Asymptotic number of invertible matrices with integer entries - MathOverflow

Asymptotic number of invertible matrices with integer entries

2012-06-18T18:21:21Z

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?

Answer by Igor Rivin for Asymptotic number of invertible matrices with integer entries

2012-06-18T18:54:14Z

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).

Answer by Gerry Myerson for Asymptotic number of invertible matrices with integer entries

2012-06-19T02:12:18Z

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.