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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?

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    $\begingroup$ 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). $\endgroup$
    – Ofir Gorodetsky
    Commented Sep 3 at 17:54
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    $\begingroup$ 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 $\endgroup$
    – Asaf
    Commented Sep 3 at 17:59
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    $\begingroup$ 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. $\endgroup$
    – Ofir Gorodetsky
    Commented Sep 3 at 21:34

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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-Rudick-Sarnak and Blomer-Lutsko references.

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