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More (shifted) runners, less loneliness
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+En) 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 En≥⌊ n/287⌋, which together with the elementary bound En≤ n−1 implies that En grows linearly in n. We also show that En≥1 for every n≥95.
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BibTeX
@misc{poliakova2026more,
title = {{More (shifted) runners, less loneliness}},
author = {Daria Poliakova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23952},
eprint = {2609.23952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23952},
}
RIS
TY - UNPB
TI - More (shifted) runners, less loneliness
AU - Daria Poliakova
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23952
UR - https://arxiv.org/abs/2609.23952
AB - 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≥⌊ n/287⌋, which together with the elementary bound E_n≤ n−1 implies that E_n grows linearly in n. We also show that E_n≥1 for every n≥95.
ER -