8
$\begingroup$

I'm looking for the asymptotic order of growth of the number of points in algebraic groups, such as $\mathrm{SL}_n(\mathbb{Z})$, of height/norm at most $X$, i.e. all entries are at most $X$ in absolute value. (One can ask the same question for other equivalent norms, such as the spectral $L^2$-norm, without changing the behavior very much; but note that I do NOT want to order matrices by the length of a word in a predetermined generating set, as Rivin does to skirt this question.)

For $\mathrm{SL}_2(\mathbb{Z})$, the rows must be successive terms of a Farey sequence of order at most $X$ (up to permutations and sign changes), giving $\asymp X^2$ matrices. What about higher $\mathrm{SL}_n$ (or other groups such as $\mathrm{Sp}_{2n}(\mathbb{Z})$ or the split orthogonal group)? Are there well-known nontrivial lower and upper bounds?

Improve this question
$\endgroup$
3
  • 6
    $\begingroup$ If the norm is the Euclidean one, this goes back to Duke-Rudnick-Sarnak, see Example 1.6 and Equation (1.14), see "Density of integer points on affine homogeneous varieties" (Duke Math. J. 71, No. 1, 143-179, 1993). $\endgroup$ Commented Sep 3, 2024 at 17:54
  • 3
    $\begingroup$ The more general/standard approach nowadays is via homogeneous dynamics, in particular the Eskin-McMullen argument (also Benoist-Oh)... Probably everything one possibly wants (and more) appears in the book by Gorodnik-Nevo $\endgroup$ Commented Sep 3, 2024 at 17:59
  • 4
    $\begingroup$ Let me also add that the state of the art - in terms of the error term in the asymptotics - is Blomer and Lutsko's recent paper "Hyperbolic lattice point counting in unbounded rank" (J. Reine Angew. Math. 812, 257-274, 2024). They also cite the following paper of Gorodnik, Nevo and Yehoshua: "Counting lattice points in norm balls on higher rank simple Lie groups" (Math. Res. Lett. 24, No. 5, 1285-1306, 2017), which seems to be a good reference for the kind of math that Asaf mentioned. All these works are still about the Euclidean norm. $\endgroup$ Commented Sep 3, 2024 at 21:34

1 Answer 1

Reset to default
10
$\begingroup$

The order of growth is $$ c_n X^{n^2 - n}, $$ where $c_n$ is a constant, explicit if we're working with the Euclidean norm. Thanks to Ofir Gorodetsky for pointing out the Duke-Rudnick-Sarnak and Blomer-Lutsko references.

Improve this answer
$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.