Determinant values on lattices

We study the distribution of determinant values on lattices in $\operatorname{M}_n(\mathbb R)$ for $n\ge 2$. Let $Λ<\operatorname{M}_n(\mathbb R)$ be a lattice whose elements have algebraic entries. We prove that if $\det (Λ)$ is not contained in a scalar multiple of $\mathbb Z$, then for every $a<b