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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!

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    $\begingroup$ I wouldn't be surprised if $c_{n,k}$ is uniformly bounded in $k$ for every fixed $n$. $\endgroup$ Oct 2, 2014 at 7:39
  • $\begingroup$ @GregMartin, that's also my guess. I am wondering if there is any heuristic way to see this, and in particular, what kind of bounds/asymptotics one should be aiming for. $\endgroup$
    – Pig
    Oct 2, 2014 at 15:05
  • $\begingroup$ Hmmm, does the constant $c_{n,k}$ arise in the proof as a cross-sectional measure of a particular compact region in $\Bbb R^{n^2}$? or a perturbation thereof by a multiplicative function of $k$? If so, that would be a way to approach the uniform upper bound. $\endgroup$ Oct 2, 2014 at 17:07

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