Coprime Mappings and Lonely Runners

TL;DR

This paper proves an approximate version of the lonely runner conjecture for large sets with bounded maximum velocity, using a novel result on the existence of coprime mappings between large integer sets.

Contribution

It introduces a new coprime mapping theorem for large integer intervals and applies it to an approximate case of the lonely runner conjecture.

Findings

Established existence of coprime mappings for large intervals

Proved an approximate version of the lonely runner conjecture for large n

Used coprime mappings to advance understanding of runner separation

Abstract

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.