Lonely Runner Conjecture. General case.

We consider (LRC) .

Conjecture Suppose n>1 runners having distinct constant speeds v_k > 0 start at a common point (origin) and run laps on a circular track with circumference 1. Then, there is a time when no runner is closer than \frac{1}{n+1} from the origin.

W.l.o.g. we can assume v_k > v_m \forall k>m.

One can formulate LRC as follows. Suppose l_k \in \mathbb{N}_0 is the number of whole laps (including 0) runner k passed on a track, than \exists t \in \mathbb{R}_+ and \exists l_k | l_k+\frac{ 1 }{n+1} \leq v_k t \leq l_k+ \frac{n}{n+1}.

Case n=1, two runners is trivial. At time t=\frac{1}{2 v_1} runner 1 is exactly distance \frac{1}{2} (l_1= 0).

Case n=2, three runners is a special case.

We start with 4 inequalities
\left\{ \begin{array}{ll}  \frac{l_1+1/3}{v_1} \leq t, &  t \leq \frac{l_1+2/3}{v_1} \\  \frac{l_2+1/3}{v_2} \leq t, & t \leq \frac{l_2+2/3}{v_2}  \end{array} \right.

Since t \in \mathbb{R}_+ we can eliminate it if all combination of left parts on the left columns are smaller than any right parts in the right column of the table. e.g.
\left\{ \begin{array}{l}  \frac{l_1+1/3}{v_1} \leq \frac{l_1+2/3}{v_1} \\  \frac{l_1+1/3}{v_1} \leq \frac{l_2+2/3}{v_2} \\  \frac{l_2+1/3}{v_2} \leq \frac{l_1+2/3}{v_1}\\  \frac{l_2+1/3}{v_2} \leq \frac{l_2+2/3}{v_2}  \end{array} \right.

The first and the last inequality are trivially correct. From the second and third inequality we would like to express l_2
\frac{v_2}{v_1} \left( l_1 + \frac{1}{3} \right) -\frac{2}{3} \leq l_2 \leq  \frac{v_2}{v_1} \left( l_1 + \frac{2}{3} \right) -\frac{1}{3}.
In other words,
\frac{v_2}{v_1} \left( l_1 + \frac{1}{3} \right) -\frac{2}{3} \leq l_2 \leq  \frac{v_2}{v_1} \left( l_1 + \frac{1}{3} \right) -\frac{2}{3} + \frac{1}{3}\frac{v_2}{v_1} +\frac{1}{3}.

There are 2 sub-cases.

  1. Sub-case \frac{v_2}{v_1} \geq 2.
    \frac{1}{3}\frac{v_2}{v_1} +\frac{1}{3} \geq 1 and \exists l_2 \in \mathbb{N}_0 satisfying inequality
  2. Sub-case \frac{v_2}{v_1} < 2 .
    In this case l_1=l_2=0 lead to (remeber that \frac{v_2}{v_1} > 1 )
    \frac{1}{3} \frac{ v_2}{v_1} - \frac{2}{3} \leq 0 \leq \frac{2}{3} \frac{v_2}{v_1} - \frac{1}{3}.

Case n=3, four runners is an illustration for a general case.

We start with 6 inequalities
\left\{ \begin{array}{ll} \frac{l_1+1/4}{v_1} \leq t, &  t \leq \frac{l_1+3/4}{v_1} \\  \frac{l_2+1/4}{v_2} \leq t, &  t \leq \frac{l_2+3/4}{v_2} \\  \frac{l_3+1/3}{v_3} \leq t, &  t \leq \frac{l_3+3/4}{v_3}  \end{array} \right.

Let’s express l_2 in terms of l_1
\frac{v_2}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} \leq l_2 \leq \frac{v_2}{v_1} \left( l_1 + \frac{3}{4} \right) -\frac{1}{4}.
Rewriting it we obtain
\frac{v_2}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} \leq l_2 \leq  \frac{v_2}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} + \frac{1}{2}\frac{v_2}{v_1} +\frac{1}{2}.
\frac{1}{2}\frac{v_2}{v_1} +\frac{1}{2} >1, since \frac{v_2}{v_1} > 1. In other words, \forall l_1 \exists l_2 \in \mathbb{N}_0 satisfying inequality.

Now, let express l_3 in terms of l_1 and l_2
\left\{ \begin{array}{l}  \frac{v_3}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} \leq l_3 \leq  \frac{v_3}{v_1} \left( l_1 + \frac{3}{4} \right) -\frac{1}{4} \\  \frac{v_3}{v_2} \left( l_2 + \frac{1}{4} \right) -\frac{3}{4} \leq l_3 \leq  \frac{v_3}{v_2} \left( l_2 + \frac{3}{4} \right) -\frac{1}{4}. \end{array}\right.

We can express l_2 with inequalities obtained earlier
\left\{ \begin{array}{l}  \frac{v_3}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} \leq l_3 \leq  \frac{v_3}{v_1} \left( l_1 + \frac{3}{4} \right) -\frac{1}{4} \\ \frac{v_3}{v_2} \left( \frac{v_2}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} + \frac{1}{4} \right) -\frac{3}{4} \leq l_3 \leq  \frac{v_3}{v_2} \left( \frac{v_2}{v_1} \left( l_1 + \frac{3}{4} \right) -\frac{1}{4} + \frac{3}{4} \right) -\frac{1}{4}.  \end{array}\right.
Collecting terms we obtain
\left\{ \begin{array}{l} \frac{v_3}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} \leq l_3 \leq  \frac{v_3}{v_1} \left( l_1 + \frac{3}{4} \right) -\frac{1}{4} \\  \frac{v_3}{v_1} \left( l_1 + \frac{1}{4} \right) -\frac{3}{4} -\frac{1}{2} \frac{v_3}{v_2} \leq l_3 \leq  \frac{v_3}{v_1} \left( l_1 + \frac{3}{4} \right)-\frac{1}{4}+ \frac{1}{2} \frac{v_3}{v_2}.  \end{array}\right.

We can see that first inequality is always stronger that the second one (meaning that if l_3 satisfies first inequalities it will satisfy second inequalities). But the first inequalities are the same as previous inequalities for l_2 with relabelling. Therefore, \forall l_1 \exists l_2, l_3 \in \mathbb{N}_0 satisfying initial inequalities and LRC holds.

General case

We start with 2 n inequalities

\left\{ \begin{array}{lll}  \frac{l_k+\frac{1}{n+1}}{v_k} \leq t, &  t \leq \frac{l_k+\frac{n}{n+1}}{v_k} &  k=1..n  \end{array} \right.

Now we expressing l_{k+m_1}, m_1 \geq 1 in terms of l_k and l_{k+m_2}, m_2 > m_1 in terms of l_{k+m_1} and l_{k}.
\frac{v_{k+m_1}}{v_k} \left( l_k + \frac{1}{n+1} \right) -\frac{n}{n+1} \leq l_{k+m_1} \leq  \frac{v_{k+m_1}}{v_k} \left( l_k + \frac{n}{n+1} \right) -\frac{1}{n+1}.
\frac{v_{k+m_2}}{v_{k+m_1}} \left( l_{k+m_1} + \frac{1}{n+1} \right) -\frac{n}{n+1} \leq l_{k+m_2} \leq  \frac{v_{k+m_2}}{v_{k+m_1}} \left( l_{k+m_1} + \frac{n}{n+1} \right) -\frac{1}{n+1}.
\frac{v_{k+m_2}}{v_k} \left( l_k + \frac{1}{n+1} \right) -\frac{n}{n+1} \leq l_{k+m_2} \leq  \frac{v_{k+m_2}}{v_k} \left( l_k + \frac{n}{n+1} \right) -\frac{1}{n+1}.

Now we can substitute first inequalities into the second ones.
\frac{v_{k+m_2}}{v_{k+m_1}} \left( \frac{v_{k+m_1}}{v_k} \left( l_k + \frac{1}{n+1} \right) -\frac{n}{n+1} + \frac{1}{n+1} \right) -\frac{n}{n+1}  \leq l_{k+m_2} \leq  \frac{v_{k+m_2}}{v_{k+m_1}} \left(\frac{v_{k+m_1}}{v_k} \left( l_k + \frac{n}{n+1} \right) -\frac{1}{n+1} + \frac{n}{n+1} \right) -\frac{1}{n+1}.
Rearranging terms we obtain
\frac{v_{k+m_2}}{v_k} \left( l_k + \frac{1}{n+1} \right) -\frac{n}{n+1}  - \frac{n-1}{n+1} \frac{v_{k+m_2}}{v_{k+m_1}}  \leq l_{k+m_2} \leq  \frac{v_{k+m_2}}{v_k} \left( l_k + \frac{n}{n+1} \right) -\frac{1}{n+1}  +\frac{n-1}{n+1} \frac{v_{k+m_2}}{v_{k+m_1}}.
Compare it to
\frac{v_{k+m_2}}{v_k} \left( l_k + \frac{1}{n+1} \right) -\frac{n}{n+1}  \leq l_{k+m_2} \leq  \frac{v_{k+m_2}}{v_k} \left( l_k + \frac{n}{n+1} \right) -\frac{1}{n+1}.
The second inequality is always stronger.

Therefore, we left with the set of inequalities (when they are satisfied LRC holds when LRC holds they are satisfied!!! )
\frac{v_{k}}{v_1} \left( l_1 + \frac{1}{n+1} \right) -\frac{n}{n+1}  \leq l_{k} \leq \frac{v_{k}}{v_1} \left( l_1 + \frac{n}{n+1} \right) -\frac{1}{n+1},\ \forall k=2..n.

Lets rewrite it again
\frac{v_{k}}{v_1} \left( l_1 + \frac{1}{n+1} \right) -\frac{n}{n+1} \leq l_{k} \leq \frac{v_{k}}{v_1} \left( l_1 + \frac{1}{n+1} \right) -\frac{n}{n+1} + \frac{n-1}{n+1} \left( \frac{v_k}{v_1} +1 \right)

\frac{n-1}{n+1} \left( \frac{v_k}{v_1} +1 \right) > 2 \frac{n-1}{n+1} = 1+ \frac{n-3}{n+1} \geq 1 if n \geq 3 .

It is not clear from this why \frac{1}{n+1} is tight distance.

It is clear why it is not clear. The above just tell that with speed ratio more the one we can always find a lap for faster runner. The tightness comes from from ratios less than one. Most probably from the interactions or the ratios coming from the same pairs.

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