The structure of Lonely Runner spectra

TL;DR

This paper investigates the structure of the Lonely Runner spectrum, revealing that the accumulation points of the spectrum in n dimensions are exactly the spectrum in n-1 dimensions, offering insights into the geometric properties related to the conjecture.

Contribution

The paper characterizes the accumulation points of the Lonely Runner spectrum, showing they correspond exactly to the spectrum in one lower dimension, without directly proving the conjecture.

Findings

Accumulation points of $\\mathcal{S}(n)$ are exactly $\\mathcal{S}(n-1)$.

Provides structural understanding of the spectrum related to the Lonely Runner problem.

Does not resolve the conjecture but offers geometric insights.

Abstract

For each subtorus $T$ of $(\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the (infimal) $L^\infty$-distance from $T$ to the point $(1/2,\ldots, 1/2)$. The $n$-th Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture, we study the structure of the sets $\mathcal{S}(n)$. The main purpose of this note is to show that the set of accumulation points of $\mathcal{S}(n)$ is precisely $\mathcal{S}(n-1)$.