Asymptotic number of invertible matrices with integer entries - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T19:29:55Z http://mathoverflow.net/feeds/question/99929 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99929/asymptotic-number-of-invertible-matrices-with-integer-entries Asymptotic number of invertible matrices with integer entries Kofi 2012-06-18T18:21:21Z 2012-06-19T06:36:18Z <p>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 $M_r$. </p> <p><strong>Question</strong>: Does $p(r)$ possess an asymptotic expansion in $r$ as $r \rightarrow \infty$, and if yes, what is it?</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/99929/asymptotic-number-of-invertible-matrices-with-integer-entries/99933#99933 Answer by Igor Rivin for Asymptotic number of invertible matrices with integer entries Igor Rivin 2012-06-18T18:54:14Z 2012-06-19T03:49:48Z <p>You actually care about the number of <em>singular</em> matrices (which is the difference between the number of invertible matrices and the number of unrestricted matrices). This has been studied: see </p> <p>Author Yonatan R. Katznelson Title: Integral Matrices of Fixed Rank Journal: proceedings of the AMS, 120(3) 1994</p> <p><strong>ADDITION</strong> It would be useful to adjoin my comments to @Gerry's answer: </p> <p>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:</p> <p>Newman, Morris(1-UCSB) Counting modular matrices with specified Euclidean norm. J. Combin. Theory Ser. A 47 (1988), no. 1, 145–149. </p> <p>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</p> <p>Eskin, Alex(1-PRIN); McMullen, Curt(1-CA) Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71 (1993), no. 1, 181–209. </p> <p>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).</p> http://mathoverflow.net/questions/99929/asymptotic-number-of-invertible-matrices-with-integer-entries/99962#99962 Answer by Gerry Myerson for Asymptotic number of invertible matrices with integer entries Gerry Myerson 2012-06-19T02:12:18Z 2012-06-19T02:12:18Z <p>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): </p> <p>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. </p> <p>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. </p>