On the Time for a Runner to Get Lonely

The Lonely Runner Conjecture asserts that if n runners with distinct constant speeds run on the unit circle R /Z starting from 0 at time 0, then each runner will at some time t > 0 be lonely in the sense that she/he will be separated by a distance at least 1/n from all the others at time t. In investigating the size of t, we show that an upper bound for t in terms of a certain number of rounds (which, in the case where the lonely runner is static, corresponds to the number of rounds of the slowest non-static runner) is equivalent to a covering problem in dimension n - 2. We formulate a conjecture regarding this covering problem and prove it to be true for n = 3, 4, 5, 6. Then, we use our method of proof to demonstrate that the Lonely Runner Conjecture with Free Starting Points is satisfied for n = 3, 4. Finally, we show that the so-called gap of loneliness in one round (with respect to the Lonely Runner Conjecture), where we have m 1 runners including one static runner, is bounded from below by 1/(2m - 1) for all integer m >= 2.

Automated summary

This article investigates the Lonely Runner Conjecture, proving its validity in certain cases and presenting an upper bound related to the number of rounds. It also explores a conjecture regarding a covering problem.