Georgia Institute of Technology College of Sciences

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 $B$.