### Lonely runner problem

Suppose runners run, each with constant, but different, speed around a circular track of circumference 1. For each runner, is there a time at which that runner is at least from the other runners, as measured along the track?

This is known to be true for .

The problem is related to number theory problems of Diophantine approximation; see Bohman *et al* (2001) and Renault (2004), below. In this context the lonely runner problem was originally formulated in the following way (here the runners’ speeds are all taken to be integers, and a designated runner is assumed to have speed 0):

Given positive integers , is there a real number such that:

?.

Here, is the fractional part of the real number .

Could some insight as to a possible counterexample be obtained from computation?

Alternatively, can we try to use computation to get insight into the density of times for which there is a specified lonely runner?

For example, the following *Mathematica*® code:

**Needs[“LinearRegression`”];
integerbound = 2000;
k = 8;
T = Sort[Table[RandomInteger[{1, integerbound}], {i, 1, k – 1}]]
timebound = 1000;
lonelytimes = {};
howmany = 1000;
Do[
t = RandomReal[{0, timebound}];
A = Table[FractionalPart[t*T[[i]]], {i, 1, k – 1}];
If[And[1/k <= Min[A], Max[A] <= (k – 1)/k],
lonelytimes = {lonelytimes, t}, lonelytimes = lonelytimes],
{j, 1, howmany}]
lonelytimes = Sort[Flatten[lonelytimes]];
Print[“Proportion t = “, N[Length[lonelytimes]/howmany]*100, “%”]
ListPlot[lonelytimes]
Regress[lonelytimes, x, x]**

produces 7 random positive integers in the range [1,2000] and finds the percentage of times , for 1000 uniformly random samples with , for which .

A sample run gave:

Integer set ={41, 754, 850, 984, 1056, 1250, 1541}

Proportion t = 12.3%

The sorted list of t values was almost linear with rank:

meaning that the t values obtained were approximately uniformly spread over the interval (0,1000).

### References & reading

- Lonely Runner Conjecture
- Tom Bohman, Ron Holzman, Daniel J. Kleitman (2001) Six Lonely Runners. Electronic Journal of Combinatorics, vol8(2) (six_lonely_runners)
- Jérôme Renault (2004) View-obstruction: a shorter proof for 6 lonely runners. Discrete Mathematics, 287, 93-101. (view-obstruction_a_shorter_proof_for_6_lonely-runners)
- J. Barajas and O. Serra (2008 ) The Lonely Runner with Seven Runners, The Electronic Journal of Combinatorics, Volume 15(1).(the_lonely_runner_with_seven_runners)
- Daphne Liu. From Rainbow to the Lonely Runner. Department of Mathematics, California State Univ., Los Angeles, January 24, 2000. (from_rainbow-to_the_lonely_runner)