Combinatorial Theory

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 $$\mathrm{ML}(v_1,\dots ,v_n)=\underset{t\in \mathbb{R}}{max}\underset{1\le i\le n}{min}\Vert t{v}_i\Vert .$$

The Lonely Runner Conjecture asserts that $\mathrm{ML}(v_1,\dots ,v_n)\ge 1/(n+1)$ 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 $\mathrm{ML}(v_1,\dots ,v_n)=s/(ns+1)$ for some $s\in \mathbb{N}$ or $\mathrm{ML}(v_1,\dots ,v_n)\ge 1/n$. This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for $n\le 3$; 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