Optimal strong approximation conjecture for $\operatorname{SL}_{N}(\mathbb{Z})$

Let $N$ be a positive integer, let $\Gamma=\operatorname{SL}_{N}(\mathbb{Z})$, let $q\in\mathbb{Z}_{>0}$, and set

$$G_q=\operatorname{SL}_{N}(\mathbb{Z}/q\mathbb{Z}),\qquad \pi_q:\Gamma\to G_q$$

to be the quotient map. For a matrix $\gamma$, write $\lVert\gamma\rVert_\infty$ for the infinity norm of its coordinates.

Optimal strong approximation conjecture. For every $\epsilon>0$, as $q\to\infty$, there exists a set $Y\subset G_q$ with

$$|Y|\geq |G_q|\left(1-o_\epsilon(1)\right),$$

such that for every $y\in Y$ there exists $\gamma\in\Gamma$ satisfying

$$\lVert\gamma\rVert_\infty\leq q^{(N^2-1)/(N^2-N)+\epsilon},\qquad \pi_q(\gamma)=y.$$

This is the natural higher-rank analogue of Sarnak's optimal strong approximation theorem. The exponent is suggested by the comparison between the growth rate of bounded elements in $\operatorname{SL}_N(\mathbb{Z})$ and the size of $\operatorname{SL}_N(\mathbb{Z}/q\mathbb{Z})$; the paper's abstract proves the corresponding projective-action result for $N=3$, while this general formulation is presented as the extension one is led to consider.