Some remarks on the lonely runner conjecture

Terence Tao


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).


Lonely runner conjecture; Bohr sets

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Contributions to Discrete Mathematics. ISSN: 1715-0868