Lonely runner conjecture

The lonely runner conjecture asserts that among finitely many runners moving at distinct constant speeds around a unit-length track, each runner is, at some time, at least the reciprocal of their number away from every other runner.

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Circular distance: The shorter of the two arc lengths between points on a circle. A runner is lonely when every relative position has circular distance at least the required threshold.
Uniform circular motion: Motion along a circular path at constant speed. The conjecture assumes every runner follows this simple motion model.
Jörg Wills: A German mathematician whose work included geometry of numbers and the lonely runner problem. Wills formulated the problem that became the conjecture.
Three-runners problem: The lonely runner conjecture restricted to three runners with distinct speeds. This case is proved and illustrates the required one-third separation.
Lonely runner graph: A graph encoding which subsets of runners can be simultaneously lonely under specified speed relations. Graph formulations organize the combinatorial constraints behind larger cases.
Modular arithmetic: Arithmetic in which numbers differing by a fixed modulus are treated as equivalent. Positions on a unit track are computed modulo one.
Relative velocity: The rate at which the separation between two moving objects changes. A runner’s position relative to another changes at the difference of their speeds.
Geometry of numbers: The study of lattice points and their relation to geometric sets. Wills’s question arose from geometric methods concerning lattices.
Four-runners problem: The lonely runner conjecture restricted to four runners with distinct speeds. This case is also proved, with threshold one quarter of the track.
Toric dynamics: The study of dynamical systems on spaces built from products of circles. The runners’ joint motion is a linear flow on a torus.