On the time for a runner to get lonely

Abstract

The Lonely Runner Conjecture asserts that if $n$ runners with distinct constant speeds run on the unit circle $\mathbb{R}/\mathbb{Z}$ 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 Positions 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 $m\geq 2$.