Cusick–Wills Lonely Runner Conjecture

Let $u_1,\ldots,u_{k+1}$ be distinct real numbers, and write $\lVert x\rVert$ for the distance from $x$ to the closest integer:

$$\lVert x\rVert=\min(x-\lfloor x\rfloor,\lceil x\rceil-x).$$

Cusick–Wills conjecture. For each $i\in\{1,\ldots,k+1\}$, there is a real number $t$ such that

$$\lVert t u_i-tu_j\rVert\geq\frac{1}{k+1}$$

for all $j\ne i$. Equivalently, among $k+1$ runners with distinct constant speeds on a unit-length track, each runner is at distance at least $1/(k+1)$ from every other runner at some time. The conjecture is the standard formulation of the Lonely Runner Conjecture, and the source does not specify its resolution.