Quantitative Oppenheim in signature $(2,2)$ via determinant values

We give a new proof of the quantitative Oppenheim theorem in signature $(2,2)$, originally proved by Eskin-Margulis-Mozes, by recasting the problem as one about determinant values on lattices in $\operatorname{M}_2(\mathbb R)$.

The determinant $\det\begin{pmatrix}x&y\\ z&w\end{pmatrix}=xw-yz$ is a quadratic form of signature $(2,2)$, and every real quadratic form of this signature is obtained from it by a real linear change of variables.

For every Diophantine lattice $\Lambda<\operatorname{M}_2(\mathbb R)$ that is not determinant-rational, for every $a<b$, we prove an asymptotic formula for $$ \#\{v\in\Lambda:\|v\|<T,\ a<\operatorname{det} v<b\}. $$ The main term has a nonsingular contribution proportional to $(b-a)T^2$ and, when $0\in(a,b)$, a possible singular contribution from rational isotropic planes.

The proof follows the modified-height and avoidance strategy, but in the $2\times2$ case the representation theory and sublevel estimates are elementary, and the rational isotropic planes form a finite collection for a non-determinant-rational lattice.