I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$.
$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \text{rank}(A)=k ,\|A\| \leq T\} = C \cdot T^{kn}+O(T^{kn-1}).$
The approach in the paper is very combinatorial and it mentions that the result is orthogonal to the then recent results of Duke, Rudnik, Sarnak [2].
Since the last three decades, there is now a large plethora of literature about lattice point counts on homogeneous varieties using analytic number theory, circle method or homogeneous dynamics.
My question is
- Is it still true that this result is out of reach of the state of the art results on lattice point counts on homogeneous varieties?
- My particular interest is whether this result has been generalized to considering $M_{m \times n}(\mathcal{O}_K)$ matrices of a given rank $r$ and bounding it by a suitable norm. Has this generalized problem been considered anywhere?
[1] Katznelson, Yonatan R. "Integral matrices of fixed rank." Proceedings of the American mathematical society 120.3 (1994): 667-675.
[2] Duke, W., Z. Rudnick, and P. Sarnak. "Density of integer points on affine homogeneous varieties." Duke Mathematical Journal 71.1 (1993): 143.
