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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?

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Me thinks 2/(2r+1) is closer to p(r) if you ask in your example that the inverse is also an integer matrix. Otherwise, my reading wold give p(r) as 2r/(2r+1). Gerhard "Ask Me About System Design" Paseman, 2012.06.18 –  Gerhard Paseman Jun 18 '12 at 19:10
    
Oh, and I changed p(r) from percentage to fraction to be consistent with your example, otherwise I need to multiply by 100. Gerhard "One Could Also Say Proportion" Paseman, 2012.06.18 –  Gerhard Paseman Jun 18 '12 at 19:14
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I don't deny that $p(r)\to0$, but I don't see the relation to the alleged density of the invertible matrices. –  Gerry Myerson Jun 19 '12 at 1:52

2 Answers 2

You actually care about the number of singular matrices (which is the difference between the number of invertible matrices and the number of unrestricted matrices). This has been studied: see

Author Yonatan R. Katznelson Title: Integral Matrices of Fixed Rank Journal: proceedings of the AMS, 120(3) 1994

ADDITION It would be useful to adjoin my comments to @Gerry's answer:

The OP is NOT asking for enumeration of matrices in $SL(n, \mathbb{Z}),$ but rather for the cardinality of the intersection of $M^n(\mathbb{Z}) \cap GL(n, \mathbb{C}).$ On the other hand, the first asymptotic result for $SL(2, Z)$ I am aware of (using theta functions, with no error term) is given by Morris Newman:

Newman, Morris(1-UCSB) Counting modular matrices with specified Euclidean norm. J. Combin. Theory Ser. A 47 (1988), no. 1, 145–149.

I am unaware of the Selberg reference. However, the Newman result was generalized by Duke, Rudnick, Sarnak in Duke, W.(1-RTG); Rudnick, Z.(1-STF); Sarnak, P.(1-STF) Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), no. 1, 143–179. (the authors were unaware of Newman's work), with full asymptotics, and in a companion paper, a "softer" result was derived by Eskin and McMullen by ergodic-theoretic methods in the very well-known paper

Eskin, Alex(1-PRIN); McMullen, Curt(1-CA) Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71 (1993), no. 1, 181–209.

The paper of Yonatan Katznelson cited above is a sort of an off-shoot of Duke/Rudnick/Sarnak (Katznelson was a student of Sarnak, and I believe the paper was a part of his thesis).

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@Igor. How do you know the exact content of the OP ? I personally interpret it as the asymptotics of those integer matrices such that $\det M=\pm1$, rather than $\det M\ne0$. The latter situation is not that appealing. –  Denis Serre Jun 19 '12 at 6:39
    
@Denis: I am reading what the OP wrote: $p(r)$ goes to $0$ as $r\rightarrow \infty,$ as invertible matrices are dense in the set of all matrices. Now, this does not actually make sense as written, but I take it to mean that (s)he he is using $p(r)$ to refers to the proportion of non-invertible matrices, hence my interpretation. In any case, my original answer together with the addition answers both questions. –  Igor Rivin Jun 19 '12 at 14:37

Quoting from the review, by Graham Everest, of Christian Roettger, Counting invertible matrices and uniform distribution, J. Théor. Nombres Bordeaux 17 (2005), no. 1, 301–322, MR2152226 (2006c:11135):

Write $h(A)$ for the largest coefficient in absolute value of a $2\times2$ matrix with integer entries. The "hyperbolic circle problem" asks how many such matrices $A$ in SL$_2({\bf Z})$ have $h(A)\lt t$ as $t\to\infty$. The answer is an asymptotic formula with main term $Ct^2$ for some explicit constant $C\gt0$. The best known error is of shape $O(t^{{2\over3}+\epsilon})$ which was obtained by Selberg.

No citation for the Selberg result is given. Anyway, this suggests that even for the case $n=2$ an asymptotic expansion will not be easy to come by.

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The case of $SL(2, Z)$ is analyzed in a paper of Morris Newman's from around 1990 -- notice that this is NOT the question the OP is asking, since he cares about matrices invertible over the reals, not over $\mathbb{Z}$ The Newman result is generalized greatly in the very well-known paper of W. Duke, Z. Rudnick, and P. Sarnak, and the paper of Katznelson I cite is a sort of a follow-up (Katznelson was a student of Sarnak's at the time, and this was his thesis). –  Igor Rivin Jun 19 '12 at 3:34
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OP asks about "invertible matrices in $M_r$." I take that to mean matrices in $M_r$ with inverses in $M_r$. You don't. Which one of us is right is unclear, especially in light of the $1/(2r+1)$ OP gets in the case $n=1$, which is not consistent with either interpretation. Only OP knows what was intended, and until OP clarifies, we don't really know what OP wants. –  Gerry Myerson Jun 19 '12 at 5:49

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