$\check {N}AMN$ -coordinates

The product map $\check {N} \times A \times M \times N \to G$ is a diffeomorphism onto a Zariski open neighborhood of $e$ . The same is true if we permute $\check {N},A,M,N$ . This fact is used to prove the next two lemmas.

Proof. Let $\varepsilon \gt 0$ be sufficiently small so that the image of the diffeomorphism given by the product map $N \times \check {N}\times A \times M \to G$ contains $\check {N}_\varepsilon N_\varepsilon$ . Then by the implicit function theorem, there are smooth functions $x,y,a,m$ defined on $\check {N}_\varepsilon \times N_\varepsilon$ such that

\begin{align*} hn=x(h,n)y(h,n)a(h,n)m(h,n) \in N_{O(\varepsilon )}\check {N}_{O(\varepsilon )}A_{O(\varepsilon )}M_{O(\varepsilon )}.\end{align*}

Note that $y(e,n)=e$ and if $h \in \check {N}_{\varepsilon _1}$ , then $y(h,n)\in y(e,n)N_{O(\varepsilon _1)} = \check {N}_{O(\varepsilon _1)}$ . Similarly, if $n \in N_{\varepsilon _2}$ , then $x(h,n) \in N_{O(\varepsilon _2)}$ . This proves (1).

For (2), since $\check {U}$ is bounded, for a $\varepsilon \gt 0$ that is sufficiently small the image of the product map $M \times A \times N \times \check {N} \to G$ contains $\check {U}N_{O(\varepsilon )}$ . A similar application of the implicit function theorem finishes the proof.

Proof. Let $\varepsilon \gt 0$ , $g_1 \in G_\varepsilon$ and $g=manh\in MAN\check {U}$ . For $\varepsilon$ that is sufficiently small, we may write $g_1 = m_1a_1n_1h_1 \in M_{O(\varepsilon )}A_{O(\varepsilon )}N_{O(\varepsilon )}\check {N}_{O(\varepsilon )}$ since the product map $M \times A \times N \times \check {N} \to G$ is a diffeomorphism onto an open neighborhood of $e$ . Let $m'=mm_1\in mM_{O(\varepsilon )}$ , $a'=aa_1 \in aA_{O(\varepsilon )}$ , $n'=(m_1a_1)^{-1}n(m_1a_1) \in nN_{O(\varepsilon )}$ and $h'=(m_1a_1)^{-1}h(m_1a_1) \in h\check {N}_{O(\varepsilon )}$ so that $gg_1 = m'a'n'h'n_1h_1$ . Note that $h'n_1 \in \check {U}\check {N}_{O(\varepsilon )}N_{O(\varepsilon )}$ and $\check {U}\check {N}_{O(\varepsilon )} \subset \check {N}$ is bounded. Then by Lemma2.1(2), if $\varepsilon$ is sufficiently small, then we can write $h'n_1 = m_2a_2n_2h''$ where $m_2a_2n_2 \in M_{O(\varepsilon )}A_{O(\varepsilon )}N_{O(\varepsilon )}$ and $h'' \in h\check {N}_{O(\varepsilon )}$ . Then

\begin{align*} gg_1 = m'a'n'h'n_1h_1 = m'a'n'm_2a_2n_2h''h_1= m''a''n''n_2h''h_1 \end{align*}

where $m''=m'm_2 \in mM_{O(\varepsilon )}$ , $a''=a'a_2\in aA_{O(\varepsilon )}$ and $n'' = (m_2a_2)^{-1}n'(m_2a_2) \in nN_{O(\varepsilon )}$ . This completes the proof of (1).

The proof of (2) is similar and (3) can be deduced from (1) to (2) in a similar manner.

Henceforth, let $\Gamma \lt G$ be a Zariski-dense discrete subgroup.

Limit set, limit cone and holonomy group

Let $m_{\mathcal{F}}$ denote the unique $K$ -invariant probability measure on $\mathcal{F}$ . The limit set $\Lambda \subset \mathcal{F}$ of $\Gamma$ is defined by

\begin{align*} \Lambda = \{\xi \in \mathcal{F} : \exists \{\gamma _n\}_{n \in \mathbb N} \subset \Gamma\ \textrm{such that}\ (\gamma _n)_*m_{\mathcal{F}} \xrightarrow {n \to \infty } D_\xi \} \end{align*}

where $D_\xi$ denotes the Dirac measure at $\xi$ . It is the unique $\Gamma$ -minimal subset of $\mathcal{F}$ [Reference BenoistBen97].

Any $g\in G$ can be written as the commuting product $g=g_hg_e g_u$ where $g_h$ is hyperbolic, $g_e$ is elliptic and $g_u$ is unipotent. The hyperbolic component $g_h$ is conjugate to a unique element $\exp \lambda (g) \in A^+$ and $\lambda (g)$ is called the Jordan projection of $g$ . When $\lambda (g)\in \operatorname {int} {\mathfrak a}^+$ , $g\in G$ is called loxodromic in which case $g_u$ is necessarily trivial and $g_e$ is conjugate to an element $m(g)\in M$ which is unique up to conjugation in $M$ . We call its conjugacy class $[m(g)]\in [M]$ the holonomy of $g$ .

The limit cone $\mathcal{L} = \mathcal{L}_{\Gamma}$ of $\Gamma$ is the smallest closed cone containing the Jordan projection $\lambda (\Gamma )$ . It is convex and with a non-empty interior [Reference BenoistBen97] which we denote by $\textrm {int}\;\mathcal{L}$ . We note that $\lambda (g^{-1}) = \mathsf{i}(\lambda (g))$ for all $g \in G$ and hence $\mathcal{L} = \mathsf{i}(\mathcal{L})$ .

The holonomy group of $\Gamma$ is the closed subgroup $M_{\Gamma} \lt M$ generated by all of the holonomies in $\Gamma$ . By [Reference Guivarc’h and RaugiGR07, Corollary 1.10], $M_{\Gamma}$ is a normal subgroup of $M$ of finite index. In particular if $M$ is connected, then $M_{\Gamma} = M$ . If $G$ is of rank one, we always have $M = M_{\Gamma}$ for any Zariski-dense $\Gamma$ , since either $M$ is connected or $G=\textrm {SL}_2{\mathbb R}$ and $M_{\Gamma} = M = \{\pm \big (\begin{smallmatrix}1 & 0\\0 & 1 \end{smallmatrix}\big )\}$ [Reference Choi and GoldmanCG93, Lemma 2]. In general, there are examples of Zariski-dense subgroups with $M_{\Gamma} \ne M$ (e.g., Hitchin representations [Reference LabourieLab06, Theorem 1.5]).

For $g \in G$ , let $\mu (g)$ denote the Cartan projection of $g$ , that is, $\mu (g) \in \mathfrak{a}^+$ is the unique element in $\mathfrak{a}^+$ such that

\begin{align*} g \in K\exp (\mu (g))K.\end{align*}

We note that $\mu (g^{-1}) = \mathsf{i}(\mu (g))$ for all $g \in G$ .

Conformal measures

The Iwasawa cocycle $\sigma : G \times \mathcal{F} \to \mathfrak{a}$ is the map which assigns to each $(g,kM) \in G \times \mathcal{F}$ the unique element $\sigma (g,kM) \in \mathfrak{a}$ such that $gk \in Ka_{\sigma (g,\xi )}N$ . The $\mathfrak{a}$ -valued Busemann function $\beta : \mathcal{F} \times G \times G \to \mathfrak{a}$ is defined by

\begin{align*} \beta _\xi (g_1,g_2) = \sigma (g_1^{-1}, \xi ) - \sigma (g_2^{-1}, \xi ) \end{align*}

for all $g_1,g_2 \in G$ and $\xi \in \mathcal{F}$ .

Definition 2.3 (Growth indicator). The growth indicator $ $ $\psi _{\Gamma} : \mathfrak{a}^+ \to {\mathbb R} \cup \{-\infty \}$ of $\Gamma$ is defined by

\begin{align*} \psi _{\Gamma} (w) = \|w\| \inf _{\textrm{open cones }\mathcal{C}\ni w} \tau _{\mathcal{C}} \quad \textrm{for all non-zero}\ w\ \in\ \mathfrak{a}^+ \end{align*}

where $\tau _{\mathcal{C}}$ is the abscissa of convergence of the series $t \mapsto \sum _{\gamma \in \Gamma , \mu (\gamma ) \in \mathcal{C}} e^{-t\|\mu (\gamma )\|}$ . We set $\psi _{\Gamma} (0)=0$ .

It is concave, upper-semicontinuous, and satisfies $\psi _{\Gamma} |_{\textrm {int}{\mathcal L}}\gt 0$ [Reference QuintQui02, Theorem 4.2.2]. By [Reference QuintQui02, Lemma 3.1.1], when $\psi _{\Gamma} (w)\gt 0$ (i.e., $w \in \textrm {int}\mathcal{L}$ ), we have

\begin{align*} \psi _{\Gamma} (w) = \|w\|\inf _{\textrm{open cones }\mathcal{C} \ni w}\limsup _{T\to \infty }\tfrac {1}{T} \log \#\{\gamma \in \Gamma : \mu (\gamma ) \in \mathcal{C}, \; \; \|\mu (\gamma )\| \le T\}.\end{align*}

Given a closed subgroup $\Delta \lt G$ , a Borel probability measure $\nu$ on $\mathcal{F}$ is called a $\Delta$ -conformal measure $ $ if there exists $\psi \in \mathfrak{a}^*$ such that for any $\gamma \in \Delta$ and $\xi \in \mathcal{F}$ ,

\begin{align*} \frac {d\gamma _*\nu }{d\nu }(\xi ) = e^{\psi (\beta _\xi (e,\gamma ))}\end{align*}

where $\gamma _*\nu (Q) = \nu (\gamma ^{-1}Q)$ for any Borel subset $Q \subset \mathcal{F}$ . In that case, we call $\nu$ a $(\Delta ,\psi )$ -conformal measure $ $ .

Generalized BMS measures

We recall the definitions of generalized Bowen–Margulis–Sullivan measures using the Hopf parametrization of $G/M$ . There is a unique open $G$ -orbit in $\mathcal{F} \times \mathcal{F}$ given by

(2.3) \begin{align} \mathcal{F}^{(2)} = G.(e^+,e^-) \subset \mathcal{F}\times \mathcal{F}. \end{align}

If $(x,y) \in \mathcal{F}^{(2)}$ , then we say that $x$ and $y$ are in general position. The Hopf parametrization is a diffeomorphism $G/M \to \mathcal{F}^{(2)} \times \mathfrak{a}$ defined by

For a pair $(\nu _{\psi _1}, \nu _{\psi _2})$ of $(\Gamma , \psi _1)$ - and $(\Gamma , \psi _2)$ -conformal measures on $\mathcal F$ , the generalized Bowen–Margulis–Sullivan measure $\mathsf m=\mathsf{m}_{\nu _{\psi _1},\nu _{\psi _2}}$ is defined on $G/M \cong \mathcal{F}^{(2)} \times \mathfrak{a}$ by

(2.4) \begin{align} {\mathsf{m}}_{\nu _{\psi _1},\nu _{\psi _2}}(gM) = e^{\psi _1\left (\beta _{g^+}(e,g)\right ) + \psi _2\left (\beta _{g^-}(e,g)\right )} \, d\nu _{\psi _1}(g^+) \, d\nu _{\psi _2}(g^-) \, dw. \end{align}

The measure $\mathsf m$ is left $\Gamma$ -invariant and right $A$ -quasi-invariant. It is $A$ -invariant if and only if $\psi _2=\psi _1\circ {\operatorname {i}}$ . The measure $\mathsf m$ descends to a measure on $\Gamma \backslash G/M$ and by lifting it using the Haar probability measure on $M$ , we also obtain a measure on $\Gamma \backslash G$ . Abusing notation, we also denote it by ${\mathsf{m}} =\mathsf{m}_{\nu _{\psi _1},\nu _{\psi _2}}$ as well.

Local mixing

We recall the local mixing theorem for the Haar measure on $\Gamma \backslash G$ which will be used in Section 4. Let $dx$ denote the right $G$ -invariant measure on $\Gamma \backslash G$ induced by the Haar measure on $G$ . Given an inner product $\langle \cdot ,\cdot \rangle _*$ on $\mathfrak{a}$ , let $I: \ker \psi _{\mathsf v} \to {\mathbb R}$ be defined by

(3.2) \begin{align} I(u) = \langle u, u\rangle _* - \frac {\langle u, \mathsf{v} \rangle _*^2}{\langle \mathsf{v}, \mathsf{v}\rangle _*}\quad \textrm{for all}\ u \in \ker \psi _{\mathsf v}. \end{align}

Theorem 3.3 ([Reference Chow and SarkarCS23, Theorem 1.3] and [Reference Edwards, Lee and OhELO22, Theorem 3.4]). There exist $\kappa _{\mathsf{v}} \gt 0$ and an inner product $\langle \cdot , \cdot \rangle _*$ on $\mathfrak{a}$ such that for any $u \in \ker \psi _{\mathsf v}$ and $\phi _1, \phi _2 \in C_{\textrm{c}}(\Gamma \backslash G)$ , we have

\begin{align*} \lim _{t \to +\infty } t^{\frac {{r} - 1}{2}}e^{(2\rho - \psi _{\mathsf v})(t\mathsf{v} + \sqrt {t}u)} \int _{\Gamma \backslash G} \phi _1(xa_{t\mathsf{v} + \sqrt {t}u}) \phi _2(x) \, dx \\ =\frac {\kappa _{\mathsf{v}}e^{-I(u)}}{|m_{\mathcal{X}_{\mathsf v}}|} \sum _{Z} \mathsf{m}^{\textrm{BR}}_{\mathsf v}\bigr |_{Z\check {N}}(\phi _1)\cdot \mathsf{m}^{\textrm{BR}_\star }_{\mathsf v}\bigr |_{ZN}(\phi _2) \end{align*}

where the sum is taken over all $A$ -ergodic components $Z$ of $\mathsf{m}_{\mathsf v}$ .

Moreover, there exist $\eta _{\mathsf{v}}\gt 0$ and $s_{\mathsf{v}}\gt 0$ such that for all $\phi _1, \phi _2 \in C_{\textrm{c}}(\Gamma \backslash G)$ , there exists $D_{\mathsf v}\gt 0$ depending continuously on $\phi _1$ and $\phi _2$ such that for all $(t,u) \in (s_{\mathsf{v}},\infty ) \times \ker \psi _{\mathsf v}$ such that $t\mathsf{v}+\sqrt {t}u \in \mathfrak{a}^+$ , we have

\begin{align*} \left |t^{\frac {{r} - 1}{2}}e^{(2\rho - \psi _{\mathsf v})(t\mathsf{v} + \sqrt {t}u)} \int _{\Gamma \backslash G} \phi _1(xa_{t\mathsf{v} + \sqrt {t}u}) \phi _2(x) \, dx\right | \le D_{\mathsf{v}} e^{-\eta _{\mathsf{v}} I(u)}. \end{align*}

Proof of Theorem4.2 assuming Theorem4.5

Choose $f_1 \in C_{\textrm{c}}(\ker \psi _{\mathsf v})$ with $\int f_1(u)\, du=1$ . In view of (4.3), the function

\begin{align*} f= \unicode {x1D7D9}_{{\mathcal{X}_{\mathsf v}}} \otimes f_1\end{align*}

can be considered as a function in $C_{\textrm{c}}(\Omega )$ and

\begin{align*} \mathsf{m}_{\mathsf v}(f) = |m_{\mathcal{X}_{\mathsf v}}|\int f_1(u)\, du = |m_{\mathcal{X}_{\mathsf v}}|.\end{align*}

Therefore, for every $[\gamma ] \in [\Gamma _{\textrm{prim}}]$ ,

\begin{align*} \int _{C(\gamma )} f = \psi _{\mathsf v}(\lambda (\gamma ))\int f_1(u)\, du = \psi _{\mathsf v}(\lambda (\gamma )).\end{align*}

By applying Theorem4.5 to this function $f$ , we obtain (4.1) with $[\Gamma ]$ replaced with $[\Gamma _{\textrm{prim}}]$ :

(4.4) \begin{align} \sum _{ [\gamma ] \in [\Gamma _{\textrm{prim}}],\, \lambda (\gamma ) \in {\mathbb{T}}_{T,b}} \varphi (m(\gamma )) \sim \frac {\kappa _{\mathsf{v}}}{\delta _{\mathsf v} } \int _{\mathsf{K}} e^{\delta _{\mathsf v} b(u)} \, du \cdot \int _{M_{\Gamma} }\varphi \, dm \cdot \frac {e^{\delta _{\mathsf v} T}}{T^{({r}+1)/2}}. \end{align}

We claim that this asymptotic remains true when $[\Gamma _{\textrm{prim}}]$ is replaced with $[\Gamma ]$ . To see that, we first take $\varphi \equiv 1$ and $b \equiv 0$ and obtain

(4.5) \begin{align} \#\{[\gamma ]\in [\Gamma _{\textrm{prim}}]:\lambda (\gamma ) \in {\mathbb{T}}_T\} \sim \frac {\kappa _{\mathsf{v}}}{\delta _{\mathsf v} [M:M_{\Gamma} ]}\int _{\mathsf{K}}e^{\delta _{\mathsf v} b(u)}\,du \cdot \frac {e^{\delta _{\mathsf v} T}}{T^{({r}+1)/2}}. \end{align}

Suppose that $\mathsf{K}$ is a convex set containing $0$ . Observe that if $\lambda (\gamma _0^j) \in {\mathbb{T}}_{T}({\mathsf v},\mathsf{K},b)$ , then $\lambda (\gamma _0) = \frac {1}{j}\lambda (\gamma _0^j) \in {\mathbb{T}}_{T/j}({\mathsf v},\frac {1}{j}\mathsf{K},b) \subset {\mathbb{T}}({\mathsf v},\mathsf{K},b)$ by the convexity of $\mathsf K$ . ThereforeFootnote ⁵

\begin{align*} \#\{[\gamma ]\in [\Gamma ]:\lambda (\gamma ) \in {\mathbb{T}}_T\} - \#\{[\gamma ]\in [\Gamma _{\textrm{prim}}]:\lambda (\gamma ) \in {\mathbb{T}}_T\} \\ \ll T \#\{[\gamma ]\in [\Gamma _{\textrm{prim}}]:\lambda (\gamma ) \in {\mathbb{T}}_{T/2}\} \ll e^{\delta _{\mathsf v} T/2}. \end{align*}

Identifying $\ker \psi _{\mathsf v}$ with ${\mathbb R}^{r-1}$ , we next consider the case when $\mathsf{K}$ is a box $\prod _{i=1}^{r-1} [a_i,b_i]$ . Set $B_j = [0,b_1] \times \cdots \times [0,a_j] \times \cdots \times [0,b_{r-1}]$ and note that for any $f\in C(\ker \psi _{\mathsf v})$ , we have

(4.6) \begin{align} \int _{\prod _{i=1}^{r-1} [a_i,b_i]}f = \int _{\prod _{i=1}^{r-1} [0,b_i]}f - \sum _{j=1}^{r-1}\int _{B_j}f + (r-2)\int _{\prod _{i=1}^{r-1} [0,a_i]}f. \end{align}

Theorem4.2 now follows for $\mathsf{K}=\prod _{i=1}^{r-1} [a_i,b_i]$ by applying Theorem4.2 to each box containing $0$ on the right-hand side of (4.6).

For general $\mathsf{K}$ , using the hypothesis that the boundary of $\mathsf{K}$ has zero Lebesgue measure, we can approximate $\mathsf{K}$ above and below by boxes and apply Theorem4.2 to the boxes.

Proof of Theorem1.4

We now deduce Theorem1.4 from Theorem4.2. Fix $\varepsilon \gt 0$ , $w \in \mathfrak{a}$ and conjugation-invariant Borel subset $\Theta \subset M$ with smooth boundary. Recall that ${\mathbb{T}}={\mathbb{T}}({\mathsf v}, {\varepsilon }, w)=\{u+w\in {\mathfrak a}^+: \|u-\mathbb R {\mathsf v}\| \le {\varepsilon }\}$ . Let

\begin{align*} B_T=\{v \in {\mathbb{T}}: \|v\| \le T\}.\end{align*}

Then we want an asymptotic for

\begin{align*} \#\{[\gamma ]\in [\Gamma ]: \lambda (\gamma ) \in B_T, \, m(\gamma ) \in \Theta \}.\end{align*}

Decomposing $w$ as an element in the direct sum $\mathfrak{a}=\ker \psi _{\mathsf v}\oplus {\mathbb R}{\mathsf v}$ , we may assume without loss of generality that $w \in \ker \psi _{\mathsf v}$ . Note that the set $B_T$ is not a truncated tube as defined at the beginning of Section 4. Let

\begin{align*} \mathsf{K} = \{u \in \ker \psi _{\mathsf v}: \|u-w-{\mathbb R}{\mathsf v}\| \le \varepsilon \}\end{align*}

so that

\begin{align*} {\mathbb{T}}={\mathbb{T}}({\mathsf v},\mathsf{K}) = \{t{\mathsf v}+u \in \mathfrak{a}^+: t\in {\mathbb R}, \, u \in \mathsf{K}\}.\end{align*}

For $u\in \mathsf{K}$ , define $b(u)\in {\mathbb R}$ as the unique number such that $u + b(u){\mathsf v}$ is orthogonal to $\mathsf v$ with respect to the inner product inducing the norm $\|\cdot \|$ . We claim that for fixed $\varepsilon '\gt 0$ , when $T$ is sufficiently large, we have

(4.7) \begin{align} {\mathbb{T}}_{T,b-\varepsilon '} \subset B_T \subset {\mathbb{T}}_{T,b}. \end{align}

Suppose that $v \in {\mathbb{T}}_{T,b-\varepsilon '}$ . Then $v=t{\mathsf v}+u$ for some $u \in \mathsf{K}$ and $t \le T+b(u)-\varepsilon '$ . We have

\begin{align*} & \|t{\mathsf v}+u\| = \|(t-b(u)){\mathsf v} + (u+b(u){\mathsf v})\| \\ & = \sqrt {(t-b(u))^2+\|u+b(u){\mathsf v}\|^2} \le \sqrt {(T-\varepsilon ')^2 + \|u+b(u){\mathsf v}\|^2} \le T \end{align*}

where the last inequality holds when $T$ is sufficiently large since $u+b(u){\mathsf v}$ is bounded. This shows that $v \in B_T$ .

Now suppose that $v \in B_T$ . Then $v=t{\mathsf v}+u$ for some $u \in \mathsf{K}$ and $t \in {\mathbb R}$ with $\|t{\mathsf v}+u\| \le T$ . We have

\begin{align*} t-b(u)\le \|t{\mathsf v}+u\| = \sqrt {(t-b(u))^2 + \|u+b(u){\mathsf v}\|^2} \le T\end{align*}

and hence $t \le T+b(u)$ and $v \in {\mathbb{T}}_{T,b}$ . This proves (4.7).

It follows from (4.7) that

\begin{align*} \#\{[\gamma ]\in [\Gamma ]:\lambda (\gamma ) \in {\mathbb{T}}_{T,b}\} \le \#\{[\gamma ]\in [\Gamma ]: \lambda (\gamma ) \in B_T, \, m(\gamma ) \in \Theta \} \\ \le \#\{[\gamma ]\in [\Gamma ]:\lambda (\gamma ) \in {\mathbb{T}}_{T,b}\}. \end{align*}

Applying Theorem4.2 to ${\mathbb{T}}_{T,b-\varepsilon '}$ and ${\mathbb{T}}_{T,b}$ and then taking $\varepsilon ' \to 0$ , we conclude that

\begin{align*} \lim _{T\to \infty } \frac {T^{(r-1)/2}}{e^{\delta _{\mathsf v} T}}\#\{[\gamma ]\in [\Gamma ]: \lambda (\gamma ) \in B_T, \, m(\gamma ) \in \Theta \} = \kappa _{\mathbb{T}} \cdot \operatorname {Vol}_M(\Theta \cap M_{\Gamma} ). \end{align*}

where

(4.8) \begin{align} \kappa _{\mathbb{T}}= \frac {\kappa _{\mathsf{v}}}{\delta _{\mathsf v}}\int _{\mathsf{K}}e^{\delta _{\mathsf v} b(u)} \, du . \end{align}

Counting in $\check {N}AMN$ -coordinates

Let $\nu$ and $\nu _{{\operatorname {i}}}$ denote the unique $(\Gamma , \psi _{{\mathsf v}})$ - and $(\Gamma ,\psi _{{\operatorname {i}}({\mathsf v})})$ -conformal measures on $\mathcal F$ respectively. Let $\operatorname {Vol}_M$ denote the Haar probability measure on $M$ . Fix bounded Borel sets

\begin{align*} \check {\Xi } \subset \check {N},\;\; \Xi \subset N,\;\; \Theta \subset M\end{align*}

with non-empty relative interiors and null boundaries:

\begin{align*} \nu (\partial \check {\Xi } e^+)=\nu _{\mathsf{i}}(\partial \Xi ^{-1}e^-)=\operatorname {Vol}_M(\partial \Theta )=0.\end{align*}

For $T \gt 0$ , consider the following subset of $\check {N}A^+MN$ :

(5.1) \begin{align} S_{T,b}=S_{T,b}(\check {\Xi },\Xi ,{\mathbb{T}},\Theta ) = \check {\Xi }\exp ({\mathbb{T}}_{T,b}) \Theta \Xi . \end{align}

Define the measures $\tilde {\nu }$ on $\check {N}$ and $\tilde {\nu }_{\mathsf{i}}$ on $N$ by

\begin{align*} d\tilde {\nu }(h) = e^{\psi _{\mathsf v}(\beta _{h^+}(e,h))} \, d\nu (h^+) ; \quad d\tilde {\nu }_{\mathsf{i}}(n) = e^{(\psi _{\mathsf v}\circ \mathsf{i})(\beta _{n^-}(e,n))}\, d\nu _{\mathsf{i}}(n^-). \end{align*}

In this subsection, we prove an asymptotic for $\# \Gamma \cap S_{T,b}$ .

Proposition 5.1. We have, as $T\to \infty$ ,

(5.2) \begin{align} \#(\Gamma \cap S_{T,b}) \sim c({\mathsf v}, b) \tilde {\nu }(\check {\Xi })\tilde {\nu }_{\mathsf{i}}(\Xi ^{-1})\operatorname {Vol}_M(\Theta )\frac {e^{\delta _{\mathsf v} T}}{T^{({r}-1)/2}} . \end{align}

For a sufficiently small ${\varepsilon }\gt 0$ , consider the following $\varepsilon$ -approximations of $S_{T,b}$ :

\begin{align*} S_{T,b,\varepsilon }^- = \bigcap _{g_1,g_2 \in G_\varepsilon }g_1S_{T,b}g_2; \quad S_{T,b,\varepsilon }^+ = \bigcup _{g_1,g_2 \in G_\varepsilon }g_1S_{T,b}g_2. \end{align*}

In the next lemma, we state a property of the $\check {N}AMN$ -coordinates that we will use to prove Lemma5.2 which bounds the sets $S_{T,b,\varepsilon }^\pm$ by product subsets of $\check {N}A^+MN$ that approximate $S_{T,b}$ .

Proof. We prove the half of the lemma involving $-$ signs; the second half involving the $+$ signs is similar.

For $T\gt 0$ , let ${\mathbb{T}}_{T,\varepsilon }^-$ (respectively $\check {\Xi }_{\varepsilon }^-,\Xi _{\varepsilon } ^-,\Theta _{\varepsilon }^-$ ) be the intersection of ${\mathbb{T}}_{T,b}$ (respectively $\check {\Xi }_{\varepsilon },\Xi _{\varepsilon } ,\Theta _{\varepsilon }$ ) and the complement of an $O(\varepsilon )$ -neighborhood of its exterior, or more explicitly,

\begin{align*} {\mathbb{T}}_{T,\varepsilon }^- = {\mathbb{T}}_{T,b} - (\partial {\mathbb{T}}_{T,b}+\mathfrak{a}_{O(\varepsilon )}).\end{align*}

To check (1), let $g\in \check N_{O(\varepsilon )}\check {\Xi }_{\varepsilon }^- \exp ({\mathbb{T}}_{T,\varepsilon }^-) M_{O(\varepsilon )} \Theta ^-_{\varepsilon } \Xi _{\varepsilon }^-N_{O(\varepsilon )}$ . By Lemma2.2(3), it follows that for all $g_1,g_2 \in G_\varepsilon$ ,

\begin{align*} g_1^{-1}gg_2^{-1} \in \check N_{O(\varepsilon )}\check {\Xi }_{\varepsilon }^- A_{O(\varepsilon )}\exp ({\mathbb{T}}_{T,\varepsilon }^-) M_{O(\varepsilon )} \Theta ^-_{\varepsilon } \Xi _{\varepsilon }^-N_{O(\varepsilon )} \subset S_{T,b}.\end{align*}

This implies that $g \in S_{T,b,\varepsilon }^-$ , which establishes (1). It is clear that (2) is satisfied. Moreover, since $\nu (\partial \check {\Xi }e^+) = 0$ , we have

\begin{align*} |\nu (\check {\Xi } e^+) - \nu (\check {\Xi }_{\varepsilon }^-e^+)| \le \nu (\check {N}_{O(\varepsilon )}\partial \check {\Xi }e^+) \xrightarrow {\varepsilon \to 0} 0.\end{align*}

Similarly for $\Xi _{\varepsilon } ^-$ and $\Theta _{\varepsilon }^-$ and hence (3) is satisfied.

An integral over tubes

It is possible to generalize Lemma5.2 by replacing the truncated tubes ${\mathbb{T}}_{T,b}$ with a sequence of sufficiently nice compact subsets of $\mathfrak{a}$ and then attempt to obtain the counting as in Proposition5.1 following the well-known approaches of [Reference Eskin and McMullenEM93, Reference Oh and ShahOS13, Reference Mohammadi and OhMO15], etc. However, in carrying out this in this higher rank and infinite volume setting, putting

(5.4) \begin{align} L({\mathbb{T}}_0)=\frac {\kappa _{\mathsf{v}}}{|m_{\mathcal{X}_{\mathsf v}}|}\int _{t\mathsf{v}+\sqrt {t}u \in {\mathbb{T}}_{0}}e^{\delta _{\mathsf v} t}e^{-I(u)} \, dt \, du \end{align}

where $\kappa _{\mathsf v}$ and $I(u)$ are as in Theorem3.3, we need to control the asymptotic of $L({\mathbb{T}}_{T,b})$ and $L ({\mathbb{T}}_{T,\varepsilon }^\pm )$ as $T\to \infty$ as in Lemma5.4 where the shape of the tubes plays an important role. We will need the following technical lemma in the proof of Lemma5.4.

Lemma 5.3. Define $f_T:\ker \psi _{\mathsf v} \to \mathbb R$ by

where $R_T(u) = \{t\ge 0 : t \le T+b(\sqrt {t}u), \sqrt {t}u \in \mathsf{K} \}$ . We have

(5.5) \begin{align} \lim _{T\to \infty }f_T(u)= \left\{\begin{array}{ll} \frac {1}{\delta _{\mathsf v}}e^{\delta _{\mathsf v} b(u)} & \textrm{if $u\in{\text{int}}\,\mathsf{K}$} \\[5pt] 0 & \textrm{if $u\in\ker\psi_{\mathsf{v}}-\mathsf{K}$} . \end{array}\right.\end{align}

Proof. To compute the desired limit, we will first describe the values in $R_T(u)$ . For convenience, let

\begin{align*} & M_b=\max b; \qquad m_b=\min b; \\[5pt] & J_{u}= \{s\ge 0 : su \in \mathsf{K}\} \quad \textrm{and} \quad J_{u}^2 = \{s^2: s \in J_{u}\} \quad \textrm{for } u\in \ker \psi _{\mathsf v}. \end{align*}

Then

\begin{align*} \begin{aligned} R_T(u/\sqrt {T}) & = \{t \ge 0: t \le T+b(\sqrt {\tfrac {t}T}u), \, \sqrt {\tfrac {t}T}u \in \mathsf{K}\} \\[5pt] & = \{t\ge 0: t \le T+b(\sqrt {\tfrac {t}T}u), \, t \in T J_{u}^2\} \\[5pt] & \subset \left [0,\min \left \{T+M_b, \left (\sup J_{u}^2\right )T\right \}\right ] \end{aligned} \end{align*}

First, fix $u \in \textrm {int} \mathsf{K}$ . Then

\begin{align*} s_u:= \sup \{0\le s\le 1: s u \notin \mathsf K\}\lt 1.\end{align*}

Assume that $T$ is sufficiently large so that

\begin{align*} s_u T \lt T+m_b\lt T+M_b\lt \left (\sup J_{u}^2\right )T.\end{align*}

Observe that $[ s_u T , T+m_b] \subset R_T(u/\sqrt {T})$ and $R_T(u/\sqrt {T}) \cap (T+M_b, \infty ) = \emptyset .$ Note that since $s_u \lt 1$ , we have

(5.6) \begin{align} \frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} \int _0^{s_uT} e^{\delta _{\mathsf v} t}\, dt \le \frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} e^{\delta _{\mathsf v} s_uT} \end{align}

and hence the quantity on the left-hand side goes to $0$ as $T \to \infty$ .

Next we investigate which values in $[T+m_b, T+M_b]$ are in $R_T(u/\sqrt {T})$ . Fix $\varepsilon \gt 0$ . Using continuity of $b$ , we note that for all $T$ sufficiently large and for all $\eta \in (\varepsilon , b(u) - m_b)$ , we have

\begin{align*} T+b(u)-\eta \lt T+b\left (\sqrt {\tfrac {T+b(u)-\eta }T}u\right )\end{align*}

and henceFootnote ⁶

(5.7) \begin{align} [s_u T , T+b(u) - o_T(1)] \subset R_T(u/\sqrt {T}). \end{align}

Similarly, we note that for all $T$ sufficiently large and for all $\eta \in (\varepsilon , M_b - b(u))$ , we have

\begin{align*} T+b(u)+\eta \gt T+b\left (\sqrt {\tfrac {T+b(u)+\eta }T}u\right )\end{align*}

and hence

(5.8) \begin{align} R_T(u/\sqrt {T})\cap [T+b(u)+o_T(1) , \infty ) = \emptyset . \end{align}

Also note that

(5.9) \begin{align} \frac {e^{- I(u)/T}}{e^{\delta _{\mathsf v} T}} \int _{T+b(u)-o_T(1)}^{T+b(u)+o_T(1)} e^{\delta _{\mathsf v} t}\, dt = \frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} e^{\delta _{\mathsf v} b(u)}(e^{o_T(1)}-e^{-o_T(1)}) \end{align}

goes to $0$ as $T \to \infty$ . Summarizing (5.6)–(5.9), we have shown that to compute $\lim _{T\to \infty }f_T(u)$ , we can replace $R_T(u/\sqrt {T})$ with $[s_T, T+b(u)-o_T(1)]$ to get

\begin{align*} \lim _{T\to \infty }f_T(u) & = \lim _{T\to \infty }\frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} \int _{R_T(u/\sqrt {T})} e^{\delta _{\mathsf v} t}\, dt \\[5pt] & = \lim _{T\to \infty }\frac {1}{\delta _{\mathsf v} e^{\delta _{\mathsf v} T}}e^{- I(u)/T} (e^{\delta _{\mathsf v}(T+b(u)-o_T(1))} - o_T(e^{\delta _{\mathsf v} T})) \\[5pt] & = \frac {1}{\delta _{\mathsf v}}e^{\delta _{\mathsf v} b(u)}. \end{align*}

Next, fix $u \notin \mathsf{K}$ . Since $\mathsf{K}$ is closed, there exists $\eta \gt 0$ such that $(1-\eta ,1+\eta )\cap J_u = \emptyset$ . Then for all sufficiently large $T$ , we have

\begin{align*} R_T(u/\sqrt {T}) = \Big\{t\ge 0: t \le T+b\Big(\sqrt {\tfrac {t}T}u\Big), \, t \in T J_{u}^2\Big\} \subset [0,(1-\eta )^2T]. \end{align*}

Then

\begin{align*} f_T(u) \le \frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} \int _0^{(1-\eta )^2T} e^{\delta _{\mathsf v} t}\, dt \le \frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} e^{\delta _{\mathsf v} (1-\eta )^2T}\end{align*}

and hence $\lim _{T\to \infty }f_T(u)=0$ .

Lemma 5.4. We have

(5.10) \begin{align} \lim \limits _{T\to \infty }\frac {1}{e^{\delta _{\mathsf v} T} T^{(1-{r})/2}}\int _{t\mathsf{v}+\sqrt {t}u \in {\mathbb{T}}_{T,b}}e^{\delta _{\mathsf v} t} e^{- I(u)} \, dt \, du =\frac {1}{\delta _{\mathsf v}}\int _{\mathsf{K}} e^{\delta _{\mathsf v} b(u)} \, du. \end{align}

Proof. We may assume ${\mathbb{T}}_{T,b} = \{t{\mathsf v}+u:0\le t\le T+b(u), \, u \in \mathsf{K}\}$ as ${\mathbb{T}}_{T,b}- \mathfrak{a}^+$ is contained in a fixed compact set independent of $T$ . Then

\begin{align*} R_T(u) = \{t\ge 0 : t \le T+b(\sqrt {t}u), \sqrt {t}u \in \mathsf{K} \}.\end{align*}

By changing the variable $\sqrt {T}u$ to $u$ , we have

\begin{align*} \begin{aligned}[b] & \frac {1}{ e^{\delta _{\mathsf v} T}T^{(1-{r})/2}}\int _{t\mathsf{v}+\sqrt {t}u \in {\mathbb{T}}_{T,b}}e^{\delta _{\mathsf v} t} e^{- I(u)} \, dt \, du \\[5pt] & = \frac {1}{ e^{\delta _{\mathsf v} T}T^{(1-{r})/2}}\int _{\ker \psi _{\mathsf v}}e^{- I(u)} \int _{R_T(u)} e^{\delta _{\mathsf v} t}\, dt \, du \\[5pt] & = \int _{\ker \psi _{\mathsf v}}\frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} \int _{R_T(u/\sqrt {T})} e^{\delta _{\mathsf v} t}\, dt \, du\\[5pt] & =\int _{\ker \psi _{\mathsf v}} f_T. \end{aligned} \end{align*}

Set

\begin{align*} \mathsf A_T= \int _{u \in \textrm {int} \mathsf{K}} f_T, \;\; \mathsf B_T= \int _{u\in \textrm{hull} (\mathsf{K}\cup \{0\}) -\mathsf K} f_T \quad \textrm{and}\quad \mathsf C_T= \int _{u\notin \textrm{hull}(\mathsf{K} \cup \{0\})} f_T \end{align*}

where $\textrm{hull}\mathsf K\cup \{0\}$ means the convex hull of $\mathsf K \cup \{0\}$ . By hypothesis that $\partial \mathsf K$ has measure zero, we have $\int _{\ker \psi _{\mathsf v}} f_T= \mathsf A_T + \mathsf B_T + \mathsf C_T$ .

We obtain the asymptotics of $\mathsf A_T$ , $\mathsf B_T$ and $\mathsf C_T$ below.

2.1. Asymptotic of $\mathsf A_T$

Since

\begin{align*} \begin{aligned} R_T(u/\sqrt {T}) & = \Big\{t \ge 0: t \le T+b\Big(\sqrt {\tfrac {t}T}u\Big), \, \sqrt {\tfrac {t}T}u \in \mathsf{K}\Big\} \\ & = \Big\{t\ge 0: t \le T+b\Big(\sqrt {\tfrac {t}T}u\Big), \, t \in T J_{u}^2\Big\} \\ & \subset \left [0,\min \left \{T+M_b, \left (\sup J_{u}^2\right )T\right \}\right ]\\[12pt] \end{aligned} \end{align*}

where $J_{u}= \{s\ge 0 : su \in \mathsf{K}\}$ , we have for all $u \in \ker \psi _{\mathsf v}$ ,

\begin{align*} f_T(u) \le \frac {1}{e^{\delta _{\mathsf v} T}} \int _0^{T+M_b} e^{\delta _{\mathsf v} t}\, dt \le \frac {1}{\delta _{\mathsf v}}e^{\delta _{\mathsf v} M_b}. \end{align*}

Since $\mathsf K$ is bounded and the integral in (5) is uniformly bounded for all $T$ we can apply the Lebesgue dominated convergence theorem for the sequence $f_T|_{\operatorname {int} \mathsf K}$ to compute the asymptotic of $\mathsf A_T$ using Lemma5.3:

\begin{align*} \lim _{T\to \infty } {\mathsf A_T}= \int _{u \in \textrm {int} \mathsf{K}} \lim _{T\to \infty }f_T( u) du = \frac {1}{\delta _{\mathsf v}}\int _{u \in \textrm {int} \mathsf{K}}e^{\delta _{\mathsf v} b(u)} \, du. \end{align*}

Hence it suffices to show that $\mathsf B_T+\mathsf C_T$ converges to $0$ as $T\to \infty$ .

Asymptotic of $\mathsf B_T$

For $u\in\textrm{hull} (\mathsf{K} \cup \{0\})$ , we still have the constant upper bound in (5). Hence we can also apply the Lebesgue dominated convergence theorem and Lemma5.3 to compute

\begin{align*} \lim _{T\to \infty } \mathsf{B}_T = \int _{u\in \textrm{hull} (\mathsf{K}\cup \{0\}) -\mathsf K}\lim _{T\to \infty }f_T(u) \, du = \int _{u\in \textrm{hull} (\mathsf{K}\cup \{0\}) -\mathsf K}0 \, du=0 \end{align*}

Asymptotic of $\mathsf C_T$

Fix $u\notin \textrm{hull}(\mathsf K\cup \{0\})$ . Then for all $t \ge 1$ , $tu\notin \mathsf{K}$ otherwise $u$ would lie on the segment with endpoints $0$ and $tu$ and be in $\textrm{hull}(\mathsf{K} \cup \{0\})$ . In particular, we have

\begin{align*} j_{u}: = \sup ( J_{u})= \sup \{0\le s\le 1 : su \in \mathsf{K}\} \lt 1. \end{align*}

Define

\begin{align*} g(u):={ 2\sqrt {\delta _{\mathsf v} I(u)(1-j_{u}^2)}} .\end{align*}

Using $R_T(u/\sqrt {T}) \subset TJ_{u}^2 \subset T[0,j_{u}^2]$ , we get for all $T\gt 0$ ,

\begin{align*} f_T(u) =\frac {1}{e^{\delta _{\mathsf v} T}}e^{- I(u)/T} \int _{R_T(u/\sqrt {T})} e^{\delta _{\mathsf v} t}\, dt \le \tfrac {1}{\delta _{\mathsf v}}e^{- I(u)/T}e^{-\delta _{\mathsf v} T(1-j_{u}^2)} \le \tfrac {1}{\delta _{\mathsf v}}e^{- g(u)}, \end{align*}

where the last inequality uses the fact that $a+b \ge 2\sqrt {ab}$ for all $a,b \ge 0$ . Note that the function $g(u)$ is radially increasing with at least a linear rate: for all $r\gt 1$ , we have

\begin{align*} g(ru)={2 \sqrt {\delta _{\mathsf v} I(r u)(1-j_{r u}^2)}} =2r\sqrt {\delta _{\mathsf v} I(u)(1-j_{u}^2/r^2)} \ge r g(u),\end{align*}

so the function $u\mapsto e^{-g(u)}$ is $L^1$ -integrable on $\ker \psi _{\mathsf v} -\textrm{hull}(\mathsf{K} \cup \{0\})$ . Hence we apply the Lebesgue dominated convergence theorem to compute the asymptotic of $\mathsf C_T$ using Lemma5.3:

\begin{align*} \lim _{T\to \infty } \mathsf{C}_T = \int _{u\notin \textrm{hull} (\mathsf{K}\cup \{0\})}\lim _{T\to \infty }f_T(u) \, du = \int _{u\notin \textrm{hull} (\mathsf{K}\cup \{0\})}0 \, du=0. \end{align*}

This finishes the proof of Lemma5.4.

We are now ready to give the proof of Proposition5.1.

Proof of Proposition5.1

We first introduce some notation. Given a bounded Borel subset $B$ of $G$ , define the counting function $F_B: G \times G \to \mathbb N$ by

\begin{align*} F_B(g,h) = \sum _{\gamma \in \Gamma } 1_B(g^{-1}\gamma h).\end{align*}

Note that $ F_{S_{T,b}}(e,e) = \#\Gamma \cap S_{T,b}$ . The function $F_B$ is $\Gamma$ -invariant in both arguments so it descends to a function on $\Gamma \backslash G \times \Gamma \backslash G$ which we still denote by $F_B$ . For $F_1,F_2 :\Gamma \backslash G \times \Gamma \backslash G \to {\mathbb R}$ , let

\begin{align*} \langle F_1,F_2 \rangle = \int _{\Gamma \backslash G \times \Gamma \backslash G}F_1(x_1,x_2)F_2(x_1,x_2) \, dx_1 \, dx_2\end{align*}

when the integral makes sense where $dx_1,dx_2$ are both the Haar measure on $\Gamma \backslash G$ . For $\varepsilon \gt 0$ smaller than the injectivity radius of $\Gamma$ at $e$ , we fix a nonnegative function $\psi _\varepsilon \in C^\infty (G)$ with $\operatorname {supp} \psi _\varepsilon \subset G_\varepsilon$ and $\int _G \psi _\varepsilon \, dg=1$ . Let $\Psi _\varepsilon \in C^\infty (\Gamma \backslash G)$ be defined by $\Psi _\varepsilon (\Gamma g) = \sum _{\gamma \in \Gamma } \psi _\varepsilon (\gamma g)$ for all $g \in G$ .

Observe that we have

(5.11) \begin{align} \langle F_{S_{T,b,\varepsilon }^-}, \Psi _\varepsilon \otimes \Psi _\varepsilon \rangle \le F_{S_{T,b}}(e,e) \le \langle F_{S_{T,b,\varepsilon }^+}, \Psi _\varepsilon \otimes \Psi _\varepsilon \rangle . \end{align}

Given a bounded Borel subset $B \subset G$ , we define $f_{B}:\check {N}\times MN \to {\mathbb R}$ by

\begin{align*} f_{B}(h,mn) = \tfrac {\kappa _{\mathsf{v}}}{|m_{\mathcal{X}_{\mathsf v}}|}\int _{h^{-1}a_{t\mathsf{v}+\sqrt {t}u}mn \in B}e^{\delta _{\mathsf v} t}e^{-I(u)} \, dt \, du. \end{align*}

Continuing the notation (5.4), set

\begin{align*} L({\mathbb{T}}_{T,b})=\tfrac {\kappa _{\mathsf{v}}}{|m_{\mathcal{X}_{\mathsf v}}|}\int _{t\mathsf{v}+\sqrt {t}u \in {\mathbb{T}}_{T,b}}e^{\delta _{\mathsf v} t}e^{-I(u)} \, dt \, du\end{align*}

which is the value of $f_{S_{T,b}}(h,mn)$ when it is non-zero.

We denote by

\begin{align*} & I_1: MAN\check {N} \to \mathfrak{a}, & J_1: MAN\check {N} \to \check {N}, \\ & I_2: A\check {N}MN \to \mathfrak{a}, & J_2: A\check {N}MN \to MN \end{align*}

the natural projection maps. For a bounded subset $B \subset G$ , define

\begin{align*} \tilde f_{B}(g_1,g_2) = f_{B} {(J_1(g_1^{-1}),J_2(g_2^{-1})) } e^{\psi _{\mathsf v}(I_1(g_1^{-1})-I_2(g_2^{-1}))} \end{align*}

for any $g_i\in G$ such that $J(g_i^{-1})$ and $I(g_i^{-1})$ are defined for $i=1,2$ .

The first step is to rewrite $\langle F_{S_{T,b,\varepsilon }^\pm }, \Psi _\varepsilon \otimes \Psi _\varepsilon \rangle$ by decomposing the Haar measure on $G$ using $\check {N}AMN$ coordinates and separating the expected main term from local mixing from the error term. Set $Q_{S_{T,b,\varepsilon }^\pm }$ to be

(5.12) \begin{align} \int _{K/M\times K}\int _{G_\varepsilon \times G_\varepsilon } \tilde f_{S_{T,b,\varepsilon }^\pm }(g_1^{-1} k_1,g_2^{-1} k_2) \cdot \psi _\varepsilon (g_1) \psi _\varepsilon (g_2) \, dg_1 \, dg_2 \, d\nu (k_1^+) \, d\nu _{\mathsf{i}}(k_2^+). \end{align}

Then as in [Reference Chow and FrommCF23, Lemmas 5.4–5.8], we have

\begin{align*} & \langle F_{S_{T,b,\varepsilon }^\pm }, \Psi _\varepsilon \otimes \Psi _\varepsilon \rangle \\ & = Q_{S_{T,b,\varepsilon }^\pm } +\int _{h^{-1}a_{t\mathsf{v}+\sqrt {t}u}mn \in S_{T,b,\varepsilon }^\pm }e^{\delta _{\mathsf v} t}E_\varepsilon (t,u,h,mn) \, dt \, du \, dh \, dm \, dn \end{align*}

where

\begin{align*} E_\varepsilon (t,u,h,mn) = t^{({r}-1)/2}e^{2\rho (t\mathsf{v}+\sqrt {t}u)-\delta _{\mathsf v} t}\int _{\Gamma \backslash G} \Psi _\varepsilon (xh)\Psi _\varepsilon (xa_{t\mathsf{v}+\sqrt {t}u}mn) \, dx \\ - \tfrac {\kappa _{\mathsf{v}}e^{-I(u)}}{|m_{\mathcal{X}_{\mathsf v}}|}\mathsf{m}^{\textrm{BR}_\star }_{\mathsf v}\bigr (h^{-1}.\Psi _\varepsilon )\mathsf{m}^{\textrm{BR}}_{\mathsf v}\bigr ((mn)^{-1}.\Psi _\varepsilon ). \end{align*}

In view of (5.12) and Lemma5.2, we only need to consider $k_1 \in K$ such that $J_1(k_1^{-1}g_1)$ is in a bounded set. Similarly for $k_2$ . Then by Lemma2.2(1)-(2), we get

\begin{align*} \psi _{\mathsf v}(I_1(k_1^{-1}g_1))=\psi _{\mathsf v}(I_1(k_1^{-1}))+O(\varepsilon ); \quad \psi _{\mathsf v}(I_2(k_2^{-1}g_2))=\psi _{\mathsf v}(I_2(k_2^{-1}))+O(\varepsilon ) \end{align*}

and

(5.13) \begin{align} \begin{aligned}[b] & J_1(k_1^{-1}g_1)^{-1} \in \check {N}_{O(\varepsilon )} J_1(k_1^{-1})^{-1}; & J_2(k_2^{-1} g_2) \in M_{O(\varepsilon )}J_2(k_2^{-1})N_{O(\varepsilon )}. \end{aligned} \end{align}

Let ${\mathbb{T}}_{T,\varepsilon }^- = {\mathbb{T}}({\mathsf v},\mathsf{K}_\varepsilon ^-,T,b_\varepsilon ^-) \subset {\mathbb{T}}_{T,b}$ , $\check {\Xi }_{\varepsilon }^- \subset \check {\Xi }$ , $\Xi _{\varepsilon } ^- \subset \Xi$ and $\Theta _{\varepsilon }^- \subset \Theta$ be as in Lemma5.2. For convenience, let

\begin{align*} S_{T,b_\varepsilon ^-}:=\check {\Xi }_{\varepsilon }^- \exp ({\mathbb{T}}_{T,\varepsilon }^-) \Theta _{\varepsilon }^- \Xi _{\varepsilon } ^-.\end{align*}

By Lemma5.2(1) and (5.13), for all $g_1,g_2\in G_\varepsilon$ we have

\begin{align*} f_{S_{T,b,\varepsilon }^-}(J_1(k_1^{-1}g_1),J_2(k_2^{-1}g_2)) \ge f_{S_{T,b_\varepsilon ^-}}(J_1(k_1^{-1}),J_2(k_2^{-1})).\end{align*}

It now follows that

(5.14) \begin{align} Q_{S_{T,b,\varepsilon }^-} \ge \int _{K/M\times K} (1+O(\varepsilon )) \tilde f_{S_{T,b_\varepsilon ^-}}(k_1, k_2) \, d\nu (k_1^+) \, d\nu _{\mathsf{i}}(k_2^+). \end{align}

Using Lemma5.2(2)–(3), we observe thatFootnote ⁷

(5.15) \begin{align} \begin{aligned}[b] & \int _{K/M\times K} (f_{S_{T,b}} - f_{S_{T,b_\varepsilon ^-}})(J_1(k_1^{-1}),J_2(k_2^{-1})) \, d\nu (k_1^+)\, d\nu _{\mathsf{i}}(k_2^+) \\ & =L({\mathbb{T}}_{T,b})O\left (\nu \left ((\check {\Xi }- \check {\Xi }_{\varepsilon }^-)e^+\right ) + \nu _{\mathsf{i}}\left ((\Xi ^{-1}- (\Xi _{\varepsilon } ^-)^{-1})e^- \right ) + \operatorname {Vol}_M\left (\Theta -\Theta _{\varepsilon }^-\right ) \right ) \\ & \qquad \qquad \qquad + (L({\mathbb{T}}_{T,b}) - L({\mathbb{T}}_{T,\varepsilon }^-))\nu (\check {\Xi }_{\varepsilon }^-)\nu ((\Xi _{\varepsilon } ^-)^{-1})\operatorname {Vol}_M(\Theta ^-) \\ & = L({\mathbb{T}}_{T,b})o_\varepsilon (1)+O(L({\mathbb{T}}_{T,b}) - L({\mathbb{T}}_{T,\varepsilon }^-)) \\ & = L({\mathbb{T}}_{T,b})o_\varepsilon (1) \end{aligned} \end{align}

where we note that $L({\mathbb{T}}_{T,b}) - L({\mathbb{T}}_{T,\varepsilon }^-) = L({\mathbb{T}}_{T,b})o_\varepsilon (1)$ by Lemma5.4.Footnote ⁸

Combining (5.14) and (5.15) yields

(5.16) \begin{align} Q_{S_{T,b,\varepsilon }^-} \ge \int _{K/M\times K} (1+O(\varepsilon )) \tilde f_{S_{T,b}}(k_1, k_2) d\nu (k_1^+) \, d\nu _{\mathsf{i}}(k_2^+) + L({\mathbb{T}}_{T,b})o_\varepsilon (1). \end{align}

A similar argument shows that

(5.17) \begin{align} Q_{S_{T,b,\varepsilon }^+} \le \int _{K/M\times K} (1+O(\varepsilon )) \tilde f_{S_{T,b}}(k_1, k_2) d\nu (k_1^+) \, d\nu _{\mathsf{i}}(k_2^+) + L({\mathbb{T}}_{T,b})o_\varepsilon (1) \end{align}

and we conclude that

(5.18) \begin{align} Q_{S_{T,b,\varepsilon }^\pm } = \int _{K/M\times K} (1+O(\varepsilon )) \tilde f_{S_{T,b}} \, d\nu (k_1^+) \, d\nu _{\mathsf{i}}(k_2^+) + L({\mathbb{T}}_{T,b})o_\varepsilon (1). \end{align}

Considering $k_1 \in K$ such that $k_1^{-1} = ma_wnh^{-1} \in MAN\check {N},$ we have $J_1(k_1^{-1}) = h^{-1}$ , $h^+ = k_1M$ , $I_1(k_1^{-1}) = w = \beta _{h^+}(e,h)$ and hence

\begin{align*} \int _{{k_1^+\in K/M ,\, (J_1(k_1^{-1}))^{-1} \in \check {\Xi }}} e^{\psi _{\mathsf v}(I_1(k_1^{-1}))} \, d\nu (k_1^+) = \int _{h \in \check {\Xi }} e^{\psi _{\mathsf v}(\beta _{h^+}(e,h))} \, d\nu (h^+) = \tilde {\nu }(\check {\Xi }). \end{align*}

Similarly, considering $k_2 \in K$ such that $k_2^{-1} = a_whm^{-1}n^{-1} \in A\check {N}MN,$ we have $J_2(k_2^{-1}) = m^{-1}n^{-1} \in MN$ , $k_2^- = n^-$ , $I_2(k_2^{-1}) =w=-i(\beta _{n^-}(e,n))$ and hence

\begin{align*} \int _{{k_2 \in K ,\, J_2(k_2^{-1}) \in \Theta \Xi }} e^{-\psi _{\mathsf v}(I_2(k_2^{-1}))}d\nu _{\mathsf{i}}(k_2^+) & = \int _{nm \in \Xi ^{-1}\Theta ^{-1}}e^{(\psi _{\mathsf v}\circ \mathsf{i})(\beta _{n^-}(e,n))}\, d\nu _{\mathsf{i}}(n^-) \, dm \\ & = \tilde {\nu }_{\mathsf{i}}(\Xi ^{-1})\operatorname {Vol}_M(\Theta ). \end{align*}

Thus, we obtain

\begin{align*} & \langle F_{S_{T,b,\varepsilon }^\pm }, \Psi _\varepsilon \otimes \Psi _\varepsilon \rangle \\ & = (1+O(\varepsilon ))L({\mathbb{T}}_{T,b})\tilde {\nu }(\check {\Xi })\tilde {\nu }_{\mathsf{i}}(\Xi ^{-1})\operatorname {Vol}_M(\Theta ) \\ & \quad +\int _{h^{-1}a_{t\mathsf{v}+\sqrt {t}u}mn \in S_{T,b,\varepsilon }^\pm }e^{\delta _{\mathsf v} t}E_\varepsilon (t,u,h,mn) \, dt \, du \, dh \, dm \, dn +L({\mathbb{T}}_{T,b})o_\varepsilon (1). \end{align*}

For the error term $E_\varepsilon (t,u,h,mn)$ , we claim that

(5.19) \begin{align} \lim \limits _{T\to \infty }\frac {1}{e^{\delta _{\mathsf v} T} T^{(1-{r})/2}}\int _{h^{-1}a_{t\mathsf{v}+\sqrt {t}u}mn \in S_{T,b,\varepsilon }^\pm }e^{\delta _{\mathsf v} t}E_\varepsilon (t,u,h,mn) \, dt \, du \, dh \, dm \, dn=0. \end{align}

To prove (5.19), note that since $\check {\Xi } \subset \check {N}$ and $\Xi \subset N$ are bounded, by Theorem3.3, there exist positive constants $\eta _{\mathsf{v}}, D_{\mathsf{v}}$ and $s_{\mathsf{v}}$ such that

\begin{align*} |E_\varepsilon (t,u,h,mn)| \le D_{\mathsf{v}} e^{-\eta _{\mathsf{v}} I(u)}\end{align*}

for all $(t,u) \in (s_{\mathsf{v}},\infty ) \times \ker \psi _{\mathsf{v}}$ and $h,mn$ such that $h^{-1}a_{t\mathsf{v}+\sqrt {t}u}mn \in S_{T,b,\varepsilon }^\pm$ . Then (5.19) follows by using similar reasoning as in Lemma5.4 and the fact that for fixed $u \in \ker \psi _{\mathsf v}$ , $\lim \limits _{t\to \infty }E_\varepsilon (t,u,h,mn)=0$ .

Now using Lemma5.4 and (5.11), taking $T\to \infty$ and then $\varepsilon \to 0$ , we conclude that

\begin{align*} \#(\Gamma \cap S_T) \sim \frac {\kappa _{\mathsf{v}}}{\delta _{\mathsf v}|m_{\mathcal{X}_{\mathsf v}}|}\int _{\mathsf{K}} e^{\delta _{\mathsf v} b(u)} \, du \cdot \tilde {\nu }(\check {\Xi })\tilde {\nu }_{\mathsf{i}}(\Xi ^{-1})\operatorname {Vol}_M(\Theta ).\end{align*}

This finishes the proof of Proposition5.1.

Counting via flow boxes

We use flow boxes as in [Reference MargulisMar04] and [Reference Margulis, Mohammadi and OhMMO14].

Definition 5.5 ( $\varepsilon$ -flow box at $g_0$ ). Given $g_0 \in G$ and $\varepsilon \gt 0$ , the $\varepsilon$ - $g_0$ is defined by

\begin{align*} \mathcal{B}(g_0,\varepsilon )=g_0(\check {N}_\varepsilon N \cap N_\varepsilon \check {N} AM)M_\varepsilon A_\varepsilon .\end{align*}

We denote the projection of $\mathcal{B}(g_0,\varepsilon )$ into $\Gamma \backslash G/M$ by $\tilde {\mathcal{B}}(g_0,\varepsilon )$ .

For $g_0\in G$ and $T, \varepsilon \gt 0$ , we denote

(5.20) \begin{align} \mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta ) = \mathcal{B}(g_0,\varepsilon ){\mathbb{T}}_{T,b}\Theta \mathcal{B}(g_0,\varepsilon )^{-1}. \end{align}

Our next goal is to obtain an asymptotic for $\# \Gamma \cap \mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ .

Proposition 5.6. Let $g_0 \in G$ . For all sufficiently small $\varepsilon \gt 0$ , we have

\begin{align*} \#(\Gamma \cap \mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )) =c({\mathsf v}, b)\left ( \frac {\mathsf{m}_{\mathsf v}(\tilde {\mathcal{B}}(g_0,\varepsilon ))}{b_{{r}}(\varepsilon )}\operatorname {Vol}_M(\Theta ) (1+O(\varepsilon )) +o_T(1)\right )\frac {e^{\delta _{\mathsf v} T}}{ T^{({r}-1)/2}} \end{align*}

where $c({\mathsf v},b)$ is as in ( 4.2 ) and $b_r(\varepsilon )$ denotes the volume of the Euclidean $r$ -ball of radius $\varepsilon$ .

The asymptotic in Proposition5.6 will be deduced from Proposition5.1. Using another wavefront-type lemma argument, we show the sets $S_{T,b}$ and $\mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta )$ are approximately the same in the following precise sense.

Lemma 5.7. There exists $C\gt 0$ such that for all sufficiently small $\varepsilon \gt 0$ and for all sufficiently large $T,T'$ with $T\gt T'$ , we have

\begin{align*} & \mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta ) - \mathcal{V}_{T',b}(e,\varepsilon ,{\mathbb{T}},\Theta ) \\ & \subset S_{T,b_\varepsilon ^+}(\check {N}_{\varepsilon + O(\varepsilon e^{-C T'})},(N_{\varepsilon + O(\varepsilon e^{-C T'})})^{-1},{\mathbb{T}}^+_{\varepsilon },\Theta ^+_{\varepsilon }) \\ & \qquad - S_{T'-O(\varepsilon ),b_\varepsilon ^+}(\check {N}_{\varepsilon + O(\varepsilon e^{-C T'})},(N_{\varepsilon + O(\varepsilon e^{-CT'})})^{-1},{\mathbb{T}}^+_{\varepsilon },\Theta ^+_{\varepsilon }) \end{align*}

where $\mathsf{K}_\varepsilon ^+$ and $b_\varepsilon ^+$ are defined by the equation ${\mathbb{T}}_{T,b} +\mathfrak{a}_{O(\varepsilon )} = {\mathbb{T}}_T({\mathsf v},\mathsf{K}_\varepsilon ^+,b_\varepsilon ^+)$ , ${\mathbb{T}}_\varepsilon ^+ := {\mathbb{T}}({\mathsf v},\mathsf{K}_\varepsilon ^+)$ , and $\Theta ^+_\varepsilon = \bigcup _{m_1,m_2 \in M_{O(\varepsilon )}}m_1\Theta m_2$ .

Proof. Since ${\mathsf v}\in \operatorname {int} {\mathfrak a}^+$ , there exists $T_0\gt 0$ sufficiently large such that the linear forms $\psi _{\mathsf v}$ and $\alpha \in \Phi ^+$ are all positive on ${\mathbb{T}} - {\mathbb{T}}_{T_0,b}$ . Therefore there exists a constant $C\gt 0$ such that

(5.21) \begin{align} C\cdot \psi _{\mathsf v} (w) \le \min _{\alpha \in \Phi ^+}\alpha (w) \end{align}

for all $w\in \mathfrak{a}^+$ in the $O(\varepsilon )$ -neighborhood of ${\mathbb{T}}- {\mathbb{T}}_{T_0,b}$ . By (2.2) and (5.21), for all $w\in \mathfrak{a}^+$ in the $O(\varepsilon )$ -neighborhood of ${\mathbb{T}}-{\mathbb{T}}_{T_0,b}$ , we have

(5.22) \begin{align} a_{-w}N_\varepsilon a_w \subset N_{\varepsilon e^{-C\psi _{\mathsf v}(w)}}. \end{align}

Similarly, we have

(5.23) \begin{align} a_{w}\check {N}_\varepsilon a_{-w} \subset \check {N}_{\varepsilon e^{-C\psi _{\mathsf v}(w)}}. \end{align}

Let $T\gt T'\gt T_0$ and ${\mathbb{T}}_{T,\varepsilon }^+ = {\mathbb{T}}_T({\mathsf v},\mathsf{K}_\varepsilon ^+,b_\varepsilon ^+)$ . Let

\begin{align*} g \in \mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta ) -\mathcal{V}_{T',b}(e,\varepsilon ,{\mathbb{T}},\Theta ).\end{align*}

Write $g=g_1amg_2^{-1}$ where $g_1,g_2 \in \mathcal{B}(e,\varepsilon )$ , $a \in \exp ({\mathbb{T}}_{T,b} - {\mathbb{T}}_{T',b})$ and $m \in \Theta$ . Since $g_1, g_2 \in \mathcal{B}(e,\varepsilon )$ , by Lemma2.1(1) we can write $g_1 = h_1n_1m_1a_1 \in \check {N}_\varepsilon N_{O(\varepsilon )}M_\varepsilon A_\varepsilon$ and $g_2 = n_2h_2m_2a_2 \in N_\varepsilon \check {N}_{O(\varepsilon )}M_{O(\varepsilon )}A_{O(\varepsilon )}$ . Then

\begin{align*} g=h_1n_1m_1a_1ma (m_2a_2)^{-1}h_2^{-1}n_2^{-1}. \end{align*}

Let $a' = aa_1a_2^{-1}$ , $m'=m_1mm_2^{-1}$ and $n_3 = (m'a')^{-1}n_1m'a'$ . Then $a' \in {\mathbb{T}}^+_{T,\varepsilon } - {\mathbb{T}}^+_{T'-O(\varepsilon ),\varepsilon }$ , $m' \in \Theta _\varepsilon ^+$ and $n_3\in N_{O(\varepsilon e^{-C T'})}$ by (5.22). By Lemma2.1(1), we can write $n_3h_2^{-1} = m_3a_3h_4n_4 \in M_{O(\varepsilon )}A_{O(\varepsilon )}\check {N}_{O(\varepsilon )}N_{O(\varepsilon e^{-C T'})}$ . Then

\begin{align*} g &= h_1m'a'n_3h_2^{-1}n_2^{-1} = h_1m'a'm_3a_3h_4n_4n_2^{-1} \\ &= h_5m''a''n_5 \in \check {N}_{\varepsilon +O(\varepsilon e^{-C T'})}\Theta ^+_\varepsilon \exp ({\mathbb{T}}^+_{T,\varepsilon } - {\mathbb{T}}^+_{T'-O(\varepsilon ),\varepsilon })N_{\varepsilon +O(\varepsilon e^{-C T'})}. \end{align*}

where $a'' = a'a_3$ , $m''=m'm_3$ , $n_5 =n_4n_2^{-1}\in N_{\varepsilon +O(\varepsilon e^{-C T'})}$ and $h_5 = h_1(m''a'')h_4(m''a'')^{-1} \in \check {N}_{\varepsilon +O(\varepsilon e^{-C T'})}$ . This completes the proof.

Proof of Proposition5.6

It suffices to consider the case $g_0=e$ . Note that the boundaries $\partial N_\varepsilon$ and $\partial \check {N}_\varepsilon$ are proper real algebraic subvarieties of $\mathcal{F}$ and hence $\nu (\partial \check {N}_\varepsilon ) = \nu _{\mathsf{i}}(\partial N_\varepsilon ) = 0$ by [Reference Kim and OhKO25, Theorem 1.1]. We have a trivial inclusion $S_{T,b}(\check {N}_\varepsilon ,N_\varepsilon ^{-1},{\mathbb{T}},\Theta ) \subset \mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta )$ . By Proposition5.1, we have

(5.24) \begin{align} \#(\Gamma \cap \mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta )) \ge c({\mathsf v},b)\left (\tilde {\nu }(\check {N}_\varepsilon )\tilde {\nu }_{\mathsf{i}}(N_\varepsilon )\operatorname {Vol}_M(\Theta ) + o_T(1)\right )\frac {e^{\delta _{\mathsf v} T}}{T^{({r}-1)/2}}. \end{align}

By [Reference Chow and FrommCF23, Lemma 5.20], we have

(5.25) \begin{align} \mathsf{m}_{\mathsf v}(\tilde {\mathcal{B}}(e,\varepsilon )) = (1+O(\varepsilon )) b_{{r}}(\varepsilon )\tilde {\nu }(\check {N}_\varepsilon )\tilde {\nu }_{\mathsf{i}}(N_\varepsilon ). \end{align}

Then using (5.24) and (5.25), we get

\begin{align*}& \#(\Gamma \cap \mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta )) \\& \ge c({\mathsf v},b) \left ( \frac {\mathsf{m}_{\mathsf v}(\tilde {\mathcal{B}}(e,\varepsilon ))}{b_{{r}}(\varepsilon )} \operatorname {Vol}_M(\Theta ) (1+O(\varepsilon ))+o_T(1)\right )\frac {e^{\delta _{\mathsf v} T}}{T^{({r}-1)/2}}. \end{align*}

It remains to establish the reverse inequality. By Lemma5.7, we have

\begin{align*} &\mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta ) - \mathcal{V}_{T/2,b}(e,\varepsilon ,{\mathbb{T}},\Theta ) \\ &\subset S_{T,b_\varepsilon ^+}(\check {N}_{\varepsilon + O(\varepsilon e^{-C T/2})},(N_{\varepsilon + O(\varepsilon e^{-C T/2})}^-)^{-1},{\mathbb{T}}^+_\varepsilon ,\Theta ^+_\varepsilon ) \end{align*}

and

\begin{align*} \mathcal{V}_{T/2,b}(e,\varepsilon ,{\mathbb{T}},\Theta ) \subset S_{T/2,b_\varepsilon ^+}(\check {N}_{O(\varepsilon )},(N_{O(\varepsilon )})^{-1},{\mathbb{T}}^+_\varepsilon ,\Theta ^+_\varepsilon ). \end{align*}

Then using Proposition5.1 in the above inclusions, we get

(5.26) \begin{align} \begin{aligned}[b] & \#(\Gamma \cap \mathcal{V}_{T,b}(e,\varepsilon ,{\mathbb{T}},\Theta )) -\#(\Gamma \cap \mathcal{V}_{T/2,b}(e,\varepsilon ,{\mathbb{T}},\Theta )) \\ & \le c({\mathsf v},b_\varepsilon ^+) \left (\tilde {\nu }(\check {N}_{\varepsilon + O(\varepsilon e^{-C T/2})})\tilde {\nu }_{\mathsf{i}}(N_{\varepsilon + O(\varepsilon e^{-C T/2})})\operatorname {Vol}_M(\Theta _\varepsilon ^+) + o_T(1)\right )\frac {e^{\delta _{\mathsf v} T}}{T^{({r}-1)/2}} \\ & = c({\mathsf v},b)(\tilde {\nu }(\check {N}_{\varepsilon })\tilde {\nu }_{\mathsf{i}}(N_{\varepsilon })\operatorname {Vol}_M(\Theta )(1+O(\varepsilon )) + o_T(1))\frac {e^{\delta _{\mathsf v} T}}{T^{({r}-1)/2}} \\ & =c({\mathsf v},b)\left (\frac {\mathsf{m}_{\mathsf v}(\tilde {\mathcal{B}}(e,\varepsilon ))}{b_{{r}}(\varepsilon )}\operatorname {Vol}_M(\Theta ) (1+O(\varepsilon )) + o_T(1)\right )\frac {e^{\delta _{\mathsf v} T}}{ T^{({r}-1)/2}} \end{aligned}. \end{align}

Moreover,

(5.27) \begin{align} \begin{aligned}[b] & \#(\Gamma \cap \mathcal{V}_{T/2,b}(e,\varepsilon ,{\mathbb{T}},\Theta )) \\ & \qquad \qquad \le c({\mathsf v},b)(\tilde {\nu }(\check {N}_{\varepsilon })\tilde {\nu }_{\mathsf{i}}(N_{\varepsilon })\operatorname {Vol}_M(\Theta _\varepsilon ^+) + o_T(1))\frac {e^{\delta _{\mathsf v} T/2}}{(T/2)^{({r}-1)/2}}. \end{aligned} \end{align}

Combining (5.26) and (5.27), we obtain the desired inequality.

Application of a closing lemma

For $g_0\in G$ and $T, \varepsilon \gt 0$ , let

\begin{align*} \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )=\{gamg^{-1}:g\in \mathcal{B}(g_0,\varepsilon ),am\in {\mathbb{T}}_{T,b} \Theta \}.\end{align*}

Note that the sets $\mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ and $\mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ are similar but the latter consists only of loxodromic elements and the former does not. We relate $\Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ to $\Gamma \cap \mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ by using the following closing lemma for regular directions.

Lemma 5.8 (Reference Chow and FrommCF23, Lemma 2.7). There exists $s_0\gt 0$ for which the following holds. Let $\varepsilon \gt 0$ be sufficiently small and $g_0\in G$ . Suppose there exist $g_1,g_2\in \mathcal{B}(g_0,\varepsilon )$ and $\gamma \in G$ such that

(5.28) \begin{align} g_1 \tilde {a}_\gamma \tilde {m}_\gamma = \gamma g_2 \end{align}

for some $\tilde {m}_\gamma \in M$ and $\tilde {a}_\gamma \in A$ with

(5.29) \begin{align} s=\min _{\alpha \in \Phi ^+} \alpha (\log \tilde {a}_\gamma )\ge s_0. \end{align}

Then there exist $g\in \mathcal{B}(g_0,\varepsilon + O(\varepsilon e^{-s}))$ , $a_\gamma \in A$ and $m_\gamma \in M$ such that

\begin{align*} \gamma =ga_\gamma m_\gamma g^{-1}.\end{align*}

Moreover, $a_\gamma \in \tilde {a}_\gamma A_{O(\varepsilon )}$ and $m_\gamma \in \tilde {m}_\gamma M_{O(\varepsilon )}$ .

The next Lemma5.9 is a precise formulation of the property that the sets $\Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ and $\Gamma \cap \mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ are approximately the same. The proof of Lemma5.9 uses the above closing Lemma5.8. The essential reason why we are able to use this closing lemma is because only a finite volume part of the tube $\mathbb{T}$ is too close to the walls of the Weyl chamber to apply the closing lemma. Excluding this finite volume, we obtain a comparison between $\Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ and $\Gamma \cap \mathcal{V}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ .

Lemma 5.9. There exists $C \gt 0$ such that for all sufficiently large $T,T'$ with $T\gt T'$ , we have

\begin{align*} \Gamma \cap \big(\mathcal{V}_{T,b_\varepsilon ^-}(g_0,\varepsilon -O(\varepsilon e^{-CT'})),{\mathbb{T}}^-_\varepsilon ,\Theta ^-_\varepsilon )- \mathcal{V}_{T',b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )\big ) \subset \Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta ) \end{align*}

where $\mathsf{K}_\varepsilon ^-, b_\varepsilon ^-$ are defined by the equation $\bigcap _{w \in \mathfrak{a}_{O(\varepsilon )}}({\mathbb{T}}_{T,b}+w) = {\mathbb{T}}_T({\mathsf v},\mathsf{K}_\varepsilon ^-,b_\varepsilon ^-)$ , ${\mathbb{T}}_\varepsilon ^- = {\mathbb{T}}({\mathsf v},\mathsf{K}_\varepsilon ^-)$ and $\Theta ^-_\varepsilon = \bigcap _{g_1,g_2 \in M_{O(\varepsilon )}}g_1\Theta g_2$ .

Proof. As in Lemma5.7, since ${\mathsf v}\in \operatorname {int} {\mathfrak a}^+$ , there exists $T_0\gt 0$ sufficiently large such that the linear forms $\psi _{\mathsf v}$ and $\alpha \in \Phi ^+$ are all positive on ${\mathbb{T}} - {\mathbb{T}}_{T_0,b}$ . Therefore there exists a constant $C\gt 0$ such that

\begin{align*} C\psi _{\mathsf v} (w) \le \min _{\alpha \in \Phi ^+}\alpha (w)\end{align*}

for all $w\in \mathfrak{a}^+$ in some $\varepsilon _0$ -neighborhood of ${\mathbb{T}}- {\mathbb{T}}_{T_0,b}$ .

For $T\gt 0$ , let ${\mathbb{T}}_{T,\varepsilon }^- = {\mathbb{T}}_T({\mathsf v},\mathsf{K}_\varepsilon ^-,b_\varepsilon ^-)$ . Fix $T\gt T'\gt T_0$ and assume $T'$ is sufficiently large so that if $w \in {\mathbb{T}} - {\mathbb{T}}_{T',b}$ , then

\begin{align*} \min _{\alpha \in \Phi ^+}\alpha (w) \gt CT'\gt s_0\end{align*}

where $s_0$ is as in Lemma5.8. Suppose

\begin{align*} \gamma \in \Gamma \cap \big(\mathcal{V}_{T,b_\varepsilon ^-}(g_0,\varepsilon -O(\varepsilon e^{-CT'})),{\mathbb{T}}^-_\varepsilon ,\Theta ^-_\varepsilon )- \mathcal{V}_{T',b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )\big).\end{align*}

Then $\gamma = g_1\exp (w)mg_2$ where $g_1,g_2 \in \mathcal{B}(g_0,\varepsilon -O(\varepsilon e^{-CT'})))$ , $w \in {\mathbb{T}}^-_{T,\varepsilon }-{\mathbb{T}}_{T',b}$ , and $m \in \Theta ^-_\varepsilon$ . By Lemma5.8, we have $\gamma = g\exp (w')m'g^{-1}$ for some

\begin{align*} g \in \mathcal{B}\big(g_0,\varepsilon - O(\varepsilon e^{-CT'})+O(\varepsilon e^{-\min _{\alpha \in \Phi ^+}\alpha (w)})\big) \subset \mathcal{B}(g_0,\varepsilon ),\end{align*}

$w' \in w A_{O(\varepsilon )}$ and $m'\in mM_{O(\varepsilon )}$ . It follows that $w\in {\mathbb{T}}_{T,b}$ and $m' \in \Theta$ , so $\gamma \in \Gamma \cap \mathcal{W}_{T,b}$ $(g_0,\varepsilon ,{\mathbb{T}},\Theta )$ .

In the next Lemma5.10, we show that there are just as many primitive elements in $\Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,\Theta )$ as nonprimitive elements. This is what allows us to consider only primitive elements in $\Gamma$ as in the joint equidistribution Theorem4.5. The proof uses Lemma5.9 and Proposition5.6 to get an estimate for $\#\Gamma _{\textrm{prim}} \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,\Theta )$ .

Lemma 5.10. Suppose $g_0 \in G$ with $\Gamma g_0M \in \operatorname {supp}\mathsf{m}_{\mathsf v}$ and $\operatorname {Vol}_M(\Theta ) \gt 0$ . Then for all sufficiently small $\varepsilon \gt 0$ and sufficiently large $T$ , we have

\begin{align*} \#\Gamma \cap (\mathcal{W}_{T,b}(g_0,\varepsilon ,\Theta )-\mathcal{W}_{2T/3,b}(g_0,\varepsilon ,\Theta ))\le \#\Gamma _{\textrm{prim}}\cap \mathcal{W}_{T,b}(g_0,\varepsilon ,\Theta ).\end{align*}

Proof. Let $\Gamma _{\textrm{prim}^k}=\{\sigma ^k:\sigma \in \Gamma _{\textrm{prim}}\}$ . We observe that

\begin{align*} & \#\Gamma _{\textrm{prim}}\cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )& \\ & =\#\Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )-\#\left (\bigcup _{k\ge 2}\Gamma _{\textrm{prim}^k}\cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )\right ) \\ & \ge \#\Gamma \cap \mathcal{W}_{T,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta )-\#\left (\bigcup _{k\ge 2}\Gamma \cap \mathcal{W}_{T/k,\hat {b}}(g_0,\varepsilon ,\hat {{\mathbb{T}}},\sqrt [k]{\Theta })\right ) \end{align*}

where $\sqrt [k]{\Theta }: = \{m \in M : m^k \in \Theta \}$ and $\hat {{\mathbb{T}}}$ is the essential tube obtained from $\mathbb{T}$ by replacing $\mathsf{K}$ with the convex hull of $\mathsf{K}\cup \{0\}$ and $\hat {b}$ is any continuous extension of $b$ to the convex hull of $\mathsf{K} \cup \{0\}$ . It suffices to show that for all sufficiently large $T$ , we have

(5.30) \begin{align} \#\left (\bigcup _{k\ge 2}\Gamma \cap \mathcal{W}_{T/k,\hat {b}}(g_0,\varepsilon ,\hat {{\mathbb{T}}},\sqrt [k]{\Theta })\right ) \le \#\Gamma \cap \mathcal{W}_{2T/3,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta ). \end{align}

Since $\mathcal{W}_{T/k,\hat {b}}(g_0,\varepsilon ,\hat {{\mathbb{T}}},\sqrt [k]{\Theta }) \subset \mathcal{V}_{T/k,\hat {b}}(g_0,\varepsilon ,\hat {{\mathbb{T}}},\sqrt [k]{\Theta })$ and $\Gamma \cap \mathcal{W}_{T/k,\hat {b}}(g_0,\varepsilon ,\hat {{\mathbb{T}}},\sqrt [k]{\Theta })$ is empty when $T/k$ is sufficiently small, using Proposition5.6 we get

(5.31) \begin{align} \#\left (\bigcup _{k\ge 2}\Gamma \cap \mathcal{W}_{T/k,\hat {b}}(g_0,\varepsilon ,\hat {{\mathbb{T}}},\sqrt [k]{\Theta })\right )=O\left (T\frac {e^{ \delta _{\mathsf v} T/2}}{T^{({r}-1)/2}}\right)\!. \end{align}

Using Lemma5.9, Proposition5.6 and $\operatorname {Vol}_M(\Theta ) \gt 0$ , we have

(5.32) \begin{align} \#\Gamma \cap \mathcal{W}_{2T/3,b}(g_0,\varepsilon ,{\mathbb{T}},\Theta ) \ge O\left (\frac {e^{2\delta _{\mathsf v} T/3}}{T^{({r}-1)/2}}\right)\!. \end{align}

The inequality (5.30) now follows from (5.31) and (5.32).