Correlation Among Runners and Some Results on the Lonely Runner Conjecture

2016/03/18 by Guillem Perarnau, Oriol Serra · 2 citations
Mathematics · Psychology · #Mathematical Dynamics and Fractals #Advanced Topology and Set Theory #Limits and Structures in Graph Theory #Conjecture #Lonely runner conjecture #Moment (physics) #Mathematics #Point (geometry) #Diophantine equation #Set (abstract data type) #Interval (graph theory) #Constant (computer programming) #Loneliness #Unit circle #Combinatorics #Computer science #Collatz conjecture #Psychology #Geometry #Physics #Social psychology #Classical mechanics

paper · pdf · doi:10.37236/5123

published in The Electronic Journal of Combinatorics 23(1) (Electronic Journal of Combinatorics)

openalex publication_date 2016/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

The Lonely Runner Conjecture, posed independently by Wills and by Cusick, states that for any set of runners running along the unit circle with constant different speeds and starting at the same point, there is a time where all of them are far enough from the origin. We study the correlation among the time that runners spend close to the origin. By means of these correlations, we improve a result of Chen on the gap of loneliness. In the last part, we introduce dynamic interval graphs to deal with a weak version of the conjecture thus providing a new result related to the invisible runner theorem of Czerwinski and Grytczuk.

Citations

• A Solution to the Lonely Runner Conjecture for Almost All Points
• Flows, View Obstructions, and the Lonely Runner
• Random runners are very lonely
• View-obstruction problems
• Untere Schranken f�r zwei diophantische Approximations-Funktionen
• The Lonely Runner with Seven Runners
• Six Lonely Runners
• The lonely runner problem for many runners

Cited by

• Lonely runner conjecture [wikipedia]