An Improved Lower Bound for the de Bruijn--Erdős Consecutive Gap Problem

Let $(x_n{)}_{n\ge 1}$ 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 $r\ge 1$, 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 $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{M_n^{(r)}}{m_n^{(r)}}\ge 1+\frac{1}{r}\mathrm{.}$$ The case $r=1$ is sharp and gives the classical factor $2$. The cases $r\ge 2$ remain much less understood. We prove the improved lower bound $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{M_n^{(r)}}{m_n^{(r)}}\ge 1+\frac{r}{r^2-1}(r\ge 2)\mathrm{.}$$ In particular, for two consecutive intervals the lower bound becomes $5\mathrm{/}3$, improving the de Bruijn–Erdős bound $3\mathrm{/}2$.