# Distribution of boundary points of expansion and application to the lonely runner conjecture

## TL;DR

This paper investigates the distribution of boundary points in expansion and applies findings to the lonely runner problem, establishing lower bounds on distances between runners under specific conditions.

## Contribution

It introduces new bounds on runner distances based on boundary point distribution, extending the understanding of the lonely runner conjecture for up to eight runners.

## Key findings

- Established a lower bound for runner distances involving a constant D(n).
- Proved that with up to eight runners, mutual distances exceed a specific constant multiple of π/7.
- Linked boundary point distribution to the lonely runner problem's distance constraints.

## Abstract

In this paper, we study the distribution of the 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 $$ ||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)\pi}{k-1} $$ for all $i=1,\ldots,k-1$ and where $1>\mathcal{D}(n)>0$ 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 around 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 $$ ||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{C\pi}{7} $$ for some constant $1>C>0$ for all $1\leq i\leq 7$.