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Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants

Romanos Diogenes Malikiosis, Francisco Santos, Matthias Schymura

TL;DR

This work casts the Lonely Runner Conjecture (LRC) and its shifted variant (sLRC) into a lattice-zonotope framework and proves a finite-checking reduction with a bound of $\sum_{S\subseteq[n]} v_S> \binom{n+1}{2}^{n-1}$, improving Tao’s $n^{O(n^2)}$ to roughly $n^{2n}$. It introduces the Lonely Vector Problem (LVP) and a cosimple zonotope approach, deriving dimension-dependent finite checks for sLRC and its generalizations without requiring LVP in every setting. In dimension two, the authors establish the cosimple-generalized sLRC and prove a complete 2D classification showing no counterexamples beyond the known $P_{2,5}$ obstruction, thereby supporting the 2D case of the conjectures. In dimension three, they prove a volume bound of 195 for potential counterexamples, with explicit width-dependent bounds (e.g., $w=4$ gives $\mathrm{vol}\le 195$, $w=3$ sectors bounded by $\le 80$ for sLRZ), enabling a computational path toward proving sLRC for five runners. Collectively, the results provide a robust geometric-route to finite verification of LRC variants via zonotope projections, covering radii, and Gale duality, sharpening the finite-computation approach and opening the door to full computational proofs of the conjectures in small dimensions and beyond.

Abstract

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to $n+1$ runners it suffices to consider positive integer velocities in the order of $n^{O(n^2)}$. Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to $\binom{n+1}{2}^{n-1} \le n^{2n}$ are enough. We prove the same finite-checking result, with the same bound, for the more general \emph{shifted} Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the \emph{Lonely Vector Problem} (LVP), about sumsets of $n$ rational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with $n$ generators, a class of lattice zonotopes that we introduce. In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to $195$.

Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants

TL;DR

This work casts the Lonely Runner Conjecture (LRC) and its shifted variant (sLRC) into a lattice-zonotope framework and proves a finite-checking reduction with a bound of , improving Tao’s to roughly . It introduces the Lonely Vector Problem (LVP) and a cosimple zonotope approach, deriving dimension-dependent finite checks for sLRC and its generalizations without requiring LVP in every setting. In dimension two, the authors establish the cosimple-generalized sLRC and prove a complete 2D classification showing no counterexamples beyond the known obstruction, thereby supporting the 2D case of the conjectures. In dimension three, they prove a volume bound of 195 for potential counterexamples, with explicit width-dependent bounds (e.g., gives , sectors bounded by for sLRZ), enabling a computational path toward proving sLRC for five runners. Collectively, the results provide a robust geometric-route to finite verification of LRC variants via zonotope projections, covering radii, and Gale duality, sharpening the finite-computation approach and opening the door to full computational proofs of the conjectures in small dimensions and beyond.

Abstract

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to runners it suffices to consider positive integer velocities in the order of . Using the zonotopal reinterpretation of the conjecture due to the first and third authors (2017) we here drastically improve this result, showing that velocities up to are enough. We prove the same finite-checking result, with the same bound, for the more general \emph{shifted} Lonely Runner Conjecture (sLRC), except in this case our result depends on the solution of a question, that we dub the \emph{Lonely Vector Problem} (LVP), about sumsets of rational vectors in dimension two. We also prove the same finite-checking bound for a further generalization of sLRC that concerns cosimple zonotopes with generators, a class of lattice zonotopes that we introduce. In the last sections we look at dimensions two and three. In dimension two we prove our generalized version of sLRC (hence we reprove the sLRC for four runners), and in dimension three we show that to prove sLRC for five runners it suffices to look at velocities adding up to .

Paper Structure

This paper contains 21 sections, 53 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. The Lonely Runner Conjectures
  3. Zonotopal restatements of LRC and sLRC
  4. Preliminaries
  5. Volume and lattice points in lattice zonotopes
  6. Concepts from the Geometry of Numbers
  7. Projections of Lonely Runner Zonotopes, and a finite checking result for LRC
  8. Projections of Lonely Runner Zonotopes
  9. Proof of Theorem \ref{['thm:main-points']}
  10. The Lonely Vector Problem, and a finite checking result for sLRC
  11. Proof of Theorem \ref{['thm:mainslrc-points']}
  12. Small cases of the LVP
  13. Cosimple configurations and a generalization of sLRC, with a finite checking result
  14. Proof of Theorem \ref{['thm:finite-checking-cosimple-conditional']}
  15. Remarks on cosimple zonotopes
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists an absolute and explicitly computable constant $C > 0$, such that the following assertions are equivalent for every natural number $n \geq 1$:

Figures (1)

  • Figure 3: The covering radius of the parallelogram $Q \cong P_{2,5}$ equals $3/5$: The left picture shows that $Q$ contains a translation of the square $[0,5/3]^2$; hence $\mu(P_{2,5}) \le 3/5$. For the equality, consider the right picture, where we scale down $Q$ by $3/5$ about its center, so that the axes-parallel square in it becomes a lattice unit square with its vertices in the boundary of $(3/5)Q$. Since any smaller dilation will fail to contain points from $\mathbb{Z}^2$, we have that $\mu \cdot P_{2,5} + \mathbb{Z}^{2}$ does not cover $\mathbb{R}^2$ for any $\mu < 3/5$

Theorems & Definitions (106)

  • Conjecture A: Lonely Runner Conjecture
  • Theorem 1.1: tao
  • Theorem A
  • Conjecture B: Shifted Lonely Runner Conjecture
  • Definition 1.2
  • Theorem B
  • Remark 1.3
  • Definition 1.4
  • Conjecture A': Equivalent to Conjecture \ref{['lrc']}
  • Proposition 1.5: zonorunners
  • ...and 96 more