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Lonely Runner Conjecture is the following problem, and the conjecture was proven to be true for $k\leq 7$:

Let $V$ be the set of $k$ distinct positive integers with $v_1<v_2<...<v_k$ Then $$\exists t\in [0,1]\quad \forall v_i\in V\quad \|tv_i\|\geq \frac{1}{k+1}$$

where $||x||$ denotes the distance from $x$ to the nearest integer.

It is known that for $k+1$ runners, one can assume the following WLOG:

  • $\gcd(v_1,...,v_k)=1$
  • $(k+1)|v_1$
  • $V$ is neither the set of arithmetic/geometric progression nor prime numbers.

Although it wasn't stated in any paper, it is trivially verifiable that one can also assume the following WLOG:

  • $\forall r\in\mathbb{Z}$ s.t. $2\leq r\leq k+1$, $\exists v_i\in V$ s.t. $r|v_i$.
  • $\forall r\in\mathbb{Z}$ s.t. $k+2\leq r\leq 2k+1$ and $2|r$, $\exists v_i\in V$ s.t. $r|v_i$.

What can I further assume about $V$?

The first three assumptions were stated in View-obstruction: a shorter proof for 6 lonely runners by Jérôme Renault, which is also a nice introduction to LRC along with the corresponding Wikipedia page (Lonely runner conjecture). The lonely runner with seven runners by Barajas is also a relevant reference.

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    $\begingroup$ You may wish to add a link to some background on the LRC, for readers not familiar with it $\endgroup$
    – Yemon Choi
    May 12, 2015 at 22:54
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    $\begingroup$ It might be worth including in your list that we can assume the speeds are integers, which at least is not obvious to me. It is proved in Section 4 here: combinatorics.org/ojs/index.php/eljc/article/view/v8i2r3 $\endgroup$
    – Sam Hopkins
    May 12, 2015 at 23:16
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    $\begingroup$ In response to @YemonChoi, this previous MO question has some references: Not-lonely runners. Will Sawin's answer establishes only "logarithmic distance between upper and lower bounds." $\endgroup$
    – Joseph O'Rourke
    May 12, 2015 at 23:43
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    $\begingroup$ You are probably interested in this recent blog post of Terry Tao as well: terrytao.wordpress.com/2015/05/13/… $\endgroup$
    – Sam Hopkins
    May 15, 2015 at 13:44
  • $\begingroup$ @Math.StackExchange I see that $k+1$ divides (w.l.o.g.) one the $v_i$'s, but why should it divide the smallest member $v_1$? $\endgroup$
    – Peter Mueller
    Jul 3, 2018 at 14:37

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