Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem

Abstract

We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number \(x\), let \(\Vert x \Vert\) denote the distance from \(x\) to the nearest integer. For each set of positive integer speeds \(v_1, \dots, v_n\), we define the associated maximum loneliness to be $$\operatorname{ML}(v_1, \dots, v_n)=\max_{t \in \mathbb{R}}\min_{1 \leq i \leq n} \Vert tv_i \Vert.$$The Lonely Runner Conjecture asserts that \(\operatorname{ML}(v_1, \dots, v_n) \geq 1/(n+1)\) for all choices of \(v_1, \dots, v_n\). We make the stronger conjecture that for each choice of \(v_1, \dots, v_n\), we have either \(\operatorname{ML}(v_1, \dots, v_n)=s/(ns+1)\) for some \(s \in \mathbb{N}\) or \(\operatorname{ML}(v_1, \dots, v_n) \geq 1/n\). This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for \(n \leq 3\); and confirming it for \(n=4\) and \(n=6\) in the case where one speed is much faster than the rest.Mathematics Subject Classifications: 11K60 (primary), 11J13, 11J71, 52C07