We study the distribution of orbits of a lattice $\Gamma\leq\SL(3,\R)$ in the moduli space $X_{2,3}$ of covolume one rank-two discrete subgroups in $\R^3$. Each orbit is dense, and our main result is the limiting distribution of these orbits with respect to norm balls, where the norm is given by the sum of squares. Specifically, we consider  $\Gamma_T=\{\gamma\in\Gamma:\|\gamma\|\leq T\}$ and show that, for any fixed $x_0\in X_{2,3}$ and $\varphi\in C_c(X_{2,3})$,

$$\lim_{T\to\infty}\frac{1}{\#\Gamma_T}\sum_{\gamma\in\Gamma_T}\varphi(x_0\cdot\gamma)=\int_{X_{2,3}}\varphi(x)d \tilde\nu_{x_0}(x),$$

where $\tilde\nu_{x_0}$ is an explicit probability measure on $X_{2,3}$ depending on $x_0$. To prove our result, we use the duality principle developed by Gorodnik and Weiss which recasts the above problem into the problem of computation of certain volume estimates of growing skewed balls in $H$ and proving ergodic theorems of the left action of the skewed balls on $\SL(3,\mathbb{R})/\Gamma$. The ergodic theorems are proven by applying theorems of Shah building on the linearisation technique. The main contribution of the paper is the application of the duality principle in the case where $H$ has infinitely many non-compact connected components.

Consider the moduli space $X_{2,3}$ of homothety classes of rank-two discrete subgroups with orientation in $\R^{3}$. The group $\SL(3,\R)$ acts transitively on $X_{2,3}$, so that $X_{2,3}\cong H\backslash \SL(3,\R)$, where $H\leq\SL(3,\R)$ is a closed subgroup. It turns out that for each $x_{0}\in X_{2,3}$ and each lattice subgroup $\Gamma \leq \SL(3,\R)$, the orbit $ x_{0}\cdot \Gamma $ is dense in $X_{2,3}$. We compute the limiting distribution of those orbits with respect to norm balls, where the norm is given by $\|g\|^2=\sum g_{i,j}^2$, $g\in\SL(3,\R)$, the sum of squares of the entries of $g$. Namely, we consider $\Gamma_{T}=\{\gamma\in\Gamma : \|\gamma\| \leq T \}$, and we show that for all $\varphi\in C_c(X_{2,3})$

\to\infty} \frac{1}{\#\Gamma_{T}}\sum_{\gamma\in \Ga_T}\varphi(x_{0}\cdot \gamma)=c_{\Ga}\int_{X_{2,3}}\varphi(x)d\nu_{x_{0}}(x)$$

for some explicit measure $\nu_{x_0}$ on $X_{2,3}$ which depends on $x_{0}$ and constant $c_{\Ga}$ which depends on $\Gamma$. To prove our result, we use the general recipe for applying the duality principle developed by Gorodnik and Weiss which reduces the above problem to computation of volumes of growing skewed balls in H and proving ergodic theorems of the left H-action on $\SL(3,\R)/\Gamma$. The main novelty of our work is that we apply the duality principle in the case that H has infinitely many connected components.

§ INTRODUCTION

§.§ The problem set-up and statement of the main result

In this paper we study the asymptotic distributional properties of the action of lattice subgroups of the special linear group on a moduli space of discrete subgroups of the Euclidean space. Such a research direction is in particular motivated by <cit.> and the more recent work <cit.>  studying random walks on such spaces. We will study the dynamics with respect to the action of growing norm balls.

We proceed to describe our specific setting. We say that $\La\subset\R^3$ is a 2-lattice, if $\La$ is the $\Z$-Span of a tuple of linearly independent vectors  ${v_1,v_2}\in\R^3$, that is,
For $\La$ we let
$$\cov(\Lambda):=\sqrt{\det(\langle v_i,v_j\rangle)},$$

which is the area of a fundamental parallelogram of $\Lambda$.
The unimodular 2-lattices $\La\subset\R^3$ are those $\La$ with $\cov(\La)=1$.
Next, we recall the definition of the shape of $\La$, a notion that was extensively studied in e.g <cit.>, which refined the classical work of Schmidt <cit.>.
We view $\R^3$ as row vectors, and for a unimodular 2-lattice $\La\subset\R^3$, let $w\in\mathbb{S}^2$ such that $w\perp \La$. We choose $\rho \in \SO(3,\R)$ such that $w \rho=e_3:=(0,0,1)$, and we define

\begin{equation}\label{definition of shape}
\s(\La,w):=\La \rho\begin{bmatrix}
\SO(2,\R) & 0 \\
0 &  1
\end{bmatrix},
\end{equation}

which is independent of the choice of $\rho$. By identifying the unimodular 2-lattices $\La\subset {e}_3^\perp$ with $X_2$, we identify $\s(\Lambda,w)$ as a point in $X_2\slash \SO(2,\R)$. Note that in general

A more intrinsic definition of a shape of a discrete subgroup, see e.g. <cit.>, is defined by the equivalence class under the equivelence relation of scaling and rotations. When defining shape in this way, one gets a point in $X_2/\text{O}(2,\R)$, and it captures slightly less information. Our definition (<ref>) is mainly motivated by the definition in <cit.>, which produces a point in the more familiar space $X_2/\SO(2,\R)$.

We consider
and note that $\s$ defined in (<ref>) yields a map
$$\s:X_{2,3}\to X_2/\SO(2,\R).$$

We define a right $\SL(3,\R)$ action on $X_{2,3}$ using the usual right matrix multiplication by
\begin{equation} \label{action of SL on X_2,3}
(\La,w). g:=\left(\frac{\La g}{\sqrt{\cov(\La g)}}, \frac{w (^tg^{-1})}{\|w (^tg^{-1})\|} \right),~g\in \SL(3,\R),~w\in \mathbb{S}^2,
\end{equation}
where $\|\cdot\|$ is the usual Euclidean norm.
To motivate our main result, we note the following statement.

Let $\Ga\leq\SL(3,\R)$ be a lattice and fix $x_0\in X_{2,3}$. Then $x_0.\Gamma$ is dense.

TODO: write how it can be proven and give reference to Dani

It turns out that for each lattice subgroup $\Ga\leq \SL(3,\R)$ and each $x_0\in X_{2,3}$, the orbit $x_0\Ga$  is dense in $X_{2,3}$. This follows for example by <cit.>, and a different proof of this fact is obtained by applying a duality argument. As we explain below $X_{2,3}$ is a homogeneous space identified with $H\backslash G$, and by Proposition 1.5 of <cit.>, it follows that all $H$-orbits in $G/\Ga$ are dense.

Our goal in this paper will be to compute the limiting distribution of $\Ga$ orbits in $X_{2,3}$ with respect to growing Hilbert-Schmidt norm balls. Namely, let $\|g\|=\sqrt{\text{Trace}(^tgg)}=\sqrt{\sum_{ij}(g_{ij}^2)}$ be the Hilbert-Schmidt norm of $g\in\SL(3,\R)$, let
$$\Ga_T:=\{\ga \in \Ga:\|\ga\|\leq T\},$$
and for
$$x_0:=(\La_0,w_0)\in X_{2,3},$$
consider the probability measures
$$\mu_{T,x_0}:=\frac{1}{\#\Ga_T}\sum_{\ga\in \Ga_T}\delta_{x_0.\gamma},~T>0.$$

Our main result is that the probability measures $\mu_{T,x_0}$ converge as $T\to\infty$ to a certain probability measure $\tilde\nu_{x_0}$ depending on ${x_0}\in X_{2,3}$ which we describe now. We observe that $X_{2,3}$ has a natural projection to $\mathbb{S}^2$ defined by
which endows $X_{2,3}$ with a fiber-bundle structure, where the fibers are isomorphic to

To define the measure $\tilde\nu_{x_0}$, we define a measure $\mu_{x_0,w}$ on each fiber $\pi_{\perp}^{-1}(w)$, and we integrate those measures by the unique rotation invariant probability measure $\mu_{\Sph^2}$ on $\Sph^2$. We note that the measures $\mu_{x_0,w}$ have a slightly surprising form; they are a combination of a $\SL(2,\R)$-invariant measure with a density involving the Hilbert-Schmidt norm of operators which we define now.

For an operator $T$ from a hyperplane $U\subset \R^3$ to another hyperplane $V\subset \R^3$, we define

\begin{equation}\label{defining hilbert-schmits for op. from hyp. to hyp.}
\|T\|^2_{\text{HS}}:=\|Tu_1\|^2+\|Tu_2\|^2,
\end{equation}

where $\{u_1,u_2\}$ is an orthonormal basis of $U$, and where the norm on the right hand side is the usual Euclidean norm on $\R^3$. We note that this norm is independent of the choice of an orthonormal basis $\{u_1,u_2\}$. Moreover, this norm is bi-$\SO(3,\R)$ invariant in the following sense. If $\rho_1,\rho_2\in\SO(3,\R)$, then $\rho_2 \circ T \circ \rho_1:\rho_1^{-1}U\to \rho_2V$ satisfies
$$\|\rho_2 \circ T \circ \rho_1\|_{\text{HS}}=\|T\|_{\text{HS}}.$$

For an ordered tuple of linearly independent vectors $B=(u_1,u_2)\in\R^3\times\R^3$ we define the linear map $T_B:\text{Span}_{\R}  \{e_1,e_2\} \to\text{Span}_{\R}\{u_1,u_2\}$, by sending $e_1\mapsto u_1$ and $e_2\mapsto u_2$. Now fix unimodular 2-lattice $\Lambda_0\subset\R^3$, and let $\mathscr{B}_0$ be an ordered tuple of linearly independent vectors forming a $\Z$-basis for $\La_0 $. We define for an arbitrary unimodular 2-lattice $\La\subset\R^3$,
\begin{equation}\label{defining psi_La_0}
\Psi_{\La_0} (\La):=\sum_{\Span_\Z{\mathscr{B}}=\La}\frac{1}{\|T_{\mathscr{B}}\circ T_{\mathscr{B}_0}^{-1}\|_{\text{HS}}^4}.
\end{equation}

We note that $\Psi_{\La_0}$ is independent of the choice of basis $\mathscr{B}_0$ (see (<ref>)), and we observe that by bi-$\SO(3,\R)$ invariance of the Hilbert-Schmidt norm that the values of the function $\Psi_{\La_0}(\La)$ only depends on the shapes of $\La$ and $\La_0$.

By identifying $\pi_\perp^{-1}(e_3)$ with $\SL(2,\Z)\backslash \SL(2,\R)$, we obtain the $\SL(2,\Z)\backslash \SL(2,\R)$ invariant measure $\mu_{e_3}$ supported on $\pi_\perp^{-1}(e_3)$ scaled such that the measure $\mu_{x_0,e_3}$ defined by
$$\mu_{x_0,e_3}(f):=\int_{\pi^{-1}_\perp(e_3)}f(\La,e_3)\Psi_{\La_0}(\La)d\mu_{e_3}(\La),~f\in C_c(X_{2,3}),$$
is a probability measure. Then, we obtain the measure supported on $\pi_\perp^{-1}(w), \text{ for } w\in \mathbb{S}^2$, by choosing $\rho_w\in \SO(3,\R)$ such that $w=e_3 \rho_w$, and defining,
which is the push-forward of the right translation by $\rho_w$ via the right action of $\SO(3,\R)$ on $X_{2,3}$ defined in (<ref>). Note that $\mu_{x_0,w}$ is independent of the choice of $\rho_w$. Finally, we define $\tilde\nu_{x_0}$ by

The main goal of this paper is to prove the following:

Let $\Ga\leq\SL(3,\R)$ be a lattice and fix ${x_0}\in X_{2,3}$. Then, $\mu_{T,{x_0}}$ converges in the weak-* topology to $\tilde\nu_{x_0}$ as $T\to\infty$. In other words,
for all $f\in C_c(X_{2,3})$, we have

\begin{equation}
\lim_{T
\to\infty} \frac{1}{\#\Gamma_{T}}\sum_{\gamma\in \Ga_T}f(x_{0}\cdot \gamma)=\int_{X_{2,3}}f(x)d \tilde\nu_{x_{0}}(x).
\end{equation}

Now consider the map
$$(\s\times\pi_\perp):X_{2,3}\to  X_2\slash \SO(2,\R)\times \Sph^2,$$
defined by
We observe that that the push-forward of $\nu_{x_0}$ by $(\s\times\pi_\perp)$ is given by

Let $\Ga\leq\SL(3,\R)$ be a lattice and fix ${x_0}\in X_{2,3}$. Then, the probability measures on $X_2\slash \SO(2,\R)\times\Sph^2  $ given by
$$(\s\times\pi_\perp)_*\mu_{T,{x_0}}=\frac{1}{\#\Ga_T}\sum_{\|\ga\|\leq T}\delta_{(\s\times\pi_\perp)({x_0}.\gamma)},~T>0,$$

converge in the weak-* topology to the probability measure $(\s\times\pi_\perp)_*(\nu_{x_0})=\s_*\mu_{x_0,e_3}\otimes\mu_{\mathbb{S}^2}$ as $T\to\infty$.

§.§ Connection to homogeneous dynamics - the duality principle

To simplify notation, we denote $G:=\SL(3,\R)$. We note that the $G$-action on $X_{2,3}$ is transitive, and we observe that the stabilizer subgroup of the base point $(\Span_\Z\{(1,0,0),(0,1,0)\},(0,0,1))$ is

\begin{equation}
H=\left \{\begin{bmatrix}
t^{-\frac{1}{2}}\ga & 0 \\
(x,y) &  t
\end{bmatrix}:t> 0, \ga \in \SL(2,\Z), (x,y)\in \R^2 \right\}.
\end{equation}

Then we obtain the identification $H \backslash G \cong X_{2,3}$.

This connects our problem to the study of distribution of orbits of closed subgroups in homogeneous spaces. In general, let $G$ be a connected Lie group, $\Ga$ be a lattice in $G$ and $H$ be a closed subgroup of $G$ (or more generally for two closed subgroups $H_1$ and $H_2$). The duality principle allows us to connect the equidistributional properties of the $\Ga$-orbits on $X$ to the equidistributional
properties of the $H$-orbits in the dual action of $H$ on $G/\Ga$, see Section 1.7 of <cit.> for an extensive exposition of the existing literature on this principle.  A general recipe for applying the duality principle was developed in <cit.> and <cit.>.  Our approach in this paper uses a theorem of Gorodnik and Weiss (<cit.>), since it allows to prove equidistribution for every starting point.

A novelty of our work is the application of the duality principle for a closed subgroup $H$ with infinitely many non-compact connected components. We note that the case when $H$ is connected algebraic was studied in great generality in <cit.>, and the case where $H$ is discrete was studied in <cit.> and in greater generality in <cit.>.

To begin with, we shall study the invariant measure on $H$ and introduce an equidistribution theorem on $G/\Ga$.

§.§ An equidistribution theorem on $G/\Ga$ with respect to the expanding skew balls in $H$

We first identify $X_
{2,3}$ in $\R^3$ with a $G$-homogeneous space:

The space of homothety classes of rank two lattices of $\R^3$ can be identified with a homogeneous space $H \backslash G$ for some closed subgroup $H$ of $G$.

We view the space $X_{2,3}$ of rank two discrete subgroups of $\R^3$,  as the set of equivalence classes, each of which contains a unimodular rank two sub-lattice. Let $G$ act on the row vectors in $\R^3$ (on the right). Then the stabilizer subgroup $H$ in $G$ of the  equivalence class containing the unimodular lattice $\Lambda_0:=\Span_{\Z}\{(1,0,0), (0,1,0) \}$ is

\begin{equation}
H=\left \{\begin{bmatrix}
t^{-\frac{1}{2}}\ga & 0 \\
(x,y) &  t
\end{bmatrix}:t> 0, \ga \in \SL(2,\Z), (x,y)\in \R^2, \text{ the sign of }\det(\ga) \text{ same as } t \right\}.
\end{equation}

On the other hand, $G$-action is clearly transitive. Therefore by the orbit stabilizer theorem

\begin{equation*}
H \backslash G \cong X_{2,3}
\end{equation*}

Let us first describe the left invariant measure on $H$. Put

\begin{equation}
\SL_{2,1}(\Z):=\begin{bmatrix}
\SL(2,\Z) & 0 \\
0 &  1
\end{bmatrix}, A=\left \{\begin{bmatrix}
t^{-\frac{1}{2}}I_2 & 0 \\
0 & t
\end{bmatrix}:t>0 \right\}, U:=\begin{bmatrix}
I_2 & 0 \\
\mathbb{\R}^2 & 1
\end{bmatrix}
\end{equation}

Then the group $H$ has the decomposition:

$$H=U\rtimes(\SL_{2,1}(\Z)\times A)=U\rtimes(A \times \SL_{2,1}(\Z)).$$

Notice that any element in $H$ can be uniquely represented as $u\ga a$, where $u\in U$, $\ga \in \SL_{2,1}(\Z)$, $a \in A$ (note that $a$ commutes with $\ga$),

where the semidirect product is given by conjugation:

\begin{equation}
(u_1,\ga_1,a_1)\cdot (u_2,\ga_2, a_2):= (u_1 \ga_1 a_1 u_2 (\ga_1 a_1)^{-1}, \ga_1 \ga_2, a_1 a_2)
\end{equation}

The left Haar measure $\mu$ on $H$ can be given by the following proposition:

For an integrable function $f=f(x)$ on $H$, write $x=u_v \ga a_t$, then under the parametrization

\begin{equation}
\ga:=\begin{bmatrix}
\ga & 0 \\
0 &  1
\end{bmatrix} \footnote{Here and henceforth we shall abuse the notation $\ga$, allowing it to represent both the $2\times 2$ and $3 \times 3$ matrices whenever its meaning is evident from the context.},
t^{-\frac{1}{2}}I_2 & 0 \\
0 & t
\end{bmatrix},
I_2 & 0 \\
v & 1
\end{bmatrix},
\end{equation}

where $\ga\in \SL(2,\Z),t>0, v=(x,y)\in \R^2$, we have

\begin{align}
\int_H f(x)d\mu(x)=& \int_{\SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} f(u_va_t\ga) dv \frac{1}{t^4} dt d\ga\\
=&\sum_{\ga\in \SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} f(u_v a_t\ga) dv \frac{1}{t^4} dt,
\end{align}

where the measure $dv \frac{1}{t^4} dt d\ga=dx dy \frac{1}{t^4} dt d\ga$ is left $H$-invariant.

To check the left invariance, fix $\ga_0 \in \SL_{2,1}(\Z)$, $a_{t_0}:=\begin{bmatrix}
t_0^{-\frac{1}{2}}I_2 & 0 \\
0 & t_0
\end{bmatrix}, t_0\ne 0$ and $u_{v_0}:=\begin{bmatrix}
I_2 & 0 \\
v_0 & 1
\end{bmatrix}$, $v_0:=(x_0,y_0)\in \R^2$, then

\begin{align*}
u_{v_0} a_{t_0} \ga_0 \cdot u_v a_t \ga
=& u_{v_0} [a_{t_0} \ga_0 u_v (a_{t_0} \ga_0)^{-1}] a_{t_0} \ga_0 a_t \ga \\
=& u_{v_0} u_{t_0^{\frac{3}{2}} v\ga_0^{-1}}a_{t_0} a_t \ga_0 \ga \\
=& u_{v_0+t_0^{\frac{3}{2}} v\ga_0^{-1}}a_{t_0 t} \ga_0 \ga
\end{align*}

It follows that

\begin{align*}
& \int_{\SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} L_{u_{v_0} a_{t_0} \ga_0}[f(ua\ga) ] dx dy \frac{1}{t^4}dt d\ga \\
=&  \int_{\SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} f(u_{v_0} a_{t_0} \ga_0ua\ga)  dx dy \frac{1}{t^4}dt d\ga \\
=&  \int_{\SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} f(u_{v_0+t_0^{\frac{3}{2}} v\ga_0^{-1}}a_{t_0 t} \ga_0 \ga) dx dy \frac{1}{t^4}dt  d\ga
\end{align*}

By the change of variable

\begin{equation*}
v:=(x,y)\mapsto v':=(x',y')=v_0+t_0^{\frac{3}{2}}v\ga_0^{-1}, t\mapsto t':=t_0t, \ga \mapsto \ga':=\ga_0 \ga,
\end{equation*}

and noticing that $\ga_0$ (with determinant $1$) does not contribute to the Jacobian, the last line of the equation above becomes

\begin{equation*}
\int_{\SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} f(u_{v'}a_{t'}\ga') (t_0^{-\frac{3}{2}})^2 dx' dy' \frac{t_0^4}{t'^4}d(\frac{t'}{t_0})  d\ga'=    \int_{\SL_{2,1}(\Z)}\int_0^{\infty}\int_{\R^2} f(u_{v'}a_{t'}\ga') dx' dy' t'^4 dt'  d\ga'
\end{equation*}

This verifies the invariance of measure. Finally, by the uniqueness of Haar measure, the measure $d\ga$ on $\SL_{2,1}(\Z)$ is the counting measure and therefore the out-most integral reduces to the sum.

Let $\|\cdot\|$ denote the Hilbert-Schmidt norm on matrices. Namely $\|A\|:=\sqrt{\text{Trace}(A^TA)}$, or the square root of the sum of squares of all entries of the matrix $A$.

For any subgroup $L$ of $G=\SL(3,\R)$, let

\begin{equation}\label{meaning of the subscript T}
L_T:=\{g\in L: \|g\|<T\}.
\end{equation}

Following <cit.>, for $g_1,g_2,g \in G$ and $T>0$, we define the so-called “skewed balls" as follows:

\begin{equation}
H_T[g_1,g_2]:=\{h\in H: \|g_1hg_2\|<T \}.
\end{equation}

\begin{equation}
V_{\ga,T}[g_1,g_2]:=\{h\in U A \ga: \|g_1hg_2\|<T \},
\end{equation}

then it follows that

\begin{equation}\label{decomposition of H_T into V_T}
H_T[g_1,g_2]=\bigsqcup_{\ga \in \SL_{2,1}(\Z)}V_{\ga,T}[g_1,g_2].
\end{equation}

For $F\in C_c(G/\Ga)$ and $g_1,g_2\in G$, we consider the measure defined by the integral

\begin{equation}\label{definition of measure funtional}
\phi_{\mu}(F,T)=\phi_{\mu}(F,T,g_1,g_2):=\frac{1}{\mu \left(H_T[g_1,g_2] \right)}\int_{H_T[g_1,g_2]}F(h^{-1}g_1\Gamma)d\mu(h).
\end{equation}

In view of Theorem 2.3 in <cit.>, the following equidistribution theorem plays the central role in the proof of Theorem <ref>:

Let $H$ and $H_T$ be defined as above, and let $\mu=\mu_H$ denote the left Haar measure on $H$, and $\mu_X$ denote the normalized $G$-invariant probability measure on $X=G/\Ga$. For any $f\in C_c(X)$, let $\phi_{\mu}(F,T)$ be defined as in <ref> with $g_1, g_2\in G$ arbitrary. Then

\begin{equation}
\lim_{T\to \infty} \phi_{\mu}(F,T)= \int_{X}F(x)d\mu_X(x).
\end{equation}

In other words, the limiting measure is the $G$-invariant probably measure on $X$.

For each point $Hg_0\in H\backslash G$, there exists a measure $\nu_{Hg_0}$ (not $G$-invariant) on $H\backslash G$ such that for any compactly supported function $\varphi \in C_c(H\backslash G)$, we have

\begin{equation}
\lim_{T\to \infty}\frac{1}{\mu(H_T)}\int_{G_T}\varphi(Hg_0g)dm(g)=\int_{H\backslash G} \varphi d\nu_{Hg_0}.
\end{equation}

§.§ Structure of the paper and further work

* The outline of our proof of Theorem <ref> is as follows. We will first give an estimate for the volume of $H_T[g_1,g_2]$ (Section 2). This estimate will help us to show the unipotent invariance of limiting measure (Section 3), which in turn opens the gate to the application of Ratner theory. In particular, a dichotomy theorem (Theorem <ref>) due to Shah plays the central role here. To exclude the first outcome of the dichotomy theorem, we use a lemma (Lemma <ref>) on the expansion of the norms of vectors under the action of unipotent subgroups. This allows us to prove the non-escape of mass on the space of one-point compactification of $G/\Ga$ and the $G$-invariance will follow from Ratner's measure classification theorem (<cit.>, <cit.>).

* In Section 5 we prove Theorem <ref> as a consequence of the duality principle stated in Theorem 2.2 to Corollary 2.4 in <cit.>.

We remark that despite that the theorem is stated from the case of rank-two lattices in $\R^3$. Our work in progress is to generalize the results to higher ranks and dimensions. The main difficulty lies in the estimtate of the volumes of balls as well as the expansion of norms under unipotent actions.

§ ACKNOWLEDGEMENTS

The authors would like to thank Nimish Shah for helpful discussions and generous sharing of his ideas. We would also like to thank Uri Shapira and Barak Weiss for their comments on the manuscript.

§ ESTIMATE OF THE HAAR MEASURE GROWTH OF SKEWED BALLS IN $H$

For $g_1, g_2 \in G$, by using Iwasawa decomposition (for block-lower-triangular matrix), we may assume

\begin{equation}\label{matrix representation of g1 and g2}
g_1:=k_1\begin{bmatrix} G_1 & 0 \\G_3 & G_4 \end{bmatrix}, g_2:=\begin{bmatrix} H_1 & 0 \\H_3 & H_4 \end{bmatrix} k_2
\end{equation}

where $k_1,k_2 \in \SO(3,\R)$, $G_1,H_1\in \R^{2\times 2}$, $G_3, H_3\in \R^{2\times 1}$ and $G_4, H_4\in \R$.

The main goal of this section is to prove the following:
$H_T[g_1,g_2]:=\{h\in H: \|g_1hg_2\|<T \}$ and $V_{\ga,T}[g_1,g_2]:=\{h\in U A \ga: \|g_1hg_2\|<T \}$. Then

\begin{equation}
V_{\ga,T}[g_1,g_2]=\frac{\pi}{G_4^2 |\det(H_1)|}\frac{T^6}{6\|G_1\ga H_1\|^4}+O_{g_1,g_2}(T^4),
\end{equation}

\begin{equation}
H_T[g_1,g_2]=\frac{\pi}{G_4^2 |\det(H_1)|}\frac{T^6}{6}\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|G_1\ga H_1\|^4}+O_{g_1,g_2}(T^4).
\end{equation}

Moreover, if $g_1$ and $g_2$ lie in a bounded subset of $G$, then the implied constant in $O_{g_1,g_2}$ is bounded.

In particular, for $g_1=g_2=e$:

\begin{equation}
\mu(H_T)=\frac{\pi}{6}\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga\|^4} T^6+O(T^4).
\end{equation}

Recall that our $H_T[g_1,g_2]$ is the disjoint union of
$V_{\ga,T}[g_1,g_2]:=\{h\in U A \ga: \|g_1hg_2\|<T \}$. We will first evaluate $V_{\ga,T}[g_1,g_2]$ and then take the sum over $\SL(2,\Z)$. Since $k_1,k_2$  preserve the Hilbert-Schmidt norm, we may assume $k_1$ and $k_2$ are identity matrices.

Every matrix in $H$ can be written as

\begin{equation*}
\begin{bmatrix}
I_2 & 0 \\
v & 1
\end{bmatrix}
\begin{bmatrix}
t^{-\frac{1}{2}}I_2 & 0 \\
0 & t
\end{bmatrix}
\begin{bmatrix}
\ga & 0 \\
0 &  1
\end{bmatrix}
t^{-\frac{1}{2}}\ga & 0 \\
t^{-\frac{1}{2}}v\ga &  t
\end{bmatrix},
\end{equation*}

where $t\ne 0, \ga \in \SL(2,\Z), v=(x,y)\in \R^2$.
By definition $h=\begin{bmatrix}
t^{-\frac{1}{2}}\ga & 0 \\
t^{-\frac{1}{2}}v\ga & t
\end{bmatrix} \in H_T[g_1,g_2]$ if and only if $\|g_1hg_2\| \le T$.

\begin{align*}
g_1hg_2=&\begin{bmatrix} G_1 & 0 \\G_3 & G_4 \end{bmatrix}
\begin{bmatrix}
t^{-\frac{1}{2}}\ga & 0 \\
t^{-\frac{1}{2}}v\ga &  t
\end{bmatrix}
\begin{bmatrix} H_1 & 0 \\H_3 & H_4 \end{bmatrix}\\
t^{-\frac{1}{2}}G_1\ga & 0 \\
t^{-\frac{1}{2}}G_3\ga+t^{-\frac{1}{2}}G_4v\ga &  G_4t
\end{bmatrix}\begin{bmatrix} H_1 & 0 \\H_3 & H_4 \end{bmatrix}\\
t^{-\frac{1}{2}}G_1\ga H_1 & 0 \\
t^{-\frac{1}{2}}G_3\ga H_1+t^{-\frac{1}{2}}G_4v\ga H_1 + G_4 H_3t &  G_4H_4t
\end{bmatrix}
\end{align*}

Upon taking the sum of squares and rearranging terms, we conclude that $\|g_1hg_2\| \le T$ is equivalent to

\begin{equation}\label{equation defining the disk of integration}
\|G_3\ga H_1+G_4 v\ga H_1+ G_4 H_3 t^{\frac{3}{2}}\|^2 \le -G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2
\end{equation}

In view of (<ref>), let

\begin{equation}\label{the notation D for the disk}
D=D_{\ga, t}:=\{v\in \R^2: \|G_3\ga H_1+G_4 v\ga H_1 + G_4 H_3 t^{\frac{3}{2}}\|^2 \le -G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2\}.
\end{equation}

\begin{equation}
\mu \left(V_{\ga,T}[g_1,g_2]\right):=\int_0^{\infty}\int_{\R^2} \mathbf{1}_{V_{\ga,T}[g_1,g_2]}(u_va_t\ga) dv \frac{1}{t^4}dt=\int_0^{\infty}m(D_{\ga,t})\frac{1}{t^4}dt,
\end{equation}

where $m(D_{\ga,t})$ is the Lebesgue measure of $D_{\ga,t}$ with the understanding that it is zero for $t$ with $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2<0$.

We  observe that $D_{\ga,t}$ is the interior of an ellipse on $\R^2$, whose area is

\begin{equation}
\pi \frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|}.
\end{equation}

The maximal value of $-G_4^2 H_4^2 t^3+tT^2$ for $t>0$ is $M_{T}:=\frac{2T^3}{3\sqrt{3}G_4 H_4}$, attained at $t_0:=\frac{T}{\sqrt 3 G_4H_4}$. It is easy to see that $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2$ has two positive roots which we denote by $a=a_{\ga,T}$ and $b=b_{\ga,T}$ with $0<a<b<\frac{T}{G_4H_4}$, whenever $\|G_1\ga H_1\|^2<M_T$. Moreover, $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2$ is positive on $(a,b)$. Observe that we only have to consider those $\ga$ with $\|G_1\ga H_1\|^2<M_T$ and $0<a<t_0<b<\sqrt{3}t_0$.

Since $-G_4^2 H_4^2 a^3+aT^2-\|G_1\ga H_1\|^2=0$,

\begin{equation}\label{range for a}
a=\frac{\|G_1\ga H_1\|^2}{T^2-G_4^2H_4^2 a^2}\in
\left(\frac{\|G_1\ga H_1\|^2}{T^2}, \frac{3\|G_1\ga H_1\|^2}{2T^2}\right).
\end{equation}

In particular,

\begin{equation}
a=\Theta \left( \frac{\|G_1\ga H_1\|^2}{T^2} \right) \label{estimate for a in all range}
\end{equation}

Meanwhile by Taylor expansion,

\begin{align}
a=&\frac{\|G_1\ga H_1\|^2}{T^2}\frac{1}{1-\frac{G_4^2H_4^2 a^2}{T^2}} \nonumber\\
=&\frac{\|G_1\ga H_1\|^2}{T^2}\left(1+\frac{G_4^2H_4^2 a^2}{T^2}+\cdots \right) \label{taylor expansion for a}
\end{align}

axis lines=center,
xmax = 4,
xmin = -1,
ymax = 4,

[domain=0.2:3,samples=250, thick, blue] -x^3+3*x^2-1
node [font=, pos=0.5, above]$f(t)=-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2$;
[red,label=[yshift=0.1ex]135:$a$,circle,fill,inner sep=1.2pt] at (axis cs:0.6527,0) ;
[red,label=[yshift=0.1ex]45:$b$,circle,fill,inner sep=1.2pt] at (axis cs:2.88,0) ;
[red,label=[yshift=0.1ex]45:,circle,fill,inner sep=1.2pt] at (axis cs:2,3) ;
[label=[yshift=0.1ex]180:$M_T$,circle,fill=red,inner sep=1.2pt] at (axis cs:0,3) ;
[label=[yshift=0.1ex]270:$t_0$,circle,fill=red,inner sep=1.2pt] at (axis cs:2,0) ;
const plot,
empty line=jump,
const plot,
empty line=jump,

Estimating the roots of the polynomial.

Moreover, for the $\ga$ with smaller norms, it follows from (<ref>) that

\begin{equation}\label{estimate for a in smaller range}
a=\frac{\|G_1\ga H_1\|^2}{T^2}\left(1+O_{g_1,g_2}\left(\frac{1}{T^2} \right)\right), \text{ whenever }\|G_1\ga H_1\|\le T.
\end{equation}

Now we compute the measure of $H_T[g_1,g_2]$. By Proposition <ref> with $f=\mathbf{1}_{H_T[g_1,g_2]}$,

\begin{align}
\mu \left(H_T[g_1,g_2] \right)
=& \sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T}} \mu \left(V_{\ga,T}[g_1,g_2]\right)\\
= &\sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T}} \frac{\pi }{G_4^2 |\det(H_1)|}\int_a^b \frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{t^4}dt \nonumber\\
=&\sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T}} \frac{\pi }{G_4^2 |\det(H_1)|}\int_a^b -\frac{G_4^2 H_4^2}{t}+\frac{T^2}{t^3}-\frac{\|G_1\ga H_1\|^2}{t^4}dt
\end{align}

\begin{align*}
&\int_a^b -\frac{G_4^2 H_4^2}{t}+\frac{T^2}{t^3}-\frac{\|G_1\ga H_1\|^2}{t^4}dt\\
=& \left(-G_4^2 H_4^2 \log(t)-\frac{T^2}{2t^2}+\frac{\|G_1\ga H_1\|^2}{3t^3}\right)\Bigg|_{t=a}^{t=b}\\
=& \left(-G_4^2 H_4^2 \log(b)-\frac{T^2}{2b^2}+\frac{\|G_1\ga H_1\|^2}{3b^3}\right)-\left(-G_4^2 H_4^2 \log(a)-\frac{T^2}{2a^2}+\frac{\|G_1\ga H_1\|^2}{3a^3}\right)
\end{align*}

Since $0<a<t_0<b<\sqrt{3}t_0:=\frac{T}{G_4H_4}$, $\|G_1\ga H_1\|^2<M_T=\frac{2T^3}{3\sqrt{3}G_4 H_4}$ and by (<ref>), the expression above is asymptotically

\begin{equation}\label{before plugging in a}
\frac{T^2}{2a^2}-\frac{\|G_1\ga H_1\|^2}{3a^3}+O_{g_1,g_2}(\log(T)).
\end{equation}

Now by (<ref>), the equation (<ref>) has the estimate

\begin{equation}\label{estimate for V ga T}
\Theta \left(\frac{T^6}{\|G_1\ga H_1\|^4} \right),
\end{equation}

for all $\ga$.

In view of (<ref>), if furthermore $\|G_1\ga H_1\|\le T$, then (<ref>) becomes

\begin{align}
=&\frac{T^6}{2\|G_1\ga H_1\|^4}\left(1+O_{g_1,g_2}\left(\frac{1}{T^2} \right)\right)^{-2}-\frac{T^6}{3\|G_1\ga H_1\|^4}\left(1+O_{g_1,g_2}\left(\frac{1}{T^2} \right)\right)^{-3}\nonumber\\
=&\frac{T^6}{2\|G_1\ga H_1\|^4}(1-O_{g_1,g_2}(\frac{1}{T^2})+\cdots)^2-\frac{T^6}{3\|G_1\ga H_1\|^4}(1-O_{g_1,g_2}(\frac{1}{T^2})+\cdots)^3 \nonumber\\
=&\frac{T^6}{6\|G_1\ga H_1\|^4}-\frac{1}{\|G_1\ga H_1\|^4}O_{g_1,g_2}\left(T^4\right)
\end{align}

It follows that

\begin{align}
\mu \left(H_T[g_1,g_2] \right)
= & \frac{\pi}{G_4^2 |\det(H_1)|}\sum_{\substack{\ga\in \SL(2,\Z), \nonumber\\ \|G_1\ga H_1\|\le T}}\left(\frac{T^6}{6\|G_1\ga H_1\|^4}+\frac{1}{\|G_1\ga H_1\|^4}O\left(T^4\right)\right) \nonumber\\
+&\frac{\pi }{G_4^2 |\det(H_1)|}\sum_{\substack{\ga\in \SL(2,\Z),\\ T<\|G_1\ga H_1\|<\sqrt{M_T}}}   O\left(\frac{T^6}{\|G_1\ga H_1\|^4} \right) \nonumber \\
=&  \frac{\pi}{G_4^2 |\det(H_1)|}\frac{T^6}{6}\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|G_1\ga H_1\|^4}+O_{g_1,g_2}(T^4), \label{measure of H_T}
\end{align}

where the last line of the equation is the consequence of the lemma below.

For any invertible $2\times 2$ matrices $A$ and $B$ and positive numbers $\alpha>2$ and $K$, we have

\begin{equation}\label{head estimate for the sum over SL2}
\sum_{\substack{\ga\in \SL(2,\Z),\\ \|A\ga B\|\le K}}\frac{1}{\|A\ga B\|^{\alpha}}
\iffalse
\begin{cases}
O_{A,B}(\log(K)) & \text{if } \alpha=2\\
O_{A,B}(K^{2-\alpha}) & \text{if } \alpha\ne 2
\end{cases}.
\fi
\end{equation}

is convergent as $K\to \infty$ and we have the tail estimate:

\begin{equation}\label{tail estimate for the sum over SL2}
\sum_{\substack{\ga\in \SL(2,\Z),\\ \|A\ga B\|> K}}\frac{1}{\|A\ga B\|^{\alpha}}=O_{A,B}(K^{2-\alpha})
\end{equation}

Let $0=\tau_0<\tau_1<\tau_2<\tau_3 \cdots$ be an enumeration of $\|A\ga B\|$ with $\ga \in \SL(2,\Z)$. For $i=1,2,3\cdots$ and $\tau>0$, let

\begin{equation}
N(\tau):=\#\{\ga\in \SL(2,\Z):\|A\ga B \|\le \tau\}
\end{equation}

and for $K>0$ $M(K):=\max\{i:N(\tau_i)\le K\}$. Observe that $M(K)\gtrsim
K$ and  $N(\tau_i)-N(\tau_{i-1})=\#\{\ga\in \SL(2,\Z):\|A\ga B \|=\tau_i\}$.

It follows from <cit.> (see Appendix B for computation details) that for $A=B=I_2$

\begin{equation}
\end{equation}

Since for operator norms we have the inequality
$$\|A^{-1}\|^{-1}_{\text{op}}\|\ga \|_{\text{op}}\|B^{-1}\|^{-1}_{\text{op}} \le \|A\ga B\|_{\text{op}}\le \|A\|_{\text{op}}\|\ga\|_{\text{op}}\|B\|_{\text{op}},$$

and all norms on finite-dimensional spaces are equivalent, we have for general $A$ and $B$, $N(\tau)\asymp O_{A,B}(\tau^2)$. For $K>0$,

\begin{align}
\sum_{\substack{\ga\in \SL(2,\Z),\\ \|A\ga B\|\ge K}}\frac{1}{\|A\ga B\|^{\alpha}}
=&\sum_{i=M(K)}^{\infty}\frac{N(\tau_i)}{\tau_i^\alpha}-\sum_{i=1}^{\infty}\frac{N(\tau_{i})}{\tau_{i+1}^\alpha}\tag{sum is convergence when $\alpha>2$}\\
= &\sum_{i=M(K)}^{\infty}\int_{\tau_i}^{\tau_{i+1}}\frac{\alpha N(\tau)}{\tau^{\alpha+1}}d\tau\\
%     =&\sum_{i=1}^{M-1}\left(\frac{N(\tau_i)}{\tau_i^\alpha}-\frac{N(\tau_{i})}{\tau_{i+1}^\alpha}\right)+\frac{N(\tau_{M})}{\tau_M^{\alpha}}\\
=&\int_{\tau_{M(K)}}^{\infty}\frac{\alpha N(\tau)}{\tau^{\alpha+1}}d\tau\\
\lesssim_{A,B} &\int_{\tau_{M(K)}}^{\infty}\frac{\alpha c }{\tau^{\alpha-1}}d\tau+\int_{\tau_{M(K)}}^{\infty}\frac{\alpha O_{\eta}(\tau^{\frac{5}{3}+\eta}) }{\tau^{\alpha+1}}d\tau\\
%    \asymp &\int_{\tau_1}^{K}\frac{\alpha c }{\tau^{\alpha-1}}d\tau+\frac{N(\tau_{M})}{\tau_M^{\alpha}}
\asymp & \tau_{M(K)}^{2-\alpha} \le K^{2-\alpha}
%    \asymp & \tau_K^{2-\alpha}
\end{align}

This gives (<ref>). The proof for (<ref>) is similar.

The following properties follow immediately from (<ref>):

For any bounded subset $B\subset G$ and any $\e>0$, there are $T_0$ and $\de>0$ such that for all $T>T_0$ and all $g_1,g_2 \in B$ we have:

\begin{equation}
\mu \left(H_{(1+\de)T}[g_1,g_2] \right)\le (1+\e)\mu \left(H_T[g_1,g_2] \right).
\end{equation}

For any $g_1,g_2 \in G$. the limit

\begin{equation}
\alpha(g_1,g_2):=\lim_{T\to \infty}\frac{\mu \left(H_T[g_1^{-1},g_2] \right)}{\mu \left(H_T \right)},
\end{equation}

exists and is positive and finite.

It follows from Proposition <ref> that

\begin{equation}\label{definition and computation of alpha}
\alpha(g_1^{-1},g_2):=\lim_{T\to \infty}\frac{\mu \left(H_T[g_1,g_2] \right)}{\mu \left(H_T \right)}=\frac{1}{G_4^2 |\det(H_1)|}\frac{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|G_1\ga H_1\|^4}}{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga \|^4}}.
\end{equation}

§ THE $U$-INVARIANCE OF LIMITING MEASURE

Let $F\in C_c(G/\Ga)$, for $g_1,g_2\in G$ with matrix representation as in (<ref>), recall the measure defined via the funtional

\begin{equation}
\phi_{\mu}(F,T)=\phi_{\mu}(F,T,g_1,g_2):=\frac{1}{\mu \left(H_T[g_1,g_2] \right)}\int_{H_T[g_1,g_2]}F(h^{-1}g_1\Gamma)d\mu(h)
\end{equation}

By Banach-Alaoglu theorem, for any fixed, $\phi_{\mu}(\cdot,T), T>0$ form a sequencially compact set in weak-star topology. By passing to a subsequence if necessary and with an abuse of notations, henceforth we assume that $\lim_{T\to \infty}\phi_{\mu}(F,T)$ exists.

The main goal of this section is to prove that the limiting measure is $U$-invariant:

$\lim_{T\to \infty}\phi_{\mu}(F,T)$ is left-invariant under $U$, where $U:=\begin{bmatrix}
I_2 & 0 \\
\mathbb{\R}^2 & 1
\end{bmatrix}$. Specifically, for any $u_0\in U$, define $L_{u_0}(F)(x):=F(u_0^{-1}x)$, then

\begin{equation}
\lim_{T\to \infty}\left(\phi_{\mu}(F,T)-\phi_{\mu}(L_{u_0} F,T)\right)=0.
\end{equation}

In view of (<ref>), we can decompose $\phi_{\mu}(F,T)$ as the following convex linear combination:

\begin{equation}
\phi_{\mu}(F,T)= \sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T}}\frac{\mu \left(V_{\ga,T}[g_1,g_2] \right)}{\mu \left(H_T[g_1,g_2] \right)}\cdot \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} F(h^{-1}g_1\Gamma) d\mu(h).
\end{equation}

We will first prove that individual terms are $U$-invariant. The following elementary lemma will be needed:

Let $f:\R^{d}\to \R$ be a bounded continuous function. Let $E\subset \R^{d}$ be a ellipsoid with surface area $S$, then for $y\in \R^{d}$, with $\|y\|$ less than the shortest radius of $E$, we have the estimate:

\begin{equation}
\left|\int_{E}[f(v)-f(y+v)]dv\right| \le C_f \|y\|S,
\end{equation}

where $C_f$ is a constant depending on $f$.

\begin{align*}
\left|\int_{B(x,r)}[f(v)-f(y+v)]dv\right|
= &\left|\int_{B(x,r)}f(v)dv - \int_{B(x+y,r)}f(v)dv\right|\\
\le & \int_{B(x,r)\triangle
\le & C_f \|y\|S, \tag{$f$ is bounded}
\end{align*}

where the last line follows from the Theorem 1 of <cit.>.

For any $\ga\in \SL(2,\Z)$ with $\|G_1\ga H_1\|^2<M_T$, let

\begin{equation}
\phi_{\ga,\mu}(F,T):=\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} F(h^{-1}g_1\Gamma) d\mu(h).
\end{equation}

For any $u_0\in U$, define $L_{u_0}(F)(x):=F(u_0^{-1}x)$, then

\begin{equation}
\lim_{T\to \infty}\left(\phi_{\ga,\mu}(F,T)-\phi_{\ga,\mu}(L_{u_0} F,T)\right)=0.
\end{equation}

Assume that $F\in C_c(G/\Ga)$ is bounded by $M_F$.
Recall from Proposition(<ref>) that

\begin{equation}\label{estimate for the measure of V_T}
\mu \left(V_{\ga,T}[g_1,g_2] \right)=\Theta \left(\frac{T^6}{\|G_1\ga H_1\|^4} \right),
\end{equation}

Fix $u_0:=u_{v_0}=U:=\begin{bmatrix}
I_2 & 0 \\
v_0 & 1
\end{bmatrix}\in U$. The integral

\begin{align*}
&\left| \int_{V_{\ga,T}[g_1,g_2]} F(h^{-1}g_1\Gamma) d\mu(h)-\int_{V_{\ga,T}[g_1,g_2]} F(u_0^{-1}h^{-1}g_1\Gamma) d\mu(h) \right |\\
=&\left|\int_{V_{\ga,T}[g_1,g_2]} \left(F(h^{-1}g_1\Gamma) - F(u_0^{-1}h^{-1}g_1\Gamma) \right) d\mu(h)\right |\\
=&\left|\int_{a}^b \int_{D} \left(F(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma) - F(u_{v_0}^{-1}\ga^{-1} a_t^{-1} u_v^{-1}g_1\Gamma) \right) dv \frac{1}{t^4}dt\right |\\
=&\left|\int_{a}^b \int_{D} \left(F(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma) - F(\ga^{-1} (\ga u_{v_0}^{-1}\ga^{-1}) a_t^{-1} u_v^{-1}g_1\Gamma) \right) dv \frac{1}{t^4}dt\right |\\
=&\left|\int_{a}^b \int_{D} \left(F(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma) - F(\ga^{-1}a_t^{-1} [a_t\ga u_{v_0}^{-1}\ga^{-1}a_t^{-1}]  u_v^{-1}g_1\Gamma) \right) dv \frac{1}{t^4}dt\right |\\
=&\left|\int_{a}^b \int_{D} \left(F(\ga^{-1} a_t^{-1} u_{-v} g_1\Gamma) - F(\ga^{-1}a_t^{-1} u_{-t^{\frac{3}{2}}v_0\ga^{-1}-v}  g_1\Gamma) \right) dv \frac{1}{t^4}dt\right |\\
=&\left| \left(\int_a^c+\int_c^b \right) \int_{D} \left(F(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma) - F(\ga^{-1}a_t^{-1} u_{-t^{\frac{3}{2}}v_0\ga^{-1}-v}  g_1\Gamma) \right) dv \frac{1}{t^4}dt\right |,\tag{note that $u_v^{-1}=u_{-v}$}
\end{align*}

where, as before, $a=a_{\ga,T}$ and $b=b_{\ga,T}$ are two positive roots of the cubic equation $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2=0$ and satisfy $0<a<t_0<b<\sqrt{3}t_0$ with $t_0:=\frac{T}{\sqrt 3 G_4H_4}$. Here

\begin{equation}\label{the truncation c}
c=c_{\ga,T}=\frac{\|G_1\ga H_1\|^{1.5}}{T^{1.5}}
\end{equation}

is a parameter for the separation of integral. The domain $D$ is as defined in (<ref>), whose area is $\pi \frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|}$.

The second integral $\int_c^b$ has the estimate

\begin{align*}
&\left| \int_c^b  \int_{D} \left(F(\ga^{-1} a_t^{-1} u_{-v} g_1\Gamma) - F(\ga^{-1}a_t^{-1} u_{-t^{\frac{3}{2}}v_0\ga^{-1}-v}  g_1\Gamma) \right) dv \frac{1}{t^4}dt \right|\\
\le &  \int_c^b  \int_{D} 2M_F dv \frac{1}{t^4}dt \\
= & 2M_F\int_c^b \pi \frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|} \frac{1}{t^4}dt\\
=& 2M_F\frac{\pi}{G_4^2 |\det(H_1)|} \left[\left(-G_4^2 H_4^2 \log(b)-\frac{T^2}{2b^2}+\frac{\|G_1\ga H_1\|^2}{3b^3}\right)-\left(-G_4^2 H_4^2 \log(c)-\frac{T^2}{2c^2}+\frac{\|G_1\ga H_1\|^2}{3c^3}\right)\right]
\end{align*}

It is easy to see that $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2$ has two positive roots which we denote by $a=a_{\ga,T}$ and $b=b_{\ga,T}$ with $0<a<b<\frac{T}{G_4H_4}$. Moreover, $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2$ is positive on $(a,b)$.

The maximal value of $-G_4^2 H_4^2 t^3+tT^2$ for $t>0$ is $M=M_{T}:=\frac{2T^3}{3\sqrt{3}G_4 H_4}$, attained at $t_0:=\frac{T}{\sqrt 3 G_4H_4}$. Observe that we only have to consider those $\ga$ with $\|G_1\ga H_1\|^2<M_T$ and $0<a<t_0<b<\sqrt{3}t_0$.
By our choice of $c$ and the bounds for $b$,  after normalization by $V_{\ga,T}[g_1,g_2]=\Theta\left(\frac{T^6}{\|G_1\ga H_1\|^4} \right)$, each one of the six terms in the last line of equation will vanish as $T\to \infty$, provided that $\ga$ is fixed.

Now it remains to estimate the first integral $\int_a^c$.

\begin{align*}
&\left| \int_a^c  \int_{D} \left(F(\ga^{-1} a_t^{-1} u_{-v} g_1\Gamma) - F(\ga^{-1}a_t^{-1} u_{-t^{\frac{3}{2}}v_0\ga^{-1}-v}  g_1\Gamma) \right) dv \frac{1}{t^4}dt \right|\\
\le &  C_F \int_a^c  \left( \|-t^{\frac{3}{2}}v_0\ga^{-1}\| \sqrt{\frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|}}\right) \frac{1}{t^4}dt  \tag{by Lemma \ref{lemma on symmetric difference}}\\
= &  C_F \int_a^c  \left( \|-t^{-\frac{5}{2}}v_0\ga^{-1}\| \sqrt{\frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|}}\right) dt .
\end{align*}

Observe that for fixed $\ga$, our choice of $c=\frac{\|G_1\ga H_1\|^{1.5}}{T^{1.5}}>>a\asymp \frac{\|G_1\ga H_1\|^{2}}{T^{2}}$ as $T\to \infty$. So the equation above can be bounded from above by (recall that $M_T=\frac{2T^3}{3\sqrt{3}G_4 H_4}$ is the maximum of $-G_4^2 H_4^2 t^3+tT^2$ for $t>0$.)

\begin{align*}
& (c-a)\cdot C_F\left( \|-a^{-\frac{5}{2}}v_0\ga^{-1}\| \sqrt{\frac{\frac{2T^3}{3\sqrt{3}G_4 H_4}-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|}}\right) \\
= & O_{F, g_1, g_2}(T^{-1.5})O_{F, g_1, g_2}(T^5\cdot T^{\frac{3}{2}})\\
= & O_{F, g_1, g_2}(T^5),
\end{align*}

and again, this term goes to zero as $T\to \infty$, after normalization by $V_{\ga,T}[g_1,g_2]=\Theta \left(\frac{T^6}{\|G_1\ga H_1\|^4} \right)$.

Then Theorem <ref> follows from the following elementary lemma from calculus, with $a_{\ga,T}=\phi_{\ga,\mu}(F,T), b_{\ga,T}=\phi_{\ga,\mu}(L_{u_0}F,T)$ and $c_{\ga,T}=\frac{\mu \left(V_{\ga,T}[g_1,g_2] \right)}{\mu \left(H_T[g_1,g_2] \right)}=\Theta_{g_1,g_2}\left(\frac{1}{\|\ga\|^4}\right)$:

Let $a_{\ga,T},b_{\ga,T},c_{\ga,T}$ be three bounded (in both $n$ and $T$) real sequences with $\ga \in \SL(2,\Z)$, and $c_{\ga,T}, T\in \R_+$. Assume

* For any $T>0$, we have $\sum_{\ga} c_{\ga,T}=1$;

* For any $\e>0$, there exists $N_{\e}$, such that $\sum_{\|\ga\|>N_{\e}} c_{\ga,T}<\e$ for all $T>0$;

* For any $\ga$, we have $\lim_{T\to \infty}(a_{\ga,T}-b_{\ga,T})=0$,

$$\lim_{T\to \infty}\sum_{\ga} c_{\ga,T}(a_{\ga,T}-b_{\ga,T})=0.$$

Assume $a_{\ga,T},b_{\ga,T}$ are bounded by $B$. For any $\e>0$, choose $N_{\e}$, such that $\sum_{\|\ga\|>N_{\e}} c_{\ga,T}<\e$ for all $T>0$. Then

\begin{align*}
\limsup_{T\to \infty}\left| \sum_n c_{\ga,T}(a_{\ga,T}-b_{\ga,T})\right|
\le&\limsup_{T\to \infty}\left(\left|\sum_{\|\ga\|\le N_{\e}} c_{\ga,T}(a_{\ga,T}-b_{\ga,T})\right|+\left|\sum_{\|\ga\|> N_{\e}} c_{\ga,T}(a_{\ga,T}-b_{\ga,T})\right|\right)\\
\le & \limsup_{T\to \infty}\left|\sum_{\|\ga\|\le N_{\e}} c_{\ga,T}(a_{\ga,T}-b_{\ga,T})\right|+2B\e\\
\le& 2B\e
\end{align*}

Letting $\e \to 0$, we are done.

§ PROOF OF THEOREM <REF>: THE $G$-INVARIANCE OF LIMITING MEASURE ON $G/\GA$.

Our goal of this section and the next is to generalize Theorem <ref> from $U$ to $G$. In contrast to the proof of the $U$-invariance in the previous section, our approach here is much more convoluted, which involves the linearization techniques as well as some representation results (cf. <cit.> and <cit.>).

To begin with, let $\overline{X} = \overline{G/\Ga}$ denote the one-point compactification of $X=G/\Ga$. Our plan is to show that if $\lim_{T\to \infty} \phi_{\mu}(\cdot,T)=\eta(\cdot)$ (recall this is by applying the Banach-Alaoglu theorem and passing to a subsequence [recall if any convergent subsequence of a bounded sequence converges to the same limit, then so does the original bounded sequence.]) for some normalized measure $\eta$ on $\overline{X}$, then

* $\eta(\infty)=0$, and

* $\eta$ is $G$-invariant.

As a result we must have $\eta|_X=\mu_X$.

To prove the first statement, we shall utilize the following theorem due to Shah <cit.> (also cf. <cit.>) on the divergence of polynomial trajectories. For a general connected semisimple Lie group $G$ without compact factors and $\Ga$ a lattice in $G$. Let $\frak g$ be the Lie algebra of $G$. For positive integers $d$ and $n$, denote by
$\mathcal {P}_{d,n}(G)$ the set of functions $q: \R^n \to G$ such that for any $a, b \in \R^n$, the map

\begin{equation}
\tau\in \R \mapsto \Ad(q(\tau a+b))\in \frak{g}
\end{equation}

is a polynomial of degree at most $d$ with respect to some basis of $\frak g$.

Let $V_G = \sum_{i=1}^{\dim \frak g} \bigwedge^i \frak{g}$. There is a natural action of $G$ on $V_G$ induced from the adjoint
representation (in other words, we are considering a representation $\pi:G \to \text{GL}(V_G)$ but sometimes omit the symbol $\pi$). Fix a norm on $V_G$. For a Lie subgroup $H$ of $G$ with Lie algebra $\frak{h}$, take a unit vector $p_H \in \bigwedge^{\dim \frak h} \frak{h}$.

With notations above, there exist closed subgroups $U_i (i=1,2,...,l)$ such that each $U_i$ is the unipotent radical of a parabolic subgroup, $U_i\Ga$ is compact in $X=G/\Ga$ and for any $d,n\in \N$, $\e,\de>0$, there exists a compact set $C \subset G/\Ga$ such that for any $q \in \mathcal{P}_{d,n}(G)$ and a bounded open convex set $D\subset \R^n$, one of the following holds:

* there exist $\ga'\in \Ga$ and $i=1,...,l$ such that $\sup_{v\in D}\|q(v)\ga'.p_{U_i}\| \le \de$;

* $m(v\in D:q(v)\Ga \notin C)< \e m(D)$, where $m$ is the Lebesgue measure on $\R^n$.

For $g_1\in G=\SL(3,\R)$ and $\ga' \in \Ga$, we shall consider the following family of maps: Let $n=2, t>0$, $v\in \R^2$ and define $q_t(v):=\ga a_t^{-1} u_v^{-1} g_1$ (Here $\ga \in \SL_{2,1}(\Z)$ shouldn't be confused with $\ga'\in \Ga$ above).

Our strategy is to investigate how $q_t(v)$ fails the condition 1 of the Theorem <ref> by studying the expanding phenomenon of the map $q_t(v)$. As a result, the second statement holds and an approximation argument will give $\eta(\infty)=0$.

Denote by $\mathcal{H}_{\Ga}$ the family of all proper closed connected subgroups $H$ of $G$ such that $\Ga \cap H$ is a lattice in H, and $\Ad(H \cap \Ga)$ is Zariski-dense in $\Ad(H)$. We have the following important theorem:

The set $\mathcal{H}_{\Ga}$ is countable. For any $H\in \mathcal{H}_{\Ga}, \Ga.p_H$ is discrete.

Apply Theorem <ref> with $G=\SL(3,\R)$ to $q_t(v)$. The theorem gives a finite family of unipotent radicals of parabolic subgroups, denoted $U_1,...,U_l$. By Theorem <ref>, we have that the set $\Ga.p_{U_i}$ is discrete in $V_G$.

Write $V=V_G=V_0 \bigoplus V_1$, where $V_0$ is the space of vectors fixed by $G$ and $V_1$ is its $G$-invariant complement (exists because every finite-dimensional representation of a semisimple Lie group is completely reducible). Denote by $\Pi$ the projection of $V_G$ onto $V_1$ with kernel $V_0$, and note that by Lemma 17 in <cit.> that $\Pi(\Ga.p_{U_i})$ is discrete. Since $p_{U_i}$ is not fixed by $G$ (this is because the action is through conjugation and $U_i$'s are not normal subgroups in the simple group $G$), $\Pi(\Ga.p_{U_i})$ does not contain $0$. Furthermore, $\cup_{i=1}^l \Ga.p_{U_i}$ is discrete. So it follows that

\begin{equation}\label{inf of the projection is at least r}
\inf_{x\in \cup_{i=1}^l \Ga.p_{U_i}} \|\Pi(x)\|:=r>0.
\end{equation}

The union $\cup_{i=1}^l
\Ga.p_{U_i}$ is what we are looking for in the first outcome of Theorem <ref>. The following lemma will play a crucial role in proving $\eta(\infty)=0$:

Let $U_1,...,U_l$ are as above. Fix $(a_1,a_2)\in \R^2$ and $\beta=\frac{1}{2}t^{0.4}$. Then for any $\de>0$, there exist $t_0>0$ such that for any $0<t<t_0$,

\begin{equation}\label{expansion inequality for applying shah dichotomy}
\sup_{v\in D(\beta)+(a_1,a_2)} \|a_t^{-1}u_v^{-1}g_1.x\| >\de,
\end{equation}

for any $x\in \cup_{i=1}^r \Ga.p_{U_i}$,
where $D(\beta)$ is the disk of radius $\beta$ in $\R^2$ centered at the origin.

§.§ Proof of Lemma <ref>

The idea of the proof is to decompose the elements in $AU$ into elements in two subgroups of $\SL(3,\R)$ isomorphic to $\SL(2,\R)$ and then use the representation theory of $\SL(2,\R)$ and Lemma <ref> below to show find the expansion property for each. Note that a similar approach was used in <cit.>.

Observe that

\begin{align}
\begin{bmatrix}
\sqrt t & 0 & 0 \\
0 & \sqrt t & 0 \\
0 & 0 & \frac{1}{t}
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-z & -y & 1
\end{bmatrix} \nonumber\\
\begin{bmatrix}
\sqrt t & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & \frac{1}{\sqrt t}
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-\frac{y}{\sqrt{t}} & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & \sqrt t & 0 \\
0 & 0 & \frac{1}{\sqrt{t}}
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & -y & 1
\end{bmatrix} \label{one order}\\
\begin{bmatrix}
1 & 0 & 0 \\
0 & \sqrt t & 0 \\
0 & 0 & \frac{1}{\sqrt{t}}
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & -\frac{y}{\sqrt{t}} & 1
\end{bmatrix}
\begin{bmatrix}
\sqrt t & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & \frac{1}{\sqrt t}
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-z & 0 & 1
\end{bmatrix}\label{the other order}
\end{align}

It follows that the action of $a_t^{-1}u_v^{-1}$ on a vector through can be identified with two consecutive actions of elements in

\begin{equation}
\SL^{(1)}(2,\R):=
\left \{\begin{bmatrix}
1 & 0 & 0 \\
0 & a & b \\
0 & c & d
\end{bmatrix}:ad-bc=1 \right \}
\end{equation}

\begin{equation}
\SL^{(2)}(2,\R):=
\left \{\begin{bmatrix}
a & 0 & b \\
0 & 1 & 0 \\
b & 0 & d
\end{bmatrix}:ad-bc=1 \right \},
\end{equation}

each isomorphic to $\SL(2,\R)$. For example $\begin{bmatrix}
1 & 0 & 0 \\
0 & \sqrt t & 0 \\
0 & 0 & \frac{1}{\sqrt{t}}
\end{bmatrix}$ can be identified with $\begin{bmatrix}
\sqrt t & 0 \\
0 & \frac{1}{\sqrt{t}}
\end{bmatrix}$ and $\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-z & 0 & 1
\end{bmatrix}$
can be identified with
1  & 0 \\
-z & 1
\end{bmatrix}$.

Now we return to the study of the study of $a_t^{-1}u_v^{-1}w$, where $w=g_1.x$ with $x\in \cup_{i=1}^l \Ga.p_{U_i}\subset{V_G}$ as above. We consider the two following decompositions of $V_1$ (recall that this is the invariant completement of $V_0$ of fixed vectors in $V$ by $G$),

\begin{align}
V_1 &= V_{0}^{(1)} \oplus V_{1}^{(1)} \\
&= V_{0}^{(2)} \oplus V_{1}^{(2)} \label{decompositive of vs w.r.t. SL2}
\end{align}

where $V_0^{(i)}$ is the subspace of fixed vectors of the action of $\SL^{(i)}(2,\R)$ for $i=1,2$ and $V_1^{(i)}$ is its invariant complement. Note that since $G$ is generated by $\SL^{(1)}(2,\R)$ and $\SL^{(2)}(2,\R)$, see Lemma <ref>, we must have[Alternatively, note that the group $UA$ is epimorphic, see e.g. <cit.>.] $V_{0}^{(1)} \cap V_{0}^{(2)}=\{0\}$. Later on, we will also decompose $V_{1}^{(i)}$ into irreducible components with respect to the restriction of the representation $\pi$ to $\SL^{(i)}(2,\R)$, respectively.

For any $w_1\in V_1, \|w_1\|\ge r$ for some $r>0$, we write

\begin{equation}    w_1=w_{0}^{(1)}+w_{1}^{(1)}=w_{0}^{(2)}+w_{1}^{(2)}
\end{equation}

where $w_{i}^{(j)} \in V_{i}^{(j)}$ from above. We expect the non-fixed components, $w_{1}^{(1)}$ and $w_{1}^{(2)}$ to be not simultaneously small. In fact we have:

$\inf_{w_1\in V_1, \|w_1\|\ge r} \left( \|w_{1}^{(1)}\|+\|w_{1}^{(2)}\| \right) = Lr$ for some $L>0$

Indeed, using the fact that the sphere of radius $r$ is compact and that $V_{0}^{(1)} \cap V_{0}^{(2)}=\{0\}$, we have
$$\inf_{w_1\in V_1, \|w_1\|= 1} \left( \|w_{1}^{(1)}\|+\|w_{1}^{(2)}\| \right) = L > 0.$$

By scaling, it is easy to see for $r \ge 1$,

\begin{align*}
\inf_{w_1\in V_1, \|w_1\|= r} \left( \|w_{1}^{(1)}\|+\|w_{1}^{(2)}\| \right)
%    =& \frac{r}{1}\inf_{w_1\in V_1, \|w_1\|= r} \left( \|w_{1}^{(1)}\|+\|w_{1}^{(2)}\| \right) \\
= & r \inf_{w_1\in V_1, \|w_1\|= 1} \left( \|w_{1}^{(1)}\|+\|w_{1}^{(2)}\| \right) \tag{by linearity and scaling} \\
=& Lr>0.
\end{align*}

It follows from this claim that for any vector that cannot be fixed by $G$, then its non-$\SL^{(i)}(2,\R)$-fixed component must be bounded away from zero by at least $\frac{Lr}{2}>0$ for either $i=1$ or $i=2$.

Therefore, the study of the expanding phenomenon on $a_t^{-1}u_v^{-1}w$ can be transformed into the study of representation theory of $\SL(2,\R)$. Recall that if $\pi$ is an $(n+1)$-dimensional irreducible representation of $\SL(2,\R)$ and $\pi'$ is the induced Lie algebra representation, then there exist a basis $v_0,v_1,...,v_n$ such that

\begin{align}
&\pi'(H)(v_i)=(n-2i)v_i, i=0,1,...,n;\\
&\pi'(X)(v_i)=i(n-i+1)v_{i-1}, i=0,1,...,n;\\
&\pi'(Y)(v_i)=v_{i+1}, i=0,1,...,n~(v_{n+1}=0).
\end{align}

$H = \begin{bmatrix}
1 & ~~0\\
0 & -1
\end{bmatrix},
X = \begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}$, and
$Y = \begin{bmatrix}
0 & 0\\
1 & 0
\end{bmatrix}$ form a generating set of $\frak{sl}(2,\R)$.

Under the matrix Lie group-Lie algebra correspondence,

\[\pi \left(\begin{bmatrix}
\sqrt{t} & 0\\
0 & \frac{1}{\sqrt{t}}
\end{bmatrix} \right)=\pi(\exp(\log(\sqrt{t})H))\]
has the matrix representation

\begin{equation}\label{matrix rep of a_t under the basis}
\begin{bmatrix}
\sqrt{t}^n & & &\\
& \sqrt{t}^{n-2} & & \\
& & \ddots &\\
& & & \sqrt{t}^{-n}
\end{bmatrix}
\end{equation}

\[\pi \left(\begin{bmatrix}
1 & 0\\
-y & 1
\end{bmatrix} \right)=\pi(\exp(-yY))\]

has the matrix representation

\begin{equation}\label{matrix rep of u_y under the basis}
\begin{bmatrix}
1 & & &\\
p_{21}(y) & 1 & & \\
\vdots   & \ddots & \ddots &\\
p_{nn}(y) & \cdots &  p_{n1}(y)  &  1
\end{bmatrix}
\end{equation}

where $p_{kl}(y)$ is a monomial in $y$ of degree $l$.
Both matrices are under the basis $v_0,v_1,...,v_n$. Observe that $\R v_n$ is the fixed subspace of $\pi \left(\begin{bmatrix}
1 & 0\\
-y & 1
\end{bmatrix}\right)$.

The following important lemma allows to prove the expansion of vectors in the representation space by the group $UA$, see (<ref>).

Let $V$ be a finite dimensional vector space with a norm $\|\cdot\|$, $\frak{n}$ be a nilpotent Lie algebra of $\End(V)$ with a basis $\{b_i:i=1,...,m\}$, and $N=\exp(\frak{n})\subset \GL(V)$ be the Lie group of $\frak{n}$. Define a map $\Theta:\R^m \to N$:

\begin{equation}
\Theta(t_1,...,t_m)=\exp \left( \sum_{i=1}^m t_ib_i \right), (t_1,....,t_m)\in \R^m.
\end{equation}

This is a polynomial map since $N$ is nilpotent.

For $\beta>0$, put
$D_{\exp}(\beta)=\Theta([0,\beta]\times \cdots \times [0,\beta])$.

\begin{equation}
W=\{v\in V:\frak{n}.v=0\}=\{v\in V:N.v=v\}.
\end{equation}

Denote by $\pr_W$ the orthogonal projection on $W$ with respect to some scalar product on $V$. Then there exists a constant $c_0>0$ such that for any $\beta \in (0,1)$ and $v\in V$,

\begin{equation}
\sup_{n\in D_{\exp}(\beta)}\|\pr_W(n.v)\| \ge c_0 \beta^d\|v\|,
\end{equation}

where $d$ is the degree of the polynomial map $\Theta$.

In our case, $V=V_G$, $N=
\pi \left(\begin{bmatrix}
1 & 0\\
\R & 1
\end{bmatrix}\right)$ (identified with $\pi \left(\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 &\R & 1
\end{bmatrix}\right))$ or $\pi \left(\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
\R & 0 & 1
\end{bmatrix}\right))$, $m=1$, $d=n=\dim V$ and $W=\R v_n$. Note that (keep in mind that for convenience we omit the representation symbol $\pi$):

\begin{align}
& \left \|\begin{bmatrix}
\st & 0 \\
0 & \frac{1}{\st}
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
-y & 1
\end{bmatrix}.w_1 \right \| \nonumber \\
\ge & C \left \| \pr_W\left(\begin{bmatrix}
\st & 0 \\
0 & \frac{1}{\st}
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
-y & 1
\end{bmatrix}.w_1 \right) \right \| \tag{by the boundedness of the linear operator $\pr_W$}\\
= & C \left \| \begin{bmatrix}
\st & 0 \\
0 & \frac{1}{\st}
\end{bmatrix}.
\pr_W\left(
\begin{bmatrix}
1 & 0 \\
-y & 1
\end{bmatrix}.w_1 \right) \right \| \tag{since the matrix \eqref{matrix rep of a_t under the basis} commutes with $\pr_W=\pr_{\R e_n}$}\\
=& C \frac{1}{\st^n} \left \|
\pr_W\left(
\begin{bmatrix}
1 & 0 \\
-y & 1
\end{bmatrix}.w_1 \right) \right \| \tag{see the last component of \eqref{matrix rep of a_t under the basis}}\\
=& C \frac{1}{\st^n} \left \|
\pr_W\left(
\begin{bmatrix}
1 & 0 \\
-(y-a_1) & 1
\end{bmatrix}\begin{bmatrix}
1 & 0 \\
-a_1 & 1
\end{bmatrix}.w_1 \right) \right \|
\end{align}

By Lemma <ref>, for any $a_1\in \R$, we have that with $\beta=\frac{1}{2}t^{0.4}$, there exists $C'>0$ and $-y\in [a_1,a_1+\beta]$

\begin{align}
\left \|
\pr_W\left(
\begin{bmatrix}
1 & 0 \\
-y & 1
\end{bmatrix}.w_1 \right) \right \| \ge C'\beta^n\|w_1\|.
\end{align}

Namely there exist $C'>0$ such that for some $-y\in [a_1,a_1+\beta]$

\begin{align}\label{expansion inequality}
\left \|\begin{bmatrix}
\st & 0 \\
0 & \frac{1}{\st}
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
-y & 1
\end{bmatrix}.w_1 \right \| \ge C' \frac{1}{t^{0.1n}}\|w_1\|.
\end{align}

In general, if the $\SL^{(i)}(2,\R)$-representation is not irreducible, we may choose a basis for the non-fixed component $V_{1}^{(i)}$ and decompose it into irreducible representations. By taking the sup-norm with respect to the coefficients of a vector under this basis, we can again recover the inequality (<ref>).

Now for $w=g_1.x$, under (<ref>) we have $\|w\|\ge \|g_1^{-1}\|^{-1}_{op}\|x\|\ge C''\|g_1^{-1}\|^{-1}_{op} \Pi(x) \ge C''\|g_1^{-1}\|^{-1}_{op} r,$ and the claim above guarantees that we have such an expansion for $a_t^{-1}u_v^{-1}.w$ by choosing either the order (<ref>) or the order (<ref>) for the decomposition of $a_t^{-1}u_v^{-1}$.

§.§ Proof of $\eta{(\infty)}=0$

Therefore, for any $x\in \cup_{i=1}^r \Ga.p_{U_i}$,

\begin{equation}
\lim_{t\to 0} \sup_{v\in D(\frac{1}{2}\st^{0.4})} \|a_t^{-1}u_v^{-1}g_1.x\| =\infty.
\end{equation}

\begin{align}
\phi_{\ga,\mu}(F,T):=& \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} F(h^{-1}g_1\Gamma) d\mu(h)\\
=& \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a}^b \int_{D} F \left(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma \right) dv \frac{1}{t^4}dt
\end{align}

\begin{equation}\label{definition of D}
D=D_{\ga, t, T}:=\{v\in \R^2: \|G_3\ga H_1+G_4 v\ga H_1+G_4 H_3 t^{\frac{3}{2}}\|^2 \le -G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2\},
\end{equation}

and $0<a<b$ are two positive roots of $-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2:=-Bt^3+tT^2-f_1(\ga)$.

$D$ is the elliptic disk centered at $-G_4^{-1}G_3 -H_3H_1^{-1}\ga^{-1}t^{\frac{3}{2}} \in \R^2$ and it contains the disk with the same center whose radius is equal to the shorter radius of the ellipse, which is
$$C_{g_1,g_2,\ga}\sqrt{ -G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2},$$

for some constant $C_{g_1,g_2,\ga}>0$.

The  displacement of the center from the origin has two parts: one part ($-G_4^{-1}G_3$) is constant and the other part ($-H_3H_1^{-1}\ga^{-1}t^{\frac{3}{2}}$) involves $t$ for which we have to handle separately.

Our next step is to find a sub-interval of $ (a,b)$, in which we have $D \supset D(2\beta)-G_4^{-1}G_3=D(t^{0.4})-G_4^{-1}G_3$ for sufficiently large $T$ and in the complement of that sub-interval, the integral above will vanish as $T\to \infty$ (Here $D(2\beta)\subset \R^2$ is the disk centered at O with radius $2\beta$ which shouldn't be confused with $D_{\exp}$ from Lemma <ref>). Using the elementary geometry fact that  if there are two disks, then one disk contains the other if the difference of their radii is greater than the distance between their centers, and simplifying coefficients, we are actually interested in finding $t$ satisfying

\begin{equation}
\sqrt{-Bt^3+tT^2-f_1(\ga)} \gtrsim
c_1t^{\frac{3}{2}}+2\beta= c_1t^{\frac{3}{2}}+t^{0.4},
\end{equation}

for sufficiently large $T$.

Take $t=a+\de$ with $\de>0$. Since $a\asymp \frac{1}{T^2}$ is a root of $-Bt^3+tT^2-f_1(\ga)$, the equation above is equivalent to

\begin{equation}\label{important inequality determininig the range for t}
\de T^2 \gtrsim
\end{equation}

Recall from (<ref>) we used the truncation

\begin{equation}
c=c_{\ga,T}=\frac{\|G_1\ga H_1\|^{1.5}}{T^{1.5}}
\end{equation}

for the integral (this will serve as the upper bound below). Take

\begin{equation}\label{definition of delta T}
\de_T=\frac{\|G_1\ga H_1\|^{2.5}}{T^{2.5}}.
\end{equation}

It is easy to see that for $t\in (a+\de_T, c)$, (<ref>) holds as $T\to \infty$, since the left hand side is asymptotically $\frac{1}{T^{0.5}}$. Moreover,

\begin{align}
& \lim_{T\to \infty}\left|\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a}^{a+\de_T} \int_{D} F \left(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma \right) dv \frac{1}{t^4}dt \right| \nonumber \\
\le & \lim_{T\to \infty} \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a}^{a+\de_T} \pi \frac{-G_4^2 H_4^2 t^3+tT^2-\|G_1\ga H_1\|^2}{G_4^2 |\det(H_1)|}  \frac{1}{t^4}dt \nonumber \\
=& 0 \tag{integrate three parts separately and recall $\mu \left(V_{\ga,T}[g_1,g_2] \right) \asymp_{\ga} T^6$}.
\end{align}

and by our computation in Section 3, we also have

\begin{align}
\lim_{T\to \infty} \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{c}^{b} \int_{D} F \left(\ga^{-1} a_t^{-1} u_v^{-1} g_1\Gamma \right) dv \frac{1}{t^4}dt=0.
\end{align}

For the integration in the range between $a+\de_T$ and $c$, noticing that by Lemma <ref> with $(a_1,a_2)=-G_4^{-1}G_3$,
the first outcome in Theorem <ref> fails as $T\to \infty$ and therefore the second outcome holds. Here $D \supset D(\beta)-G_4^{-1}G_3$ is the open convex set involved. Let $\epsilon>0$, and let $C\subset G/\Ga$ be a compact set such that

\begin{equation*}
m(v\in D:q_t(v)\Ga \notin C)< \e m(D),
\end{equation*}

whenever $t\in (a+\de_T,c)$.
Taking the complement,

\begin{equation}\label{the second outcome of Shah theorem}
m(v\in D:q_t(v)\Ga \in C) > (1-\e) m(D).
\end{equation}

Now take $F\in C_c(G/\Ga)$ with $\mathbf{1}_C \le F \le 1$. As $T\to \infty$,

\begin{align*}
&\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} F( h^{-1}g_1\Gamma) d\mu(h)\\
\ge& \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} F \left(q_t(v)\Ga \right) dv \frac{1}{t^4}dt\\
\ge& \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} \mathbf{1}_C \left(q_t(v)\Ga \right) dv \frac{1}{t^4}dt \\
\ge& \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} (1-\e)m(D) \frac{1}{t^4}dt \tag{by \eqref{the second outcome of Shah theorem}}
\end{align*}

By the computation above we know

\begin{align*}
&\lim_{T\to \infty}\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} (1-\e)m(D) \frac{1}{t^4}dt \\
=&  \lim_{T\to \infty}\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a}^{b} \int_{D} (1-\e)m(D) \frac{1}{t^4}dt \\
=& 1-\e
\end{align*}

\begin{equation}
\liminf_{T\to \infty}\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} F( h^{-1}g_1\Gamma) d\mu(h)\ge 1-\e
\end{equation}

and thus

\begin{equation}
\limsup_{T\to \infty}\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} \mathbf{1}_{\Supp (F)^c}( h^{-1}g_1\Gamma) d\mu(h)\le \e
\end{equation}

Recall that each coefficient $\frac{\mu \left(V_{\ga,T}[g_1,g_2] \right)}{\mu \left(H_T[g_1,g_2] \right)} \asymp \frac{1}{\|\ga\|^4}$ in the convex linear combination, and therefore by (<ref>) the sum is absolutely convergent. Choose $A_{\e}>0$ so that

\begin{equation*}
\sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T},\\
\|\ga\|>A_{\e}}\frac{\mu \left(V_{\ga,T}[g_1,g_2] \right)}{\mu \left(H_T[g_1,g_2] \right)}\cdot \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} \mathbf{1}_{\Supp (F)^c}(h^{-1}g_1\Gamma) d\mu(h)<\e
\end{equation*}

\begin{align*}
\eta(\infty) \le & \limsup_{T\to \infty}\phi_{\mu}(\mathbf{1}_{\Supp (F)^c},T) \\
\le & \limsup_{T\to \infty} \sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T\\ \|\ga\|\le A_{\e}}}\frac{\mu \left(V_{\ga,T}[g_1,g_2] \right)}{\mu \left(H_T[g_1,g_2] \right)}\cdot \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} \mathbf{1}_{\Supp (F)^c}(h^{-1}g_1\Gamma) d\mu(h)+\e\\
\le & 2\e.
\end{align*}

Therefore $\eta(\infty)=0$.

§.§ Proof of $G$-invariance

Recall $U=\begin{bmatrix}
I_2 & 0\\
\R^2 & 1
\end{bmatrix}$. For a closed subgroup
$H$ of $G$, denote

\begin{align}
&N(H,U):=\{g\in G:Ug\subset gH\},\\
&S(H,U):=\cup_{H'\subsetneq H, H'\in \mathcal{H}_{\Ga}}N(H',U).
\end{align}

\begin{equation}
Y:=\bigcup_{H\in \mathcal{H}_{\Ga}}N(H,U)\Ga=\bigcup_{H\in \mathcal{H}_{\Ga}}[N(H,U)-S(H,U)]\Ga \subset G/\Ga,
\end{equation}

where $\mathcal{H}_
{\Ga}$ was defined above Theorem <ref>.
The equality holds since for any $g\in N(H,U)$, if $g$ is also in $S(H,U)$, then $g$ must belong to $N(H',U)$ for some $H'\subsetneq H$ (note for Lie subgroups, this condition means $\dim H' < \dim H$) and $H'\in \mathcal{H}_{\Ga}$. Since $H'$ has strictly lower dimension, by repeating this argument we see eventually, $g$ will fall into some $N(\tilde{H}, U)-S(\tilde{H}, U)$ (with $S(\tilde{H}, U)$ possibly empty when $\tilde{H}$ has minimal dimension).

It follows from Dani’s generalization of Borel density theorem and Ratner’s topological
rigidity that $Y$ precisely form the set of points in $G/\Ga$ with non-dense orbit (Lemma 19.4 <cit.>).
Specifically, $y \in Y$ if and only if the orbit $Uy$ is not dense in $G/\Ga$.

A proof similar to that in the previous section shows that $Y$ is also a $\eta$-null set:

By the discreteness of $\mathcal{H}_{\Ga}$ (Theorem <ref>), it suffices to show

\begin{equation}
\eta([N(H,U)-S(H,U)]\Ga)=0,
\end{equation}

for all $H\in \mathcal{H}_{\Ga}$. Since $[N(H,U)-S(H,U)]\Ga$ is a countable union of compact subsets in $G/\Ga$ (See <cit.> Proposition 3.1), it suffices to show $\eta(C)=0$ for any compact subset $C$ of $[N(H,U)-S(H,U)]\Ga$.

The main tool in the proof is the following consequence of Prososition 5.4 in <cit.>:

Let $d,n \in \N, \e>0, H\in \mathcal{H}_{\Ga}$. For any compact set $C \subset [N(H, U)-S(H,U)]\Ga$, there exists a compact set $F \subset V_G$ such that for any neighborhood $\Phi$ of $F$ in $V_G$, there exists a
neighborhood $\Psi$ of $C$ in $G/\Ga$ such that for any $q \in \mathcal{P}_{d,n}(G)$ and a bounded open convex set
$D \subset \R^n$, one of the following holds:

* There exist $\ga\in \Ga$ such that $q(D)\ga.p_H \subset \Phi$.

* $m(t\in D: q(t)\Ga \in \Psi)<\e m(D)$, where $m$ is the Lebesgue measure on $\R^n$.

Fix $\e>0$. Now apply Theorem <ref> to $q_t(v)$, which gives a compact set $F\subset V_G$. Let $\Phi$ be a compact neighborhood of $F$ in $V_G$ and $\Psi \supset C$ be the neighborhood of $C$ given by the theorem. By the same argument as in the Section 4.3 (with Lemma <ref> applied), we have that for $t\in (a+\de_T, c)$,

\begin{equation*}
m(v\in D:q_t(v)\Ga \in \Psi)<\e m(D),
\end{equation*}

whenever $t\in (a+\de_T,c)$. Here $D$ is the elliptic disc defined in (<ref>). $c$ and $\delta_T$ are defined in (<ref>) and (<ref>), respectively.

Now let $f\in C_c(G/\Ga)$ be such that $\mathbf{1}_C \le f \le 1$ and $\Supp(f)\subset \Psi$, it follows that

\begin{align*}
& \limsup_{T\to \infty}\frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} f( h^{-1}g_1\Gamma) d\mu(h)\\
=& \limsup_{T\to \infty} \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a}^{b} \int_{D} f \left(q_t(v)\Ga \right) dv \\
=& \limsup_{T\to \infty} \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} f \left(q_t(v)\Ga \right) dv \frac{1}{t^4}dt\\
\le& \limsup_{T\to \infty} \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} \mathbf{1}_{\Psi} \left(q_t(v)\Ga \right) dv \frac{1}{t^4}dt \\
\le& \limsup_{T\to \infty} \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)} \int_{a+\de_T}^{c} \int_{D} \e m(D) \frac{1}{t^4}dt \\
\le & \e,
\end{align*}

and by the argument of truncating the convex linear combination as we did in the end of subsection <ref>.

\begin{align*}
\eta(C)
\le & \limsup_{T\to \infty} \phi_{\mu}(\mathbf{1}_{C},T) \\
\le & \limsup_{T\to \infty} \sum_{\substack{\ga\in \SL(2,\Z),\\ \|G_1\ga H_1\|^2<M_T\\ \|\ga\|\le A_{\e}}}\frac{\mu \left(V_{\ga,T}[g_1,g_2] \right)}{\mu \left(H_T[g_1,g_2] \right)}\cdot \frac{1}{\mu \left(V_{\ga,T}[g_1,g_2] \right)}\int_{V_{\ga,T}[g_1,g_2]} \mathbf{1}_{C}(h^{-1}g_1\Gamma) d\mu(h)+\e\\
\le & 2\e.
\end{align*}

Hence $\eta(C)=0$ and thus $\eta(Y)=0$.

Now we perform the ergodic decomposition of the $U$-invariant measure $\eta$. By Theorem 2.2 of <cit.>, each ergodic component of $\eta$ is either $G$-invariant or supported on $Y\cup \{\infty\}$. But we have just prove that $\eta(Y\cup \{\infty\})=0$. Therefore $\eta$ is $G$-invariant.

§ PROOF OF THEOREM <REF>: THE LIMITING MEASURE ON $H\BACKSLASH G$.

We need to prove for all $\varphi\in C_c(X_{2,3})$, we have

\begin{equation}
\lim_{T
\to\infty} \frac{1}{\#\Gamma_{T}}\sum_{\gamma\in \Ga_T}\varphi(x_{0}\cdot \gamma)=\int_{X_{2,3}}\varphi(x)d\tilde\nu_{x_{0}}(x)
\end{equation}

for some  measure $\tilde\nu_{x_0}$ on $X_{2,3}$ which depends on $x_{0}$.

The first thing we need to do is to construct a measure $\nu_{x_0}$ on $H\backslash G$. Since $G$ is unimodular while $H$ is not (the measure $\mu$ we defined is not right invariant), there is no $G$-invariant Haar measure on $H\backslash G$.

Nevertheless, it is still possible to find a lift $Y\subset G$ of $H\backslash G$ such that $H \times Y \to G$ is a Borel isomorphism and decompose the Haar measure on $G$ to obtain a measure on $Y$. Here

We follow Section 2.5 of <cit.>.
First, we provide an explicit measurable section
$$\sigma:H\backslash G \to Y\subset G,$$
and then we define a measure $\nu_Y$ on $Y$ such that
\begin{equation} \label{unfolding haar measure on G using section}
dm=d\mu d\nu_Y,
\end{equation}
where $\mu$ is the left Haar measure on $H$ and $m$ is a Haar measure on $G$.

To define the section,  let $\mathcal{F}_2 \subset \begin{bmatrix}
\SL(2,\R) & 0\\
0 & 1
\end{bmatrix}$ denote a measurable fundamental domain of

\begin{equation}
\begin{bmatrix}
\SL(2,\Z) & 0\\
0 & 1
\end{bmatrix} \bigg\backslash
\begin{bmatrix}
\SL(2,\R) & 0\\
0 & 1
\end{bmatrix} \bigg\slash  \begin{bmatrix}
\SO(2,\R) & 0\\
0 & 1
\end{bmatrix}
\cong
\SL(2,\Z)\backslash \SL(2,\R) /\SO(2,\R),
\end{equation}

and consider
$$Y:=\mathcal{F}_2\cdot \SO(3,\R).$$

[In fact, since $\begin{bmatrix}
\SO(2,\R) & 0\\
0 & 1
\end{bmatrix}$ is contained in $\SO(3,\R)$, $Y$ remains the same even if we take $\mathcal{F}_2$ to be the fundamental domain of $\begin{bmatrix}
\SL(2,\Z) & 0\\
0 & 1
\end{bmatrix} \bigg\backslash  \begin{bmatrix}
\SL(2,\R) & 0\\
0 & 1
\end{bmatrix}$.]
Then, we claim that the product map $H \times Y \to G$ is a Borel isomorphism, which defines a section $\sigma$ identifying $H\backslash G$ with $Y$. The surjectivity is clear from the block-wise Iwasawa decomposition. We only verify the injectivity here:

Suppose $h_1g_1s_1=h_2g_2s_2$ where $h_i\in H, g_i\in \mathcal{F}_2$ and $s_i\in \SO(3,\R)$. Then

* & 0 \\
* & *
\end{bmatrix}\ni g_2^{-1} h_2^{-1}h_1 g_1=s_2 s_1^{-1} \in \SO(3,\R),$$

and it follows from the definition of $\SO(3,\R)$ that both side must lie in $\begin{bmatrix}\SO(2,\R) & 0\\0 & 1 \end{bmatrix}$. So $h_2^{-1}h_1=g_2s_2s_1^{-1}g_1^{-1}\in \begin{bmatrix}\SL(2,\R) & 0\\0 & 1 \end{bmatrix}$. But $\begin{bmatrix}\SL(2,\R) & 0\\0 & 1 \end{bmatrix} \cap H= \begin{bmatrix}\SL(2,\Z) & 0\\0 & 1 \end{bmatrix}$. Hence $\ga g_1s_1=g_2s_2$ for some $\ga \in \begin{bmatrix}
\SL(2,\Z) & 0 \\0 & 1
\end{bmatrix}$. It follows that $g_2^{-1}\ga g_1=s_2 s_1^{-1}\in \begin{bmatrix}\SL(2,\R) & 0 \\0 & 1 \end{bmatrix} \cap \SO(3,\R)=\begin{bmatrix}\SO(2,\R) & 0 \\0 & 1 \end{bmatrix}$. Hence $g_1=g_2, s_1=s_2$ by the definition of fundamental domain. and $h_1=h_2$.

Next, let $T:=
\left\{ \begin{bmatrix}
t^{-\frac{1}{2}}\SL(2,\R) & 0\\
\R^2 & t
\end{bmatrix}:t>0 \right\}$ and $S:=\SO(3,\R)$. On $T$ we define the Haar measure in a similar way as the left Haar measure on $H$ (cf. Proposition <ref>). Below $d\tau=dv \frac{dt}{t^4} dg$, where $dg$ is the Haar measure on $\SL(2,\R)$ defined through standard Iwasawa decomposition, under which $\SO(2,\R)$ has volume $2\pi$. $d\rho$ is the standard Haar measure on $\SO(3,\R)$ that gives the total volume $8\pi^2$. Then, by using Theorem 8.32 of <cit.>, we unfold a Haar measure on $G$ as follows

\begin{align}
&\int_G f(g)dm(g)\\
=&\int_S \int_T f(\tau \rho) d\tau d\rho \\
=&\int_S\int_{\SL(2,\R)} \int_0^{\infty}\int_{\R^2} f(u_v a_t g \rho)    dv \frac{dt}{t^4}dg d\rho\\
=&\int_S\sum_{\ga \in \SL(2,\Z)}\int_{\mathcal{F}_2}\int_{\SO(2,\R)}  \int_0^{\infty}\int_{\R^2} f(u_v a_t \ga g\rho'  \rho)    dv \frac{dt}{t^4}dg  d\rho'd\rho\\
=&2\pi \int_S\sum_{\ga \in \SL(2,\Z)}\int_{\mathcal{F}_2} \int_0^{\infty}\int_{\R^2}  f(u_v a_t \ga g\rho)    dv \frac{dt}{t^4}dg d\rho\\
=&2\pi \int_S\sum_{\ga \in \SL(2,\Z)} \int_0^{\infty}\int_{\R^2} \int_{\mathcal{F}_2}  f(u_v a_t \ga g\rho)  dg  dv \frac{dt}{t^4}d\rho\\
=& 2\pi\int_H \int_S\int_{\mathcal{F}_2}  f(h g\rho)  dg d\rho  d\mu(h)\\
%   =&\text{Vol}(\SO(2,\R))^2 \int_H \int_{\frac{\mathcal{F}_2 \times S}{K}}  f(h gK) d(gK)  dh
%   =&\text{Vol}(\SO(2,\R))^2\int_H \int_{\mathcal{F}_2 S}  f(h g) dg  dh\\
=& \int_H \int_{\mathcal{F}_2}\int_S  f(h g\rho) 2\pi d\rho dg  d\mu(h)
%   =& \int_H \int_{\mathcal{F}_2 S}  f(h g) (2\pi dg) dh
\end{align}

Then it follows that the measure $\nu_Y$ on $Y$ by
$$d\nu_Y:=2\pi d\rho dg,$$

satisfies (<ref>).

Fix $Hg_0\in H\backslash G$. By identifying $H \backslash G$ with $Y$, we define a measure on $H \backslash G$ via

\begin{equation}
\end{equation}

where $\alpha(\cdot,\cdot)$ is given in (<ref>).

Then it follows from Theorem 2.3 of <cit.> that for any compactly supported function $\varphi \in C_c(H\backslash G)$, we have

\begin{equation}
\lim_{T\to \infty}\frac{1}{\mu(H_T)}\int_{G_T}\varphi(Hg_0g)dm(g)=\int_{H\backslash G} \varphi d\nu_{Hg_0}.
\end{equation}

In view of Theorem 2.2 and Corollary 2.4 in <cit.> (duality principle), an immediate consequence of Theorem <ref> is the following

Fix $g_0\in G$. For any compactly supported $\varphi \in C_c(H\backslash G)$,

\begin{equation}
\lim_{T\to \infty}\frac{1}{\mu(H_T)}\sum_{\ga \in \Ga_T}\varphi(Hg_0.\ga)= \int_{H\backslash G}\varphi(Hg)d\nu_{Hg_0}(Hg).
\end{equation}

Now we would like to replace the normalization factor $\mu(H_T)$ by $\# \Ga_T$.

By Theorem 1.7 of <cit.>,

\begin{equation}
\lim_{T\to \infty}\frac{m(G_T)}{\# \Ga_T}=m(G/\Ga).
\end{equation}

On the other-hand, by Lemma 6 of <cit.> (we choose the standard parameterization so that the volumes of $\SO(2,\R)$ and $\SO(3,\R)$ are $2\pi$ and $8\pi^2$ respectively),

\begin{equation}
m(G_T)\sim 8\pi^2\cdot \frac{\pi^2}{24}T^6=\frac{\pi^4}{3}T^6.
\end{equation}

Hence, by Proposition <ref>,

\begin{equation}
c_{\Ga}:=\lim_{T\to \infty}\frac{\mu(H_{T})}{\#\Ga_T}=\lim_{T\to \infty}\frac{\mu(H_{T})}{m(G_T)}\frac{m(G_T)}{\#\Ga_T}=
\frac{m(G/\Ga)}{2\pi^3}\sum_{\ga \in \SL(2,\Z)}\frac{1}{\|\ga \|^4},
\end{equation}

which shows that
$$\lim_{T\to \infty}\frac{1}{\#\Ga_T}\sum_{\ga \in \Ga_T}\varphi(Hg_0.\ga)= c_\Ga\int_{H\backslash G}\varphi(Hg)d\nu_{Hg_0}(Hg).$$

Now we compute $\nu_{Hg_0}(H\backslash G)$.  For $g_0,y \in Y$, we consider the decompositions

\begin{equation} \label{decompositions of g_0 and y in Y}
g_0:=\begin{bmatrix} G_1 & 0 \\0 & 1 \end{bmatrix}\rho_0, y:=\begin{bmatrix} H_1 & 0 \\0 & 1 \end{bmatrix} \rho,
\end{equation}

where $G_1,G_2\in\mathcal{F}_2$ and $\rho_0,\rho\in\SO(3,\R).$ Hence $\alpha$ in (<ref>) takes a more simpler form (note $g_0^{-1}:=\rho_0^{-1}\begin{bmatrix} G_1^{-1} & 0 \\0 & 1 \end{bmatrix}$)

\begin{align*}
\alpha(g_0,y)
=& \frac{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|G_1^{-1}\ga H_1\|^4}}{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga \|^4}}.
%    =& C_{\alpha}(g_0^{-1},y)\frac{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|G_1^{-1}\ga H_1\|^4}+\sum_{\ga\in \SL_{- 1}(2,\Z)}\frac{1}{\|G_1^{-1}\ga H_1\|^4}}{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga \|^4}}.
\end{align*}

Using the invariance of the $\SO(3,\R)$ invariance of the Hilbert-Schmidt norm, we compute

\begin{align*}
&m(G/\Ga)\nu_{Hg_0}(H\backslash G)\\
%   =& 8\pi^2 \int_{\mathcal{F}_2}2\pi dg\\
=&\frac{8\pi^2}{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga \|^4}} \sum_{\ga\in \SL(2,\Z)}\int_{\SL(2,\Z)\backslash \SL(2,\R)/\SO(2,\R)}\frac{1}{\|G_1^{-1}\ga H_1\|^4} dH_1 \\
=&\frac{4\pi}{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga \|^4}} \int_{\SL(2,\R)}\frac{1}{\|G_1^{-1}g\|^4}  dg\\
=&\frac{4\pi}{\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga \|^4}} \int_{\SL(2,\R)}\frac{1}{\|g\|^4}  dg.
\end{align*}

To compute the above integral, we consider the $\SL(2,\R)$ Iwasawa decomposition,

\begin{equation*}
1 & x\\
0 & 1
\end{bmatrix}\begin{bmatrix}
a & 0\\
0 & \frac{1}{a}
\end{bmatrix}k.
\end{equation*}

Noticing that  $k$ preserves the Euclidean $2$-norm, the integral above is equal to

\begin{align*}
\int_{\SL(2,\R)}\frac{1}{\|g\|^4}  dg=&2\pi\int_{\R_+}\int_{\R} \frac{1}{\left( a^2+\frac{1}{ a^2}+\frac{x^2}{a^2} \right)^2} \frac{dxda}{a^3}\\
=&\pi\int_{\R_+}\int_{\R} \frac{1}{\left( y+\frac{1}{ y}+\frac{x^2}{y} \right)^2} \frac{dxdy}{y^2}\tag{let $a=\sqrt{y}$}\\
=&\pi\int_{\R_+}\int_{\R} \frac{1}{\left(1+x^2+y^2 \right)^2} dxdy\\
\end{align*}

\begin{equation}
\nu_{Hg_0}(H\backslash G)=\frac{2\pi^3}{m(G/\Ga)\sum_{\ga\in \SL(2,\Z)}\frac{1}{\|\ga\|^4}}
\end{equation}

and thus

$$c_{\Gamma}\nu_{Hg_0}(H\backslash G)=1.$$

Therefore, $\tilde\nu_{Hg_0}:=c_{\Ga}\nu_{Hg_0}$ is a probability measure, and we conclude

\begin{equation}
\lim_{T\to \infty}\frac{1}{\# \Ga_T}\sum_{\ga \in \Ga_T}\varphi(Hg_0.\ga)= \int_{H\backslash G}\varphi(Hg)d\tilde \nu_{Hg_0}(g).
\end{equation}

We express $\tilde\nu_{Hg_0}$ as
$$d\tilde\nu_{Hg_0}=\Phi_{Hg_0}(y)dg d\rho,$$
$$\Phi_{Hg_0}(y):=\sum_{\ga\in \GL(2,\Z)}\frac{1}{\|G_1^{-1}\ga H_1\|^4},$$
$\Phi_{Hg_0}(y)dg$ is a probability measure on $\mathcal{F}_2$, and $d\rho$ is a probability measure on $\SO(3,\R)$. Now we interpret the coefficient $\Phi_{Hg_0}(y)$
in terms of rank-two discrete subgroups in $\R^3$. Let $(\La_0,w_0)$ and $(\La,w)$ denote the oriented rank-two discrete subgroups of $\R^3$ corresponding to $g_0$ and $y$, respectively. Let $e_1,e_2$ be the canonical basis of $\R^2$, written in terms of row vectors:

Then by (<ref>) it follows that
$$\mathscr{B}_0:=\{(e_1G_1,0)\rho_0,(e_2G_1,0)\rho_0 \},$$
$$\mathscr{B}:=\{(e_1\ga H_1,0)\rho,(e_2\ga H_1,0)\rho\}$$
form  $\Z$-bases of $\La_0$ and $\La$, respectively.

Consider the following linear maps between two-dimensional subspaces of $\R^3$:

\begin{equation}
T_{\mathscr{B}_0}: \Span_{\R}\{(e_1,0),(e_2,0)\} \to \Span_{\R}\{\mathscr{B}_0\}, (e_i,0)\mapsto (e_iG_1,0)\rho_0, i=1,2
\end{equation}

\begin{equation}
T_{\mathscr{B}}: \Span_{\R}\{(e_1,0),(e_2,0)\} \to \Span_{\R}\{\mathscr{B}\}, (e_i,0)\mapsto (e_i\ga H_1,0)\rho, i=1,2.
\end{equation}

Recall from the introduction (see (<ref>)) the Hilbert-Schmidt norm   $\|T\|_{\text{HS}}$ of an operator $T$ from the two-dimensional subspace $U\subset \R^3$ to the two-dimensional subspace $V\subset \R^3$, which is computed by choosing an orthonormal basis $\{u_1,u_2\}$ of $U$ and evaluating

\begin{equation}
\|T\|^2_{\text{HS}}:=\|Tu_1\|^2+\|Tu_2\|^2,
\end{equation}

where the norm on the right hand side is the usual Euclidean norm on $\R^3$. Using the orthonormal basis $\{(e_1,0)\rho_0 , (e_2,0)\rho_0\}$ of $\Span_\R\{\mathcal{B}_0\}$, we get
$$\|T_{\mathscr{B}}\circ T_{\mathscr{B}_0}^{-1}\|_{\text{HS}}^2=\|G_1^{-1}\ga H_1\|^2,$$

which shows

\begin{equation}\label{intrinsic interpretation of the sum}
\Phi_{g_0}(y)=\sum_{\ga\in \GL(2,\Z)}\frac{1}{\|G_1^{-1}\ga H_1\|^4}=\sum_{\Span_{\Z}\mathscr{B}=\La}\frac{1}{\|T_{\mathscr{B}}\circ T_{\mathscr{B}_0}^{-1}\|_{\text{HS}}^4}=\Psi_{\La_0}(\La),
\end{equation}

where $\Psi_{\La_0}(\La)$ defined in (<ref>).
It's now straight forward to verify that the measure $\tilde\nu_{x_0}$ for $x_0=(\Span_\Z\{e_1,e_2\},e_3)\cdot g_0$ defined before Theorem <ref> is identified with $\tilde\nu_{H_{g_0}}$:

Let $X_2=\SL(2,\Z)\backslash\SL(2,\R)$. For any $f\in C_c(X_{2,3})$, we have

\begin{align}
\tilde\nu_{x_0}(f)
:=&\int_{\mathbb S^2}\int_{\pi_{\perp}^{-1}(w)}f((\La,w),\rho_w) \Phi_{\La_0}(\La)d((\rho_w)_*\mu_{e_3})(\La)d\mu_{\mathbb S^2}(w)\\
=&\int_{\mathbb S^2}\int_{\pi_{\perp}^{-1}(e_3)}f((\La,e_3),\rho_w) \Psi_{\La_0}(\La)d\mu_{e_3}(\La)d\mu_{\mathbb S^2}(w)\\
=&\int_{\mathbb S^2}\int_{X_2}f(\Z^2\times 0,\eta\rho_w) \Psi_{\La_0}(\Z^2\times 0)d\eta d\mu_{\mathbb S^2}(w)\\
=&\frac{1}{\vol(\SO(2,\R))}\int_{\mathbb S^2}\int_{\SO(2,\R)}\int_{X_2}f(\Z^2\times 0,\eta k \rho_w) \Psi_{\La_0}(\Z^2\times 0)d\eta dk d\mu_{\mathbb S^2}(w)\\
=&\frac{1}{\vol(\SO(2,\R))}\int_{ \SO(3,\R)}\int_{X_2}f(\Z^2\times 0,\eta\rho) \Psi_{\La_0}(\Z^2\times 0)d\eta d\rho\\
=&\frac{1}{\vol(\SO(2,\R))}\int_{ \SO(3,\R)}\int_{\mathcal F_2}\int_{\SO(2,\R)}f(\Z^2\times 0,\eta\rho k) \Psi_{\La_0}(\Z^2\times 0)d\eta d\rho dk\\
=&\int_{ \SO(3,\R)}\int_{\mathcal F_2}f(\Z^2\times 0,\eta\rho) \Psi_{\La_0}(\Z^2\times 0)d\eta d\rho
\end{align}

The last line is the same as $\int_{H\backslash G}\varphi(Hg)d\tilde\nu_{Hg_0}(g)=\int_{H\backslash G}\varphi(Hg)d\tilde c_{\Ga}\nu_{Hg_0}(g)$ (note that both $\tilde \nu_{x_0}$ and $\tilde \nu_{Hg_0}$ are probability measures).
This proves Theorem <ref>.

§ TWO COPIES OF $\SL(2,\R)$ GENERATES $\SL(3,\R)$

\[
a & 0 & b\\
0 & 1 & 0\\
c & 0 & d
\end{bmatrix} : ad-bc=1\right\} ,
\]

\[
G_{2}:=\left\{ \begin{bmatrix}
1 & 0 & 0\\
0 & a & b\\
0 & c & d
\end{bmatrix}: ad-bc=1\right\} .
\]

Let $G$ be a closed subgroup of $\SL(3,\R)$ which includes
$G_{1}$ and $G_{2}$. Then we claim that $G=\SL(3,\R)$.

Let us first show that

\[
G\supseteq G_{3}:=\left\{ \begin{bmatrix}
a & b & 0\\
c & d & 0\\
0 & 0 & 1
\end{bmatrix}: ad-bc=1\right\}.
\]

Observe that

\begin{align*}
1 & 0 & 0\\
0 & 1 & 0\\
0 & -s_{2} & 1
\end{bmatrix}\begin{bmatrix}
1 & 0 & s_{1}\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}  \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & s_{2} & 1
\end{bmatrix}\begin{bmatrix}
1 & 0 & -s_{1}\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\\
1 & 0 & s_{1}\\
0 & 1 & 0\\
0 & -s_{2} & 1
\end{bmatrix}  \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & s_{2} & 1
\end{bmatrix}\begin{bmatrix}
1 & 0 & -s_{1}\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\\
1 & s_{1}s_{2} & s_{1}\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}  \begin{bmatrix}
1 & 0 & -s_{1}\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\\
= & \begin{bmatrix}
1 & s_{1}s_{2} & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}.
\end{align*}

So for all $s\in\R$, we have $\begin{bmatrix}
1 & s & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\in G$.
By applying the matrix transpose to the above, we get that for all
$r\in\R$, $\begin{bmatrix}
1 & 0 & 0\\
r & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\in G$.
Observe that
Let $\epsilon$ be non-zero close to zero, and note that for $r=\frac{e^{t}-1}{\epsilon}$
and $s=\frac{-r}{1+r\epsilon}$ we get

\begin{align*}
\begin{bmatrix}
1 & r\\
0 & 1
\end{bmatrix}\begin{bmatrix}
1 & 0\\
\epsilon & 1
\end{bmatrix}\begin{bmatrix}
1 & s\\
0 & 1
\end{bmatrix} & =\begin{bmatrix}
1+r\epsilon & r+s+rs\epsilon\\
\epsilon & 1+s\epsilon
\end{bmatrix}\\
& =\begin{bmatrix}
e^{t} & 0\\
\epsilon & e^{-t}
\end{bmatrix}.
\end{align*}

So, for all $\epsilon>0$ and $t\in\R$ we have

\[
\begin{bmatrix}
e^{t} & 0 & 0\\
\epsilon & e^{-t} & 0\\
0 & 0 & 1
\end{bmatrix}\in G
\]

and since $G$ is closed, we get by taking $\epsilon\to0$ that

\[
\begin{bmatrix}
e^{t} & 0 & 0\\
0 & e^{-t} & 0\\
0 & 0 & 1
\end{bmatrix}\in G.
\]

The above show that in fact $G_{3}\leq G$. It follows that
that the full diagonal
group and the unipotent upper and lower triangular subgroups are all contained in $G$. By using the dimension matching argument on Lie algebras, we see their product form a neighborhood of identity in $\SL(3,\R)$.

Therefore, $G=\SL(3,\R)$ follows from the fact that a neighborhood of identity of a Lie group generates the whole group.

§ ESTIMATES ON THE NUMBER OF $\SL(2,\Z)$ POINTS IN SKEWED BALLS

\[
B_{\tau}=\left\{ g\in\SL(2,\R):\left|\left|g\right|\right|\leq\tau\right\} .
\]

From <cit.> Example 4.3, we have the following estimate

\[
N(\tau):=\left|\SL(2,\Z)\cap B_{\tau}\right|=\frac{\text{vol}(B_{\tau})}{\text{vol}(\SL(2,\R)/\SL(2,\Z))}+O_{\eta}(\text{vol}(B_{\tau})^{5/6+\eta}).
\]

We now compute $\text{vol}(B_{\tau})$. First, recall that
the Haar measure on $\SL(2,\R)$ is expressed by

\[
\nu(f)=\int f(n_{x}a_{t}k_{\theta})\frac{1}{e^{t}}d\theta\ dt\ dx,
\]

where $n_{x}=\begin{bmatrix}
1 & x\\
& 1
\end{bmatrix},$ $a_{t}=\begin{bmatrix}
e^{t/2} & 0\\
& e^{-t/2}
\end{bmatrix},\ k_{\theta}=\begin{bmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{bmatrix}$.

It follows from the bi-invariance of the Haar measure that

\[
\text{vol}(B_{\tau})=\text{vol}\left\{ g\in\SL(2,\R):\left|\left|g\right|\right|\leq\tau\right\} .
\]

and since the norm is $k_{\theta}$ invariant, we get

\[
\text{vol}\left\{ g\in\SL(2,\R):\left|\left|g\right|\right|\leq\tau\right\} =2\pi\text{vol}\left\{ \begin{bmatrix}
e^{t/2} & xe^{-t/2}\\
& e^{-t/2}
\end{bmatrix}:\left|\left|\begin{bmatrix}
e^{t/2} & xe^{-t/2}\\
& e^{-t/2}
\end{bmatrix}\right|\right|\leq\tau\right\} .
\]

We have

\[
e^{t}+\left(x^{2}+1\right)e^{-t}\leq\tau^{2}\iff x^{2}\leq e^{t}\left(\tau^{2}-2\cosh t\right).
\]

By Fubini's theorem, we see that

\[
\text{vol}\left\{ \begin{bmatrix}
e^{t/2} & xe^{-t/2}\\
& e^{-t/2}
\end{bmatrix}:\left|\left|\begin{bmatrix}
e^{t/2} & xe^{-t/2}\\
& e^{-t/2}
\end{bmatrix}\right|\right|\leq\tau\right\}
=\int_{\cosh t\leq\tau^{2}/2}\underbrace{\frac{2\sqrt{\tau^{2}-2\cos ht}}{e^{t/2}}}_{=f(t)}dt=I_{+}+I_{-},
\]

where $I_{+}=\int_{t\geq0,\ \cosh t\leq\tau^{2}/2}f(t)dt$, $I_{-}=\int_{t\leq0,\ \cosh t\leq\tau^{2}/2}f(t)dt$.

* Computation of $I_{+}$: The function $\cosh t$ is injective
on $\R_{\geq0}$. and its inverse $\text{\ensuremath{\cosh}}^{-1}:[1,\infty)\to\R_{\geq0}$
has the following expression

\[
\text{\ensuremath{\cosh}}^{-1}(\alpha)=\ln(\alpha+\sqrt{\alpha^{2}-1}).
\]

We have the following identities,

\[
\sinh(\text{cosh}^{-1}(\alpha))=\sqrt{\alpha^{2}-1},
\]

We put $\alpha=\cosh t,$ then $d\alpha=\sinh t\ dt$, which gives

\[
\]

Put $\beta=\frac{\alpha}{\tau^{2}}$, then

\[
\]

* Computation of $I_{-}$: The function $\cosh t$ is injective
on $\R_{\leq0}$ and its inverse $\text{\ensuremath{\cosh}}^{-1}:[1,\infty)\to\R_{\leq0}$
has the following expression

\[
\text{\ensuremath{\cosh}}^{-1}(\alpha)=-\ln(\alpha+\sqrt{\alpha^{2}-1}).
\]

We have the following identities,

\[
\sinh(\text{cosh}^{-1}(\alpha))=-\sqrt{\alpha^{2}-1},
\]

We put $\alpha=\cosh t,$ then $d\alpha=\sinh t\ dt$, which gives,

\[
\]

Put $\beta=\frac{\alpha}{\tau^{2}}$, then

\[
\]

\[
\]

\[
\]

Then as $\tau\to\infty$, we get that

\[
\]

To summarize, the above shows that $\text{vol}(B_{\tau})=c'\tau^{2}+O(\frac{1}{\tau}),$

\begin{equation}
N(\tau)=c\tau^{2}+O_{\eta}(\tau^{5/3+\eta}).\label{eq:lattice points in SL_2}
\end{equation}

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

journaltitleInventiones mathematicae
titleInteger points on spheres and their orthogonal lattices

titleLimit distributions of orbits of unipotent flows and values of quadratic forms

journaltitleIsrael Journal of Mathematics
titleOrbits of euclidean frames under discrete linear groups

London Mathematical Society

journaltitleCompositio Mathematica
titleEquidistribution of primitive rational points on expanding horospheres

journaltitleErgodic Theory and Dynamical Systems
titleLattice action on the boundary of SL$(n,\mathbb R )$

titleStationary measures for $\mathrm{SL}_2(\mathbb{R})$-actions on homogeneous bundles over flag varieties

journaltitleJournal für die reine und angewandte Mathematik (Crelles Journal)
titleCounting lattice points

journaltitleGeometric and Functional Analysis
titleErgodic theory and the duality principle on homogeneous spaces

journaltitleGAFA Geometric And Functional Analysis
titleDistribution of lattice orbits on homogeneous varieties

journaltitleJournal of Modern Dynamics
titleDIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS

titleLie Groups: Beyond an Introduction, 2nd Edition

journaltitleErgodic Theory and Dynamical Systems
titleOn the space of ergodic invariant measures of unipotent flows

journaltitleJ. Korean Math. Soc
titleLattice action on finite volume homogeneous spaces

Annals of Mathematics

journaltitleAnnals of Mathematics
titleOn Raghunathan's Measure Conjecture

journaltitleGeometric and Functional Analysis
titleDynamics on the space of 2-lattices in 3-space

journaltitleMonatshefte für Mathematik
titleThe distribution of sublattices of $\mathbb Z_m$

journaltitleAdvances in Geometry
titleAn upper bound on the volume of the symmetric difference of a body and a congruent copy

journaltitleProceedings of the Indian Academy of Sciences - Mathematical Sciences
titleLimit distributions of expanding translates of certain orbits on homogeneous spaces

journaltitleDuke Mathematical Journal
titleLimit distributions of polynomial trajectories on homogeneous spaces

journaltitleErgodic Theory and Dynamical Systems
titleOn actions of epimorphic subgroups on homogeneous spaces

journaltitleTranslations of Mathematical Monographs, AMS
titleDynamical Systems on Homogeneous Spaces