Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture
This result advances a long-standing combinatorial number theory conjecture for a small number of runners, providing a computational proof for the case of 5 runners.
The authors prove the shifted Lonely Runner Conjecture for 5 runners and identify exactly three primitive tight instances, only two of which are tight for the original conjecture. The proof is computational, using a rephrasing in terms of covering radii of zonotopes and an upper bound on velocities.
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.