The lonely runner problem

The ‘Lonely Runner’ Problem Only Appears Simple

Take a group of runners circling a track at unique, constant paces. Answering the question of how many will always end up running alone, no matter their speed, has vexed mathematicians for decades.

WIREDPaulina Rowińska

Lonely runner conjecture - Wikipedia

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The conjecture then states that, for any collection v₁,v₂,…,v_n−1 of positive, distinct speeds, there exists some time t > 0 such that

1/n ≤ frac(v_it) ≤ 1 − 1/n.

Consider the time τ_i = 1/v_i it takes each runner to complete one circuit of the unit track. If the times τ_i are all rational (or at least if they are all commensurate with one another) then they have a least common multiple, and the ratio of that least common multiple to the greatest common divisor of the track times has a prime factorization. A solution to the Chinese Remainder Theorem will then put all the runners but one in their places to leave that runner “lonely” at some time t > 0, and at every multiple of the least common multiple of track times after that.

Number Theory - The Chinese Remainder Theorem

The Chinese Remainder Theorem

Chinese Remainder Theorem - Algorithms for Competitive Programming

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If not all of the times τ_i are commensurate with one another, then the runners can be partitioned into classes, such that all the runners of each class are running at speeds that are commensurable, that is, in rational ratios to each other in the same class, but no two runners of different classes are running at commensurable speeds.

The Chinese Remainder Theorem may then be solved for each commensurable class of runners, and these solutions may be combined by applying Dirichlet’s Diophantine Approximation Theorem to prove the existence of a time when each runner is “lonely” in the general case.

Dirichlet’s diophantine approximation theorem | Bulletin of the Australian Mathematical Society | Cambridge Core

Dirichlet’s diophantine approximation theorem - Volume 16 Issue 2

Cambridge CoreT.W. Cusick

Dirichlet’s diophantine approximation theorem

SUNY Research ConnectCusick, T. W.

On Dirichlet’s approximation theorem - Monatshefte für Mathematik

Though we cannot improve on the upper bound in Dirichlet’s approximation theorem,Kaindl has shown that the upper bound can be lowered fromt n tot n −t n−1−t n−2−...−t−1, if we admit equality. We show thatKaindl’s upper bound is lowest possible in this case. The result is then generalized to linear forms.

SpringerLinkFranz Langmayr

It appears unfortunately as though we are being “played” by the gentlemen of the district with their various propositions and conjectures at the running track. Maybe we are missing something here, or, it is a trick question.