PreprintCC BY-NC-ND 4.0arXiv
Near optimal gap in the Lonely Runner Conjecture
Abstract
For a set V of n distinct positive integers, let ML(V)=maxt∈ℝ/ℤminv∈ V|vt|, where |x| is the distance from x to the nearest integer. We prove that there is an absolute constant c>0 such that for every n≥2 and every V, ML(V)>(1+c)/2n. The proof combines a refinement of the large GCD argument of Green and Walker with a new physical spillover inequality for interval multiplicities.
Related
BibTeX
@misc{hu2026near,
title = {{Near optimal gap in the Lonely Runner Conjecture}},
author = {Xiyu Hu and Yifan Jing},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23790},
eprint = {2610.23790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23790},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Near optimal gap in the Lonely Runner Conjecture
AU - Xiyu Hu
AU - Yifan Jing
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23790
UR - https://arxiv.org/abs/2610.23790
AB - For a set V of n distinct positive integers, let ML(V)=max_t∈ℝ/ℤmin_v∈ V|vt|, where |x| is the distance from x to the nearest integer. We prove that there is an absolute constant c>0 such that for every n≥2 and every V, ML(V)>(1+c)/2n. The proof combines a refinement of the large GCD argument of Green and Walker with a new physical spillover inequality for interval multiplicities.
ER -