for every sufficiently large $r$, \[ \limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr) \geq c\sqrt{\log r}, \qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr) \geq c\sqrt{\log r}, \] and \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{\log r}{100r}. \] Thus all three asymptotic conjectures made by de Bruijn and Erdős in 1949 are resolved. The ratio bound matches the upper bound of Clément and Steinerberger up to an absolute constant and answers a question of Brethouwer. The proofs compare interval counts at nearby times. Pointwise control leads to a one-dimensional sequence-discrepancy argument for the ratio, while averaged control and Halász's planar $L^1$ discrepancy theorem give the two one-sided conclusions.
CommentsThe earlier version only resolved one of the three limits conjectured by de Bruijn and Erdős
