Lonely runner conjecture: Difference between revisions

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{{short description|Unsolved problem in mathematics}}

{{unsolved|mathematics|Is the lonely runner conjecture true for every number of runners?}}

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 <math>n</math> runners on a track of unit length, with constant speeds all distinct from one another, will each be ''lonely'' at some time—at least <math>1/n</math> 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 7seven runners or lessfewer, but remains unsolved in the general case remains unsolved. Implications of the conjecture include solutions to view -obstruction problems and bounds on properties, related to [[chromatic number]]s, of certain graphs.

==Formulation==

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Consider <math>n</math> runners on a circular track of unit length. At the initial time <math>t=0</math>, 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 <math>t</math> if they are at a distance (measured along the circle) of at least <math>1/n</math> 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 <math>n-1</math> other runners, with nonzero speeds, are considered.{{efn|Some authors use the convention that <math>n</math> is the number of non-stationary runners, and thus the conjecture is that the gap of loneliness is at most <math>1/(n+1)</math>.{{sfn|Tao|2018}} }} The moving runners may be further restricted to ''positive'' speeds only: by symmetry, runners with speeds <math>x</math> and <math>-x</math> 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 shethey can be made stationary by subtracting hertheir speed from all runners, leaving herthem with zero speed. The conjecture then states that, for any collection <math>v_1,v_2,...\dots,v_{n-1}</math> of positive, distinct speeds, there exists some time <math>t>0</math> such that

<math display="block">\frac{1}{n}\leq \operatorname{frac}(v_it)\leq 1-\frac{1}{n}\qquad (i=1,...\dots,n-1),</math>

where <math>\operatorname{frac}(x)</math> denotes the [[fractional part]] of <math>x</math>.{{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 <math>i</math>th runner at time <math>t</math>, 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 <math>\operatorname{frac}(vt)=3/4</math>.}} 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. His conjecture concerns how well multiple real numbers could be simultaneously approximated with a single denominator.

== Implications ==

[[File:View-obstruction problem for squares.svg|thumb|alt=A series of red squares and a green line, with slope 2, narrowly hitting the squares.|Squares of side length 1/3 placed at every half-integer coordinate in the positive quadrant obstruct any ray from the origin in(besides those lying on thatan directionaxis). Any smaller side length will leave small gaps.]]

Suppose <math>C</math> is a {{mvar|n}}-[[hypercube]] of side length <math>s</math> in {{mvar|n}}-dimensional space (<math>n\geq2</math>). InfinitelyPlace manya copiescentered copy of <math>C</math>, all scaled by some factor <math>\alpha</math>, are placed at pointsevery atpoint nonnegativewith [[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 <math>C</math>, 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 <math>\alphas<(n-1)/(n+1)</math>, 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 <math>1/3</math> in side length will leave gaps, as shown, and squares with side length <math>1/3</math> 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 <math>G</math> on the set of integers, and using some finite set <math>D</math> of positive integer distances, has an edge between <math>x,y\in\mathbb{Z}</math> if and only if <math>|x-y|\in D</math>. For example, if <math>D=\{2\}</math>, every consecutive pair of even integers, and of odd integers, is adjacent, all together forming two [[connected component (graph theory)|connected component]]s. A {{mvar|k}}-''regular coloring'' of the integers with step <math>\lambda\in\mathbb{R}</math> assigns to each integer <math>n</math> one of <math>k</math> colors based on the [[Modular arithmetic#Residue systems|residue]] of <math>\lfloor \lambda n\rfloor</math>modulo <math>k</math>. For example, if <math>\lambda=0.5</math>, the coloring repeats every <math>2k</math> integers and each pair of integers <math>2m, 2m+1</math> are the same color. The lonely runner conjecture implies that the minimumTaking <math>k=|D|+1</math>, forthe whichlonely runner conjecture implies <math>G</math> admits a proper {{mvar|k}}-regular coloring (i.e., each node is colored differently than its adjacencies) for ''some'' step value is at most <math>|D|+1</math>.{{sfn|Barajas|Serra|2009|p=5688}} For example, <math>(k,\lambda)=(2,0.5)</math> generates a proper coloring on the distance graph generated by the example <math>D=\{2\}</math>. (<math>k</math> is known as the ''regular chromatic number'' of <math>D</math>.)

Given a [[directed graph]] <math>G</math>, a [[nowhere-zero flow]] on <math>G</math> associates a positive value <math>f(e)</math> to each edge <math>e</math>, such that the flow outward from each node is equal to the flow inward. Suppose <math>f</math> is further restricted to positive integers. The lonely runner conjecture implies that, if <math>fG</math> attainshas a nowhere-zero flow with at most <math>k</math> differentdistinct integer values, then <math>G</math> has a nowhere-zero flow with values only in <math>\{1,2,\ldots,k\}</math> (possibly after reversing the directions of some arcs of <math>G</math>). This result was proven for <math>k\geq 5</math> 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}}

==Known results==

For a given setup of runners, let the ''gap of loneliness''{{sfn|Perarnau|Serra|2016}} <math>\delta</math> bedenote the smallest of the runners' maximum distances of loneliness, and the ''gap of loneliness''{{sfn|Perarnau|Serra|2016}} <math>\delta_n</math> denote the minimum <math>\delta</math> across all setups with <math>n</math> runners. TheIn conjecturethis thennotation, the conjecture asserts that <math>\delta_n\geq 1/n</math>, a bound which, if correct, cannot be improved. For example, if the runner to be lonely is stationary and speeds <math>v_i=i</math> are chosen, then there is no time at which hethey isare strictly more than <math>1/n</math> units away from all others, showing that <math>\delta_n \leq 1/n</math>.{{efn|Let the lonely runner be fixed at 0. For sake of contradiction, suppose there exists <math>t</math> such that <math>\{v_it\}\in (1/n, 1-1/n)</math> for all <math>i</math>. By the pigeonhole principle, there exist distinct <math>i</math> and <math>j</math> such that <math>\{v_it\}\leq \{v_jt\} < \{v_it\}+1/n</math> But <math>\|v_j-v_i\|=v_k</math> for some <math>k</math>, so either <math>\{v_kt\}=\{v_jt\}-\{v_it\}<1/n</math> or <math>\{v_kt\}=1-(\{v_jt\}-\{v_it\})>1-1/n</math>, a contradiction.{{sfn|Bohman|Holzman|Kleitman|2001|p=2}} }} Alternatively, this conclusion iscan abe corollaryquickly ofderived from the [[Dirichlet approximation theorem]]. AFor <math>n\geq 2</math> a simple lower bound <math>\delta_n\geq 1/(2n-2)</math> may be obtained via a coveringprobability 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 <math>n</math> runners with integer speeds, it is true for <math>n</math> numbersrunners with real speeds.{{sfn|Bohman|Holzman|Kleitman|2001|pp=12–13}}

=== Tighter bounds ===

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

<math display=block>\delta_n\geq \frac{1}{2n-4+o(1)}.</math>

{{harvtxt|Tao|2018}} proved the current best known asymptotic result: for sufficiently large <math>n</math>,

<math display="block">\delta_n\geq \frac{1}{2n-2}+\frac{c\log n}{n^2(\log\log n)^2}</math>

for some constant <math>c>0</math>. He also showed that the full conjecture is implied by proving the conjecture for integer speeds of size <math>n^{O(n^2)}</math> (see [[big O notation]]). This implication theoretically allows proving the conjecture for a given <math>n</math> 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. IfFor sufficiently large <math>n\geq 33</math>, it holds true andif

<math display="block">\frac{v_{i+1}}{v_i}\geq 1 + \frac{22\log(n-1)}{n-1} \qquad (i=1,...\dots,n-12),.</math>

thenIn other words, the minimumconjecture gapholds oftrue lonelinessfor islarge <math>1/n</math> if the speeds grow quickly enough. InIf otherthe wordsconstant 22 is replaced with 33, then the conjecture holds true for <math> n\geq 3316343</math> if the speeds grow quickly enough.{{sfn|Dubickas|2011|p=27}} A slightly strongersimilar result for sufficiently large <math>n\geq 40</math> only requires a similar assumption on the firstfor <math> i =\lfloor n/2122 \rfloor-1,\dots,n-2</math> speeds.{{sfn|Czerwiński|2018|p=1302}} In a similar fashion but unconditionallyUnconditionally on <math>n</math>, the conjecture is true if <math>v_{i+1}/v_i\geq 2</math> for all <math> i</math>.{{sfn|Barajas|Serra|2009}}

=== For specific {{mvar|n}} ===

{{Anchor|For specific n}}

The conjecture is true for <math>n\leq 7</math> runners. The proofs for <math>n\leq 3</math> are elementary; the <math>n=4</math> 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 <math>n=5</math>, <math>n=6</math>, and <math>n=7</math> cases were settled in 1984, 2001 and 2008, respectively. The first proof for <math>n=5</math> was computer-assisted., Allbut 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 <math>n=6</math>.}}

For some <math>n</math>, there exist sporadic examples with a maximum separation of <math>1/n</math> besides the example of <math>v_i=i</math> given above.{{sfn|Bohman|Holzman|Kleitman|2001|p=2}} For <math>n=5</math>, the only otherknown example (up to shifts and scaling) is <math>\{0,1,3,4,7\}</math>; for <math>n=6</math> the only known example is <math>\{0,1,3,4,5,9\}</math>; and for <math>n=8</math> the onlyknown exampleexamples isare <math>\{0,1,4,5,6,7,11,13\}</math> and <math>\{0,1,2,3,4,5,7,12\}</math>.{{sfn|Bohman|Holzman|Kleitman|2001|p=3}} All solutions for <math>n\leq 20</math> reaching exactly <math>1/n</math> in separation are known through exhaustive computer search, and thereThere exists an explicit infinite family of extremalsuch examplessporadic 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 <math>v_i</math>, either <math>\delta = s/(sns(n-1)+1)</math> for some positive integer <math>s</math>,{{efn|Taking <math>s=1</math> yields the lonely runner conjecture.}} or <math>\delta \geq 1/(n-1)</math>, where <math>\delta</math> is that setup's gap of loneliness. He confirmed this conjecture for <math>n\leq 34</math> 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 <math>n \geq 3</math> there is a positive integer <math>N</math> such that for any collection <math>v_1,v_2,\dots,v_{n-1}</math> of positive, distinct speeds, there exists some time <math>t>0</math> such that <math>\operatorname{frac}(v_it)\in [1/n,1-1/n]</math> for <math>i=1, \dots,n-1</math> with

<math display="block">t \leq \frac{N}{\operatorname{min} (v_1,\dots, v_{n-1})}.</math>

Rifford confirmed this conjecture for <math>n=3,4,5,6</math> and showed that the minimal <math>N</math> in each case is given by <math>N=1</math> for <math>n=3,4,5</math> and <math>N=2</math> for <math>n=6</math>. The latter result (<math>N=2</math> for <math>n=6</math>) shows that if we consider six runners starting from <math>0</math> at time <math>t=0</math> with constant speeds <math>v_0,v_1,\dots,v_{5}</math> with <math>v_0=0</math>

and <math>v_1,\dots,v_{5}</math> distinct and positive then the static runner is separated by a distance at least <math>1/6</math> from the others during the first two rounds of the slowest non-static runner (but not necessary during the first round).

=== Other results ===

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{{refend}}

== External links ==

*[http://gardenwww.irmacsopenproblemgarden.sfu.caorg//?q=op/lonely_runner_conjecture Article in the Open Problem Garden] no. 4, 551–562.

{{DEFAULTSORT:Lonely Runner Conjecture}}