Determinant values on lattices

Abstract

We study the distribution of determinant values on lattices in $M_d(\mathbb{R})$ for $d\ge 2$. Let $\hat{I}›<M_d(\mathbb{R})$ be a lattice whose elements all have algebraic entries. We prove that if $\det (\hat{I}›)$ is not contained in a scalar multiple of $\mathbb{Z}$, then for every $a<b$, $$\#\{v\in \hat{I}›:\Vert v\Vert <T,a<\det v<b,\det v\ne 0\}\sim \frac{C_d}{\mathrm{covol}(\hat{I}›)}(b-a)T^{d(d-1)}$$ as $T\to \infty$, where $\Vert \cdot \Vert$ 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 $(2,2)$.

142 pages