Lonely runner conjecture

Unsolved problem in mathematics:

Is the lonely runner conjecture true for every number of runners?

(more unsolved problems in mathematics)

In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that $n$ runners, with constant speeds all distinct from one another, will each be lonely at some time—at least $1/n$ units away from all others.

The conjecture was first posed in 1967 by German mathematician Jörg M. Wills, in purely number-theoretic terms, and independently in 1974 by T. W. Cusick; its illustrative and now-popular formulation dates to 1998. The conjecture is known to be true for 7 runners or less, but remains unsolved in the general case. Implications of the conjecture include solutions to view obstruction problems and bounds on the chromatic number of certain graphs.

Formulation

Example of a case of the conjecture with n=6 runners. Runners colored black have not yet been lonely. The white arcs, of length 2/n, indicate that a runner is currently lonely. Yellow runners have experienced loneliness.

Consider $n$ runners on a circular track of unit length. At the initial time $t=0$, 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 at time $t$ if they are at a distance (measured along the circle) of at least $1/n$ 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, some simplifications are made. The runner to be lonely is stationary at 0 (with zero speed), and therefore $n-1$ other runners, with nonzero speeds, are considered. The moving runners may be further restricted to positive speeds only: by symmetry, runners with speeds $x$ and $-x$ have the same distance from 0 at all times. Proving the result for any stationary runner implies the general result for all runners, since she can be made stationary by subtracting her speed from all runners, leaving her with zero speed. The conjecture then states that, for any collection $v_{1},v_{2},...,v_{n-1}$ of positive, distinct speeds, there exists some time $t>0$ such that $${\frac {1}{n}}\leq \operatorname {frac} (v_{i}t)\leq 1-{\frac {1}{n}}\qquad (i=1,...,n-1),$$ where $\operatorname {frac} (x)$ denotes the fractional part of $x$. Interpreted visually, if the runners are running counterclockwise, the middle of the inequality is the distance from the origin to the $i$th runner at time $t$, measured counterclockwise. Wills' conjecture was part of his work in Diophantine approximation, the study of how closely fractions can approximate irrational numbers. His conjecture concerns how well multiple real numbers could be simultaneously approximated with a single denominator.

Implications

Squares of side length 1/3 placed at every half-integer coordinate in the positive quadrant obstruct any ray from the origin in that direction. Any smaller side length will leave small gaps.

Suppose $C$ is a n-hypercube in n-dimensional space ($n\geq 2$). Infinitely many copies of $C$, all scaled by some factor $\alpha$, are placed at points at nonnegative half-integer coordinates. A ray from the origin into the positive orthant (in other words, directed positively in every dimension) may either miss all of the copies of $C$ or hit at least one copy. Cusick (1973) made an independent formulation of the lonely runner conjecture in this context; the conjecture implies that there are gaps if and only if $\alpha <(n-1)/(n+1)$. For example, squares any smaller than $1/3$ in side length will leave gaps, as shown.

In graph theory, a distance graph $G$ on the set of integers, and using some finite set $D$ of positive integer distances, has an edge between $x,y\in \mathbb {Z}$ if and only if $|x-y|\in D$. For example, if $D=\{2\}$, every consecutive pair of even integers, and of odd integers, is adjacent, all together forming two connected components. A k-regular coloring of the integers with step $\lambda \in \mathbb {R}$ assigns to each integer $n$ one of $k$ colors based on the residue of $\lfloor \lambda n\rfloor$modulo $k$. For example, if $\lambda =0.5$, the coloring repeats every $2k$ integers and each pair of integers $2m,2m+1$ are the same color. The lonely runner conjecture implies that the minimum $k$ for which $G$ admits a proper k-regular coloring (i.e., each node is colored differently than its adjacencies) for some step value is at most $|D|+1$. For example, $(k,\lambda )=(2,0.5)$ generates a proper coloring on the distance graph generated by the example $D=\{2\}$. ($k$ is known as the regular chromatic number of $D$.)

Given a directed graph $G$, a nowhere-zero flow on $G$ associates a positive value $f(e)$ to each edge $e$, such that the flow outward from each node is equal to the flow inward. Suppose $f$ is further restricted to positive integers. The lonely runner conjecture implies that, if $f$ attains at most $k$ different values, then $G$ has a nowhere-zero flow with values only in $\{1,2,\ldots ,k\}$. This result was proven for $k\geq 5$ with separate methods, and because the smaller cases of the lonely runner conjecture are settled, the full theorem is proven.

Known results

For a given setup of runners, let the gap of loneliness $\delta$ be the smallest of the runners' maximum distances of loneliness, and $\delta _{n}$ denote the minimum $\delta$ across all setups with $n$ runners. The conjecture then asserts that $\delta _{n}\geq 1/n$, a bound which, if correct, cannot be improved. For example, if the runner to be lonely is stationary and speeds $v_{i}=i$ are chosen, then there is no time at which he is strictly more than $1/n$ units away from all others, showing that $\delta _{n}\leq 1/n$. Alternatively, this conclusion is a corollary of the Dirichlet approximation theorem. A simple lower bound $\delta _{n}\geq 1/2n$ may be obtained via a covering argument.

The conjecture can be reduced to restricting the runners' speeds to positive integers: If the conjecture is true for $n$ runners with integer speeds, it is true for $n$ numbers with real speeds.

Tighter bounds

Slight improvements on the lower bound $1/2n$ are known. Chen & Cusick (1999) showed for $n\geq 5$ that if $2n-5$ is prime, then $\delta _{n}\geq {\tfrac {1}{2n-5}}$, and if $4n-9$ is prime, then $\delta _{n}\geq {\tfrac {2}{4n-9}}$. Perarnau & Serra (2016) showed unconditionally for sufficiently large $n$ that $$\delta _{n}\geq {\frac {1}{2n-4+o(1)}}.$$

Tao (2018) proved the current best known asymptotic result: for sufficiently large $n$, $$\delta _{n}\geq {\frac {1}{2n}}+{\frac {c\log n}{n^{2}(\log \log n)^{2}}}$$ for some constant $c>0$. He also showed that the full conjecture is implied by proving the conjecture for integer speeds of size $n^{O(n^{2})}$ (see big O notation). This implication theoretically allows proving the conjecture for a given $n$ by checking a finite set of cases, but the number of cases grows too quickly to be practical.

The conjecture has been proven under specific assumptions on the runners' speeds. If $n\geq 33$ and $${\frac {v_{i+1}}{v_{i}}}\geq 1+{\frac {22\log(n-1)}{n-1}}\qquad (i=1,...,n-1),$$ then the minimum gap of loneliness is $1/n$. In other words, the conjecture holds true for $n\geq 33$ if the speeds grow quickly enough. A slightly stronger result for $n\geq 40$ only requires a similar assumption on the first $\lfloor n/21\rfloor$ speeds. In a similar fashion but unconditionally on $n$, the conjecture is true if $v_{i+1}/v_{i}\geq 2$.

For specific n

The conjecture is true for $n\leq 7$ runners. The proofs for $n\leq 3$ are elementary; the $n=4$ case was established in 1972. The $n=5$, $n=6$, and $n=7$ cases were settled in 1984, 2001 and 2008, respectively. The first proof for $n=5$ was computer-assisted. All have since been proved with elementary methods.

For some $n$, there exist sporadic examples with a maximum separation of $1/n$ besides the example of $v_{i}=i$ given above. For $n=5$, the only other example (up to shifts and scaling) is $\{0,1,3,4,7\}$; for $n=6$ the only example is $\{0,1,3,4,5,9\}$; and for $n=8$ the only example is $\{0,1,4,5,6,7,11,13\}$. All solutions for $n\leq 20$ reaching exactly $1/n$ in separation are known through exhaustive computer search, and there exists an explicit infinite family of extremal examples.

Kravitz (2021) formulated a sharper version of the conjecture that addresses near-equality cases. More specifically, he conjectures that for a given set of speeds $v_{i}$, either $\delta =s/(sn+1)$ for some positive integer $s$, or $\delta \geq 1/(n-1)$, where $\delta$ is that setup's gap of loneliness. He confirmed this conjecture for $n\leq 3$ and a few special cases.

Other results

A much stronger result exists for randomly chosen speeds: using the stationary-runner convention, if $n$ and $\varepsilon >0$ are fixed and $n-1$ runners with nonzero speeds are chosen uniformly at random from $\{1,2,\ldots ,k\}$, then $P(\delta \geq 1/2-\varepsilon )\to 1$ as $k\to \infty$. In other words, runners with random speeds are likely at some point to be "very lonely"—nearly $1/2$ units from the nearest other runner. The full conjecture is true if "loneliness" is replaced with "almost aloneness", meaning at most one other runner is within $1/n$ of a given runner. The conjecture has been generalized to an analog in algebraic function fields.

Notes and references

Works cited

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• Bienia, Wojciech; Goddyn, Luis; Gvozdjak, Pavol; Sebő, András; Tarsi, Michael (January 1998). "Flows, view obstructions, and the lonely runner". Journal of Combinatorial Theory, Series B. 72 (1): 1–9. doi:10.1006/jctb.1997.1770.
• Betke, U.; Wills, J. M. (1972). "Untere schranken für zwei diophantische approximations-funktionen". Monatshefte für Mathematik. 76 (3): 214. doi:10.1007/BF01322924. S2CID 122549668.
• Bohman, Tom; Holzman, Ron; Kleitman, Dan (February 2001). "Six lonely runners". The Electronic Journal of Combinatorics: R3–R3. doi:10.37236/1602.
• Chen, Yong-Gao; Cusick, T. W. (January 1999). "The view-obstruction problem for n-dimensional cubes". Journal of Number Theory. 74 (1): 126–133. doi:10.1006/jnth.1998.2309.
• Chow, Sam; Rimanić, Luka (January 2019). "Lonely runners in function fields". Mathematika. 65 (3): 677–701. doi:10.1112/S002557931900007X.
• Cusick, T. W. (1973). "View-obstruction problems". Aequationes Mathematicae. 9 (2–3): 165–170. doi:10.1007/BF01832623. S2CID 122050409.
• —— (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.
• ——; Pomerance, Carl (1984). "View-obstruction problems, III". Journal of Number Theory. 19 (2): 131–139. doi:10.1016/0022-314X(84)90097-0.
• Czerwiński, Sebastian (2012). "Random runners are very lonely". Journal of Combinatorial Theory, Series A. 119 (6): 1194–1199. arXiv:1102.4464. doi:10.1016/j.jcta.2012.02.002. S2CID 26415692.
• ——; Grytczuk, Jarosław (September 2008). "Invisible runners in finite fields". Information Processing Letters. 108 (2): 64–67. doi:10.1016/j.ipl.2008.03.019.
• Czerwiński, Sebastian (May 2018). "The lonely runner problem for lacunary sequences". Discrete Mathematics. 341 (5): 1301–1306. doi:10.1016/j.disc.2018.02.002.
• Dubickas, A. (2011). "The lonely runner problem for many runners". Glasnik Matematicki. 46: 25–30. doi:10.3336/gm.46.1.05.
• Perarnau, Guillem; Serra, Oriol (March 2016). "Correlation among runners and some results on the lonely runner conjecture". The Electronic Journal of Combinatorics. 23 (1): P1.50. doi:10.37236/5123.
• Goddyn, L.; Wong, Erick B. (2006). "Tight instances of the lonely runner" (PDF). Integers. 6 (A38). Retrieved 1 May 2022.
• Kravitz, N. (2021). "Barely lonely runners and very lonely runners: a refined approach to the Lonely Runner Problem". Combinatorial Theory. 1. arXiv:1912.06034. doi:10.5070/C61055383.
• Renault, J. (2004). "View-obstruction: A shorter proof for 6 lonely runners". Discrete Mathematics. 287 (1–3): 93–101. doi:10.1016/j.disc.2004.06.008.
• Tao, Terence (31 December 2018). "Some remarks on the lonely runner conjecture". Contributions to Discrete Mathematics. 13: No 2 (2018). doi:10.11575/cdm.v13i2.62728.
• Wills, Jörg M. (1967). "Zwei sätze über inhomogene diophantische approximation von irrationalzehlen". Monatshefte für Mathematik. 71 (3): 263–269. doi:10.1007/BF01298332.

External links

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