The structure of Lonely Runner spectra

For each subtorus $T$ of $(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^n$, let $D(T)$ denote the (infimal) $L^{\mathrm{\infty }}$-distance from $T$ to the point $(1\mathrm{/}2,\dots ,1\mathrm{/}2)$. 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 $(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n)\subseteq [0,1\mathrm{/}2-1\mathrm{/}(n+1)]$. 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 $\mathcal{S}(n-1)$.