Year: 2026
Type: preprint
Abstract: 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 $n\geq95$.
Source: arXiv (Cornell University)
Author: Poliakova, Daria
Institution: Universität Hamburg
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W7214009968
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