# Equidistribution of primitive lattices in $\mathbb{R}^{n}$

Tal Horesh IST Austria<EMAIL_ADDRESS>Yakov Karasik Technion, Israel,
<EMAIL_ADDRESS>
###### ?abstractname?

We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their
covolume tends to infinity, with respect to certain parameters of such
lattices. These parameters include, for example, the subsapce that a lattice
spans, namely its projection to the Grassmannian; its homothety class; and its
equivalence class modulo rescaling and rotation, often referred to as a
_shape_. We add to a prior work of Schmidt by allowing sets in the spaces of
parameters that are general enough to conclude joint equidistribution of these
parameters. In addition to the primitive $d$-lattices $\Lambda$ themselves, we
also consider their orthogonal complements in $\mathbb{Z}^{n}$,
$\Lambda^{\perp}$, and show that the equidistribution occurs jointly for
$\Lambda$ and $\Lambda^{\perp}$. Finally, our asymptotic formulas for the
number of primitive lattices include an explicit error term.

######  ?contentsname?

  1. 1 Introduction
  2. 2 Spaces of lattices
    1. 2.1 Spaces of homothety classes of $d$-lattices.
    2. 2.2 Factor lattices and the space of pairs.
    3. 2.3 Relation between the spaces of lattices and their measures
  3. 3 Counting lattices: Main theorem
  4. 4 Refined Iwasawa components in $\operatorname{SL}_{n}\left(\mathbb{R}\right)$.
    1. 4.1 Refining the Iwasawa decomposition of $\operatorname{SL}_{n}\left(\mathbb{R}\right)$
    2. 4.2 Parameterizations of the RI components
    3. 4.3 RI decomposition of the Haar measure on $\operatorname{SL}_{n}\left(\mathbb{R}\right)$
    4. 4.4 Interpretation of the RI components of $g\in\operatorname{SL}_{n}\left(\mathbb{R}\right)$
  5. 5 Sets of representatives for the spaces in Theorem 3.2
  6. 6 Correspondence between integral matrices and primitive lattices
  7. 7 Neglecting lattices up the cusp: reduction to counting in compact sets
  8. 8 Almost a proof of the main theorem
  9. 9 Proof of technical counting proposition
  10. A Dual lattices
  11. B Factor lattices

###  1 Introduction

The aim of this paper is to extend classical counting and equidistribution
results for _primitive vectors_ to their higher rank counterparts: _primitive
lattices_. A primitive vector is an $n$-tuple of integers
$(a_{1},\ldots,a_{n})$ with $\gcd(a_{1},\ldots,a_{n})=1$, and the the set of
primitive vectors in $\mathbb{R}^{n}$ is denoted by
$\mathbb{Z}_{\operatorname{prim}}^{n}$. We can associate to each vector $0\neq
v\in\mathbb{R}^{n}$ the discrete subgroup that it spans, $\mathbb{Z}v$;
following this logic, a rank $d$ ($1\leq d\leq n$) analog for a vector is a
_lattice of rank $d$_ in $\mathbb{R}^{n}$, namely

$\Lambda=\mathbb{Z}v_{1}\oplus\cdots\oplus\mathbb{Z}v_{d}$

where $v_{1},\ldots,v_{d}\in\mathbb{R}^{n}$ are linearly independent. We will
refer to it briefly as a _$d$ -lattice_. We say that a $d$-lattice $\Lambda$
is _integral_ if $\Lambda\subset\mathbb{Z}^{n}$, and _primitive_ if
$\Lambda=V\cap\mathbb{Z}^{n}$, where $V$ is a $d$-dimensional rational
subspace of $\mathbb{R}^{n}$. For example, a primitive $1$-lattice is simply
all the integral points on a rational line, or, equivalently, it is
$\mathbb{Z}v$ where $v$ is a primitive vector.

Questions about counting primitive vectors date back to the days of Gauss and
Dirichlet, e.g. with the _Gauss Class Number problem_. In the 20th century,
questions about equidistribution of integral vectors began to arise, with the
principal example being _Linnik-type problems_ [Lin68, EH99, Duk03, Duk07,
ELMV11, BO12, EMV13]. These questions and others generalize naturally to
primitive lattices, as we now turn to describe.

###### The Primitive Circle Problem.

The well known Gauss Circle Problem concerns the asymptotic number of integral
vectors up to (euclidean) norm $X>0$. The analogous question for primitive
vectors, namely the asymptotic amount of primitive vectors up to norm $X$, is
often referred to as the _primitive circle problem_ [Now88, ZC99, Wu02]. In
lattices, the role of a norm is played by the _covolume_ : the covolume of
$\Lambda$, denoted $\operatorname{covol}(\Lambda)$, is the volume of a
fundamental paralelopiped for $\Lambda$ in the linear space

$V_{\Lambda}:=\mbox{span}_{\mathbb{R}}(\Lambda).$

Thus, the _primitive circle problem for lattices_ is to estimate the
asymptotics of

$\\#\\{\text{primitive $d$-lattices in $\mathbb{R}^{n}$ of covolume up to
$X$}\\}$ (1.1)

as $X\to\infty$. Note that for $1$-lattices, the notions of norm and covolume
coincide: $\operatorname{covol}(\mathbb{Z}v)=\left\|v\right\|$, hence when
$d=1$ the above recovers the “original” primitive circle problem. Schmidt
[Sch68] showed that the amount in (1.1) equals

$c_{d,n}X^{n}+O(X^{n-\max\\{\frac{1}{d},\frac{1}{n-d}\\}}),$

where

$c_{d,n}=\frac{1}{n}{n\choose
d}\cdot\frac{\prod_{i=n-d-1}^{n}\operatorname{Leb(B_{i})}}{\prod_{j=1}^{d}\operatorname{Leb(B_{j})}}\cdot\frac{\prod_{i=2}^{d}\zeta\left(i\right)}{\prod_{j=n-d+1}^{n}\zeta\left(j\right)},$

Leb is the Lebesgue measure, and $B_{i}$ is the unit ball in $\mathbb{R}^{i}$.
We remark that the optimal exponent in the error term of the circle problem
(primitive or not) is established only in dimensions $n\geq 4$. As far as the
authors are aware, this ($d=1$, $n\geq 4$) is the only case where an optimal
error exponent is known for the lattices circle problem (primitive or not).

###### Linnik-type problems.

This is a unifying name for questions on the distribution of the projections
of integral vectors to the unit sphere, i.e. of $v/\left\|v\right\|$ when
$v\in\mathbb{Z}^{n}\text{ or $\mathbb{Z}_{\operatorname{prim}}^{n}$}$. Viewing
the unit sphere as the space of oriented lines in $\mathbb{R}^{n}$, the
analogous object when considering $d$-lattices would be the _Grassmannian_ of
oriented $d$-dimensional subspaces in $\mathbb{R}^{n}$, denoted
$\operatorname{Gr}(d,n)$ (and defined explicitly in Section 2). Accordingly,
we will view our lattices as carrying an orientation, which simplifies our
discussion on the technical level but has no effect on the results. In
particular, the two-to-one correspondence between primitive vectors and
primitive $1$-lattices (arising from the fact that $v$ and $-v$ span the same
lattice) becomes a one-to-one correspondence between primitive vectors and
oriented primitive $1$-lattices. The _average Linnik problem for primitive
lattices_ is to study the distribution of the (oriented) spaces $V_{\Lambda}$
in $\operatorname{Gr}(d,n)$ as $\operatorname{covol}(\Lambda)\leq X\to\infty$.
Note that $V_{\Lambda}$ are exactly the rational subspaces in
$\operatorname{Gr}(d,n)$.

###### Shapes of orthogonal lattices.

More recently, with the rise of dynamical approaches in number theory, another
type of equidistribution questions for primitive vectors arose. To a primitive
vector $v$ we associate the $(n-1)$-lattice $v^{\perp}\cap\mathbb{Z}^{n}$,
referred to as as the _orthogonal lattice_ of $v$, where $v^{\perp}$ is the
orthogonal hyperplane to $v$. Several recent papers (by Marklof [Mar10], Aka
Einsiedler and Shapira [AES16b, AES16a], Einsiedler, Mozes, Shah and Shapira
[EMSS16], Einsiedler Rï¿œhr and Wirth [ERW17]) studied the equidistribution of
_shapes_ of the orthogonal lattices to primitive vectors as their norm tends
to infinity, where the shape of a lattice is its similarity class modulo
rotation and homothety. The space of shapes of $d$-lattices is a double coset
space of $\operatorname{SL}_{d}(\mathbb{R})$, denoted $\mathcal{X}_{d}$ and
defined explicitly in Section 2, and the aforementioned papers show (among
other things) that the shapes of the orthogonal lattices to
$v\in\mathbb{Z}_{\operatorname{prim}}^{n}$ equidistribute in
$\mathcal{X}_{n-1}$ as $\left\|v\right\|\to\infty$ w.r.t. a uniform measure
arriving from the Haar measure on $\operatorname{SL}_{n-1}(\mathbb{R})$. In
the works of Einsiedler et al. they in fact show that the shapes of
$v^{\perp}\cap\mathbb{Z}^{n}$ equidistribute in $\mathcal{X}_{n-1}$ _jointly_
with the directions $v/\left\|v\right\|$ in $\mathbb{S}^{n-1}$.

Just like for (primitive) vectors, orthogonal lattices can be defined for
(primitive) lattices as well: for a primitive $d$-lattice $\Lambda$, we let:

$\Lambda^{\perp}:=V_{\Lambda}^{\perp}\cap\mathbb{Z}^{n},$

where $V_{\Lambda}^{\perp}$ is the orthogonal complement of $V_{\Lambda}$.
Note that $\Lambda^{\perp}$ is primitive by definition, and that it has rank
$n-d$. Also note that $\Lambda\mapsto\Lambda^{\perp}$ defines a bijection
between primitive lattices of ranks $d$ and $n-d$. This bijetion extends to a
bijection between oriented lattices, with a natural choice of orientation on
the orthogonal lattice (Definition 2.2).

One could then ask about the equidistribution of shapes of the orthogonal
lattices $\Lambda^{\perp}$ to primitive lattices $\Lambda$, where the one
dimensional case $\Lambda=\mathbb{Z}v$ recovers the question studied in the
aforementioned papers about the equidistribution of shapes of
$v^{\perp}\cap\mathbb{Z}^{n}$.

Equidistribution of a sequence in a finite-volume space can be deduced from
counting in “sufficiently general” subsets of this space. Indeed, we will
count in subsets that have _controlled boundary_ (Definition 3.1).

######  Theorem 1.1 (A consequence of Theorems 3.2 and 3.4).

Assume that $\Phi\subseteq\operatorname{Gr}(d,n)$ and
$\mathcal{E}\times\mathcal{F}\subseteq\mathcal{X}_{d}\times\mathcal{X}_{n-d}$
are BCS’s. Then, the number of primitive $d$-lattices $\Lambda$ with
$\operatorname{covol}(\Lambda)\leq X$, $V_{\Lambda}\in\Phi$ and
$(\operatorname{shape}\left(\Lambda\right),\operatorname{shape}\left(\Lambda^{\perp}\right))\in\mathcal{E}\times\mathcal{F}$
is

$\frac{\operatorname{vol}(\mathcal{E})\operatorname{vol}(\mathcal{F})\operatorname{vol}(\Phi)}{n\cdot\prod_{i=2}^{n}\zeta\left(i\right)}\Upsilon(d)\Upsilon(n-d)\cdot
X^{n}+O_{\epsilon}(X^{n-\frac{\tau_{n}n}{\beta}+\epsilon})$

for every $\epsilon>0$, where
$\tau_{n}=\left\lceil\left(n-1\right)/2\right\rceil/4n^{2}$,

$\Upsilon(d):=\frac{d!\prod_{i=2}^{d}\operatorname{Leb(B_{i})}}{\left[\operatorname{SO}_{d}\left(\mathbb{R}\right):Z\left(\operatorname{SO}_{d}\left(\mathbb{R}\right)\right)\right]}$

(in the denominator is the index of the center of
$\operatorname{SO}_{d}\left(\mathbb{R}\right)$, which is either $1$ or $2$),
and

$\beta=\begin{cases}2\left(n-d-1\right)\left(n^{2}-1\right)+n^{2}&\text{when
$\mathcal{E}$ is bounded}\\\
2\left(d-1\right)\left(n^{2}-1\right)+n^{2}&\text{when $\mathcal{F}$ is
bounded}\\\
2\left(\max\left\\{d,n-d\right\\}-1\right)\left(n^{2}-1\right)+n^{2}&\text{when
neither of $\mathcal{E},\mathcal{F}$ is bounded}\\\ 1&\text{when both
$\mathcal{E},\mathcal{F}$ are bounded}\end{cases}.$

In the above theorem, $\operatorname{vol}$ stands for a uniform Radon measure
(that is unique up to multiplication by a positive scalar and will be defined
in Section 2), so it implies the joint uniform distribution of the directions,
shapes, and shapes of the orthogonal lattices of primitive lattices as their
covolume tends to infinity. In particular, it is quite interesting to observe
that the shapes of $\Lambda$ and of $\Lambda^{\perp}$ are independent
parameters, meaning that there is no way to know the shape of
$\Lambda^{\perp}$ given the shape of $\Lambda$, even though the latter lattice
determines the first.

We remark that the equidistribution of shapes of primitive $d$-lattices was
shown in a non-quantitative form by Schmidt in [Sch98].

###### Counting $d$-lattices: Beyond shapes.

There exists a wide body of work on equidistribution problems in the space

$\mathcal{L}_{n}:=\operatorname{SL}_{n}(\mathbb{R})/\operatorname{SL}_{n}(\mathbb{Z}),$

which is the space of unimodular (i.e. with covolume one and positive
orientation) full lattices in $\mathbb{R}^{n}$, as well as in the space
$\mathcal{X}_{n}$ of their shapes. The restriction to covolume one is due to
the fact that the space $\mathcal{L}_{n}$ (and therefore $\mathcal{X}_{n}$)
has finite volume, while the space of all lattices,
$\operatorname{GL}_{n}(\mathbb{R})/\operatorname{GL}_{n}(\mathbb{Z})$, does
not. Comparing the spaces $\mathcal{L}_{n}$ and $\mathcal{X}_{n}$, the space
$\mathcal{L}_{n}$ naturally contains “more information” than
$\mathcal{X}_{n}$, which is obtained by modding $\mathcal{L}_{n}$ by
rotations. This brings up the question of whether one can define a space of
“unimodular $d$-lattices in $\mathbb{R}^{n}$” so as to consider the unimodular
$d$-lattices in $\mathbb{R}^{n}$ without modding out by rotations. In Section
2 we introduce two such spaces, which are homogeneous spaces of
$\operatorname{SL}_{n}(\mathbb{R})$, but do not carry an
$\operatorname{SL}_{n}(\mathbb{R})$-invariant measure. However, they do
support a natural measure which is invariant under a direct product of two
subgroups of $\operatorname{SL}_{n}(\mathbb{R})$. We will prove a stronger
statement than Theorem 1.1, namely Theorem 3.2, where we count primitive
$d$-lattices according to their projections to these more refined spaces, from
which we conclude Theorem 1.1, of counting primitive $d$-lattices according to
their projections to $\mathcal{X}_{d}$.

###### Organization of the paper.

The paper is organized as follows: In Section 2 we define all the spaces and
notions related to lattices that appear throughout this work. In Section 3 we
state Theorem 3.2, and explain how it implies Theorem 1.1; then the rest of
the paper is devoted to proving Theorem 3.2. In Section 4 we define
coordinates on $\operatorname{SL}_{n}(\mathbb{R})$ that are a refinement of
the Iwasawa coordinates, and study their properties. In Section 5 we identify
subsets of $\operatorname{SL}_{n}(\mathbb{R})$ that are sets of
representatives for all the quotient spaces that appear in Theorem 3.2. In
Section 6, we associate to each primitive $d$-lattice a unique element in
$\operatorname{SL}_{n}(\mathbb{Z})$, transferring the problem of counting
primitive $d$-lattices to the problem of counting points of
$\operatorname{SL}_{n}(\mathbb{Z})$ in a family of increasing sets in
$\operatorname{SL}_{n}(\mathbb{R})$. These sets are not compact, even though
the number of lattice points in each of the sets is finite. In Section 7, we
reduce to counting in a family of compact subsets, obtaining a “classical”
counting lattice point problem in $\operatorname{SL}_{n}(\mathbb{R})$. In
Section 8 we state the solution of this counting problem (Proposition 8.1),
and prove Theorem 3.2 based on it. In Section 9 we prove Proposition 8.1 using
a method developed by Gorodnik and Nevo in [GN12], for counting lattice points
in semi-simple Lie groups. In the appendix, we expand further on the notions
introduced in Section 2, as some of them are new, and others are not new but
possibly unfamiliar to the reader.

######  Acknowledgement.

This work was done when both authors were visiting IST Austria. The first
author was being supported by EPRSC grant EP/P026710/1, and the second author
had a great time there and is grateful for the hospitality. The appendix to
this paper is largely based on a mini course the first author had given at IST
in February 2020.

###  2 Spaces of lattices

We begin by defining explicitly the spaces $\mathcal{X}_{d}$ and
$\operatorname{Gr}(d,n)$ appearing in Theorem 1.1. An oriented subspace on
$\mathbb{R}^{n}$ is a subspace with a sign attached, and the Grassmannian
$\operatorname{Gr}(d,n)$ is the set of all $d$-dimensional oriented subspaces
of $\mathbb{R}^{n}$. It can also be defined as the following quotients

$\operatorname{Gr}(d,n)=\operatorname{SO}_{n}\left(\mathbb{R}\right)/\left\\{\left[\begin{smallmatrix}\operatorname{SO}_{d}\left(\mathbb{R}\right)&0_{d,n-d}\\\
0_{n-d,d}&\operatorname{SO}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]\right\\}=\operatorname{SL}_{n}\left(\mathbb{R}\right)/\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{R}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d,d}&\operatorname{SL}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right],$

where $P_{d}$ is the group of upper triangular matrices of determinant $1$
with positive diagonal entries. A coset of $\operatorname{SO}_{n}$ (or
$\operatorname{SL}_{n}$) represents an oriented $d$-dimensional subspace $V$
if the first $d$ columns of the matrices in this coset span $V$ with the
corresponding orientation.

Recall that the shape of (an oriented) lattice is its equivalent class modulo
homothety and rotation. The space of shapes of (oriented) $d$-lattices is

$\mathcal{X}_{d}=\operatorname{SO}_{d}\left(\mathbb{R}\right)\backslash\operatorname{SL}_{d}\left(\mathbb{R}\right)/\operatorname{SL}_{d}\left(\mathbb{Z}\right)\simeq
P_{d}/\operatorname{SL}_{d}\left(\mathbb{Z}\right).$

Both $\operatorname{Gr}(d,n)$ and $\mathcal{X}_{d}$ are equipped with natural
measures that decent from the Haar measure on ambient group, and are unique up
to a positive scalar – for $\operatorname{Gr}(d,n)$ it is the measure that is
invariant under $\operatorname{SO}_{n}\left(\mathbb{R}\right)$, and for
$\mathcal{X}_{d}$ it is the measure that is obtained from the Haar measure on
$\operatorname{SL}_{n}(\mathbb{R})$. The specific normalizations we choose for
these measures are given in Section 2.3.

We now proceed to define a space of $d$-lattices in $\mathbb{R}^{n}$ that
encodes both their shapes, and their directions (i.e. their projections to the
Grassmannian).

####  2.1 Spaces of homothety classes of $d$-lattices.

A _unimodular_ lattice is an oriented lattice with positive orientation and
covolume one. Recall our notation for the space of rank $n$ unimodular
lattices

$\mathcal{L}_{n}=\operatorname{SL}_{n}(\mathbb{R})/\operatorname{SL}_{n}(\mathbb{Z}),$

where a matrix in $\operatorname{SL}_{n}\left(\mathbb{R}\right)$ lies in the
coset that represents a full unimodular lattice in $\mathbb{R}^{n}$ if its
columns span this lattice. As the space of shapes $\mathcal{X}_{n}$ is
obtained from $\mathcal{L}_{n}$ by modding by $\operatorname{SO}_{n}$, the
space $\mathcal{L}_{n}$ is more refined, containing not only the information
about the shape of a lattice, but also about its position in $\mathbb{R}^{n}$.
To define the analogous space for $d$-lattices in $\mathbb{R}^{n}$, notice
that since $\mathcal{L}_{n}$ consists of a unique representative from any
equivalence class of $n$-lattices in $\mathbb{R}^{n}$ modulo homothety, one
can identify $\mathcal{L}_{n}$ with the space of such equivalence classes. The
space of homothety classes of oriented $d$-lattices inside $\mathbb{R}^{n}$ is

$\mathcal{L}_{d,n}:=\operatorname{SL}_{n}\left(\mathbb{R}\right)/\left(\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{SL}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]\times\left\\{\left[\begin{smallmatrix}\alpha^{\frac{1}{d}}\operatorname{I}_{d}&0_{d\times
n-d}\\\ 0_{n-d\times
d}&\alpha^{-\frac{1}{n-d}}\operatorname{I}_{n-d}\end{smallmatrix}\right]:\alpha>0\right\\}\right),$

where a matrix in $\operatorname{SL}_{n}\left(\mathbb{R}\right)$ lies in the
coset that represents an equivalence class of a $d$-lattice in
$\mathbb{R}^{n}$ if and only if its first $d$ columns span a positive scalar
multiplication of this lattice, with the corresponding orientation. The need
to mod out by the block-scalar group follows from the fact that the first $d$
columns of a matrix in $\operatorname{SL}_{n}$ span a lattice that is hardly
ever unimodular, so one needs to mod out by the covolume (hence an element in
this quotient space is an equivalence class of $d$-lattices up to homothety).
However, modding out the block-scalar group comes with a price, which is that
the space $\mathcal{L}_{d,n}$ does not carry an
$\operatorname{SL}_{n}(\mathbb{R})$-invariant measure. To fix this flaw, we
claim (Remark 4.1) that

$\mathcal{L}_{d,n}\simeq\operatorname{SO}_{n}\left(\mathbb{R}\right)\left[\begin{smallmatrix}P_{d}(\mathbb{R})&0_{d\times
n-d}\\\ 0_{n-d\times
d}&\operatorname{I}_{n-d}\end{smallmatrix}\right]/\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&0_{d\times
n-d}\\\ 0_{n-d\times
d}&\operatorname{SO}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right],$

and observe that, while the quotiented manifold is not a group, it is
diffeomorphic to the group
$\operatorname{SO}_{n}\left(\mathbb{R}\right)\times\left[\begin{smallmatrix}P_{d}(\mathbb{R})&0_{d\times
n-d}\\\ 0_{n-d\times d}&\operatorname{I}_{n-d}\end{smallmatrix}\right]$.
Therefore, $\mathcal{L}_{d,n}$ carries a Radon measure that is invariant under
this group, and unique up to a positive scalar. This measure is the natural
one in the sense that, as we shall see below, it projects to the uniform
measures on $\mathcal{L}_{d}$ and $\mathcal{X}_{d}$. With respect to it,
$\mathcal{L}_{d,n}$ has finite volume (which wouldn’t have been the case if we
hadn’t modded out by the covolume, just like in the case of
$\mathcal{L}_{n}$).

####  2.2 Factor lattices and the space of pairs.

Recall that a lattice $\Lambda$ is primitive if it is of the form
$\Lambda=\mathbb{Z}^{n}\cap V_{\Lambda}$; this is equivalent to the fact that
any basis of $\Lambda$ can be completed to a basis of $\mathbb{Z}^{n}$.
Theorem 1.1 consists of a joint equidistribution result for primitive lattices
$\Lambda$ and their orthogonal complements $\Lambda^{\perp}=\mathbb{Z}^{n}\cap
V_{\Lambda}^{\perp}$. It is a consequence of the stronger Theorem 3.2 below,
in which $\Lambda^{\perp}$ is replaced by another $\left(n-d\right)$-lattice
in $V_{\Lambda}^{\perp}$:

######  Definition 2.1.

The _factor lattice_ $\Lambda^{\pi}$ of a primitive $d$-lattice $\Lambda$ is
the projection of $\mathbb{Z}^{n}$ to $V_{\Lambda}^{\perp}$.

The factor lattice $\Lambda^{\pi}$ is isometric to the quotient
$\mathbb{Z}^{n}/\Lambda$ (Proposition B.3), so one should think of
$\Lambda^{\pi}$ as a realization of $\mathbb{Z}^{n}/\Lambda$ inside
$\mathbb{R}^{n}$. Notice that, like $\Lambda^{\perp}$, $\Lambda^{\pi}$ is a
full lattice inside $V_{\Lambda}^{\perp}$, and in particular is of rank $n-d$.
The relation between $\Lambda^{\perp}$ and $\Lambda^{\pi}$ is that they are
_dual_ to one another (for the definition of dual lattices, see [Cas71, I.5],
or Appendix A in the present paper. For the duality of $\Lambda^{\perp}$ and
$\Lambda^{\pi}$, see Claim B.5). It holds that

$\operatorname{covol}(\Lambda^{\pi})=\operatorname{covol}(\Lambda)^{-1}=\operatorname{covol}(\Lambda^{\perp})^{-1}$

([Sch67, Sch68]; see also Proposition B.4 and Corollary B.6 in the Appendix).
Since we view $d$-lattices as carrying an orientation, we need to define an
orientation on $\Lambda^{\pi}$ and $\Lambda^{\perp}$, which is done as
follows.

######  Definition 2.2.

Let $L$ be a full lattice in $V_{\Lambda}^{\perp}$. A basis $\mathbf{C}$ for
$L$ is positively oriented if $\det(\mathbf{B}|\mathbf{C})=1$ for a positively
oriented basis $\mathbf{B}$ of $\Lambda$.

The last space we introduce is the space of pairs of oriented lattices
$(\Lambda,L)$ such that (i) $\Lambda$ is a $d$-lattice, (ii) $L$ is a full
lattice in $V_{\Lambda}^{\perp}$ (hence is of rank $n-d$ and its orientation
is given in Definition 2.2), and (iii)
$\operatorname{covol}(\Lambda)\operatorname{covol}(L)=1$. In fact, it is the
space of homothety classes of such pairs, where $(\Lambda,L)$ is equivalent to
$(\alpha^{\frac{1}{d}}\Lambda,\alpha^{-\frac{1}{n-d}}L)$ for every $\alpha>0$.
This space is given as the quotient

$\text{$\mathcal{P}_{d,n}$}:=\operatorname{SL}_{n}\left(\mathbb{R}\right)/\left(\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{SL}_{n-d}\left(\mathbb{Z}\right)\end{smallmatrix}\right]\times\left\\{\left[\begin{smallmatrix}\alpha^{\frac{1}{d}}\operatorname{I}_{d}&0_{d\times
n-d}\\\ 0_{n-d\times
d}&\alpha^{-\frac{1}{n-d}}\operatorname{I}_{n-d}\end{smallmatrix}\right]:\alpha>0\right\\}\right).$

Note the similarity to the definition of $\mathcal{L}_{d,n}$ above; here also,
modding out the block-scalar group results in having to
$\operatorname{SL}_{n}(\mathbb{R})$-invariant measure, whih can be fixed by
observing (Remark 4.1) that

$\mathcal{P}_{d,n}\simeq\operatorname{SO}_{n}\left(\mathbb{R}\right)\left[\begin{smallmatrix}P_{d}(\mathbb{R})&0_{d\times
n-d}\\\ 0_{n-d\times
d}&P_{n-d}\end{smallmatrix}\right]/\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&0_{d\times
n-d}\\\ 0_{n-d\times
d}&\operatorname{SL}_{n-d}\left(\mathbb{Z}\right)\end{smallmatrix}\right],$

and therefore $\mathcal{P}_{d,n}$ carries a Radon measure that is invariant
under the group
$\operatorname{SO}_{n}\left(\mathbb{R}\right)\times\left[\begin{smallmatrix}P_{d}(\mathbb{R})&0_{d\times
n-d}\\\ 0_{n-d\times d}&P_{n-d}\end{smallmatrix}\right]$, and is unique up to
a positive scalar.

More details on dual lattices and factor lattices can be found in the
appendix.

####  2.3 Relation between the spaces of lattices and their measures

While the spaces $\mathcal{L}_{d}$, $\mathcal{X}_{d}$ and
$\operatorname{Gr}(d,n)$ are well known, the spaces $\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}$, as far as we are aware, make their first appearance here.
It therefore seems appropriate to explain how $\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}$ add to the more “familiar” spaces $\mathcal{L}_{d}$,
$\mathcal{X}_{d}$ and $\operatorname{Gr}(d,n)$. It is not hard to see that
$\mathcal{P}_{d,n}$ projects canonically to $\mathcal{L}_{d,n}$, which in turn
projects canonically to $\mathcal{X}_{d}$ and $\operatorname{Gr}(d,n)$. In
fact, one could also map $\mathcal{L}_{d,n}\to\mathcal{L}_{d}$ and
$\mathcal{P}_{d,n}\to\mathcal{L}_{n-d}$, but these projections are not
cannonical, and depend on a choice of coordinates on the $d$-dimensional
(resp. $n-d$ dimensional) vector spaces in $\mathbb{R}^{n}$. Such a choice of
coordinates will be introduced in Section 4.1, defining the non-canonical
identifications

$\displaystyle\mathcal{L}_{d,n}\;\leftrightarrow$
$\displaystyle\;\mathcal{L}_{d}\times\operatorname{Gr}(d,n),$ (2.1)
$\displaystyle\mathcal{P}_{d,n}\;\leftrightarrow$
$\displaystyle\;\mathcal{L}_{d,n}\times\mathcal{L}_{n-d}.$

The full picture is depicted in the following commutative diagram of
projections:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.82135pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{P}_{d,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.82135pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.82222pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}{{}}{}{{}{{}}{}{{}{{}{{}{{}{{}}}}}}}}}}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}}}}}{\hbox{\kern-8.49214pt\raise-68.60884pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\hbox{\kern-1.76802pt\lower-2.42363pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
39.24449pt\raise-68.59996pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-1.49007pt\lower-2.6038pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}}}}}}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}{{}}{}{{}{{}}{}{{}{{}}{}{{}}{}{{}}{}{{}}}}}}{\hbox{\kern
85.26042pt\raise 10.49861pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-2.64075pt\lower-1.42351pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
34.82135pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{L}_{d,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
78.90295pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.86215pt\raise-29.82222pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}{{}}{}{{}{{}}{}{{}{{}{{}{{}{{}}}}}}}}}}}{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}}{}{{}{{}{{}{{}{{}{{}}{}{{}{{}}{}{{}{{}}}}}}}}}}{\hbox{\kern
53.0676pt\raise-68.99785pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
1.81046pt\lower-2.39214pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
78.90295pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{Gr}(d,n)}$}}}}}}}{\hbox{\kern-10.19635pt\raise-39.65553pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{L}_{n-d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-68.59996pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
37.27979pt\raise-39.65553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{L}_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
44.86215pt\raise-68.59996pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern
0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{X}_{n-d}}$}}}}}}}{\hbox{\kern
36.65479pt\raise-78.43327pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{X}_{d}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
(2.2)

######  Notation 2.3.

Given an oriented $d$-lattice $\Lambda<\mathbb{R}^{n}$ we denote its homothety
class by $\left[\Lambda\right]\in\mathcal{L}_{d,n}$, and its shape by
$\operatorname{shape}\left(\Lambda\right)\in\mathcal{X}_{d}$. Given a pair
$(\Lambda,L)$ of lattices in orthogonal subspaces where $\Lambda$ is primitive
of rank $d$, we denote its homothety class by
$\left[(\Lambda,L)\right]\in\mathcal{P}_{d,n}$. The image of $\Lambda$ in
$\mathcal{L}_{d}$ (resp. of $L$ in $\mathcal{L}_{n-d}$) is denoted
$\left\llbracket\Lambda\right\rrbracket$ (resp. $\left\llbracket
L\right\rrbracket$).

With these notations, Diagram (2.2) becomes:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
22.6806pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left[(\Lambda,L)\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}}{}{{}{{}}{}{{}{{}}{}{{}{{}{{}{{}}}}}}}}}}{\hbox{\kern
16.09456pt\raise 10.49861pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}}{}{{}{{}{{}}{}{{}}{}{{}{{}}}}}}{\hbox{\kern
116.19028pt\raise 6.84926pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.7639pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
60.18063pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}{{}}}}}}{\hbox{\kern-8.95923pt\raise-68.5014pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}{{}}{}{{}{{}{{}}{}{{}{{}{{}}{}{{}{{}}}}}}}}}}}{\hbox{\kern
5.36148pt\raise-5.50136pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}{{}}}}}}{\hbox{\kern
59.41313pt\raise-68.5014pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
60.18063pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\left[\Lambda\right]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
78.68065pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
116.18066pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
69.43063pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.43063pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{}{{}{{}{{}{{}{{}}{}{{}{{}{{}}{}{{}{{}{{}}{}{{}{{}}}}}}}}}}}{\hbox{\kern
74.50275pt\raise-5.50136pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{}{{}{{}}{}{{}}{}{{}{{}}{}{{}}{}{{}}{}{{}{{}}{}{{}{{}}}}}}{\hbox{\kern
78.38986pt\raise-68.5014pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{{}{}{{}}{{}{}{}{}}{}}}}\ignorespaces{}{\hbox{\kern
116.18066pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{V_{\Lambda}}$}}}}}}}{\hbox{\kern-13.90279pt\raise-39.5pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\left\llbracket
L\right\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-44.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-68.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
55.45842pt\raise-39.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\left\llbracket\Lambda\right\rrbracket\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
69.43063pt\raise-44.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
69.43063pt\raise-68.5pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{shape}\left(L\right)}$}}}}}}}{\hbox{\kern
46.6806pt\raise-79.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{shape}\left(\Lambda\right)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

######  Remark 2.4.

For completeness, we provide an explicit description of the above maps. The
projection $\mathcal{P}_{d,n}\to\mathcal{L}_{d,n}$ is the one of modding by
$\left[\begin{smallmatrix}\operatorname{I}_{d}&0_{d\times n-d}\\\ 0_{n-d\times
d}&\operatorname{SL}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]$ \
from the right, and the projection
$\mathcal{L}_{d,n}\to\operatorname{Gr}(d,n)$ is modding by
$\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{R}\right)&0_{d\times
n-d}\\\ 0_{n-d\times d}&\operatorname{I}_{n-d}\end{smallmatrix}\right]$ from
the right. The projections $\mathcal{L}_{i}\to\mathcal{X}_{i}$ for $i=d,n-d$
are given by modding from the left by
$\operatorname{SO}_{i}\left(\mathbb{R}\right)$. The projections
$\mathcal{P}_{d,n}\to\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}\to\mathcal{X}_{n-d}$ , as mentioned, depend on a choice of
coordinates and will be explicated in Remark 4.2. The projection
$\mathcal{P}_{d,n}\to\mathcal{X}_{d}$ is modding from the left by
$\operatorname{SO}_{n}\left(\mathbb{R}\right)\left[\begin{smallmatrix}\operatorname{I}_{d}&0_{d\times
n-d}\\\ 0_{n-d\times d}&P_{n-d}\end{smallmatrix}\right]$, and similarly for
$\mathcal{P}_{d,n}\to\mathcal{X}_{n-d}$.

The projections depicted in Diagram (2.2) are critical to understanding why
the measures we introduced on $\mathcal{L}_{d,n}$ and $\mathcal{P}_{d,n}$ are
indeed the natural uniform measures on these spaces; as we shall see below
(Lemma 2.5), with a consistent choice of normalizations, these measures
project to the uniform measures on $\mathcal{L}_{d}$, $\mathcal{X}_{d}$ and
$\operatorname{Gr}(d,n)$.

###### Normalizations of the measures.

As explained above, all the spaces considered are endowed with invariant Radon
measures that are unique up to a positive scalar, and have finite volume with
respect to these measures. Hence, by fixing the volume of the entire space, a
unique measure is determined. This is what we now turn to do, where all the
unique measures obtained are denoted by $\operatorname{vol}$ (the space in
question should be understood from the context).

In accordance with the fact that
$\operatorname{SO}_{d}\left(\mathbb{R}\right)/\operatorname{SO}_{d-1}\left(\mathbb{R}\right)$
is diffeomorphic to the sphere $\mathbb{S}^{d-1}$, we choose a Haar measure on
the special orthogonal group such that
$\operatorname{Haar}(\operatorname{SO}_{d}\left(\mathbb{R}\right))=\operatorname{Leb(\mathbb{S}^{d-1})}\cdot\operatorname{Haar}(\operatorname{SO}_{d-1}\left(\mathbb{R}\right))$,
namely:

$\operatorname{Haar}(\operatorname{SO}_{d}\left(\mathbb{R}\right))=\prod_{i=1}^{d-1}\operatorname{Leb(\mathbb{S}^{i})}.$

We refer to [Gar14] for the fact that, with our standard choice (Section 4.3)
of Haar measure on $\operatorname{SL}_{d}\left(\mathbb{R}\right)$,
$\operatorname{vol}(\operatorname{SL}_{d}\left(\mathbb{R}\right)/\operatorname{SL}_{d}\left(\mathbb{Z}\right))=\prod_{i=2}^{d}\zeta\left(i\right)\cdot\prod_{i=1}^{d-1}\operatorname{Leb(\mathbb{S}^{i})}/\operatorname{Haar}(\operatorname{SO}_{d}\left(\mathbb{R}\right))$,
where $\zeta$ is the Riemann Zeta function. Thus,

$\operatorname{vol}(\mathcal{L}_{d})=\prod_{i=2}^{d}\zeta\left(i\right).$

Similarly, in accordance with the fact that
$\operatorname{SO}_{n}\left(\mathbb{R}\right)/(\operatorname{SO}_{d}\left(\mathbb{R}\right)\times\operatorname{SO}_{n-d}\left(\mathbb{R}\right))$
is diffeomorphic to the Grassmannian $\operatorname{Gr}(d,n)$, we choose to
normalize the volume of the Grassmannian as
$\operatorname{Haar}(\operatorname{SO}_{n}\left(\mathbb{R}\right))/(\operatorname{Haar}(\operatorname{SO}_{d}\left(\mathbb{R}\right))\operatorname{Haar}(\operatorname{SO}_{n-d}\left(\mathbb{R}\right)))$,
namely

$\operatorname{vol}(\operatorname{Gr}(d,n))=\frac{\prod_{i=1}^{n-1}\operatorname{Leb(\mathbb{S}^{i})}}{\prod_{i=1}^{d-1}\operatorname{Leb(\mathbb{S}^{i})}\prod_{i=1}^{n-d-1}\operatorname{Leb(\mathbb{S}^{i})}}=\frac{\prod_{i=n-d}^{n-1}\operatorname{Leb(\mathbb{S}^{i})}}{\prod_{i=1}^{d-1}\operatorname{Leb(\mathbb{S}^{i})}}.$

Finally, recall from Theorem 1.1 that

$\Upsilon(d)=\frac{\operatorname{Haar}(\operatorname{SO}_{d}\left(\mathbb{R}\right))}{\left[\operatorname{SO}_{d}\left(\mathbb{R}\right):Z\left(\operatorname{SO}_{d}\left(\mathbb{R}\right)\right)\right]},$

where $Z$ denotes the center. Note that the denominator is $1$ if $d$ is odd
and $2$ if it is even.

As we shall now prove (Lemma 2.5), fixing the volumes of $\mathcal{L}_{d}$ and
$\operatorname{Gr}(d,n)$ determines the volumes of $\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}$ in correspondence with the identifications in (2.1). In
fact,

$\displaystyle\operatorname{vol}(\mathcal{L}_{d})\;=$
$\displaystyle\;\operatorname{vol}(\mathcal{X}_{d})\Upsilon(d),$ (2.3)
$\displaystyle\operatorname{vol}(\mathcal{L}_{d,n})=$
$\displaystyle\;\operatorname{vol}(\mathcal{L}_{d})\operatorname{vol}(\operatorname{Gr}(d,n)),$
$\displaystyle\operatorname{vol}(\mathcal{P}_{d,n})=$
$\displaystyle\;\operatorname{vol}(\mathcal{L}_{d,n})\operatorname{vol}(\mathcal{L}_{n-d}).$

###### Interactions between the measures.

We claim that the measures we introduced on $\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}$ are the products of uniform measures in a manner that
corresponds to the identifications in (2.1). For example, if
$\Psi\subseteq\mathcal{L}_{d,n}$ projects to the sets
$\widetilde{\mathcal{E}}\subseteq\mathcal{L}_{d}$ and
$\Phi\subseteq\operatorname{Gr}(d,n)$, then
$\operatorname{vol}(\Psi)=\operatorname{vol}(\widetilde{\mathcal{E}})\operatorname{vol}(\Phi)$.
This (and in particular, (2.3)) is a consequence of the following lemma.

######  Lemma 2.5.

  1. 1.

Assume $\widetilde{\mathcal{E}}\subseteq\mathcal{L}_{n}$ is the lift of
$\mathcal{E}\subseteq\mathcal{X}_{n}$. If $\mathcal{E}$ is a BCS then
$\widetilde{\mathcal{E}}$ is, and
$\text{$\operatorname{vol}(\widetilde{\mathcal{E}})$}=\operatorname{vol}(\mathcal{E})\Upsilon(n)$.

  2. 2.

Assume $\Psi\subseteq\mathcal{L}_{d,n}$ is the product to
$\widetilde{\mathcal{E}}\subseteq\mathcal{L}_{d}$ and
$\Phi\subseteq\operatorname{Gr}(d,n)$ (e.g. if $\Psi$ is the inverse image of
$\widetilde{\mathcal{E}}$). If $\widetilde{\mathcal{E}}$ and $\Phi$ are BCS’s,
then so is $\Psi$, and
$\operatorname{vol}(\Psi)=\operatorname{vol}(\widetilde{\mathcal{E}})\operatorname{vol}(\Phi)$.

  3. 3.

Assume $\varXi\subseteq\mathcal{P}_{d,n}$ is the product of
$\widetilde{\mathcal{E}}\subseteq\mathcal{L}_{d}$,
$\widetilde{\mathcal{F}}\subseteq\mathcal{L}_{n-d}$ and
$\Phi\subseteq\operatorname{Gr}(d,n)$. If $\widetilde{\mathcal{E}}$,
$\widetilde{\mathcal{F}}$ and $\Phi$ are BCS’s, then so is $\varXi$, and
$\operatorname{vol}(\varXi)=\operatorname{vol}(\widetilde{\mathcal{E}})\operatorname{vol}(\widetilde{\mathcal{F}})\operatorname{vol}(\Phi)$.

  4. 4.

Assume $\varXi\subseteq\mathcal{P}_{d,n}$ is the product of
$\Psi\subseteq\mathcal{L}_{d,n}$ and
$\widetilde{\mathcal{F}}\subseteq\mathcal{L}_{n-d}$. If $\Psi$ and
$\widetilde{\mathcal{F}}$ are BCS’s, then so is $\Psi$, and
$\operatorname{vol}(\varXi)=\operatorname{vol}(\Psi)\operatorname{vol}(\widetilde{\mathcal{F}})$

###### ?proofname? .

Part 1 is proved in [HK19, Lemma 3.8], Part 3 is a consequence of Parts 2 and
4, and Parts 2 and 4 are a consequence of [HK20, Propositions 6.15 and 6.16].
∎

###  3 Counting lattices: Main theorem

Theorem (1.1) stated in the Introduction is in fact a consequence of the more
general Theorem 3.2, which we now introduce. It our main result, which we
prove throughout the rest of the paper. This theorem concerns the
equidistribution of $\left[(\Lambda,\Lambda^{\pi})\right]\in\mathcal{P}_{d,n}$
for primitive $d$-lattices $\Lambda$, implying the equidistribution of the
projections of $\left[(\Lambda,\Lambda^{\pi})\right]$ to all the spaces below
$\mathcal{P}_{d,n}$, e.g.
$(\left[\Lambda\right],\left\llbracket\Lambda^{\pi}\right\rrbracket)\in\mathcal{L}_{d,n}\times\mathcal{L}_{n-d}$,
$(\operatorname{shape}\left(\Lambda\right),\operatorname{shape}\left(\Lambda^{\pi}\right))\in\mathcal{X}_{d}\times\mathcal{X}_{n-d}$
, etc. Then Theorem (1.1) follows from the fact that the equidistribution
still holds when replacing $\Lambda^{\pi}$ by its dual, $\Lambda^{\perp}$,
which is the content of Theorem (3.4).

Let us provide more details. First of all, for the counting to hold, we
require that $\left[(\Lambda,\Lambda^{\pi})\right]$ and its projections to all
the spaces below it fall in sets that satisfy the following regularity
condition:

######  Definition 3.1.

A subset $B$ of an orbifold $\mathcal{M}$ will be called _boundary
controllable set_ , or a BCS, if for every $x\in\mathcal{M}$ there is an open
neighborhood $U_{x}$ of $x$ such that $U_{x}\cap\partial B$ is contained in a
finite union of embedded $C^{1}$ submanifolds of $\mathcal{M}$, whose
dimension is strictly smaller than $\dim\mathcal{M}$. In particular, $B$ is a
BCS if its (topological) boundary consists of finitely many subsets of
embedded $C^{1}$ submanifolds.

A few more notions before we can state our main theorem. We refer to the left
component in $\mathcal{X}_{d}\times\mathcal{X}_{n-d}$,
$\mathcal{L}_{d}\times\mathcal{L}_{n-d}$ and
$\mathcal{P}_{d,n}=\\{\left[(\Lambda,L)\right]\\}$ as the $d$ component, and
to the right component in these spaces as the $n-d$ component. We say that the
$d$ (resp. $n-d$) component of a set in one of these spaces is _bounded_ if
the image of this set under the projection to $\mathcal{X}_{d}$ (resp.
$\mathcal{X}_{n-d}$) is bounded. Notice that this is well defined since the
projections to the shape spaces are canonical; they are also proper, so a set
in $\mathcal{P}_{d,n}$ (or $\mathcal{L}_{d,n-d}\times\mathcal{L}_{n-d}$, or
$\mathcal{L}_{d}\times\mathcal{L}_{n-d}$…) is bounded if and only if both its
$d$ and $n-d$ components are bounded.

######  Theorem 3.2.

Assume that $\Phi\subseteq\operatorname{Gr}(d,n)$,
$\mathcal{E}\times\mathcal{F}\subseteq\mathcal{X}_{d}\times\mathcal{X}_{n-d}$,
$\widetilde{\mathcal{E}}\times\widetilde{\mathcal{F}}\subseteq\mathcal{L}_{d}\times\mathcal{L}_{n-d}$,
$\Psi\subseteq\mathcal{L}_{d,n}$ and $\varXi\subseteq\mathcal{P}_{d,n}$ are
BCS’s. Then, for every $\epsilon>0$:

  1. 1.

The number of primitive $d$-lattices $\Lambda$ of covolume up to $X$ with
$V_{\Lambda}\in\Phi$ and
$(\operatorname{shape}\left(\Lambda\right),\operatorname{shape}\left(\Lambda^{\pi}\right))\in\mathcal{E}\times\mathcal{F}$
is

$\frac{\operatorname{vol}(\mathcal{E})\operatorname{vol}(\mathcal{F})\operatorname{vol}(\Phi)}{n\cdot\prod_{i=2}^{n}\zeta\left(i\right)}\Upsilon(d)\Upsilon(n-d)\cdot
X^{n}+\text{error term}.$

  2. 2.

The number of primitive $d$-lattices $\Lambda$ of covolume up to $X$ with
$V_{\Lambda}\in\Phi$ and
$(\left\llbracket\Lambda\right\rrbracket,\left\llbracket\Lambda^{\pi}\right\rrbracket)\in\widetilde{\mathcal{E}}\times\widetilde{\mathcal{F}}$
is

$\frac{\operatorname{vol}(\widetilde{\mathcal{E}})\operatorname{vol}(\widetilde{\mathcal{F}})\operatorname{vol}(\Phi)}{n\cdot\prod_{i=2}^{n}\zeta\left(i\right)}\cdot
X^{n}+\text{error term}.$

  3. 3.

The number of primitive $d$-lattices $\Lambda$ of covolume up to $X$ with
$\left[\Lambda\right]\in\Psi$ and
$\left\llbracket\Lambda^{\pi}\right\rrbracket\in\widetilde{\mathcal{F}}$ is

$\frac{\operatorname{vol}(\Psi)\operatorname{vol}(\widetilde{\mathcal{F}})}{n\cdot\prod_{i=2}^{n}\zeta\left(i\right)}\cdot
X^{n}+\text{error term}.$

In particular, the number of primitive $\Lambda$ with
$\operatorname{covol}(\Lambda)\leq X$ whose homothety class lies inside a BCS
$\Psi\subseteq\mathcal{L}_{d,n}$ is
$\frac{\operatorname{vol}(\Psi)}{\prod_{i=n-d+1}^{n}\zeta\left(i\right)}\cdot\frac{X^{n}}{n}$,
plus an error term.

  4. 4.

The number of primitive $d$-lattices $\Lambda$ of covolume up to $X$ with
$\left[(\Lambda,\Lambda^{\pi})\right]\in\varXi$ is

$\frac{\operatorname{vol}(\varXi)}{n\cdot\prod_{i=2}^{n}\zeta\left(i\right)}\cdot
X^{n}+\text{error term}.$

The error term is
$O_{\epsilon}(X^{n-\frac{\tau_{n}n}{2\left(n-d-1\right)\left(n^{2}-1\right)+n^{2}}+\epsilon})$
with $\tau_{n}=\left\lceil\left(n-1\right)/2\right\rceil/4n^{2}$ when the $d$
component is bounded,
$O_{\epsilon}(X^{n-\frac{\tau_{n}n}{2\left(d-1\right)\left(n^{2}-1\right)+n^{2}}+\epsilon})$
when the $n-d$ component is bounded,
$O_{\epsilon}(X^{n-\frac{\tau_{n}n}{2\left(\max\left\\{d,n-d\right\\}-1\right)\left(n^{2}-1\right)+n^{2}}+\epsilon})$
when neither of them is bounded and $O_{\epsilon}(X^{n-n\tau_{n}+\epsilon})$
when both of them are bounded.

######  Remark 3.3.

  1. 1.

Regarding the aforementioned work of Schmidt: our leading constants in Theorem
3.2 coincide with Schmidt’s constant $c_{d,n}$ ([Sch98, Thm 2], [Sch15, Thm
1.2]) when the measures on the appearing subspaces are normalized to
probability measures. In fact, as can be computed from the chosen
normalizations of the volumes of the spaces that appear in Theorem 3.2, along
with the fact that
$\operatorname{Leb(\mathbb{S}^{i})}=\left(i+1\right)\operatorname{Leb(B_{i+1})}$,
the relation between Schmidt’s constant and our chosen normalizations is given
by

$c_{d,n}=\frac{\operatorname{vol}(\operatorname{Gr}(d,n))}{2}\cdot\frac{\operatorname{vol}(\mathcal{L}_{d})\operatorname{vol}(\mathcal{L}_{n-d})}{\operatorname{vol}(\mathcal{L}_{n})}.$

The $\frac{1}{2}$ factor is due to the fact that the lattices we count carry
an orientation, so every non-oriented lattice is counted twice. We also note
that a set that is BCS is Jordan measurable, and indeed Schmidt (in [Sch98])
provides an example for how the asymptotic formula for number of $d$-lattices
with shapes in $\mathcal{E}$ fails when $\mathcal{E}$ is not Jordan
measurable.

  2. 2.

The case of $d=n-1$ was also obtained in [Mar10], using a dynamical approach,
as well as in [HK19]. It was also considered in [AES16b, AES16a, EMSS16,
ERW17, Ber19] in a more delicate setting.

  3. 3.

Primitive $d$-lattices are in one-to-one correspondence with rational
subspaces in $\mathbb{R}^{n}$. These spaces are the rational points on the
Grassmanian variety: the projective variety consisting of all the
$d$–dimensional spaces in $\mathbb{R}^{n}$. Therefore, the aforementioned
result of Schmidt can be read as the counting of rational points up to a
bounded height in the Grassmannian variety (the height being the covolume of
the unique primitive lattice in the space). As such, it provides yet another
example where the Manin conjecture [FMT89, Pey95] on counting rational points
in varieties holds. The more refined counting we suggest in Theorem 3.2 (Part
4) plays a key role in the intensive study of rational points on the
Grassmannian conducted in [BHW21]. In this paper, the authors confirm a
modification to the Manin conjecture suggested by Peyre [Pey17], and obtain an
equidistribution result for the integral lattices in the rational tangent
bundle of the Grassmannian.

Note that Theorem (1.1) is just Part 1 of Theorem 3.2, only with
$\Lambda^{\perp}$ instead of $\Lambda^{\pi}$. As we have already mentioned,
these lattices are dual to one another; we denote the dual of a lattice
$\Lambda$ by $\Lambda^{*}$. Then Theorem (1.1) is obtained from Theorem 3.2
and the following.

######  Theorem 3.4.

A version of Theorem 3.2 holds when replacing
$\left(\Lambda,\Lambda^{\pi}\right)$ by each of
$\left(\Lambda,\Lambda^{\perp}\right)$,
$\left(\Lambda^{*},\Lambda^{\perp}\right)$,
$\left(\Lambda^{*},\Lambda^{\pi}\right)$.

Indeed, note that in all of the pairs above, the right-hand lattice spans the
orthogonal subspace to the one spanned by the left-hand lattice. In other
words, their homothety classes are elements in $\mathcal{P}_{d,n}$.

###### Proof of Theorem 3.4.

Note that the three pairs $\left(\Lambda,\Lambda^{\perp}\right)$,
$\left(\Lambda^{*},\Lambda^{\perp}\right)$ and
$\left(\Lambda^{*},\Lambda^{\pi}\right)$ are obtained from the pair in Theorem
3.2 $\left(\Lambda,\Lambda^{\pi}\right)$ by taking the dual of one of
$\Lambda,\Lambda^{\pi}$ or of both. According to Proposition A.8 and
Proposition A.9, taking the dual of one of the lattices in a pair or of both
is a measure preserving auto-diffeomorphism of $\mathcal{P}_{d,n}$. ∎

###  4 Refined Iwasawa components in
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$.

####  4.1 Refining the Iwasawa decomposition of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$

Set $G=G_{n}:=\operatorname{SL}_{n}\left(\mathbb{R}\right)$ and let $G=KAN$ be
the Iwasawa decomposition of $G$, meaning that $K=K_{n}$ is
$\operatorname{SO}_{n}\left(\mathbb{R}\right)$, $A=A_{n}$ is the diagonal
subgroup in $G$, and $N=N_{n}$ is the subgroup of upper unipotent matrices. We
also let $P_{n}=A_{n}N_{n}$. Consider the following isomorphic copy of
$\operatorname{SL}_{d}\left(\mathbb{R}\right)\times\operatorname{SL}_{n-d}\left(\mathbb{R}\right)$
inside $G$,

$G^{\prime\prime}:=\left[\begin{array}[]{c|c}G_{d}&0_{d,n-d}\\\ \hline\cr
0_{n-d,d}&G_{n-d}\end{array}\right]=\left[\begin{array}[]{c|c}\operatorname{SL}_{d}\left(\mathbb{R}\right)&0_{d,n-d}\\\
\hline\cr
0_{n-d,d}&\operatorname{SL}_{n-d}\left(\mathbb{R}\right)\end{array}\right].$

Write $G^{\prime\prime}=K^{\prime\prime}A^{\prime\prime}N^{\prime\prime}$ for
the Iwasawa decomposition of $G^{\prime\prime}$, namely

$\displaystyle K^{\prime\prime}$ $\displaystyle:=$ $\displaystyle K\cap
G^{\prime\prime}=\left[\begin{array}[]{c|c}K_{d}&0_{d,n-d}\\\ \hline\cr
0_{n-d,d}&K_{n-d}\end{array}\right]=\left[\begin{array}[]{c|c}\operatorname{SO}_{d}\left(\mathbb{R}\right)&0_{d,n-d}\\\
\hline\cr
0_{n-d,d}&\operatorname{SO}_{n-d}\left(\mathbb{R}\right)\end{array}\right],$
$\displaystyle A^{\prime\prime}$ $\displaystyle:=$ $\displaystyle A\cap
G^{\prime\prime}=\left[\begin{array}[]{c|c}A_{d}&0_{d,n-d}\\\ \hline\cr
0_{n-d,d}&A_{n-d}\end{array}\right],$ $\displaystyle N^{\prime\prime}$
$\displaystyle:=$ $\displaystyle N\cap
G^{\prime\prime}=\left[\begin{array}[]{c|c}N_{d}&0_{d,n-d}\\\ \hline\cr
0_{n-d,d}&N_{n-d}\end{array}\right].$

Let

$P^{\prime\prime}:=A^{\prime\prime}N^{\prime\prime}\;\text{and}\;Q:=KP^{\prime\prime}$

(note that $Q$ is not a group, but it is a smooth manifold). To complete the
definition of the Refined Iwasawa decomposition, we define
$K^{\prime},A^{\prime},N^{\prime}$ that complete
$K^{\prime\prime},A^{\prime\prime},N^{\prime\prime}$ to $K$, $A$ and $N$
respectively. Let

$N^{\prime}:=\left[\begin{array}[]{c|c}\operatorname{I}_{d}&\mathbb{R}^{d,n-d}\\\
\hline\cr 0_{n-d,d}&\operatorname{I}_{n-d}\end{array}\right],\quad
A^{\prime}:=\left[\begin{array}[]{c|c}a^{\frac{1}{d}}\operatorname{I}_{d}&0_{d,n-d}\\\
\hline\cr
0_{n-d,d}&a^{-\frac{1}{n-d}}\operatorname{I}_{n-d}\end{array}\right],$

and observe that $N=N^{\prime\prime}N^{\prime}$,
$A=A^{\prime\prime}A^{\prime}$, and that $A^{\prime}$ is a one-parameter
subgroup of $A$ which commutes with $G^{\prime\prime}$. Often we would like to
restrict to the upper $d\times d$ (resp. lower
$\left(n-d\right)\times\left(n-d\right)$) block of $G^{\prime\prime}$, hence
we denote

$G_{d}^{\prime\prime}:=\left[\begin{array}[]{c|c}G_{d}&0_{d,n-d}\\\ \hline\cr
0_{n-d,d}&\operatorname{I}_{n-d}\end{array}\right]=\left[\begin{array}[]{c|c}\operatorname{SL}_{d}\left(\mathbb{R}\right)&0_{d,n-d}\\\
\hline\cr 0_{n-d,d}&\operatorname{I}_{n-d}\end{array}\right]$

(resp.
$G_{n-d}^{\prime\prime}:=\left[\begin{array}[]{c|c}\operatorname{I}_{d}&0_{d,n-d}\\\
\hline\cr 0_{n-d,d}&G_{n-d}\end{array}\right]$). Similarly for
$K^{\prime\prime}$,$P^{\prime\prime}$ and $A^{\prime\prime}$.

Fix a transversal $K^{\prime}$ of the diffeomorphism
$K/K^{\prime\prime}\to\operatorname{Gr}(d,n)$, meaning that
$K=K^{\prime}K^{\prime\prime}$; later (Proposition 5.3) we will restrict to a
specific choice of a transversal, with some desirable properties. Then $Q$ is
also $K^{\prime}G^{\prime\prime}$, and we let

$Q_{d}:=K^{\prime}G_{d}^{\prime\prime}.$

Note that $Q=Q_{d}G_{n-d}^{\prime\prime}$. Then the Refined Iwasawa (or RI,
for short) decomposition is given by

$G=K^{\prime}K^{\prime\prime}A^{\prime\prime}A^{\prime}N^{\prime\prime}N^{\prime}=K^{\prime}G^{\prime\prime}A^{\prime}N^{\prime}=KP^{\prime\prime}A^{\prime}N^{\prime}=QA^{\prime}N^{\prime}=Q_{d}G_{n-d}^{\prime\prime}A^{\prime}N^{\prime}.$

######  Remark 4.1.

In particular, the two quotients with whom we expressed $\mathcal{L}_{d,n}$
and $\mathcal{P}_{d,n}$ in the Introduction indeed agree. For
$\mathcal{L}_{d,n}$, we have that
$\operatorname{SL}_{n}\left(\mathbb{R}\right)=KP^{\prime\prime}A^{\prime}N^{\prime}=KP_{d}^{\prime\prime}P_{n-d}^{\prime\prime}A^{\prime}N^{\prime}$,
so

$\mathcal{L}_{d,n}=\operatorname{SL}_{n}\left(\mathbb{R}\right)/G_{d}^{\prime\prime}\left(\mathbb{Z}\right)G_{n-d}^{\prime\prime}A^{\prime}N^{\prime}=\operatorname{SL}_{n}\left(\mathbb{R}\right)/G_{d}^{\prime\prime}\left(\mathbb{Z}\right)K_{n-d}^{\prime\prime}P_{n-d}^{\prime\prime}A^{\prime}N^{\prime}\simeq
KP_{d}^{\prime\prime}/G_{d}^{\prime\prime}\left(\mathbb{Z}\right)K_{n-d}^{\prime\prime}.$

Similarly,

$\mathcal{P}_{d,n}=\operatorname{SL}_{n}\left(\mathbb{R}\right)/G^{\prime\prime}\left(\mathbb{Z}\right)N^{\prime}A^{\prime}\simeq
KP^{\prime\prime}/G^{\prime\prime}\left(\mathbb{Z}\right).$

######  Remark 4.2.

The choice of a transversal $K^{\prime}$ determines projections
$\mathcal{L}_{d,n}\to\mathcal{L}_{d}$ and
$\mathcal{P}_{d,n}\to\mathcal{L}_{n-d}$: both these maps are induced by

$\displaystyle\operatorname{SL}_{n}(\mathbb{R})$ $\displaystyle\to$
$\displaystyle\operatorname{SL}_{n}(\mathbb{R})$ $\displaystyle
k^{\prime}g^{\prime\prime}a^{\prime}n^{\prime}$ $\displaystyle\mapsto$
$\displaystyle g^{\prime\prime}a^{\prime}n^{\prime}.$

In particular, for every $d$ lattice $\Lambda$ and every $\left(n-d\right)$
lattice $L$ in $V_{\Lambda}^{\perp}$, the projections
$\left\llbracket\Lambda\right\rrbracket\in\mathcal{L}_{d}$ and
$\left\llbracket L\right\rrbracket\in\mathcal{L}_{n-d}$ from Notation 2.3 are
now determined.

####  4.2 Parameterizations of the RI components

Our use of the RI decomposition of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$ is motivated by the fact that,
as we shall see in Section 5, the components appearing in this decomposition
or some special subsets of theirs are in a sense isomorphic to the homogeneous
spaces of $\operatorname{SL}_{n}\left(\mathbb{R}\right)$ that we mentioned in
the Introduction. Specifically, we shall state (Proposition 5.3) that certain
subsets of $G_{d}$, $P_{d}$, $Q_{d}$ and $Q$ are parameterized by the spaces
$\mathcal{L}_{d}$, $\mathcal{X}_{d}$, $\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}$; in particular, subsetes of
$G^{\prime\prime}=\left[\begin{smallmatrix}G_{d}&0\\\
0&G_{n-d}\end{smallmatrix}\right]$ and
$P^{\prime\prime}=\left[\begin{smallmatrix}P_{d}&0\\\
0&P_{n-d}\end{smallmatrix}\right]$ are parameterized by
$\mathcal{L}_{d}\times\mathcal{L}_{n-d}$ and
$\mathcal{X}_{d}\times\mathcal{X}_{n-d}$ respectively. Also, it is clear that
$K^{\prime}$ and $N^{\prime}$ can be parameterized $\operatorname{Gr}(d,n)$
and $\mathbb{R}^{d\left(n-d\right)}$; the parameterization for
$K^{\prime}\subset K$ by the Grassmannian will be the inverse of the
restriction of the quotient map, and a choice of an isomorphism for
$N^{\prime}$ is arbitrary (but fixed). We let $k_{U}^{\prime}$ denote the
element of $K^{\prime}$ that corresponds to an oriented $d$-dimensional
subspace $U$ in $\operatorname{Gr}(d,n)$.

When $D$ is a subset of a space that is parameterized by a RI component $S$
(or a subset of it), we let $S_{D}$ denote the image of $D$ in $S$ under this
parameterization. For example, $D\subset\mathbb{R}^{d\left(n-d\right)}$, then
$N_{D}^{\prime}$ denotes its image in $N^{\prime}$. Finally, the groups
$A,A^{\prime\prime},A^{\prime}$ are clearly isomorphic to $\mathbb{R}^{n-1}$,
$\mathbb{R}^{d-1}\times\mathbb{R}^{n-d-1}$ and $\mathbb{R}$ respectively. We
choose the following parameterizations: $a=\operatorname{diag}(a_{1},\ldots
a_{n})$ will be written as
$a_{\underline{h}}=a_{\left(h_{1},\ldots,h_{n-1}\right)}$ if

$\left(a_{1},a_{2},\ldots,a_{n-1},a_{n}\right)=(e^{-h_{1}/2},e^{\left(h_{1}-h_{2}\right)/2},\ldots,e^{\left(h_{n-2}-h_{n-1}\right)/2},e^{h_{n-1}/2}).$

Accordingly, we write $A^{\prime\prime}\ni
a^{\prime\prime}=a_{\underline{s},\underline{w}}^{\prime\prime}$ with
$\underline{s}=\left(s_{1},\ldots,s_{d-1}\right)\in\mathbb{R}^{d-1}$ and
$\underline{w}=\left(w_{1},\ldots,w_{n-d-1}\right)\in\mathbb{R}^{n-d-1}$ if
$a^{\prime\prime}=\left[\begin{smallmatrix}a_{\underline{s}}\in A_{d}\\\
&a_{\underline{w}}\in A_{n-d}\end{smallmatrix}\right]$. For an element in
$A^{\prime}$, we let

$a_{t}^{\prime}:=\left[\begin{smallmatrix}e^{\frac{t}{d}}\operatorname{I}_{d}&0\\\
0&e^{-\frac{t}{n-d}}\operatorname{I}_{n-d}\end{smallmatrix}\right].$

####  4.3 RI decomposition of the Haar measure on
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$

It is well known (e.g. [Kna02, Prop. 8.43]) that a Haar measure on
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$ can be decomposed according to
the Iwasawa components of $\operatorname{SL}_{n}\left(\mathbb{R}\right)$. Let
us extend this to a Refined Iwasawa decomposition of the Haar measure on
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$.

For every $S\subset G=\operatorname{SL}_{n}\left(\mathbb{R}\right)$ appearing
as a component in the Iwasawa or Refined Iwasawa decompositions of $G$, we let
$\mu_{S}$ denote a measure on $S$ as follows: $\mu_{K},\mu_{N}$ are Haar
measures, and so do $\mu_{K^{\prime\prime}}$, $\mu_{N^{\prime\prime}}$,
$\mu_{P^{\prime\prime}}$,
$\mu_{G^{\prime\prime}},\mu_{G},\mu_{G_{i}},\mu_{K_{i}},\mu_{N_{i}}$ and
$\mu_{N^{\prime}}$ for $i=d,n-d$. The measures $\mu_{N}$,
$\mu_{N^{\prime}},\mu_{N_{i}}$ and $\mu_{N^{\prime\prime}}$ are Lebesgue; as
$N=N^{\prime\prime}\ltimes N^{\prime}$ and all three groups are unimodular,
$\mu_{N}=\mu_{N^{\prime\prime}}\times\mu_{N^{\prime}}$. We assume that the
Haar measures $\mu_{K}$ and $\mu_{K^{\prime\prime}}$ are normalized such that
$\mu_{K^{\prime\prime}}\left(K^{\prime\prime}\right)=\mu_{K_{d}}\left(K_{d}\right)\mu_{K_{n-d}}\left(K_{n-d}\right)$,
where (as we have mentioned in Section 1)
$\mu_{K_{i}}\left(K_{i}\right)=\prod_{j=1}^{i-1}\operatorname{Leb(\mathbb{S}^{j})}$.
Since $K^{\prime}$ parameterizes $\operatorname{Gr}(d,n)=K/K^{\prime\prime}$,
we can endow it with a measure $\mu_{K^{\prime}}$ that is the pullback of a
$K$-invariant Radon measure on $K/K^{\prime\prime}$ so that
$\mu_{K}=\mu_{K^{\prime}}\times\mu_{K^{\prime\prime}}$.

The measures $\mu_{A},\mu_{A^{\prime}},\mu_{A^{\prime\prime}}$ are Radon
measures such that

$\mu_{A^{\prime}}=e^{nt}dt,\quad\mu_{A_{i}}=\prod_{j=1}^{i}e^{-h_{j}}dh_{j}$

and

$\mu_{A^{\prime\prime}}=\mu_{A_{d}^{\prime\prime}}\times\mu_{A_{n-d}^{\prime\prime}},\quad\mu_{A}=\mu_{A^{\prime}}\times\mu_{A^{\prime\prime}}.$

Note that these measures are non-Haar. We use the measures on
$A_{i},A^{\prime\prime}$ to determine a (normalization of the) Haar measure on
$G_{i}$:

$\mu_{G_{i}}=\mu_{K_{i}}\times\mu_{A_{i}}\times\mu_{N_{i}};\quad\mu_{G^{\prime\prime}}=\mu_{G_{d}^{\prime\prime}}\times\mu_{G_{n-d}^{\prime\prime}}.$

Since $Q$ is diffeomorphic to the group $K\times P^{\prime\prime}$, we endow
it with the Haar measure on this group:
$\mu_{Q}=\mu_{K}\times\mu_{P^{\prime\prime}}$. Since
$\mu_{K}=\mu_{K^{\prime}}\times\mu_{K^{\prime\prime}}$, we also have that also
$\mu_{Q}=\mu_{K^{\prime}}\times\mu_{G^{\prime\prime}}$. Since the product map
$K^{\prime}\times G_{d}^{\prime\prime}\to
Q_{d}=K^{\prime}G_{d}^{\prime\prime}$ is a Borel bijection, we may define
$\mu_{Q_{d}}=\mu_{K^{\prime}}\times\mu_{G_{d}^{\prime\prime}}$.

All in all we have the following:

$\displaystyle\begin{array}[]{c}\mu_{G}=\mu_{K^{\prime}}\times\mu_{K^{\prime\prime}}\times\mu_{A^{\prime\prime}}\times\mu_{A^{\prime}}\times\mu_{N^{\prime\prime}}\times\mu_{N^{\prime}}=\mu_{K^{\prime}}\times\mu_{G^{\prime\prime}}\times\mu_{A^{\prime}}\times\mu_{N^{\prime}}\\\
=\mu_{Q}\times\mu_{A^{\prime}}\times\mu_{N^{\prime}}=\mu_{Q_{d}}\times\mu_{G_{n-d}^{\prime\prime}}\times\mu_{A^{\prime\prime}}\times\mu_{N^{\prime\prime}},\\\
\\\
\mu_{G^{\prime\prime}}=\mu_{G_{d}^{\prime\prime}}\times\mu_{G_{n-d}^{\prime\prime}},\>\mu_{P^{\prime\prime}}=\mu_{P_{d}^{\prime\prime}}\times\mu_{P_{n-d}^{\prime\prime}},\>\mu_{K^{\prime\prime}}=\mu_{K_{d}^{\prime\prime}}\times\mu_{K_{n-d}^{\prime\prime}},\>\mu_{A^{\prime\prime}}=\mu_{A_{d}^{\prime\prime}}\times\mu_{A_{n-d}^{\prime\prime}}.\end{array}$
(4.8)

To abbreviate, we will use $\mu$ instead of $\mu_{G}$.

####  4.4 Interpretation of the RI components of
$g\in\operatorname{SL}_{n}\left(\mathbb{R}\right)$

Like the Iwasawa decomposition, the Refined Iwasawa decomposition induces
coordinates on $\operatorname{SL}_{n}\left(\mathbb{R}\right)$. The RI
components of an element $g\in\operatorname{SL}_{n}\left(\mathbb{R}\right)$
encode certain information regarding the lattices spanned by its columns, as
explained in the proposition below. To state it, we extend the definition of
primitiveness and of factor lattices from $d$-lattices in $\mathbb{Z}^{n}$, to
$d$-lattices in any full lattice of $\mathbb{R}^{n}$.

######  Definition 4.3.

Assume that a $d$-lattice $\Lambda$ is contained inside a full lattice
$\Delta<\mathbb{R}^{n}$. We say that $\Lambda$ is _primitive_ inside (or with
respect to) $\Delta$ if $\Lambda=\Delta\cap V_{\Lambda}$. In other words, if
there is no subgroup of $\Delta$ that lies inside $V_{\Lambda}$ and properly
contains $\Lambda$. Given a $d$-lattice $\Lambda$ that is primitive inside
$\Delta$, we define the _factor lattice of $\Lambda$ (w.r.t. $\Delta$)_,
denoted $\Lambda^{\pi,\Delta}$, as the orthogonal projection of $\Delta$ into
the space $(V_{\Lambda})^{\perp}$. When $\Lambda$ is primitive inside
$\mathbb{Z}^{n}$, we omit the explicit mentioning of $\mathbb{Z}^{n}$, and say
just that $\Lambda$ is primitive. Accordingly, we denote $\Lambda^{\pi}$, and
say that it is the factor lattice of $\Lambda$.

Let us also introduce the following notation.

######  Notation 4.4.

For $g\in\operatorname{SL}_{n}\left(\mathbb{R}\right)$ and
$d\in\left\\{1,\ldots,n-1\right\\}$, let $\Lambda_{g}$ denote the lattice
spanned by the columns of $g$, and let $\Lambda_{g}^{d}$ denote the lattice
spanned by the first $d$ columns of $g$. Let
$\Lambda_{g}^{\underleftarrow{d}}$ denote the lattice spanned by the last $d$
columns of $\Lambda_{g}$. Finally, given a lattice $\Lambda$ in
$\mathbb{R}^{n}$, recall that $V_{\Lambda}$ denotes the linear space spanned
by $\Lambda$.

######  Proposition 4.5.

Let $g\in\operatorname{SL}_{n}\left(\mathbb{R}\right)$ and $1\leq d\leq n-1$.
Denote $\Lambda=\Lambda_{g}^{d}$, the lattice spanned by the first $d$ columns
of $g$, and $\Lambda^{\sharp}=\Lambda^{\pi,\Lambda_{g}}$, the factor lattice
of $\Lambda$ w.r.t. $\Lambda_{g}$. Write $g=kan=qa_{t}^{\prime}n^{\prime}$
with
$q=k_{U}^{\prime}g^{\prime\prime}=k_{U}^{\prime}k^{\prime\prime}a_{\underline{s},\underline{w}}^{\prime\prime}n^{\prime\prime}$.
Let $g^{\prime\prime}=\left[\begin{smallmatrix}g_{d}&0\\\
0&g_{n-d}\end{smallmatrix}\right]$,
$g_{d}^{\prime\prime}=\left[\begin{smallmatrix}g_{d}&0\\\
0&\operatorname{I}_{n-d}\end{smallmatrix}\right]$,
$g_{n-d}^{\prime\prime}=\left[\begin{smallmatrix}\operatorname{I}_{d}&0\\\
0&g_{n-d}\end{smallmatrix}\right]$, and similarly for $p^{\prime\prime}$. The
RI components of $g$ represent parameters related to $\Lambda$ in the
following way:

$\begin{array}[]{cccccccccc}(i)&U&=&V_{\Lambda}&&(i)^{\sharp}&U^{\perp}&=&V_{\Lambda^{\sharp}}\\\
(ii)&e^{t}&=&\operatorname{covol}(\Lambda)&&(ii)^{\sharp}&e^{-t}&=&\operatorname{covol}(\Lambda^{\sharp})\\\
(iii)&e^{\frac{it}{d}-\frac{s_{i}}{2}}&=&\operatorname{covol}(\Lambda^{i})&&(iii)^{\sharp}&e^{-\frac{jt}{n-d}-\frac{w_{j}}{2}}&=&\operatorname{covol}(\left(\Lambda^{\sharp}\right)^{j})\\\
(iv)&\Lambda_{q}^{d}&\in&\left[\Lambda\right]&&(iv)^{\sharp}&\Lambda_{q}^{\underleftarrow{n-d}}&\in&\left[\Lambda^{\sharp}\right]\\\
(v)&\Lambda_{g_{d}}&\in&\left\llbracket\Lambda\right\rrbracket&&(v)^{\sharp}&\Lambda_{g_{n-d}}&\in&\left\llbracket\Lambda^{\sharp}\right\rrbracket\\\
(vi)&\Lambda_{p_{d}}&\in&\operatorname{shape}\left(\Lambda\right)&&(vi)^{\sharp}&\Lambda_{p_{n-d}}&\in&\operatorname{shape}\left(\Lambda^{\sharp}\right)\end{array}$

for every $1\leq i\leq d$ and $1\leq j\leq n-d$, and: $(vii)$
$(\Lambda_{q}^{d},\Lambda_{q}^{\underleftarrow{n-d}})\in\left[(\Lambda_{g}^{d},(\Lambda_{g}^{d})^{\pi,\Lambda_{g}})\right]$.

###### ?proofname?.

Since the columns of $k$ are obtained by performing the Gram-Schmidt
orthogonalization procedure on the columns of $g$, we have that the first $d$
columns of $k$ span $V_{\Lambda}$. Since
$k^{\prime}=k\left(k^{\prime\prime}\right)^{-1}$ where
$k^{\prime\prime}\in\left[\begin{smallmatrix}\operatorname{SO}_{d}\left(\mathbb{R}\right)&0\\\
0&\operatorname{SO}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]$,
then the first $d$ columns of $k^{\prime}$ span (and are in fact an
orthonormal basis to) the same space as the first $d$ columns of $k$, which is
$V_{\Lambda}$. This proves $(i)$ and $(i)^{\sharp}$, by definition of
orientation on $V_{\Lambda}^{\perp}$.

Write $g(n^{\prime})^{-1}(a^{\prime})^{-1}=k^{\prime}g^{\prime\prime}=q$;
right multiplication by an element of $N^{\prime}$ does not change the first
$d$ columns of $g$, and right multiplication by $(a^{\prime})^{-1}$ multiplies
each of these columns by $e^{-\frac{t}{d}}$. This means that
$e^{-\frac{t}{d}}\Lambda=\Lambda_{q}^{d}$, proving $\left(iv\right)$, and
$\left(v\right)$, $\left(vi\right)$ directly follow. Also, notice that
$\Lambda_{q}^{d}$ has covolume one, since it is a rotation of the unimodular
$\Lambda_{g_{d}^{\prime\prime}}^{d}$ ; then, by considering the covolumes of
the lattices on both sides, we obtain $e^{-t}\operatorname{covol}(\Lambda)=1$,
proving $\left(ii\right)$.

Considering $g^{\prime\prime}$, it is clear that the lattice
$\Lambda_{g_{n-d}}$ is the factor lattice of $\Lambda_{g_{d}}$ w.r.t.
$\Lambda_{g^{\prime\prime}}$. Rotating it by left multiplication by
$k^{\prime}$, we have that the lattice $\Lambda_{q}^{\underleftarrow{n-d}}$ is
the factor lattice of $\Lambda_{q}^{d}$ w.r.t. $\Lambda_{q}$. But since
$q=g(n^{\prime})^{-1}(a^{\prime})^{-1}$, we may also say that
$\Lambda_{q}^{\underleftarrow{n-d}}$ is the factor lattice of
$\Lambda_{g(n^{\prime})^{-1}(a^{\prime})^{-1}}^{d}<V_{\Lambda}$ w.r.t.
$\Lambda_{g(n^{\prime})^{-1}(a^{\prime})^{-1}}$, namely it is the projection
of $\Lambda_{g(n^{\prime})^{-1}(a^{\prime})^{-1}}$ to $V_{\Lambda}^{\perp}$.
Noticing that this projection kills the contribution of $(n^{\prime})^{-1}$,
as well as the first $d$ columns of $g(n^{\prime})^{-1}(a^{\prime})^{-1}$, we
remain only with the projection of the lattice spanned by the last $n-d$
columns of $g$, on which $(a^{\prime})^{-1}$ acts as multiplication by
$e^{-\frac{t}{n-d}}$. In other words, $\Lambda_{q}^{\underleftarrow{n-d}}$ is
in fact $e^{-\frac{t}{n-d}}\Lambda^{\sharp}$. This proves
$\left(iv\right)^{\sharp}$, from which $\left(v\right)^{\sharp}$ and
$\left(vi\right)^{\sharp}$ follow, and then similarly to how we proved
$\left(ii\right)$ we also obtain $\left(ii\right)^{\sharp}$111The fact that
$\operatorname{covol}(\Lambda^{\sharp})=\operatorname{covol}(\Lambda_{g})/\operatorname{covol}(\Lambda)$
is also proved in the Appendix, Proposition B.4.. Since $\Lambda_{q}^{d}$ and
$\Lambda_{q}^{\underleftarrow{n-d}}$ span orthogonal subspaces and are both
unimodular, then $\left(iv\right)$ and $\left(iv\right)^{\sharp}$ imply
$(vii)$.

It is well known that if $g=kan$ and
$a=\operatorname{diag}(\alpha_{1},\ldots,\alpha_{r})$, then
${\scriptstyle\prod_{1}^{i}}\alpha_{j}=\operatorname{covol}(\Lambda_{g}^{i})$.
Since the lattice $\Lambda_{g_{d}^{\prime\prime}}^{d}$ is a rotation of
$e^{-t/d}\Lambda$, it has the same covolume and partial covolumes; writing
$g_{d}^{\prime\prime}=k_{d}^{\prime\prime}a_{d}^{\prime\prime}n_{d}^{\prime\prime}$,
we have that the product of the first $1\leq i\leq d$ entries of
$a_{d}^{\prime\prime}$ is
$e^{-\frac{it}{d}}\operatorname{covol}(\Lambda_{g_{d}^{\prime\prime}}^{i})=e^{-\frac{it}{d}}\operatorname{covol}(\Lambda_{g}^{i})$.
On the other hand, since we have
$a_{d}^{\prime\prime}=\operatorname{diag}(e^{-\frac{s_{1}}{2}},e^{\frac{s_{1}-s_{2}}{2}},\ldots,e^{\frac{s_{d-2}-s_{d-1}}{2}},e^{\frac{s_{d-1}}{2}},1,\ldots,1)$,
comparing the products of the first $i$ elements implies
$\operatorname{covol}(\Lambda^{i})\cdot
e^{-\frac{it}{d}}=e^{-\frac{s_{i}}{2}}$ and proves $\left(iii\right)$. Doing
the same for
$a_{n-d}^{\prime\prime}=\operatorname{diag}(1,\ldots,1,e^{-\frac{w_{1}}{2}},e^{\frac{w_{1}-w_{2}}{2}},\ldots,e^{\frac{w_{n-d-2}-w_{n-d-1}}{2}},e^{\frac{w_{n-d-1}}{2}})$,
while recalling that $\Lambda_{g_{n-d}^{\prime\prime}}^{\underleftarrow{n-d}}$
is a rotation of $e^{\frac{t}{n-d}}\Lambda^{\sharp}$, implies
$e^{\frac{jt}{n-d}}\operatorname{covol}((\Lambda^{\sharp})^{j})=e^{-\frac{w_{j}}{2}}$
for every $1\leq j\leq n-d$ and proves $\left(iii\right)^{\sharp}$. ∎

###  5 Sets of representatives for the spaces in Theorem 3.2

In this section we find “isomorphic” copies of the spaces
$\mathcal{X}_{d}\times\mathcal{X}_{n-d}$,
$\mathcal{L}_{d}\times\mathcal{L}_{n-d}$, $\mathcal{L}_{d,n}$,
$\mathcal{P}_{d,n}$ and $\operatorname{Gr}(d,n)$ inside the RI components of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$, namely subsets of the RI
components that are parameterized by these spaces. These will be in fact sets
of representatives for the cosets that are the elements of these spaces, that
are carefully chosen so that they imitate the geometry of the spaces as much
as possible; for example, they will have the property that under the quotient
map (that is one to one on a set of representatives), a BCS in the space will
correspond to a BCS in the set. In [HK20, Section 6] we have defined such
“good” sets of representatives, and named them _spread models_. For the
convenience of the reader, we omit the definition, and instead list here the
properties of spread models that will be used in this work.

######  Proposition 5.1 (see [HK20, Propositions 6.5, 6.7 and 6.10]).

Suppose that a Lie group $H$ acts properly and smoothly on a manifold
$\mathcal{M}$ and $F\subset\mathcal{M}$ is a spread model for the action of
$H$. For measurable sets $B\subseteq\mathcal{M}/H$ and $\mathcal{B}\subseteq
H$ the following holds:

  1. 1.

If $B$ and $\mathcal{B}$ are BCS, then so does
$\mathcal{M}_{B}\cdot\mathcal{B}\subseteq\mathcal{M}$, where $\mathcal{M}_{B}$
is a full set of representatives for $B$ inside $F$; in the case where $H$ is
also discrete, then $B$ is BCS iff $\mathcal{M}_{B}$ is BCS.

  2. 2.

if $\mu_{H}$ and $\mu_{\mathcal{M}}$ are Radon measures on their respective
spaces, then there is a unique Radon measure $\mu_{\mathcal{M}/H}$ on
$\mathcal{M}/H$ so that

$\mu_{\mathcal{M}}\left(\mathcal{M}_{B}\cdot\mathcal{B}\right)=\mu_{\mathcal{M}/H}\left(B\right)\cdot\mu_{H}\left(\mathcal{B}\right).$

The spread model we use for the space of shapes $\mathcal{X}_{i}$ ($i=d,n-d$)
is a fundamental domain for the action of
$\operatorname{SL}_{i}\left(\mathbb{Z}\right)$ on
$\operatorname{SO}_{i}\left(\mathbb{R}\right)\backslash\operatorname{SL}_{i}\left(\mathbb{R}\right)\cong
P_{i}$, whose construction is essentially due to Siegel and is made explicit
in [Gre93, Sch98, HK20, Section 7]. In the case of $i=2$, where $P_{2}$ is
diffeomorphic with the hyperbolic upper half plane, this fundamental domain is
the well known set depicted in Figure 1.

?figurename? 1: $F_{2}$, a fundamental domain for
$\operatorname{SL}_{2}\left(\mathbb{Z}\right)$ in $P_{2}$ (the hyperbolic
upper half plane).

######  Notation 5.2.

We denote this fundamental domain in $P_{i}$ by $F_{i}$. Also, we denote by
$\widetilde{F_{i}}$ the fundamental domain in
$\operatorname{SL}_{i}\left(\mathbb{R}\right)$ for
$\operatorname{SL}_{i}\left(\mathbb{Z}\right)$ that is obtained as

$\widetilde{F_{i}}=\bigcup_{z\in F_{i}}K_{z}\cdot z,$

([HK20, Theorem 7.10 and Proposition 7.13]) where for every $z\in F_{i}$,
$K_{z}\subset\operatorname{SO}_{i}\left(\mathbb{R}\right)$ is a fundamental
domain for the finite group of rotations in
$\operatorname{SO}_{i}\left(\mathbb{R}\right)$ that preserve $\Lambda_{z}$.

Indeed, we will use $F_{i}$ as a spread model for $\mathcal{X}_{i}$, and
$\widetilde{F_{i}}$ as a spread model for $\mathcal{L}_{i}$. Here is the full
list of spread models representing the spaces that appear in Theorem 3.2:

######  Proposition 5.3 ([HK20, Prop. 8.1]).

There exists a spread model $K^{\prime}\subset K$ for the space
$\operatorname{Gr}(d,n)\cong K^{\prime\prime}\backslash K=Q/G^{\prime\prime}$.
Moreover, the following subsets of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$ are spread models for the
corresponding spaces:

  * •

$\widetilde{F_{i}}\subset G_{i}$ for the spaces
$\mathcal{L}_{i}=G_{i}/G_{i}\left(\mathbb{Z}\right)$ for $i=d,n-d$

  * •

$F_{i}\subset P_{i}$ for the spaces $\mathcal{X}_{i}\simeq K_{i}\backslash
G_{i}/G_{i}\left(\mathbb{Z}\right)\simeq P_{i}/G_{i}\left(\mathbb{Z}\right)$

  * •

$K^{\prime}(\widetilde{F_{d}}\times\operatorname{I}_{n-d})\subset
K^{\prime}G_{d}^{\prime\prime}=Q_{d}\subset Q$ for the space
$\mathcal{L}_{d,n}\simeq
Q/G_{d}^{\prime\prime}\left(\mathbb{Z}\right)G_{n-d}^{\prime\prime}\left(\mathbb{R}\right)$

  * •

$K^{\prime}(\widetilde{F_{d}}\times\widetilde{F_{n-d}})\subset
K^{\prime}G^{\prime\prime}=Q$ for the space $\mathcal{P}_{d,n}\simeq
Q/G^{\prime\prime}\left(\mathbb{Z}\right)$

The measures we have defined on this spaces in the Introduction are the unique
measures from part 2 of Proposition 5.1.

###  6 Correspondence between integral matrices and primitive lattices

The goal of this section is to translate Theorem 3.2 into a counting problem
of integral matrices. The first step is to establish a correspondence between
primitive $d$-lattices and integral matrices in a fundamental domain of the
following discrete group of $\operatorname{SL}_{n}(\mathbb{R})$:

$\Gamma:=\left(N^{\prime}\rtimes
G^{\prime\prime}\right)\left(\mathbb{Z}\right)=\left[\begin{array}[]{cc}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&\mathbb{Z}^{d,n-d}\\\
0&\operatorname{SL}_{n-d}\left(\mathbb{Z}\right)\end{array}\right].$

######  Proposition 6.1.

There exists a bijection $\Lambda\leftrightarrow\gamma_{\Lambda}$ between
oriented primitive $d$-lattices and integral matrices in a fundamental domain
of $\operatorname{SL}_{n}\left(\mathbb{R}\right)\curvearrowleft\Gamma$, that
sends a $d$-lattice $\Lambda$ to $\gamma_{\Lambda}$, the unique integral
matrix in the fundamental domain whose first $d$ columns span $\Lambda$.

###### ?proofname?.

The direction $\Leftarrow$ is simple: given
$\gamma\in\Omega\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)$, its columns
span $\mathbb{Z}^{n}$ hence by definition its first $d$ columns span a
primitive $d$–lattice. In the opposite direction, let $\mathbf{B}$ be basis
for $\Lambda$. Since $\Lambda$ is primitive, $\mathbf{B}$ can be completed to
a basis of $\mathbb{Z}^{n}$; let
$\gamma\in\operatorname{SL}_{n}\left(\mathbb{Z}\right)$ be a matrix having
this basis in its columns, with $\mathbf{B}$ in the first $d$ columns. The
orbit $\gamma\cdot\Gamma$, whose elements consist of integral matrices having
a basis for $\Lambda$ in their first $d$ columns, meets $\Omega$ in a single
point, $\gamma_{\Lambda}$. ∎

Let us construct an explicit fundamental domain for $\Gamma$. Let

$\square:=\text{the unit cube $\left(-1/2,1/2\right]^{d\left(n-d\right)}$}$

and, recalling Notation 5.2, set

$\Omega:=K^{\prime}G^{\prime\prime}_{\widetilde{F_{d}}\times\widetilde{F_{n-d}}}A^{\prime}N^{\prime}_{\square}.$
(6.1)

It is easy to see that $\Omega$ is a fundamental domain for the right action
of $\Gamma$ on $\operatorname{SL}_{n}\left(\mathbb{R}\right)$. The next goal
is to reduce the proof of Theorem 3.2 into a problem of counting integral
matrices in subsets of $\operatorname{SL}_{n}\left(\mathbb{R}\right)$, and
specifically of $\Omega$. We begin by defining these subsets.

######  Notation 6.2.

For $T>0$, $\Phi\subseteq\operatorname{Gr}(d,n)$,
$\widetilde{\mathcal{E}}\times\widetilde{\mathcal{F}}\subseteq\mathcal{L}_{d}\times\mathcal{L}_{n-d}$,
$\mathcal{E}\times\mathcal{F}\subseteq\mathcal{X}_{d}\times\mathcal{X}_{n-d}$,
$\Psi\subseteq\mathcal{L}_{d,n}$ and $\varXi\subseteq\mathcal{P}_{d,n}$,
consider

$\Omega_{T}(\Phi,\mathcal{E},\mathcal{F})=\,\Omega\,\cap\,\left\\{g=k^{\prime}k^{\prime\prime}p^{\prime\prime}a^{\prime}n^{\prime}:\begin{matrix}k^{\prime}\in
K^{\prime}_{\Phi},a^{\prime}\in A^{\prime}_{\left[0,T\right]},\\\
p^{\prime\prime}\in
P^{\prime\prime}_{\mathcal{E}\times\mathcal{E}^{\pi}}\end{matrix}\right\\}=K^{\prime}_{\Phi}K^{\prime\prime}P^{\prime\prime}_{\mathcal{E}\times\mathcal{F}}A^{\prime}_{\left[0,T\right]}N^{\prime}_{\square},$
$\Omega_{T}(\Phi,\widetilde{\mathcal{E}},\widetilde{\mathcal{F}})=\,\Omega\cap\left\\{g=k^{\prime}g^{\prime\prime}a^{\prime}n^{\prime}:\begin{matrix}k^{\prime}\in
K^{\prime}_{\Phi},a^{\prime}\in A^{\prime}_{\left[0,T\right]},\\\
g^{\prime\prime}\in
G^{\prime\prime}_{\widetilde{\mathcal{E}}\times\widetilde{\mathcal{E}}^{\pi}}\end{matrix}\right\\}=K^{\prime}_{\Phi}G^{\prime\prime}_{\widetilde{\mathcal{E}}\times\widetilde{\mathcal{F}}}A^{\prime}_{\left[0,T\right]}N^{\prime}_{\square},$
$\Omega_{T}(\Psi,\widetilde{\mathcal{F}})=\,\Omega\,\cap\,\left\\{g=qa^{\prime}n^{\prime}:q\in
Q_{\Psi\times\widetilde{\mathcal{E}}^{\pi}},a^{\prime}\in
A^{\prime}_{\left[0,T\right]}\right\\}=(Q_{d})_{\Psi}(G_{n-d}^{\prime\prime})_{\widetilde{\mathcal{F}}}A^{\prime}_{\left[0,T\right]}N^{\prime}_{\square},$

and

$\Omega_{T}(\varXi)=\,\Omega\,\cap\,\left\\{g=qa^{\prime}n^{\prime}:q\in
Q_{\varXi},a^{\prime}\in
A^{\prime}_{\left[0,T\right]}\right\\}=Q_{\varXi}A^{\prime}_{\left[0,T\right]}N^{\prime}_{\square}.$

Now the following is immediate from Proposition 6.1 and Proposition 4.5:

######  Corollary 6.3.

Consider the correspondence $\Lambda\leftrightarrow\gamma_{\Lambda}$ between
primitive $d$–lattices and matrices in
$\Omega\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)$, and let $T>0$.

  1. 1.

For $\Phi\subseteq\operatorname{Gr}(d,n)$ and
$\mathcal{E}\times\mathcal{E}^{\pi}\subseteq\mathcal{X}_{d}\times\mathcal{X}_{n-d}$,

$\Omega_{T}(\Phi,\mathcal{E},\mathcal{F})\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)=\\{\gamma_{\Lambda}:\operatorname{covol}(\Lambda)\leq
e^{T},V_{\Lambda}\in\Phi,\operatorname{shape}\left(\Lambda\right)\in\mathcal{E},\operatorname{shape}\left(\Lambda^{\pi}\right)\in\mathcal{F}\\};$

  2. 2.

For $\Phi\subseteq\operatorname{Gr}(d,n)$ and
$\widetilde{\mathcal{E}}\times\widetilde{\mathcal{F}}\subseteq\mathcal{L}_{d}\times\mathcal{L}_{n-d}$,

$\displaystyle\Omega_{T}(\Phi,\widetilde{\mathcal{E}},\widetilde{\mathcal{F}})\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)$
$\displaystyle=\\{\gamma_{\Lambda}:\operatorname{covol}(\Lambda)\leq
e^{T}\text{,
}V_{\Lambda}\in\Phi,\left\llbracket\Lambda\right\rrbracket\in\widetilde{\mathcal{E}},\left\llbracket\Lambda^{\pi}\right\rrbracket\in\widetilde{\mathcal{F}}\\};$

  3. 3.

For $\Psi\subseteq\mathcal{L}_{d,n}$ and
$\widetilde{\mathcal{F}}\subseteq\mathcal{L}_{n-d}$,

$\Omega_{T}(\Psi,\widetilde{\mathcal{F}})\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)=\\{\gamma_{\Lambda}:\operatorname{covol}(\Lambda)\leq
e^{T}\text{,
}\left[\Lambda\right]\in\Psi,\left\llbracket\Lambda^{\pi}\right\rrbracket\in\widetilde{\mathcal{F}}\\};$

  4. 4.

For $\varXi\subseteq\mathcal{P}_{d,n}$,

$\displaystyle\Omega_{T}(\varXi)\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)$
$\displaystyle=\\{\gamma_{\Lambda}:\operatorname{covol}(\Lambda)\leq
e^{T},\left[(\Lambda,\Lambda^{\pi})\right]\in\varXi\\}.$

###  7 Neglecting lattices up the cusp: reduction to counting in compact sets

We are now at the point where we have reduced the proof of Theorem 3.2 to a
problem of counting integral matrices inside the subsets of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$ that are defined in Notation
6.2. The main obstacle in handling this counting problem, is that these sets
are not compact: while their $A^{\prime}$ component is bounded in
$\left[0,T\right]$, their $A^{\prime\prime}$ component is not bounded from
above. This section is devoted to tackling this issue, by reducing to counting
in compact subsets of $\Omega$. These subsets will be obtained by truncating
the $A^{\prime\prime}$ coordinates of the sets from Notation 6.2; truncating
the coordinates in $A_{d}^{\prime\prime}$ will result in bounding the $d$
component, and truncating the coordinates in $A_{n-d}^{\prime\prime}$ will
result in bounding the $n-d$ component:

######  Notation 7.1.

For every $\underline{S}=(S_{1},\ldots,S_{d})>\underline{0}$,
$\underline{W}=(W_{1},\dots,W_{n-d})>\underline{0}$ and a subset
$\mathcal{B}\subset G$, let $\mathcal{B}^{\underline{S},\underline{W}}$ denote
the subset

$\mathcal{B}\cap\\{g:\pi_{A^{\prime\prime}}\left(g\right)=a_{\underline{s},\underline{w}},s_{i}\leq
S_{i}\,\forall 1\leq i\leq d,w_{j}\leq W_{j}\,\forall 1\leq j\leq n-d\\},$

where $\pi_{A^{\prime\prime}}$ is the projection to the $A^{\prime\prime}$
component. If $\pi_{A_{n-d}^{\prime\prime}}(\mathcal{B})=e$, then we denote
$\mathcal{B}^{\underline{S}}$, and similarly if
$\pi_{A_{d}^{\prime\prime}}(\mathcal{B})=e$ then we denote
$\mathcal{B}^{\underline{W}}$.

Recall the fundamental domain $\Omega\subset G$ for $\Gamma$ appearing in
Formula 6.1. Accordingly with Notation 7.1, we let

$\Omega_{T}^{\underline{S},\underline{W}}:=K^{\prime}\left[\begin{smallmatrix}\widetilde{F_{d}}^{\underline{S}}\\\
&\widetilde{F_{n-d}}^{\underline{W}}\end{smallmatrix}\right]A_{T}^{\prime}N^{\prime}_{\square}.$

######  Remark 7.2.

It is not hard to see that
$\mu(\Omega_{T}-\Omega_{T}^{\underline{S},\underline{W}})$ is in
$O(e^{nT-\min\left\\{S_{\min},W_{\min}\right\\}})$, where
$S_{\min}=\min_{j}S_{j}$ and $W_{\min}=\min_{j}W_{j}$.

The goal of this section is to prove the following.

######  Proposition 7.3.

Let $\Omega$ be our fundamental domain for $\Gamma$ in $G$, and
$\underline{\sigma}=\left(\sigma_{1},\ldots,\sigma_{d-1}\right)$,
$\underline{\omega}=\left(\omega_{1},\ldots,\omega_{n-d-1}\right)$ where
$0<\sigma_{i},\omega_{i}<1$ $\forall i$. Then for every $\epsilon>0$

$\\#\left|\left(\Omega_{T}-\Omega_{T}^{\underline{\sigma}T,\underline{\infty}}\right)\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)\right|=O_{\epsilon}\left(e^{T\left(n-\sigma_{\min}+\epsilon\right)}\right),\>\\#\left|\left(\Omega_{T}-\Omega_{T}^{\underline{\infty},\underline{\omega}T}\right)\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)\right|=O_{\epsilon}\left(e^{T\left(n-\omega_{\min}+\epsilon\right)}\right)$

and in particular

$\\#\left|\left(\Omega_{T}-\Omega_{T}^{\underline{\sigma}T,\underline{\omega}T}\right)\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)\right|=O_{\epsilon}\left(e^{T\left(n-\min\left\\{\sigma_{\min},\omega_{\min}\right\\}+\epsilon\right)}\right),$

where $\sigma_{\min}=\min\sigma_{i}$, $\omega_{\min}=\min\omega_{i}$.

###### ?proofname?.

Let
$\gamma_{\Lambda}=ka_{\underline{s},\underline{w}}^{\prime\prime}a_{t}^{\prime}n\in\Omega_{T}\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)$.
In what follows, $\Lambda^{i}$ is the lattice spanned by the first $i$ columns
of $\gamma_{\Lambda}$, $(\Lambda^{\pi})^{j}<\Lambda^{\pi}$ is the lattice
spanned by the first $j$ columns of the $n\times(n-d)$ matrix obtained by
projecting the columns of $\gamma_{\Lambda}$ to $V_{\Lambda}^{\perp}$, and
$\left(\Lambda^{\perp}\right)^{k}<\Lambda^{\perp}$ is the lattice spanned by
the first $k$ columns of the matrix that represents $\Lambda^{\perp}$ inside
$\widetilde{F_{n-d}}$.

Recall that
$\gamma_{\Lambda}\in\Omega_{T}-\Omega_{T}^{\left(\underline{\sigma}T,\underline{\infty}\right)}$
if and only if $\exists i\in\left\\{1,\ldots,d-1\right\\}$ for which
$s_{i}\geq\sigma_{i}T$. This implies (using Proposition 4.5 and the fact that
$\Lambda^{i}$ is integral) that
$1\leq\operatorname{covol}(\Lambda^{i})=e^{\frac{it}{d}-\frac{\sigma_{i}t}{2}}\leq
e^{(\frac{i}{d}-\frac{\sigma_{i}}{2})T}$. For the remaining $i\neq
i^{\prime}\in\left\\{1,\ldots,d-1\right\\}$,
$1\leq\operatorname{covol}(\Lambda^{i^{\prime}})=e^{\frac{it}{d}}\leq
e^{\frac{i}{d}T}$.

By [HK19, Prop. 6.4], the number of such possible lattices $\Lambda$ is
$e^{T\left(n-\sigma+\epsilon\right)}$ where $\sigma=\sum\sigma_{i}$. Thus, the
number of $\operatorname{SL}_{n}\left(\mathbb{Z}\right)$ elements $\gamma$ in
$\Omega_{T}-\Omega_{T}^{\left(\underline{\sigma}T,\underline{\infty}\right)}$
is bounded by

$\displaystyle\\#\left|\bigcup_{\begin{subarray}{c}\underline{u}=\left(u_{1},\ldots,u_{d-1}\right)\\\
\in\left\\{0,1\right\\}^{d-1}-\left\\{0\right\\}\end{subarray}}\\{\Lambda:\forall
i,\,\operatorname{covol}(\Lambda_{v}^{i})\in[1,e^{T(\frac{i}{d}-\frac{\sigma_{i}u_{i}}{2})}]\\}\right|$
$\displaystyle=$
$\displaystyle\sum_{\underline{u}\in\left\\{0,1\right\\}^{d-1}-\left\\{0\right\\}}O_{\epsilon}(e^{T(n-\sigma+\epsilon)})=O_{\epsilon}(e^{T(n-\sigma_{\min}+\epsilon)})$

where $\sigma_{\min}=\min\left\\{\sigma_{i}\right\\}$, as $\epsilon$ is
arbitrary.

Now recall that
$\gamma_{\Lambda}\in\Omega_{T}-\Omega_{T}^{\left(\underline{\infty},\underline{\omega}T\right)}$
if and only if $\exists j\in\left\\{1,\ldots,n-d-1\right\\}$ for which
$w_{j}\geq\omega_{j}T$. By Lemma A.15 and Proposition B.5,
$\operatorname{covol}((\Lambda^{\pi})^{j})$ is proportional to the quotient
$\operatorname{covol}((\Lambda^{\perp})^{n-d-j})/\operatorname{covol}(\Lambda)$,
with a positive constant that depends only on $d$ and $n$. Then, by
Proposition 4.5 and Lemma A.15,

$\widetilde{C}e^{-t}\cdot\operatorname{covol}((\Lambda^{\perp})^{n-d-j})=\widetilde{C}\cdot\frac{\operatorname{covol}((\Lambda^{\perp})^{n-d-j})}{\operatorname{covol}(\Lambda)}\leq\operatorname{covol}((\Lambda^{\pi})^{j})=e^{-\frac{jt}{n-d}-\frac{w_{j}}{2}},$

where $\widetilde{C}>0$. Hence, up to an additive constant that becomes
negligible when $t$ is large, $w_{j}\geq\omega_{j}T$ implies

$1\leq\operatorname{covol}((\Lambda^{\perp})^{n-d-j})\leq
e^{t-\frac{jt}{n-d}-\frac{w_{j}}{2}}\leq
e^{T(1-\frac{j}{n-d}-\frac{\omega_{j}}{2})}.$

For the remaining $j\neq j^{\prime}\in\left\\{1,\ldots,n-d-1\right\\}$,
$1\leq\operatorname{covol}(\Lambda^{n-d-j^{\prime}})\leq
e^{t(1-\frac{j^{\prime}}{n-d})}\leq e^{T(1-\frac{j^{\prime}}{n-d})}$. By
[HK19, Cor. 6.4], the number of such possible lattices $\Lambda$ is
$e^{T\left(n-\omega+\epsilon\right)}$ where $\omega=\sum\omega_{i}$. Thus, by
similar considerations, the number
$\\#|(\Omega_{T}-\Omega_{T}^{\left(\underline{\infty},\underline{\omega}T\right)})\cap\operatorname{SL}_{n}\left(\mathbb{Z}\right)|$
is in $O(e^{T\left(n-\omega_{\min}+\epsilon\right)})$, where
$\omega_{\min}=\min\omega_{i}$. ∎

###  8 Almost a proof of the main theorem

In this section, we prove Theorem 3.2 based on a proposition for counting
lattice points in $\operatorname{SL}_{n}\left(\mathbb{R}\right)$ that we state
here and prove in Section 9. The statement includes a parameter $\tau$ of
lattices in Lie groups, that we now define.

In certain Lie groups, among which algebraic simple non-compact Lie groups
$G$, there exists $p\in\mathbb{N}$ for which the matrix coefficients
$\langle\pi_{G/\Gamma}^{0}u,v\rangle$ are in $L^{p+\epsilon}\left(G\right)$
for every $\epsilon>0$, with $u,v$ lying in a dense subspace of
$L_{0}^{2}\left(G/\Gamma\right)$ (see [GN09, Thm 5.6]). Let
$p\left(\Gamma\right)$ be the smallest among these $p$’s, and denote

$m\left(\Gamma\right)=\begin{cases}1&\text{if $p=2$,}\\\ 2\left\lceil
p\left(\Gamma\right)/4\right\rceil&\text{otherwise.}\end{cases}$

Define:

$\tau\left(\Gamma\right)=\frac{1}{2m\left(\Gamma\right)\left(1+\dim
G\right)}\in\left(0,1\right),$

######  Proposition 8.1.

Let $\Gamma<\operatorname{SL}_{n}\left(\mathbb{R}\right)$ be a lattice with
$\tau=\tau\left(\Gamma\right)$. Set
$\lambda_{n}=\frac{n^{2}}{2\left(n^{2}-1\right)}$. If
$\varXi\subseteq\mathcal{P}_{d,n}$ is a BCS, then the following holds for
every $0<\epsilon<\tau$:

  1. 1.

For $\underline{S}=\left(S_{1},\dots,S_{d-1}\right)$ and
$\underline{W}=\left(W_{1},\dots,W_{n-d-1}\right)$, $\mathbf{S}=\sum S_{i}$
and $\mathbf{W}=\sum W_{i}$, and every
$T\geq\frac{\mathbf{S}+\mathbf{W}}{n\lambda_{n}\tau\left(\Gamma\right)}+O_{\varXi}(1)$:

$\\#(\Omega_{T}^{\underline{S},\underline{W}}(\varXi)\cap\Gamma)=\frac{\mu(\Omega_{T}^{\underline{S},\underline{W}}(\varXi))}{\mu\left(\Gamma\backslash
G\right)}+O_{\varXi,\epsilon}(e^{\left(\mathbf{S}+\mathbf{W}\right)/\lambda_{n}}e^{nT\left(1-\tau+\epsilon\right)}).$

  2. 2.

For every $\delta\in\left(0,\tau-\epsilon\right)$ , $T\geq O_{\varXi}(1)$,
$\underline{S}\left(T\right)=\left(S_{1}\left(T\right),\dots,S_{d-1}\left(T\right)\right)$
and
$\underline{W}\left(T\right)=\left(W_{1}\left(T\right),\dots,W_{n-d-1}\left(T\right)\right)$
such that $\mathbf{S}\left(T\right)+\mathbf{W}\left(T\right)=\sum
S_{i}\left(T\right)+\sum W_{i}\left(T\right)\leq
n\delta\lambda_{n}T+O_{\varXi}(1)$:

$\\#(\Omega_{T}^{\underline{S}\left(T\right),\underline{W}\left(T\right)}(\varXi)\cap\Gamma)=\frac{\mu(\Omega_{T}^{\underline{S}\left(T\right),\underline{W}\left(T\right)}(\varXi))}{\mu\left(\Gamma\backslash
G\right)}+O_{\varXi,\epsilon}(e^{nT\left(1-\tau+\delta+\epsilon\right)}).$

When only $\underline{S}$ is bounded the condition on
$\mathbf{S}\left(T\right)+\mathbf{W}\left(T\right)$ becomes
$\mathbf{W}\left(T\right)\leq n\delta\lambda_{n}T+O_{\varXi}(1)$, and when
only $\underline{W}$ is bounded the condition becomes
$\mathbf{S}\left(T\right)\leq n\delta\lambda_{n}T+O_{\varXi}(1)$.

###### Proof of Theorem 3.2 assuming Proposition 8.1.

By Corollary 6.3, when setting $T=\log X$, the quantities we seek to estimate
in parts 1,2,3 and 4 of the theorem is in one to one correspondence with the
integral matrices in the following subsets of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)$, respectively: 1)
$\Omega_{T}(\Phi,\mathcal{E},\mathcal{F})$, 2)
$\Omega_{T}(\Phi,\widetilde{\mathcal{E}},\widetilde{\mathcal{F}})$, 3)
$\Omega_{T}(\Psi,\widetilde{\mathcal{F}})$, or 4) $\Omega_{T}(\varXi)$.
Observe that, according to Section 4.3 on the measures on the RI components,
to Proposition 5.3 on the spread models for $\mathcal{X}_{i}$,
$\mathcal{L}_{i}$, $\mathcal{L}_{d,n}$ and $\mathcal{P}_{d,n}$, to 5.1(2) for
the agreement of the measures on the lattice spaces and the measures on their
spread models, and to our choice of measure on
$\operatorname{SO}_{i}\left(\mathbb{R}\right)$, indeed the main terms in
Theorem 3.2 are the volumes of these sets, divided by the measure of
$\operatorname{SL}_{n}\left(\mathbb{R}\right)/\operatorname{SL}_{n}\left(\mathbb{Z}\right)$
which is
$\operatorname{vol}(\mathcal{L}_{n})=\prod_{i=2}^{n}\zeta\left(i\right)$.

First we claim that it is sufficient to prove part 4 of the theorem, and then
the other parts will follow. Indeed, the family
$\Omega_{T}(\Phi,\mathcal{E},\mathcal{F})$ appearing in part 1 is a special
case of the family
$\Omega_{T}(\Phi,\widetilde{\mathcal{E}},\widetilde{\mathcal{F}})$ appearing
in part 2, by taking $\widetilde{\mathcal{E}},\widetilde{\mathcal{F}}$ to be
the lifts of $\mathcal{E},\mathcal{F}$. According to Lemma 2.5(1),
$\widetilde{\mathcal{E}},\widetilde{\mathcal{F}}$ are BCS’s when
$\mathcal{E},\mathcal{F}$ are, and
$\text{$\operatorname{vol}(\widetilde{\mathcal{E}})$}=\operatorname{vol}(\mathcal{E})\Upsilon(d)$
(similarly for $\widetilde{\mathcal{F}}$ and $n-d$). In particular, the main
term provided in part 1 of the theorem for $\Phi,\mathcal{E},\mathcal{F}$
coincides with the one provided in part 2 for $\Phi$ and the lifts of
$\mathcal{E},\mathcal{F}$. Hence, part 1 is a consequence of part 2.
Similarly, due to Lemma 2.5(3) and (4), parts 2 and 3 respectively are a
consequence of part 4. We conclude that it is sufficient to prove part 4 of
the theorem.

Let $0<\epsilon<\tau_{n}$ and $0<\delta<\tau_{n}-\epsilon$. Suppose first that
the $d$ component of $\varXi\subseteq\mathcal{P}_{d,n}$ is not bounded, and
$n-d$ component is bounded. Recall
$\lambda_{n}=\frac{n^{2}}{2\left(n^{2}-1\right)}$ and let
$\underline{\sigma}_{d}=(\frac{\delta
n\lambda_{n}}{d-1}-\epsilon)\cdot\underline{1}_{d-1}$. Note that the sum of
the coordinates of $\underline{\sigma}_{d}$ is $\delta
n\lambda_{n}-(d-1)\epsilon$, which, for $T$ large enough, is smaller than
$\delta n\lambda_{n}+O_{\varXi}(1)/T$ (aiming to satisfy the condition in
Proposition 8.1). Since the $n-d$ component of $\varXi$ is bounded, there
exists $\underline{W}=\underline{W}_{\varXi}$ such that
$\Omega_{T}(\varXi)=\Omega_{T}^{\infty,\underline{W}_{\varXi}}(\varXi)$. Using
this, along with Propositions 7.3 and 8.1, we have

$\\#(\operatorname{SL}_{n}\left(\mathbb{Z}\right)\cap\Omega_{T}(\varXi))=\\#(\operatorname{SL}_{n}\left(\mathbb{Z}\right)\cap\Omega_{T}^{\infty,\underline{W}_{\varXi}}(\varXi))$
$=\\#(\operatorname{SL}_{n}\left(\mathbb{Z}\right)\cap\Omega_{T}^{\underline{\sigma}_{d}T,\underline{W}_{\varXi}}(\varXi))+O_{\epsilon}(e^{nT\left(1-\frac{\delta\lambda_{n}}{d-1}+\epsilon\right)})$
$=\mu(\Omega_{T}^{\underline{\sigma}_{d}T,\underline{W}_{\varXi}}(\varXi))+O_{\varXi,\epsilon}(e^{nT\left(1-\tau_{n}+\delta+\epsilon\right)})+O_{\epsilon}(e^{nT\left(1-\frac{\delta\lambda_{n}}{d-1}+\epsilon\right)})$

According to Remark 7.2, we may replace the measure of
$(\Omega)^{\underline{\sigma}_{d}T,\underline{W}_{\varXi}}_{T}(\varXi)$ in the
expression above with the measure of $\Omega{}_{T}(\varXi)$, since the
difference in measures is
$O(e^{nT\left(1-\frac{\delta\lambda_{n}}{d-1}\right)})$, which is swallowed in
the error term. Then we choose $\delta$ that will balance the two error terms
above, i.e. $\delta$ that satisfies:
$1-\tau_{n}+\delta=1-\frac{\delta\lambda_{n}}{d-1}$; this means
$\delta=\tau_{n}\cdot(1-\frac{\lambda_{n}}{d-1+\lambda_{n}})=\tau_{n}\cdot(1-\frac{n^{2}}{2\left(d-1\right)\left(n^{2}-1\right)+n^{2}})$.
We conclude that in the case where only the $d$ component of $\varXi$ is
bounded, then
$\\#(\operatorname{SL}_{n}\left(\mathbb{Z}\right)\cap\Omega_{T}(\varXi))=\mu(\Omega_{T}(\varXi))+O_{\varXi,\epsilon}(e^{nT(1-\frac{\tau_{n}n^{2}}{2\left(d-1\right)\left(n^{2}-1\right)+n^{2}}+\epsilon)})$.
Similarly, when only the $n-d$ component is bounded, we define
$\underline{\omega}_{n-d}:=\frac{\delta
n\lambda_{n}}{n-d-1}\cdot\underline{1}_{n-d-1}$ and the error term changes to
$O_{\varXi,\epsilon}(e^{nT(1-\frac{\tau_{n}n^{2}}{2\left(n-d-1\right)\left(n^{2}-1\right)+n^{2}}+\epsilon)})$;
when both $d$ and $n-d$ components are non bounded, the error term changes to
$O_{\varXi,\epsilon}(e^{nT(1-\frac{\tau_{n}n^{2}}{2\left(\max\left\\{d,n-d\right\\}-1\right)\left(n^{2}-1\right)+n^{2}}+\epsilon)})$.
When both $d$ and $n-d$ components are bounded, we use only (the first part
of) Proposition 8.1, and obtain an error term of
$O_{\varXi,\epsilon}(e^{nT\left(1-\tau+\epsilon\right)})$. ∎

###  9 Proof of technical counting proposition

the goal of this section is to prove Proposition 8.1. We need the following
from [GN12]:

######  Definition 9.1.

Let $G$ be a Lie group with a Borel measure $\mu$, and let
$\\{\mathcal{O}_{\epsilon}\\}_{\epsilon>0}$ be a family of identity
neighborhoods in $G$. Assume $\left\\{\mathcal{B}_{T}\right\\}_{T>0}\subset G$
is a family of measurable domains and denote

$\mathcal{B}_{T}^{+}\left(\epsilon\right):=\mathcal{O}_{\epsilon}\mathcal{B}_{T}\mathcal{O}_{\epsilon}=\bigcup_{u,v\in\mathcal{O}_{\epsilon}}u\,\mathcal{B}_{T}\,v,$
$\mathcal{B}_{T}^{-}\left(\epsilon\right):=\bigcap_{u,v\in\mathcal{O}_{\epsilon}}u\,\mathcal{B}_{T}\,v.$

The family $\left\\{\mathcal{B}_{T}\right\\}$ is _Lipschitz well-rounded
(LWR)_ with (positive) parameters $(\mathcal{C},T_{0})$ if for every
$0<\epsilon<1/\mathcal{C}$ and $T>T_{0}$:

$\mu\left(\mathcal{B}_{T}^{+}\left(\epsilon\right)\right)\leq\left(1+\mathcal{C}\epsilon\right)\>\mu\left(\mathcal{B}_{T}^{-}\left(\epsilon\right)\right).$
(9.1)

The parameter $\mathcal{C}$ is called the _Lipschitz constant_ of the family
$\left\\{\mathcal{B}_{T}\right\\}$.

The definition above allows any family
$\left\\{\mathcal{O}_{\epsilon}\right\\}_{\epsilon>0}$ of identity
neighborhoods; in this paper we shall restrict to the following:

######  Assumption 9.2.

We will assume that
$\mathcal{O}_{\epsilon}^{G}=\exp\left(B_{\epsilon}\right)$, where
$B_{\epsilon}$ is an origin-centered $\epsilon$-ball inside the Lie algebra of
$G$, and $\exp$ is the Lie exponent.

######  Remark 9.3.

We allow the case of a constant family
$\left\\{\mathcal{B}_{T}\right\\}=\mathcal{B}$: we say that $\mathcal{B}$ is a
Lipschitz well rounded _set_ (as apposed to a Lipschitz well rounded _family_)
with parameters $(\mathcal{C},\epsilon_{0})$ if
$\mu(\mathcal{B}^{+}\left(\epsilon\right))\leq(1+\mathcal{C}\epsilon)\>\mu(\mathcal{B}^{-}\left(\epsilon\right))$
for every $0<\epsilon<\epsilon_{0}$. It is proved in [HK20, Prop. 3.5] that if
a set $\mathcal{B}$ is BCS and bounded, then it is LWR.

######  Theorem 9.4 ([GN12, Theorems 1.9, 4.5, and Remark 1.10]).

Let $G$ be an algebraic simple Lie group with Haar measure $\mu$, and let
$\Gamma<G$ be a lattice. Assume that $\\{\mathcal{B}_{T}\\}\subset G$ is a
family of finite-measure domains which satisfy $\mu(\mathcal{B}_{T})\to\infty$
as $T\to\infty$. If the family $\\{\mathcal{B}_{T}\\}$ is Lipschitz well-
rounded with parameters $(C_{\mathcal{B}},T_{0})$, then $\exists T_{1}>0$ such
that for every $\delta>0$ and $T>T_{1}$:

$\\#(\mathcal{B}_{T}\cap\Gamma)-\mu(\mathcal{B}_{T})/\mu\left(G/\Gamma\right)=O_{G,\Gamma,\delta}(C_{\mathcal{B}}^{\frac{\dim
G}{1+\dim G}}\cdot\mu(\mathcal{B}_{T})^{1-\tau\left(\Gamma\right)+\delta}),$

where $\mu\left(G/\Gamma\right)$ is the measure of a fundamental domain of
$\Gamma$ in $G$ and $\tau\left(\Gamma\right)\geq\frac{1}{2\left(1+\dim
G\right)}$ is a parameter that depends on the rate of decay of the matrix
coefficients of the $G$-representation on $L_{0}^{2}\left(G/\Gamma\right)$.
The parameter $T_{1}$ is such that $T_{1}\geq T_{0}$ and for every $T\geq
T_{1}$

$C_{\mathcal{B}}^{\frac{\dim G}{1+\dim
G}}=O_{G,\Gamma}(\mu\left(\mathcal{B}_{T}\right)^{\tau\left(\Gamma\right)}).$
(9.2)

###### Proof of Proposition 8.1.

To prove part 1 using Theorem 9.4, we must show that the family
$\\{\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi)\\}_{T>0}$ is
Lipschitz well rounded. Define the map

$\begin{array}[]{cccc}r:&\operatorname{SL}_{n}\left(\mathbb{R}\right)&\to&K\times
A^{\prime}\times A^{\prime\prime}\times N^{\prime\prime}\times N^{\prime}\\\
&g=ka^{\prime}a^{\prime\prime}n^{\prime\prime}n^{\prime}&\mapsto&\left(k,a^{\prime},a^{\prime\prime},n^{\prime\prime},n^{\prime}\right)\end{array}.$

Showing that the image
$\\{r(\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi))\\}_{T>0}$
under $r$ is Lipschitz well rounded with Lipschitz constant and $T_{0}$ that
do not depend on $\underline{S},\underline{W}$, would imply, according to
Corollaries 4.3 in [HK20] and 10.5 in [HK19], that
$\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi)$ is LWR with
Lipschits constant that is
$\prec_{\varXi}e^{2\left(\mathbf{S}+\mathbf{W}\right)}$ and $T_{0}$ that does
not depend on $\underline{S},\underline{W}$. Now,

$r(\Omega^{\underline{S},\underline{W}}_{T}(\varXi))=r(Q_{\varXi^{\underline{S},\underline{W}}})\times
A_{\left[0,T\right]}^{\prime}\times N^{\prime}_{\square}$

with $r(Q_{\varXi^{\underline{S},\underline{W}}})\subset K\times
A^{\prime\prime}\times N^{\prime\prime}$. Each of the three factors in the
above direct product is LWR: $r(Q_{\varXi^{\underline{S},\underline{W}}})$ is
LWR independently of $\underline{S},\underline{W}$ by [HK19, Lemma
11.1]222Actually it was proved for $d=n-1$, but the proof is valid for any
$1<d<n$., $A_{\left[0,T\right]}^{\prime}$ is LWR by [HK19, Proposition 9.6]
and $N^{\prime}_{\square}$ is LWR by [HK20, Proposition 3.5], combined with
the fact that $\square$ is a bounded BCS. Since a direct product of LWR
families is itself LWR, we conclude that
$r(\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi))$ is LWR, and
therefore $\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi)$ is.
The error term in part 1 of the proposition is obtained from Theorem 9.4,
while recalling that the volume of
$\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi)$ in
$\operatorname{SL}_{n}(\mathbb{R})$ is of order $e^{nT}$, and that the
Lipschitz constant of this family is
$\prec_{\varXi}e^{2\left(\mathbf{S}+\mathbf{W}\right)}$. It remains to explain
the lower bound on $T$, but this is obtained from substituting the LWR
parameters of $\Omega^{\left(\underline{S},\underline{W}\right)}_{T}(\varXi)$
into Equation 9.2.

For the proof of the second part, let us compute a bound on
$\mathbf{S},\mathbf{W}$ for which the error term established in part 1 of the
proposition remains smaller than the main term. Namely, that there exists
$\gamma\in\left(0,1\right)$ for which

$(\mathbf{S}+\mathbf{W})/\lambda_{n}+\left(1-\tau\left(\Gamma\right)+\epsilon\right)\cdot
nT=\Gamma\cdot nT.$

If $\delta$ denotes the number $\gamma+\tau\left(\Gamma\right)-\epsilon-1$, we
have that $\gamma=\delta+1+\epsilon-\tau\left(\Gamma\right)$. Then $\gamma<1$
if and only if $\delta<\tau\left(\Gamma\right)-\epsilon$, where
$\tau\left(\Gamma\right)-\epsilon$ is positive since
$\tau\left(\Gamma\right)>\epsilon$. We conclude that for
$0<\delta<\tau\left(\Gamma\right)-\epsilon$ and
$\mathbf{S}+\mathbf{W}=\mathbf{S}\left(T\right)+\mathbf{W}\left(T\right)<\delta\lambda_{n}nT$,
the counting applies with an error term of order $e^{\gamma
nT}=e^{nT\left(1-\tau+\delta+\epsilon\right)}$. As for the lower bound $T_{1}$
on $T$, in part 1 we got $\mathbf{S}+\mathbf{W}\leq
n\lambda_{n}\tau\left(\Gamma\right)T+O_{\varXi}\left(1\right)$; so, combining
both bounds on $\mathbf{S}+\mathbf{W}$ we get

$\mathbf{S}+\mathbf{W}\leq\min\left\\{n\lambda_{n}\delta
T,\>n\lambda_{n}\tau\left(\Gamma\right)T\right\\}+O_{\Psi}\left(1\right)=n\lambda_{n}\delta
T+O_{\varXi}\left(1\right)$

for $T$ large enough and
$\delta\in\left(0,\tau\left(\Gamma\right)-\epsilon\right)$. This completes the
proof. ∎

## Appendix: Dual lattices, factor Lattices

This appendix is completely independent from the paper. It aims to be a source
for facts that are not necessarily unknown or hard to prove, but can be hard
to find in the literature. For example, while the concept of dual lattices
(sometimes referred to as _polar_ lattices) is well known [Cas71], and we have
also found the notes [Reg04] very useful), the concept of factor lattices, a
term coined by Schmidt in [Sch68], is much less known, and in fact we have not
found any reference for it. Another aim of this appendix is to add the
differential perspective of smooth maps between spaces of lattices. For
example, we will see that the map sending a $d$-lattice to its dual is an
auto-diffeomorphism of the space of $d$-lattices inside $\mathbb{R}^{n}$.

In what follows, if the columns of a matrix $\mathbf{B}=\mathbf{B}_{n\times
d}$ span a $d$-lattice $\Lambda<\mathbb{R}^{n}$ (resp. a subspace
$V<\mathbb{R}^{n}$), we say that $\mathbf{B}$ is a basis for the lattice
$\Lambda$ (resp. the subspace $V$). Recall that the covolume of a $d$-lattice
$\Lambda$, which is the volume of the fundamental parallelepiped for $\Lambda$
in the linear subspace that it spans, equals
$(|\det(\mathbf{B}^{\mbox{t}}\mathbf{B})|)^{1/2}$ where $\mathbf{B}$ is (any)
a basis for $\Lambda$. We will use an underscore to denote that an object is
being spanned by a set, so that $\Lambda_{\mathbf{B}}$ is the lattice spanned
(over $\mathbb{Z}$) by $\mathbf{B}$, $V_{\Lambda}$ is the linear space spanned
(over $\mathbb{R}$) by $\Lambda$, etc.

###  ?appendixname? A Dual lattices

Given a $d$-lattice $\Lambda<\mathbb{R}^{n}$, we define the _dual lattice_ of
$\Lambda$ as

$\Lambda^{*}=\left\\{y\in\mbox{span}_{\mathbb{R}}(\Lambda):\forall
x\in\Lambda,\langle x,y\rangle\in\mathbb{Z}\right\\}.$

Note that it is contained in $V_{\Lambda}$, and that a priori, it is unclear
that $\Lambda^{*}$ is indeed a lattice.

######  Proposition A.1.

The matrix $\mathbf{D}_{n\times
d}:=\mathbf{B}(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}$ is basis for
$\Lambda^{*}$.

This proposition motivates the notation $\mathbf{D}=\mathbf{B}^{*}$ for
$\mathbf{D}$ as above, as well as the name _the dual basis_ of $\mathbf{B}$.
The proof will make use of the following lemma:

######  Lemma A.2.

$\mathbf{B}$ and $\mathbf{D}$ span the same $d$-dimensional linear space in
$\mathbb{R}^{n}$, and their columns satisfy the orthonormality relation

$\left\langle b_{i},d_{j}\right\rangle=\begin{cases}1&i=j\\\ 0&i\neq
j\end{cases}$

where $(b_{i})_{i=1}^{d}$ are the columns of $\mathbf{B}$ and
$(d_{i})_{i=1}^{d}$ are the columns of $\mathbf{D}$.

###### ?proofname?.

The matrix $\mathbf{D}$ is of the form $\mathbf{B}g$ where
$g\in\operatorname{GL}_{d}\left(\mathbb{R}\right)$, so $\mathbf{B}$ and
$\mathbf{D}$ span the same linear space in $\mathbb{R}^{n}$. The
orthonormality relation follows from the fact that
$\mathbf{B}^{\mbox{t}}\mathbf{D}=\operatorname{I}_{d}$. ∎

###### Proof of Proposition A.1.

In order to show $\Lambda_{\mathbf{D}}=\Lambda_{\mathbf{B}}^{*}$, we will
prove two inclusions. For
$\Lambda_{\mathbf{D}}\supseteq\Lambda_{\mathbf{B}}^{*}$, we first note that by
Lemma A.2, $V_{\mathbf{D}}\supseteq\Lambda_{\mathbf{B}}^{*}$. As a result, an
element $y\in\Lambda_{\mathbf{B}}^{*}$ is of the form
$y=\sum_{i=1}^{d}\alpha_{i}d_{i}$, where $(d_{i})_{i=1}^{d}$ are the columns
of $\mathbf{D}$. Let $(b_{i})_{i=1}^{d}$ be the columns of $\mathbf{B}$. It
remains to show that $\alpha_{j}\in\mathbb{Z}$ for every $j$, which is the
case since

$\alpha_{j}=\sum_{i=1}^{d}\alpha_{i}\delta_{i,j}=\sum_{i=1}^{d}\alpha_{i}\left\langle
b_{i},d_{j}\right\rangle=\langle
b_{i},\sum_{i=1}^{d}\alpha_{i}d_{j}\rangle=\left\langle
b_{i},y\right\rangle\in\mathbb{Z},$

where in the second equality we applied Lemma A.2, and in the final inclusion
we used $b_{i}\in\Lambda_{\mathbf{B}}$ and $y\in\Lambda_{\mathbf{B}}^{*}$.

For the reverse inclusion
$\Lambda_{\mathbf{D}}\subseteq\Lambda_{\mathbf{B}}^{*}$, it is sufficient to
check that $\mathbf{D}\subseteq\Lambda_{\mathbf{B}}^{*}$, namely that
$\left\langle x,d_{j}\right\rangle\in\mathbb{Z}$ for every
$x\in\Lambda_{\mathbf{B}}$ and $j=1,\ldots,d$. Indeed, write
$x=\sum_{i=1}^{d}\alpha_{i}b_{i}$ where $\alpha_{i}\in\mathbb{Z}$. Then again
by Lemma A.2,

$\left\langle
x,d_{j}\right\rangle=\langle\sum_{i=1}^{d}\alpha_{i}b_{i},d_{j}\rangle=\sum_{i=1}^{d}\alpha_{i}\left\langle
b_{i},d_{j}\right\rangle=\sum_{i=1}^{d}\alpha_{i}\delta_{i,j}=\alpha_{j}\in\mathbb{Z}.$

######  Corollary A.3.

The dual lattice $\Lambda^{*}$ is a $d$-lattice, that spans the same linear
subspace as $\Lambda$.

######  Corollary A.4.

One has that $(\Lambda^{*})^{*}=\Lambda$, and
$\operatorname{covol}(\Lambda^{*})=\operatorname{covol}(\Lambda)^{-1}$.

###### ?proofname?.

Check that $\mathbf{D}(\mathbf{D}^{\mbox{t}}\mathbf{D})^{-1}=\mathbf{B}$ and
that
$(|\det(\mathbf{D}^{\mbox{t}}\mathbf{D})|)^{1/2}=1/(|\det(\mathbf{B}^{\mbox{t}}\mathbf{B})|)^{1/2}$.
∎

######  Example A.5.

The dual of $\mathbb{Z}^{n}$ is $\mathbb{Z}^{n}$, but in general the dual of
an integral (and even primitive) lattice must not be integral. For example,
the dual of $\mathbb{Z}(1,1)$ is $\mathbb{Z}(\frac{1}{2},\frac{1}{2})$. More
generally, when $\Lambda<\mathbb{Z}^{n}$, then the entries of $\Lambda^{*}$
are in the ring $\mathbb{Z}[\frac{1}{\operatorname{covol}(\Lambda)^{2}}]$. To
see this, recall that (Proposition A.1) if $\mathbf{B}$ is a basis for
$\Lambda$, then $\mathbf{B}(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}$ is a basis
for $\Lambda^{*}$. Since
$(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}=\text{adj}\left(\mathbf{B}\right)/\det(\mathbf{B}^{\mbox{t}}\mathbf{B})=\text{adj}\left(\mathbf{B}\right)/\operatorname{covol}(\Lambda)^{2}$
where $\text{adj}\left(\mathbf{B}\right)$ is the adjucate matrix of
$\mathbf{B}$, and $\text{adj}\left(\mathbf{B}\right)$ is integral since
$\mathbf{B}$ is, we get that the entries of
$(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}$ (and therefore also the entries of
$\mathbf{B}(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}$) are in
$\mathbb{Z}[\frac{1}{\operatorname{covol}(\Lambda)^{2}}]$.

Consider the following two spaces:

$\widetilde{\mathcal{L}_{d,n}}:=\operatorname{GL}_{n}\left(\mathbb{R}\right)/\left(\left[\begin{smallmatrix}\operatorname{GL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{GL}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]\right)=\text{space
of $d$-lattices inside $\mathbb{R}^{n}$}$
$\widetilde{\mathcal{P}_{d,n}}:=\operatorname{GL}_{n}\left(\mathbb{R}\right)/\left(\left[\begin{smallmatrix}\operatorname{GL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{GL}_{n-d}\left(\mathbb{Z}\right)\end{smallmatrix}\right]\right)=\begin{array}[]{c}\text{space
of pairs $\left(\Lambda,L\right)$ where $\Lambda$ a $d$-lattice in
$\mathbb{R}^{n}$}\\\ \text{and $L$ is an $n-d$ lattice in
$V_{\Lambda}^{\perp}$}\end{array}$

Let us justify why these quotients are indeed the spaces of lattices we claim
they are. A coset of a group element $g$ inside
$\widetilde{\mathcal{L}_{d,n}}$ corresponds to the $d$-lattice spanned by the
first $d$ columns of $g$. A coset of a group element $g$ inside
$\widetilde{\mathcal{P}_{d,n}}$ corresponds to the pair
$\left(\Lambda,L\right)$, where $\Lambda$ is the $d$-lattice spanned by the
first $d$ columns of $g$, and $L$ is the lattice spanned by the orthogonal
projections of the last $n-d$ columns of $g$ to $V_{\Lambda}^{\perp}$. Note
that since the columns of $g$ are independent, and the first $d$ columns span
$V_{\Lambda}$, then the projections of the last $n-d$ columns to
$V_{\Lambda}^{\perp}$ must be independent; in particular, $L$ is an
$\left(n-d\right)$-lattice. Proving that these two identifications are well
defined and bijective is an easy exercise.

######  Proposition A.6.

The map $\Lambda\mapsto\Lambda^{*}$ is a measure preserving diffeomorphism
from $\widetilde{\mathcal{L}_{d,n}}$ to itself.

###### ?proofname?.

Write a matrix $g\in\operatorname{GL}_{n}\left(\mathbb{R}\right)$, as two
rectangular matrices $g=\left[\mathbf{B}|\mathbf{C}\right]$, where
$\mathbf{B}_{n\times d}$ and $\mathbf{C}_{n-d\times n}$. Define a map from
$\operatorname{GL}_{n}\left(\mathbb{R}\right)$ to itself by
$\left[\mathbf{B}|\mathbf{C}\right]\mapsto\left[\mathbf{B}(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}|\mathbf{C}\right]$.
This map is well defined modulo the subgroup
$\left[\begin{smallmatrix}\operatorname{GL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{GL}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]$,
and so it descends to a map from $\widetilde{\mathcal{L}_{d,n}}$ to itself .
The latter maps the lattice represented by the coset of $g$,
$\Lambda=\Lambda_{\mathbf{B}}$, to
$\Lambda_{\mathbf{B}(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}}$, which is
$\Lambda^{*}$, according to Proposition A.1. Hence, the descended map from
$\widetilde{\mathcal{L}_{d,n}}$ to itself is $\Lambda\mapsto\Lambda^{*}$.
Being an involution by Corollary A.4, this map is bijective and measure
preserving. It is a diffeomorphism (into its image, which is the whole space),
since it is obtained from a map on
$\operatorname{GL}_{n}\left(\mathbb{R}\right)$ which is a diffeomorphism,
because it is algebraic and therefore smooth, and since it is its own inverse,
hence also its inverse is smooth. ∎

Following Proposition A.6, one expects that
$\left(\Lambda,L\right)\mapsto\left(\Lambda^{*},L^{*}\right)$ is a measure
preserving diffeomorphism of $\widetilde{\mathcal{P}_{d,n}}$. Proving this
fact will require the following lemma.

######  Lemma A.7.

Let $\mathbf{B}_{n\times d}$ and $\mathbf{C}_{n\times n-d}$ such that
$\mathbf{B}\cup\mathbf{C}$ is an independent set, and let
$\pi:\mathbb{R}^{n}\to V_{\mathbf{B}}^{\perp}$ be the orthogonal projection.
Let $\mathbf{D}_{n\times n-d}$ denote the last $n-d$ columns of the basis
$[\mathbf{B}|\mathbf{C}]^{*}$. Then
$\Lambda_{\mathbf{D}}=(\Lambda_{\pi\left(\mathbf{C}\right)})^{*}$.

###### ?proofname?.

We want to show that $\Lambda_{\mathbf{D}}$ is the dual of
$\Lambda_{\pi\left(\mathbf{C}\right)}$. First, since the set $\mathbf{C}$ is
independent of $V_{\mathbf{B}}$, then the set $\pi(\mathbf{C})$ is an
independent set inside $V_{\mathbf{B}}^{\perp}$. Being of size
$|\pi\left(\mathbf{C}\right)|=|\mathbf{C}|=n-d=\dim\left(V_{\mathbf{B}}^{\perp}\right)$,
then $\pi\left(\mathbf{C}\right)$ also spans $V_{\mathbf{B}}^{\perp}$.
Moreover, according to Lemma A.2 and the fact that
$[\mathbf{B}|\mathbf{C}]^{*}=[*|\mathbf{D}]$, we have that the columns of
$\mathbf{B}$ and $\mathbf{D}$ are orthogonal, meaning that
$\Lambda_{\mathbf{D}}\subset V_{\mathbf{B}}^{\perp}$. All in all
$V_{\mathbf{D}}=V_{\Lambda_{\pi(\mathbf{C})}}=V_{\mathbf{B}}^{\perp}$. To show
$\Lambda_{\mathbf{D}}\subseteq\Lambda_{\pi(\mathbf{C})}$, it suffices to show
that $\mathbf{D}\subseteq\Lambda_{\pi(\mathbf{C})}$. Namely, that for every
$c\in\mathbf{C}$ and $y\in\mathbf{D}$ it holds that
$\left\langle\pi\left(c\right),y\right\rangle\in\mathbb{Z}$. Indeed,

$\left\langle\pi\left(c\right),y\right\rangle=\left\langle
c,y\right\rangle\in\mathbb{Z}$

where the equality is due to the fact that $y\in V_{\mathbf{B}}^{\perp}$ and
the inclusion holds since $c\in\mathbf{B}|\mathbf{C}$ and
$y\in\left(\mathbf{B}|\mathbf{C}\right)^{*}$. This shows that
$\Lambda_{\mathbf{D}}\subseteq(\Lambda_{\pi\left(\mathbf{C}\right)})^{*}$. For
the reverse inclusion, let $y\in(\Lambda_{\pi\left(\mathbf{C}\right)})^{*}$.
Since in particular $y\in V_{\mathbf{D}}$, so $y=\sum\alpha_{i}d_{i}$. We need
to show that $\alpha_{i}$ are integers. indeed,

$\alpha_{i}=\sum\alpha_{i}\delta_{i,j}=\sum\alpha_{i}\langle
d_{i},\pi(c_{j})\rangle=\langle\sum\alpha_{i}d_{i},\pi(c_{j})\rangle=\langle
y,\pi(c_{j})\rangle\in\mathbb{Z}.$

∎

######  Proposition A.8.

The three maps

$\begin{array}[]{ccc}\left(\Lambda,L\right)\mapsto\left(\Lambda^{*},L\right),&\left(\Lambda,L\right)\mapsto\left(\Lambda,L^{*}\right),&\left(\Lambda,L\right)\mapsto\left(\Lambda^{*},L^{*}\right)\end{array}$

from $\widetilde{\mathcal{P}_{d,n}}$ to itself are measure preserving
diffeomorphisms.

###### ?proofname?.

Consider first the map
$\operatorname{GL}_{n}\left(\mathbb{R}\right)\to\operatorname{GL}_{n}\left(\mathbb{R}\right)$
defined in the proof of Proposition A.6. This map is well defined modulo the
subgroup
$\left[\begin{smallmatrix}\operatorname{GL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{GL}_{n-d}\left(\mathbb{Z}\right)\end{smallmatrix}\right]$,
and so it descends to a map from $\widetilde{\mathcal{P}_{d,n}}$ to itself. As
such, according to the explanation about the identification between cosets of
this subgroup in $\operatorname{SL}_{n}\left(\mathbb{R}\right)$ and elements
$\left(\Lambda,L\right)$ in $\widetilde{\mathcal{P}_{d,n}}$, the descended map
from $\widetilde{\mathcal{P}_{d,n}}$ to itself is
$\left(\Lambda^{*},L\right)\mapsto\left(\Lambda^{*},L\right)$. Similarly to
the proof of Proposition A.6, the map is a measure preserving diffeomorphism.

For the map $\left(\Lambda,L\right)\mapsto\left(\Lambda,L^{*}\right)$,
consider the map from $\operatorname{GL}_{n}\left(\mathbb{R}\right)$ to itself
given by $g=[\mathbf{B}|\mathbf{C}]\mapsto[\mathbf{B}|\mathbf{D}]$, where
$\mathbf{B}=\mathbf{B}_{n\times d}$, $\mathbf{C}=\mathbf{C}_{n\times n-d}$ and
$\mathbf{D}$ is the last $n-d$ columns of the basis
$g^{*}=[\mathbf{B}|\mathbf{C}]^{*}$. This map is algebraic and also an
involution, because by taking the last $n-d$ columns of the basis
$([\mathbf{B}|\mathbf{C}]^{*})^{*}=\mathbf{B}|\mathbf{C}$ we recover
$\mathbf{C}$. It is defined modulo the subgroup
$\left[\begin{smallmatrix}\operatorname{GL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{GL}_{n-d}\left(\mathbb{Z}\right)\end{smallmatrix}\right]$,
and according to Lemma A.7, it descends to the map
$\left(\Lambda,L\right)\mapsto\left(\Lambda,L^{*}\right)$ from
$\widetilde{\mathcal{P}_{d,n}}$ to itself. Again, it is a measure preserving
diffeomorphism. The map
$\left(\Lambda,L\right)\mapsto\left(\Lambda^{*},L^{*}\right)$ is obtained
similarly from
$g=[\mathbf{B}|\mathbf{C}]\mapsto[\mathbf{B}(\mathbf{B}^{\mbox{t}}\mathbf{B})^{-1}|\mathbf{D}]$,
where $\mathbf{B},\mathbf{C},\mathbf{D}$ are as before. ∎

##### Spaces of unimodular lattices

The space
$\mathcal{L}_{d,n}=\operatorname{SL}_{n}\left(\mathbb{R}\right)/\left(\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{SL}_{n-d}\left(\mathbb{R}\right)\end{smallmatrix}\right]\times
A^{\prime}\right)$, where

$A^{\prime}=\left\\{\left[\begin{smallmatrix}\alpha^{\frac{1}{d}}\operatorname{I}_{d}&0_{d\times
n-d}\\\ 0_{n-d\times
d}&\alpha^{-\frac{1}{n-d}}\operatorname{I}_{n-d}\end{smallmatrix}\right]:\alpha>0\right\\},$

is the space of oriented $d$-lattices in $\mathbb{R}^{n}$, up to homothety.
Similarly, the space
$\mathcal{P}_{d,n}=\operatorname{SL}_{n}\left(\mathbb{R}\right)/\left(\left[\begin{smallmatrix}\operatorname{SL}_{d}\left(\mathbb{Z}\right)&\mathbb{R}^{d,n-d}\\\
0_{n-d\times
d}&\operatorname{SL}_{n-d}\left(\mathbb{Z}\right)\end{smallmatrix}\right]\times
A^{\prime}\right)$ is the space of pairs $\left(\Lambda,L\right)$ of oriented
lattices satisfying $\operatorname{covol}(\Lambda)\operatorname{covol}(L)=1$,
where $\Lambda$ is a $d$-lattice and $L$ is an $(n-d)$-lattice in
$V_{\Lambda}^{\perp}$. More accurately, it is the space of equivalence classes
of such pairs, modulo the equivalence relation
$\left(\Lambda^{\prime},L^{\prime}\right)\sim\left(\Lambda,L\right)$ iff there
exists $\alpha>0$ such that
$\left(\Lambda^{\prime},L^{\prime}\right)=\left(\alpha^{\frac{1}{d}}\Lambda,L^{-\frac{1}{n-d}}\right)$.

These spaces, unlike $\widetilde{\mathcal{L}_{d,n}}$ and
$\widetilde{\mathcal{P}_{d,n}}$, have finite volume.

######  Proposition A.9.

Propositions A.6 and A.8 hold also when replacing
$\widetilde{\mathcal{L}_{d,n}}$ and $\widetilde{\mathcal{P}_{d,n}}$ by
$\mathcal{L}_{d,n}$ and $\mathcal{P}_{d,n}$.

Indeed, the dual of a unimodular lattice is also unimodular, by Corollary A.4;
so the map $\Lambda\mapsto\Lambda^{*}$ is an involution of
$\mathcal{L}_{d,n}$, and similarly the maps in Proposition A.8 are involutions
of $\mathcal{P}_{d,n}$. The proofs for the adaptations of Propositions A.6 and
A.8 to unimodular lattices are obtained by adjusting the proofs of these
propositions: the appearances of $\operatorname{GL}$ are replaced by
$\operatorname{SL}$, and the ambient group should be quotiented by
$A^{\prime}$ as well.

######  Remark A.10.

Since the spaces $\mathcal{L}_{d,n}$ and $\mathcal{P}_{d,n}$ are unbounded but
of finite volume, it is desireble to have a criterion for determining when a
set is compact, or alternatively, when does a sequence of elements in the
space diverge to infinity. For the more familiar space of full unimodular
lattices in $\mathbb{R}^{d}$,
$\operatorname{SL}_{d}\left(\mathbb{R}\right)/\operatorname{SL}_{d}\left(\mathbb{Z}\right)$,
which is also non-compact but of finite measure, the answer is provided by
Mahler’s Compactness Criterion [BM00, Chapter V.3]. The latter states that a
set is compact if and only if there exists $\delta>0$ such that all the
lattices in this set have shortest vector of length $>\delta$. Equivalently, a
sequence of lattices $\\{\Lambda_{m}\\}$ diverges if and only if the lengths
of the shortest vectors in $\\{\Lambda_{m}\\}$ is a squence of positive real
numbers that converges to zero. The purpose of the following is to state an
analogous criterion for compactness in the spaces $\mathcal{L}_{d,n}$ and
$\mathcal{P}_{d,n}$. For every equivalence class
$\left[\Lambda_{0}\right]\in\mathcal{L}_{d,n}$ (resp.
$\left[(\Lambda_{0},L_{0})\right]\in\mathcal{P}_{d,n}$), a _unimodular
representative_ is the unique representative
$\Lambda\in\left[\Lambda_{0}\right]$ of covolume one (resp.