The lonely runner conjecture in terms of the gap of loneliness

For $t\in\mathbf{R}/\mathbf{Z}$, let $\|t\|_{\mathbf{R}/\mathbf{Z}}$ be the distance from any representative of $t$ to the nearest integer. For an $n$-tuple of non-zero integers $(v_1,\dots,v_n)$, define

$$\delta(v_1,\dots,v_n)=\max_{t\in\mathbf{R}/\mathbf{Z}}\min\bigl(\|tv_1\|_{\mathbf{R}/\mathbf{Z}},\dots,\|tv_n\|_{\mathbf{R}/\mathbf{Z}}\bigr).$$

Let $\delta_n$ be the infimum of $\delta(v_1,\dots,v_n)$ over all $n$-tuples of distinct non-zero integers. Lonely runner conjecture. For every $n\geq 1$, one has

$$\delta_n=\frac{1}{n+1}.$$

The quantity $\delta_n$ is called the gap of loneliness. Dirichlet approximation gives the upper bound $\delta_n\leq 1/(n+1)$, so the conjecture asserts that this elementary upper bound is sharp for every $n$. It is equivalent to the runner formulation and remains open in general.