Coprime Mappings and Lonely Runners

For $x$ real, let $\{ x \}$ be the fractional part of $x$ (i.e. $\{x\} = x - \lfloor x \rfloor$). The lonely runner conjecture can be stated as follows: for any $n$ positive integers $v_1 < v_2 < \dots < v_n$ there exists a real number $t$ such that $1/(n+1) \le \{ v_i t\} \le n/(n+1)$ for $i = 1, \dots, n$. In this paper we prove that if $\epsilon >0$ and $n$ is sufficiently large (relative to $\epsilon$) then such a $t$ exists for any collection of positive integers $v_1 < v_2 < \dots < v_n$ such that $v_n < (2-\epsilon)n$. This is an approximate version of a natural next step for the study of the lonely runner conjecture suggested by Tao. The key ingredient in our proof is a result on coprime mappings. Let $A$ and $B$ be sets of integers. A bijection $f:A \to B$ is a coprime mapping if $a$ and $f(a)$ are coprime for every $a \in A$. We show that if $A,B \subset [n]$ are intervals of length $2m$ where $m = e^{ \Omega({(\log\log n)}^2)}$ then there exists a coprime mapping from $A$ to $B$. We do not believe that this result is sharp.