Six Lonely Runners

For $x$ real, let $ \{x\}$ be the fractional part of $x$ (i.e. $ \{x\} = x - \lfloor x \rfloor $). In this paper we prove the $k=5$ case of the following conjecture (the lonely runner conjecture): for any $k$ positive reals $ v_1, \dots , v_k $ there exists a real number $t$ such that $ 1/(k+1) \le \{v_it \} \le k/(k+1) $ for $ i= 1, \dots, k$.