Some remarks on the lonely runner conjecture

Abstract

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 $\mathbf{R}/\mathbf{Z}$ starting at a common time and place, then each runner will at some time be separated by a distance of at least $\frac{1}{n+1}$ from the others. In this paper we make some remarks on this conjecture. Firstly, we can improve the trivial lower bound of $\frac{1}{2n}$ slightly for large $n$, to $\frac{1}{2n}+\frac{c\log n}{n^2(\log \log n{)}^2}$ for some absolute constant $c>0$; previous improvements were roughly of the form $\frac{1}{2n}+\frac{c}{n^2}$. 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 $O(n)$.

31 pages, no figures. To appear, Contrib. Disc. Math. Some corrections (suggested by Anthony Quas) have been implemented

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