Problems on Track Runners

TL;DR

This paper explores scheduling and coverage problems for runners on a circle, demonstrating conditions under which at least one runner avoids a given arc and when all runners can be simultaneously in an arc over time.

Contribution

It establishes new results on runner scheduling and coverage, including existence proofs for schedules and simultaneous coverage with rationally independent speeds.

Findings

Existence of a schedule with k runners avoiding a specific arc at all times.

For rationally independent speeds, runners will infinitely often all be in any arc after some time.

Several related coverage problems are also analyzed.

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 $t \geq 0$, at least one of the $k$ runners is not in $A$. On the other hand, we show the following: Assume that $k$ runners $1,2,\ldots,k$, with constant rationally independent (thus distinct) speeds $\xi_1,\xi_2,\ldots,\xi_k$, 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.