Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem

2021/12/09 by Noah Kravitz · 2 citations
Decision Sciences · Mathematics · Psychology · #Game Theory and Applications #Conjecture #Combinatorics #Mathematics #Integer (computer science) #Loneliness #Geometry #Physics #Computer science #Psychology #Social psychology

paper · pdf · doi:10.5070/c61055383

published in Combinatorial Theory 1(0)

openalex publication_date 2021/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce a sharpened version of the well-known Lonely Runner Conjecture of Wills and Cusick. Given a real number \(x\), let \(\Vert x \Vert\) denote the distance from \(x\) to the nearest integer. For each set of positive integer speeds \(v₁, …, v_n\), we define the associated maximum loneliness to be ML(v₁, …, v_n)=max_{t ∈ ℝ}min_{1 ≤ i ≤ n} \Vert tv_i \Vert.The Lonely Runner Conjecture asserts that \(ML(v₁, …, v_n) ≥ 1/(n+1)\) for all choices of \(v₁, …, v_n\). We make the stronger conjecture that for each choice of \(v₁, …, v_n\), we have either \(ML(v₁, …, v_n)=s/(ns+1)\) for some \(s ∈ ℕ\) or \(ML(v₁, …, v_n) ≥ 1/n\). This view reflects a surprising underlying rigidity of the Lonely Runner Problem. Our main results are: confirming our stronger conjecture for \(n ≤ 3\); and confirming it for \(n=4\) and \(n=6\) in the case where one speed is much faster than the rest.Mathematics Subject Classifications: 11K60 (primary), 11J13, 11J71, 52C07

Citations

• Flows, View Obstructions, and the Lonely Runner
• View-obstruction problems, III
• Invisible runners in finite fields
• The Lonely Runner with Seven Runners
• View-obstruction: a shorter proof for 6 lonely runners
• On the chromatic number of circulant graphs
• Barely lonely runners and very lonely runners

Cited by

• Lonely runner conjecture [wikipedia]