We consider (LRC) .
Conjecture Suppose runners having distinct constant integer speeds 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 from the origin.
We first show trivial proves for runners.
Let denote the distance of x to the nearest integer.
Let . where the infimum is taken over all n-tuples of distinct positive integers.
In terms of the LRC state that
Without loss of generality (wlog) we can assume .
Case . Wlog we can assume are relatively prime. At time two runners are at the same distance from the origin from different sides of the lap at distances proportional to . Since they are relatively prime both runners visit all the points . The largest distance is , where means integer part.
For example, , at times runners are at positions and and the maximum distance from origin is .
Another example, , the maximum distance is .
If the maximum distance is at time .
In general the maximum distance is with the minimum for runners at speeds . Therefore, .
- First we assume simple case: , e.g .
At times runner 3 is at the origin. There is a time such that some time before it was the same distance with runner 1 and some time after it will be the same distance with runner 2. (The situation is reversed at times ). We are interested when that is happening for the time runners 1 and 2 are most distant from the origin.
For runners 1 and 3: They need to pass with speed , so the time needed is . Runner 3 passes . Therefore, the closest point is reached when is maximum.
If , around
Lemma 1. .
Now, consider the case when .
Proof. . Since nominator is greater than 0, except when
Corollary 3. .
Corollary 4. only for .
- Case: .
Lemma 5. Fix time . Let be the distance from the origin of runner 3 at the moment and be the distance of the runners 1 and 2 from the origin at the same time. The maximal distance when runners 1 and 2 or 1 and 3 equidistant around time is greater than .
Proof. Either the runner 3 is running toward or away from the origin. e.g. the distance to the origin either decrease with time or increase. Let be the speed of the runner 1 or runner 2 moving in opposite direction, so that the difference in distances is decreasing moving either forward or backward in time. Since runner 3 will cover greater distance away from the origin that other runner
Lemma 6. Fix the time when . Let , and – distance of runner 3 from the origin. .
Combining Lemma 5 and Lemma 6 shows .
Overall, only for runners .