A few more Lonely Runners

TL;DR

This paper explores the Lonely Runner Conjecture by analyzing the Lonely Runner polyhedron, identifying new instances satisfying the conjecture, and proposing a lattice point approach to prove it.

Contribution

It introduces new families of instances satisfying the conjecture and suggests that proving lattice points in superlattices suffices for the proof.

Findings

Identified new families of instances satisfying the conjecture

Proposed a lattice point approach involving superlattices

Revisited the polyhedral approach to the conjecture

Abstract

Lonely Runner Conjecture, proposed by J\"{o}rg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers $k$ and $n_1,n_2,\ldots,n_k$ there exists a positive real number $t$ such that the distance of $t\cdot n_j$ to the nearest integer is at least $\frac{1}{k+1}$, $\forall~~1\leq j\leq k$. In a recent article Beck, Hosten and Schymura described the Lonely Runner polyhedron and provided a polyhedral approach to identifying families of lonely runner instances. We revisit the Lonely Runner polyhedron and highlight some new families of instances satisfying the conjecture. In addition, we relax the sufficiency of existence of an integer point in the Lonely Runner polyhedron to prove the conjecture. Specifically, we propose that it suffices to show the existence of a lattice point of certain superlattices of the integer lattice in the Lonely Runner polyhedron.