Determinant values on lattices

We study the distribution of determinant values on lattices in $\operatorname{M}_d(\mathbb R)$ for $d\ge 2$. Let $\Lambda<\operatorname{M}_d(\mathbb R)$ be a lattice whose elements all have algebraic entries. We prove that if $\det (\Lambda)$ is not contained in a scalar multiple of $\mathbb Z$, then for every $a<b$, $$\#\{v\in\Lambda:\|v\| <T,\ a<\operatorname{det} v<b,\ \operatorname{det} v\ne0\} \sim \frac{C_d}{\operatorname{covol}(\Lambda)} (b-a)T^{d(d-1)}$$ as $T\to \infty$, where $\|\cdot\|$ is the Frobenius norm and $C_d>0$ depends only on $d$. For such a lattice, under an isotropic noncoincidence hypothesis, automatic for $d=2,3$ and satisfied for all diagonal lattices when $d\ge 4$, we also obtain an asymptotic formula for the determinant-zero lattice points. The same conclusions hold for the broader class of Diophantine lattices, under the corresponding hypotheses. For $d=2$, our result recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature $(2,2)$.