Combinatorial Theory

For a subtorus \(T \subseteq (\mathbb{R}/\mathbb{Z})^n\), let \(D(T)\) denote the \(L^\infty\)-distance from \(T\) to the point \((1/2, \ldots, 1/2)\). For a subtorus \(U \subseteq (\mathbb{R}/\mathbb{Z})^n\), define \(\mathcal{S}_1(U)\), the Lonely Runner spectrum relative to \(U\), to be the set of all values of \(D(T)\) as \(T\) ranges over the \(1\)-dimensional subtori of \(U\) not contained in the union of the coordinate hyperplanes of \((\mathbb{R}/\mathbb{Z})^n\). The relative spectrum \(\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)\) is the ordinary Lonely Runner spectrum that has been studied previously. Giri and the second author recently showed that the relative spectra \(\mathcal{S}_1(U)\) for two-dimensional subtori \(U \subseteq (\mathbb{R}/\mathbb{Z})^n\) essentially govern the accumulation points of the Lonely Runner spectrum \(\mathcal{S}_1((\mathbb{R}/\mathbb{Z})^n)\). In the present work, we prove that such relative spectra \(\mathcal{S}_1(U)\) have a very rigid arithmetic structure, and that one can explicitly find a complete characterization of each such relative spectrum with a finite calculation; carrying out this calculation for a few specific examples sheds light on previous constructions in the literature on the Lonely Runner Problem.

Mathematics Subject Classifications: 11J13, 52C07, 11J06, 11B75

Keywords: Lonely Runner conjecture, Diophantine approximation, spectra, combinatorial number theory