Lonely runner conjecture

File:Lonely runner.gif

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.{{sfn|Bohman|Holzman|Kleitman|2001|p=1}}

This visual formulation of the conjecture was first published in 1998.{{sfn|Bienia|Goddyn|Gvozdjak|Sebő|1998|p=3}} In many formulations, including the original by Jörg M. Wills,{{Sfnm|1a1=Wills|1y=1967|2a1=Bienia|2a2=Goddyn|2a3=Gvozdjak|2a4=Sebő|2y=1998}}{{sfn|Wills|1967}} 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.{{efn|Some authors use the convention that $n$ is the number of non-stationary runners, and thus the conjecture is that the gap of loneliness is at most $1/(n+1)$.{{sfn|Tao|2018}} }} 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, and so are essentially equivalent. Proving the result for any stationary runner implies the general result for all runners, since they can be made stationary by subtracting their speed from all runners, leaving them with zero speed. The conjecture then states that, for any collection $v\_1,v\_2,\backslash dots,v\_\{n-1\}$ of positive, distinct speeds, there exists some time $t>0$ such that

$$\backslash frac\{1\}\{n\}\backslash leq\; \backslash operatorname\{frac\}(v\_it)\backslash leq\; 1-\backslash frac\{1\}\{n\}\backslash qquad\; (i=1,\backslash dots,n-1),$$

where $\backslash operatorname\{frac\}(x)$ denotes the fractional part of $x$.{{sfn|Bohman|Holzman|Kleitman|2001|p=2}} Interpreted visually, if the runners are running counterclockwise, the middle term of the inequality is the distance from the origin to the $i$th runner at time $t$, measured counterclockwise.{{Efn|For example, if the origin is at a 6 o'clock position, a runner at the 9 o'clock position will have $\backslash operatorname\{frac\}(vt)=3/4$.}} This convention is used for the rest of this article. Wills' conjecture was part of his work in Diophantine approximation,{{sfnm|1a1=Wills|1y=1967|2a1=Betke|2a2=Wills|2y=1972}} the study of how closely fractions can approximate irrational numbers.

File:View-obstruction problem for squares.svg

Suppose $C$ is a {{mvar|n}}-hypercube of side length $s$ in {{mvar|n}}-dimensional space ($n\backslash geq2$). Place a centered copy of $C$ at every point with half-integer coordinates. A ray from the origin may either miss all of the copies of $C$, in which case there is a (infinitesimal) gap, or hit at least one copy. {{harvtxt|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 , ignoring rays lying in one of the coordinate hyperplanes.{{Sfn|Cusick|1974|p=1}} For example, placed in 2-dimensional space, squares any smaller than $1/3$ in side length will leave gaps, as shown, and squares with side length $1/3$ 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, a distance graph $G$ on the set of integers, and using some finite set $D$ of positive integer distances, has an edge between $x,y$ if and only if $|x-y|\backslash in\; D$. For example, if $D=\backslash \{2\backslash \}$, every consecutive pair of even integers, and of odd integers, is adjacent, all together forming two connected components. A {{mvar|k}}-regular coloring of the integers with step $\backslash lambda\backslash in\backslash mathbb\{R\}$ assigns to each integer $n$ one of $k$ colors based on the residue of $\backslash lfloor\; \backslash lambda\; n\backslash rfloor$modulo $k$. For example, if $\backslash lambda=0.5$, the coloring repeats every $2k$ integers and each pair of integers $2m,\; 2m+1$ are the same color. Taking $k=|D|+1$, the lonely runner conjecture implies $G$ admits a proper {{mvar|k}}-regular coloring (i.e., each node is colored differently than its adjacencies) for some step value.{{sfn|Barajas|Serra|2009|p=5688}} For example, $(k,\backslash lambda)=(2,0.5)$ generates a proper coloring on the distance graph generated by $D=\backslash \{2\backslash \}$. ($k$ 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. The lonely runner conjecture implies that, if $G$ has a nowhere-zero flow with at most $k$ distinct integer values, then $G$ has a nowhere-zero flow with values only in $\backslash \{1,2,\backslash ldots,k\backslash \}$ (possibly after reversing the directions of some arcs of $G$). This result was proven for $k\backslash geq\; 5$ with separate methods, and because the smaller cases of the lonely runner conjecture are settled, the full theorem is proven.{{sfn|Bienia|Goddyn|Gvozdjak|Sebő|1998}}

For a given setup of runners, let $\backslash delta$ denote the smallest of the runners' maximum distances of loneliness, and the gap of loneliness{{sfn|Perarnau|Serra|2016}} $\backslash delta\_n$ denote the minimum $\backslash delta$ across all setups with $n$ runners. In this notation, the conjecture asserts that $\backslash delta\_n\backslash geq\; 1/n$, 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 they are strictly more than $1/n$ units away from all others, showing that $\backslash delta\_n\; \backslash leq\; 1/n$.{{efn|Let the lonely runner be fixed at 0. For sake of contradiction, suppose there exists $t$ such that $\backslash \{v\_it\backslash \}\backslash in\; (1/n,\; 1-1/n)$ for all $i$. By the pigeonhole principle, there exist distinct $i$ and $j$ such that But $\backslash |v\_j-v\_i\backslash |=v\_k$ for some $k$, so either or $\backslash \{v\_kt\backslash \}=1-(\backslash \{v\_jt\backslash \}-\backslash \{v\_it\backslash \})>1-1/n$, a contradiction.{{sfn|Bohman|Holzman|Kleitman|2001|p=2}} }} Alternatively, this conclusion can be quickly derived from the Dirichlet approximation theorem. For $n\backslash geq\; 2$ a simple lower bound $\backslash delta\_n\backslash geq\; 1/(2n-2)$ may be obtained via a probability argument.{{sfn|Tao|2018|pp=2–3}}

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 $n$ runners with real speeds.{{sfn|Bohman|Holzman|Kleitman|2001|pp=12–13}}

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

$$\backslash delta\_n\backslash geq\; \backslash frac\{1\}\{2n-4+o(1)\}.$$

{{harvtxt|Tao|2018}} proved the current best known asymptotic result: for sufficiently large $n$,

$$\backslash delta\_n\backslash geq\; \backslash frac\{1\}\{2n-2\}+\backslash frac\{c\backslash log\; n\}\{n^2(\backslash log\backslash log\; n)^2\}$$

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.{{sfn|Czerwiński|2018|p=1302}}

The conjecture has been proven under specific assumptions on the runners' speeds. For sufficiently large $n$, it holds true if

$$\backslash frac\{v\_\{i+1\}\}\{v\_i\}\backslash geq\; 1\; +\; \backslash frac\{22\backslash log(n-1)\}\{n-1\}\; \backslash qquad\; (i=1,\backslash dots,n-2).$$

In other words, the conjecture holds true for large $n$ if the speeds grow quickly enough. If the constant 22 is replaced with 33, then the conjecture holds true for $n\backslash geq\; 16343$.{{sfn|Dubickas|2011|p=27}} A similar result for sufficiently large $n$ only requires a similar assumption for $i\; =\backslash lfloor\; n/22\; \backslash rfloor-1,\backslash dots,n-2$.{{sfn|Czerwiński|2018|p=1302}} Unconditionally on $n$, the conjecture is true if $v\_\{i+1\}/v\_i\backslash geq\; 2$ for all $i$.{{sfn|Barajas|Serra|2009}}

{{Anchor|For specific n}}

The conjecture is true for $n\backslash leq\; 7$ runners. The proofs for $n\backslash leq\; 3$ are elementary; the $n=4$ case was established in 1972.{{sfnm|1a1=Betke|1a2=Wills|1y=1972|1pp=215–216|2a1=Cusick|2y=1974|2p=5|ps=. Cusick's paper independently proves this result.}} 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, but all cases have since been proved with elementary methods.{{sfnm|1a1=Cusick|1a2=Pomerance|1y=1984|1p=133|2a1=Bohman|2a2=Holzman|2a3=Kleitman|2y=2001|3a1=Barajas|3a2=Serra|3y=2008a|4a1=Renault|4y=2004|ps=. Renault gives an elementary proof for $n=6$.}}

For some $n$, there exist sporadic examples with a maximum separation of $1/n$ besides the example of $v\_i=i$ given above.{{sfn|Bohman|Holzman|Kleitman|2001|p=2}} For $n=5$, the only known example (up to shifts and scaling) is $\backslash \{0,1,3,4,7\backslash \}$; for $n=6$ the only known example is $\backslash \{0,1,3,4,5,9\backslash \}$; and for $n=8$ the known examples are $\backslash \{0,1,4,5,6,7,11,13\backslash \}$ and $\backslash \{0,1,2,3,4,5,7,12\backslash \}$.{{sfn|Bohman|Holzman|Kleitman|2001|p=3}} There exists an explicit infinite family of such sporadic cases.{{sfn|Goddyn|Wong|2006}}

{{harvtxt|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 $\backslash delta\; =\; s/(s(n-1)+1)$ for some positive integer $s$,{{efn|Taking $s=1$ yields the lonely runner conjecture.}} or $\backslash delta\; \backslash geq\; 1/(n-1)$, where $\backslash delta$ is that setup's gap of loneliness. He confirmed this conjecture for $n\backslash leq\; 4$ and a few special cases.

{{harvtxt|Rifford|2022}} addressed the question of the size of the time required for a runner to get lonely. He formulated a stronger conjecture stating that for every integer $n\; \backslash geq\; 3$ there is a positive integer $N$ such that for any collection $v\_1,v\_2,\backslash dots,v\_\{n-1\}$ of positive, distinct speeds, there exists some time $t>0$ such that $\backslash operatorname\{frac\}(v\_it)\backslash in\; [1/n,1-1/n]$ for $i=1,\; \backslash dots,n-1$ with

$$t\; \backslash leq\; \backslash frac\{N\}\{\backslash operatorname\{min\}\; (v\_1,\backslash dots,\; v\_\{n-1\})\}.$$

Rifford confirmed this conjecture for $n=3,4,5,6$ and showed that the minimal $N$ in each case is given by $N=1$ for $n=3,4,5$ and $N=2$ for $n=6$. The latter result ($N=2$ for $n=6$) shows that if we consider six runners starting from $0$ at time $t=0$ with constant speeds $v\_0,v\_1,\backslash dots,v\_\{5\}$ with $v\_0=0$

and $v\_1,\backslash dots,v\_\{5\}$ distinct and positive then the static runner is separated by a distance at least $1/6$ from the others during the first two rounds of the slowest non-static runner (but not necessary during the first round).

A much stronger result exists for randomly chosen speeds: using the stationary-runner convention, if $n$ and $\backslash varepsilon>0$ are fixed and $n-1$ runners with nonzero speeds are chosen uniformly at random from $\backslash \{1,2,\backslash ldots,k\backslash \}$, then $P(\backslash delta\backslash geq\; 1/2\; -\; \backslash varepsilon)\backslash to\; 1$ as $k\backslash to\backslash infty$. 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.{{sfn|Czerwiński|2012|p=2}} 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.{{sfn|Czerwiński|Grytczuk|2008}} The conjecture has been generalized to an analog in algebraic function fields.{{sfn|Chow|Rimanić|2019}}

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{{refend}}

• [http://www.openproblemgarden.org//?q=op/lonely_runner_conjecture Article in the Open Problem Garden] no. 4, 551–562.

{{DEFAULTSORT:Lonely Runner Conjecture}}

Category:Diophantine equations

Category:Conjectures

Category:Unsolved problems in number theory