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arXiv · 2506.13379

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

David Alcántara·Francisco Criado·Francisco Santos
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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.

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David Alcántara, Francisco Criado, Francisco Santos. 2025-06-16. Covering radii of $3$-zonotopes and the shifted Lonely Runner Conjecture. https://doi.org/10.1080/10586458.2026.2651085

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