Fix numbers $m, n, k\in {\Bbb Z}\_+$ and $r\in {\Bbb R}\_+$. 


What non-trivial estimates exist for the probability that a random $m\times n$ matrix, with integer entries and with all its rows of Euclidean norm less than or equal to $r$, will have rank $k$?


I'm particularly interested in results asymptotic in the variable $r$. (I've worked out the coefficient for $m=n=2, k=1$, but my method doesn't seem to generalize.)



Distributions of ranks of random integer matrices
This is addressed in 


MR1169034 (94e:11073) 
Katznelson, Yonatan R.(1-MSRI)
Integral matrices of fixed rank. 
Proc. Amer. Math. Soc. 120 (1994), no. 3, 667–675. 


