发表机构
University of Hamburg（汉堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对被否证的平移孤独跑者猜想，给出其失效的定量界限，证明孤独度下确界中的修正项 $E_n$ 线性增长，并证明 $n\geq95$ 时 $E_n\geq1$。
AI 中文摘要
最近，Blanco、Criado 和 Santos 否证了平移的孤独跑者猜想。我们给出了随着跑者数量增长，该猜想失效的定量界限。配置的孤独度定义为随时间变化，从原点到最近跑者的距离的最大值。我们记单位圆上具有不同正整数速度且任意初始平移的 $n$ 个跑者的配置的孤独度下确界为 $1/(n+1+E_n)$。我们证明了 $E_n\ge\lfloor n/287\rfloor$，结合基本界 $E_n\leq n-1$，这蕴含 $E_n$ 随 $n$ 线性增长。我们还证明了对于每个 $n\geq95$，有 $E_n\geq1$。
英文摘要
The shifted lonely runner conjecture was recently disproved by Blanco, Criado and Santos. We give quantitative bounds on its failure as the number of runners grows. The loneliness of a configuration is the maximum, over time, of the distance from the origin to the nearest runner. We write $1/(n+1+E_n)$ for the infimum of loneliness over configurations of $n$ runners on the unit circle with distinct positive integer velocities and arbitrary initial shifts. We prove $E_n\ge\lfloor n/287\rfloor$, which together with the elementary bound $E_n\leq n-1$ implies that $E_n$ grows linearly in $n$. We also show that $E_n\geq1$ for every $n\geq95$.
Comments7 pages
