Random runners are very lonely

Suppose that k runners having different constant speeds run laps on a circular track of unit length. The Lonely Runner Conjecture states that, sooner or later, any given runner will be at distance at least 1 / k from all the other runners. We prove that, with probability tending to one, a much stronger statement holds for random sets in which the bound 1 / k is replaced by 1 / 2 − ε . The proof uses Fourier analytic methods. We also point out some consequences of our result for colouring of random integer distance graphs.