AI 中文总结
受孤独跑步者猜想启发，研究跑步者速度时变问题。通过证明得出最慢和最快跑步者在某时刻与其他跑步者距离大于\(2^{-n + 1}\)且此距离最优，还构造例子说明中间跑步者情况，得到舍恩伯格定理的非线性类似物。
AI 中文摘要
受著名的孤独跑步者猜想的启发，我们研究了跑步者速度随时间变化的变体问题。设\(n \geq 3\)个跑步者从单位圆上的同一点出发，每个跑步者\(i\in[n]\)都有一个局部可积的速度函数\(\nu_i\in L^1_{\mathrm{loc}}(\mathbb{R}_{>0})\)。假设他们的速度几乎处处严格有序，且每对跑步者之间的相对距离发散。我们证明，最慢和最快的跑步者在某个时刻与其他每个跑步者的距离都严格大于\(2^{-n + 1}\)，并且\(2^{-n + 1}\)这个距离是最优的。另一方面，我们构造了一些例子，其中每个中间跑步者在所有时刻都与另一个跑步者任意接近。因此，我们还得到了关于单位立方体中台球运动的舍恩伯格经典定理的一个精确非线性类似物。
英文摘要
Motivated by the celebrated Lonely Runner Conjecture, we study a variant in which the runners have time-dependent velocities. Let $n \geq 3$ runners start from the same point on the unit circle, where each runner $i\in[n]$ has a locally integrable velocity function $ν_i\in L^1_{\mathrm{loc}}(\mathbb{R}_{>0})$. 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.
Comments15 pages
