  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 ${\bf R}/{\bf Z}$ 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 $O(n)$.

Combinatorics