The Lonely Runner Spectrum

For distinct speeds v₁,…,\(v_{n}\), define L(t) = \(min_{i}\) ||t·\(v_{i}\)|| where ||·|| is distance to nearest integer. The lonely runner conjecture asserts \(max_{t}\) L(t) ≥ 1/(n+1). Sharper question: characterise the SET of times t achieving loneliness, and determine the measure of {t : L(t) ≥ α} for each α ∈ [0, 1/2].

The original conjecture asks about the maximum; this refinement asks about the full spectrum. Chen and others have shown the spectrum can be fractal-like for specific speed tuples. Tao's 2018 'almost all speeds' result implies the measure is generically large but says nothing about individual instances. Even n = 4 has only partial spectral classification.