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

David Alc\'antara; Francisco Criado; Francisco Santos

arxiv: 2506.13379 · v1 · submitted 2025-06-16 · 🧮 math.CO · cs.CG

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

David Alc\'antara , Francisco Criado , Francisco Santos This is my paper

Pith reviewed 2026-05-19 09:39 UTC · model grok-4.3

classification 🧮 math.CO cs.CG

keywords lonely runner conjectureshifted lonely runnercovering radiizonotopescomputational enumerationlattice polytopes

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The pith

The shifted Lonely Runner Conjecture holds for five runners, with exactly three primitive tight instances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the shifted Lonely Runner Conjecture holds when there are five runners. The proof proceeds by translating the conjecture into a problem about the covering radii of particular zonotopes and then exhaustively checking integer velocities within a known bound. It further identifies all primitive tight instances, finding three in total of which two also satisfy the original non-shifted conjecture. A supporting algorithm is developed to compute upper bounds on covering radii for rational lattice polytopes by building dyadic fundamental domains.

Core claim

The shifted Lonely Runner Conjecture holds for 5 runners. There are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture. The proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes and on an upper bound for the integer velocities to be checked, together with a new algorithm for bounding the covering radius of rational lattice polytopes based on constructing dyadic fundamental domains.

What carries the argument

The central mechanism is the rephrasing of the shifted Lonely Runner Conjecture in terms of covering radii of zonotopes, along with an algorithm that bounds these radii for rational lattice polytopes using dyadic fundamental domains.

Load-bearing premise

The upper bound on integer velocities is both correct and tight enough that checking up to that bound finds all possible counterexamples.

What would settle it

A counterexample consisting of five velocities where the minimum distance to integers is less than 1/5 but with some velocity larger than the bound used would show the result is incomplete.

Figures

Figures reproduced from arXiv: 2506.13379 by David Alc\'antara, Francisco Criado, Francisco Santos.

**Figure 1.** Figure 1: A graphical version of the proof of Proposition 1.7. The positions of the runner with velocity 3 are represented by rhomboids so that each horizontal section represents a choice s3 ∈ [−3, 3] for its starting position. description of the last covered points of the corresponding sLR zonotope, which we include in Section 4.2. Compare with Remark 4.5. 2. Zonotopal statement of the Lonely Runner Conjecture A … view at source ↗

**Figure 2.** Figure 2: States of Algorithm 2 at different dyadic depths, applied to 1 2 Z where Z is the 2-dimensional sLR zonotope with volume vector (1, 2, 4). Notice that in the last step there are two choices for each voxel. (The figure takes D = ∞ for simplicity, although in general that could make the algorithm not terminate) [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: The three sLR zonotopes with µ(Z) = 3 5 with their last covered points drawn in red (segments) and blue (points). elements, depending on whether the last covered point lies in the interior of a segment or is the end-point of it. The projection of these zonotopes onto Z 2 in the direction parallel to their last covered segments is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Projections of the three tight sLR zonotopes scaled down by µ = 3 5 in the direction of their last covered segments. The last covered segments project onto the dots in red, while the isolated last covered points project onto the dots in blue. Remark 4.5. Last covered points, considered modulo Z 3 in the tiling 3 5 Z + Z 3 , correspond bijectively to tight starting positions (s1, s2, s3, s4), considered mod… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper computationally settles the five-runner case of the shifted Lonely Runner Conjecture and counts the primitive tight instances, using a new bounding algorithm plus an external velocity bound.

read the letter

The main result is a computational verification that the shifted Lonely Runner Conjecture holds for five runners, plus an exact count of three primitive tight instances, only two of which work for the original conjecture. They reach this by recasting the problem as a covering-radius question for 3-zonotopes and running an exhaustive search over integer velocities inside a supplied bound. As a supporting tool they introduce an algorithm that bounds covering radii of rational lattice polytopes by building dyadic fundamental domains. That algorithm is the clearest piece of new machinery and looks reusable for other lattice problems. The enumeration itself is also new; earlier papers had the geometric translation and the velocity bound but did not carry out this specific computation. The central limitation is the dependence on the external upper bound for velocities from the 2024+ Malikiosis-Santos-Schymura paper. The search is complete only if that bound is both correct and tight enough to catch every possible counterexample. The current paper invokes the bound without re-deriving or independently testing it, so any gap in the earlier work would propagate here. This is targeted work for people already following the Lonely Runner Conjecture, discrepancy questions, or covering-radius problems in zonotopes. A reader who wants concrete data points for the five-runner case or a practical method for similar polytopes will get direct value. The paper deserves peer review. The computational claims are checkable in principle, the new algorithm is worth a close look, and settling a specific case of a long-open conjecture is the sort of incremental progress that merits referee time even if the result remains case-specific.

Referee Report

1 major / 2 minor

Summary. The paper proves that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners via exhaustive computational enumeration of integer velocity tuples. It rephrases the problem in terms of covering radii of 3-zonotopes (following Henze and Malikiosis 2017), supplies a new deterministic algorithm for bounding covering radii of rational lattice polytopes by constructing dyadic fundamental domains, and invokes an external upper bound on velocities (Malikiosis, Santos and Schymura 2024+) to render the search finite. The paper further reports that there are exactly three primitive tight instances, only two of which are tight for the original (non-shifted) Lonely Runner Conjecture.

Significance. If the central computational claim holds, the work resolves sLRC for five runners and classifies all primitive tight instances, constituting a concrete advance on a long-standing conjecture. The newly devised algorithm for bounding covering radii via dyadic fundamental domains is a reusable methodological contribution that strengthens the deterministic character of the enumeration and avoids post-hoc parameter fitting.

major comments (1)

[section describing the enumeration and velocity bound] The completeness of the claim that sLRC holds for 5 runners rests on the external upper bound for integer velocities supplied by Malikiosis, Santos and Schymura (2024+). The manuscript invokes this bound to truncate the enumeration but does not re-derive it or supply an independent verification that every possible counterexample lies inside the searched range; this assumption is load-bearing for the exhaustiveness argument.

minor comments (2)

[algorithm section] The description of the dyadic-fundamental-domain algorithm would benefit from an explicit statement of its termination criterion and a small worked example on a low-dimensional zonotope.
[results on tight instances] Table or list of the three primitive tight instances should include the explicit velocity vectors and the corresponding covering-radius values for direct verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the significance of the results, and the recognition of the new algorithmic contribution. We address the major comment below.

read point-by-point responses

Referee: [section describing the enumeration and velocity bound] The completeness of the claim that sLRC holds for 5 runners rests on the external upper bound for integer velocities supplied by Malikiosis, Santos and Schymura (2024+). The manuscript invokes this bound to truncate the enumeration but does not re-derive it or supply an independent verification that every possible counterexample lies inside the searched range; this assumption is load-bearing for the exhaustiveness argument.

Authors: We agree that the exhaustiveness of the enumeration depends on the cited upper bound. This bound is a theorem established in the independent work of Malikiosis, Santos and Schymura (2024+), which we invoke to guarantee that the search over integer velocity tuples is finite. Our manuscript does not re-derive the bound, as that would duplicate material from the cited paper; instead, our focus is the new deterministic algorithm for bounding covering radii of rational lattice polytopes via dyadic fundamental domains and its application to the shifted Lonely Runner Conjecture. To address the referee's concern, we will add an explicit clarifying paragraph in the introduction (or the section on the enumeration) that states the reliance on the external result, summarizes its role, and directs the reader to the proof in Malikiosis et al. (2024+). This revision will make the logical structure of the exhaustiveness argument fully transparent while preserving the paper's scope. revision: partial

Circularity Check

1 steps flagged

Minor self-citation for velocity upper bound supporting computational completeness

specific steps

self citation load bearing [Abstract]
"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+)."

The claimed completeness of the enumeration (hence the proof that sLRC holds for 5 runners and the count of tight instances) depends on the cited upper bound being both correct and sufficiently tight; this bound is justified solely by citation to prior work sharing an author (Santos) with the present paper.

full rationale

The paper establishes its main result via exhaustive enumeration over a finite range of integer velocities, using a rephrasing of sLRC as a covering-radius question (cited to Henze-Malikiosis 2017, no author overlap) together with a new algorithm for computing covering radii of rational polytopes via dyadic fundamental domains. The only self-citation is the upper bound on velocities (Malikiosis-Santos-Schymura 2024+), which limits the search space but is not itself derived from or fitted to the target conjecture within this manuscript. No equation, ansatz, or uniqueness claim reduces the result to a quantity defined by the conjecture; the enumeration and new algorithm supply independent content. This is a normal supporting self-citation rather than a load-bearing loop that forces the conclusion by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the correctness of the external velocity bound, standard facts about zonotopes and covering radii, and the new dyadic-domain algorithm; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The upper bound on integer velocities from Malikiosis, Santos and Schymura (2024+) is sufficient to capture all possible counterexamples.
Invoked to limit the computational search space.
standard math Covering radius of a rational lattice polytope can be bounded by constructing dyadic fundamental domains.
Core of the new algorithm described in the abstract.

pith-pipeline@v0.9.0 · 5656 in / 1449 out tokens · 23805 ms · 2026-05-19T09:39:39.072641+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners... relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes... and on an upper bound for the (integer) velocities

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture
math.CO 2026-03 conditional novelty 8.0

Explicit counterexamples disprove the shifted Lonely Runner Conjecture for n=5 and the Lonely Vector Property for n=12 by introducing coloopless zonotopes.
Upper Bounds on Covering Minima of Convex Bodies
math.CO 2026-01 unverdicted novelty 6.0

Two new upper bounds on covering minima of convex bodies are given from projections and intersections, shown sharp for direct sums and applied to terminal simplices to narrow a 2017 conjecture gap.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 2 Pith papers

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Matthias Beck, Serkan Ho¸ sten, and Matthias Schymura. Lonely Runner Polyhedra. Integers, 19:#A29, 13 pp, 2019

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The covering radius and a discrete surface area for non-hollow simplices

Giulia Codenotti, Francisco Santos, and Matthias Schymura. The covering radius and a discrete surface area for non-hollow simplices. Discrete & Computational Geometry , 67(1):65–111, 2022. 14https://github.com/endorh/slrc-zonotopes/blob/main/check_certificates. py COVERING RADII AND THE SHIFTED LONELY RUNNER CONJECTURE 21

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Computing the covering radius of a polytope with an application to lonely runners

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Huangfu and J

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J¨ org M. Wills. Zur simultanen homogenen diophantischen Approximation. I. Monatsh. Math. , 72:254–263, 1968

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J¨ org M. Wills. Zur simultanen homogenen diophantischen Approximation. II. Monatsh. Math. , 72:368–381, 1968. (D. Alc´ antara, F. Santos) Departmento de Matem ´aticas, Estad´ıstica y Com- putaci´on, Universidad de Cantabria, Santander, Spain Email address : david.alcantara@unican.es, francisco.santos@unican.es (F. Criado) Departamento de Matem ´aticas, C...

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[1] [1]

The lonely runner with seven runners

Javier Barajas and Oriol Serra. The lonely runner with seven runners. Electron. J. Combin., 15:#48, 18 pp. (electronic), 2008

work page 2008

[2] [2]

Lonely Runner Polyhedra

Matthias Beck, Serkan Ho¸ sten, and Matthias Schymura. Lonely Runner Polyhedra. Integers, 19:#A29, 13 pp, 2019

work page 2019

[3] [3]

The covering radius and a discrete surface area for non-hollow simplices

Giulia Codenotti, Francisco Santos, and Matthias Schymura. The covering radius and a discrete surface area for non-hollow simplices. Discrete & Computational Geometry , 67(1):65–111, 2022. 14https://github.com/endorh/slrc-zonotopes/blob/main/check_certificates. py COVERING RADII AND THE SHIFTED LONELY RUNNER CONJECTURE 21

work page 2022

[4] [4]

Computing the covering radius of a polytope with an application to lonely runners

Jana Cslovjecsek, Romanos Diogenes Malikiosis, M´ arton Nasz´ odi, and Matthias Schy- mura. Computing the covering radius of a polytope with an application to lonely runners. Combinatorica, 42(4):463–490, February 2022

work page 2022

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Cusick and C

Thomas W. Cusick and C. Pomerance. View-obstruction problems III. J. Number Theory, 19:131–139, 1984

work page 1984

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On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture

Matthias Henze and Romanos Diogenes Malikiosis. On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture. Aequationes Math., 91(2):331–352, 2017

work page 2017

[7] [7]

Huangfu and J

Q. Huangfu and J. A. J. Hall. Parallelizing the dual revised simplex method. Mathe- matical Programming Computation, 10(1):119–142, December 2018

work page 2018

[8] [8]

Lattice translates of a polytope and the frobenius problem

Ravi Kannan. Lattice translates of a polytope and the frobenius problem. Combina- torica, 12(2):161–177, June 1992

work page 1992

[9] [9]

Linearly- exponential checking is enough for the lonely runner conjecture and some of its vari- ants, 2024

Romanos Diogenes Malikiosis, Francisco Santos, and Matthias Schymura. Linearly- exponential checking is enough for the lonely runner conjecture and some of its vari- ants, 2024. https://arxiv.org/abs/2411.06903

work page arXiv 2024

[10] [10]

The Lonely Runner Conjecture turns 60, 2024

Guillem Perarnau and Oriol Serra. The Lonely Runner Conjecture turns 60, 2024. https://arxiv.org/abs/2409.20160

work page arXiv 2024

[11] [11]

On the time for a runner to get lonely

Ludovic Rifford. On the time for a runner to get lonely. Acta Appl. Math. , 180:Paper No. 15, 66, 2022

work page 2022

[12] [12]

J¨ org M. Wills. Zur simultanen homogenen diophantischen Approximation. I. Monatsh. Math. , 72:254–263, 1968

work page 1968

[13] [13]

J¨ org M. Wills. Zur simultanen homogenen diophantischen Approximation. II. Monatsh. Math. , 72:368–381, 1968. (D. Alc´ antara, F. Santos) Departmento de Matem ´aticas, Estad´ıstica y Com- putaci´on, Universidad de Cantabria, Santander, Spain Email address : david.alcantara@unican.es, francisco.santos@unican.es (F. Criado) Departamento de Matem ´aticas, C...

work page 1968