Let a 2D grid basis $\mathcal{B}(\theta_1, \theta_2,r_1, r_2)$ whose change-of-basis matrix with respect to $\mathcal{B}_0$ the canonical basis of $\mathbb{R}^2$ writes : $$P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} = \left(\begin{array}{cc} r_1\cos(\theta_1) & r_2 \cos(\theta_2) \\ r_1\sin(\theta_1) & r_2 \sin(\theta_2) \end{array}\right)$$
and the parameters of the grid satisfy $$ \begin{array}[ccc] &&\\ \theta_1 & \in & (-\frac{\pi}{2}, \frac{\pi}{2}]\\ \theta_2 & \in & [-\frac{\pi}{2}, \theta_1)\\ r_1,r_2 & > & 0 \end{array} $$ The intersections of the grid $\mathcal{B}(\theta_1, \theta_2,r_1, r_2)$ written in $\mathcal{B}_0$ is the following set $$ \left\lbrace P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\;;\;\forall z\in \mathbb{Z}^2\right\rbrace$$
Given this setting, computing the minimal distance between two intersections of this grid is equivalent to solving the following problem $$\min_{z\in\mathbb{Z}^2_*}\|P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\|_2$$
Where $\mathbb{Z}^2_*=\mathbb{Z}^2\setminus \lbrace (0,0)\rbrace$. The solution of this problem can be tricky to compute in corner cases. For example, let the grid be as follows: $$r_1 = 2\;;\;r_2=5\;;\;\theta_1 = -10^° \;;\;\theta_2 = -11^°$$ With this configuration the minimal value for $\|P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\|_2 $ is achieved for $z^*=(5,-2)$.
Finally, my question is, is there a faster method than Integer Programming to solve this problem. Particularly I would like to know if there is a way to compute $z_i^-$, $z_i^+$ such that $$\textrm{arg}\min_{z\in \mathbb{Z}^2_*}\|P_{\mathcal{B}_0}^{\mathcal{B}(\theta_1, \theta_2,r_1, r_2)} z\|_2 \subset[z_1^-,z_1^+]\times[z_2^-,z_2^+] $$