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Here's my question:

Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having determinant greater than C?

Thanks for helping!

G.

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For $C$ fixed and $B$ large this is (to first order) some constant times $B^{k^2},$ with the error term of order $O(B^{k^2-k}),$ for more see the classic paper of W. Duke, Z. Rudnick, and P. Sarnak in Duke Journal (around 20 years ago). However, this might not be the regime you are interested in.

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    $\begingroup$ What is this $n$? $\endgroup$
    – Robert Israel
    Commented Jul 24, 2014 at 15:11
  • $\begingroup$ Thanks a lot for the answer but that is actually not the regime I am interested in.. I will have a look at the paper $\endgroup$
    – g1a
    Commented Jul 24, 2014 at 18:45
  • $\begingroup$ @RobertIsrael, the $n$ is $B,$ of course :) $\endgroup$
    – Igor Rivin
    Commented Jul 24, 2014 at 23:49
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    $\begingroup$ "A mathematician is one who thinks $A$, says $B$, and writes $C$ --- but it's really $D$." $\endgroup$
    – Gerry Myerson
    Commented Jul 25, 2014 at 0:04

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