@article{10902/37801,
year = {2025},
month = {10},
url = {https://hdl.handle.net/10902/37801},
abstract = {Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to n + 1 runners it suffices to consider positive integer velocities in the order of nO(n2). Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to n2+1n−1 ≤ n2n are enough. We prove the same finite-checking result, with the same bound, for the more general shifted Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the one VectorProem(LVP), about sumsets of nrational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with n generators, a class of lattice zonotopes that we introduce. In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to 195},
organization = {R. D. Malikiosis was granted a renewed research stay by the Alexander von Humboldt Foundation for the
completion of this project; R. D. Malikiosis also acknowledges that this project is carried out within the framework of the National Recovery and Resilience Plan Greece 2.0, funded by the European Union, NextGenerationEU (Implementation body: HFRI, Project Name: HANTADS, No. 14770).
Work of F. Santos is partially supported by grant PID2022-137283NB-C21 funded by MCIN/AEI/10.13039/501100011033.},
publisher = {Cambridge University Press},
publisher = {Forum of Mathematics, Sigma, 2025, 13(e164), 1-32},
title = {Linearly exponential checking is enough for the lonely runner conjecture and some of its variants},
author = {Malikiosis, Romanos Diogenes and Santos, Francisco and Schymura, Matthias},
}