Appears in collection : Arithmetic and Diophantine Geometry, via Ergodic Theory and o-minimality

We study the value-distribution problem of det on an irrational lattice $L<M_n(\mathbb R)$: how are the values of det on L distributed on $\mathbb R$? In a recent joint work in progress with Wooyeon Kim, we obtain quantitative results toward this question; for n=2, this amounts to a quantitative version of the Oppenheim conjecture for quadratic forms of signature (2,2), as studied by Eskin-Margulis-Mozes (2005).