I am currently encountering the following arithmetical problem: given four nonzero integers $A,B,C,D$, let $\Omega=\bigg\lbrace (x,y)\in{\mathbb N_{\geq 0}}^2 \ \bigg| \ \frac{Ax+By}{Cx+Dy} \in {\mathbb Z} \ \bigg\rbrace$ and suppose that $\Omega$ is non-empty.
Let then $\mu$ be the minimum value of all ${\sf min}(x,y)$ where $(x,y)\in \Omega$.
Is it easy to compute $\mu$ from $A,B,C,D$? If not, can one find a “reasonably good” lower bound for $\mu$?
A concrete example:
If
$$ \begin{array}{lcl} A &=& 522834163445445988434458010516405 \\ B &=& -3063742572717320569341511991159738 \\ C &=& 1666455861030599542832067804101203 \\ D &=& - 9765222175513935643148512770417523 \\ \end{array} $$
can $\mu$ be computed exactly or up to a reasonable precision?