# The Equidistribution of Grids of Rings of Integers in Number Fields of
Degrees 3,4 and 5

Yuval Yifrach

###### Abstract

It was shown by M. Bhargava and P. Harron that for $n=3,4,5$, the shapes of
rings of integers of $S_{n}$-number fields of degree $n$ become
equidistributed in the space of shapes when the fields are ordered by
discriminant. Instead of shapes, we correspond grids to each number field,
which preserve more of the number fields’ data. The space of grids is a fiber
bundle over the space of shapes. We strengthen Bhargava-Harron’s result by
proving that the grids of rings of integers of $S_{n}$-number fields become
equidistributed in the space of grids.

##  1 Introduction

Let $\mathbb{F}$ be a number field of degree $n$ and let
$\mathcal{O}_{\mathbb{F}}$ denote its ring of integers. Let
$\sigma_{1},\dots,\sigma_{r}$ denote the $r$ real embeddings of $\mathbb{F}$
and $\tau_{1},\overline{\tau_{1}},\dots,\tau_{s},\overline{\tau_{s}}$ denote
the $s$ pairs of complex embeddings of $\mathbb{F}$. We define the signature
of $\mathbb{F}$ to be $(r,s)$. These embeddings induce a map
$\sigma:\mathbb{F}\rightarrow\mathbb{R}^{r}\oplus\mathbb{C}^{s}\cong\mathbb{R}^{n}$
defined by:

$\mathbb{F}\ni\alpha\mapsto(\sigma_{1}(\alpha),\dots,\sigma_{r}(\alpha),\tau_{1}(\alpha),\dots,\tau_{s}(\alpha))\in\mathbb{R}^{r}\oplus\mathbb{C}^{s}.$

We denote the image of $\mathcal{O}_{\mathbb{F}}$ under $\sigma$ by
$\Lambda(\mathbb{F})$. The space $X_{n}$ of unimodular lattices in
$\mathbb{R}^{n}$ is commonly identified with
$\operatorname{SL}_{n}(\mathbb{R})/\operatorname{SL}_{n}(\mathbb{Z})$ and is
endowed with the quotient topology. Given a lattice $\Lambda$ in
$\mathbb{R}^{n}$ we denote $\left<\Lambda\right>$ to be the corresponding
scaled unimodular lattice. It is well known that $\Lambda(\mathbb{F})$ is a
lattice in $\mathbb{R}^{n}$ and therefore $\left<\Lambda(\mathbb{F})\right>\in
X_{n}$.

It is very natural to study geometric aspects of the lattices
$\Lambda(\mathbb{F})$ as $\mathbb{F}$ varies. It is natural to expect that
unless an obvious obstacle of an algebraic origin exists, various geometric
aspects of the lattices $\Lambda(\mathbb{F})$ would behave ”randomly” as
$\mathbb{F}$ varies. The first naive attempt is to consider
$\left<\Lambda(\mathbb{F})\right>$ as a point in $X_{n}$ equipped with the
uniform ($\operatorname{SL}_{n}(\mathbb{R})$-invariant) probability measure
$m_{X_{n}}$, and expect that the countable collection
$\left<\Lambda(\mathbb{F})\right>$ equidistributes there as $\mathbb{F}$
varies in a natural way. This naive attempt fails drastically because of an
obvious ”algebraic” reason as we shall presently explain. For convenience,
assume that the signature of $\mathbb{F}$ is $(n,0)$. Then the unimodular
lattice $\left<\Lambda(\mathbb{F})\right>$ contains the vector
$v_{\overline{1}}=\left(\frac{1}{\lvert\operatorname{Disc}(\mathbb{F})\rvert}\right)^{\frac{1}{2n}}(1,\dots,1)$
and therefore by Mahler’s compactness criterion (see [7]), we have that not
only does $\Lambda(\mathbb{F})$ fail to equidistribute as $\mathbb{F}$ varies,
but in fact, any compact set in the space of lattices contains only finitely
many such lattices (in other words $\left<\Lambda(\mathbb{F})\right>$ diverges
to infinity as $\mathbb{F}$ varies).

In the paper [3], the authors show, roughly, that orthogonal to the short
vector $v_{\overline{1}}$, the geometry of the lattice
$\left<\Lambda(\mathbb{F})\right>$ does look random as $\mathbb{F}$ varies. In
this paper we consider slightly more refined ways to remedy the divergence to
infinity. That is, we identify a closed sub-homogeneous space of $X_{n}$,
which we call $Y^{\ell_{\overline{1}}}_{1}$ (the reason for this notation will
become clear later), and deform $\left<\Lambda(\mathbb{F})\right>$ into
$Y_{1}^{\ell_{\overline{1}}}$ in a manner that still preserves some of the
geometry of $\Lambda(\mathbb{F})$, and establish the equidistribution of the
resulting points in $Y^{\ell_{\overline{1}}}_{1}$ as $\mathbb{F}$ varies.

In fact, the deformation of $\left<\Lambda(\mathbb{F})\right>$ into
$Y^{\ell_{\overline{1}}}_{1}$ will depend on an ad hoc choice of a linear
direct complement $E$ of the line
$\mathbb{R}\cdot(1,\dots,1)\subset\mathbb{R}^{n}$ (in the case of signature
$(n,0)$, for the remaining signatures the vector will be replaced by a
different one) and it is interesting to note that the answer to the question
”how do the deformed lattices distribute in $Y^{\ell_{\overline{1}}}_{1}$ as
$\mathbb{F}$ varies” depends on the choice of $E$, namely, the asymptotic
behaviour is completely different for $E=V_{0}$ and for $E\neq V_{0}$ where
$V_{0}$ is the orthogonal complement of $(1,\dots,1)$.

We remark that (as will be explained in more detail below), the deformed
lattices in $Y^{\ell_{\overline{1}}}_{1}$ record some of the original
geometrical information of $\Lambda(\mathbb{F})$. For example, we show below
that one can recover the main result of [3] from our main results. As we are
building heavily on the analysis and results from [3], this last remark should
only be taken into account on the motivational level, supporting the ”natural”
nature of our deformation procedure which we shall now define.

For any line $\ell$ in $\mathbb{R}^{n}$ denote:

$Y^{\ell}\vcentcolon=\\{\Lambda\in X_{n}:\Lambda\cap\ell\text{ is a lattice in
}\ell\\},\text{ }Y_{1}^{\ell}\vcentcolon=\\{\Lambda\in
X_{n}:\Lambda\cap\ell\text{ is a unimodular lattice in }\ell\\}.$ (1.1)

######  Definition 1.1.

A Borel probability measure $\mu$ on $X_{n}$ is called homogeneous if there
exists a closed subgroup $H\leq\operatorname{SL}_{n}(\mathbb{R})$ such that
$\mu$ is invariant under $H$ and supported on an $H$ orbit. An orbit
supporting a homogeneous measure is called periodic. Periodic orbits are
closed in $X_{n}$.

Each of the subsets $Y^{\ell},Y^{\ell}_{1}$ is an orbit of a closed subgroup
of $\operatorname{SL}_{n}(\mathbb{R})$. Indeed, $Y^{\ell}$ is the orbit of the
subgroup of elements $g\in\operatorname{SL}_{n}(\mathbb{R})$ satisfying
$g\ell=\ell$ and $Y^{\ell}_{1}$ is the orbit of the subgroup of elements
$g\in\operatorname{SL}_{n}(\mathbb{R})$ satisfying $gv=v$ for every
$v\in\ell$. The orbit $Y^{\ell}$ is dense in $X_{n}$ and the orbit
$Y^{\ell}_{1}$ is closed in $X_{n}$ and in fact periodic. We denote the
homogeneous measure on $Y^{\ell}_{1}$ by $m_{Y^{\ell}_{1}}$. We will use the
following normalization group to deform elements of $Y^{\ell}$ into
$Y^{\ell}_{1}$:

######  Definition 1.2.

Given subspaces $E_{1},E_{2}\subset\mathbb{R}^{n}$ such that $E_{1}\oplus
E_{2}=\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we define a linear operator
$g_{E_{1},E_{2}}^{t}\in\operatorname{SL}_{n}(\mathbb{R})$ by
$g(x)=e^{t/dim(E_{1})}x$, $g(y)=e^{-t/dim(E_{2})}y$ for every $x\in E_{1},y\in
E_{2}$. The set $\\{g_{E_{1},E_{2}}^{t}\\}_{t\in\mathbb{R}}$ is a group, which
we shall call the $E_{1},E_{2}$ normalization group.

Define the vector
$\overline{1}\vcentcolon=(1,\dots,1,1,0,\dots,1,0)\in\mathbb{R}^{n}$ as the
image of
$(1_{\mathbb{R}},\dots,1_{\mathbb{R}},1_{\mathbb{C}},\dots,1_{\mathbb{C}})\in\mathbb{R}^{r}\oplus\mathbb{C}^{s}$
under the identification with $\mathbb{R}^{n}$. Then $\overline{1}$ is an
element of $\Lambda(\mathbb{F})$ because it is the image of the identity in
$\mathbb{F}$ under $\sigma$. Denote
$\ell_{\overline{1}}=\\{t\overline{1}:t\in\mathbb{R}\\}$ and note that
$\left<\Lambda(\mathbb{F})\right>\in Y^{\ell_{\overline{1}}}$. Let
$\left\langle\cdot,\cdot\right\rangle$ denote the standard inner product on
$\mathbb{R}^{n}$. For any subspace $E\subset\mathbb{R}^{n}$, define
$E^{\perp}=\\{x\in\mathbb{R}^{n}:\left\langle x,E\right\rangle=0\\}$ and
$V_{0}=\ell_{\overline{1}}^{\perp}$.

Let $E$ be a subspace of $\mathbb{R}^{n}$ of co-dimension 1 satisfying
$E\oplus\ell_{\overline{1}}=\mathbb{R}^{n}$ and let $t\in\mathbb{R}$ be such
that:

$q_{E}(\mathbb{F})\vcentcolon=g^{t}_{E,\ell_{\overline{1}}}\left<\Lambda(\mathbb{F})\right>\in
Y_{1}^{\ell_{\overline{1}}}$ (1.2)

($t$ will be the solution of
$e^{-t}\operatorname{cov}(\Lambda(\mathbb{F}))^{-1/n}\sqrt{n}=1$). Therefore
$q_{E}(\cdot)$ deforms an element of $Y^{\ell_{\overline{1}}}$ into an element
of $Y_{1}^{\ell_{\overline{1}}}$. The following definition parametrizes a
subset of the number fields using the discriminant.

######  Definition 1.3.

For any $T>0$ and signature $(n-2s,s)$, let $\mathcal{F}^{s}_{n}(T)$ be the
set of isomorphism classes of degree $n$ number fields $\mathbb{F}$ of
signature $(n-2s,s)$, with discriminant $\leq T$ in absolute value such that
the Galois group of the normal closure of $\mathbb{F}$ is $S_{n}$. It is well
known that $\mathcal{F}_{n}(T)$ is finite for any $T>0$.

So far we have constructed a correspondence between number fields and elements
of $Y_{1}^{\ell_{\overline{1}}}$, which are essentially lattices with some
extra properties. As appeared earlier, the aim of this paper is to analyze the
connection between number fields and lattices given by this correspondence.
More concretely, we will analyze the asymptotic distribution as
$T\rightarrow\infty$ of
$\\{q_{E}(\mathbb{F})\\}_{\mathbb{F}\in\mathcal{F}_{n}^{s}(T)}$ for the
different choices for $E$. The answer will turn out to be dependent upon the
choice of $E$ and to imply Bhargava-Harron’s main result in [3].

###  1.1 New Equidistribution Results for Number Fields of Degrees 3,4,5

In this subsection we state our new results which implement the philosophy
described in the first part of this introduction.

######  Theorem 1.4.

Let $n=3,4,5$ and let $(r,s)$ be a signature. Let $E\subset\mathbb{R}^{n}$ be
a subspace such that $E\oplus\ell_{\overline{1}}=\mathbb{R}^{n}$ and $E\neq
V_{0}$. Then:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{q_{E}(\mathbb{F})}\longrightarrow
m_{Y_{1}^{\ell_{\overline{1}}}}\text{ weakly as }T\longrightarrow\infty.$
(1.3)

######  Theorem 1.5.

Let $n=3,4,5$ and let $(n-2s,s)$ be a signature. Denote
$\operatorname{Div}(n)=\\{1\leq k\leq n:k\mid n\\}$ to be the set of divisors
of $n$. Then for any $k\in\operatorname{Div}(n)$ there exists a periodic orbit
$F^{(k)}\subset Y_{1}^{\ell_{\overline{1}}}$, $\ell=\ell(k)\in\mathbb{N}$,
disjoint closed connected subsets $F^{(k)}_{1},\dots,F^{(k)}_{\ell(k)}\subset
F^{(k)}$ and nonnegative scalars $s_{k,\ell};\ell=1,\dots,\ell(k)$ not all
zero, such that:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{q_{V_{0}}(\mathbb{F})}\longrightarrow\sum_{k\in\operatorname{Div}(n)}\sum_{\ell=1}^{\ell(k)}s_{k,\ell}\mu^{(k)}\mid_{F^{(k)}_{\ell}}\text{
weakly as }T\longrightarrow\infty.$ (1.4)

where $\mu^{(k)}$ is the homogenous measure supported on the periodic orbit
$F^{(k)}$.

The two Theorems above are the main results of this paper. In order to
motivate them we explain how each of them implies Bhargava-Harron’s main
result in [3] which is stated below as Theorem 1.6.

First we give additional background required to state [3, Theorem 1]. In the
above setting, consider the orthogonal projection of $\Lambda(\mathbb{F})$ on
$V_{0}$ and denote it by $P\Lambda(\mathbb{F})$. Fix a linear isometry $\phi$
between $V_{0}$ and $\mathbb{R}^{n-1}$ and use it to identify
$P\Lambda(\mathbb{F})$ with its image under $\phi$. From now on we think of
$P\Lambda(\mathbb{F})$ as a subset of $\mathbb{R}^{n-1}$. Since
$\overline{1}\in\Lambda(\mathbb{F})$, $P\Lambda(\mathbb{F})$ is a lattice in
$\mathbb{R}^{n-1}$. The shape of $\mathbb{F}$, denoted $s(\mathbb{F})$, is
defined as the $\operatorname{O}_{n-1}(\mathbb{R})$-orbit of the lattice
$\left<P\Lambda(\mathbb{F})\right>$ inside $X_{n-1}$. It is thus an element of
the double quotient
$\operatorname{O}_{n-1}(\mathbb{R})\setminus\operatorname{SL}_{n-1}(\mathbb{R})/\operatorname{SL}_{n-1}(\mathbb{Z})$.
This double quotient is called the space of $n-1$-dimensional shapes and is
denoted $\mathcal{S}_{n-1}$. Let $m_{\mathcal{S}_{n-1}}$ be the measure on
$\mathcal{S}_{n-1}$ coming from the Haar measure on
$\operatorname{SL}_{n-1}(\mathbb{R})$. Bhargava-Harron prove:

######  Theorem 1.6 ( [3], Theorem 1).

Let $n=3,4,5$. Then:

$\frac{1}{\lvert\cup_{s}\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\cup_{s}\mathcal{F}^{s}_{n}(T)}\delta_{s(\mathbb{F})}\longrightarrow
m_{\mathcal{S}_{n-1}}\text{ weakly as }T\longrightarrow\infty.$ (1.5)

In a nutshell, we will explain why $q_{E}(\mathbb{F})$ contains more
information than $s(\mathbb{F})$ and how to use it to deduce Bhargava-Harron’s
result from either one of Theorems 1.4,1.5. Indeed, there are natural
projections:

$Y_{1}^{\ell_{\overline{1}}}\xrightarrow{\pi_{1}}X_{n-1}\xrightarrow{\pi_{2}}\mathcal{S}_{n-1}$
(1.6)

defined for any $\Lambda\in Y_{1}^{\ell_{\overline{1}}}$ by
$\pi_{1}(\Lambda)=\left<P_{V_{0}}(\Lambda)\right>$ and for any $\Lambda\in
X_{n-1}$ by $\pi_{2}(\Lambda)=\operatorname{O}_{n-1}(\mathbb{R})\Lambda$.
Recall that $q_{E}(\mathbb{F})$ was defined as
$g^{t}_{E,\ell_{\overline{1}}}\left<\Lambda(\mathbb{F})\right>$ where
$t\in\mathbb{R}$ was chosen to make sure that
$g^{t}_{E,\ell_{\overline{1}}}\left<\Lambda(\mathbb{F})\right>\in
Y_{1}^{\ell_{\overline{1}}}$. By definition of
$g_{E,\ell_{\overline{1}}}^{t}$, $P_{V_{0}}(\Lambda(\mathbb{F}))$ and
$P_{V_{0}}(g_{E,\ell_{\overline{1}}}^{t}\left<\Lambda(\mathbb{F})\right>)$
have the same projection on $V_{0}$ up to scalar multiple. Since
$s(\mathbb{F})$ was defined as the $O_{n-1}$ orbit of this projection (after
normalizing by the covolume) we deduce that:

$s(\mathbb{F})=\pi_{2}\circ\pi_{1}(q_{E}(\mathbb{F})).$ (1.7)

Let $m_{X_{n-1}}$ be the natural measure on $X_{n-1}$ coming from
$\operatorname{SL}_{n-1}(\mathbb{R})$. Denote
$(\pi_{1})_{*}m_{Y_{1}^{\ell_{\overline{1}}}}$ and $(\pi_{2})_{*}m_{X_{n-1}}$
to be the pushforward measures. It is simple to check that:

$(\pi_{1})_{*}m_{Y_{1}^{\ell_{\overline{1}}}}=m_{X_{n-1}}\text{ and
}(\pi_{2})_{*}m_{X_{n-1}}=m_{\mathcal{S}_{n-1}}.$ (1.8)

Moreover, the probability measures
$\mu_{i}=\frac{1}{\mu(F_{i})}\mu\mid_{F_{i}}$ appearing in Theorem 1.5 will
turn to be invariant under the action of the group
$G_{0}(\ell_{\overline{1}}):=\\{g\in\operatorname{SL}_{n}(\mathbb{R}):gv=v\text{
for all }v\in\ell_{\overline{1}}\text{ and }gV_{0}=V_{0}\\}$ and the following
diagram will commute:

${G_{0}(\ell_{\overline{1}})\curvearrowright}$${F_{i}}$${\operatorname{SL}_{n-1}(\mathbb{R})\curvearrowright}$${X_{n-1}.}$$\scriptstyle{g\mapsto\phi\circ(g\mid_{V_{0}})\circ\phi^{-1}}$$\scriptstyle{\pi_{1}}$
(1.9)

We deduce that for any $i$:

$(\pi_{1})_{*}\mu_{i}=m_{X_{n-1}}$ (1.10)

which together with Equation (1.8) implies for any $i$:

$(\pi_{2}\circ\pi_{1})_{*}m_{Y_{1}^{\ell_{\overline{1}}}}=(\pi_{2}\circ\pi_{1})_{*}\mu_{i}=m_{\mathcal{S}_{n-1}}.$
(1.11)

Fix a signature $(r,s)$. By equation (1.7) for any $T>0$:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{s(\mathbb{F})}=\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{\pi_{2}\circ\pi_{1}(q_{E}(\mathbb{F}))}=(\pi_{2}\circ\pi_{1})_{*}\left(\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{q_{E}(\mathbb{F})}\right)$
(1.12)

and by Theorem 1.4, equation (1.11) and the continuity of the pushforward
operation:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{s(\mathbb{F})}\longrightarrow(\pi_{2}\circ\pi_{1})_{*}m_{Y_{1}^{\ell_{\overline{1}}}}=m_{\mathcal{S}_{n-1}}\text{
weakly as }T\longrightarrow\infty.$ (1.13)

Averaging the above equation over all $s=0,\dots,\lfloor n/2\rfloor$ recovers
Bhargava-Harron’s result. Therefore Theorem 1.4 implies Bhargava-Harron’s
Theorem 1.6. To show that Theorem 1.5 also implies Theorem 1.6 note that for
any $T>0$:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{s(\mathbb{F})}=\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{\pi_{2}\circ\pi_{1}(q_{V_{0}}(\mathbb{F}))}=(\pi_{2}\circ\pi_{1})_{*}\left(\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{q_{V_{0}}(\mathbb{F})}\right)$
(1.14)

and by Theorem 1.5, equation (1.11) and the continuity of the pushforward
operation:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{s(\mathbb{F})}\longrightarrow\sum_{i=1}^{k}s_{i}\mu(F_{i})(\pi_{2}\circ\pi_{1})_{*}\mu_{i}=m_{\mathcal{S}_{n-1}}\text{
weakly as }T\longrightarrow\infty.$ (1.15)

###  1.2 Testing the Construction on Quadratic Number Fields

Quadratic number fields are well studied objects and their rings of integers
can be fully described. Therefore, we would like to motivate our construction
by examining it for quadratic number fields $\mathbb{F}$. More specifically,
we would like to analyze the ’asymptotic portion’ of quadratic number fields
with predetermined geometric behaviour phrased using $q_{E}$. The following
theorem summerizes this analysis:

######  Theorem 1.7.

Let $n=2$ and let $(r,s)$ be a signature. Let $E$ be a 1-dimensional subspace
in $\mathbb{R}^{2}$ such that $E\neq\ell_{\overline{1}},V_{0}$. Then:

$\frac{1}{\lvert\mathcal{F}^{s}_{2}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{2}(T)}\delta_{q_{E}(\mathbb{F})}\longrightarrow
m_{Y_{1}^{\ell_{\overline{1}}}}\text{ weakly as }T\longrightarrow\infty.$
(1.16)

When $E=V_{0}$ there exists a periodic orbit $\\{p_{1},p_{2}\\}\subset
Y_{1}^{\ell_{\overline{1}}}$ and $t\in(0,1)$ such that:

$\frac{1}{\lvert\mathcal{F}^{s}_{2}(T)\rvert}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{2}(T)}\delta_{q_{E}(\mathbb{F})}\longrightarrow
t\delta_{p_{1}}+(1-t)\delta_{p_{2}}\text{ weakly as }T\longrightarrow\infty.$
(1.17)

Theorem 1.7 is a simple test case due to the complete description of rings of
integers in quadratic fields. We use this description to sketch a proof of the
theorem.

For simplicity of this exposition, we limit ourselves to the case $s=0$.
Firstly, it is easy to see that when $n=2$, $Y_{1}^{\ell_{\overline{1}}}$ is
isomorphic to the unit circle $S^{1}$ and that
$m_{Y_{1}^{\ell_{\overline{1}}}}$ is the Lebesgue measure $\lambda$ on
$S^{1}$. Secondly, let $D$ be a square free natural number (nonnegative,
because $s=0$). Given $D=1\mod 4$, denote $\omega=\frac{1+\sqrt{D}}{2}$. It is
well known that:

$\mathcal{O}_{\mathbb{Q}(\sqrt{D})}=\\{a+b\omega:a,b\in\mathbb{Z}\\}.$ (1.18)

Let $\overline{u}=(1,u)\in\mathbb{R}^{2}$ such that $u\neq-1$ and let
$E=\overline{u}^{\perp}$. Using this description,
$q_{E}(\mathbb{Q}(\sqrt{D}))$ can be calculated to be (This calculation is
carried in grated generality in the proof of Lemma 2.21 below, we refer the
reader to this proof for reference):

$q_{E}(\mathbb{Q}(\sqrt{D}))=\frac{1}{2(1+u)}\left(1+u+(1-u)\sqrt{D}\right)\operatorname{mod}1\in
S^{1}.$ (1.19)

Denote for any $T>0$ and $i=0,\dots,3$:

$\mathcal{D}_{T}(i)=\\{0<D<T:\text{ }D\text{ is a square free natural number
and }D=i\mod 4\\}.$ (1.20)

Assume further that $u\neq 1$ so that $\overline{u}\neq\overline{1}$. Then
classical asymptotic estimations on the density of square free integers (see
[9, Theorem 8.2.1]) and equation (1.19) show that:

$\frac{1}{\lvert\mathcal{D}_{T}(1)\rvert}\sum_{D\in\mathcal{D}_{T}(1)}\delta_{q_{E}(\mathbb{Q}(\sqrt{D}))}\longrightarrow\lambda\text{
as }T\longrightarrow\infty.$ (1.21)

When $D=2,3\mod 4$ the description of $\mathbb{Q}(\sqrt{D})$ is different (and
also well known), but also shows in the same way that:

$\frac{1}{\lvert\mathcal{D}_{T}(2)\cup\mathcal{D}_{T}(3)\rvert}\sum_{D\in\mathcal{D}_{T}(2)\cup\mathcal{D}_{T}(3)}\delta_{q_{E}(\mathbb{Q}(\sqrt{D}))}\longrightarrow\lambda\text{
as }T\longrightarrow\infty.$ (1.22)

Averaging equations (1.21) and (1.22) implies the first statement of Theorem
1.7.

When $\overline{u}=\overline{1}$ and $D=1\operatorname{mod}4$ is square free,
equation (1.19) says
$q_{V_{0}}(\mathbb{Q}(\sqrt{D}))=q_{E}(\mathbb{Q}(\sqrt{D}))=1/2+\mathbb{Z}$.
When $D=2,3\operatorname{mod}4$ the same calculation leading to equation
(1.19) which as we said is given in Lemma 2.21 below, says that
$q_{V_{0}}(\mathbb{Q}(\sqrt{D}))=q_{E}(\mathbb{Q}(\sqrt{D}))=\mathbb{Z}$. From
this it is immediate to deduce that for some $t\in(0,1)$ (related to the
density of $\mathcal{D}_{T}(1)$ in $\mathbb{Z}$):

$\frac{1}{\lvert\mathcal{D}_{T}(1)\cup\mathcal{D}_{T}(2)\cup\mathcal{D}_{T}(3)\rvert}\sum_{D\in\mathcal{D}_{T}(1)\cup\mathcal{D}_{T}(2)\cup\mathcal{D}_{T}(3)}\delta_{q_{V_{0}}(\mathbb{Q}(\sqrt{D}))}\longrightarrow
t\delta_{0}+(1-t)\delta_{1/2}\text{ as }T\longrightarrow\infty$ (1.23)

proving equation (1.17) and the second statement of Theorem 1.7.

##  2 The Ingredients

###  2.1 Notation

Given a natural number $n$, a ring of rank $n$ over another ring $R$ is a ring
which is also a free $R$-module. Given a ring $R$, we denote
$\operatorname{GL}_{n}(R)$ to be the space of all invertible $n\times n$
matrices with entries from $R$. The subset of $\operatorname{GL}_{n}(R)$ of
all matrices with determinant $1$ is denoted $\operatorname{SL}_{n}(R)$.
$(\mathbb{R}^{n},\left\langle\cdot,\cdot\right\rangle)$ denotes the standard
inner product structure of $\mathbb{R}^{n}$. Given a linear subspace
$E\subset\mathbb{R}^{n}$, we denote $E^{\perp}=\\{y\in\mathbb{R}^{n}:\text{
}\left\langle y,x\right\rangle=0\text{ for all }x\in E\\}$ and we denote
$P_{E}:\mathbb{R}^{n}\rightarrow E$ to be the orthogonal projection onto $E$.
$\operatorname{Vol}_{n}$ denotes the standard Lebesgue measure on
$\mathbb{R}^{n}$. Given linear subspaces $E_{1},E_{2}\subset\mathbb{R}^{n}$,
we write $\mathbb{R}^{n}=E_{1}\oplus E_{2}$ if $E_{1}+E_{2}=\mathbb{R}^{n}$
and $E_{1}\cap E_{2}=\\{0\\}$. The set of all matrices
$g\in\operatorname{GL}_{n}(\mathbb{R})$ such that $g^{t}g=gg^{t}=I$ is denoted
by $O(n)$, and $\operatorname{O}(n)\cap\operatorname{SL}_{n}(\mathbb{R})$ is
denoted $\operatorname{SO}(n)$. For a natural number $n$ we denote
$\Sigma_{n}$ to be a fundamental domain for the action of
$\operatorname{SL}_{n}(\mathbb{Z})$ on $\operatorname{SL}_{n}(\mathbb{R})$.
When additional properties of the fundamental domain are required, we will
state so explicitly. Given a locally compact topological space $X$ and Borel
measures $(\mu_{n})_{n\geq 0},\mu$ on $X$, we say that $\mu$ is a weak limit
of $\mu_{n}$ if for any continuous and compactly supported function
$f:X\rightarrow\mathbb{R}$, $\int fd\mu_{n}\longrightarrow\int fd\mu$. Given a
finite Borel measure $\mu$ on a topological space $X$, a continuous map
$\pi:X\rightarrow Y$ to another topological space $Y$, $\pi_{*}\mu$ denotes
the push-forward measure of $\mu$, which is a Borel measure on $Y$, defined by
$\pi_{*}(\mu)(A)=\mu(\pi^{-1}A)$ for any Borel set $A\subset Y$. In case $\mu$
is any measure (not necessarily finite), a Jordan measurable subset $A\subset
X$ is a Borel measurable set such that $\mu(\partial A)=0$. Given a subset
$R\subset\mathbb{R}^{m}$, we denote $N(R)$ to be the size of
$R\cap\mathbb{Z}^{m}$. Given $n\in\mathbb{N}$, we denote
$[n]=\\{1,\dots,n\\}.$ A number field is a finite dimensional field extension
$\mathbb{F}\supset\mathbb{Q}$. Given a number field $\mathbb{F}$ of degree $n$
and number $s=1,\dots,\lfloor\frac{n}{2}\rfloor$, we say that $\mathbb{F}$ is
of signature $(n-2s,s)$ if the minimal polynomial $p$ of $\mathbb{F}$ has
precisely $s$ pairs of conjugate complex roots. Given a natural numbers
$n,k,d$, a function $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$ is said to be
homogeneous of degree $d$ if $f(\lambda v)=\lambda^{d}f(v)$ for every
$v\in\mathbb{R}^{n}$ and $\lambda>0$. An $n$-dimensional lattice is a subset
of $\mathbb{R}^{n}$ given by $\mathbb{Z}$-span of a basis for
$\mathbb{R}^{n}$. The covolume of a lattice in a Euclidean space
$\Lambda\subset(\mathbb{R}^{n},\left\langle\cdot,\cdot\right\rangle)$ denoted
$\text{cov}(\Lambda)$, is defined as $\operatorname{Vol_{n}}(\mathcal{F})$ and
$\mathcal{F}$ is any fundamental domain for the additive action of $\Lambda$
on $\mathbb{R}^{n}$. Given a lattice $\Lambda\subset\mathbb{R}^{n}$,
$\Lambda^{*}$ is the dual lattice defined by
$\\{y\in\mathbb{R}^{n}:\left\langle x,y\right\rangle\in\mathbb{Z}\text{ for
all }x\in\Lambda\\}$. Given an $n$-dimensional lattice $\Lambda$, we denote
$\left<\Lambda\right>=\text{cov}(\Lambda)^{-1/n}\Lambda$. $S_{n}$ denotes the
symmetric group of order $n$. The space of unimodular lattices in
$\mathbb{R}^{n}$ is denoted $X_{n}$ and sometimes $X(\mathbb{R}^{n})$.

###  2.2 New Lattice Normalization

In this subsection we repeat in more detail the construction carried in the
introduction.
For every degree $n$ number field $\mathbb{F}$ of signature $(r,s)$ we denote
$\sigma_{1},\dots,\sigma_{r}$ to be a fixed ordering of the real embeddings
and $\tau_{1},\overline{\tau_{1}},\dots,\tau_{s},\overline{\tau_{s}}$ to be a
fixed ordering of the pairs of complex embeddings. We will use this ordering
throughout the paper. To stress this fact, we introduce:

######  Notation 2.1.

For any degree $n$ number field $\mathbb{F}$ of signature $(r,s)$,
$\\{\sigma_{1},\dots,\sigma_{r},\tau_{1},\overline{\tau_{1}},\dots,\tau_{s},\overline{\tau_{s}}\\}$
denotes a fixed arbitrary ordering of the natural embeddings of $\mathbb{F}$.

For every $\pi=(\pi_{r},\pi_{s})\in S_{r}\times S_{s}$, we denote
$\Lambda_{\pi}(\mathbb{F})$ to be the image of $\mathcal{O}_{\mathbb{F}}$
under

$(\sigma_{\pi_{r}(1)},\dots,\sigma_{\pi_{r}(r)},\tau_{\pi_{s}(1)},\dots,\tau_{\pi_{s}(s)}):\mathbb{F}\rightarrow\mathbb{R}^{r}\oplus\mathbb{C}^{s}\cong\mathbb{R}^{n}.$
(2.1)

It is well known that $\Lambda_{\pi}(\mathbb{F})$ is a lattice in
$\mathbb{R}^{n}$. Define the vector
$\overline{1}\vcentcolon=(1,\dots,1,1,0,\dots,1,0)\in\mathbb{R}^{n}$ to be the
image of
$(1_{\mathbb{R}},\dots,1_{\mathbb{R}},1_{\mathbb{C}},\dots,1_{\mathbb{C}})\in\mathbb{R}^{r}\oplus\mathbb{C}^{s}$
under the identification with $\mathbb{R}^{n}$. Let
$\left\langle\cdot,\cdot\right\rangle$ be the standard inner product on
$\mathbb{R}^{n}$, let $V_{0}=\\{x\in\mathbb{R}^{n}:\left\langle
x,\overline{1}\right\rangle=0\\}$ and denote
$\ell_{\overline{1}}=\mathbb{R}\cdot\overline{1}$. Now we carry in detail the
construction of $q_{E}(\mathbb{F})$ from the introduction.

Since the language of grids will be useful for us, we introduce the following
definition:

######  Definition 2.1.

The space of $n$-dimensional grids is defined by:

$Y_{n}=\\{\Lambda+v:\Lambda\in X_{n},v\in\mathbb{R}^{n}\\}.$ (2.2)

Any $M\in Y_{n}$ uniquely determines a lattice $\Lambda_{M}\in X_{n}$ such
that there exists $v\in\mathbb{R}^{n}$, defined up to $\Lambda_{M}$,
satisfying $M=\Lambda_{M}+v$. Denote the corresponding element in
$\mathbb{R}^{n}/\Lambda$ by $\text{vec}(M)$ and call it the translating vector
of $M$.

Note that $X(\mathbb{R}^{n})$ is embedded naturally inside $Y_{n}$ by the
identity map (taking $v=0$ in the definition of the grid). Next, we come back
to the definition of the space in which we shall have equidistribution.

######  Definition 2.2.

For any line $\ell$ in $\mathbb{R}^{n}$ define:

$Y^{\ell}(\mathbb{R}^{n})\vcentcolon=\\{\Lambda\in X_{n}:\Lambda\cap\ell\text{
is a lattice in }\ell\\}$

and

$Y_{1}^{\ell}(\mathbb{R}^{n})\vcentcolon=\\{\Lambda\in
X_{n}:\Lambda\cap\ell\text{ is a unimodular lattice in }\ell\\}$

The relation between this definition and the language of grids is the
following. For any line $\ell$ in $\mathbb{R}^{n+1}$ there is a natural
bijection from $Y_{1}^{\ell}(\mathbb{R}^{n+1})$ to $Y_{n}$. Indeed, for any
$\Lambda\in Y_{1}^{\ell}(\mathbb{R}^{n+1})$ there exists
$u(\ell,\Lambda)\notin\ell^{\perp}$ such that:

$\Lambda^{*}=\bigcup_{k\in\mathbb{Z}}\Lambda^{*}\cap\ell^{\perp}+ku(\ell,\Lambda).$
(2.3)

The vector $u(\ell,\Lambda)$ is unique up to choice of direction for $\ell$.
Fix such choice and define:

######  Definition 2.3.

The function $\rho^{\ell}_{n+1}:Y_{1}^{\ell}(\mathbb{R}^{n+1})\rightarrow
Y_{n}$ is defined by:

$Y_{1}^{\ell}(\mathbb{R}^{n+1})\ni\Lambda\mapsto(\Lambda^{*}\cap\ell^{\perp})+P_{\ell^{\perp}}(u(\ell,\Lambda))$
(2.4)

where $u(\ell,\Lambda)\in\mathbb{R}^{n+1}$ is as above.

We will not prove the following lemma:

######  Lemma 2.4.

For any $n$ and $\ell$ as above, the map $\rho_{n}^{\ell}$ is a bijection.

The submanifold $Y_{1}^{\ell}(\mathbb{R}^{n+1})$ is a periodic orbit of the
group:

$G(\ell)\vcentcolon=\\{g\in\operatorname{SL}_{n+1}(\mathbb{R}):g\mid_{\ell}=\operatorname{Id}\mid_{\ell}\\}\cong\operatorname{SL}_{n}(\mathbb{R})\ltimes\mathbb{R}^{n}\leq\operatorname{SL}_{n+1}(\mathbb{R}).$

This group also acts on $Y_{n}$ in a natural transitive way making the map
$\rho_{n+1}^{\ell}$ equivariant. Denote $m_{Y_{1}^{\ell}(\mathbb{R}^{n+1})}$
to be the Haar measure on $Y_{1}^{\ell}(\mathbb{R}^{n+1})$ coming from
periodicity of the $G(\ell)$-orbit $Y_{1}^{\ell}(\mathbb{R}^{n+1})$. The
measure $m_{Y_{1}^{\ell}(\mathbb{R}^{n+1})}$ and the pushforward measure
$m_{Y_{n}}\vcentcolon=(\rho^{\ell}_{n+1})_{*}m_{Y_{1}^{\ell}(\mathbb{R}^{n+1})}$
are thus invariant under the action of $G(\ell)$.

Now that we finished describing the probability space where equidistribution
will occur, we discuss the way to deform our lattices coming from number
fields into it. As in the introduction, the tool for this purpose is the
following family of one parameter groups:

######  Definition 2.5.

Given subspaces $E_{1},E_{2}\subset\mathbb{R}^{n}$ such that $E_{1}\oplus
E_{2}=\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we define a linear operator
$g_{E_{1},E_{2}}^{t}\in\operatorname{SL}_{n}(\mathbb{R})$ by
$g(x)=e^{t/dim(E_{1})}x$, $g(y)=e^{-t/dim(E_{2})}y$ for every $x\in E_{1},y\in
E_{2}$. The set $\\{g_{E_{1},E_{2}}^{t}\\}_{t\in\mathbb{R}}$ is clearly a
group, which we shall call the $E_{1},E_{2}$ normalization group.

Next we explain how to carry the deformation from $Y^{\ell_{\overline{1}}}$
into $Y^{\ell_{\overline{1}}}_{1}$ using the normalization groups. In
particular, we do that for the lattices coming from number fields.

######  Definition 2.6.

Let $\Lambda\in Y^{\ell_{\overline{1}}}$. Let $E\subset\mathbb{R}^{n}$ be such
that $\ell_{\overline{1}}\oplus E=\mathbb{R}^{n}$. Let $t\in\mathbb{R}$ be
such that $e^{-t}\operatorname{cov}(\Lambda)^{-1/n}\sqrt{n}=1$. We define:

$\Lambda(E)=g_{E,\ell_{\overline{1}}}^{t}(\Lambda).$ (2.5)

For every degree $n$ number field $\mathbb{F}$ of signature $(r,s)$ and
$\pi=(\pi_{r},\pi_{s})\in S_{r}\times S_{s}$ we define:

$\Lambda^{\pi}_{\mathbb{F}}(E)=(\Lambda_{\pi}(\mathbb{F}))(E).$ (2.6)

Note that by the choice of $t$ it holds that $\Lambda(E)\in
Y_{1}^{\ell_{\overline{1}}}$. Thus $\Lambda(E)\in
Y_{1}^{\ell_{\overline{1}}}(\mathbb{R}^{n})$ and in particular so is
$\Lambda^{\pi}_{\mathbb{F}}(E)$. Denote:

$\Gamma_{E}(\Lambda)=\rho_{n}^{\ell_{\overline{1}}}(\Lambda(E))$ (2.7)

and

$\Gamma_{E}^{\pi}(\mathbb{F})=\Gamma_{E}(\Lambda^{\pi}_{\mathbb{F}}(E)).$
(2.8)

We restate Theorem 1.4 from the introduction in the language of grids, because
this will be more natural for our proof later.

######  Theorem 2.7.

Let $n=3,4,5$ and let $(r,s)$ be a signature. Let $E$ be a subspace of
$\mathbb{R}^{n}$ of such that $E\oplus\ell_{\overline{1}}=\mathbb{R}^{n}$ and
$E\neq V_{0}$. Then:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\pi\in S_{r}\times
S_{s}}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{\Gamma_{E}^{\pi}(\mathbb{F})}\longrightarrow
m_{Y_{n-1}}\text{ weakly as }T\longrightarrow\infty.$ (2.9)

Next we deal with the case $E=V_{0}$. To state our next result, we need the
following definition:

######  Definition 2.8.

Let $\Lambda\in X_{n},m\in\mathbb{N}$. Define:

$Y_{n}(m)=\\{\Lambda+v\in Y_{n}:mv\in\Lambda,m^{\prime}v\notin\Lambda\text{
for all }m^{\prime}<m\\}.$ (2.10)

Note that $Y_{n}(m)$ is identified under $\rho_{n}^{\ell_{\overline{1}}}$ with
a periodic orbit of $G(\ell_{\overline{1}})$ in $X_{n}$ and thus exists unique
Borel probability measure $m_{Y_{n}(m)}$ supported on $Y_{n}(m)$ and invariant
under this action.

Now we can deal with the case $E=V_{0}$.

######  Theorem 2.9.

Let $n=3,4,5$ and let $(n-2s,s)$ be a signature. Denote
$\operatorname{Div}(n)=\\{1\leq k\leq n:k\mid n\\}$ to be the set of divisors
of $n$. Then for any $k\in\operatorname{Div}(n)$ there exists
$\ell=\ell(k)\in\mathbb{N}$, disjoint closed connected subsets
$F^{(k)}_{1},\dots,F^{(k)}_{\ell(k)}\subset Y_{n}(k)$ and nonnegative scalars
$s_{k,\ell};\ell=1,\dots,\ell(k)$ not all zero, such that:

$\frac{1}{\lvert\mathcal{F}^{s}_{n}(T)\rvert}\sum_{\pi\in S_{r}\times
S_{s}}\sum_{\mathbb{F}\in\mathcal{F}^{s}_{n}(T)}\delta_{q_{V_{0}}(\mathbb{F})}\longrightarrow\sum_{k\in\operatorname{Div}(k)}\sum_{l=1}^{\ell(k)}s_{k,\ell}m_{Y_{n}(k)}\mid_{F_{\ell}^{(k)}}\text{
weakly as }T\longrightarrow\infty.$ (2.11)

###  2.3 Basic Subsets

######  Definition 2.10.

Given a lattice
$\Lambda=\operatorname{sp}_{\mathbb{Z}}\\{w_{1},\dots,w_{n}\\}\subset\mathbb{R}^{n}$,
a fundamental domain $\Sigma$ for the action of
$\operatorname{stab}_{\operatorname{SL}_{n}(\mathbb{R})}(\Lambda)$ on
$\operatorname{SL}_{n}(\mathbb{R})$, an open and bounded subset
$S\subset\Sigma$ and $U=I_{1}e_{1}+\dots+I_{n}e_{n}\subset[0,1]^{n}$ where
$I_{i}\subset[0,1]$ is an interval (such subset will be denoted by box), we
define:

$S\times_{\Sigma}U=\\{g^{-1}(\Lambda+I_{1}w_{1}+\dots+I_{n}w_{n}):g\in
S\\}\subset Y_{n}.$

We refer to subsets of the above form as $\Lambda$-$\Sigma$-basic subsets.

######  Definition 2.11.

Given $U$ as in Definition 2.10 and an ordered basis
$w=\\{w_{1},\dots,w_{n}\\}$ of $\mathbb{R}^{n}$, denote:

$U_{w}=I_{1}w_{1}+\dots+I_{n}w_{n}.$

The following technical Lemma will not be proved.

######  Lemma 2.12.

Using the notation of Definition 2.10, the collection
$\mathcal{T}=\\{S\times_{\Sigma}U:S\subset\Sigma,\text{such that
}\sigma(\partial S)=0\text{, }U\subset[0,1]^{n}\text{ a box}\\}$ consists of
subsets of $m_{Y_{n}}$-measure-$0$ boundary which constitute a basis for the
natural topology on $Y_{n}$.

###  2.4 Bhargava’s Correspondence

Some of the main tools we use (as do Bhargava-Harron in [3]) involve
parametrizations all cubic, quartic and quintic orders which are carried in
[1, 2, 4].

######  Definition 2.13.

Let $T$ be a ring. Define $V_{T}$ to be:

  1. (a)

the space $Sym^{3}T^{2}(\otimes T)$ of binary cubic forms over $T$, if $n=3$;

  2. (b)

the space $Sym^{2}T^{3}\otimes T^{2}$ of pairs of ternary quadratic forms over
$T$, if $n=4$;

  3. (c)

the space $T^{4}\otimes\wedge^{2}T^{5}$ of quadruples of alternating quinary
$2$-forms over $T$ , if $n=5$.

For $n=3,4,5$, we set $r=r(n)=2,3,6$ respectively. Note that
$G_{T}=\operatorname{GL}_{n-1}(T)\times\operatorname{GL}_{r-1}(T)$ acts
naturally on $V_{T}$. The discriminant of an element $v\in V_{T}$ is a
polynomial of degree $d$ in the coefficients of $v$, where $d=4,12,40$ when
$n=3,4,5$ respectively (see [8]).

As explained in [3], the following Theorem can be deduced from [4, §15], [1,
Corollary 5] and [2, Corollary 3] and is Theorem 2 in [3]:

######  Theorem 2.14.

The nondegenerate (i.e. with non-zero discriminant) elements of
$V_{\mathbb{Z}}$ are in canonical bijection with isomorphism classes of pairs
$((R,\alpha),(S,\beta))$, where $R$ is a nondegenerate ring of rank $n$ and
$S$ is a rank $r$ resolvent ring of $R$, and $\alpha$ and $\beta$ are
$\mathbb{Z}$-bases for $R/\mathbb{Z}$ and $S/\mathbb{Z}$, respectively. In
this bijection, the discriminant of an element of $V_{\mathbb{Z}}$ is equal to
the discriminant of the corresponding ring $R$ of rank $n$. Moreover, under
this bijection, the action of $G_{\mathbb{Z}}$ on $V_{\mathbb{Z}}$ results in
the corresponding natural action of $G_{\mathbb{Z}}=GL_{n-1}(\mathbb{Z})\times
GL_{r-1}(\mathbb{Z})$ on $(\alpha,\beta)$. Finally, every isomorphism class of
maximal ring $R$ of rank $n$ arises in this bijection, and the elements of
$V_{\mathbb{Z}}$ yielding $R$ consists of exactly one $G_{\mathbb{Z}}$-orbit.
We denote the rings corresponding to $v\in V_{\mathbb{R}}$ by
$R_{\mathbb{Z}}(v),S_{\mathbb{Z}}(v)$ and the bases corresponding to $v$ by
$\alpha(v),\beta(v)$.

A resolvent ring of a cubic, quartic or quintic ring is some quadratic, cubic,
or sextic ring respectively, which satisfies some conditions (see [1, 2] for
more details). Since the precise definition is not required here, we choose
not to get into the details of this matter. Theorem 2.14 holds with any field
$K$ in place of $\mathbb{Z}$ with the same proofs as in [1, 2]. In particular,
taking $K=\mathbb{R}$:

######  Theorem 2.15.

There is a canonical bijection between the nondegenerate elements of
$V_{\mathbb{R}}$ and isomorphism classes of pairs $((R,\alpha),(S,\beta))$,
where $R$ is a nondegenerate ring of rank $n$ over $R$ and $S$ is the (unique)
rank $r$ resolvent ring of $R$ over $\mathbb{R}$, and $\alpha$ and $\beta$ are
$\mathbb{R}$-bases for $R/\mathbb{R}$ and $S/\mathbb{R}$, respectively.
Moreover, under this bijection, the action of $G_{\mathbb{R}}$ on
$V_{\mathbb{R}}$ results in the corresponding natural action of
$G_{\mathbb{R}}$ on $(\alpha,\beta)$. We denote the rings corresponding to
$v\in V_{\mathbb{R}}$ by $R(v),S(v)$.

######  Remark 2.16.

Theorems 2.14,2.15 are compatible in the following sense. On the one hand
$V_{\mathbb{Z}}\subset V_{\mathbb{R}}$ naturally and if $v\in V_{\mathbb{Z}}$
corresponds under Theorem 2.14 to the ring $R$, then it corresponds under
Theorem 2.15 to $R\otimes\mathbb{R}$. Moreover, given $v\in V_{\mathbb{Z}}$
the multiplication tables of the algebras $R,S$ corresponding to $v$ under
Theorem 2.14 with respect to the bases $\alpha,\beta$ are the same as the
multiplication tables of $R^{\prime},S^{\prime}$ with respect to the bases
$\alpha^{\prime},\beta^{\prime}$ which correspond to $v$ when thought of as an
element of $V_{\mathbb{R}}$. This follows from [4, §15 (1) and (2)] when
$n=3$, [1, (14),(21),(22) and (23)] when $n=4$ and by [2, (16),(17),(21) and
(22)] when $n=5$, and is written also in the top of page 5 of [3].

######  Remark 2.17.

Rank-$n$ rings $R$ over $\mathbb{R}$ are in particular Étale algebras over
$\mathbb{R}$ of rank $n$, which implies
$R\cong\mathbb{R}^{n-2k}\times\mathbb{C}^{k}$ for some $k=0,\dots,\lfloor
n/2\rfloor$. Following the notation of [3], given $k=0,\dots,\lfloor
n/2\rfloor$, we denote $V_{\mathbb{R}}^{(k)}$ to be the subset of
$V_{\mathbb{R}}$ of elements for which the corresponding ring $R$ (recall
Theorem 2.15) has the structure $\mathbb{R}^{n-2k}\times\mathbb{C}^{k}$.
Therefore, and as mentioned in [3], the nondegenerate orbits for the action of
$G_{\mathbb{R}}$ on $V_{\mathbb{R}}$ are precisely
$V_{\mathbb{R}}^{(0)},\dots,V_{\mathbb{R}}^{(\lfloor n/2\rfloor)}$. Moreover,
recalling Remark 2.16, we denote
$V_{\mathbb{Z}}^{(i)}=V_{\mathbb{R}}^{(i)}\cap V_{\mathbb{Z}}$ for
$i=1,\dots,\lfloor n/2\rfloor$.

######  Definition 2.18.

Given $v\in V_{\mathbb{Z}}$, denote by $R(v)$ the ring corresponding to it
under Theorem 2.15.

  1. (a)

We say that $v$ is irreducible if $R(v)$ is isomorphic to an order inside a
type $S_{n}$ number field;

  2. (b)

We say that $v$ is maximal if $R(v)$ is a maximal ring of rank $n$ over
$\mathbb{Z}$ (which means it cannot be embedded as a sub $\mathbb{Z}$ module
inside any other ring of rank $n$ over $\mathbb{Z}$).

Recalling the natural identification of $V_{\mathbb{Z}}$ inside
$\mathbb{Z}^{d}$ for an appropriate $d$ (depending on $n=3,4,5$), we say that
$v\in\mathbb{Z}^{d}$ is irreducible (respectively, maximal) if the
corresponding element of $V_{\mathbb{Z}}$ is irreducible (respectively,
maximal).

######  Corollary 2.19.

Let $n=3,4,5$ and $X>0$. There is $1:1$ correspondence between
$G_{\mathbb{Z}}$-orbits of irreducible maximal elements of
$V_{\mathbb{Z}}^{(s)}$ with discriminant less than $X$ and rings of integers
of isomorphism classes of signature $(r,s)$ number fields with discriminant
less than $X$. The correspondence is given by $G_{\mathbb{Z}}.v\leftrightarrow
R_{\mathbb{Z}}(v)$.

######  Definition 2.20.

Given $n=3,4,5$ we let $v_{0}\in V_{\mathbb{R}}$ and let
$u^{\perp}=E\subset\mathbb{R}^{n}$ such that $u\notin V_{0}$.

  1. (a)

For any $v\in V_{\mathbb{R}}$ let $\operatorname{MT}_{v}$ denote the
multiplication table A.1 when $n=3$, A.2 when $n=4$ and A.3 when $n=5$. Given
a rank $n$ ring $R$ and $x_{1},\dots,x_{n-1}\in R$ we write
$\operatorname{MT}_{v}(x_{1},\dots,x_{n-1})$ if $(x_{i})_{i=1}^{n-1}$
satisfies $\operatorname{MT}_{v}$;

  2. (b)

For $i=1,\dots,n-1$,
$\overline{\alpha}_{i}:V_{\mathbb{R}}\rightarrow\mathbb{R}^{n}$ denote the
continuous vector functions such that:

$\operatorname{MT}_{v}(\overline{\alpha}_{1}(v),\dots,\overline{\alpha}_{n-1}(v))$
(2.12)

for any $v\in V_{\mathbb{R}}$ whose existence is guaranteed by A.2;

  3. (c)

Given $g\in\mathbb{R}_{+}\Sigma_{n-1}\times\Sigma_{r-1}$ denote
$\Lambda_{v_{0}}(g)=\operatorname{span}_{\mathbb{Z}}\\{\overline{1},\overline{\alpha}_{1}(gv_{0}),\dots,\overline{\alpha}_{n}(gv_{0})\\}$;

  4. (d)

Given a basis $w=(w_{j})_{j=1}^{n-1}$ of $V_{0}$, we define a function
$f_{w}:\mathbb{R}_{+}\cdot(\Sigma_{n-1}\times\Sigma_{r-1})v_{0}\rightarrow
V_{0}$ by:

$f_{w}(p)=\frac{1}{\left\langle u,1\right\rangle}\sum_{i=1}^{n-1}\left\langle
u,\overline{\alpha}_{i}(g_{p}v_{0})\right\rangle w_{i},$

where for $p\in\mathbb{R}_{+}\cdot\Sigma_{n-1}v_{0}$, $g_{p}$ is the element
of $\mathbb{R}_{+}\cdot\Sigma_{n-1}\times\Sigma_{r-1}$ satisfying
$g_{p}v_{0}=p$;

  5. (e)

Given a basis $w=\\{w_{1},\dots,w_{n-1}\\}$ of $V_{0}$, we denote:

$\Lambda_{w}=\operatorname{span}_{\mathbb{Z}}\\{w_{1},\dots,w_{n-1}\\}.$
(2.13)

The main goal of this Subsection is to prove the following structural Lemma:

######  Lemma 2.21.

Let $n=3,4,5$, $i=1,\dots,\lfloor n/2\rfloor$ and let
$u^{\perp}=E\subset\mathbb{R}^{n}$ be a subspace of co-dimension 1 not
containing $\overline{1}$. Let $\Sigma_{n-1}^{\prime}$ be any fundamental
domain for the action of $\operatorname{SL}_{n-1}(\mathbb{Z})$ on
$\operatorname{SL}_{n-1}(\mathbb{R})$ and let $v_{0}\in V_{\mathbb{R}}^{(i)}$.
Then there exists a fixed basis of $V_{0}$ given by $w=(w_{j})_{j=1}^{n-1}$, a
function $\Pi:\mathcal{F}_{\infty}^{(i)}\rightarrow S_{n}$ (recall Subsection
2.1) and $g_{0}\in\operatorname{SL}_{n-1}(\mathbb{R})$ such that the
fundamental domain:

$\Sigma_{n-1}\vcentcolon=g_{0}^{t}\Sigma_{n-1}^{\prime}g_{0}^{-t}$ (2.14)

satisfies that for any $\Lambda_{w}$-$\Sigma_{n-1}$-basic subset
$S\times_{\Sigma_{n-1}}U\subset Y_{n-1}$:

$\mathcal{F}^{(i)}_{X}\cap(\Gamma_{E}^{\Pi})^{-1}(S\times_{\Sigma_{n-1}}U)\xleftrightarrow{\sim}$
(2.15) $\\{\text{irreducible, maximal points inside
}[0,X](g_{0}^{-t}S^{-1}g_{0}^{t}\times\Sigma_{r-1})v_{0}\\}\cap((f_{w}+q_{0})\operatorname{mod}\Lambda_{w})^{-1}(U_{w})$

where $q_{0}\in V_{0}$ is some fixed vector (recall Definition 2.11 for
definition of $U_{w}$). Note that the function
$\Gamma_{E}^{\Pi}:\mathcal{F}_{\infty}^{(i)}\rightarrow Y_{n-1}$ is defined by
$\mathcal{F}_{\infty}^{(i)}\ni\mathbb{F}\mapsto\Gamma_{E}^{\Pi(\mathbb{F})}(\mathbb{F})$.
Moreover, for any $\pi\in S_{n}$ the function $\Pi$ can be replaced with
$\pi\circ\Pi$ (where
$(\pi\circ\Pi)(\mathbb{F})\vcentcolon=\pi\circ(\Pi(\mathbb{F}))$).

###### Proof.

Let $i=1,\dots,\lfloor n/2\rfloor$ and pick $v_{0}\in V_{\mathbb{R}}^{(i)}$.
Pick any fundamental domain $\Sigma_{n-1}^{\prime}$ for the action of
$\operatorname{SL}_{n-1}(\mathbb{Z})$ on $\operatorname{SL}_{n-1}(\mathbb{R})$
and similarly pick $\Sigma_{r-1}$ to be a fundamental domain for the action of
$\operatorname{SL}_{r-1}(\mathbb{Z})$ on
$\operatorname{SL}_{r-1}(\mathbb{R})$. For any
$v=t(g,h)v_{0}\in\mathbb{R}_{+}\cdot(\Sigma^{\prime}_{n-1},\Sigma_{r-1})v_{0}$
recall Definiton 2.20 and denote:

$\Lambda_{v}=\Lambda_{v_{0}}(t(g,h)).$ (2.16)

Next, write:

$\Lambda_{v}=\operatorname{span}_{\mathbb{Z}}\\{\overline{\alpha}_{1}(v),\dots,\overline{\alpha}_{n-1}(v),\overline{1}\\},D_{v}=\operatorname{cov}(\Lambda_{v})^{-1/n}.$
(2.17)

This is the lattice we start with. Next we follow the steps of normalization
which are described in Equation (2.6) and find $\Lambda_{v}(E)$ and
$\Gamma_{E}(\Lambda(v))$ explicitly using $v_{0},t(g,h)$. The first step is to
normalize the covolume of $\Lambda_{v}$ by defining:

$\Lambda_{v}^{0}=\operatorname{span}_{\mathbb{Z}}\\{D_{v}\overline{\alpha}_{1}(v),\dots,D_{v}\overline{\alpha}_{n-1}(v),D_{v}\overline{1}\\}.$
(2.18)

The next step is acting on $\Lambda^{0}_{v}$ with the
$E,\mathbb{R}\cdot\overline{1}$ normalization group from Definition 2.5. To
this end, write:

$\overline{\alpha}_{i}=(\overline{\alpha_{i}})_{\perp}+c_{i}\overline{1}\text{
such that }(\overline{\alpha}_{i})_{\perp}\in E,i=1,\dots,n-1$ (2.19)

and obviously:

$D_{v}\overline{\alpha}_{i}=D_{v}(\overline{\alpha_{i}})_{\perp}+D_{v}c_{i}\overline{1}\text{
such that }(\overline{\alpha}_{i})_{\perp}\in E,i=1,\dots,n-1.$ (2.20)

Denote $\lambda_{v}=n^{-1/2}D_{v}^{-1}$. The appropriate element $g_{t}$ from
the $E,\mathbb{R}\overline{1}$ normalization group is defined uniquely by:

$g_{t}(\overline{1})=\lambda_{v}\overline{1};$
$g_{t}(e)=\lambda_{v}^{-\frac{1}{n-1}}e\text{ for every }e\in E.$

Denote also:

$\alpha_{i}^{0}\vcentcolon=\lambda_{v}^{-\frac{1}{n-1}}D_{v}(\overline{\alpha}_{i})_{\perp}+c_{i}D_{v}\lambda_{v}\overline{1}=g_{t}(D_{v}\overline{\alpha}_{i}).$
(2.21)

Let

$\beta_{i}=\frac{\lambda_{v}^{\frac{1}{n-1}}\left((\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{i-1})_{\perp}\times\overline{1}\times(\overline{\alpha}_{i+1})_{\perp}\dots\times(\overline{\alpha}_{n-1})_{\perp}\right)}{D_{v}\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle}$
(2.22)

and denote:

$\tau_{v}\vcentcolon=\frac{\alpha_{1}^{0}\times\dots\times\alpha_{n-1}^{0}}{\left\langle\alpha_{1}^{0}\times\dots\times\alpha_{n-1}^{0},D_{v}\lambda_{v}\overline{1}\right\rangle},P_{V_{0}}(\tau_{v})=\frac{P_{V_{0}}(\alpha_{1}^{0}\times\dots\times\alpha_{n-1}^{0})}{\left\langle\alpha_{1}^{0}\times\dots\times\alpha_{n-1}^{0},D_{v}\lambda_{v}\overline{1}\right\rangle}=\frac{P_{V_{0}}(u)}{\left\langle
u,1\right\rangle}+\sum_{i=1}^{n-1}c_{i}\beta_{i}.$ (2.23)

Note that we have found $\Lambda_{v}(E)$ and it is given by:

$\Lambda_{v}(E)=\operatorname{span}_{\mathbb{Z}}\\{\lambda_{v}D_{v}\overline{1},\alpha_{1}^{0},\dots,\alpha_{n-1}^{0}\\}$
(2.24)

(recall Definition 2.6). Next we find $\Gamma_{E}(\Lambda_{v})$. By Equation
(A.6):

$P_{V_{0}}\Lambda_{v}(E)=g_{0}g^{t}g_{0}^{-1}(P_{V_{0}}\Lambda_{v_{0}}(E))$
(2.25)

where:

$g_{0}=\begin{pmatrix}\mid\ &\cdots&\mid\\\
P_{V_{0}}\alpha_{1}^{0}(v_{0})&\cdots&P_{V_{0}}\alpha_{n-1}^{0}(v_{0})\\\
\mid&\cdots&\mid\end{pmatrix}$

and therefore $\beta_{i}=\beta_{i}(g)$ depends only on $g$ for every
$i=1,\dots,n-1$. We claim that $\\{\beta_{i}(g)\\}_{i=1}^{n-1}$ is a basis for
the lattice:

$\left(\Lambda_{v}(E)\right)^{*}\cap
V_{0}=\left(P_{V_{0}}\Lambda_{v}(E)\right)^{*}=\left(g_{0}g^{t}g_{0}^{-1}(P_{V_{0}}\Lambda_{v_{0}}(E))\right)^{*}$
(2.26)

where the first equality holds by known identities of the dual lattice and the
second by Equation (2.25). Indeed, by definition of the dual lattice, it
suffices to check that
$\left\langle\beta_{i},\alpha_{j}^{0}\right\rangle\in\\{\delta_{ij},-\delta_{ij}\\}$
and that $\left\langle\beta_{i},\lambda_{v}D_{v}\overline{1}\right\rangle=0$
for every $i,j=1,\dots,n-1$. Indeed, let $i=1,\dots,n-1$. Then:

$\left\langle\beta_{i},\alpha_{i}^{0}\right\rangle=\left\langle\frac{\lambda_{v}^{\frac{1}{n-1}}\left((\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{i-1})_{\perp}\times\overline{1}\times(\overline{\alpha}_{i+1})_{\perp}\dots\times(\overline{\alpha}_{n-1})_{\perp}\right)}{D_{v}\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle},\lambda_{v}^{-\frac{1}{n-1}}D_{v}(\overline{\alpha}_{i})_{\perp}+c_{i}D_{v}\lambda_{v}\overline{1}\right\rangle$
(2.27)
$=\left\langle\frac{\lambda_{v}^{\frac{1}{n-1}}\left((\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{i-1})_{\perp}\times\overline{1}\times(\overline{\alpha}_{i+1})_{\perp}\dots\times(\overline{\alpha}_{n-1})_{\perp}\right)}{D_{v}\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle},\lambda_{v}^{-\frac{1}{n-1}}D_{v}(\overline{\alpha}_{i})_{\perp}\right\rangle$
$=\left\langle\frac{\left((\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{i-1})_{\perp}\times\overline{1}\times(\overline{\alpha}_{i+1})_{\perp}\dots\times(\overline{\alpha}_{n-1})_{\perp}\right)}{\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle},(\overline{\alpha}_{i})_{\perp}\right\rangle$
$=\frac{\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{i-1})_{\perp}\times\overline{1}\times(\overline{\alpha}_{i+1})_{\perp}\dots\times(\overline{\alpha}_{n-1})_{\perp},(\overline{\alpha}_{i})_{\perp}\right\rangle}{\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle}$
$=\frac{(-1)^{i}\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle}{\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle}=(-1)^{i}$

where the first equality holds by definition of $\beta_{i},\alpha_{i}^{0}$,
the second holds by definition of the cross product, the third is elimination
of the constants, the forth is taking out the constant and the fifth holds by
two general properties of the mixed product saying that for any
$u_{1}\dots,u_{n}\in\mathbb{R}^{n}$:

$\left\langle u_{1}\times\dots\times u_{n-1},u_{n}\right\rangle=\left\langle
u_{n}\times u_{1}\dots\times u_{n-2},u_{n-1}\right\rangle.$

and that for any $\pi\in S_{n-1}$:

$u_{1}\times\dots\times
u_{n-1}=\operatorname{sgn}(\pi)u_{\pi(1)}\times\dots\times u_{\pi(n-1)}.$

Let $i\neq j$ be in $\\{1,\dots,n-1\\}$. Then:

$\left\langle\beta_{i},\alpha_{j}^{0}\right\rangle=\left\langle\frac{\lambda_{v}^{\frac{1}{n-1}}\left((\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{i-1})_{\perp}\times\overline{1}\times(\overline{\alpha}_{i+1})_{\perp}\dots\times(\overline{\alpha}_{n-1})_{\perp}\right)}{D_{v}\left\langle(\overline{\alpha}_{1})_{\perp}\times\dots\times(\overline{\alpha}_{n-1})_{\perp},\overline{1}\right\rangle},\lambda_{v}^{-\frac{1}{n-1}}D_{v}(\overline{\alpha}_{j})_{\perp}+c_{j}D_{v}\lambda_{v}\overline{1}\right\rangle=0$
(2.28)

because the element in the left side of the inner product is perpendicular to
both $\overline{1}$ and to $(\overline{\alpha}_{j})_{\perp}$. By the same
reason, $\left\langle\beta_{i},\overline{1}\right\rangle=0$ for every $i$.
Since for each $i$, $\beta_{i}(g)$ is a continuous function in $g$ and by
Equation (2.26) above, there exists some fixed basis $w_{1},\dots,w_{n-1}$ of
the lattice $(P_{V_{0}}\Lambda_{v_{0}}(E))^{*}$ such that:

$\beta_{i}(g)=g_{0}^{t}g^{-1}g_{0}^{-t}w_{i}\text{ for all }i=1,\dots,n-1.$
(2.29)

Define $\Sigma_{n-1}=g_{0}^{t}\Sigma_{n-1}^{\prime}g_{0}^{-t}$ and write
$\Lambda_{w}=\operatorname{span}_{\mathbb{Z}}\\{w_{1},\dots,w_{n-1}\\}$. We
continue to calculate $\Gamma_{E}(\Lambda(v))$. Recall that by Definition 2.3,
$\rho_{n}:X_{n}\rightarrow Y_{n-1}$ is an identification between lattices and
grids and that by Definition 2.6:
$\Gamma_{E}(\Lambda(v))=\rho_{n}(\Lambda_{v}(E)^{*})$. Write by Equation
(2.24) and by the definitions of $\beta_{i},\tau_{v}$ (recall Equations
(2.22),(2.23)):

$(\Lambda_{v}(E))^{*}=\left(\begin{pmatrix}\mid\ &\cdots&\mid&\mid\\\
\alpha_{1}^{0}&\cdots&\alpha_{n-1}^{0}&\lambda_{v}D_{v}\overline{1}\\\
\mid&\cdots&\mid&\mid\end{pmatrix}\mathbb{Z}^{n}\right)^{*}=\begin{pmatrix}\mid\
&\cdots&\mid&\mid\\\
\alpha_{1}^{0}&\cdots&\alpha_{n-1}^{0}&\lambda_{v}D_{v}\overline{1}\\\
\mid&\cdots&\mid&\mid\end{pmatrix}^{-t}\mathbb{Z}^{n}=\begin{pmatrix}\mid\
&\cdots&\mid&\mid\\\ \beta_{1}&\cdots&\beta_{n-1}&\tau_{v}\\\
\mid&\cdots&\mid&\mid\end{pmatrix}\mathbb{Z}^{n}$ (2.30)

which means that
$(\Lambda_{v}(E))^{*}=\operatorname{span}_{\mathbb{Z}}\\{\beta_{1},\dots,\beta_{n-1},\tau_{v}\\}$
and since $\beta_{i}\in V_{0}$ for every $i=1,\dots,n-1$, we can take
$u=\tau_{v}$ in Definition 2.3 of $\rho_{n}$ and deduce that the first layer
is given by:

$\Gamma_{E}(\Lambda(v))=\rho_{n}(\Lambda_{v}(E)^{*})=\operatorname{span}_{\mathbb{Z}}\\{\beta_{1},\dots,\beta_{n-1}\\}+P_{V_{0}}(\tau_{v}).$
(2.31)

We already know (see Corollary 2.19) that for any $v_{0}\in
V_{\mathbb{R}}^{(0)}$ (recall Remark 2.17) irreducible maximal points
$v=t(g,g^{\prime})v_{0}\in
V_{\mathbb{Z}}\cap\mathbb{R}_{+}\cdot(\Sigma^{\prime}_{n-1},\Sigma_{r-1})v_{0}$
correspond under $v\mapsto R_{\mathbb{Z}}(v)$ to rings of integers of
equivalence classes of type $S_{n}$ number fields $\mathbb{F}$. Denote the
number field corresponding to $v$ by $\mathbb{F}_{v}$. Recall that in Notation
2.1 we fixed (in particular) an ordering for the natural maps of
$\mathbb{F}_{v}$, whom we denote by $\sigma^{v}_{1},\dots,\sigma^{v}_{n}$. Let
$\pi(\mathbb{F}_{v})\in S_{n}$ be a permutation such that:

$(\sigma^{v}_{\pi(\mathbb{F}_{v})(1)},\dots,\sigma^{v}_{\pi(\mathbb{F}_{v})(n)})\alpha_{i}(t(g,g^{\prime})v_{0})=\overline{\alpha}_{i}(t(g,g^{\prime})v_{0});\forall
i=1,\dots,n$ (2.32)

(recall A.2 for definition of $\overline{\alpha}_{i}$) and define:

$\Pi(\mathbb{F}_{v})=\pi(\mathbb{F}_{v}).$ (2.33)

Let $v=t(g,g^{\prime})v_{0}\in
V_{\mathbb{Z}}\cap\mathbb{R}_{+}\cdot(\Sigma_{n-1}^{\prime},\Sigma_{r-1})v_{0}$
be irreducible maximal. Applying Equations (2.31) and (2.26) in particular to
$v$ shows that:

$\Gamma^{\pi}_{E}(\mathbb{F}_{v})=\Gamma_{E}(\Lambda(v))=\operatorname{span}_{\mathbb{Z}}\\{\beta_{1},\dots,\beta_{n-1}\\}+P_{V_{0}}(\tau_{v})=g_{0}^{t}g^{-1}g_{0}^{-t}(\Lambda_{v_{0}}(E)^{*}\cap
V_{0})+P_{V_{0}}(\tau_{v})$ (2.34)

which shows that $\Gamma^{\pi}_{E}(\mathbb{F}_{v})\in S\times_{\Sigma_{n-1}}U$
if and only if $g_{0}^{t}g^{-1}g_{0}^{-t}\in S$ and $P_{V_{0}}(\tau_{v})\in
g_{0}^{t}g^{-1}g_{0}^{-t}U\operatorname{mod}g_{0}^{t}g^{-1}g_{0}^{-t}\Lambda_{w}$
if and only if $g^{-1}\in g_{0}^{-t}Sg_{0}^{t}$ and:

$f_{w}(v)\vcentcolon=\frac{P_{V_{0}}(u)}{\left\langle
u,1\right\rangle}+\frac{1}{\left\langle
u,1\right\rangle}\sum_{i=1}^{n-1}\left\langle
u,\overline{\alpha}_{i}(gv_{0})\right\rangle w_{i}\in
g_{0}^{t}g^{-1}g_{0}^{-t}U\operatorname{mod}g_{0}^{t}g^{-1}g_{0}^{-t}\Lambda_{w}$
(2.35)

if and only if:

$v\in\\{\text{irreducible, maximal points inside
}[0,X](g_{0}^{t}Sg_{0}^{-t}\times\Sigma_{r-1})v_{0}\\}\cap(f_{w}\operatorname{mod}\Lambda_{w})^{-1}(U_{w})$
(2.36)

where the second equivalence holds by Equation (2.29). Note that the only
property of $\Pi$ we used is the continuity of
$\beta_{i}(g)\in\mathbb{R}^{n}$. Since we for any $\pi\in S_{n}$, the
functions $\pi(\beta_{i}(g))$ are also continuous in $g$, the same argument
shows that we could have used $\pi\circ\Pi$ as claimed. ∎

###  2.5 Weyl-Type Theorem and Verification of Analytical Properties

For Theorem 2.23 below, the following definition comes handy.

######  Definition 2.22.

Let $n,k\in\mathbb{N}$, $F:D\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$,
$\Lambda_{k}\subset\mathbb{R}^{k}$ a lattice, $\\{B_{R}\\}_{R\geq 0}$ a
monotonic increasing family of subsets of $\mathbb{R}^{n}$ and
$\Gamma\subset\mathbb{R}^{n}$ a discrete subset. We say that
$F\operatorname{mod}\Lambda_{k}(\Gamma)$ is $B_{R}$-equidistributed if the
sequence of discrete counting measures $\sigma^{D}_{R}$ of $\Gamma$ points in
$B_{R}\cap D$ satisfies:

$\lim_{R\rightarrow\infty}(F\operatorname{mod}\Lambda_{k})_{*}\sigma^{D}_{R}=\lambda_{k}$
(2.37)

in the weak sense, where
$F\operatorname{mod}\Lambda_{k}:D\rightarrow\mathbb{R}^{k}/\Lambda_{k}$ is the
composition of $F$ with the quotient map,
$(F\operatorname{mod}\Lambda_{k})_{*}\sigma^{D}_{R}$ denotes the push-forward
measure, and $\lambda_{k}$ is the Haar probability measure on the
$k$-dimensional torus $\mathbb{R}^{k}/\Lambda_{k}$. When
$\Lambda_{k}=\mathbb{Z}^{k}$ we abbreviate
$F\operatorname{mod}\Lambda_{k}(\Gamma)=F\operatorname{mod}1(\Gamma)$.

The following Theorem from [11] is the main tool in the proof of Theorem 1.5.

######  Theorem 2.23 ( [11]).

Let $n,D,k\in\mathbb{N}$. Let $S$ be a level surface of some smooth
homogeneous function
$F:\mathbb{R}^{n}\setminus\\{\overline{0}\\}\rightarrow\mathbb{R}_{+}$ (namely
$S=\\{x\in\mathbb{R}^{n}:F(x)=t\\}$ for some $t>0$), let $A\subset S$ be open
and Jordan measurable in $S$ and let
$f=(f_{1},\dots,f_{k}):\mathbb{R}^{n}\setminus\\{\overline{0}\\}\rightarrow\mathbb{R}^{k}$
be homogeneous of degree $D$ and smooth inside $\mathbb{R}_{+}\cdot A$. Let
$\Gamma\subset\mathbb{R}^{n}$ be a lattice. Assume that there exists
$v\in\Gamma$ such that:

$\left(\frac{\partial^{d}f}{\partial
v^{d}}\operatorname{mod}1\right)_{*}m_{A}\vcentcolon=\left(\frac{\partial^{d}f_{1}}{\partial
v^{d}}\operatorname{mod}1,\dots,\frac{\partial^{d}f_{k}}{\partial
v^{d}}\operatorname{mod}1\right)_{*}m_{A}\ll\lambda_{k}$

where $m_{A}$ is the $n-1$-dimensional surface area on $A$ and $\lambda_{k}$
is the Lebesgue measure on $(S^{1})^{k}$. Then:

$f\operatorname{mod}1(\Gamma)\text{ is $[0,R]\cdot A$-equidistributed}.$
(2.38)

For a subset $I\subset\mathbb{R}_{+}$, $I\cdot A$ denotes the set $\\{ta:t\in
I,a\in A\\}$.

######  Definition 2.24.

Let $S\subset\mathbb{R}^{d}$ be a level surface of some smooth homogenous
function.

  1. (a)

A subset $D_{\mathbb{Z}}\subset\mathbb{Z}^{d}$ is said to be of $S$-density 1
in $\mathbb{Z}^{d}$ if:

$\lim_{R\rightarrow\infty}\frac{\lvert(([0,R]\cdot S)\cap
D_{\mathbb{Z}})\rvert}{\operatorname{Vol}_{d}([0,1]\cdot S)R^{d}}=1;$ (2.39)

  2. (b)

A subset $C_{\mathbb{Z}}\subset\mathbb{Z}^{d}$ is said to be a finite
congruence set if it is defined by union of finitely many congruence
conditions modulo coprime numbers on the entries of $v\in C_{\mathbb{Z}}$
(namely conditions of the form $v_{i}\operatorname{mod}p=s$ where
$v=(v_{1}\dots,v_{n})$, $p,s$ are integers and $p$ is taken from a set of
coprime numbers). Similarly, a set $C_{\mathbb{Z}}\subset\mathbb{Z}^{d}$ is
said to be an infinite congruence set if it is defined by infinitely many
congruence conditions as above.

We will require a modification of Theorem 2.23:

######  Corollary 2.25.

Under the assumptions of Theorem 2.23 Equation (2.38) holds for
$C_{\mathbb{Z}}\cap D_{\mathbb{Z}}$ instead of $\Gamma$ where $D_{\mathbb{Z}}$
is any set of $S$-density $1$ in $\mathbb{Z}^{d}$ and $C_{\mathbb{Z}}$ is any
finite congruence set.

###### Proof.

By taking the average, it suffices to prove the case where $C_{\mathbb{Z}}$ is
given by precisely one condition modulo each reminder. Let $p_{1},\dots,p_{k}$
be the coprime numbers defining the congruence conditions on the first
coordinate of elements in $C_{\mathbb{Z}}$ and let $s_{1},\dots,s_{k}$ be the
corresponding reminders. Let $P=p_{1}\cdot\dots\cdot p_{k}$ and for any $i$,
$P_{i}=P/p_{i}$. Since the $p_{i}$’s are coprime, $P_{i},p_{i}$ are coprime.
By Bezout’s identity there exist integers $M_{i},m_{i}$ such that:

$M_{i}P_{i}+m_{i}p_{i}=1.$ (2.40)

Then $x_{1}\vcentcolon=\sum_{i=1}^{k}s_{i}M_{i}P_{i}$ is a solution to the
congruence system of conditions on the first entries of elements from
$C_{\mathbb{Z}}$. Since $p_{i}$’s are distinct primes, by the Chinese
remainder theorem any other solution to the system is given by $x_{1}+lP$ for
some integer $l$. Repeating this argument for any other coordinate we can
write:

$C_{\mathbb{Z}}=\\{(x_{1}+l_{1}Q_{1},\dots,x_{d}+l_{d}Q_{d}):(l_{1},\dots,l_{d})\in\mathbb{Z}^{d}\\}$
(2.41)

for some fixed $x_{i},Q_{i}\in\mathbb{Z}$. Note that there exists $c>0$ such
that for any $R>0$:

$\lvert C_{\mathbb{Z}}\cap[0,R]\cdot S\rvert\geq cR^{d}.$ (2.42)

Let $F$ satisfy the conditions of Theorem 2.23. Then the function
$G(y_{1},\dots,y_{d})\vcentcolon=F(x_{1}+y_{1}Q_{1},\dots,x_{d}+y_{d}Q_{d})$
also satisfies the conditions of Theorem 2.23 (since $x_{i},Q_{i}$ are
integers) and therefore by Equation (2.41):

$F\operatorname{mod}1(C_{\mathbb{Z}})\text{ is }[0,R]\cdot A\text{
equidistributed}.$ (2.43)

where $A$ is as in Theorem 2.23. By assumption on $D_{\mathbb{Z}}$:

$\lim_{R\rightarrow\infty}\frac{\lvert(([0,R]\cdot
S)\cap\mathbb{Z}^{d})\setminus D_{\mathbb{Z}}\rvert}{R^{d}}=0$ (2.44)

so by Equation (2.42):

$\lim_{R\rightarrow\infty}\frac{(([0,R]\cdot S)\cap C_{\mathbb{Z}})\setminus
D_{\mathbb{Z}}}{\lvert C_{\mathbb{Z}}\cap[0,R]S\rvert}=0$ (2.45)

which means that $D_{\mathbb{Z}}^{c}$ points are negligible in
$C_{\mathbb{Z}}$ so in fact:

$F\operatorname{mod}1(C_{\mathbb{Z}}\cap D_{{\mathbb{Z}}})\text{ is
}[0,R]\cdot A\text{ equidistributed}$ (2.46)

as desired. ∎

######  Lemma 2.26.

The subset $\Gamma_{d}\subset V_{\mathbb{Z}}$ of irreducible maximal lattice
points in $V_{\mathbb{Z}}$ (recall Definition 2.18) is the intersection of a
set of $S=\Sigma^{\prime}_{n-1}\times\Sigma_{r-1}v_{0}$ density $1$ in
$\mathbb{Z}^{d}$ and an infinite congruence set.

###### Proof.

By [3, Theorem 4] and the short discussion following it we know that the
irreducible points are of $S$ density 1. By [3, Section 5], the maximal points
are defined by infinitely many congruence conditions modulo prime powers. ∎

######  Corollary 2.27.

Under the assumptions of Theorem 2.23 Equation (2.38) holds for $\Gamma_{d}$
instead of $\mathbb{Z}^{d}$.

###### Proof.

Write $\Gamma_{d}=C_{d}\cap D_{d}$ where $D_{d}$ is a set of $S$ density 1 and
$C_{d}$ is some infinite congruence set. For any prime $p$ let $U_{p}$ be the
subset of $C_{d}$ where the congruence conditions involve only $p$. By [3,
Lemma 11]:

$\lvert\left([0,R]S\cap\mathbb{Z}^{d}\right)\setminus
U_{p}\rvert=O(R^{d}/p^{2})$ (2.47)

and by [3, Section 5] there exists $\alpha>0$ such that:

$\lvert[0,R]S\cap\Gamma_{d}\rvert=\alpha
R^{d}\operatorname{Vol}([0,1]S)+o(R^{d}).$ (2.48)

Let $\epsilon>0,R>0$ and let $Y_{0}$ to be later determined. Then:

$\frac{\lvert[0,R]S\cap\left(\Gamma_{d}\setminus\cup_{p<Y_{0}}U_{p}\right)\rvert}{\lvert[0,R]S\cap\left(\Gamma_{d}\setminus\cup_{p>Y_{0}}U_{p}\right)\rvert}\leq\frac{\sum_{p\geq
Y_{0}}O(R^{d}/p^{2})}{\lvert[0,R]S\cap\Gamma_{d}\rvert}=\frac{\sum_{p\geq
Y_{0}}O(R^{d}/p^{2})}{\alpha
R^{d}\operatorname{Vol}([0,1]S)+o(R^{d})}\leq\epsilon$ (2.49)

for any $Y_{0}$ large enough. Let $k,f$ be as in Theorem 2.23 and let
$W\subset\mathbb{R}^{k}/\mathbb{Z}^{k}$ be an open subset. Denote for any
$R>0$ and $Y>0$:

$N(\Gamma_{d}^{<Y},A,R)=\lvert[0,R]S\cap(f\operatorname{mod}1)^{-1}(W)\cap\left(\cup_{p<Y}U_{p}\cap\Gamma_{d}\right)\rvert$
(2.50)

and similarly define $N(\Gamma_{d}^{>Y},A,R)$ and
$N(\Gamma_{d}^{<Y},R)=N(\Gamma_{d}^{<Y},\mathbb{R}^{k}/\mathbb{Z}^{k},R)$,$N(\Gamma_{d}^{>Y},R)=N(\Gamma_{d}^{>Y},\mathbb{R}^{k}/\mathbb{Z}^{k},R)$.
Then to prove the Corollary we need to estimate for any open Jordan measurable
$W\subset\mathbb{R}^{k}/\mathbb{Z}^{k}$:

$\frac{N(\Gamma_{d}^{<Y},W,R)+N(\Gamma_{d}^{>Y},W,R)}{N(\Gamma_{d}^{<Y},R)+N(\Gamma_{d}^{>Y},R)}=\frac{\frac{N(\Gamma_{d}^{<Y},W,R)}{N(\Gamma_{d}^{<Y},R)}+\frac{N(\Gamma_{d}^{>Y},W,R)}{N(\Gamma_{d}^{<Y},R)}}{1+\frac{N(\Gamma_{d}^{>Y},R)}{N(\Gamma_{d}^{<Y},R)}}=\frac{N(\Gamma_{d}^{<Y},W,R)}{N(\Gamma_{d}^{<Y},R)}+\epsilon$
(2.51)

for all $R,Y$ large enough, by Equation (2.49). Since $\Gamma_{d}^{<Y}$ is
defined by finitely many congruence conditions, Corollary 2.25 implies that
$\frac{N(\Gamma_{d}^{<Y},W,R)}{N(\Gamma_{d}^{<Y},R)}\rightarrow\operatorname{Vol}_{k}(W)$
as $R\rightarrow\infty$. Taking $\epsilon\rightarrow 0$ we deduce:

$\frac{N(\Gamma_{d}^{<Y},W,R)+N(\Gamma_{d}^{>Y},W,R)}{N(\Gamma_{d}^{<Y},R)+N(\Gamma_{d}^{>Y},R)}\rightarrow\operatorname{Vol}_{k}(W)$
(2.52)

and the claim of Theorem 2.23 is proved. ∎

Our main result of this Section will be stated using the function $f_{n}^{E}$
guaranteed by Lemma 2.21. For convenience and clarity, we define it again:

######  Definition 2.28.

For any hyperplane $E=u^{\perp}\subset\mathbb{R}^{n}$, $v_{0}\in
V_{\mathbb{R}}^{(0)}$ and $\Sigma_{n-1}^{\prime}$ be a fundamental domain for
the action of $\operatorname{SL}_{n-1}(\mathbb{Z})$ on
$\operatorname{SL}_{n-1}(\mathbb{R})$ which we will use throughout this
Section, let $f^{E}_{n}$ denote the function corresponding to the subspace $E$
under Lemma 2.21 and let $(w_{i})_{i=1}^{n-1}$ be the fixed basis of $V_{0}$
also defined in Lemma 2.21. Explicitly, we denote:

$f_{n}^{E}(p)=\frac{1}{\left\langle
u,\overline{1}\right\rangle}\sum_{i=1}^{n-1}\left\langle
u,\overline{\alpha}_{i}(p)\right\rangle w_{i}+q_{0}$

for any $p\in\mathbb{R}_{+}\cdot\Sigma_{n-1}\times\Sigma_{r-1}v_{0}$.
Moreover, denote:

$\Lambda_{w}=\operatorname{span}_{\mathbb{Z}}\\{w_{1},\dots,w_{n-1}\\}.$
(2.53)

The following Lemma summarizes the properties of the function $f_{n}^{E}$
which are relevant for Section 3.

######  Lemma 2.29.

Let $n=3$ ($n=4$,$n=5$), let $u\in\mathbb{R}^{n}$ satisfying
$u\notin(\mathbb{R}\cdot\overline{1})\cup\overline{1}^{\perp}$ and let
$v_{0}\in V_{\mathbb{R}}^{(0)}$. Then $f_{n}^{E=u^{\perp}}$ is homogeneous of
degree $D=1$ ($D=2$,$D=4$) and for any Jordan measurable bounded open subset
$U$ of $S=\Sigma_{n-1}^{\prime}\times\Sigma_{r-1}v_{0}$ it holds that
$f_{n}^{E},U,S$ satisfy the conclusion of Theorem 2.23 for $k=n-1$. Namely:

$f_{n}^{E}\operatorname{mod}\Lambda_{w}(\Gamma_{d})\text{ is $[0,R]\cdot
U$-equidistributed}$ (2.54)

where $\Gamma_{d}\subset V_{\mathbb{Z}}$ is the set of irreducible maximal
lattice points in $V_{\mathbb{Z}}$ (recall Definition 2.18).

Since there is an overlap in the arguments of the proof for $n=3,4,5$, we
write this overlap once and then continue to prove each case separately. Let
$u\in\mathbb{R}^{n}$ such that
$u\notin(\mathbb{R}\cdot\overline{1})\cup\overline{1}^{\perp}$ and fix
$v_{0}\in V^{(0)}_{\mathbb{R}}$. Let $U\subset
S=\Sigma_{n-1}^{\prime}\times\Sigma_{r-1}v_{0}$ be open and Jordan measurable
subset. It holds (e.g. by Weyl’s criterion for uniform distribution in [10])
that (2.54) is equivalent to:

$\text{For any }v_{\mathbb{Z}}\in\Lambda_{w}\text{ }\left\langle
f_{n}^{E},v_{\mathbb{Z}}\right\rangle\operatorname{mod}1(\Gamma_{d})\text{ is
$[0,R]\cdot U$-equidistributed}.$ (2.55)

Suppose for sake of contradiction that $v_{\mathbb{Z}}\in\Lambda_{w}$ is such
that (2.55) does not hold. Denote $f_{0}=\left\langle
f_{n}^{E},v_{\mathbb{Z}}\right\rangle$. Note that by Definition 2.28 there
exist real constants $(\gamma_{1},\dots,\gamma_{n-1})\neq 0$ such that:

$f_{0}(p)=\sum_{i=1}^{n-1}\gamma_{i}\left\langle\overline{\alpha}_{i}(p),u\right\rangle$
(2.56)

for every $p\in\mathbb{R}_{+}\cdot U\subset\mathbb{R}^{d}$. By Lemma 2.26,
$\Gamma_{d}$ is the intersection of an infinite congruence set and a set of
$S$ density $1$ in $\mathbb{Z}^{d}$. Therefore, by Corollary 2.27 we may
assume that for any $v\in\Gamma_{d}$, the function $f_{0}$, $S,A$ and $v$ do
not satisfy the conditions of Theorem 2.23. Indeed, otherwise the Corollary
will ensure that (2.55) holds. Explicitly, it means that for any
$v\in\Gamma_{d}$ the measure $(\frac{\partial^{D(n)}f_{0}}{\partial
v^{D(n)}}\operatorname{mod}1)_{*}m_{A}$ is not absolutely continuous w.r.t
$\lambda$ (recall Definition 2.22). Lemma A.1 implies that there exists a
subset $B=B_{v}\subset U$ of positive $m_{U}$ measure such that
$\nabla\frac{\partial^{D(n)}f_{0}}{\partial v^{D(n)}}=0$ on $B$. By Subsection
A.2 the smooth functions $\alpha_{1},\dots,\alpha_{n-1}:\mathbb{R}_{+}\cdot
U\rightarrow\mathbb{R}^{n}$ satisfy the multiplication table A.1 when $n=3$,
A.2 when $n=4$ and A.3 when $n=5$. Therefore, by Lemma A.2 for the open set
$V=\mathbb{R}_{+}\cdot U$ and $\alpha_{1},\dots,\alpha_{n-1}$ we deduce that
$\alpha_{1},\dots,\alpha_{n-1}$ are real analytic in $V$. Since linear
combinations and derivatives of real analytic functions is again real
analytic, this shows that $f_{0}$ and
$\nabla\frac{\partial^{D(n)}f_{0}}{\partial v^{D(n)}}$ are real analytic. By
Fubini’s Theorem $[1/2,3/2]\cdot B$ has positive $d$-dimensional Lebesgue
measure so Lemma A.3 for $\nabla\frac{\partial^{D(n)}f_{0}}{\partial
v^{D(n)}}$ and $[1/2,3/2]\cdot B\subset V$ implies that:

$\nabla\frac{\partial^{D(n)}f_{0}}{\partial v^{D(n)}}=0\text{ on }V.$ (2.57)

Indeed, since $\nabla\frac{\partial^{D(n)}f_{0}}{\partial v^{D(n)}}$ is
homogeneous of degree $-1$, $\nabla\frac{\partial^{D(n)}f_{0}}{\partial
v^{D(n)}}\mid_{B}=0$ implies $\nabla\frac{\partial^{D(n)}f_{0}}{\partial
v^{D(n)}}\mid_{[1/2,3/2]\cdot B}=0$.

###### Proof of Lemma 2.29, case $n=3$.

We continue from Equation (2.57). Since $v\in\Gamma_{d}$ was arbitrary and
$\Gamma_{d}$ contains a basis for $\mathbb{R}^{d}$, we can write:

$f_{0}(a_{1},\dots,a_{4})=\sum_{i=1}^{4}\eta_{i}a_{i}$ (2.58)

for some constants $\eta_{i}\in\mathbb{R};i=1,\dots,4$ and for every
$(a_{1},\dots,a_{4})\in\mathbb{R}_{+}\cdot U$. In other words, for every
$g=\begin{pmatrix}\ r&s\\\ t&u\\\ \end{pmatrix}\in U$:

$\gamma_{1}\left\langle\overline{\alpha}_{1}(gv_{0}),u\right\rangle+\gamma_{2}\left\langle\overline{\alpha}_{2}(gv_{0}),u\right\rangle=\sum_{i=1}^{4}\eta_{i}a_{i}(gv_{0})$

where for $v\in V_{\mathbb{R}}$ we denote $v=(a_{1}(v),\dots,a_{4}(v))$. Since
$\left\langle\overline{\alpha}_{1}(gv_{0}),\overline{1}\right\rangle,\left\langle\overline{\alpha}_{2}(gv_{0}),\overline{1}\right\rangle$
are proportional to $a_{2}(gv_{0}),a_{3}(gv_{0})$ respectively (see [5,
p.35]), we get (possibly for different constants $\gamma_{i},\eta_{i}$):

$\gamma_{1}\left\langle
P_{V_{0}}\overline{\alpha}_{1}(gv_{0}),u\right\rangle+\gamma_{2}\left\langle
P_{V_{0}}\overline{\alpha}_{2}(gv_{0}),u\right\rangle=\sum_{i=1}^{4}\eta_{i}a_{i}(gv_{0}).$
(2.59)

Write $\kappa_{i}=\left\langle
P_{V_{0}}\overline{\alpha}_{i}(v_{0}),u\right\rangle;i=1,2$, then as follows
from Subsection A.2:

$\left\langle P_{V_{0}}\overline{\alpha}_{1}(gv_{0}),u\right\rangle=(ru-
st)(\kappa_{1}r+\kappa_{2}s),$ (2.60) $\left\langle
P_{V_{0}}\overline{\alpha}_{2}(gv_{0}),u\right\rangle=(ru-
st)(\kappa_{1}t+\kappa_{2}u).$ (2.61)

By the first equation at the bottom of [5, p.38] we get:

$\sum_{i=1}^{4}\eta_{i}a_{i}(gv_{0})=\eta_{1}(a_{0}r^{3}+b_{0}r^{2}s+c_{0}rs^{2}+d_{0}s^{3})+\eta_{2}(3a_{0}r^{2}t+2b_{0}rst+c_{0}s^{2}t+b_{0}r^{2}u+2c_{0}rsu+3d_{0}s^{2}u)$
$+\eta_{3}(3a_{0}rt^{2}+2b_{0}rtu+b_{0}st^{2}+c_{0}ru^{2}+2c_{0}stu+3d_{0}su^{2})+\eta_{4}(a_{0}t^{3}+b_{0}t^{2}u+c_{0}tu^{2}+d_{0}u^{3})$

Assume $a_{0}\neq 0$. Then since neither of $r^{3},r^{2}t,rt^{2},t^{3}$ appear
in Equations (2.60),(2.61), Equation (2.59) implies that
$\eta_{i}=0;i=1,\dots,4$, a contradiction. Similarly, assuming that $d_{0}\neq
0$, since $s^{3},s^{2}u,su^{2},u^{3}$ do not appear in Equations (2.60),(2.61)
we get the same contradiction. Since
$\operatorname{Disc}(v_{0})=b_{0}^{2}c_{0}^{2}-4a_{0}c_{0}^{3}-4b_{0}^{3}d_{0}-27a_{0}^{2}d_{0}^{2}+18a_{0}b_{0}c_{0}d_{0}\neq
0$ by assumption, we find that $b_{0},d_{0}\neq 0,a_{0}=d_{0}=0$. The
coefficients of $rst,r^{2}u$ in the right hand side of (2.59) are
$2\eta_{2}b_{0},\eta_{2}b_{0}$ respectively. On the left hand side they are
$-\gamma_{1}\kappa_{1},\gamma_{1}\kappa_{1}$ respectively. This implies that
$-\gamma_{1}\kappa_{1}=2\gamma_{1}\kappa_{1}$ thus $\kappa_{1}\gamma_{1}=0$.
Similarly, we show that $\kappa_{2}\gamma_{2}=0$. Note that by our choice of
$u$, $\kappa_{i}=\left\langle
P_{V_{0}}\overline{\alpha}_{i}(v_{0}),u\right\rangle\neq 0;i=1,2$ so that
$\gamma_{1}=\gamma_{2}=0$, a contradiction. ∎

###### Proof of Lemma 2.29, case $n=4$.

We continue from Equation (2.57). Since $v\in\Gamma_{d}$ was arbitrary and
$\Gamma_{d}$ contains a basis for $\mathbb{R}^{d}$, we can write:

$f_{0}(a_{1},\dots,a_{4})=\sum_{i,j=1}^{6}\eta_{i,j}a_{i}(gv_{0})a_{j}(gv_{0})$
(2.62)

where $a_{i}(v);i=1,\dots,12$ are the coefficients of an element $v\in
V_{\mathbb{R}}$. Fix $v_{0}=(A_{0},B_{0})\in V^{(0)}_{\mathbb{R}}$ where
$A_{0},B_{0}$ is a pair of symmetric $3\times 3$ matrices (see [5, p.46]).
Write:

$h\vcentcolon=A_{0}=\begin{pmatrix}a_{0}&d_{0}/2&f_{0}/2\\\
d_{0}/2&b_{0}&e_{0}/2\\\ f_{0}/2&e_{0}/2&c_{0}\end{pmatrix}.$ (2.63)

For a general $v\in V_{\mathbb{R}}^{(0)}$ write $v=(A(v),B(v))$ where
$A(v),B(v)$ is a pair of symmetric $3\times 3$ matrices (see [5, p.46]).
Equation (2.62) is equivalent to:

$\gamma_{1}\left\langle\overline{\alpha}_{1}(gv_{0}),u\right\rangle+\gamma_{2}\left\langle\overline{\alpha}_{2}(gv_{0}),u\right\rangle+\gamma_{3}\left\langle\overline{\alpha}_{3}(gv_{0}),u\right\rangle=\sum_{i,j=1}^{6}\eta_{i,j}a_{i}(gv_{0})a_{j}(gv_{0}).$

Since
$\left\langle\overline{\alpha}_{i}(gv_{0}),\overline{1}\right\rangle,i=1,2,3$
can each be written as some linear combination of
$a_{i}(gv_{0})a_{j}(gv_{0});i,j=1,\dots,6$ (see [5, p. 41-42]), we get
(possibly for different constants):

$\gamma_{1}\left\langle
P_{V_{0}}\overline{\alpha}_{1}(gv_{0}),u\right\rangle+\gamma_{2}\left\langle
P_{V_{0}}\overline{\alpha}_{2}(gv_{0}),u\right\rangle+\gamma_{3}\left\langle
P_{V_{0}}\overline{\alpha}_{3}(gv_{0}),u\right\rangle=\sum_{i,j=1}^{6}\eta_{i,j}a_{i}(gv_{0})a_{j}(gv_{0}).$
(2.64)

Write $\kappa_{i}=\left\langle
P_{V_{0}}\overline{\alpha}_{i}(v_{0}),u\right\rangle;i=1,2,3$ and
$g=\begin{pmatrix}-&v_{1}&-\\\ -&v_{2}&-\\\ -&v_{3}&-\end{pmatrix}$ then it
follows from Subsection A.2:

$\left\langle
P_{V_{0}}\overline{\alpha}_{1}(gv_{0}),u\right\rangle=\det(g)\left\langle
v_{1},\overline{\kappa}\right\rangle,$ (2.65) $\left\langle
P_{V_{0}}\overline{\alpha}_{2}(gv_{0}),u\right\rangle=\det(g)\left\langle
v_{2},\overline{\kappa}\right\rangle,$ (2.66) $\left\langle
P_{V_{0}}\overline{\alpha}_{3}(gv_{0}),u\right\rangle=\det(g)\left\langle
v_{3},\overline{\kappa}\right\rangle.$ (2.67)

where $\overline{\kappa}=(\kappa_{1},\kappa_{2},\kappa_{3})$. If $g\in U$, by
[5, p.46]:

$A(gv_{0})=gA_{0}g^{t}=\begin{pmatrix}-&v_{1}&-\\\ -&v_{2}&-\\\
-&v_{3}&-\end{pmatrix}\begin{pmatrix}a_{0}&d_{0}/2&f_{0}/2\\\
d_{0}/2&b_{0}&e_{0}/2\\\
f_{0}/2&e_{0}/2&c_{0}\end{pmatrix}\begin{pmatrix}\textemdash&\mid&\mid\\\
v_{1}&v_{2}&v_{3}\\\ \mid&\mid&\mid\end{pmatrix}=\begin{pmatrix}-&v_{1}&-\\\
-&v_{2}&-\\\ -&v_{3}&-\end{pmatrix}\begin{pmatrix}\left\langle
h_{1},v_{1}\right\rangle&\left\langle h_{1},v_{2}\right\rangle&\left\langle
h_{1},v_{3}\right\rangle\\\ \left\langle h_{2},v_{1}\right\rangle&\left\langle
h_{2},v_{2}\right\rangle&\left\langle h_{2},v_{3}\right\rangle\\\ \left\langle
h_{3},v_{1}\right\rangle&\left\langle h_{3},v_{2}\right\rangle&\left\langle
h_{3},v_{3}\right\rangle\end{pmatrix}.$ (2.68)

A general entry of the matrix above will be denoted by
$(i,j)\vcentcolon=\left\langle v_{i},(\left\langle
h_{1},v_{j}\right\rangle,\left\langle h_{2},v_{j}\right\rangle,\left\langle
h_{3},v_{j}\right\rangle)\right\rangle$. Therefore, by Equations
(2.64),(2.65),(2.66) and (2.67) we get:

$\det(g)(\gamma_{1}\left\langle
v_{1},\overline{\kappa}\right\rangle+\gamma_{2}\left\langle
v_{2},\overline{\kappa}\right\rangle+\gamma_{3}\left\langle
v_{3},\overline{\kappa}\right\rangle)=\sum_{i_{1},j_{1},i_{2},j_{2}=1,\dots,6}\eta_{i_{1},i_{2},j_{1},j_{2}}(i_{1},j_{1})(i_{2},j_{2}).$
(2.69)

Write $v_{i}=(v_{i1},v_{i2},v_{i3})$ for every $i=1,2,3$. Assume without loss
of generality that $\kappa_{1}\neq 0$ and $\gamma_{1}\neq 0$ (it will be clear
from the argument why this assumption preserves the generality). Consider the
coefficients $c_{1},c_{2}$ of $v_{11}^{2}v_{22}v_{33}$ and
$v_{11}^{2}v_{23}v_{32}$ respectively on the left hand side of Equation
(2.69). By definition of $\det(g)$ and Equation (2.69) it holds that
$c_{1}=-c_{2}=\kappa_{1}\gamma_{1}\neq 0$. However, direct calculation shows
that the coefficient of $v_{11}^{2}v_{22}v_{33}$ on the right hand side of
Equation $(\ref{H})$ is
$\eta_{1,1,2,3}h_{11}h_{23}+\eta_{1,3,1,2}h_{13}h_{12}$. The same calculation
shows that $\eta_{1,1,2,3}h_{11}h_{32}+\eta_{1,3,1,2}h_{12}h_{13}$ is the
coefficient of $v_{11}^{2}v_{23}v_{32}$. However, $h$ is symmetric (recall
Equation (2.63)) so these two expressions are equal. Under the assumption that
Equation (2.69) poses we got that on the one hand $c_{1}=-c_{2}$ and on the
other $c_{1}=c_{2}$. The inevitable conclusion that $c_{1}=c_{2}=0$
contradicts our assumption that $\kappa_{1},\gamma_{1}\neq 0$. Indeed, the
choice $\kappa_{1},\gamma_{1}\neq 0$ did not affect the generality: we would
have otherwise onsidered the coefficients of
$v_{i,j}^{2}v_{kl}v_{mp},v_{i,j}^{2}v_{kp}v_{ml}$ for some other appropriate
$i,j,k,l,m,p\in\\{1,2,3\\}$. ∎

###### Proof of Lemma 2.29, case $n=5$.

We continue from Equation (2.57). Denote:

$I=\\{(i,j,k):i,j=1,\dots,5,k=1,\dots,4\\}$ (2.70)

and for $i_{0}=(i,j,k)\in I$ denote:

$A(i_{0})=A^{(k)}_{i,j}.$ (2.71)

Since $v\in\mathbb{Z}^{d}$ was arbitrary, we can write:

$f_{0}(A^{(1)},\dots,A^{(4)})=\sum_{i_{1},\dots,i_{5}\in
I}\eta_{i_{1},\dots,i_{5}}A(i_{1})A(i_{2})A(i_{3})A(i_{4})A(i_{5})$ (2.72)

Write $g_{4}^{t}=\begin{pmatrix}\ \mid&\mid&\mid&\mid\\\
v_{1}&v_{2}&v_{3}&v_{4}\\\
\mid&\mid&\mid&\mid\end{pmatrix},g_{5}^{t}=\begin{pmatrix}\
\mid&\mid&\mid&\mid&\mid\\\ u_{1}&u_{2}&u_{3}&u_{4}&u_{5}\\\
\mid&\mid&\mid&\mid&\mid\end{pmatrix}$ and $g=(g_{4},g_{5})$. By [5, p. 52] we
know that:

$g\cdot v_{0}=\left(g_{4}\begin{bmatrix}g_{5}A^{(1)}g_{5}^{t}\\\
g_{5}A^{(2)}g_{5}^{t}\\\ g_{5}A^{(3)}g_{5}^{t}\\\
g_{5}A^{(4)}g_{5}^{t}\end{bmatrix}\right)$ (2.73)

where we abuse the notation $g=(g_{4},g_{5})$. Now, given $u$ as in the
conditions of the Lemma we write by A.2:

$\left\langle
P_{V_{0}}(\alpha_{i}(gv_{0}),u\right\rangle=\det(g_{4})\left(\sum_{j=1}^{4}v_{ij}\left\langle
P_{V_{0}}(\alpha_{j}(v_{0})),u\right\rangle\right),$ (2.74) $\left\langle
P_{\mathbb{R}\cdot\overline{1}}\alpha_{i}(gv_{0}),u\right\rangle=\frac{1}{n}\left\langle\alpha_{i}(gv_{0}),\overline{1}\right\rangle\left\langle
u,\overline{1}\right\rangle.$ (2.75)

Equation (2.72) and the above two equations imply:

$\sum_{i_{1},\dots,i_{5}\in
I}\eta_{i_{1},\dots,i_{5}}\left(A(i_{1})A(i_{2})A(i_{3})A(i_{4})A(i_{5})\right)(gv_{0})=f_{0}(A^{(1)}(gv_{0}),A^{(2)}(gv_{0}),A^{(3)}(gv_{0}),A^{(4)}(gv_{0}))$
$=\left\langle\overline{\alpha}_{i}(gv_{0}),1\right\rangle+\frac{\det{(g_{4})}}{\left\langle
u,1\right\rangle}\sum_{j=1}^{4}v_{ij}\left\langle
P_{V_{0}}(\alpha_{j}(v_{0})),u\right\rangle+q_{0}$ (2.76)

and so:

$\sum_{i_{1},\dots,i_{5}\in
I}\eta_{i_{1},\dots,i_{5}}\left(A(i_{1})A(i_{2})A(i_{3})A(i_{4})A(i_{5})\right)(gv_{0})-\left\langle\overline{\alpha}_{i}(gv_{0}),1\right\rangle=\frac{\det{(g_{4})}}{\left\langle
u,1\right\rangle}\sum_{j=1}^{4}v_{ij}\left\langle
P_{V_{0}}(\alpha_{j}(v_{0})),u\right\rangle+q_{0}$ (2.77)

which, by Equation (2.73) is a contradiction, the left hand side of the above
equation has non-trivial dependence in $g_{5}$ while the right hand side
doesn’t. ∎

##  3 Proofs

In this Section we prove Theorems 1.5 and 1.4 (in this order) by proving their
equivalent formulations, namely Theorems 2.7 and 2.9 repectively. Let
$n=3,4,5$ and fix fundamental domains $\Sigma_{n-1}^{\prime},\Sigma_{r-1}$ for
the actions of $\operatorname{SL}_{n-1}(\mathbb{Z})$ and
$\operatorname{SL}_{r-1}(\mathbb{Z})$ on $\operatorname{SL}_{n-1}(\mathbb{R})$
and $\operatorname{SL}_{n-1}(\mathbb{R})$ respectively. Since we use it in the
proof, we remind the reader of Definition 2.28 by repeating it once again:

######  Definition.

For any hyperplane $E=u^{\perp}\subset\mathbb{R}^{n}$, let $f^{E}_{n}$ denote
the function corresponding to the subspace $E$ under Lemma 2.21. Explicitly,
we denote:

$f_{n}^{E}(p)=\frac{1}{\left\langle
u,\overline{1}\right\rangle}\sum_{i=1}^{n-1}\left\langle
u,\overline{\alpha}_{i}(p)\right\rangle w_{i}+q_{0}$

for any $p\in\mathbb{R}_{+}\cdot(\Sigma_{n-1}\times\Sigma_{r-1})v_{0}$ (recall
that in Definition 2.28 $(w_{i})_{i=1}^{n-1}$ was some fixed basis of
$V_{0}$).

Before proving the main Theorems we make a reduction that will serve us both
in the proof of Theorem 2.7: First, for convenience of this discussion, given
a choice of orderings namely $\Pi:\mathcal{F}^{(0)}_{\infty}\rightarrow S_{n}$
(recall Subsection 2.1 for the definition of $\mathcal{F}^{(0)}_{\infty}$),
write $\operatorname{ED}(\Pi)$ if (recall Subsection 2.2 for the definition of
$\Gamma^{\Pi(\mathbb{F})}_{E}$):

$\Gamma^{\Pi(\mathbb{F})}_{E}\text{ is equidistributed in }Y_{n-1}\text{ when
}\mathbb{F}\in\mathcal{F}_{n}\text{ are ordered by discriminant}.$

For $n=3,4,5$, if there exists $\Pi:\mathcal{F}_{\infty}^{(0)}\rightarrow
S_{n}$ such that $\operatorname{ED}(\Pi)$ and for any $\pi\in S_{n}$,
$\operatorname{ED}(\pi\circ\Pi)$ where
$(\pi\circ\Pi)(\mathbb{F})\vcentcolon=\pi\circ(\Pi(\mathbb{F}))$ then
averaging the counting measures for each $\pi$ proves the equidistribution
statement in Theorem 2.7. Since we find such $\Pi$ using Lemma 2.21, the extra
property $\operatorname{ED}(\pi\circ\Pi)$ for any $\pi\in S_{n}$ will hold by
the last part of this Lemma with the same argument. We continue with this
reduction in mind.

###### Proof of Theorem 2.7.

Let $\overline{1}\neq u\in\mathbb{R}^{n}$ such that $u\notin V_{0}$ and denote
$E=u^{\perp}$. Fix $i=1,\dots,\lfloor n/2\rfloor$, pick $v_{0}\in
V_{\mathbb{R}}^{(i)}$ and let $\Pi:\mathcal{F}_{X}^{(0)}\rightarrow S_{n}$ and
$\Sigma_{n-1}$ be the function and fundamental domain (for the action of
$\operatorname{stab}_{\operatorname{SL}_{n}(\mathbb{R})}(\Lambda_{w})$ on
$\operatorname{SL}_{n-1}(\mathbb{R})$) which correspond to $E$, $v_{0}$ and
$i$ by Lemma 2.21. It suffices to prove the claim for any signature separately
because then the claim follows by taking the average. Since our discussion
(particularly Theorem 2.21) holds for any $i$, we assume without the loss of
generality that $i=0$ and we are in the totally real case. Abbreviate
$f_{E}=f_{n}^{E}$. For an open bounded and Jordan measurable subset
$S\subset\Sigma_{n-1}$ and $V=I_{1}w_{1}+\dots+I_{n-1}w_{n-1}$ where
$I_{i}\subset[0,1]$ are open intervals, let $A=S\times_{\Sigma_{n-1}}V\subset
Y_{n-1}$ be a $\Lambda_{w}$-$\Sigma_{n-1}$-basic subset (recall Definition
2.10). From here on, we abbreviate ”open bounded and Jordan measurable” to
OBJOM.
For every $\epsilon>0$ let $\Sigma_{r-1}^{\epsilon}\subset\Sigma_{r-1}$ be
OBJOM such that:

$\operatorname{Vol}\left([0,1]\cdot(S\times\Sigma_{r-1})v_{0}\right)\geq\operatorname{Vol}\left([0,1]\cdot(S\times\Sigma_{r-1}^{\epsilon})v_{0}\right)\geq(1-\epsilon)\operatorname{Vol}\left([0,1]\cdot(S\times\Sigma_{r-1})v_{0}\right).$
(3.1)

For every $T>0$ denote:

$S_{T}^{\epsilon}=[0,T]\cdot\left(S\times\Sigma_{r-1}^{\epsilon}v\right)v_{0},S_{T}=[0,T]\cdot\left(S\times\Sigma_{r-1}\right)v_{0};$
(3.2) $S_{T,\mathbb{Z}}^{\epsilon}=S_{T}^{\epsilon}\cap\\{\text{irreducible
maximal points}\\},S_{T,\mathbb{Z}}=S_{T}\cap\\{\text{irreducible maximal
points}\\};$ (3.3)
$L_{T}=[0,T]\cdot\left(\Sigma_{n-1}\times\Sigma_{r-1}\right)v_{0},L_{T,\mathbb{Z}}=L_{T}\cap\\{\text{irreducible
maximal points}\\};$ (3.4) $p_{T}^{\epsilon}(V)=\frac{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\cap f_{E}^{-1}(V)\rvert}{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}.$ (3.5)

Repeating the proof of Theorem 5 in [3] word by word but replacing the shape
function (denoted as $q$ in [3, p. 7]) with the lattice function
$l:V_{\mathbb{R}}^{(0)}\rightarrow X_{n-1}$ defined for every
$v=t(v)(g_{1}(v),g_{2}(v))v_{0}\in\mathbb{R}_{+}\cdot\Sigma_{n-1}\times\Sigma_{r-1}v_{0}$
by:

$l(v)=g_{0}g_{1}(v)^{t}g_{0}^{-1}\Lambda_{w},$ (3.6)

we get:

$\lvert S_{T,\mathbb{Z}}\rvert=\operatorname{Vol}(S_{T})+o(T^{d}),\lvert
L_{T,\mathbb{Z}}\rvert=\operatorname{Vol}(L_{T})+o(T^{d}).$ (3.7)

Given a subset $B\subset Y_{n-1}$ and $T>0$ denote $\mu_{T}(B)$ to be the
proportion of totally real number fields $\mathbb{F}$ with discriminant less
than $T$ that satisfy $\Gamma_{E}^{\pi}(\mathbb{F})\in B$ (recall Subsection
2.2). Invoke Lemma 2.21 to deduce that for any $T>0$:

$\mu_{T}(A)=\frac{\lvert S_{T,\mathbb{Z}}\cap f_{E}^{-1}(V)\rvert}{\lvert
L_{T,\mathbb{Z}}\rvert}=\frac{\lvert S_{T,\mathbb{Z}}\cap
f_{E}^{-1}(V)\rvert}{\lvert S^{\epsilon}_{T,\mathbb{Z}}\cap
f_{E}^{-1}(V)\rvert}\cdot\frac{\lvert S^{\epsilon}_{T,\mathbb{Z}}\cap
f_{E}^{-1}(V)\rvert}{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}\cdot\frac{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}{\lvert
S_{T,\mathbb{Z}}\rvert}\cdot\frac{\lvert S_{T,\mathbb{Z}}\rvert}{\lvert
L_{T,\mathbb{Z}}\rvert}.$ (3.8)

We estimate every element in the above product separately. First, we deal with
$p_{T}^{\epsilon}(V)=\frac{\lvert S_{T,\mathbb{Z}}^{\epsilon}\cap
f_{E}^{-1}(V)\rvert}{\lvert S_{T,\mathbb{Z}}^{\epsilon}\rvert}$. By
application of Lemma 2.29 for the OBJOM set
$\left(S\times\Sigma_{r-1}^{\epsilon}\right)v_{0}$:

$p_{T}^{\epsilon}(V)\rightarrow\operatorname{Vol}_{n-1}(V)\text{ as
}T\rightarrow\infty.$ (3.9)

By Equation (3.7):

$\frac{\lvert S_{T,\mathbb{Z}}\rvert}{\lvert
L_{T,\mathbb{Z}}\rvert}\rightarrow\frac{\operatorname{Vol}(S_{1})}{\operatorname{Vol}(L_{1})}\text{
as }T\rightarrow\infty$ (3.10)

and by the calculation in the proof of [3, Proposition 12] applied for
$l^{-1}(S)\cap\\{\operatorname{Disc}(v)\leq 1\\}$ instead of
$\mathcal{R}_{1,W}$:

$\frac{\lvert S_{T,\mathbb{Z}}\rvert}{\lvert
L_{T,\mathbb{Z}}\rvert}\rightarrow m_{X_{n-1}}(S)\text{ as
}T\rightarrow\infty$ (3.11)

(recall Subsection 2.2 for definition of $m_{X_{n-1}}$, the Haar measure on
the space of lattices). Since $S_{1}^{\epsilon}$ is bounded we can use [3,
Lemma 6] on it to deduce that for any $T>0$:

$\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert=\operatorname{Vol}(S_{T}^{\epsilon})+o(T^{d})$
(3.12)

which implies, by Equations (3.7) and (3.1) that:

$\frac{\lvert S_{T,\mathbb{Z}}\rvert}{\lvert
S^{\epsilon}_{T,\mathbb{Z}}\rvert}=1-\epsilon+o_{T}(1).$ (3.13)

Consequently, by dividing the space into its disjoint intersections with
$f^{-1}_{E}(V),f^{-1}_{E}(V^{c})$:

$\left(\lvert S_{T,\mathbb{Z}}\cap f_{E}^{-1}(V)\rvert-\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap f_{E}^{-1}(V)\rvert\right)+\left(\lvert
S_{T,\mathbb{Z}}\cap f_{E}^{-1}(V^{c})\rvert-\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap
f_{E}^{-1}(V^{c})\rvert\right)\leq\epsilon\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert+\lvert S_{T,\mathbb{Z}}^{\epsilon}\rvert
o_{T}(1)$

which implies, since both terms in parenthesis in the above equation are non-
negative:

$\frac{\lvert S_{T,\mathbb{Z}}\cap f_{E}^{-1}(V)\rvert}{\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap f_{E}^{-1}(V)\rvert}-1\leq\epsilon\frac{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}{\lvert S_{T,\mathbb{Z}}^{\epsilon}\cap
f_{E}^{-1}(V)\rvert}+\frac{\lvert S_{T,\mathbb{Z}}^{\epsilon}\rvert}{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\cap
f_{E}^{-1}(V)\rvert}o_{T}\left(1\right)=\epsilon
p^{\epsilon}_{T}(V)+p_{T}^{\epsilon}(V)o_{T}(1)$ (3.14)

but now using (3.9):

$\frac{\lvert S_{T,\mathbb{Z}}\cap f_{E}^{-1}(V)\rvert}{\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap
f_{E}^{-1}(V)\rvert}=1+\epsilon\operatorname{Vol}(V)(1+o_{T}(1)).$ (3.15)

Using (3.11),(3.13),(3.9) and (3.15) and taking $\epsilon\rightarrow 0$ we
deduce:

$\mu_{T}(A)\rightarrow
m_{X_{n-1}}(S)\operatorname{Vol}(V)=m_{Y_{n-1}}(A)\text{ as
}T\rightarrow\infty$ (3.16)

where the last equality holds by Fubini’s Theorem and Definition 2.10. To
deduce the above limit for any other Jordan measurable subset $A\subset
Y_{n-1}$ invoke Theorem B.3. Note that this proves the desired claim, which is
about the weak convergence of $\mu_{T}$ to $\mu_{Y_{n-1}}$. ∎

###### Proof of Theorem 2.9.

The reader may find it useful to recall the definition of the corresponding
grid which appears in Subsection 2.2. By Lemma 2.21, there exists a fixed
basis of $V_{0}$ given by $w=(w_{j})_{j=1}^{n-1}$, a function
$\Pi:\mathcal{F}_{\infty}^{(0)}\rightarrow S_{n}$ (recall Subsection 2.1) and
$g_{0}\in\operatorname{SL}_{n-1}(\mathbb{R})$ such that:

$\Sigma_{n-1}\vcentcolon=g_{0}^{t}\Sigma_{n-1}^{\prime}g_{0}^{-t}$ (3.17)

satisfies that for any $\Lambda_{w}$-$\Sigma_{n-1}$-basic subset
$S\times_{\Sigma_{n-1}}U\subset Y_{n-1}$:

$\mathcal{F}^{(0)}_{X}\cap(\Gamma_{V_{0}}^{\Pi})^{-1}(S\times_{\Sigma_{n-1}}U)\xleftrightarrow{1:1}$
(3.18) $\\{\text{irreducible, maximal points inside
}[0,X](g_{0}^{-t}S^{-1}g_{0}^{t}\times\Sigma_{r-1})v_{0}\\}\cap((f_{w}+q_{0})\operatorname{mod}\Lambda_{w})^{-1}(U_{w})$

where $q_{0}\in V_{0}$ is some fixed vector (recall Definition 2.11 for
definition of $U_{w}$). In this case:

$f_{w}(p)=\frac{1}{\left\langle\overline{1},\overline{1}\right\rangle}\sum_{i=1}^{n-1}\left\langle\overline{1},\overline{\alpha}_{i}(g_{p}v_{0})\right\rangle
w_{i}=\frac{1}{n}\sum_{i=1}^{n-1}\left\langle\overline{1},\overline{\alpha}_{i}(g_{p}v_{0})\right\rangle
w_{i}.$ (3.19)

By the discussion carried in [5, p.35 for $n=3$,p.41 for $n=4$ and p.50 for
$n=5$] for every $i=1,\dots,n-1$:

$\left\langle\overline{1},\overline{\alpha}_{i}(g_{p}v_{0})\right\rangle\in\mathbb{Z}\text{
if }p\in V_{\mathbb{Z}}\text{ and moreover
}\left\langle\overline{1},\overline{\alpha}_{i}(g_{p}v_{0})\right\rangle\text{
is a polynomial in the coefficients of }p.$ (3.20)

Let $\Lambda_{w}=\operatorname{span}_{\mathbb{Z}}\\{w_{1},\dots,w_{n-1}\\}$
and denote:

$D=\\{\sum_{i=1}^{n-1}\frac{j_{i}}{n}w_{i}\operatorname{mod}\Lambda_{w}:j_{i}=1,\dots,n\\}.$
(3.21)

By Equations (3.18)-(3.20) for every totally real $S_{n}$-number field
$\mathbb{F}$ of degree $n$:

$\Gamma_{V_{0}}^{\Pi}(\mathbb{F})\in\Sigma_{n-1}\times_{\Sigma_{n-1}}D.$
(3.22)

Let $m_{\Sigma_{n-1}}$ be the restriction of
$m_{\operatorname{SL}_{n-1}(\mathbb{R})}$ to $\Sigma_{n-1}$ and let $k$ be a
divisor of $n$. Recall Definition 2.8 to note that if $\zeta\in D$ is such
that $k\zeta\in\Lambda_{w}$ and $(k-1)\zeta\notin\Lambda_{w}$ then:

$\operatorname{SL}_{n-1}(\mathbb{R})\times_{\Sigma_{n-1}}\\{\zeta\\}=Y_{n}(k).$
(3.23)

Denote also

$F_{\zeta}:=\overline{\Sigma_{n-1}\times_{\Sigma_{n-1}}\\{\zeta\\}}\subset
Y_{n}(k).$ (3.24)

By Equation (3.20)
$f_{i}(p)\vcentcolon=\left\langle\overline{1},\overline{\alpha}_{i}(g_{p}v_{0})\right\rangle$
is an integer polynomial in the coefficients of $p$. Therefore there exists a
congruence set $C(\zeta)$ with congruences modulo $k$ such that:

$f_{w}(p)=\zeta\iff p\in C(\zeta)\iff\Gamma_{V_{0}}^{\Pi}(p)\in F_{\zeta}.$
(3.25)

Let $\mu_{T}$ be defined for any $T>0$ as in the proof of Theorem 1.4. To
prove Theorem 2.9 it will suffice to find $\alpha_{\zeta}>0$ such that for any
OBJOM $S\subset\Sigma_{n-1}$, $A_{\zeta,S}=S\times_{\Sigma_{n-1}}\\{\zeta\\}$
satisfies:

$\mu_{T}(A_{\zeta,S})\rightarrow\alpha_{\zeta}m_{X_{n-1}}(S)=m_{Y_{n}(k)}\mid_{F_{\zeta}}(A_{\zeta,S})\text{
as }T\rightarrow\infty$ (3.26)

by (3.25). Let $\alpha_{\zeta}$ be the density of $C(\zeta)$ inside
$V_{\mathbb{Z}}$, namely:

$\alpha_{\zeta}=\lim_{T\rightarrow\infty}\frac{\lvert C(\zeta)\cap
V_{\mathbb{Z}}\cap B_{T}(0)\rvert}{T^{d}}$ (3.27)

which also satisfies since $S_{T}^{\epsilon}$ is an OBJOM:

$\frac{\lvert S^{\epsilon}_{T,\mathbb{Z}}\cap
f_{w}^{-1}(\\{\zeta\\})\rvert}{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}=\frac{\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap C(\zeta)\rvert}{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}\rightarrow\alpha_{\zeta}\text{ as
}T\rightarrow\infty.$ (3.28)

Then precisely as in Equation (3.8) it holds that:

$\mu_{T}(A_{\zeta,S})=\frac{\lvert S_{T,\mathbb{Z}}\cap
f_{w}^{-1}(\\{\zeta\\})\rvert}{\lvert L_{T,\mathbb{Z}}\rvert}=\frac{\lvert
S_{T,\mathbb{Z}}\cap f_{w}^{-1}(\\{\zeta\\})\rvert}{\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap
f_{w}^{-1}(\\{\zeta\\})\rvert}\cdot\frac{\lvert
S^{\epsilon}_{T,\mathbb{Z}}\cap f_{w}^{-1}(\\{\zeta\\})\rvert}{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}\cdot\frac{\lvert
S_{T,\mathbb{Z}}^{\epsilon}\rvert}{\lvert
S_{T,\mathbb{Z}}\rvert}\cdot\frac{\lvert S_{T,\mathbb{Z}}\rvert}{\lvert
L_{T,\mathbb{Z}}\rvert}.$ (3.29)

where the second term converges, by Equation (3.28), to $\alpha_{\zeta}$ as
$T\rightarrow\infty$ and the remaining terms are the same as in the proof of
Theorem 1.4. Therefore $\mu_{T}(A_{\zeta,S})$ converges to
$\alpha_{\zeta}m_{X_{n-1}}(S)$ as $T\rightarrow\infty$. We therefore choose:

$\\{F^{(k)}_{\ell}\\}=\\{F_{\zeta}:\zeta\in D\text{ is such that
}k\zeta\in\Lambda_{w},(k-1)\zeta\notin\Lambda_{w}\\}$ (3.30)

and conclude by Equations (3.22) and (3.25). ∎

##  Appendix A Calculations

###  A.1 Multiplication Tables

For convenience we include explicitly the multiplication tables that define
Bhargava’s correspondence in [1, 2]. They are copied from [5, p. 35,39-40].

####  A.1.1 Case $n=3$

Let $v\in V_{\mathbb{R}}$ be given by $v=(a,b,c,d)$. The multiplication table
corresponding to $v$ is:

$\alpha_{1}\alpha_{2}=-ad$ $\alpha_{1}^{2}=-ac+b\alpha_{1}-a\alpha_{2}$ (A.1)
$\alpha_{2}^{2}=-bd+d\alpha_{1}-c\alpha_{2}.$

####  A.1.2 Case $n=4$

Let $v\in V_{\mathbb{R}}$ be given by
$v=(a_{11}x^{2}+a_{22}y^{2}+a_{33}z^{2}+a_{12}xy+a_{13}xz+a_{23}yz,b_{11}x^{2}+b_{22}y^{2}+b_{33}z^{2}+b_{12}xy+b_{13}xz+b_{23}yz)$.
The multiplication table corresponding to $v$ is:

$\alpha_{1}^{2}=h_{11}+g_{11}\alpha_{1}+f_{11}\alpha_{2}+e_{11}\alpha_{3}$
$\alpha_{2}^{2}=h_{22}+g_{22}\alpha_{1}+f_{22}\alpha_{2}+e_{22}\alpha_{3}$
$\alpha_{3}^{2}=h_{33}+g_{33}\alpha_{1}+f_{33}\alpha_{2}+e_{33}\alpha_{3}$
(A.2) $\alpha_{1}\alpha_{2}=h_{12}+e_{12}\alpha_{3}$
$\alpha_{1}\alpha_{3}=h_{13}+f_{13}\alpha_{2}+e_{13}\alpha_{3}$
$\alpha_{2}\alpha_{3}=h_{23}+g_{23}\alpha_{1}+f_{23}\alpha_{2}+e_{23}\alpha_{3}.$

####  A.1.3 Case $n=5$

Let $v\in V_{\mathbb{R}}$ be given by $v=(A_{1},\dots,A_{4})$ where $A_{i}$
are $5\times 5$ skew symmetric matrices. The multiplication table
corresponding to $v$ is:

$\alpha_{i}\alpha_{j}=c_{ij}^{0}+\sum_{k=1}^{4}c^{k}_{ij}\alpha_{k}$ (A.3)

for some constants $c_{ij}^{k}$ which are some polynomials in the entries of
$(A_{1},\dots,A_{4})$ and appearing in [5, p. 51].

###  A.2 Construction of a Basis

Let $n=3,4,5$ and fix $v_{0}\in V_{\mathbb{R}}^{(0)}$. Let
$\overline{\alpha}_{1}(v_{0}),\dots,\overline{\alpha}_{n}(v_{0})\in\mathbb{R}^{n},\overline{\alpha}_{n}(v_{0})=\overline{1}$
be a set of vectors such that
$\operatorname{MT}_{v_{0}}(\overline{\alpha}_{1},\dots,\overline{\alpha}_{n-1})$
with the pointwise product on $\mathbb{R}^{n}$.

For any $g=\begin{pmatrix}\mid\ &\cdots&\mid\\\
\overline{g_{1}}&\cdots&\overline{g_{n-1}}\\\ \mid&\cdots&\mid\end{pmatrix}$
and $h\in\operatorname{SL}_{r-1}(\mathbb{R})$, write
$v=t(g,h)v_{0};t\in\mathbb{R}$ in a unique way (it is indeed unique by [3,
Section 3]) and for any $i=1,\dots,n-1$, denote:

$\overline{\alpha}_{i,V_{0}}(v)=t\left\langle\overline{g_{i}},P_{V_{0}}\overline{\alpha}_{i}(v_{0})\right\rangle.$
(A.4)

We know that there exist vectors
$\overline{\alpha}_{1}(v),\dots,\overline{\alpha}_{n-1}(v)\in\mathbb{R}^{n}$
such that
$\operatorname{MT}_{v}(\overline{\alpha}_{1}(v),\dots,\overline{\alpha}_{n-1}(v))$
and:

$P_{V_{0}}\overline{\alpha}_{i}(v)=\overline{\alpha}_{i,V_{0}}(v)\text{ for
}i=1,\dots,n-1$ (A.5)

using the last sentence of [3, Theorem 3] and as is used implicitly in [5, p.
39 for $n=3$, p.44 for $n=4$ and p.52 for $n=5$]. Therefore by Equations
(A.4),(A.5):

$\operatorname{span}_{\mathbb{Z}}\\{P_{V_{0}}\overline{\alpha}_{1}(v),\dots,P_{V_{0}}\overline{\alpha}_{n-1}(v)\\}=g_{0}g^{t}g_{0}^{-1}\operatorname{span}_{\mathbb{Z}}\\{P_{V_{0}}\overline{\alpha}_{1}(v_{0}),\dots,P_{V_{0}}\overline{\alpha}_{n-1}(v_{0})\\}$
(A.6)

where:

$g_{0}=\begin{pmatrix}\mid\ &\cdots&\mid\\\
P_{V_{0}}\overline{\alpha}_{1}(v_{0})&\cdots&P_{V_{0}}\overline{\alpha}_{n-1}(v_{0})\\\
\mid&\cdots&\mid\end{pmatrix}.$

For any $i=1,\dots,n-1$ write:

$\overline{\alpha}_{i}(v)=P_{V_{0}}\overline{\alpha}_{i}(v)+\operatorname{trace}(\overline{\alpha}_{i}(v))\overline{1}.$
(A.7)

Since the action of
$\mathbb{R}\cdot\operatorname{SL}_{n-1}\times\operatorname{SL}_{r-1}(\mathbb{R})$
on $V_{\mathbb{R}}$ is smooth and since
$v\mapsto\operatorname{trace}(\overline{\alpha}_{i}(v))$ is smooth by direct
computation of the trace in the multiplication table (see A.1), we deduce that
the functions $v\mapsto\overline{\alpha}_{i}(v)$ are smooth for
$i=1,\dots,n-1$.

###  A.3 Additional Lemmas

In this part of the Appendix we state several supporting results for
Subsection 2.5.

######  Lemma A.1.

Let $d\in\mathbb{N}$. Let $S$ be a level surface of some smooth homogeneous
function (as in Theorem 2.23) on $\mathbb{R}^{d}$ and let $A\subset S$ be
bounded, open and Jordan measurable in $S$. Let $F:A\rightarrow\mathbb{R}$ be
smooth. If $\nabla F(x)\neq 0$ for $m_{A}$ a.e. $x\in A$ (where $m_{A}$ is the
surface measure on $A$) then $F_{*}m_{A}\ll\lambda_{\mathbb{R}}$ where
$\lambda_{\mathbb{R}}$ is the Lebesgue measure on $\mathbb{R}$.

###### Proof.

We first prove the claim when there exists $c_{0}>0$ such that $\lVert\nabla
F\rVert>c_{0}$ on $A$ (and therefore on $\overline{A}$ because $F$ is smooth).
In this case, by compactness of $\overline{A}$ we can find $c>0$ and
$\epsilon_{0}>0$ such that for any $\epsilon<\epsilon_{0}$ it holds that
$\lambda_{\mathbb{R}}(F(B_{\epsilon}^{A}(x)))\geq c\lVert\nabla F(x)\rVert
m_{A}(B_{\epsilon}^{A}(x))\geq cc_{0}m_{A}(B_{\epsilon}^{A}(x))$ where
$B_{\epsilon}^{A}$ is an (e.g. Euclidean) open ball of radius $\epsilon$ in
$A$. By standard estimation argument, it will hold that
$F_{*}m_{A}\ll\lambda_{\mathbb{R}}$. Next we prove the Lemma’s claim in the
general context. Define the closed subset of $A$ by $E=\\{x\in A:\nabla
F(x)=0\\}$ and let $E_{n}\supset E$ be a sequence of open subsets of $A$ such
that $m_{A}(E_{n})\rightarrow 0$ as $n\rightarrow\infty$. For any $n$, denote
$\mu_{n}=\mathbbm{1}_{E_{n}^{c}}m_{A}$ to be a sequence of measures on $A$
such that $\mu_{n}\rightarrow m_{A}$ weakly as $n\rightarrow\infty$. Let
$S\subset\mathbb{R}$ be a $\lambda_{\mathbb{R}}$-null set. By compactness of
$\overline{A}\cap E_{n}^{c}$, there exists $c_{n}>0$ such that $\lVert\nabla
F\rVert\geq c_{n}$ on $\overline{A}\cap E_{n}^{c}$ so by what we proved
before: $F_{*}\mu_{n}\ll\lambda_{\mathbb{R}}$ for every $n$. Since
$F_{*}\mu_{n}\rightarrow F_{*}m_{A}$ weakly as $n\rightarrow\infty$ we deduce
$F_{*}m_{A}\ll\lambda_{\mathbb{R}}$ as desired. ∎

######  Lemma A.2.

Let $n=3,4,5$, $d=4,12,40$, respectively and let $U\subset
V^{(0)}_{\mathbb{R}}$ be an open subset. Assume that
$\alpha_{1},\dots,\alpha_{n-1}:U\rightarrow\mathbb{R}^{n}$ are smooth and
satisfy (A.1) if $n=3$, (A.2) if $n=4$ and (A.3) if $n=5$. Then
$\alpha_{1},\dots,\alpha_{n-1}$ are real analytic on $U$.

###### Proof.

Let $v_{0}\in U$ and denote
$\phi:\mathbb{R}\times\operatorname{SL}_{n-1}(\mathbb{R})\times\operatorname{SL}_{r-1}(\mathbb{R})\rightarrow
V^{(0)}_{\mathbb{R}}$ the isomorphism given by $(t,g,h)\mapsto t(g,h)v_{0}$.
By its definition, the action of
$\mathbb{R}\times\operatorname{SL}_{n-1}(\mathbb{R})\times\operatorname{SL}_{r-1}(\mathbb{R})$
on $V_{\mathbb{R}}^{(0)}$ is by polynomials in the coefficients of
$(t,g,h)\in\mathbb{R}\times\operatorname{SL}_{n-1}(\mathbb{R})\times\operatorname{SL}_{r-1}(\mathbb{R})$
(see [5, p. 38,47,52] or [1], [2] for definition of this action) so that the
map $\phi$ is real analytic. Therefore, by the inverse functions theorem for
real analytic functions (see [6, Theorem 1.8.1]), so is the map $\phi^{-1}$.
By Equation (A.7) it is evident that for any $i=1,\dots,n-1$ the function
$\overline{\alpha}_{i}\circ\phi$ is real analytic. Indeed, the equation says:

$\overline{\alpha}_{i}(\phi(t,g,h))=\overline{\alpha}_{i}(t(g,h)v_{0})=P_{V_{0}}\overline{\alpha}_{i}(t(g,h)v_{0})+\operatorname{trace}(\overline{\alpha}_{i}(t(g,h)v_{0}))\overline{1}=t\left\langle\overline{g_{i}},P_{V_{0}}\overline{\alpha}_{i}(v_{0})\right\rangle+P(t,g,h)$
(A.8)

where $g=\begin{pmatrix}\mid\ &\cdots&\mid\\\
\overline{g_{1}}&\cdots&\overline{g_{n-1}}\\\ \mid&\cdots&\mid\end{pmatrix}$
and $P$ is some polynomial in $t$ and the entries of $g,h$ by direct
calculation from A.1 to find $\operatorname{trace}(\overline{\alpha}_{i}(v))$
in terms of $v$ and [5, p. 38,47,52] to write $v=t(g,h)v_{0}$ in terms of
$t,g,h$. Altogether we deduce that the function:

$\overline{\alpha}_{i}=(\overline{\alpha}_{i}\circ\phi)\circ\phi^{-1}$ (A.9)

is the composition of two real analytic functions and therefore real analytic.
∎

######  Lemma A.3 ( [6], Chapter 3).

Let $d\in\mathbb{N}$. If a real analytic function
$F:U\subset\mathbb{R}^{d}\rightarrow\mathbb{R}$ vanishes on a set of positive
Lebesgue measure in the open set $U\subset\mathbb{R}^{n}$ then $F\mid_{U}=0$
identically.

##  Appendix B Some Topology

For this Section, fix a lattice
$\Lambda=\operatorname{sp}_{\mathbb{Z}}\\{w_{1},\dots,w_{n}\\}\subset\mathbb{R}^{n}$
and a fundamental domain $\Sigma$ for the action of
$\operatorname{stab}_{\operatorname{SL}_{n}(\mathbb{R})}(\Lambda)$ on
$\operatorname{SL}_{n}(\mathbb{R})$. We abbreviate ’$\Sigma$-$\Lambda$ basic
subsets’ to ’basic subsets’. Also, for this Section only we denote
$\mu\vcentcolon=m_{Y_{n-1}}.$

######  Lemma B.1.

Let $W\subset Y_{n}$ be a measurable set with $\mu(\partial W)=0$. Then for
every $\epsilon>0$ there exist $B_{1},\dots,B_{m}\subset W$ disjoint basic
subsets, such that $\sum_{i=1}^{m}\mu(B_{i})\geq\mu(W)-\epsilon$.

###### Proof.

We may assume that $W$ is bounded, since otherwise we may intersect $W$ with
large enough bounded ball in $Y_{n}$. Let $U\supset\partial W$ be open such
that $\mu(U)\leq\epsilon$. For any $x\in\overline{W\setminus U}$, let
$B_{x}\subset\text{int}(W)$ be a Basic subset. Take a finite sub-cover of
$\overline{W\setminus U}$, $(B_{i})_{i=1}^{m}$. Then clearly the $B_{i}$’s are
included in $W$ (they are not necessarily disjoint), and:

$\mu\left(\bigcup_{i=1}^{m}B_{i}\right)\geq\mu(W)-\mu(W\cap
U)\geq\mu(W)-\mu(U)\geq\mu(W)-\epsilon.$

Finally, note that the intersection of every two basic subsets can be
estimated by finite union of disjoint basic subsets with arbitrarily small
defect, so we may also assume that the $B_{i}$’s are disjoint. ∎

The following Lemma is a Corollary of Lemma 2.12 and the definition of basic
subsets.

######  Lemma B.2.

Let $S\subset Y_{n}$ be bounded and satisfy $\mu(S)\leq\epsilon$ for some
$\epsilon>0$. Then there exist basic subsets $(D_{i})_{i=1}^{k}$ such that
$S\subset\bigcup_{i=1}^{k}D_{i}$ and $\sum_{i=1}^{k}\mu(D_{i})\leq 2\epsilon.$

######  Theorem B.3.

Let $W\subset Y_{n-1}$ be measurable with $\mu$-zero boundary and let $E$ be
as in the proof of Theorem 1.4 (see 3). Then
$\lim_{T\rightarrow\infty}\mu_{T}(W)=\mu(W)$.

###### Proof.

We may assume that $W$ is bounded, since otherwise we can intersect $W$ with a
large enough bounded ball in $Y_{n-1}$. Let $\epsilon>0$ and using the Lemma
B.1, find $B_{1},\dots,B_{m}$, a collection of disjoint basic subsets,
satisfying the conclusion of the Lemma. Then by Lemma B.1 and Lemma B.2
applied for $W\setminus\bigcup_{i=1}^{m}B_{i}$ resulting in a collection
$(D_{i})_{i=1}^{k}$ of basic subsets, we get:

$\lvert\mu(W)-\mu_{T}(W)\rvert\leq\lvert\mu(\bigcup_{i=1}^{m}B_{i})-\mu_{T}(\bigcup_{i=1}^{m}B_{i})\rvert+\lvert\mu(W\setminus\bigcup_{i=1}^{m}B_{i})-\mu_{T}(W\setminus\bigcup_{i=1}^{m}B_{i})\rvert$
$\leq\lvert\mu(\bigcup_{i=1}^{m}B_{i})-\mu_{T}(\bigcup_{i=1}^{m}B_{i})\rvert+\epsilon+\mu_{T}(\bigcup_{i=1}^{k}D_{i})\leq\lvert\mu(\bigcup_{i=1}^{m}B_{i})-\mu_{T}(\bigcup_{i=1}^{m}B_{i})\rvert+\sum_{i=1}^{k}\mu_{T}(D_{i})+\epsilon,$

implying, by Lemma B.1, Equation (3.16) and since the $B_{i}$’s are disjoint,
that for any $\epsilon>0$,
$\limsup_{T\rightarrow\infty}\lvert\mu(W)-\mu_{T}(W)\rvert\leq 3\epsilon$, as
required. ∎

## References

  * [1] Manjul Bhargava.  Higher composition laws. III. The parametrization of quartic rings.  Ann. of Math. (2), 159(3):1329–1360, 2004.
  * [2] Manjul Bhargava.  Higher composition laws. IV. The parametrization of quintic rings.  Ann. of Math. (2), 167(1):53–94, 2008.
  * [3] Manjul Bhargava and Piper Harron.  The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields.  Compos. Math., 152(6):1111–1120, 2016.
  * [4] B. N. Delone and D. K. Faddeev.  The theory of irrationalities of the third degree.  Translations of Mathematical Monographs, Vol. 10. American Mathematical Society, Providence, R.I., 1964.
  * [5] Piper Harron.  The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering.  2016\.
  * [6] Steven G. Krantz and Harold R. Parks.  A primer of real analytic functions.  Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston, MA, second edition, 2002.
  * [7] K. Mahler.  On lattice points in $n$-dimensional star bodies. I. Existence theorems.  Proc. Roy. Soc. London Ser. A, 187:151–187, 1946.
  * [8] M. Sato and T. Kimura.  A classification of irreducible prehomogeneous vector spaces and their relative invariants.  Nagoya Math. J., 65:1–155, 1977.
  * [9] Harold N. Shapiro.  Introduction to the theory of numbers.  A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983.
  * [10] Hermann Weyl.  Über die Gleichverteilung von Zahlen mod. Eins.  Math. Ann., 77(3):313–352, 1916.
  * [11] Yuval Yifrach.  A note about weyl equidistribution theorem, 2022.