  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
$\mathcal{S}(n-1)$.
