Some remarks on the lonely runner conjecture
Terence Tao · Contributions to Discrete Mathematics · 2018
The lonely runner conjecture of Wills and Cusick, in its most popular formulation, asserts that if $n$ runners with distinct constant speeds run around a unit circle $\R/\Z$ starting at a common time and place, then each runner will at some time be separated by a distance of at least 1/(n+1) from the others. In this paper we make some remarks on this conjecture. Firstly, we can improve the trivial lower bound of 1/(2n) slightly for large n, to (1/(2n)) + (c \log n)/(n^2 (\log\log n)^2) for some absolute constant c>0; previous improvements were roughly of the form (1/(2n)) + c/n^2. Secondly, we show that to verify the conjecture, it suffices to do so under the assumption that the speeds are integers of size n^{O(n^2)}. We also obtain some results in the case when all the velocities are integers of size O(n).