Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture

TL;DR

This paper proves the shifted Lonely Runner Conjecture for five runners using computational methods and zonotope covering radii, identifying unique tight instances and developing a new bounding algorithm.

Contribution

It establishes the shifted Lonely Runner Conjecture for five runners, characterizes primitive tight instances, and introduces an algorithm for bounding covering radii of rational lattice polytopes.

Findings

sLRC holds for 5 runners

Exactly 3 primitive tight instances identified

Developed an algorithm for bounding covering radii

Abstract

We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes (Henze and Malikiosis, 2017), and on an upper bound for the (integer) velocities to be checked (Malikiosis, Santos and Schymura, 2024+). As a tool for the proof, we devise an algorithm for bounding the covering radius of rational lattice polytopes, based on constructing dyadic fundamental domains.