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[1] and calculating the chromatic number of distance graphs and circulant graphs.[2] The conjecture was given its picturesque name by L. Goddyn in 1998.[3]
Formulation
Consider k 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 at time t if they are at a distance of at least 1/k from every other runner at time t. The lonely runner conjecture states that each runner is lonely at some time.
A convenient reformulation of the conjecture is to assume that the runners have integer speeds,[4] 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 − 1 positive integers with greatest common divisor 1,
where ||x|| denotes the distance of real number x to the nearest integer.
Known results
| k | year proved | proved by | notes |
|---|---|---|---|
| 1 | - | - | trivial: ; any |
| 2 | - | - | trivial: |
| 3 | - | - | trivial: with zero speed of the runner to be lonely, it happens within the first round of the slower runner |
| 4 | 1972 | Betke and Wills;[5] Cusick[6] | - |
| 5 | 1984 | Cusick and Pomerance;[7] Bienia et al.[3] | - |
| 6 | 2001 | Bohman, Holzman, Kleitman;[8] Renault[9] | - |
| 7 | 2008 | Barajas and Serra[2] | - |
Dubickas[10] shows that for a sufficiently large number of runners for speeds the lonely runner conjecture is true if .
Czerwiński[11] shows that, with probability tending to one, a much stronger statement holds for random sets in which the bound is replaced by .
Notes
- ^ T. W. Cusick (1973). "View-Obstruction problems". Aequationes Mathematicae. 9 (2–3): 165–170. doi:10.1007/BF01832623.
- ^ a b J. Barajas and O. Serra (2008). "The lonely runner with seven runners". Electronic Journal of Combinatorics. 15: R48.
- ^ a b W. Bienia; et al. (1998). "Flows, view obstructions, and the lonely runner problem". Journal of Combinatorial Theory, Series B. 72: 1–9. doi:10.1006/jctb.1997.1770.
- ^ This reduction is proved, for example, in section 4 of the paper by Bohman, Holzman, Kleitman.
- ^ Betke, U.; Wills, J. M. (1972). "Untere Schranken für zwei diophantische Approximations-Funktionen". Monatshefte für Mathematik. 76 (3): 214. doi:10.1007/BF01322924.
- ^ T. W. Cusick (1974). "View-obstruction problems in n-dimensional geometry". Journal of Combinatorial Theory, Series A. 16 (1): 1–11. doi:10.1016/0097-3165(74)90066-1.
- ^ Cusick, T.W.; Pomerance, Carl (1984). "View-obstruction problems, III". Journal of Number Theory. 19 (2): 131–139. doi:10.1016/0022-314X(84)90097-0.
- ^ Bohman, T.; Holzman, R.; Kleitman, D. (2001), "Six lonely runners", Electronic Journal of Combinatorics, 8 (2)
- ^ Renault, J. (2004). "View-obstruction: A shorter proof for 6 lonely runners". Discrete Mathematics. 287: 93–101. doi:10.1016/j.disc.2004.06.008.
- ^ Dubickas, A. (2011). "The lonely runner problem for many runners". Glasnik Matematicki. 46: 25–30. doi:10.3336/gm.46.1.05.
- ^ Czerwiński, S. (2012). "Random runners are very lonely". Journal of Combinatorial Theory, Series A. 119: 1194–1199. arXiv:1102.4464. doi:10.1016/j.jcta.2012.02.002.