Coprime Mappings and Lonely Runners
Tom Bohman, Fei Peng

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 real, let be the fractional part of (i.e. ). The lonely runner conjecture can be stated as follows: for any positive integers there exists a real number such that for . In this paper we prove that if and is sufficiently large (relative to ) then such a exists for any collection of positive integers such that . 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 and be sets of integers. A bijection is a coprime mapping if and are coprime for every . We show that if …
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Taxonomy
TopicsLimits and Structures in Graph Theory · Mathematical Dynamics and Fractals
