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So, I've been reading about the lonely runner conjecture, but I have to admit that my knowledge of diophantine approximation is so limited that I wouldn't be able to properly explain the term.

So let's consider the simplest case, which is $n$ "real" "linear" runners in a circle ($=\mathbb R/\mathbb Z$).

Let me try and view this problem from the angle of equidistribution on tori; I don't know a lot about this subject either, but I think I remember having read about this in a book by Tao when the library was still open (ie. before Corona). So suppose we got $n$ runners with speeds $v_1, \ldots, v_n$. Choose a $t > 0$ such that $α := t (v_1, \ldots, v_n)$ has only irrational components. Then the sequence $x_n := nα$ should be equidistributed on the $n$-torus, and therefore we should be able to approximate any configuration of runners by choosing a point within the $n$-torus and using equidistribution.

I'm only vaguely familiar with the case of polynomials, but this should in principle generalise to polynomial functions under certain assumptions on the leading coefficient, right?

Is this approximately how one would approach the problem from that angle? Is there any reference?

I'd be enormously grateful for any comments on this matter.

Thank you very much in advance.

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  • $\begingroup$ Full disclosure: I just looked it up, and the results on equidistribution should be approximately correct. $\endgroup$
    – Cloudscape
    Commented Feb 3, 2022 at 2:00
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    $\begingroup$ It's not good enough for the components to be irrational, e.g., $(\sqrt2,\sqrt2)$ has irrational components but its multiples clearly show no equidistribution on the torus. I think you need linear independence over the rationals. A good reference on equidistribution is the Kuipers & Niederreiter book, Uniform Distribution Modulo One. $\endgroup$
    – Gerry Myerson
    Commented Feb 3, 2022 at 2:23
  • $\begingroup$ Of course. It's getting late; sorry about that. $\endgroup$
    – Cloudscape
    Commented Feb 3, 2022 at 2:23
  • $\begingroup$ @GerryMyerson I'm not saying the Tao book is BS, it's just that I didn't remember it quite so well. $\endgroup$
    – Cloudscape
    Commented Feb 3, 2022 at 14:04

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A corrected version of this argument is contained in Section 4 of Six Lonely Runners by Bohman, Holzman, and Kleitman.

Their argument shows, using equidistribution, that the case of the lonely runner conjecture with rational speeds implies the general case.

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    $\begingroup$ Ah, THIS is Kronecker's theorem. Thanks a lot! $\endgroup$
    – Cloudscape
    Commented Feb 3, 2022 at 9:48
  • $\begingroup$ I just skimmed through some dynamical systems notes and saw that I should have remembered Kronecker's theorem. $\endgroup$
    – Cloudscape
    Commented Feb 3, 2022 at 10:41

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