In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely units away from all others.

The conjecture was first posed in 1967 by German mathematician runners or fewer, but the general case remains unsolved. Implications of the conjecture include solutions to view-obstruction problems and bounds on properties, related to chromatic numbers, of certain graphs.

Formulation

, all runners are at the same position and start to run; the runners' speeds are constant, all distinct, and may be negative. A runner is said to be lonely from every other runner. The lonely runner conjecture states that each runner is lonely at some time, no matter the choice of speeds.

This visual formulation of the conjecture was first published in 1998. In many formulations, including the original by Jörg M. Wills, other runners, with nonzero speeds, are considered. The moving runners may be further restricted to positive denotes the fractional part., measured counterclockwise. This convention is used for the rest of this article.

Wills' conjecture was part of his work in Diophantine approximation, the study of how closely fractions can approximate irrational numbers: Dirichlet's approximation theorem positive integers, and the lonely runner conjecture states that it cannot.

Implications

is a -hypercube in at every point with half-integer, in which case there is a (infinitesimal) gap, or hit at least one copy. , ignoring rays lying in one of the coordinate hyperplanes. or greater will obstruct every ray that is not parallel to an axis. The conjecture generalizes this observation into any number of dimensions.

In graph theory, every consecutive pair of even integers, and of odd integers, is adjacent, all together forming two connected componentss. A -regular coloring colors based on the residue admits a proper -regular coloring (i.e., each node is colored differently than its adjacencies) for some step value. is known as the regular chromatic number.)

Given a directed graph, a nowhere-zero flow with separate methods, and because the smaller cases of the lonely runner conjecture are settled, the full theorem is proven.

Known results

denote the smallest of the runners' maximum distances of loneliness, and the gap of loneliness. Alternatively, this conclusion can be quickly derived from the Dirichlet approximation theorem may be obtained via a probability argument.

runners with real speeds.

Tighter bounds

are known. .

(see big O notation). . by checking a finite set of cases, but the number of cases grows too quickly to be practical.

...

For specific n

case was established in 1972. have since been proved with elementary methods. Building upon results concerning "finite checking" by which were later improved by , case. This method was subsequently and independently extended by runners, by runners, and by runners.

given above.. There exists an explicit infinite family of such sporadic cases.

, and a few special cases.

from the others during the first two rounds of the slowest non-static runner (but not necessarily during the first round).

Other results

. In other words, runners with random units from the nearest other runner. of a given runner. The conjecture has been generalized to an analog in algebraic function fields.

modelled the conjecture through a polyhedron-dimensional space, the lonely runner polyhedron is with distinct positive integer entries.

Notes and references

Notes

Citations

Works cited

External links

• Article in the Open Problem Garden no. 4, 551–562.