Let $\|\cdot \|$ be some matrix norm on the space of $n \times n$ matrices. Denote $$ M(r) := \{ A \in \mathrm{Mat}_{n \times n}(\mathbb{Z}) \mid \| M \| \leq r \}.$$ Denote by $p(r)$ the percentage of invertible matrices in $M_r$.

Question: Does $p(r)$ possess an asymptotic expansion in $r$ as $r \rightarrow \infty$, and if yes, what is it?

Of course, this does depend on the norm used and the dimension. Taking in the simplest case $n=1$ (thus eliminating the question about which norm to take), one gets $p(r) = 1/(2r + 1)$. Of course, in general, $p(r) \longrightarrow 0$ as $r \rightarrow \infty$ as the invertible matrices are dense in the set of all matrices.

Of course, it should be easy to check for example, how many of the matrices that only have the numbers $-10, \dots, 10$ as entries are invertible. But what about an asymptotic series? Did someone think about this?

Let $\|\cdot \|$ be some matrix norm on the space of $n \times n$ matrices. Denote $$ M(r) := \{ A \in \mathrm{Mat}_{n \times n}(\mathbb{Z}) \mid \| M \| \leq r \}.$$ Denote by $p(r)$ the fraction of invertible matrices in $M_r$.

Question: Does $p(r)$ possess an asymptotic expansion in $r$ as $r \rightarrow \infty$, and if yes, what is it?

Of course, this does depend on the norm used and the dimension. Taking in the simplest case $n=1$ (thus eliminating the question about which norm to take), one gets $p(r) = 2/(2r + 1)$. Of course, in general, $p(r) \longrightarrow 0$ as $r \rightarrow \infty$ as the invertible matrices are dense in the set of all matrices.

Of course, it should be easy to check for example, how many of the matrices that only have the numbers $-10, \dots, 10$ as entries are invertible. But what about an asymptotic series? Did someone think about this?