Random runners are very lonely

2011/02/22 by Sebastian Czerwiński, Czerwiński, Sebastian · 3 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1102.4464

openalex publication_date 2011/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that k runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least 1/k from all the other runners. We prove that, with probability tending to one, a much stronger statement holds for random sets in which the bound 1/k is replaced by \thinspace 1/2-ε . The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs.

Cited by

• Riesz products and the Lonely Runner Conjecture: A wider gap of loneliness
• Chromatic numbers of rank-two Abelian Cayley graphs
• Higher-dimensional generalization of Youngs' theorem and circular colorings

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