The finite-group formulation of the Lonely Runner Conjecture

Let $k$ be a positive integer, let $S\subset\mathbb{N}$ have size $k$, and for a natural number $n$ let $\mathbb{Z}_n$ denote the cyclic group of order $n$. For an integer $m$, set

$$B=\pm\{0,1,\ldots,m\},\qquad m=\left\lceil\frac{n}{k+1}\right\rceil-1.$$

The finite-group formulation. There exist a natural number $n$ and an element $x\in\mathbb{Z}_n$ such that

$$xS\cap B=\varnothing.$$

This is presented as an equivalent formulation of the Lonely Runner Conjecture in terms of dilates in cyclic groups. The general conjecture remains open.