The Lonely Runner Conjecture turns 60

TL;DR

This survey reviews the 60-year history of the Lonely Runner Conjecture, highlighting progress, tools, and connections despite it remaining unproven, and provides a comprehensive overview of related research developments.

Contribution

It offers a panoramic overview of the conjecture's development, emphasizing key contributions, methods, and the evolving landscape of related mathematical problems.

Findings

Steady progress with partial results and new tools

Connections to various problems and applications

The conjecture remains open after 60 years

Abstract

The Lonely Runner Conjecture originated in Diophantine approximation is turning 60. Even if the conjecture is still widely open, the flow of partial results, innovative tools and connections to different problems and applications has been steady on its long life. This survey attempts to give a panoramic view of the status of the problem, trying to highlight the contributions of the many papers that it has originated.