Single Lonely Runner Conjecture

Establish that for every integer n ≥ 1 and every set V of n pairwise distinct real speeds, there exists an index i ∈ {1,…,n} and a real time t such that the distance of t(vj − vi) to the nearest integer is at least 1/n for all j ≠ i. In other words, show that at least one runner among n becomes lonely at distance at least 1/n from every other runner at some time.

Background

This is a weaker variant of the Lonely Runner Conjecture proposed by Joel Spencer. It asks for the existence of at least one runner achieving the conjectured isolation distance, rather than guaranteeing it for all runners.

The conjecture relates to results on invisible runners, where removing certain speeds can improve the loneliness gap, and to structural properties of runner interactions.

References

Conjecture 27 (Single LR Conjecture). For every n ∈ N and every n-set V of pairwise distinct real numbers, there exists i ∈ [n] and t ∈ R such that mj=it(v −jv ) i n . (21)

— The Lonely Runner Conjecture turns 60 (2409.20160 - Perarnau et al., 30 Sep 2024) in Conjecture 27, Section 10.1 (Invisible runners)