Problems on Track Runners

Abstract

Consider the circle $C$ of length 1 and a circular arc $A$ of length $\ell \in (0,1)$. It is shown that there exists $k=k(\ell )\in \mathbb{N}$, and a schedule for $k$ runners along the circle with $k$ constant but distinct positive speeds so that at any time $t\ge 0$, at least one of the $k$ runners is not in $A$. On the other hand, we show the following: Assume that $k$ runners $1,2,\dots ,k$, with constant rationally independent (thus distinct) speeds ${\xi }_1,{\xi }_2,\dots ,{\xi }_k$, run clockwise along a circle of length $1$, starting from arbitrary points. For every circular arc $A\subset C$ and for every $T>0$, there exists $t>T$ such that all runners are in $A$ at time $t$. Several other problems of a similar nature are investigated.

9 pages, 1 figure