  In this paper we study the distribution of boundary points of expansion. As
an application, we say something about the lonely runner problem. We show that
given $k$ runners $\mathcal{S}_i$ round a unit circular track with the
condition that at some time
$||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$
for all $i=1,2\ldots,k-2$, then at that time we have
\begin{align}||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)\pi}{k-1}\nonumber
\end{align}for all $i=1,\ldots, k-1$ and where $\mathcal{D}(n)>0$ is a constant
depending on the degree of a certain polynomial of degree $n$. In particular,
we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners
running round a unit circular track with distinct constant speed and the
additional condition
$||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$
for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual
distance must satisfy the lower
bound\begin{align}||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{\pi}{7C\sqrt{3}}\nonumber
\end{align}for some constant $C>0$ for all $1\leq i \leq 7$.
