Lonely Runner Conjecture

Establish that for every set of n distinct positive integers w_1,w_2,\dots,w_n and every runner i\in[n], there exists a time t\in\mathbf{R} such that \(\min_{j\neq i}\lVert t(w_i-w_j)\rVert_{\mathbf{T}}\geq 1/n\), equivalently proving the Lonely Runner Conjecture for arbitrary collections of distinct constant speeds.

Background

The paper introduces the Lonely Runner Conjecture as a longstanding problem originating in Diophantine approximation and view-obstruction problems. In its runner formulation, n runners with distinct constant speeds move around the unit circle from a common starting point, and each runner is required to attain a time at which its distance from every other runner is at least 1/n. The paper notes that the formulation for distinct real speeds is equivalent to the stated formulation for distinct positive integer speeds.

The paper does not resolve this conjecture in general. Its main theorem instead proves a polynomial improvement over the trivial lower bound for the maximum loneliness, showing that for n distinct positive integer speeds the bound is at least $1/(2n)+1/n^{5/3+o(1)}$.