Lonely Runner Conjecture (original version)

Prove that for every integer n ≥ 1, any index i ∈ {1,…,n}, and any set of pairwise distinct real speeds v1,…,vn, there exists a real time t such that the distance of t(vj − vi) to the nearest integer is at least 1/n for all j ≠ i. Equivalently, show that for each runner i among n runners on the unit circle with distinct constant speeds, there is a time at which runner i is at distance at least 1/n from every other runner.

Background

The Lonely Runner Conjecture, originating in Diophantine approximation and geometric view–obstruction problems, asserts a universal lower bound on the isolation distance each runner must eventually achieve. The survey adopts two standard formulations: the original with distinct speeds and a reduced form omitting a stationary runner, both equivalent in intent.

The conjecture remains widely open and has spurred numerous equivalent formulations (e.g., in terms of Diophantine gaps, view–obstruction parameters, and covering radii of zonotopes), partial results for small n, and many specialized cases.