Sarnak–Xue lattice-point counting conjecture

Let $G=\operatorname{SO}(n,1)^\circ$, let $\rho=\tfrac{n-1}{2}$, and let $\Gamma<G(\mathbb{Z})$ satisfy $[G(\mathbb{Z}):\Gamma]<\infty$. Fix $o\in\mathbb H^n$, let $d$ be hyperbolic distance, and define

$$N(T,q):=\#\{\gamma\in\Gamma(q):d(\gamma(o),o)\le T\}.$$

Sarnak–Xue conjecture. For any $T\gg1$ and $q\gg1$,

$$N(T,q)\ll_\epsilon\frac{e^{(n-1+\epsilon)T}}{[\Gamma:\Gamma(q)]}+e^{\rho T}$$

for any $\epsilon>0$, with implied constant independent of $T$ and $q$. This estimate is a uniform lattice-point bound implying the multiplicity conjecture; it was proved in the cited discussion for cocompact arithmetic subgroups of $\operatorname{SO}(n,1)$ when $n=2,3$, and remains open in general.