Amending the Lonely Runner Spectrum Conjecture

Abstract

Let $||x||$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1,\dots ,v_n$, we define its maximum loneliness, also known as the gap $\hat{I}´$, to be $$ML(v_1,\dots ,v_n)=\underset{t\in \mathbb{R}}{\max }\underset{1\le i\le n}{\min }||t{v}_i||.$$ The Loneliness Spectrum Conjecture, recently proposed by Kravitz (2021), asserts that $$\exists s\in \mathbb{N},\text{ML}(v_1,\dots ,v_n)=\frac{s}{sn+1}\text{ or }\text{ML}(v_1,\dots ,v_n)\ge \frac{1}{n}.$$ We disprove the Loneliness Spectrum Conjecture for $n=4$ with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for $n=4$ whenever there exists a pair of speeds with a common factor of at least $3$ and also prove some related results.

23 pages