Some remarks on the lonely runner conjecture

TL;DR

This paper discusses the lonely runner conjecture, providing improved bounds on the minimum separation distance and reducing the problem to a finite set of integer speeds, advancing understanding of the conjecture's validity.

Contribution

It improves the lower bound for large n and shows that verifying the conjecture can be limited to a finite set of integer speeds, simplifying the problem.

Findings

Improved the lower bound to rac{1}{2n} + rac{c \, ext{log} n}{n^2 ( ext{log} ext{log} n)^2} for large n.

Reduced the verification of the conjecture to integer speeds of size n^{O(n^2)}.

Provided results for integer speeds of size O(n).

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