Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants

TL;DR

This paper improves the bounds on velocities needed to verify the Lonely Runner Conjecture and its variants, reducing the complexity of finite checks and extending results to higher dimensions and generalized settings.

Contribution

It drastically improves velocity bounds for checking the conjecture, introduces a zonotopal reinterpretation, and extends results to generalized and higher-dimensional cases.

Findings

Velocities up to n^{2n} suffice for the LRC with n+1 runners.

Finite-checking bounds are established for the shifted LRC and zonotope generalizations.

In dimension two, the conjecture is verified for four runners; in dimension three, for five runners with velocities summing to 195.

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 $n^{O(n^2)}$. 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 $\binom{n+1}{2}^{n-1} \le n^{2n}$ are enough. We prove the same finite-checking result, with the same bound, for the more general \emph{shifted} Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the \emph{Lonely Vector Problem} (LVP), about sumsets of $n$ rational 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 $195$.