Lonely runners in real life: Sharp bounds for time-dependent velocities

Abstract

Motivated by the celebrated Lonely Runner Conjecture, we study a variant in which the runners have time-dependent velocities. Let $n\ge 3$ runners start from the same point on the unit circle, where each runner $i\in [n]$ has a locally integrable velocity function $\hat{I}{½}_i\in L_{\mathrm{loc}}^1({\mathbb{R}}_{>0})$. Assume that their velocities are strictly ordered almost everywhere and that the relative distance between every pair diverges. We prove that each of the slowest and fastest runners is at a distance strictly larger than $2^{-n+1}$ from every other runner at some time. Moreover, we show that the distance $2^{-n+1}$ is optimal. On the other hand, we construct examples in which every intermediate runner remains arbitrarily close to another runner at all times. As a consequence, we also obtain a sharp nonlinear analogue of a classical theorem of Schoenberg on billiard ball motion in the unit cube.

15 pages