Let be $x_1,x_2,...x_k>\delta \gt 0$ Real numbers. Consider the box in $\mathbb R^k$ given by $B_0 =[0,x_1-\delta]\times ... \times [0, x_k -\delta]$. Let $n_1, n_2,...,n_k \in \mathbb Z$, we define the box $B_{n_1,...,n_k}$ as $(n_1 x_1,...,n_k x_k)+B_0$ that is, $B_{n_1,...,n_k}$ it is a traslation of $B_0$ of $(n_1 x_1,...,n_k x_k)$ units. Consider $B=\cup_{(n_1, n_2,...,n_k)\in \mathbb Z^k} B_{n_1,...,n_k}$, so B is formed by infinitely copies of $B_0$ in $\mathbb R^k$.
We define $r(t)=(t,t,...t)\in \mathbb R^k$, clearly $r(0)\in B_0$.
Define $gap(B)=\min _{t \geq 0} [r(t)\in B\setminus B_0]-\max_{t \geq 0} [r(t)\in B_0] $. The gap function measures the "distance" accross the line $r(t)$ between $B_0$ and its nearest copy.
Are there an explicit formula for $gap(B)$ in terms of $x_1,x_2,...,x_k, \delta$?