Take $A,B,C,D$ pairwise coprime with $$n<A,B,C,D<2n$$ $$ n/8<|A−B|,|C−D|,|A−C|,|A−D|,|B−C|,|B−D|<n/4$$ and consider the null space of $\begin{bmatrix}A&B\end{bmatrix}\otimes \begin{bmatrix}C&D\end{bmatrix}$ spanned by $3\times 4$ matrix $$N=\begin{bmatrix} \begin{bmatrix}1&0\end{bmatrix}\otimes \begin{bmatrix}-D&C\end{bmatrix}\\ \begin{bmatrix}-B&A\end{bmatrix}\otimes \begin{bmatrix}1&0\end{bmatrix}\\ \begin{bmatrix}-B&A\end{bmatrix}\otimes \begin{bmatrix}-D&C\end{bmatrix}\\ \end{bmatrix}$$

Denote the $\Bbb Q$-linear space spanned by rows of $N$ by $T_{A,B,C,D}\subseteq\Bbb Q^4$ and denote the set of non-zero $\Bbb Z$ vectors in $T_{A,B,C,D}$ by $T_{A,B,C,D}[\Bbb Z]^\star$.

For a $\Bbb Z$ vector $v$ denote $\|v\|_\infty$ to be largest coordinate by magnitude.

Consider the quantity $$\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty.$$

1. Is there a name for this and how is $\min_{v\in T_{A,B,C,D}[\Bbb Z]^\star}\|v\|_\infty$ distributed as a function of $A,B,C,D$ chosen with the constraints above?

2. What is its average value?