Lonely runner conjecture

In number theory, and especially the study of diophantine approximation, the lonely runner conjecture is a conjecture originally due to J. M. Wills in 1967. Applications of the conjecture are widespread in mathematics; they include view obstruction problems [cite journal
author = T. W. Cusick
title = View-Obstruction problems
journal = Aequationes Math.
volume = 9
pages = 165-170
year = 1973 ] and calculating the chromatic number of distance graphs and circulant graphs [cite journal
author = J. Barajas and O. Serra
title = The lonely runner with seven runners
journal = The Electronic Journal of Combinatorics
volume = 15
pages = R48
year = 2008 ] . The conjecture was given its picturesque name by L. Goddyn in 1998 [cite journal
author = W. Bienia and others
title = Flows, view obstructions, and the lonely runner problem
journal = Journal of combinatorial theory series B
volume = 72
pages = 1-9
year = 1998] .

The conjecture

Consider "k" + 1 runners on a circular track of unit length. At "t" = 0, all runners are at the same position and start to run; the runners' speeds are pairwise distinct. A runner is said to be "lonely" if she is at distance of at least 1/("k" + 1) from each other runner. The lonely runner conjecture states that every runner gets lonely at some time.

A convenient reformulation of the problem is to assume that the runners have integer speeds, not all divisible by the same prime; the runner to be lonely has zero speed. The conjecture then states that for any set "D" of "k" positive integers with gcd 1, there exists a real "t" such that

: $forall\; din\; Dquad\; ||td||\; >\; frac\{1\}\{k+1\},$

where ||"x"|| denotes the distance of real number "x" to the nearest integer.

Known results

Notes