# Tori Approximation of Families of Diagonally Invariant Measures

Omri Nisan Solan  Yuval Yifrach

###### Abstract

We approximate any portion of any orbit of the full diagonal group $A$ in the
space of unimodular lattices in $\mathbb{R}^{n}$ using a fixed proportion of a
compact $A$-orbit. Using those approximations for the appropriate sequence of
orbits, we prove the existence of non-ergodic measures which are also weak
limits of compactly supported $A$-invariant measures. In fact, given any
countably many $A$-invariant ergodic measures, our methods show that there
exists a sequence of compactly supported periodic $A$-invariant measures such
that the ergodic decomposition of its weak limit has these measures as factors
with positive weight. Using the same methods, we prove that any compactly
supported $A$-invariant and ergodic measure is the weak limit of the
restriction of different compactly supported periodic measures to a fixed
proportion of the time. In addition, for any $c\in(0,1]$ we find a sequence of
compactly supported periodic $A$-invariant measures that converge weakly to
$cm_{X_{n}}$ where $m_{X_{n}}$ denotes the Haar measure on $X_{n}$. In
particular, we prove the existence of partial escape of mass for compact
$A$-orbits. These results give affirmative answers to questions posed by
Shapira in [17]. Our proofs are based on a modification of Shapira’s proof in
[17] and on a generalization of a construction of Cassels, as well as on
effective equidistribution estimates of Hecke neighbors by Clozel, Oh and
Ullmo, and a number theoretic construction of a special number field.

##  1 Introduction

Let $(X_{n},d_{X_{n}})$ denote the metric space of unimodular lattices in
$\mathbb{R}^{n}$ and let $A<\operatorname{SL}_{n}(\mathbb{R})$ denote the
subgroup of diagonal matrices with positive diagonal inside the group of real
matrices with determinant 1. Among the ergodic $A$-invariant probability
measures on $X_{n}$, those which are supported on compact orbits stand out due
to their connection to number theory. Indeed, there is a surjective map
between the set of full $\mathbb{Z}$-modules inside totally real number fields
and the set of compact $A$-orbits in $X_{n}$. For any totally real degree $n$
number field $K$, let $\sigma_{i}:K\hookrightarrow\mathbb{R};i=1,\dots,n$ be
some ordering of the natural embeddings of $K$ and let
$\sigma=(\sigma_{1},\dots,\sigma_{n}):K\rightarrow\mathbb{R}^{n}$ denote their
concatenation. The compact orbit corresponding to a full module $M\subset K$
is then the $A$-orbit of the unimodular lattice
$\operatorname{cov}(\sigma(M))^{-1/n}\sigma(M)$ where
$\operatorname{cov}(\Lambda)$ denotes the co-volume of a lattice $\Lambda$.
This orbit is indeed compact, and every compact $A$-orbit is given in this
manner. Denote:

$\mathcal{M}(X_{n})=\\{\text{finite measures on $X_{n}$}\\};$
$\mathcal{M}(X_{n})^{A}_{e}=\\{\text{ergodic }A\text{-invariant probability
measures on $X_{n}$}\\};$ $\mathcal{M}(X_{n})^{A}_{c}=\\{\text{ergodic
}A\text{-invariant probability measures on $X_{n}$ supported on compact
orbits}\\}.$

We endow $\mathcal{M}(X_{n})$ with the weak-$\ast$ topology, and study the
closure $\overline{\mathcal{M}(X_{n})^{A}_{c}}$ in it. It is known that apart
from measures of compact orbits, $m_{x_{n}},0\in\mathcal{M}(X_{n})$. The Haar
measure is obtained by applying ergodicity to the result of Benoist and Oh
[2], or more directly using Shapira and Zheng’s [18]. The trivial measure is
obtained by Shapira [17]. In this paper, we show that $\mathcal{M}(X_{n})$
contains several families of natural and important measures, thus answering
positively the discussion in [10, §1.5] and negatively Conjecture [10,
Conjecture 1.11].

###  1.1 Measures with Predetermined Ergodic Factors

The first family of measures we consider contains non-ergodic measures. Given
any finitely many elements
$\mu_{1},\dots,\mu_{k}\in\mathcal{M}(X_{n})^{A}_{e}$, we find a measure in
$\mu\in\overline{\mathcal{M}(X_{n})^{A}_{c}}$ such that the ergodic
decomposition of $\mu$ has positive weight on each of the $\mu_{i}$’s for
$i=1,\dots,k$. In [17, p.4 Q1, Q2], Shapira asks whether non-ergodic measures
and measures with a given periodic $A$-invariant ergodic factor can be
obtained as weak limits of $\mathcal{M}(X_{n})^{A}_{c}$ elements. The
following Theorem shows in particular that this is indeed the case:

######  Theorem 1.1.

Let $\mu_{1},\mu_{2},\dots,\mu_{k}\in\mathcal{M}(X_{n})^{A}_{e}$ be a finite
sequence of $A$-invariant ergodic measures. Then there exists
$\mu\in\overline{\mathcal{M}(X_{n})^{A}_{c}}$ such that the ergodic
decomposition of $\mu$ has positive weights on the $\mu_{i}$’s. In particular
for any $\mu\in\mathcal{M}(X_{n})^{A}_{c}$ there exists
$\tilde{\mu}\in\overline{\mathcal{M}(X_{n})^{A}_{c}\setminus\\{\mu\\}}$ such
that the ergodic decomposition of $\tilde{\mu}$, contains $\mu$ with positive
weight.

######  Remark 1.2 (The weights in the ergodic decomposition).

For a single measure $\mu_{1}$, we get an explicit lower bound on the weight,
namely $\mu\geq\left(\frac{\lfloor
n/2\rfloor}{4n^{2}(n^{2}-1)}\right)^{n-1}\mu_{1}$. More generally, for a
finite sequence of measures $\mu_{1},\dots,\mu_{k}$ we get
$\mu\geq\Theta(k^{1-n})\mu_{i}$.

######  Remark 1.3 (Countably many measures).

Using the same technics one can prove the same result for countably many
measures. For possibly infinite sequence $\mu_{1},\mu_{2},\dots$ of measures
and every sequence of positive constants $c_{1}+c_{2}+\cdots=1$ we can we can
construct $\mu$ with $\mu\geq\Theta(c^{n-1}_{i})\mu_{i}$.

###  1.2 Partial Escape of Mass and Entire Mass Approximations

The next measures we will find in $\overline{\mathcal{M}(X_{n})^{A}_{c}}$ are
measures with total mass strictly between $0$ and $1$. In his paper [17, p.4,
Q2], Shapira proves that the zero measure is contained in
$\overline{\mathcal{M}(X_{n})^{A}_{c}}$ and poses the question of whether
measures with total mass strictly between $0$ and $1$ also lie there. In [8]
David and Shapira find a sequence of orbits with a lower bound on the escape
of mass, but did not prove that the limit is not zero. We answer Shapira’s
question in the affirmative in two ways. The first is the following corollary
of Theorem 1.1:

######  Corollary 1.4.

For every sequence
$\mu_{1},\mu_{2},\dots,\mu_{k},\dots\in\mathcal{M}(X_{n})^{A}_{e}$ there
exists $\mu\in\overline{\mathcal{M}(X_{n})^{A}_{c}}$ such that
$\mu(X_{n})\in(0,1)$ and the ergodic decomposition of $\mu$ has positive
weights on $\mu_{i}$ for all $i$.

The next family of measures we find inside the closure of the compactly
supported periodic measures is the family $\\{cm_{X_{n}}:c\in(0,1]\\}$ where
$m_{X_{n}}$ is the Haar probability measure on $X_{n}$. This family of
measures is substantially different from the measure families we approximated
so far since we can describe their entire ergodic decomposition (which is
simply $cm_{X_{n}}$) rather than describe some of the factors and have no
control over the remaining factors. In [18], they show that
$m_{X_{n}}\in\overline{\mathcal{M}(X_{n})^{A}_{c}}$. Here we prove a more
general theorem.

######  Theorem 1.5.

Let $c\in(0,1]$. Then $cm_{X_{n}}\in\overline{\mathcal{M}(X_{n})^{A}_{c}}$.

In particular, the above theorem gives another answer to Shapira’s question
regarding the partial escape of mass.

######  Remark 1.6 (Volume of the compact orbits).

Every compact orbit $Ax$ has a volume
$\operatorname{Vol}(Ax)=\operatorname{Vol}(\mathbb{R}^{n-1}/\log\operatorname{stab}_{A}(x))$
and a corresponding order $R_{x}$. The compact orbits we construct for the
proof of Theorem 1.1 have volume $O(\log\operatorname{Disc}(R_{x}))$. As
written, our proof of Theorem 1.5 doesn’t provide estimates on the volumes. In
Remark 5.13, we argue that similarly to the proof of Theorem 1.5, one can
choose the approximating compact orbits $Ax$ to have volume
$O(\operatorname{Disc}(R_{x})^{c^{\prime}})$ for some $c^{\prime}>0$.

The estimate of the regulator in Remark 1.6 that provides a sequence of orbits
with volume polynomial in the discriminant of the corresponding order and
partial escape of mass, contradicts [10, Conjecture 1.11], which states that
every such sequence of orbits must converge to an algebraic measure. The
spirit of the conjecture is not disproved, and one could hope that the
conjecture can be corrected by considering only partial escape of mass. We do
not believe this is the case for general orbits. Instead, we provide the
following conjecture:

######  Conjecture 1.7.

Let $Ax_{i}\subset X_{n}$ be a sequence of compact orbits corresponding to
orders $R_{i}$. Suppose that
$\operatorname{Vol}(Ax_{i})>\operatorname{disc}(Ax_{i})^{\varepsilon}$ for
some $\varepsilon>0$. Here $\operatorname{disc}(Ax_{i})$ is the discriminant
of the orbit $A_{x_{i}}$. It is defined as in [10]. Then
$\lim_{i\to\infty}\mu_{Ax_{i}}$ is a combination of countably many periodic
algebraic orbits of groups $H_{j}\supsetneq A$, with total mass bounded from
below as a function of $\varepsilon$. Moreover, if
$K_{i}=R_{i}\otimes\mathbb{Q}$, and
$(\operatorname{disc}(R_{i})/\operatorname{disc}K_{i})<\operatorname{disc}(R_{i})^{\delta}$
for some $\delta=\delta(\varepsilon)$, then $\lim_{i\to\infty}\mu_{Ax_{i}}$ is
algebraic. We suppose the same will hold for general arithmetic $G/\Gamma$
with the appropriate generalization of
$\operatorname{disc}(R_{i})/\operatorname{disc}K_{i}$. The reader may compare
this discussion to a similar discussion carried out in [10, Corollary 1.7].

###  1.3 Method of Proof

We will first show a sketch of the proof of Theorem 1.1, for the particular
case $k=1$ where we approximate a single measure $\mu$. We will first show how
to do that depending on a conjecture. Then we will show how to overcome the
need to use the conjecture.

######  Conjecture 1.8.

Let $K$ be a number field. For every $I\leq\mathcal{O}_{K}$, we get a measure
$\mu_{I}$ on a compact orbit associated to $I$ (See Definition 2.6). The
measure $\mu_{I}$ depends only on the class of $I$ in the class group ${\rm
Cl}_{\mathcal{O}_{K}}$. It is believed that

$\displaystyle\mu_{K}:=\frac{1}{\\#{\rm Cl}_{\mathcal{O}_{K}}}\sum_{I\in{\rm
Cl}_{\mathcal{O}_{K}}}\mu_{I}\xrightarrow{\operatorname{disc}(K)\to\infty}m_{X_{n}},$
(1.1)

where $m_{X_{n}}$ is the Haar measure on $X_{n}$, and $\operatorname{disc}(K)$
is the discriminant of $K$. Moreover, we expect that
$d(\mu_{K},m_{X_{n}})=O(\operatorname{disc}(K)^{-\star})$, for $\star>0$.

Another ingredient of the proof is a construction of a number field with very
small units compared to its discriminant. A construction of Shapira [17]
following Cassels [4] is a series of number fields $K$ whose unit groups are
generated by elements $u\in\mathcal{O}_{K}^{\times}$ such that
$\log|u|=O(\log{\rm disc}(K))$ (which is exceptionally small compared to the
bound given by the class number formula: $\log|u|=O({\rm disc}(K)^{1/{d-1}})$,
provided that the unit lattice is sufficiently balanced). Applying Conjecture
1.8 to these number fields, we get that that the different measures $\mu_{I}$
for $I\in{\rm Cl}_{\mathcal{O}_{K}}$ are equidistributed. Using this
equidistribution we can approximate a generic point $x\in X_{n}$ of our
desired ergodic measure $\mu$, namely there is $I\in{\rm
Cl}_{\mathcal{O}_{K}}$ such that $d(x,x^{\prime})=O({\rm disc}^{-\star}(K))$
for some $x^{\prime}\in\operatorname{supp}\mu_{I}$. Hence for
$r=\theta(\log({\rm disc}(K)))$ ball in $A$, $a\in B_{A}(r)$ we have that $ax$
is close to $ax^{\prime}\in\operatorname{supp}(\mu_{I})$. On the other hand,
the orbit $ax^{\prime}$ is not much larger. The periods are controlled by the
size of the units, which are again $O(\log({\rm disc}(K)))$. This would imply
that we can approximate every ball in an $A$-orbit with measures $\mu_{I}$ for
$I$ in the class group of the special number field $K$.

To overcome the need to use Conjecture 1.8, we prove a weaker form on a
special collection of orders. We take a measure $\mu_{I}$ as above and apply
to it a $p$-Hecke operator $T_{p}$. On the one hand, Hecke operators are known
to have well-behaved equidistribution properties (See Clozel, Oh and Ullmo [5]
for an effective equidistribution result on Hecke operators and [2, 1] for
other results using equidistribution of Hecke operations applied to closed
orbits). We expect to have $f(T_{p}\mu_{I},m_{X_{n}})=O(p^{-\star})$. On the
other hand, the resulting measure $T_{p}\mu_{I}$ is the average of measures on
compact orbits, corresponding to modules of a sub-order
$R=\mathbb{Z}+p\mathcal{O}_{K}\subset\mathcal{O}_{K}$ of index
$[\mathcal{O}_{K}:R]=p^{d-1}$.

While this guarantees that the orbits would be equidistributed, to bound the
sizes of the orbits we tweak the construction of $K$. Not only
$\mathcal{O}_{K}$ should have units of logarithmic size, but also $R$. This
guarantees that $T_{p}\mu_{I}$ is an average of measures of logarithmically
small compact orbits, and enables the proof to work without Conjecture 1.8.

######  Remark 1.9.

Conjecture 1.8 seems very complicated. It was proven in case $n=2$ in [9]. A
non-effective version of it is proven for $n=3$ under some splitting
restriction in [11]. The proof uses a deep and nontrivial number theoretic
result, namely, the subconvexity estimate of $L$ functions of cubic fields.

The proof of Theorem 1.5 goes along similar lines. We start with a compact
orbit $Ax_{0}$, constructed using the techniques used in [17]. As in [17],
this orbit will lie in the cusp for a density $1$ proportion of its lifetime.
To achieve a measure with partial escape of mass, we apply two Hecke
operators. The first Hecke operator is responsible for pulling an appropriate
proportion of the orbit out of the cusp. The second operator is used to get
equidistribution from the section of the orbit outside of the cusp (this can
be done with one operator, see Remark 5.13). However, this is not a single
$A$-orbit. Instead, we prove by ergodicity that at least one of the $A$-orbits
that constitute the Hecke operators applied to the orbit has to be almost
equidistributed as well.

######  Remark 1.10 (Comparison to [2] and [18]).

Plugging $c=1$ to Theorem 1.5 yield a constructions identical to the one in
[2] (with extra steps which are irrelevant for the $c=1$ case), the Hecke
operator of a compact orbit. We obtain its equidistribution and then use
ergodicity to find one compact orbit in the Hecke operator which
equidistributes. The construction in [18] is of the same kind, however, they
show a way to sample a specific compact orbit that equidistributes, not using
ergodicity in the same fashion.

###  1.4 Further Research

The first natural improvement of Theorem 1.1 is the full approximation.

######  Open Question 1.11.

Let $\mu\in\mathcal{M}(X_{n})^{A}_{c}$. Can it be the limit measure of other
measures in $\mathcal{M}(X_{n})^{A}_{c}$? In other words, is
$\mu\in\overline{\mathcal{M}(X_{n})^{A}_{c}\setminus\\{\mu\\}}$?

It can be seen that simply improving the bounds we give in this paper cannot
give a positive answer to this question, for the following reason. If $n=3$,
$Ay$ is a compact orbit and $Ax$ is an orbit. We can consider the set
$B=\\{a\in A:d_{X_{3}}(ax,Ay)<\delta\\}$. The connected components of $B$ are
roughly hexagons. It can be seen that two such hexagons $H_{1},H_{2}$ cannot
have $R$ long parts of the boundaries which are $\delta R$ close to one
another, for some $\delta>0$ and all $R>0$ sufficiently large.

Focusing of a different aspect of this work, we would like to find even less
trivial limits of large $A$-orbits in the following sence:

######  Open Question 1.12.

What are the possible limits $\lim_{i\to\infty}\mu_{Ax_{i}}$ where $Ax_{i}$ is
a compact orbit corresponding to an order $R_{i}$ satisfying that
$\operatorname{reg}(R_{i})>\operatorname{disc}(R_{i})^{\varepsilon}$? Can it
be a non-ergodic $A$-invariant probability measure? Can it contain as an
ergodic component a compact orbit uniform measure $\mu_{Ay}$?

Another possible direction for improvement is by working in different
homogenous spaces. Many questions can be asked, and here we choose to focus on
a single homogenous space:

######  Open Question 1.13.

Let $B$ be an $n^{2}$-dimensional central simple algebra over $\mathbb{Q}$
which splits over $\mathbb{R}$. Let $\mathcal{O}_{B}$ be an order in $B$. Let
$\Gamma=\operatorname{SL}_{1}(\mathcal{O}_{B})\subseteq\operatorname{SL}_{n}(\mathbb{R})$
a lattice. There are many compact orbits in
$\operatorname{SL}_{n}(\mathbb{R})/\Gamma$, coming from degree-$n$ totally
real number fields $K\subset B$. What are the possible limits of these compact
orbits? Can Haar measure be obtained? Can you extend Theorem 1.1 to this
setting? Can you compute an explicit non-Haar limit measure (in contrast to
Theorem 1.1 where we have a control only over a small portion of the measure)?

###  1.5 Acknowledgments

The second author would like to express his deep gratitude to Uri Shapira for
his support and encouragement. Without them, this paper would not exist. We
thank Uri Shapira for bringing the main questions answered in this paper into
our attention and for many intriguing discussions with him which contributed a
lot to this paper. Moreover, the first author thanks Andreas Wieser and Elon
Lindenstrauss for many fruitful discussions. The second author acknowledges
the support of ISF grants number 871/17. This work has received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant Agreement No. 754475). This work is
part of the first author’s Ph.D. thesis.

##  2 Notation and Preliminaries

######  Definition 2.1 ($O$-notations).

For two real functions $f,g$ on a set $A$ we write $f\ll g$ if there exists a
constant $C$ independent on the parameters of $f$ and $g$ such that $|f|\leq
Cg$ on $A$. The notation $O(g)$ will refer to some implicit function $f$ which
satisfies $f\ll g$. The notation $\Theta(g)$ will refer to some implicit
function $f$ which satisfies $g\ll f\ll g$. Whenever $r$ is a parameter going
to $0$ or $\infty$, the notation $o_{r}(g)$ will refer to some implicit
function $f$ which satisfies $f\ll g\cdot h$, for some implicit function $h\to
0$ as $r$ goes to $0$ or $\infty$ respectively. In this case, we sometimes
write $f=O(g)$. Whenever $r$ is a parameter going to either $\infty$ or $0$
and $f$

Fix $\lVert\cdot\rVert$ to denote the supremum norm on $\mathbb{R}^{n}$. Given
a lattice $\Lambda\subset\mathbb{R}^{n}$ we use $\text{cov}(\Lambda)$ to
denote the co-volume of $\Lambda$. Let $X_{n}$ denote the space of unimodular
lattices in $\mathbb{R}^{n}$ and let $d(\cdot,\cdot)$ denote the metric on
$X_{n}$ coming from the operator norm on linear operators on
$(\mathbb{R}^{n},\lVert\cdot\rVert)$. Let
$\mathbb{R}^{n-1}_{0}=\\{v\in\mathbb{R}^{n}:\sum_{i}v_{i}=0\\}$ and we abuse
notations and define
$\exp=\exp\circ\operatorname{diag}:\mathbb{R}^{n-1}_{0}\to A$ the standard
parametrization. We denote by $m_{X_{n}}$ probability measure on
$X_{n}=\operatorname{SL}_{n}(\mathbb{R})/\operatorname{SL}_{n}(\mathbb{Z})$
coming from the Haar measure on $\operatorname{SL}_{n}(\mathbb{R})$. We now
present formally the relation between compact $A$-orbits in $X_{n}$ and
subgroups of number fields. Denote by
$L,U\subset\operatorname{SL}_{n}(\mathbb{R})$ the subgroups of lower and upper
triangular matrices with diagonal $1$.

######  Definition 2.2 (Space of Measures).

Let $\mathcal{M}(X_{n})$ denote the space of finite measures on $X_{n}$
endowed with the topology induced by $\mu_{k}\rightarrow\mu$ if for any $f\in
C_{c}(X_{n})$ it holds that $\mu_{k}(f)\rightarrow\mu(f)$. We define a metric
on $X_{n}$ which induces this topology by letting, for any
$\mu_{1},\mu_{2}\in\mathcal{M}(X_{n})$:

$d(\mu_{1},\mu_{2})=\sup_{\epsilon>0}\sup\\{\epsilon\lvert\int fd\mu_{1}-\int
fd\mu_{2}\rvert:f:\mathcal{X}_{n}\rightarrow\mathbb{R}:\text{ f is
$1$-Lipschitz and is supported on }K_{\epsilon}\\}$ (2.1)

where $\mathcal{K}_{\epsilon}=\\{x\in X_{n}:\lambda_{1}(x)>\epsilon\\}$.

The following definition is relevant to §4.

######  Definition 2.3 (Special subgroups of
$\operatorname{SL}_{n}(\mathbb{R})$).

Let

$w_{0}=\left(\frac{n+1-2i}{2}\right)_{i=1}^{n}\in\mathbb{R}^{n-1}_{0},\qquad
a_{t}=\exp(tw_{0})\text{ for all }t\in\mathbb{R},$ (2.2)

and note that

$\displaystyle
L=\\{g\in\operatorname{SL}_{n}(\mathbb{R}):a_{t}ga_{-t}\xrightarrow{t\to\infty}I\\},\quad
U=\\{g\in\operatorname{SL}_{n}(\mathbb{R}):a_{-t}ga_{t}\xrightarrow{t\to\infty}I\\},$

that is, $L,U$ are the contracting and expanding horospheres with respect to
$a_{t}$.

######  Definition 2.4.

For every degree $n$, totally real number field $K$ denote by ${\rm Lat}_{K}$
the set of free $\mathbb{Z}$-modules of rank $n$ in $K$, where we identify two
lattices $\Lambda_{1},\Lambda_{2}\subset K$ if $\Lambda_{1}=k\Lambda_{2}$ for
some $k\in K^{\times}$. For every rank $n$, $\mathbb{Z}$-module
$\Lambda\subseteq K$ consider the lattice
$x_{\Lambda}:=\sigma(\Lambda)/\text{cov}(\sigma(\Lambda))\in X_{n}$, where
$\sigma_{i}:K\hookrightarrow\mathbb{R};i=1,\dots,n$ is some ordering of the
natural embeddings of $K$ and let
$\sigma=(\sigma_{1},\dots,\sigma_{n}):K\rightarrow\mathbb{R}^{n}$ denote their
concatenation. Denote by $\mathcal{O}_{\Lambda}=\\{k\in
K:k\Lambda\subseteq\Lambda\\}$. This is a ring. Denote by
$\mathcal{O}_{\Lambda}^{\times,>0}=\\{u\in\mathcal{O}_{\Lambda}^{\times}:\sigma_{i}(u)>0:i=1,\dots,n\\}.$
For every $U\subseteq\mathcal{O}_{K}^{\times,>0}$ denote
$A_{U}=\\{\operatorname{diag}(\sigma_{1}(u),\sigma_{2}(u),...,\sigma_{n}(u)):u\in
U\\}$.

######  Theorem 2.5.

Fix $d\geq 0$. For every totally real number field $K$, $[\Lambda]\in{\rm
Lat}_{K}$, and choice of the ordering of the real embeddings of $K$, the
orbits $Ax_{\Lambda}$ is compact, independent of the representative
$\Lambda\subset K$ of $[\Lambda]\in{\rm Lat}_{K}$ and this is a one to one
parametrization of all compact $A$-orbits in $X_{n}$.

This theorem is equivalent to [16], apart from the part of this
parametrization being one-to-one, whsich is Folklore and we do not use in this
paper.

######  Definition 2.6.

For every point $x\in X_{n}$ such that the orbit $Ax$ is compact, denote
$\mu_{Ax}$ the $A$-invariant measure probability on $Ax$. For every
$[\Lambda]\in{\rm Lat}_{K}$ denote $\mu_{\Lambda}=\mu_{Ax_{\Lambda}}$.

##  3 Effective Approximations

In this section, we prove an effective approximation theorem of points in
$X_{n}$ by points of compact orbits. Formally, the goal is to prove the
following lemma.

######  Lemma 3.1.

For every compact set $K\subset X_{n}$ there exists a sequence
$t_{k}\xrightarrow{k\to\infty}\infty$ such that for every $y\in X_{n}$ there
exists a sequence $(y_{k})_{k}\subset X_{n}$ such that
$d(y_{k},y)<\exp\left(-\frac{\lfloor n/2\rfloor}{4n(n^{2}-1)}t_{k}\right)$
such that $y_{k}$ is stabilized by a subgroup $\exp(\Lambda_{k})\subseteq A$
where $\Lambda_{k}\subseteq\mathbb{R}^{n-1}_{0}$ is generated by vectors
$(v^{(i)}_{k})_{i=1}^{n-1}$ such that
$\left|\left(v_{k}^{(i)}\right)_{j}-(1-\delta_{ij}n)t_{k}\right|=O(1)$.

######  Remark 3.2.

Lemma 3.1 implies in particular that $Ax_{k}$ is compact with total volume
$n^{n-3/2}t^{n-1}_{k}+O(t_{k}^{n-2})$ in the $\mathbb{R}^{n-1}_{0}$
parametrization, and in addition that for every $v\in\mathbb{R}^{n-1}_{0}$ and
for $y,y_{k}$ as in the lemma, we have:

$\displaystyle d((\exp v).y_{k},(\exp v).y)\leq O\left(\exp\left(\max_{1\leq
i,j\leq n}|v_{i}-v_{j}|-\frac{\lfloor
n/2\rfloor}{4n(n^{2}-1)}t_{k}\right)\right).$ (3.1)

Since the set $B_{0}=\\{v\in\mathbb{R}^{n}_{0}:\max_{1\leq i,j\leq
n}|v_{i}-v_{j}|\leq 1\\}$ has volume
$\operatorname{vol}(B_{0})=\frac{1}{\sqrt{n}}$, we deduce that a portion of
$\left(\frac{1}{n}\frac{\lfloor
n/2\rfloor}{4n(n^{2}-1)}\right)^{n-1}+O(1/t_{k})$ of the orbit $Ay_{k}$
approximates the the orbit $Ay$.

The proof of Lemma 3.1 will be composed of two components. The first is an
approximation of lattices via Hecke neighbors. We will define Hecke neighbors
in Definition 3.4 and obtain a good approximation by them in Lemma 3.6. The
second component generates infinitely many points in $X_{n}$ with compact
$A$-orbits, with a control on the geometry of their stabilizers in $A$ and the
stabilizers of their sublattices of a given index. This gives us the arsenal
of points on which we apply Lemma 3.6 to deduce Lemma 3.1.

###  3.1 Hecke Density

The source of our good approximation comes from the quantitative version of
the equidistribution of Hecke neighbors. There are many references to the
equidistribution of Hecke neighbors such as [5, Theorem 1.1], [13, Theorem
3.7], [12, Theorem 1.2]. In this section we will cite the result of Laurent
Clozel, Hee Oh and Emmanuel Ullmo [5] and deduce the approximation result we
need.

######  Definition 3.3 (Function space).

Let $L^{2}_{0}(X_{n})=\\{f\in
L^{2}(X_{n},m_{x_{n}}):\int_{X_{n}}fdm_{X_{n}}=0\\}$ be the Hilbert space of
$L^{2}$ function on $X_{n}$ with $0$ mean.

######  Definition 3.4 (Definition of $p$-Hecke neighbors and Hecke operator).

For every sequence of integers $k_{1}\leq k_{2}\leq...\leq k_{n}$ consider:

$a=\frac{1}{p^{(k_{1}+\dots+k_{n})/n}}\operatorname{diag}(p^{k_{1}},p^{k_{2}},\dots,p^{k_{n}})\in\operatorname{SL}_{n}(\mathbb{R}).$

For every $x=g\operatorname{SL}_{n}(\mathbb{Z})\in X_{n}$ denote
$T_{a}(x)=g\operatorname{SL}_{n}(\mathbb{Z})a\operatorname{SL}_{n}(\mathbb{Z})$.
This set is is finite since $a\operatorname{SL}_{n}(\mathbb{Z})a^{-1}$ is
commensurable to $\operatorname{SL}_{n}(\mathbb{Z})$. The size
$\\#T_{a}(x)=\\#(\operatorname{SL}_{n}(\mathbb{Z})a\operatorname{SL}_{n}(\mathbb{Z})/\operatorname{SL}_{n}(\mathbb{Z}))$
depends only on $k_{1},...,k_{n}$ and not $x$. Equivalently,

$T_{a}(x)=\left\\{\frac{1}{\sqrt[n]{\text{cov}(x^{\prime})}}x^{\prime}:x^{\prime}\subseteq
x\text{ with
}x/x^{\prime}\cong\mathbb{Z}/p^{k_{1}}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p^{k_{n}}\mathbb{Z}\right\\}.$

For every function $f\in L^{2}(X_{n})$ define the Hecke action on functions

$T^{\rm F}_{a}(f)(x)=\frac{1}{\\#T_{a}(x)}\sum_{x^{\prime}\in
T_{a}(x)}f(x^{\prime})\in L_{0}^{2}(X_{n}).$

For every point $x\in X_{n}$, define the Hecke action on measures

$T_{a}^{\rm M}(x):=\frac{1}{\\#T_{a}(x)}\sum_{x^{\prime}\in
T_{a}(x)}\delta_{x^{\prime}}$

and for every measure $\mu$ on $X_{n}$ define

$T_{a}^{\rm M}(\mu):=\int_{X_{n}}T_{a}^{\rm M}(x)d\mu(x).$

The following theorem is a particular case of [5, Theorem 1.1], specialized
for $\operatorname{SL}_{n}$ as in [5, Example 5.1].

######  Theorem 3.5.

For every prime $p$ and $k_{1}\leq k_{2}\leq...\leq
k_{n},a\in\operatorname{SL}_{n}(\mathbb{R})$ as in Definition 3.4, the
operator norm of $\left.T^{\rm F}_{a}\right|_{L^{2}_{0}(X_{n})}$ is bounded
by:

$\left\|\left.T^{\rm F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|\leq\prod_{i\leq
n/2}\frac{1}{p^{(k_{n+1-i}-k_{i})/2}}\frac{(k_{n+1-i}-k_{i})(p-1)+(p+1)}{p+1}\leq
p^{-\frac{1}{2}\sum_{i=1}^{n/2}(k_{n+1-i}-k_{i})}\cdot C(k_{1},...,k_{n}),$

where $C(k_{1},...,k_{n})$ depends polynomially on $k_{1},...,k_{n}$.

######  Lemma 3.6.

For every compact subset $K\subset X_{n}$ there exists $C=C(K)>0$ such that
for any $x,y\in K$ and $p,k_{1}\leq k_{2}\leq...\leq
k_{n},a\in\operatorname{SL}_{n}(\mathbb{R})$ as in Definition 3.4, there
exists a Hecke neighbor $x^{\prime}\in T_{a}(x)$ such that:

$d\left(x^{\prime},y\right)\leq C(K)\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|^{1/(n^{2}-1)},$ (3.2)

where $\alpha>0$ is some constant depending only on $n$.

###### Proof.

Recall the right invariant Riemannian metric
$d_{\operatorname{SL}_{n}(\mathbb{R})}$ on
$\operatorname{SL}_{n}(\mathbb{R})$, and its descend to $X_{n}$, the metric
$d_{X_{n}}$. Let $r_{0}<\min(\operatorname{inj}(y),\operatorname{inj}(y))$,
where $\operatorname{inj}(x)$ is the injectivity radius of $x$, that is, the
maximal radius $r$ such the translation map $g\mapsto gx$ is injective on
$B_{\operatorname{SL}_{n}(\mathbb{R})}(I;r)$, itself being the radius $r$ ball
around the identity in $\operatorname{SL}_{n}(\mathbb{R})$. Let
$f_{x}=\chi_{B_{X_{n}}(x;r_{0})}$ be the indicator of a radius $r_{0}$ ball of
$x$. Then
$\int_{X_{n}}f_{x}dm_{X_{n}}=\operatorname{vol}(B_{\operatorname{SL}_{n}(\mathbb{R})}(I;r_{0}))=\Theta(r_{0}^{n^{2}-1})$,
and a similar equality holds for $f_{y}=\chi_{B_{X_{n}}(y;r_{0})}$. Denote by
$v_{r_{0}}=\operatorname{vol}(B_{\operatorname{SL}_{n}(\mathbb{R})}(I;r_{0}))$.
Then $\tilde{f}_{x}=f_{x}-v_{r_{0}}\in L^{2}_{0}(x_{n})$ and
$\tilde{f}_{y}=f_{y}-v_{r_{0}}\in L^{2}_{0}(x_{n})$ have norm
$\|f_{x}\|^{2}=\|f_{y}\|^{2}=v_{r_{0}}(1-v_{r_{0}})^{2}+(1-v_{r_{0}})v_{r_{0}}^{2}=(1-v_{r_{0}})v_{r_{0}}$.
Consider $\left\langle\tilde{f}_{x},T^{\rm
F}_{a}(\tilde{f}_{y})\right\rangle$. On the one hand, it is at most
$\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|\|f_{x}\|\|f_{y}\|=\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|(1-v_{r_{0}})v_{r_{0}}$. On the other
hand, consider the set:

$T_{a}^{-1}(B_{X_{n}}(y,r_{0}))=\\{x^{\prime}\in X_{n}:T_{a}(x^{\prime})\cap
B_{X_{n}}(y,r_{0})\neq\emptyset\\}.$

and note that $T^{\rm
F}_{a}(\tilde{f}_{y})|_{(T_{a}^{-1}(B_{X_{n}}(y,r_{0})))^{c}}\equiv-
v_{r_{0}}$. Assume that $T_{a}^{-1}(B_{X_{n}}(y,r_{0}))\cap
B_{X_{n}}(x,r_{0})=\emptyset$, and get that $T^{\rm
F}_{a}(f_{y})|_{B_{X_{n}}(x,r_{0})}\equiv-v_{r_{0}}$. It follows that:

$\displaystyle\left\langle\tilde{f}_{x},T^{\rm
F}_{a}(\tilde{f}_{y})\right\rangle$
$\displaystyle=\int_{X_{n}}\tilde{f}_{x}T^{\rm
F}_{a}(\tilde{f}_{y})dm_{X_{n}}=\int_{X_{n}}f_{x}T^{\rm
F}_{a}(\tilde{f}_{y})dm_{X_{n}}=\int_{B_{X_{n}}(x,r_{0})}T^{\rm
F}_{a}(\tilde{f}_{y})dm_{X_{n}}$
$\displaystyle=v_{r_{0}}\cdot(-v_{r_{0}})=-v_{r_{0}}^{2}.$

We deduce that

$v_{r_{0}}^{2}\leq\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|\cdot\|\tilde{f}_{x}\|\cdot\|\tilde{f}_{y}\|=\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|(1-v_{r_{0}})v_{r_{0}},$

and hence $v_{r_{0}}\leq\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|$. Using this logic in reverse, we
deduce that if $v_{r_{0}}>\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|$ then
$T_{a}^{-1}(B_{X_{n}}(y,r_{0}))\cap B_{X_{n}}(x,r_{0})\neq\emptyset$, that is,
there exist $x^{\prime}=g_{0}x,y^{\prime}=g_{1}y$ such that $y^{\prime}\in
T_{a}(x^{\prime})$ and $g_{0},g_{1}\in
B_{\operatorname{SL}_{n}(\mathbb{R})}(I,r_{0})$. Thus
$g_{0}^{-1}y^{\prime}=g_{0}^{-1}g_{1}y\in
g_{0}^{-1}T_{a}(x^{\prime})=T_{a}(x)$. On the other hand,
$g_{0}^{-1}y^{\prime}=g_{0}^{-1}g_{1}y$ satisfies
$d_{X_{n}}(g_{0}^{-1}g_{1}y,y)\leq 2r_{0}$.

Altogether, we have proved that if $\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|<r_{0}\leq\min(\operatorname{inj}(x),\operatorname{inj}(y))$
then there exists $x^{\prime\prime}=g_{0}^{-1}g_{1}y$ with
$x^{\prime\prime}\in T_{a}(x)$ and $d_{X_{n}}(x^{\prime\prime},y)\leq 2r_{0}$.
Now, let $K\subset X_{n}$ be a compact set. Denote the minimum of the
injectivity radius on $K$ by $r_{K}>0$. If $v_{r_{K}}>\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|$, then we can find
$r_{0}=\Theta(\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|^{1/(n^{2}-1)})$ with $r_{0}<r_{K}$
and $v_{r_{0}}>\left\|\left.T^{\rm F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|$,
and the desired follows. If $v_{r_{K}}\leq\left\|\left.T^{\rm
F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|$, then the desired follows for
$C(K)=\frac{\operatorname{diam}(K)}{(v_{r_{K}})^{1/(n^{2}-1)}}$. ∎

######  Remark 3.7 (Composition of Hecke operators).

The composition of Hecke operators is a linear combination of different Hecke
operators. This follows from the double-coset description, as the
multiplication of two finite double-cosets is again a double-coset. We mention
explicitly the case of the operators $a,a^{\prime}$ defined with
$(-k,0,0,\dots,0),(-l,0,0,\dots,0)$ respectively, where $0\leq k\leq l$. The
composition is

$T^{\rm F}_{a}\circ T^{\rm F}_{a^{\prime}}=T^{\rm F}_{a^{\prime}}\circ T^{\rm
F}_{a}=\frac{p^{n-1}(p-1)}{p^{n}-1}T^{\rm
F}_{a_{k+l}}+\sum_{i=1}^{l-1}\frac{p^{n-1}-1}{p^{n}-1}\cdot\frac{p-1}{p^{i}}\cdot
T^{\rm F}_{a_{k+l-i,i}}+\frac{p^{n-1}-1}{p^{n}-1}\cdot\frac{1}{p^{l-1}}T^{\rm
F}_{a_{k,l}}$

Where $a_{k^{\prime},l^{\prime}}$ corresponds to
$-k^{\prime}\leq-l^{\prime}\leq 0\leq 0\leq\dots\leq 0$. A similar equality
hold for $T^{\rm M}$.

###  3.2 Construction of special number field

######  Theorem 3.8.

For every prime number $p\equiv 1\mod 2n$ sufficiently big as a function of
$n$, one can find a totally real number field $K$ with the following
properties:

  1. (a)

The unit group $\mathcal{O}_{K}^{\times}$ contains units $u_{1},...,u_{n}$
with $u_{1}u_{2}\cdots u_{n}=1$.

  2. (b)

One can order the real embeddings $\sigma_{1},...,\sigma_{n}:K\to\mathbb{R}$
such that $\sigma_{i}(u_{j})>0$ for all $1\leq i,j\leq n$, and

$\displaystyle\log\sigma_{i}(u_{j})=\begin{cases}-2d(d-1)\log p+O(1),&\text{if
}i=j,\\\ 2d\log p+O(1),&\text{if }i\neq j.\end{cases}$ (3.3)

  3. (c)

The units $u_{i}$ lie in the ring $\mathbb{Z}+p\mathcal{O}_{K}$.

###### Proof.

Since $p\equiv 1\mod 2n$, the polynomial $x^{n}+1$ has a $n$ different
solutions mod $p$. By Hensel’s Lemma (See [7]), it has $n$ solutions mod
$p^{n}$, call them
$a_{0}^{\prime},...,a_{n-1}^{\prime}\in\\{1,...,p^{n}-1\\}$. Let
$a_{i}=a_{i}^{\prime}+2ip^{n}$ for $i=0,\dots,n-1$, and note that $p^{n}+1\leq
a_{i+1}-a_{i}\leq 3p^{n}$ for all $0\leq i\leq d-2$. consider now the
polynomial $R(x)=\frac{(px-a_{1})(px-a_{2})\cdots(px-a_{n})-1}{p^{n}}$. It has
integer coefficients since by assumption,
$\sum_{i_{1}<...<i_{k}}a_{i_{1}}\cdots a_{i_{k}}=0\mod{p^{n}}$ for all
$k=1,...,n-1$ and $a_{1}a_{2}\cdots a_{n}=(-1)^{n}\mod{p^{n}}$.

To control the real embeddings of the resulting number field, we will
approximate the real roots of $R$.

By Taylor’s theorem (see [14, §20.3]) applied at $a_{i_{0}}/p$ we have

$R(x)=R(a_{i_{0}}/p)+R^{\prime}(a_{i_{0}}/p)(x-a_{i_{0}}/p)+\frac{1}{2}R^{\prime\prime}(x^{\prime})(x-a_{i_{0}}/p)^{2},$

for some $x^{\prime}\in[a_{i_{0}}/p,x]$ (this interval notation does not
assume $a_{i_{0}}/p\leq x$). Note that

$R(a_{i_{0}}/p)=-\frac{1}{p^{n}},\qquad R^{\prime}(a_{i_{0}}/p)=\prod_{j\neq
i_{0}}(a_{i_{0}}/p-a_{j}/p)=(-1)^{n-i_{0}}\alpha_{n-i_{0}}p^{n^{2}-2n+2},$

for some $\alpha_{i_{0}}\geq 1$, with $\alpha_{i_{0}}=\Theta(1)$, and for all
$x^{\prime}\in[-p^{n-1},2np^{n-1}]$ we have

$|R^{\prime\prime}(x^{\prime})|=\left|\sum_{1\leq k<l\leq n}\prod_{j\neq
k,l}(x-a_{k}/p)\right|=O(p^{n^{2}-3n+2}).$

That implies that for all $x\in[-p^{n-1},2np^{n-1}]$ we have

$\displaystyle
R(x)=-\frac{1}{p^{n}}+(-1)^{n-i_{0}}\alpha_{i_{0}}p^{n^{2}-2n+1}(x-a_{i_{0}}/p)+O(p^{n^{2}-3n+2})(x-a_{i_{0}}/p)^{2},$
(3.4)

Choose $x_{i_{0}}^{\prime}$ with
$x_{i_{0}}^{\prime}-a_{i_{0}}/p=2\frac{(-1)^{n-i_{0}}}{p^{n^{2}-n+1}\alpha_{i_{0}}}$.
Note that $|x_{i_{0}}^{\prime}-a_{i_{0}}/p|<1$, and hence
$x_{i_{0}}^{\prime}\in[-p^{n-1},2np^{n-1}]$, Then

$R(x_{i_{0}}^{\prime})=-\frac{1}{p^{n}}+(-1)^{n-i_{0}}\alpha_{i_{0}}p^{n^{2}-2n+1}(x_{i_{0}}^{\prime}-a_{i_{0}}/p)+O(p^{-n^{2}-n})=\frac{1}{p^{n}}+O(p^{-n^{2}-n}).$

If $p$ is sufficiently large as a function of $n$ then
$R(x_{i_{0}}^{\prime})>0$. Consequently, due to the mean value theorem, there
is $x_{i_{0}}\in[a_{i_{0}}/p,x_{i_{0}}^{\prime}]$ with $R(x_{i_{0}})=0$. By
Eq. (3.4) we deduce that:

$\displaystyle
0=R(x_{i_{0}})=-\frac{1}{p^{n}}+(-1)^{n-i_{0}}\alpha_{i_{0}}p^{n^{2}-2n+1}(x_{i_{0}}-a_{i_{0}}/p)+O(p^{n^{2}-3n+2})(x_{i_{0}}-a_{i_{0}}/p)^{2}.$
(3.5)

This implies:

$\displaystyle\frac{1}{p^{n}}$
$\displaystyle=(x_{i_{0}}-a_{i_{0}}/p)\left((-1)^{n-i_{0}}\alpha_{i_{0}}p^{n^{2}-2n+1}+O(p^{n^{2}-3n+2})(x_{i_{0}}-a_{i_{0}}/p)\right)$
$\displaystyle=(x_{i_{0}}-a_{i_{0}}/p)(-1)^{n-i_{0}}\alpha_{i_{0}}p^{n^{2}-2n+1}\left(1+O\left(p^{-n^{2}}\right)\right),$

and hence,

$x_{i_{0}}=a_{i_{0}}/p+\frac{(-1)^{n-i_{0}}}{p^{n^{2}-n+1}\alpha_{i_{0}}}+O\left(\frac{1}{p^{2n^{2}-n+1}}\right).$

Since $|a_{i_{0}}/p-x_{i_{0}}|<1$ we deduce that all $x_{i}$ are different,
and hence these are the distinct roots of $R$. Since $|a_{i}-a_{j}|\geq p^{n}$
for all $i\neq j$ it follows that $|a_{i}/p-x_{j}|\geq p^{n-1}-1>1$. It
follows that $R=\prod_{i=1}^{d}(x-x_{i})$ is irreducible, as if
$R_{0}=\prod_{i\in I}(x-x_{i})$ is an integer polynomial $i_{0}\notin I$ then
in the ring $\mathbb{Z}[x]/(R_{0})$ the element $px-a_{i_{0}}$ is a unit, but
it’s norm in $\mathbb{Z}[x]/(R_{0})$ is $\prod_{i\in I}(px_{i}-a_{i_{0}})$,
which is greater the $1$ in absolute value.

The number field $K=\mathbb{Q}[\alpha]$ where $R(\alpha)=0$ has units
$u_{i}^{\prime}=p\alpha-a_{i}$. Let $u_{i}=(u_{i}^{\prime})^{2}$. The real
embeddings of $K$ are $\sigma_{i}:\alpha\mapsto x_{i}$, sends has
$\log\sigma_{i}(u_{j})=2\log|px_{i}-a_{j}|$. Now Eq. (3.3) follows from
properties of $x_{i}$. ∎

###  3.3 Application of Hecke operator to compact orbits

In this section, we introduce the following lemma, which describes how to
verify that a Hecke operator splits a compact orbit into sufficiently many
compact diagonal orbits.

######  Lemma 3.9.

Let $K$ be a totally real number field of degree $n$, let
$x_{\mathcal{O}_{K}}\in X_{n}$ the point with compact orbit corresponding to
the $\mathbb{Z}$-module $\mathcal{O}_{K}$ and let $p,k_{1}\leq\dots\leq
k_{n},a\in\operatorname{SL}_{n}(\mathbb{R})$ as in Definition 3.4. Let
$U\subseteq\mathcal{O}_{K}^{\times,>0}$ be a subgroup and $A_{U}\subset A$ the
corresponding diagonal subgroup as in Definition 2.4. If
$U\subseteq\mathbb{Z}+p^{k}\mathcal{O}_{K}$ and $0\leq k_{1}\leq k_{n}\leq k$
then $A_{U}\subseteq\operatorname{stab}_{A}(x^{\prime})$ for every
$x^{\prime}\in T_{a}(x_{\mathcal{O}_{K}})$.

###### Proof.

Let $u\in U$. Since $u\in\mathbb{Z}+p^{k}\mathcal{O}_{K}$, there is
$m\in\mathbb{Z}$ such that $u\equiv m\mod{p^{k}}$. Thus for every
$\bar{b}\in\mathcal{O}_{K}/(p^{k})$ we have $u\bar{b}=m\bar{b}$, that is, the
multiplication by $u$ action on $\mathcal{O}_{K}/(p^{k})$ is in fact a
multiplication by a scalar. Hence, the element
$a_{u}=\operatorname{diag}(\sigma_{1}(u),\sigma_{2}(u),\dots,\sigma_{k}(u))\in
A_{U}$ preserves $x_{\mathcal{O}_{K}}$, $p^{k}x_{\mathcal{O}_{K}}$ and acts on
the quotient by multiplication by the scalar $m$. This implies that $a_{u}$
acts on the set of mid-groups
$\\{\Lambda:p^{k}x_{\mathcal{O}_{K}}\subseteq\Lambda\subseteq
x_{\mathcal{O}_{K}}\\}$ trivially. Indeed, there is an isomorphism

$\\{\Lambda:p^{k}x_{\mathcal{O}_{K}}\subseteq\Lambda\subseteq
x_{\mathcal{O}_{K}}\\}\cong\\{\bar{\Lambda}\subseteq
x_{\mathcal{O}_{K}}/p^{k}x_{\mathcal{O}_{K}}\\},$

and $u$ acts on the right-hand side as a multiplication by a scalar, which
preserves all groups. This implies that $a_{u}$ preserves all Hecke neighbors,
by the second construction (See Definition 3.4). ∎

######  Remark 3.10.

Lemma 3.9 remains true if we replace $x_{\mathcal{O}_{K}}$ by $x_{\Lambda}$
for some $\Lambda\in{\rm Lat}_{K}$ and $U$ by a subgroup of
$\mathcal{O}_{\Lambda}$ satisfying
$U\subseteq\mathbb{Z}+p^{k}\mathcal{O}_{\Lambda}$. The proof is similar.

######  Corollary 3.11.

There is a compact subset $C\subset X_{n}$ such that for any prime $p\equiv
1\mod{2n}$ large enough as a function of $n$, there exists $x_{p}\in C$, and a
lattice $\Lambda\subset\mathbb{R}^{n-1}_{0}$ such that,

  1. _1._

For every Hecke operator $T_{a}$ with $0\leq k_{1}\leq\dots\leq k_{n}\leq 1$
and for every element $y\in T_{a}(x)$ we have that $\exp(\Lambda)\subseteq A$
stabilize $y$.

  2. _2._

The lattice $\Lambda$ is generated by vectors $(v^{(i)})_{i=1}^{n-1}$
satisfying $\left|\left(v^{(i)}\right)_{j}-(2n-2n^{2}\delta_{ij})\log
p\right|=O(1)$.

###### Proof.

Fix $p\equiv 1\mod{2n}$ sufficiently big for Theorem 3.8. Let $K$ be the
number field constructed in Theorem 3.8, and $u_{1},\dots,u_{n}$ the
corresponding units. Let $(\sigma_{i})_{i=1}^{n}$ be the real embeddings of
$K$. Let $\Lambda$ be the subgroup of $\mathbb{R}^{n-1}_{0}$ generated by
$(\log\sigma_{i}(u_{j}))_{i=1}^{n}$ for $j=1,\dots,n-1$. The bounds on $u_{j}$
in Theorem 3.8 imply Part 2. Consider the point $x_{\mathcal{O}_{K}}\in
X_{n}$. It follows from [19, Proposition A.1] that there is a compact set
$C\subseteq X_{n}$ which intersects every $A$-orbit. Hence there is
$x_{p}=mx_{\mathcal{O}_{K}}\in C$ for some $m\in A$. Since the
$\operatorname{SL}_{n}(\mathbb{R})$ action on $X_{n}$ commutes with $T_{a}$,
to prove Part 1 it is sufficient to prove that $\exp(\Lambda)\subseteq A$
stabilize $y$ for every $y\in T_{a}(x_{\mathcal{O}_{K}})$. This follows from
Lemma 3.9 and Property 3 of the unit $u_{1},...,u_{k}$. ∎

###  3.4 Proof of Lemma 3.1

The points we construct are indexed by primes $p\equiv 1\mod{2n}$. Fix such a
prime $p$. Denote $t_{p}^{\prime}=2n\log p$ and let $x_{p}$ and $\Lambda$ be
as in Corollary 3.11 with $t_{p}^{\prime}$ instead of $t_{p}$. Note that
Corollary 3.11 implies the exact bounds on a set of generators of $\Lambda$ we
need. Let $k_{1}=\dots=k_{\lfloor n/2\rfloor}=0,k_{{\lfloor
n/2\rfloor}+1}=...=k_{n}=1$, and $a\in A$ as in Definition 3.4. We claim that
for some $y_{p}\in T_{a}x_{p}$ we have $d(y,y_{p})\ll\exp(-\frac{\lfloor
n/2\rfloor}{4n(n^{2}-1)}t_{p}^{\prime})$. Indeed, by Theorem 3.5 we have
$\left\|\left.T^{\rm F}_{a}\right|_{L^{2}_{0}(X_{n})}\right\|\leq p^{-\lfloor
n/2\rfloor/2}$. Thus by Lemma 3.6, there is $y_{p}\in T_{a}(x_{p})$ with

$d(y_{p},y)=O(p^{-\frac{\lfloor
n/2\rfloor}{2(n^{2}-1)}})=O\left(\exp\left(t_{p}^{\prime}\cdot\frac{\lfloor
n/2\rfloor}{4n(n^{2}-1)}\right)\right).$

We now choose $t_{p}=t_{p}^{\prime}=O(1)$ to be smaller then $t_{p}$ to ensure

$d(y_{p},y)<\exp\left(t_{p}\cdot\frac{\lfloor
n/2\rfloor}{4n(n^{2}-1)}\right).$

This preserves the bounds on the generators of $\Lambda$.

Note that the implicit constant was provided by Theorem 3.5 and depends only
on the compact set containing $y$.∎

##  4 Approximation of Invariant Measures

In this section, we prove Theorem 1.1. We approximate measures by
approximating their generic points. When we want to approximate a finite
collection of measures simultaneously, we need to find regions of $A$-orbits
that approximate all measures on subregions. The following lemma is our tool
for this purpose:

######  Lemma 4.1 (Diagonal closing lemma).

For every compact set $K\subseteq X_{d}$ there are a compact sets
$K_{A}\subset A,K_{L}\subset L,K_{U}\subset U$, such that for every
$x_{0},x_{1}\in K$ there are $k_{A}\in K_{A},k_{L}\in K_{L},k_{U}\in K_{U}$
such that $k_{U}k_{A}k_{L}x_{0}=x_{1}$. Here $A,L,U$ are defined as in
Definition 2.3.

###### Proof.

First, we will show that for every $x_{0},x_{1}\in X_{d}$ we have $x_{0}\in
LAUx_{1}$. Since $U$ is the expanding horosphere of $a_{t}$, almost every
point in $Ux_{1}$ is generic with respect to the forward $a_{t}$ action and
the Haar measure $m_{X_{n}}$. Choose a $a_{t}$-generic point point
$u_{0}x_{1}\in Ux_{1}$. The $LU$-decomposition implies that the product map
$U\times A\times L\to\operatorname{SL}_{n}(\mathbb{R})$ is one to one. Since
$U\times A\times L$ and $\operatorname{SL}_{n}(\mathbb{R})$ are $n^{2}-1$
dimensional, this implies that $UAL\subseteq\operatorname{SL}_{n}(\mathbb{R})$
is open. Hence $UALx_{0}$ is an open set in $X_{n}$, and hence of positive
measure. Hence:

$\frac{1}{T}\int\nolimits_{0}^{T}\chi_{a_{t}u_{0}x_{1}\in
UALx_{0}}dt\xrightarrow{T\to\infty}m_{X_{n}}(UALx_{0})>0$ (4.1)

and hence for some $t>0$ we have $a_{t}u_{0}x_{1}\in UALx_{0}$. Since $AU=UA$
we deduce that $x_{0}\in LAUx_{1}$.

Let
$\|\cdot\|_{A}:A\to[0,\infty),\|\cdot\|_{L}:L\to[0,\infty),\|\cdot\|_{U}:U\to[0,\infty)$
be the distance from the identity. Define by $f:K^{2}\to[0,\infty)$ the map
attaching

$f(x_{1},x_{2})=\inf\\{\max(\|a\|_{A},\|l\|_{L},\|u\|_{U}):a\in A,l\in L,u\in
U:x_{0}=laux_{1}\\}.$

Since $LAU$ is open and the multiplication map $L\times A\times U\to LU$ is a
homeomorphism, if we have $laux_{0}=x_{1}$ for $a\in A,l\in L,u\in U$, then
for every sufficiently close $\hat{x}_{0}\sim x_{0},\hat{x}_{1}\sim x_{1}$
there are $\hat{a}\in a,\hat{l}\in L,\hat{u}\in U$ arbitrarily close to
$a,l,u$ such that $\hat{u}\hat{l}\hat{a}\hat{x}_{0}=\hat{x}_{1}$. This implies
that $f$ is upper semicontinuous. In particular, for every $(x_{0},x_{1})\in
K^{2}$ we get that $f$ is bounded in a neighborhood of $(x_{1},x_{2})$. Since
$K^{2}$ is compact this implies that $f$ is bounded everywhere, as desired. ∎

######  Definition 4.2.

For $v,v^{\prime}\in\mathbb{R}^{n-1}_{0}$, we say that $v\prec v^{\prime}$ if
for all $i=1,\dots,n-1$ we have $v_{i+1}-v_{i}\leq
v_{i+1}^{\prime}-v_{i}^{\prime}$. For every $v_{0}\in\mathbb{R}^{n-1}_{0}$
with $0\prec v_{0}$, define a _box_
$S_{v_{0}}=\\{v\in\mathbb{R}^{n-1}_{0}:0\prec v\prec v_{0}\\}$. A _boxed map_
is a map $f:S_{v_{0}}\to X_{d}$ of the form $f(v)=\exp(v).x_{0}$ for some
$x_{0}\in X_{n}$. For every compact set $K\subset X_{n}$, a boxed map
$f_{0}:S_{v_{0}}\to X_{d}$ is said to be _$K$ -bounded_ if $f(0),f(v_{0})\in
K$. Recall that $w_{0}=(\frac{d+1-2i}{2})_{i=1}^{n}$.

In the following lemma, we fix $K,K^{\prime}$ and let $R\to\infty$. The
constants in the $O$-notations depend on $K,K^{\prime}$ and hold as
$R\to\infty$.

######  Corollary 4.3 (Gluing boxes).

Let $K\subset X_{n}$ be compact and $K\subset K^{\prime}\subset X_{n}$ a
compact neighborhood. In this corollary all $O$ notations depend on
$K,K^{\prime}$. Let $R>0$. Fix any two $K$-bounded boxed maps
$f_{1}:S_{v_{1}}\to X_{n},f_{2}:S_{v_{2}}\to X_{n}$ with $v_{1},v_{2}\succ
Rw_{0}$. Then there exists a $K^{\prime}$-bounded boxed map $f:S_{v_{3}}\to
X_{n}$ with

  1. _( 1)_

$v_{3}=v_{1}+v_{2}+O_{K,K^{\prime}}(1)$;

  2. _( 2)_

There are two points $w_{1},w_{2}\in S_{v_{3}}$ with $\|w_{1}\|=o_{R}(R)$,
$\|w_{2}-v_{1}\|=o_{R}(R)$ and $0<\rho=o_{R}(R)$ we have that for all $i=1,2$
and $v\in S_{v_{i}-2\rho w_{0}}$,

$d_{X_{n}}\left(f_{i}(\rho w_{0}+v),f(w_{i}+v)\right)=o_{R}(1).$

See Figure 1 for a visualization of these conditions.

$f_{1}$$f_{2}$$f$$S_{v_{1}}+w_{1}$$S_{v_{2}}+w_{2}$$S_{v_{3}}$$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}w_{1}+v_{1}$$w_{1}$$w_{2}+v_{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}$$~{}~{}w_{2}$$0$$v_{3}$$v_{1}$$~{}~{}~{}~{}~{}~{}~{}~{}v_{1}+v_{2}$
Figure 1: Plot of the boxes in Corollary 4.3. One can see $S_{v_{3}}$, and in
it $S_{v_{1}}+w_{1},S_{v_{2}}+w_{2}$, and in them the regions approximated by
$f_{1},f_{2}$.

###### Proof.

Let $K_{A}\subset A,K_{L}\subset L,K_{U}\subset U$ as in Lemma 4.1,
constructed for $K$. Assume $K_{A}=\exp(K_{0})$ for
$K_{0}\subseteq\\{w\in\mathbb{R}^{n-1}_{0}:|w_{i}-w_{i+1}|<C_{0}\text{ for all
}i=1,\dots,n-1\\}$. Note that

$\displaystyle\text{for all~{}~{}}T>0,v\geq Tw_{0},u\in K_{U}\text{~{}~{}we
have~{}~{}}d(\exp(-v)u\exp(v),I)=O(\exp(-T)),$ (4.2)

and similarly,

$\text{for all~{}~{}}T>0,v\geq Tw_{0},l\in K_{L}\text{~{}~{}we
have~{}~{}}d(\exp(v)l\exp(-v),I)=O(\exp(-T)).$

Apply Lemma 4.1 for $f_{1}(v_{1}),f_{2}(0)$, we get that there are
$v_{0}\in\mathbb{R}^{n-1}_{0},k_{l}\in K_{L},k_{u}\in K_{U}$ such that

$|(v_{0})_{i}-(v_{0})_{i+1}|<C_{0}\text{ for all }i=1,\dots,n-1,$

and

$k_{l}\exp(v_{0})k_{u}f_{1}(v_{1})=f_{2}(v_{2}).$

Let $\rho=\sqrt{R}$, $v_{3}=v_{1}+v_{2}+v_{0}$,
$x_{0}=\exp(-v_{1})k_{u}\exp(v_{1})f_{1}(0)$, $f:S_{v_{3}}\to X_{n}$ defined
by $f(v)=\exp(v)x_{0}$ for all $v\in S_{v_{3}}$, $v_{1}=\rho w_{0}$ and
$v_{2}=v_{1}+\rho w_{0}+v_{0}$. By Eq. (4.2) we get that
$d(\exp(-v_{1})k_{u}\exp(v_{1}),I)=O(\exp(-R))=o_{R}(1)$. hence we deduce that
for $R$ sufficiently big, $x_{0}\in K^{\prime}$.

Moreover, for all $v\in S_{v_{1}-2\rho w_{0}}$ we have

$\displaystyle f(v_{0}+v)$
$\displaystyle=\exp(v_{0}+v)x_{0}=\exp(v_{0}+v-v_{1})k_{u}\exp(v_{1}-(v_{0}+v))f_{1}(v_{0}+v).$

Since $v\leq v_{1}-2\rho w_{0},v_{0}=\rho w_{0}$ we get that
$v_{1}-(v_{0}+v)\geq\rho w_{0}$, and hence

$d\left(\exp(v_{0}+v-v_{1})k_{u}\exp(v_{1}-(v_{0}+v)),I\right)=o_{R}(1),$

which implies that $d(f(v_{0}+v),f_{1}(\rho w_{0}+v))=o_{R}(1)$.

Note that

$f(v_{3})=\exp(v_{1}+v_{2}+v_{0})x_{0}=\exp(v_{2})\exp(v_{0})k_{u}f_{1}(v_{1})=\exp(v_{2})k_{l}^{-1}f_{2}(0)=\exp(v_{2})k_{l}^{-1}\exp(-v_{2})f_{2}(v_{2}).$

From here the proof of the remaining conditions for $f_{2}$ and $f(v_{3})$ are
symmetric to the approximation of $f_{1}$ and $f_{1}(0)$. ∎

######  Corollary 4.4.

For any $\mu_{1},\dots,\mu_{k}$ ergodic $A$-invariant measures on $X_{n}$ and
any $R>0$ there exists $x_{R}\in X_{n}$ such that for every $i=1,\dots,k$:

$\lim_{R\rightarrow\infty}\frac{1}{\operatorname{vol}(S_{Rw_{0}})}\left(v\mapsto\exp\left(v+iRw_{0}\right).x_{R}\right)_{*}m_{\mathbb{R}^{n-1}_{0}}\mid_{S_{Rw_{0}}}=\mu_{i}$
(4.3)

In addition, the set $\\{x_{R}:R>0\\}$ is pre-compact.

###### Proof.

Let $K$ be a compact set with $\mu_{i}(K)>0$ for each $i$, and for each
$i=1,\dots,k$ choose $y_{i}\in K$ to be a generic point for the $A$ action on
$\mu_{i}$. By ergodicity of $\mu_{i}$:

$\lim_{R\to\infty}\frac{1}{\operatorname{vol}(S_{Rw_{0}})}\int_{S_{Rw_{0}}}\chi_{\exp(v)y_{i}\in
K}dm_{\mathbb{R}^{n-1}_{0}}=\mu_{i}(K)>0.$

Denote

$\rho_{i}=\sup\\{\rho>0:\exp(v)y_{i}\notin K,\forall v\in Rw_{0}-S_{\rho
w_{0}}\\}.$

Since $\int_{Rw_{0}-S_{\rho w_{0}}}\chi_{\exp(v)y_{i}\in
K}dm_{\mathbb{R}^{n-1}_{0}}=0$, we get that
$\frac{\rho_{i}}{R}\xrightarrow{R\to\infty}0$. By the Definition of
$\rho_{i}$, there is $v_{i}\in Rw_{0}-S_{\rho_{i}w_{0}}$ such that
$\exp(v_{i})y_{i}\in K$. Since $\|v_{i}-R_{i}\|=O(\rho_{i})=o_{R}(R)$ we
deduce that

$\frac{1}{\operatorname{vol}(S_{v_{i}})}\left(v\mapsto\exp(v)y_{i}\right)_{*}m_{\mathbb{R}^{n-1}_{0}}\mid_{S_{v_{i}}}\xrightarrow{R\to\infty}\mu_{i}.$

The corollary follows now from iteratively applying Corollary 4.3 to glue the
boxed maps $(v\mapsto\exp(v)y_{i})|_{S_{v_{i}}}$ to one boxed map $f$, using
an increasing set of compact neighborhoods $K\subset K_{1}\subset
K_{2}\subset\dots\subset K_{k}$. ∎

As a corollary of Lemma 3.1 and Corollary 4.4 we can prove Theorem 1.1:

###### Proof of Theorem 1.1.

Let $\mu_{1},\mu_{2},\dots,\mu_{k}$ be ergodic $A$-invariant measures on
$X_{n}$. For any $R>0$, let $x_{R}\in X_{n}$ be the point guaranteed by
Corollary 4.4. Let $K$ the compact closure of $\\{x_{R}:R>0\\}$.

Lemma 3.1 guarantees a sequence $(t_{m})_{m=0}^{\infty}$ such that
$t_{m}\xrightarrow{m\to\infty}\infty$ such that for all $R>0,m\geq 0$ there is
$y_{R,m}\in X_{n}$ such that $d(x_{R},y_{R,m})<e^{-\alpha t_{m}},$ where
$\alpha=\frac{\lfloor n/2\rfloor}{4n(n^{2}-1)}>0$, and $y_{R,m}$ is stabilized
by a subgroup $\exp(\Lambda_{R,m})\subseteq A$ where
$\Lambda_{R,m}\subseteq\mathbb{R}^{n-1}_{0}$ is generated by vectors
$(v^{(i)}_{R,m})_{i=1}^{n-1}$ such that
$\left|\left(v_{R,m}^{(i)}\right)_{j}-(1-\delta_{ij})t_{m}\right|=O(1)$.
Choose $R_{m}=\frac{\alpha t_{m}}{k(n-1)}-\sqrt{t_{m}}$ so that all $v\in
S_{kR_{m}w_{0}}$ we have $d(\exp(v)x_{R_{m}},\exp(v)y_{R_{m},m})=o_{m}(1)$.
This implies that

$\lim_{m\rightarrow\infty}\frac{1}{\operatorname{vol}(S_{R_{m}w_{0}})}\left(v\mapsto\exp\left(v+iR_{m}w_{0}\right).y_{R_{m},m}\right)_{*}m_{\mathbb{R}^{n-1}_{0}}\mid_{S_{R_{m}w_{0}}}=\mu_{i}.$
(4.4)

The choice of $R_{m}$ guarantees that $S_{Rw_{0}}$ injects into
$\mathbb{R}^{n-1}_{0}/\Lambda_{R_{m},m}$ and hence Eq. (4.4) implies that
$\lim_{m\to\infty}\mu_{Ay_{R_{m},m}}$ contains $\mu_{i}$ as an ergodic
component. ∎

##  5 Entire Mass Approximations

First, we prove Corollary 1.4 of Theorem 1.1 and [17, Theorem 1.1].

###### Proof of Corollary 1.4.

Let $\mu_{1},\dots,\mu_{k}\in\mathcal{M}(X_{n})^{A}_{e}$. By [17, Theorem 1.1]
we can find $(\nu_{m})_{m}\subset\mathcal{M}(X_{n})^{A}_{c}$ such that
$\nu_{m}\rightarrow 0$ weakly. By Remark 1.2 of Theorem 1.1, for any $m$ there
exists $\mu^{(m)}\in\overline{\mathcal{M}}(X_{n})^{A}_{c}$ such that
$\nu_{m},\mu_{1},\dots,\mu_{k}$ all appear in $\mu^{(m)}$’s ergodic
decomposition with coefficients bounded below by $\Theta(k^{2-n})$. Any weak
limit $\mu^{(\infty)}$ of $(\mu^{(m)})_{m}$ will belong to
$\overline{\mathcal{M}}(X_{n})^{A}_{c}$ and satisfy, since $\nu_{m}\rightarrow
0$, that at least $\Theta(k^{2-n})$ of the mass of $\mu^{(m)}$ escapes, namely
$\mu^{(\infty)}(X_{n})\leq 1-\Theta(k^{2-n})$. On the other hand, for
$i=1,\dots,k$, $\mu_{i}$ will still appear in the ergodic decomposition of
$\mu^{(\infty)}$ with coefficient bounded below by $\Theta(k^{2-n})$ as
desired. ∎

The remainder of this section will be dedicated to the proof of Theorem 1.5.
The proof of this theorem will follow the same lines as the proof of Theorem
1.1: We will first construct a special number field and prove some result on
its units. Then we will connect these results to Hecke operators and use the
properties of Hecke operators to complete the proof. In contrast to the method
in the previous section, here we will fix $p$ to be the minimal prime which is
congruent to $1$ mod $2n$.

###  5.1 Distribution of Hecke neighbors of points close to the cusp

The following corollary analyzes the distribution properties of Hecke
operators where we begin with a lattice high in the cusp. For that, we will
use a result that follows from Theorem 3.5 but appeared first at [6, Théorème
1.2].

######  Corollary 5.1.

Fix $x\in X_{n}$, and a prime number $p$. For $k_{1}\leq k_{2}\leq...\leq
k_{n},a\in\operatorname{SL}_{n}(\mathbb{R})$ as in Definition 3.4 we have:

$T_{a}^{\rm M}(x)\xrightarrow{k_{n}-k_{1}\to\infty}m_{x_{n}}.$ (5.1)

######  Remark 5.2.

The convergence is uniform on compact sets. This follows from the fact that
$T_{a}(gx)=gT_{a}(x)$ for all $g\in\operatorname{SL}_{n}(\mathbb{R})$, and the
result for the particular case $x=\mathbb{Z}^{n}\in X_{n}$.

Recall the definition of Minkowski’s successive minima $\lambda_{i}(x)$ for
$x\in X_{n},i=1,\dots,n$ from [3, Chapter VIII]. They satisfy that a closed
set $U\subseteq X_{n}$ is compact if and only if $\inf_{x\in
U}\lambda_{1}(x)>0$ if and only if $\sup_{x\in U}\lambda_{n}(x)<\infty$.

######  Theorem 5.3.

Fix a prime number $p$. Denote by
$a_{p}=\operatorname{diag}(p^{-(n-1)/n},p^{1/n},p^{1/n},\dots,p^{1/n})$. For
every point $x\in X_{n}$ and $x^{\prime}\in T_{a_{p}}(x)$ we have

$\displaystyle p^{1/n}\lambda_{1}(x)\geq\lambda_{1}(x^{\prime})$ (5.2)

In addition, for every $\delta>0$ there exists a compact set $C(\delta)$ such
that the following holds. Let $(x_{i})_{i=1}^{\infty}$ be a sequence of
lattices and let $p_{i}\xrightarrow{i\to\infty}\infty$ be an prime sequence.
Assume that

$\displaystyle p_{i}^{\frac{1}{n}}\lambda_{1}(x_{i})\geq 1.$ (5.3)

Then

$\displaystyle\liminf_{i\to\infty}T^{\rm
M}_{a_{p_{i}}}(x_{i})(C(\delta))>1-\delta.$ (5.4)

######  Remark 5.4.

Although we state no-escape-of-mass in Eq. (5.4), one can prove an
equidistribution result: If we replace Eq. (5.3) by

$\displaystyle p_{i}^{\frac{1-\varepsilon}{n}}\lambda_{1}(x_{i})\geq 1.$ (5.5)

for some $\varepsilon>0$ we get that

$\displaystyle T^{\rm M}_{a_{p_{i}}}(x_{i})\xrightarrow{i\to\infty}m_{X_{n}}.$

This result could simplify the proof of Theorem 1.5, but its proof is too
complicated and can be avoided. The proof uses the fact that if
$k=k^{\prime}+s$ then $T_{a^{k}}^{\rm M}$ is a large component of $T^{\rm
M}_{a^{k^{\prime}}}\circ T^{\rm M}_{a^{s}}$, and a $p$-adic interpretation of
Hecke operators.

######  Remark 5.5.

With Eq. (5.5) instead of (5.5), Theorem 5.3 could be extended for The hecke
operators $T_{a_{p_{i}}^{k_{i}}}$, with $p_{i}^{k_{i}}$ in the place of
$p_{i}$. For different Hecke operators there are other thresholds, stated in
terms of different Minkowski successive minima. The equidistribution result
holds as well.

###### Proof of Theorem 5.3.

The bound on $\lambda_{1}$ of every Hecke neighbor follows from the definition
of Hecke operators. Since $x\subset p^{-1/n}x^{\prime}$ we get
$\lambda_{1}(x^{\prime})\leq p^{1/n}\lambda_{1}(x)$.

Let $x\in X_{n}$ be a lattice with $\lambda_{1}(x)\geq p^{-\frac{1}{n}}$ and
$x^{\prime}$ be a random point in $T_{a^{k}}(x)$. We show that for every
$\delta>0$ there is a compact set $C\subset X_{n}$ depending only on $\delta$
such that $\mathbb{P}(x^{\prime}\in C)\geq 1-\delta$ provided that $p$ is
sufficiently large as a function of $\delta$.

Let $C_{r}=\\{y\in X_{n}:\lambda_{1}(y)\geq r\\}$ for some $1>r>0$ which will
be chosen later as a function of $\delta$. To bound
$\mathbb{P}(x^{\prime}\notin C_{r})$, we will bound:

$\displaystyle\mathbb{E}(\\#(x^{\prime}\cap
B(r)\setminus\\{0\\}))\geq\mathbb{P}(x^{\prime}\notin C_{r}).$

Here $B(r)$ is the radius $r$ ball in $\mathbb{R}^{n}$. Since
$p^{-(n-1)/n}x^{\prime}\subseteq x$, we have:

$\displaystyle\mathbb{E}(\\#(x^{\prime}\cap B(r)\setminus\\{0\\}))=\sum_{v\in
B(r)\cap p^{-(n-1)/n}x\setminus\\{0\\}}\mathbb{P}(v\in x^{\prime}).$ (5.6)

To analyze the probability $\mathbb{P}(p^{-(n-1)/n}v\in x^{\prime})$ note that
$px\subseteq p^{(n-1)/n}x^{\prime}$ and satisfies
$p^{(n-1)/n}x/px=\mathbb{Z}/p$. . In particular, if $v\in px$ then
$\mathbb{P}(v\in x)p^{(n-1)/n}=1$. Otherwise, if $v\in(x\setminus px)\cap
p^{(n-1)/n}x^{\prime}$ then $v$ determines $x^{\prime}$ by the formuala
$p^{(n-1)/n}x^{\prime}=px+w\mathbb{Z}$, and hence

$\mathbb{P}(v\in
x^{\prime})=\frac{1}{\\#T_{a_{p}}(x)}=\frac{p-1}{p^{n}-1}<\frac{1}{p^{n-1}}$

. Using these estimates we get:

$\displaystyle\sum_{v\in B(r)\cap
p^{-(n-1)/n}x\setminus\\{0\\}}\mathbb{P}(v\in x^{\prime})\leq\sum_{v\in
B(p^{(n-1)/n}r)\cap x\setminus px}\frac{1}{p^{n-1}}+\sum_{v\in
B(p^{(n-1)/n}r)\cap px\setminus\\{0\\}}1.$ (5.7)

The assumption on $\lambda_{1}(x)$ implies that the second term in the right
hand side of Eq. (5.7) vanishes. To bound the first term, we need to analyze
$\\#B(p^{(n-1)/n}r)\cap x$. The following claim is a different wording of [15,
Lemma 3.5].

######  Claim 5.6.

For every lattice $x$, $R>0$ we have that if $R>\lambda_{1}(x)$ we have

$\\#\left(B(R)\cap x\setminus
0\right)\asymp\max_{i=1}^{n}\frac{R^{i}}{\lambda_{1}(x)\cdots\lambda_{i}(x)}$

By Minkowski’s theorem ([3, Chapter VIII, Theorem V]),
$\lambda_{1}(x)\cdots\lambda_{n}(x)\asymp\text{cov}(x)$. In particular,
$\lambda_{n}(x)\ll p^{(n-1)/n}$. Thus,

$\displaystyle\\#B(p^{(n-1)/n}r)\cap x$
$\displaystyle\leq\max_{i=1}^{n}\frac{\left(p^{(n-1)/n}r\right)^{i}}{\lambda_{1}(x)\cdots\lambda_{i}(x)}=\prod_{i=1}^{n}\max\left(\frac{p^{(n-1)/n}r}{\lambda_{i}(x)},1\right)$
(5.8)
$\displaystyle\asymp\prod_{i=1}^{n}\max\left(p^{(n-1)/n}r,\lambda_{i}(x)\right)\ll
p^{(n-1)/n}r\cdot(p^{(n-1)/n})^{n-1}=rp^{n-1},$

provided that $r\frac{1}{p^{(n-1)/n}}$.

Altogether Eqs. (5.6), (5.7) and (5.8) implies that

$\displaystyle\mathbb{P}(x^{\prime}\notin C_{r})\ll r.$

For every $\delta>0$ choose $r(\delta)>0$ sufficiently small so that we have
$\mathbb{P}(x^{\prime}\in C_{r(\delta)})>1-\delta$ for all
$p\geq\delta^{-n/(n-1)}$. ∎

######  Definition 5.7 (Shapira’s orbit).

Let $0<\eta<\frac{1}{2n}$ fixed and $M>0$ be a size parameter. Let $0\leq
a_{1}<\dots<a_{n}\leq M$ be integers with $a_{i+1}-a_{i}\geq\eta M$, say,
$a_{i}=\lfloor iM/n\rfloor$. Let $P(z)=(z-a_{1})(z-a_{2})\cdots(z-a_{n})-1$
and $K=\mathbb{Q}[\alpha]$ where $P(\alpha)=0$. Let $x_{\mathbb{Z}[\alpha]}$
as in Definition 2.4.

The theorem we state is an accumulation of results and computations given in
Shapira [17].

######  Definition 5.8.

For every $\pi\in S_{n}$ (the permutation group), denote by
$F_{\pi}\subseteq\mathbb{R}_{0}^{n-1}$ the set of vectors
$F_{\pi}=\\{v\in\mathbb{R}^{n-1}_{0}:v_{\pi(i)}\geq
v_{\pi(i+1)}-1:i=1,\dots,n\\}$. Here we use cyclic index notations and
$\pi(n+1)=\pi(1)$. The set $F_{\pi}$ is compact. Note that $F_{\pi}$ is well
defined for $\pi\in S_{n}/C_{n}$ where $C_{n}$ is the cyclic group of
rotations.

We recall a result by Shapira.

######  Theorem 5.9 (Shapira [17]).

There is a bound $M_{0}(\eta)$ such that for all $M>M_{0}(\eta)$ the following
occurs. Let $x_{\mathbb{Z}[\alpha]}$ as constracted above. It is stabilized by
$\exp\Lambda$ where $\Lambda$ is generated by $v_{1},...,v_{n-1},v_{n}$
satisfying $v_{1}+...+v_{n}=0$ and

$\left|\left(v^{(i)}\right)_{j}-(2-2n\delta_{ij})\log M\right|=O(1).$

Moreover there is a finite collection of points
$P_{0}\subseteq\mathbb{R}^{n-1}_{0}/\Lambda$ and a map $\pi:P_{0}\to
S_{n}/C_{n}$ such

$\frac{\operatorname{vol}\left(\bigsqcup_{p\in
P_{0}}p+(1-o_{M}(1))\log(M)F_{\pi(p)}\right)}{\operatorname{vol}(\mathbb{R}^{n-1}_{0}/\Lambda)}=1-o_{M}(1),$

and for all $p\in P_{0},v\in(1-o_{M}(1))\log(M)F_{\pi(p)}$ one has

$d_{X_{n}}(\exp(p+v)x_{\mathbb{Z}[\alpha]},\exp(v)\mathbb{Z}^{n})=o_{M}(1).$

Denote by $\operatorname{min-co}:\mathbb{R}^{n-1}_{0}\to(-\infty,0]$ the
minimum of the coordinates function, and note that
$\min_{F_{\pi}}(\operatorname{min-co})=-\frac{d-1}{2}$. This function is
important since

$\displaystyle\lambda_{1}(\exp(v)\mathbb{Z}^{n})=\exp(\operatorname{min-
co}(v))\quad\text{for all}\quad v\in\mathbb{R}^{n-1}_{0}.$ (5.9)

Let $m_{F_{\pi}}$ be the uniform probability measure on $F_{\pi}$ for all
$\pi\in S_{n}$. Note that $(\operatorname{min-co})_{*}m_{F_{\pi}}$ is
independent of $\pi\in S_{n}$. We use the following corollary of Shapira’s
result, which follows Eq. (5.9):

######  Corollary 5.10.

In the setting of Theorem 5.9, we have

$\displaystyle d\left(\int\delta_{\frac{1}{\log
M}\log\lambda_{1}(x^{\prime})}d\mu_{\mathbb{Z}[\alpha]}(x^{\prime}),(\operatorname{min-
co})_{*}m_{F_{1}}\right)=o_{M}(1).$

###### Proof of Theorem 1.5.

Let $c\in(0,1]$ and let $M>0$ be large enough. From now on, every measure in
our construction will depend on $M$. Since $m_{F_{1}}$ is absolutely
continuous with respect to Lebesgue, there is $\eta>0$ such that
$m_{F_{1}}(\operatorname{min-co}^{-1}[-\eta,0])=c$. Let $p$ be the first prime
number $p>M^{n\eta}$. By the Prime Number Theorem, $p=M^{\eta n+O(1/\log M)}$.
Let $q$ be a the first prime number with $q>\log M$. Again by the Prime Number
Theorem , $q=M^{O_{M}(\log\log M/\log M)}$. Altogether,

$\displaystyle pq=M^{\eta n+O(\log\log M/\log M)}.$ (5.10)

Let $a_{p},a_{q}$ as in Theorem 5.3 contracted for $p,q$ respectively. Let
$x_{\mathbb{Z}[\alpha]}=x_{\mathbb{Z}[\alpha]}(M)$ be as in Definition 5.7.
The measure

$\nu_{0}=T^{\rm M}_{a_{q}}(T^{\rm M}_{a_{p}}(\mu_{Ax_{\mathbb{Z}[\alpha]}})),$

is supported on several compact orbits which we will now describe. Consider
the set

$B=T_{a_{q}}(T_{a_{p}}(x_{\mathbb{Z}[\alpha]}))\subset X_{n}.$

Define a partition

$B=B_{1}\sqcup B_{2}\sqcup\dots\sqcup B_{r}$ (5.11)

by saying that two points $y_{0},y_{1}\in B$ are in the same set $B_{i}$ if
$Ay_{0}=Ay_{1}$. Note that $B$ is invariant under the
$\operatorname{stab}_{A}(x_{\mathbb{Z}[\alpha]})$, and hence so are $B_{i}$.
For every $x=ax_{\mathbb{Z}[\alpha]}\in Ax_{\mathbb{Z}[\alpha]}$ define
$T_{B_{i}}(x)=aB_{i}$. The element $a$ is well defined up to
$\operatorname{stab}_{A}(x_{\mathbb{Z}[\alpha]})$, and since $B_{i}$ is
invariant to this ambiguity, $T_{B_{i}}(x)$ is well defined.

Define $T_{B_{i}}^{\rm M}(x)=\frac{1}{\\#T_{B_{i}}(x)}\sum_{y\in
T_{B_{i}}(x)}\delta_{y}$. We deduce that

$\nu_{0}=\sum_{i=1}^{r}\beta_{i}T_{B_{i}}^{\rm
M}(\mu_{x_{\mathbb{Z}[\alpha]}}),$

for some $\beta_{i}>0$ with $\sum_{i=1}^{r}\beta_{i}=1$ (See Remark 5.12 for
further discussion on $(\beta_{i})_{i=1}^{r}$). We will now analyze $\nu_{0}$
and $T_{B_{i}}^{\rm M}(\mu_{x_{\mathbb{Z}[\alpha]}})$.

Analysis of Hecke-like operators on different parts of the compact orbit:
Distinguish two parts of $Ax_{\mathbb{Z}[\alpha]}$:

$\displaystyle(Ax_{\mathbb{Z}[\alpha]})^{-}=\\{x^{\prime}\in
Ax_{\mathbb{Z}[\alpha]}:\lambda_{1}(x^{\prime})\leq\exp((-1-\varepsilon)\eta\log
M)\\}$ (5.12) $\displaystyle(Ax_{\mathbb{Z}[\alpha]})^{+}=\\{x^{\prime}\in
Ax_{\mathbb{Z}[\alpha]}:\lambda_{1}(x^{\prime})\geq\exp((-1+\varepsilon)\eta\log
M)\\}.$ (5.13)

Then by Corollary 5.10 we get that
$\mu_{\mathbb{Z}[\alpha]}((Ax_{\mathbb{Z}[\alpha]})^{-})=1-c+O(\varepsilon)+o_{M}(1)$
and
$\mu_{\mathbb{Z}[\alpha]}((Ax_{\mathbb{Z}[\alpha]})^{+})=c+O(\varepsilon)+o_{M}(1)$.
Consequently, we get that

$\displaystyle\mu_{\mathbb{Z}[\alpha]}(Ax_{\mathbb{Z}[\alpha]}\setminus((Ax_{\mathbb{Z}[\alpha]})^{+}\cup(Ax_{\mathbb{Z}[\alpha]})^{-}))=O(\varepsilon)+o_{M}(1).$
(5.14)

Analysis on $(Ax_{\mathbb{Z}[\alpha]})^{-}$: We will now analyze the escape of
mass for the measures $T_{B_{i}}^{\rm M}(\mu_{x_{\mathbb{Z}[\alpha]}})$. For
all $x^{\prime}\in(Ax_{\mathbb{Z}[\alpha]})^{-}$, Theorem 5.3 implies that
$(pq)^{1/n}\lambda_{1}(x^{\prime})\geq\lambda_{1}(y^{\prime})$ for all
$y^{\prime}\in T_{a_{q}}(T_{a_{p}}(x^{\prime}))$. By Eqs. (5.12) and (5.10),
we get an upper bound of

$\displaystyle\lambda_{1}(y^{\prime})\leq M^{-\varepsilon+O(\log\log M/\log
M)}.$

Hence if $\varepsilon$ is sufficiently big as a function of $M$, say,
$\varepsilon=\frac{1}{\log\log M}$, we get that for all $i=1,\dots,r$,

$\displaystyle\int_{(Ax_{\mathbb{Z}[\alpha]})^{-}}\int\chi_{\mathcal{K}_{\varepsilon}}d(T^{\rm
M}_{B_{i}}(x^{\prime}))d\mu_{\mathbb{Z}[\alpha]}(x^{\prime})=\int_{(Ax_{\mathbb{Z}[\alpha]})^{-}}\int\chi_{\mathcal{K}_{\varepsilon}}dT^{\rm
M}_{a_{q}}(T^{\rm
M}_{a_{p}}(x^{\prime}))d\mu_{\mathbb{Z}[\alpha]}(x^{\prime})=0.$ (5.15)

Here $\mathcal{K}_{\varepsilon}$ is defined as in Definition 2.2.

Analysis on $(Ax_{\mathbb{Z}[\alpha]})^{+}$: Let
$x^{\prime}\in(Ax_{\mathbb{Z}[\alpha]})^{+}$. By Eqs. (5.13) and the
definition of $p$ we get that

$\lambda_{1}(x^{\prime})\geq p^{-(1-\varepsilon)/n}.$

By Theorem 5.3, we get that $T^{\rm M}_{a_{p}}(x^{\prime})$ has no escape of
mass as $M\to\infty$. By Corollary 5.1, we get that

$\displaystyle T^{\rm M}_{a_{q}}(T^{\rm
M}_{a_{p}}(x^{\prime}))\xrightarrow{M\to\infty}m_{X_{n}}.$ (5.16)

######  Definition 5.11 (Spaces of measures and ergodic decomposition).

Let $X_{n}^{*}=X_{n}\sqcup\\{*\\}$ denote the one point compactification of
$X_{n}$. This is a compact metric space. Hence, by Banach Alaoglu’s Theorem,
the space of probability measures on $X_{n}^{*}$, namely
$\mathcal{M}(X_{n}^{*})$, is again a compact metric space. The space of
$A$-invariant probability measures, denoted $\mathcal{M}(X_{n}^{*})^{A}$ is a
closed subset, hence again a compact metric space. Again, by Banach Alaoglu,
the space of of probability measures on $\mathcal{M}(X_{n}^{*})^{A}$, namely
$\mathcal{M}(\mathcal{M}(X_{n}^{*})^{A})$, is again a compact metric space. We
interpret $m_{X_{n}},\nu_{0},T_{B_{i}}^{\rm M}(\mu_{Ax_{\mathbb{Z}[\alpha]}})$
as points in $\mathcal{M}(X_{n}^{*})^{A}$. Let
$\omega=\sum_{i=1}^{r}\beta_{i}\delta_{T_{B_{i}}^{\rm
M}(\mu_{Ax_{\mathbb{Z}[\alpha]}})}\in\mathcal{M}(\mathcal{M}(X_{n}^{*})^{A})$.
This is the ergodic decomposition of $\nu_{0}$, and it satisfies $\int
d\omega=\nu_{0}$.

In these terms we will analyze the limiting behavior we got earlier. Denote

$\lim_{M\to\infty}\nu_{0}=\vec{\nu}_{0},\qquad\lim_{M\to\infty}\omega=\vec{\omega}.$

Eq. (5.15) implies that $\omega$ is supported on measures giving mass at most
$c+o_{M}(1)$ to $\mathcal{K}_{\varepsilon}$. Taking $M$ to infinite we get
that $\vec{\omega}$ is supported on measures giving mass at most $c$ to
$X_{n}$, that is, giving mass at least $1-c$ to the additional point $*$. In
formula,

$\displaystyle\vec{\omega}\left(\left\\{\mu\in\mathcal{M}(X_{n}^{*}):\mu(\\{*\\})\geq
1-c\right\\}\right)=1.$ (5.17)

Hence

$\displaystyle(1-c)\delta_{*}\leq\int
d\vec{\omega}=\lim_{M\to\infty}\int\omega=\vec{\nu}_{0}.$ (5.18)

On the other hand, Eq (5.16) implies that the part of $\nu_{0}$ coming from
the application of $T_{a_{q}}^{\rm M}\circ T_{a_{p}}^{\rm M}$ on
$(Ax_{\mathbb{Z}[\alpha]})^{+}$ equidistributes as $M\to\infty$. This implies
that $\vec{\nu}_{0}\geq cm_{X_{n}}$. Together with (5.18) we deduce that

$\displaystyle\vec{\nu}_{0}=cm_{X_{n}}+(1-c)\delta_{*}.$ (5.19)

This is the ergodic decomposition of $\vec{\nu}_{0}$, which implies that
$\vec{\omega}$, whose integral is $\vec{\nu}_{0}$ is supported on measures of
the form $c^{\prime}m_{X_{n}}+(1-c^{\prime})\delta_{*}$ for some $c^{\prime}$.
By Eq. (5.17) we deduce that almost surly $c^{\prime}\geq c$. Since the
integral $\int d\omega=cm_{X_{n}}+(1-c)\delta_{*}$, we deduce that
$\vec{\omega}=\delta_{cm_{X_{n}}+(1-c)\delta_{*}}$. In particulat,

$\min_{i=1}^{r}d_{\mathcal{M}(X^{*}_{n})}(T_{B_{i}}^{\rm
M}(\mu_{Ax_{\mathbb{Z}[\alpha]}}),cm_{X_{n}}+(1-c)\delta_{*})=\min_{\mu\in\operatorname{supp}(\omega)}d_{\mathcal{M}(X^{*}_{n})}(\mu,\operatorname{supp}(\vec{\omega}))\xrightarrow{M\to\infty}0,$

as desired. ∎

######  Remark 5.12.

The constants $\beta_{i}$ can be computed, $\beta_{i}=\frac{\\#B_{i}}{\\#B}$.
However, this computation needs a disjointness result, $T_{a_{q}}(y)\cap
T_{a_{q}}(y^{\prime})=\emptyset$ for all $y\neq y^{\prime}\in
T_{p}(x_{\mathbb{Z}[\alpha]})$. This is not needed in the proof.

######  Remark 5.13 (Variants on the proof).

Using one Hecke operator: In the proof above we use the composition of the
Hecke operators $T_{a_{q}}^{\rm M}\circ T_{a_{p}}^{\rm M}$. Had we proved the
stronger version of Theorem 5.3 as suggested in Remark 5.4, we could use
$T_{a_{p}}^{\rm M}$ directly.

Obtaining a lower bound on the volume: We sketch a way to perturb the above
proof to obtain a lower bound on the volume of the orbits which is polynomial
in the discriminant. This lower bound will come from showing that many of the
$B_{i}$’s defined in Eq. 5.11 can be made of polynomial size in the
discriminant. The $B_{i}$’s correspond to orbits of action of
$\mathcal{O}_{K}^{\times}$ on $\mathbb{P}((\mathbb{Z}/p\mathbb{Z})^{n})$. To
see that the orbits can be made large, one can use the number field Artin’s
conjecture Artin’s Conjecture, which implies that typically one of the
$B_{i}$’s is very large. However, we can sketch a rigorous argument ensuring
that most of the $B_{i}$’s are large. For that, let $r$ be the smallest prime
such that $r=1\mod 2n$ and choose the $a_{i}$’s in Definition 5.7 such that
$a_{i}\mod r$ are the roots of $x^{n}+1\mod r$. Moreover, choose the $a_{i}$’s
such that $a_{1}\cdot a_{2}\cdots a_{n}\not\equiv(-1)^{n}\mod r^{2}$. This
choice of $a_{i}$ ensures that corresponding the number field $K$ is totally
ramified at $r$. This can be used to show that the units in
$\mathcal{O}_{K}^{\times}$ generate a group of size $\Theta(r^{nm})$ inside
$(\mathcal{O}_{K}/r^{m}\mathcal{O}_{K})^{\times}$ for all $m$. Now we replace
the Hecke operators $T_{a_{p}}$ in the proof by $T_{a_{r}^{m}}$. Then, the
lower bound on the orbit sizes implies the lower bound on the size of many
$B_{i}$’s and therefore also the bound on the volumes.

## References

  * [1] Menny Aka and Uri Shapira.  On the evolution of continued fractions in a fixed quadratic field.  J. Anal. Math., 134(1):335–397, 2018.
  * [2] Yves Benoist and Hee Oh.  Equidistribution of rational matrices in their conjugacy classes.  GAFA Geometric And Functional Analysis, 17(1):1–32, 2007.
  * [3] J. W. S. Cassels.  An introduction to the geometry of numbers.  Springer Science & Business Media, 2012.
  * [4] JWS Cassels.  The product of n inhomogeneous linear forms in n variables.  Journal of the London Mathematical Society, 1(4):485–492, 1952.
  * [5] Laurent Clozel, Hee Oh, and Emmanuel Ullmo.  Hecke operators and equidistribution of Hecke points.  Invent. Math., 144(2):327–351, 2001.
  * [6] Laurent Clozel and Emmanuel Ullmo.  Équidistribution des points de hecke. contributions to automorphic forms, geometry, and number theory, 193–254, 2004.
  * [7] Keith Conrad.  Hensel’s lemma.  Unpublished notes, 2015.
  * [8] Ofir David and Uri Shapira.  Dirichlet shapes of unit lattices and escape of mass.  International Mathematics Research Notices, 2018(9):2810–2843, 2018.
  * [9] W. Duke.  Hyperbolic distribution problems and half-integral weight maass forms.  Inventiones mathematicae, 92(1):73–90, 1988.
  * [10] Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, and Akshay Venkatesh.  Distribution of periodic torus orbits on homogeneous spaces.  Duke Mathematical Journal, 148(1):119 – 174, 2009.
  * [11] Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, and Akshay Venkatesh.  Distribution of periodic torus orbits and duke’s theorem for cubic fields.  Annals of mathematics, pages 815–885, 2011.
  * [12] Alex Eskin and Hee Oh.  Ergodic theoretic proof of equidistribution of Hecke points.  Ergodic Theory Dynam. Systems, 26(1):163–167, 2006.
  * [13] Wee Teck Gan and Hee Oh.  Equidistribution of integer points on a family of homogeneous varieties: a problem of Linnik.  Compositio Math., 136(3):323–352, 2003.
  * [14] M. Kline.  Calculus: an intuitive and physical approach.  Courier Corporation, 1998.
  * [15] O. N. Solan L. Liao, R. Shi and N. Tamam.  Hausdorff dimension of weighted singular vectors in $\mathbb{R}^{2}$.  JEMS, 22(3):833–875, 2019.
  * [16] C. McMullen.  Minkowski’s conjecture, well-rounded lattices and topological dimension.  Journal of the American Mathematical Society, 18(3):711–734, 2005.
  * [17] Uri Shapira.  Full Escape of Mass for the Diagonal Group.  International Mathematics Research Notices, 2017(15):4704–4731, 07 2016.
  * [18] Uri Shapira and Cheng Zheng.  Translates of s-arithmetic orbits and applications.  arXiv preprint arXiv:2107.05017, 2021.
  * [19] G. Tomanov and B. Weiss.  Closed orbits for actions of maximal tori on homogeneous spaces.  Duke Mathematical Journal, 119(2):367–392, 2003.