Wills–Cusick Lonely Runner Conjecture

Let $n$ runners with distinct constant speeds run around the unit circle, starting at a common time and place. Identify the unit circle with $\mathbf{T}=\mathbf{R}/\mathbf{Z}$, and let $\lVert\cdot\rVert_{\mathbf{T}}$ denote the distance to the nearest integer. Equivalently, let $w_1,w_2,\dots,w_n\in\mathbf{N}$ be any $n$ distinct positive integers. Lonely Runner Conjecture. For each $i\in[n]$, there exists a time $t\in\mathbf{R}$ such that

$$\min_{j\neq i}\lVert t(w_i-w_j)\rVert_{\mathbf{T}}\geqslant \frac{1}{n}.$$

The conjecture was raised independently by Wills and Cusick and is a central problem in Diophantine approximation and view-obstruction theory. The original formulation allows distinct real speeds; the source notes that this is equivalent to the integer formulation. The paper discusses improved lower bounds toward the conjectured value.