Barely lonely runners and very lonely runners

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 $v_1, \ldots, v_n$, we define the associated maximum loneliness to be $\text{ML}(v_1, \ldots, v_n)=\max_{t \in \mathbb{R}}\min_{1 \leq i \leq n} \Vert tv_i \Vert$. The Lonely Runner Conjecture asserts that $\text{ML}(v_1, \ldots, v_n) \geq 1/(n+1)$ for all choices of $v_1, \ldots, v_n$. If the Lonely Runner Conjecture is true, then the quantity $1/(n+1)$ is the best possible, for there are known equality cases with $\text{ML}(v_1, \ldots, v_n)=1/(n+1)$. A natural but (to our knowledge) hitherto unasked question is: If $v_1, \ldots, v_n$ satisfy the Lonely Runner Conjecture but are not an equality case, must $\text{ML}(v_1, \ldots, v_n)$ be uniformly bounded away from $1/(n+1)$? We conjecture that, contrary to what one might expect, this question has an affirmative answer that reflects an underlying rigidity of the problem. More precisely, we conjecture that for each choice of $v_1, \ldots, v_n$, we have either $\text{ML}(v_1, \ldots, v_n)=s/(ns+1)$ for some $s \in \mathbb{N}$ or $\text{ML}(v_1, \ldots, v_n) \geq 1/n$. Our main results are: confirming this 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. We also obtain a number of related results.