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

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