In Duke-Rudnick-Sarnak 93, Density of integer points on affine homogeneous varieties, one of the consequences is the following,
Consider the variety $V_{n,k} = \{A \in Mat_n(\mathbb{Z}): det(A) = k\}$, and let $N(T,V) = \{A \in V_{n,k}: \|A\| \leq T\}$, where the norm is for example the square root of sum of squares of entries of $A$.
D-R-S proved in particular, that as $T \to \infty$, $$N(T,V) \sim c_{n,k} T^{n^2-n}$$ for some constant $c_{n,k}$ depending on $n$ and $k$.
My question is then, what is the behaviour of $N(T,V)$ if we allow $k,T$ to go to infinity at the same time? I am happy with just a good upper bound which is uniform in both $k$ and $T$.
Thanks!