The structure of Lonely Runner spectra

Abstract For each closed 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 directly, we study the qualitative structure of the sets $\mathcal{S}(n)$ via their accumulation points. This project brings into the picture the analogues of $\mathcal{S}(n)$ where 1-dimensional subtori are replaced by k -dimensional subtori or k -dimensional subgroups.