  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
\thinspace $1/2-\varepsilon $. The proof uses Fourier analytic methods. We also
point out some consequences of our result for colouring of random integer
distance graphs.
