Integral Matrices of Fixed Rank
Yonatan R. Katznelson · Proceedings of the American Mathematical Society · 1994
Asymptotic formulæare derived for the number of $n \times m$ matrices of fixed rank $k$ with rational integral coefficients that are contained in a Euclidean ball of radius $T$ in ${{\mathbf {R}}^{n \times m}}$. It is assumed that $n \geqslant m > k \geqslant 1$ are fixed, and the asymptotics are valid as $T$ tends to infinity. The methods used are elementary.