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Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem
 Author(s): Kravitz, Noah
 et al.
Abstract
We introduce a sharpened version of the wellknown 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
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