I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n<m\leq k$.

$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \text{rank}(A)=k ,\|A\| \leq T\} = C \cdot T^{kn}+O(T^{kn-1}).$

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

1. 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?
2. 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.

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

1. 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?
2. 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.