# A measure estimate in geometry of numbers and improvements to Dirichlet’s
theorem

Dmitry Kleinbock  Brandeis University, Waltham MA, USA, 02454-9110
<EMAIL_ADDRESS>,  Andreas Strömbergsson  Department of Mathematics,
Uppsala University, Box 480, SE-75106, Uppsala, SWEDEN<EMAIL_ADDRESS>and  Shucheng Yu  Department of Mathematics, Uppsala University, Box 480,
SE-75106, Uppsala, SWEDEN<EMAIL_ADDRESS>
###### Abstract.

Let $\psi$ be a continuous decreasing function defined on all large positive
real numbers. We say that a real $m\times n$ matrix $A$ is $\psi$-Dirichlet if
for every sufficiently large real number $t$ one can find
$\bm{p}\in{\mathbb{Z}}^{m}$,
$\bm{q}\in{\mathbb{Z}}^{n}\smallsetminus\\{\bm{0}\\}$ satisfying
$\|A\bm{q}-\bm{p}\|^{m}<\psi({t})$ and $\|\bm{q}\|^{n}<{t}$. This property was
introduced by Kleinbock and Wadleigh in 2018, generalizing the property of $A$
being Dirichlet improvable which dates back to Davenport and Schmidt (1969).
In the present paper, we give sufficient conditions on $\psi$ to ensure that
the set of $\psi$-Dirichlet matrices has zero or full Lebesgue measure. Our
proof is dynamical and relies on the effective equidistribution and doubly
mixing of certain expanding horospheres in the space of lattices. Another main
ingredient of our proof is an asymptotic measure estimate for certain compact
neighborhoods of the critical locus (with respect to the supremum norm) in the
space of lattices. Our method also works for the analogous weighted problem
where the relevant supremum norms are replaced by certain weighted quasi-
norms.

D.K. was supported by NSF grant DMS-1900560. A.S. and S.Y. were supported by
the Knut and Alice Wallenberg Foundation

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main results
    3. 1.3 Discussion of the proof of Theorem 1.2
  2. 2 Some preliminaries for Theorem 1.3
    1. 2.1 Haar measure and coordinates
    2. 2.2 A small parameter for the lower bound
    3. 2.3 Hajós’s Theorem and its consequences
  3. 3 Proof of the lower bound
  4. 4 Proof of the upper bound
    1. 4.1 Bounds on diagonal entries
    2. 4.2 A technical choice of lattice representatives
    3. 4.3 Bounds on off-diagonal symmetric pairs
    4. 4.4 Proof of Proposition 4.2
  5. 5 Measure estimates of the thickenings
    1. 5.1 Proof of the upper bound
    2. 5.2 Proof of the lower bound
  6. 6 Some preliminaries for Theorem 1.2
    1. 6.1 Dynamical interpretation of weighted $\psi$-Dirichlet matrices
    2. 6.2 Effective equidistribution and doubly mixing for certain $g_{s}$-translates
    3. 6.3 Smooth approximations and estimates on norms
  7. 7 Proof of Theorem 1.2
    1. 7.1 Application of effective equidistribution
    2. 7.2 The convergence case
    3. 7.3 The divergence case

##  1\. Introduction

###  1.1. Background

Let $m,n$ be two positive integers and let $M_{m,n}({\mathbb{R}})$ be the
space of $m$ by $n$ real matrices. The starting point of our work is the
following higher dimensional generalization of the classical Dirichlet’s
Diophantine approximation theorem, see e.g. [3, §1.5].

######  Theorem 1.1.

For any $A\in M_{m,n}({\mathbb{R}})$ and $t>1$, there exists
$(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times({\mathbb{Z}}^{n}\smallsetminus\\{\bm{0}\\})$
satisfying the following system of inequalities:

(1.1)
$\displaystyle\|A\bm{q}-\bm{p}\|^{m}\leq\frac{1}{t}\quad\textrm{and}\quad\|\bm{q}\|^{n}<t.$

Here $\|\cdot\|$ denotes the supremum norm on ${\mathbb{R}}^{m}$ and
${\mathbb{R}}^{n}$ respectively.

A natural question to ask is whether one can improve (1.1) by replacing $1/t$
by a smaller function, that is, consider the following system of inequalities:

(1.2)
$\displaystyle\|A\bm{q}-\bm{p}\|^{m}<\psi(t)\quad\textrm{and}\quad\|\bm{q}\|^{n}<t$

where $\psi$ is a positive, continuous, decreasing function which decays to
zero at infinity. Historically there have been two directions to pursue in
this regard: looking for solvability of (1.2) for an unbounded set of $t>0$
vs. for all large enough $t$. The former is sometimes referred to as
asymptotic approximation, and has culminated in definitive results such as the
Khintchine-Groshev theorem. In this paper we are interested in the latter,
less studied set-up of uniform approximation. Following the definition in
Kleinbock and Wadleigh [26], we say that an $m$ by $n$ real matrix $A$ is
$\psi$-Dirichlet if the system of inequalities (1.2) has solutions in
$(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times({\mathbb{Z}}^{n}\smallsetminus\\{\bm{0}\\})$
for all sufficiently large $t$. It is clear that $A\in M_{m,n}({\mathbb{R}})$
is $\psi$-Dirichlet if and only if $A+A^{\prime}$ is $\psi$-Dirichlet for any
$A^{\prime}\in M_{m,n}({\mathbb{Z}})$. Thus with slight abuse of notation, we
denote by $\operatorname{\mathbf{DI}}_{m,n}(\psi)\subset
M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$ the set of $\psi$-Dirichlet matrices.

Let $\psi_{1}(t)=1/t$. The problem of improving Dirichlet’s theorem was
initiated by Davenport and Schmidt [11, 10] where they showed that the set

(1.3)
$\displaystyle\operatorname{\mathbf{DI}}_{m,n}:=\bigcup_{0<c<1}\operatorname{\mathbf{DI}}_{m,n}(c\psi_{1})$

of Dirichlet improvable matrices is of Lebesgue measure zero, while having
full Hausdorff dimension $mn$. More recently, Kleinbock and Mirzadeh [23,
Theorem 1.5] showed that for any fixed $0<c<1$, the Hausdorff dimension of
$\operatorname{\mathbf{DI}}_{m,n}(c\psi_{1})$ is strictly smaller than $mn$.
There have also been extensive studies on the Hausdorff dimensions of the
(even smaller) set of the singular matrices,

$\displaystyle\textbf{Sing}_{m,n}:=\bigcap_{0<c<1}\operatorname{\mathbf{DI}}_{m,n}(c\psi_{1}).$

After a series of breakthrough work, it is now known that the Hausdorff
dimension of $\textbf{Sing}_{m,n}$ is $mn-\tfrac{mn}{m+n}$ whenever
$\max\\{m,n\\}>1$; see [4, 5, 14, 8, 9].

On the other hand, for a general decreasing function $\psi$ with $t\mapsto
t\psi(t)$ increasing, Kleinbock and Wadleigh proved a zero-one law for the
Lebesgue measure of $\operatorname{\mathbf{DI}}_{1,1}(\psi)$ depending on the
divergence or convergence of a certain series involving $\psi$ [26, Theorem
1.8]. See also [13] for the relevant dimension theory of
$\operatorname{\mathbf{DI}}_{1,1}(\psi)$, [24] for a similar zero-one law with
the supremum norm replaced by the Euclidean norm and [27, 16] for analogous
results in the inhomogeneous setting.

The arguments in [26] rely on the theory of continued fractions and are not
applicable for higher dimensions. Nevertheless, for general dimensions,
building on ideas from [7, 20], a dynamical approach was proposed in [26, §4],
reformulating the problem as a shrinking target problem, which asks whether a
generic orbit in a dynamical system hits a given sequence of shrinking targets
infinitely often. To describe this dynamical interpretation, let us first fix
some notation. Let $d=m+n$ and let
$X_{d}:=\text{SL}_{d}({\mathbb{R}})/\text{SL}_{d}({\mathbb{Z}})$ be the
homogeneous space which parameterizes the space of unimodular lattices in
${\mathbb{R}}^{d}$ via $g\text{SL}_{d}({\mathbb{Z}})\leftrightarrow
g{\mathbb{Z}}^{d}$. We note that $\text{SL}_{d}({\mathbb{R}})$ acts on $X_{d}$
naturally via the regular action: $g\Lambda=gh{\mathbb{Z}}^{d}$ for any
$g\in\text{SL}_{d}({\mathbb{R}})$ and $\Lambda=h{\mathbb{Z}}^{d}\in X_{d}$.
For any $s\in{\mathbb{R}}$, let $a_{s}$ be the diagonal matrix

$\displaystyle a_{s}:=\left(\begin{matrix}e^{s/m}I_{m}&0\\\
0&e^{-s/n}I_{n}\end{matrix}\right)\in\text{SL}_{d}({\mathbb{R}}).$

Let $\Delta:X_{d}\to[0,\infty)$ be the function defined by

(1.4)
$\displaystyle\Delta(\Lambda):=\sup_{\bm{v}\in\Lambda\smallsetminus\\{\bm{0}\\}}\log\Bigl{(}\frac{1}{\|\bm{v}\|}\Bigr{)}.$

Finally, let us denote

(1.5)
$\displaystyle\mathcal{Y}:=\left\\{\Lambda_{A}:=\left(\begin{matrix}I_{m}&A\\\
0&I_{n}\end{matrix}\right){\mathbb{Z}}^{d}\in X_{d}\>:\>A\in
M_{m,n}({\mathbb{R}})\right\\}.$

The submanifold $\mathcal{Y}\subset X_{d}$ can be naturally identified with
the $mn$-dimensional torus $M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$ via
$\Lambda_{A}\leftrightarrow A\in M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$.
Throughout the paper, we denote by Leb the probability Lebesgue measure on
$\mathcal{Y}\cong M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$; for simplicity of
notation, for any function $f$ on ${\mathcal{Y}}$ we will abbreviate the space
average
$\int_{{\mathcal{Y}}}f(\Lambda_{A})\operatorname{d}\\!{\,}\textrm{Leb}(A)$ by
either $\textrm{Leb}(f)$ or
$\int_{{\mathcal{Y}}}f(\Lambda_{A})\,\operatorname{d}\\!{A}$.

It was shown in [26, Proposition 4.5] that for any given $\psi$ as above,
there exists a unique continuous function
$r=r_{\psi}:[s_{0},\infty)\to(0,\infty)$ such that

(1.6) $A\in M_{m,n}({\mathbb{R}})$ is not $\psi$-Dirichlet $\Leftrightarrow$
$a_{s}\Lambda_{A}\in\Delta^{-1}[0,r(s)]$ for an unbounded set of $s>s_{0}$.

This equivalence is usually called the Dani Correspondence. In view of this
interpretation, our task is to find conditions which ensure that for almost
every (or almost no) $A\in M_{m,n}({\mathbb{R}})$, the orbit
$\\{a_{s}\Lambda_{A}\\}_{s>s_{0}}$ hits the shrinking target
$\Delta^{-1}[0,r(s)]$ for an unbounded set of $s$-values. We note that this
was also the strategy used in [20] giving a dynamical proof of the classical
Khintchine-Groshev Theorem, where the relevant shrinking targets are certain
cusp neighborhoods in $X_{d}$. For our case, by Mahler’s compactness
criterion, the shrinking targets $\Delta^{-1}[0,r(s)]$ are compact
neighborhoods of the critical locus $\Delta^{-1}\\{0\\}$, whose explicit
description is given by Hajós’s Theorem [12] (cf. Theorem 2.3 below). The fact
that these shrinking targets are compact sets causes new difficulties compared
to the situation in [20], see the discussion in Section 1.3.

###  1.2. Main results

In the present paper, based on the dynamical interpretation described above,
we give sufficient conditions on $\psi$ implying that
$\operatorname{\mathbf{DI}}_{m,n}(\psi)$ is of zero or full Lebesgue measure.
In fact, with some modifications, our arguments also work for the analogous
weighted problem where the supremum norms in (1.2) are replaced by certain
weighted quasi-norms, as introduced in [17]. We thus prove our main result in
that generality. We first introduce the relevant notation.

Let $\bm{\alpha}\in{\mathbb{R}}^{m}$ and $\bm{\beta}\in{\mathbb{R}}^{n}$ be
two weight vectors, that is

$\displaystyle\bm{\alpha}=(\alpha_{1},\ldots,\alpha_{m})\in({\mathbb{R}}_{>0})^{m}\quad\textrm{and}\quad\bm{\beta}=(\beta_{1},\ldots,\beta_{n})\in({\mathbb{R}}_{>0})^{n}$

with $\sum_{i}\alpha_{i}=\sum_{j}\beta_{j}=1$. We say that $A\in
M_{m,n}({\mathbb{R}})$ is $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet if the
system of inequalities

(1.7)
$\displaystyle\|A\bm{q}-\bm{p}\|_{\bm{\alpha}}<\psi(t)\quad\textrm{and}\quad\|\bm{q}\|_{\bm{\beta}}<t$

has solutions in
$(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times({\mathbb{Z}}^{n}\smallsetminus\\{\bm{0}\\})$
for all sufficiently large $t$. Here

$\displaystyle\|\bm{x}\|_{\bm{\alpha}}:=\max\left\\{|x_{i}|^{1/\alpha_{i}}\>:\>1\leq
i\leq
m\right\\}\quad\textrm{and}\quad\|\bm{y}\|_{\bm{\beta}}:=\max\left\\{|y_{j}|^{1/\beta_{j}}\>:\>1\leq
j\leq n\right\\}$

are the two quasi-norms associated with $\bm{\alpha}$ and $\bm{\beta}$
respectively. Again it is easy to see that $A\in M_{m,n}({\mathbb{R}})$ is
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet if and only if $A+A^{\prime}$ is
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet for any $A^{\prime}\in
M_{m,n}({\mathbb{Z}})$, and we denote by
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)\subset
M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$ the set of
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet matrices. We note that when
$\bm{\alpha}=(\tfrac{1}{m},\ldots,\tfrac{1}{m})$ and
$\bm{\beta}=(\tfrac{1}{n},\ldots,\tfrac{1}{n})$, then
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)=\operatorname{\mathbf{DI}}_{m,n}(\psi)$.

We now state our main result which gives sufficient conditions on $\psi$
determining when $\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$
is of full or zero Lebesgue measure.

######  Theorem 1.2.

Fix $m,n\in{\mathbb{N}}$ and two weight vectors
$\bm{\alpha}\in{\mathbb{R}}^{m}$ and $\bm{\beta}\in{\mathbb{R}}^{n}$. Let
$d=m+n$, and let

$\displaystyle\varkappa_{d}=\frac{d^{2}+d-4}{2}\quad\textrm{and}\quad\lambda_{d}=\frac{d(d-1)}{2}.$

Let $t_{0}>0$ and let $\psi:[t_{0},\infty)\to{(0,\infty)}$ be a continuous,
decreasing function such that

(1.8) the function $t\mapsto t\psi(t)$ is increasing

and

(1.9) $\displaystyle\psi(t)<\psi_{1}(t)=1/t\quad\textrm{for all $t\geq
t_{0}$}.$

Let $F_{\psi}(t):=1-t\psi(t)$. If the series

(1.10) $\displaystyle\sum_{k\geq
t_{0}}k^{-1}F_{\psi}(k)^{\varkappa_{d}}\log^{\lambda_{d}}\left(\frac{1}{F_{\psi}(k)}\right)$

converges, then $\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$ is
of full Lebesgue measure. Conversely, if the series (1.10) diverges, and

(1.11) $\displaystyle\liminf_{t_{1}\to\infty}\frac{\sum_{t_{0}\leq k\leq
t_{1}}k^{-1}F_{\psi}(k)^{\varkappa_{d}}\log^{\lambda_{d}+1}\left(\frac{1}{F_{\psi}(k)}\right)}{\left(\sum_{t_{0}\leq
k\leq
t_{1}}k^{-1}F_{\psi}(k)^{\varkappa_{d}}\log^{\lambda_{d}}\left(\frac{1}{F_{\psi}(k)}\right)\right)^{2}}=0,$

then $\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$ is of zero
Lebesgue measure.

######  Remark 1.

When $m=n=1$, Theorem 1.2 is not new; in fact [26, Theorem 1.8] is stronger in
the sense that it gives a tight zero-one law without the extra assumption
(1.11). We believe that an analogous tight zero-one law should also hold for
general dimensions $m,n$, i.e. that Theorem 1.2 should hold with the
assumption (1.11) removed. See Remark 8 for a discussion of why assumption
(1.11) is needed in our proof.

######  Remark 2.

The function $F_{\psi}(t)=1-t\psi(t)$ encodes $\psi$ via
$\psi(t)=\frac{1-F_{\psi}(t)}{t}$. In view of the assumptions (1.8) and (1.9),
$F_{\psi}$ is a decreasing function and takes values in $(0,1)$. In
particular, the limit $\lim_{t\to\infty}F_{\psi}(t)$ exists and lies in
$[0,1)$.

If $\lim_{t\to\infty}F_{\psi}(t)>0$, then the conclusion of Theorem 1.2
follows from the work of Kleinbock and Weiss [28]. Indeed, in this case the
series (1.10) diverges, and for any fixed $\beta$ we have $\sum_{t_{0}\leq
k\leq
t_{1}}k^{-1}F_{\psi}(k)^{\varkappa_{d}}\log^{\beta}\bigl{(}\frac{1}{F_{\psi}(k)}\bigr{)}\asymp\log
t_{1}$ as $t_{1}\to\infty$, so that also the assumption (1.11) holds;
moreover, $\lim_{t\to\infty}F_{\psi}(t)>0$ implies that there exists some
$c\in(0,1)$ such that $\psi(t)<c/t$, implying that
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)\subset\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(c\psi_{1})$;
and by [28, Theorem 1.4],
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(c\psi_{1})$ is a null
set.

######  Remark 3.

Let us give some explicit examples to illustrate our results. We note that
each function $\psi$ appearing below is strictly decreasing on
$[t_{0},\infty)$ for $t_{0}$ sufficiently large.

  * (1)

Let $\psi(t)=\frac{1-c(\log t)^{-\tau}}{t}$ ($\Leftrightarrow
F_{\psi}(t)=c(\log t)^{-\tau}$) for some $c>0$ and $\tau\geq 0$. In this case
the series (1.10) diverges if and only if $\tau\leq\frac{1}{\varkappa_{d}}$.
It is also easy to check that condition (1.11) is satisfied whenever
$\tau\leq\frac{1}{\varkappa_{d}}$. Hence Theorem 1.2 implies that for such
$\psi$, $\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$ is of full
measure if $\tau>\frac{1}{\varkappa_{d}}$, and of zero measure if
$\tau\leq\frac{1}{\varkappa_{d}}$.

  * (2)

Let $\psi(t)=\frac{1-c(\log t)^{-1/\varkappa_{d}}(\log\log t)^{-\tau}}{t}$
($\Leftrightarrow F_{\psi}(t)=c(\log t)^{-1/\varkappa_{d}}(\log\log
t)^{-\tau}$) for some $c>0$ and $\tau\in{\mathbb{R}}$. In this case the series
(1.10) diverges if and only if $\tau\leq\frac{\lambda_{d}+1}{\varkappa_{d}}$,
while the condition (1.11) is satisfied if and only if
$\tau<\frac{\lambda_{d}}{\varkappa_{d}}$. Hence Theorem 1.2 implies that for
such $\psi$, $\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$ is of
full measure if $\tau>\frac{\lambda_{d}+1}{\varkappa_{d}}$, and of zero
measure if $\tau<\frac{\lambda_{d}}{\varkappa_{d}}$. However, for $\tau$ in
the range
$\frac{\lambda_{d}}{\varkappa_{d}}\leq\tau\leq\frac{\lambda_{d}+1}{\varkappa_{d}}$,
Theorem 1.2 gives no information (although we believe that
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$ is of zero measure
also for these $\tau$; cf. Remark 1). We point out that in the special case
$\tau=\frac{\lambda_{d}}{\varkappa_{d}}$, the quotient in (1.11) remains
bounded as $t_{1}\to\infty$; in this case our method of proof allows us to
conclude that at least the set
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$ is not of full
Lebesgue measure; see Remark 16.

######  Remark 4.

Let us also point out that the assumption (1.9) is imposed only to avoid
making the statement of Theorem 1.2 unnecessarily complicated (since otherwise
$F_{\psi}(t)$ could be negative and then the series (1.10) is not well-
defined). Indeed, if (1.9) fails but $\psi$ satisfies the other assumptions in
Theorem 1.2, then $\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)$
is certainly of full Lebesgue measure. This is true since in this case after
possibly enlarging $t_{0}$, we have $\psi(t)\geq 1/t\geq\frac{1-(\log
t)^{-\tau}}{t}$ for all $t\geq t_{0}$ and $\tau\geq 0$, and therefore
$\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}(\psi)\supset\operatorname{\mathbf{DI}}_{\bm{\alpha},\bm{\beta}}\left({t\mapsto}\frac{1-(\log
t)^{-\tau}}{t}\right)$, where the last set is of full Lebesgue measure
whenever $\tau>\frac{1}{\varkappa_{d}}$ by Remark 3(1).

One of the main ingredients in our proof of Theorem 1.2 is a measure estimate
in geometry of numbers, which we believe is of independent interest. Let
$\mu_{d}$ be the unique left $\text{SL}_{d}({\mathbb{R}})$-invariant
probability measure on
$X_{d}=\text{SL}_{d}({\mathbb{R}})/\text{SL}_{d}({\mathbb{Z}})$. We are
interested in the sets $\Delta^{-1}[0,r]$ in $X_{d}$, as $r\to 0^{+}$. As we
have discussed, these sets shrink toward the critical locus
$\Delta^{-1}\\{0\\}$ as $r\to 0^{+}$, and by Hajós’s Theorem [12] (cf. Theorem
2.3 below), the set $\Delta^{-1}\\{0\\}$ has a simple explicit description as
a finite union of compact submanifolds of positive codimension
$\frac{d^{2}+d}{2}-1=\varkappa_{d}+1$ in $X_{d}$. In particular this implies
that $\mu_{d}\bigl{(}\Delta^{-1}\\{0\\}\bigr{)}=0$ and
$\mu_{d}\left(\Delta^{-1}[0,r]\right)\to 0$ as $r\to 0^{+}$. The following
theorem gives an asymptotic estimate on the exact rate of convergence in the
limit just mentioned.

######  Theorem 1.3.

We have

(1.12)
$\displaystyle\mu_{d}\left(\Delta^{-1}[0,r]\right)\asymp_{d}r^{\frac{(d-1)(d+2)}{2}}\log^{\frac{d(d-1)}{2}}\Bigl{(}\frac{1}{r}\Bigr{)}=r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)},\qquad\textrm{as
$r\to 0^{+}$}.$

Our proof of Theorem 1.3 proceeds by bounding the sets $\Delta^{-1}[0,r]$ from
above and below by more explicit sets whose Haar measure we can estimate
directly. In the proof of the upper bound we make crucial use of Hajós’s
Theorem. We remark that Hajós’s proof (from 1941) of the theorem, which
settled a conjecture of Minkowski from 1896, is surprisingly complicated, with
the first step being a translation of the question into an algebraic statement
about factorizations of finite abelian groups (see also [36] for a nice
presentation). It seems difficult to extend this proof in any direct way from
the case of $\Delta^{-1}\\{0\\}$ to deduce restrictions on the sets
$\Delta^{-1}[0,r]$ which are sufficiently strong to imply the desired upper
bound on $\mu_{d}\left(\Delta^{-1}[0,r]\right)$. Instead we apply Hajós’s
Theorem, in combination with a compactness argument, to obtain a convenient
containment relation for $\Delta^{-1}[0,r]$ valid for all sufficiently small
$r$ (see Lemma 4.5 and Remark 11). This initial restriction serves as the
starting point for our analysis where we use direct, geometric arguments to
derive further, $r$-dependent restrictions on $\Delta^{-1}[0,r]$ for $r$
small, strong enough to finally imply the desired upper bound on
$\mu_{d}\left(\Delta^{-1}[0,r]\right)$.

######  Remark 5.

In the case $d=2$, the following explicit formula holds [37, p. 74]:

$\displaystyle\mu_{2}\left(\Delta^{-1}[0,r]\right)=\begin{cases}\frac{12}{\pi^{2}}\bigl{(}e^{-2r}+2r-1+\Psi(e^{-2r})\bigr{)}&\text{if
}\>0\leq r\leq\frac{1}{2}\log 2,\\\\[3.0pt]
1-\frac{12}{\pi^{2}}e^{-2r}&\text{if }\>r\geq\frac{1}{2}\log 2,\end{cases}$

where the function $\Psi:(0,1]\to{\mathbb{R}}$ is defined by $\Psi(1)=0$ and
$\Psi^{\prime}(x)=(x^{-1}-1)\log(x^{-1}-1)$. It follows that in this case we
have an explicit asymptotic expansion sharpening (1.12):

$\displaystyle\mu_{2}\left(\Delta^{-1}[0,r]\right)\sim\frac{24}{\pi^{2}}r^{2}\log\Bigl{(}\frac{1}{r}\Bigr{)}+\frac{12(3-2\log
2)}{\pi^{2}}r^{2}-\frac{16}{\pi^{2}}r^{3}\log\Bigl{(}\frac{1}{r}\Bigr{)}+\cdots\qquad\text{as
}\>r\to 0^{+}.$

The explicit formula for $\mu_{2}\left(\Delta^{-1}[0,r]\right)$, stated in a
different notation, was independently obtained in [29] using a different
method.

######  Remark 6.

Theorem 1.3 is also relevant for the study of the Hausdorff dimension of the
set $\operatorname{\mathbf{DI}}_{m,n}(c\psi_{1})$. As we have mentioned,
Kleinbock and Mirzadeh recently proved that the Hausdorff dimension of
$\operatorname{\mathbf{DI}}_{m,n}(c\psi_{1})$ is less than $mn$ for every
$0<c<1$ [23, Theorem 1.5]. They derived this as an application of their main
result, [23, Theorem 1.2], which gives an explicit upper bound on the
Hausdorff dimension of a certain kind of dynamically defined subsets in the
space $X_{d}$. It seems that by using Theorem 1.3 (cf. also Theorem 5.1
below), together with a further analysis of the quantities appearing in [23,
Theorem 1.2], it should be possible to sharpen the conclusion of [23, Theorem
1.5] into a bound of the form

$\displaystyle\dim_{H}\bigl{(}\operatorname{\mathbf{DI}}_{m,n}(c\psi_{1})\bigr{)}<mn-\delta(1-c)^{\varkappa_{d}}\log^{\lambda_{d}-1}\bigl{(}(1-c)^{-1}\bigr{)}$

for all $c<1$ sufficiently near $1$, where $d=m+n$ and $\delta>0$ is a
constant which only depends on $m,n$.

###  1.3. Discussion of the proof of Theorem 1.2

We next give a more detailed outline of our proof of Theorem 1.2. For
simplicity of presentation, we will only focus on the special case when
$\bm{\alpha}=(\tfrac{1}{m},\ldots,\tfrac{1}{m})$ and
$\bm{\beta}=(\tfrac{1}{n},\ldots,\tfrac{1}{n})$; we comment in Remark 9 below
on the modifications needed to treat general weights.

We start from the Dani Correspondence, (1.6), and discretize the shrinking
target problem which appears there by introducing the following thickened
targets:

$\displaystyle B_{k}:=\bigcup_{0\leq
s<1}a_{-s}\Delta^{-1}[0,r(k+s)],\qquad\textrm{for any integer}\ k>s_{0}.$

It follows from this definition that for any $\Lambda\in X_{d}$,
$a_{k}\Lambda\in B_{k}$ if and only if $a_{s}\Lambda\in\Delta^{-1}[0,r(s)]$
for some $k\leq s<k+1$. In particular, by (1.6), $A\in M_{m,n}({\mathbb{R}})$
is not $\psi$-Dirichlet if and only if $a_{k}\Lambda_{A}\in B_{k}$ for
infinitely many integers $k$. For any $k>s_{0}$ let us define

$\displaystyle
E_{k}:=\left\\{\Lambda_{A}\in{\mathcal{Y}}\>:\>a_{k}\Lambda_{A}\in
B_{k}\right\\}.$

Then, in view of the previous discussion and the identification
$\mathcal{Y}\cong M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$, we have

$\operatorname{\mathbf{DI}}^{c}_{m,n}(\psi)=\limsup_{k\to\infty}E_{k}.$

Hence, by the Borel-Cantelli lemma,

$\displaystyle\left\\{\begin{array}[]{ll}\sum_{k}\textrm{Leb}(E_{k})<\infty&\Longrightarrow\textrm{Leb}\left(\operatorname{\mathbf{DI}}^{c}_{m,n}(\psi)\right)=0;\\\
\sum_{k}\textrm{Leb}(E_{k})=\infty\ \&\ \textrm{``quasi-independence
conditions''}&\Longrightarrow\textrm{Leb}\left(\operatorname{\mathbf{DI}}^{c}_{m,n}(\psi)\right)=1.\end{array}\right.$

We thus need to understand when the sum $\sum_{k}\textrm{Leb}(E_{k})$ diverges
or converges, respectively. It follows from our definitions that

$\textrm{Leb}(E_{k})=\int_{{\mathcal{Y}}}\chi_{B_{k}}(a_{k}\Lambda_{A})\,\operatorname{d}\\!{A}.$

It is well known that the $a_{s}$-translates $a_{s}{\mathcal{Y}}$
equidistribute in $X_{d}$ as $s\to\infty$; since our shrinking target $B_{k}$
varies in the parameter $k$, we require an effective version of this fact.
Such a result was first proved by Kleinbock and Margulis [19, Proposition
2.4.8] building on Margulis’s original thickening argument [31] and the
exponential mixing of the diagonal flow $\\{a_{s}\\}_{s\in{\mathbb{R}}}$. Here
we use the following explicit version (see Corollary 6.4 below): there exists
$\delta>0$ such that for any $f\in C_{c}^{\infty}(X_{d})$ and any $s>0$,

(1.13)
$\displaystyle\int_{{\mathcal{Y}}}f(a_{s}\Lambda_{A})\,\operatorname{d}\\!{A}=\mu_{d}(f)+O\big{(}e^{-\delta
s}{\mathcal{N}}(f)\big{)},$

where the norm ${\mathcal{N}}(\cdot)$ is the maximum of a Lipschitz norm and a
Sobolev $L^{2}$-norm (see Section 6.2). By approximating
$\\{\chi_{B_{k}}\\}_{k>s_{0}}$ from above and below by smooth functions and
applying (1.13) together with an explicit bound on the norm
${\mathcal{N}}(\cdot)$ (see Lemma 6.5), it follows that (see Lemma 7.1)

$\displaystyle\sum_{k}\textrm{Leb}(E_{k})=\infty\qquad\Longleftrightarrow\qquad\sum_{k}\mu_{d}\left(B_{k}\right)=\infty.$

Furthermore, it is not difficult to see from Theorem 1.3 that the series
$\sum_{k}\mu_{d}\left(B_{k}\right)$ diverges if and only if the series in
(1.10) diverges (see Theorem 5.1 and Lemma 6.1). This in particular settles
the convergence case of Theorem 1.2.

For the divergence case, in addition to the assumption that the series in
(1.10) diverges (which implies that $\sum_{k}\textrm{Leb}(E_{k})=\infty$), one
also needs to establish a certain quasi-independence condition, see (7.8).
Roughly speaking, we need to show that the quantities

$\left|\textrm{Leb}\left(E_{i}\cap
E_{j}\right)-\textrm{Leb}(E_{i})\textrm{Leb}(E_{j})\right|,\qquad i\neq
j>s_{0}$

are small on average. Here note that

$\displaystyle\textrm{Leb}\left(E_{i}\cap
E_{j}\right)=\int_{{\mathcal{Y}}}\chi_{B_{i}}(a_{i}\Lambda_{A})\chi_{B_{j}}(a_{j}\Lambda_{A})\,\operatorname{d}\\!{A}.$

We now apply the effective doubly mixing for the $a_{s}$-translates
$\\{a_{s}{\mathcal{Y}}\\}_{s>0}$. This result is due to Kleinbock-Shi-Weiss
[25, Theorem 1.2]; we use a more explicit version due to Björklund-Gorodnik
[2, Corollary 2.4] which states that for any $f_{1},f_{2}\in
C_{c}^{\infty}(X_{d})$ and any $s_{1},s_{2}>0$,

(1.14)
$\displaystyle\int_{{\mathcal{Y}}}f_{1}(a_{s_{1}}\Lambda_{A})f_{2}(a_{s_{2}}\Lambda_{A})\,\operatorname{d}\\!{A}=\mu_{d}(f_{1})\mu_{d}(f_{2})+O\left(e^{-\delta
D(s_{1},s_{2})}\mathcal{N}(f_{1}){\mathcal{N}}(f_{2})\right),$

where $D(s_{1},s_{2}):=\min\\{s_{1},s_{2},|s_{1}-s_{2}|\\}$. Combining this
result with (1.13) we get

$\displaystyle\left|\int_{{\mathcal{Y}}}f_{1}(a_{s_{1}}\Lambda_{A})f_{2}(a_{s_{2}}\Lambda_{A})\,\operatorname{d}\\!{A}-\textrm{Leb}(f_{1})\textrm{Leb}(f_{2})\right|\ll
e^{-\delta
D(s_{1},s_{2})}\prod_{i=1}^{2}\max\left\\{\mathcal{N}(f_{i}),\mu_{d}(f_{i})\right\\}.$

Finally, by approximating $\\{\chi_{B_{k}}\\}_{k>s_{0}}$ from below by smooth
functions, applying the above estimate (together with a trivial estimate when
$D(s_{1},s_{2})$ is small, see (7.13)) and the bounds on the norm
$\mathcal{N}(\cdot)$ (see Lemma 6.5), we show that the divergence of the
series in (1.10) together with the additional technical assumption (1.11),
implies that the required quasi-independence condition (7.8) is satisfied,
thus concluding the proof of the divergence case of Theorem 1.2.

We end our discussion with a few remarks.

######  Remark 7.

Our argument should be compared to that of Kleinbock and Margulis [20], where
the shrinking targets are certain cusp neighborhoods: In [20] the relevant
shrinking target problem is first solved for the case of $a_{s}$-orbits
starting at $\mu_{d}$-generic points in the ambient space $X_{d}$; for this
task it suffices to use, in place of (1.13) and (1.14) respectively, the
invariance of the measure $\mu_{d}$ and the exponential mixing of the
$a_{s}$-flow. Then by an approximation argument [20, §8.7], the shrinking
target property for $\mu_{d}$-generic points in $X_{d}$ is shown to imply the
same property for generic points in the submanifold ${\mathcal{Y}}$. A key
observation in this approximation step is that all shrinking targets, by
virtue of being cusp neighborhoods, remain essentially unaffected by
perturbations from a fixed neighborhood of the identity in the neutral leaf of
the $a_{s}$-flow, i.e. the centralizer of the $a_{s}$-flow in
$\text{SL}_{d}({\mathbb{R}})$. This, however, is no longer the case in our
setting, with the shrinking targets being compact sets. This is why we take
the more direct approach using effective equidistribution and doubly mixing of
the $a_{s}$-translates of ${\mathcal{Y}}$, that is, (1.13) and (1.14).

One potential advantage of this more direct approach is that if (1.13) could
be refined by replacing the measure Leb by a natural measure on some
submanifold of ${\mathcal{Y}}$, then by mimicking our analysis, one could
establish the $\psi$-Dirichlet property for generic points in that
submanifold, for any $\psi$ such that (1.10) converges. See Remark 14 below
for a discussion of the application along these lines of a recent effective
equidistribution result obtained by Chow and Yang [6].

We note that the use of equidistribution of $a_{s}$-translates of
${\mathcal{Y}}$ in the study of the Dirichlet improvability problem is not
new; it has been applied several times in the more well-studied setting of
Dirichlet improvable vectors and matrices. For $\min\\{m,n\\}=1$ and
${\mathcal{I}}\subset{\mathcal{Y}}$ being an analytic curve in ${\mathcal{Y}}$
satisfying certain explicit conditions, Shah [33, Theorem 1.2] proved that the
$a_{s}$-translates of ${\mathcal{I}}$ equidistribute in $X_{d}$ with respect
to $\mu_{d}$ as $s\to\infty$. Shah’s proof relies on Ratner’s classification
of measures invariant under unipotent flows [32], and his equidistribution
theorem is not effective; still it suffices for the deduction of the fact that
generic points on the curve ${\mathcal{I}}$ are Dirichlet non-improvable, that
is, lie outside of the set (1.3). (This is so since in this case, the relevant
“shrinking” target is in fact a fixed set of positive measure.) Shah’s results
have been generalized and strengthened in various directions [34, 35, 38, 18].
In a recent breakthrough of Khalil and Luethi [15], the authors refined (1.13)
(for the case when $n=1$) by replacing Leb with a certain fractal measure,
from which they deduced a complete analogue of Khintchine’s theorem with
respect to this fractal measure.

######  Remark 8.

Another difficulty, which also stems from the fact that our targets are
shrinking compact sets, is the fact that the norm ${\mathcal{N}}(\cdot)$
unavoidably grows (polynomially) for the smooth functions approximating the
shrinking targets from above and below (see Lemma 6.5). While the impact of
this polynomial growth of the norm can be eliminated in the convergence case
due to the exponential decay in the parameter $s$ (i.e. the factor $e^{-\delta
s}$ in the error term in (1.13)), it causes serious problems in the divergence
case, and this is exactly why we need to impose the extra assumption (1.11).
Let us here also note that this assumption (1.11) can be rephrased in terms of
the measure of the shrinking targets as follows:

$\displaystyle\liminf_{s_{1}\to\infty}\frac{\sum_{s_{0}<k\leq
s_{1}}\mu_{d}(B_{k})\log\left(\tfrac{1}{\mu_{d}(B_{k})}\right)}{\left(\sum_{s_{0}<k\leq
s_{1}}\mu_{d}(B_{k})\right)^{2}}=0.$

######  Remark 9.

In order to extend the argument outlined above to the case of general weight
vectors $\bm{\alpha}$ and $\bm{\beta}$, we have to consider a more general
one-parameter flow $\\{g_{s}\\}_{s>0}\subset\text{SL}_{d}({\mathbb{R}})$
associated to $\bm{\alpha}$ and $\bm{\beta}$ (see (5.1)), and use a dynamical
interpretation of $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet matrices which
involves this $g_{s}$-flow and generalizes (1.6); see Proposition 6.2 and
Remark 13. We therefore need analogous effective equidistribution and doubly
mixing results for the $g_{s}$-translates of ${\mathcal{Y}}$. Fortunately,
such more general (and considerably more difficult) effective results are
known to hold, thanks to the work of Kleinbock-Margulis [21, Theorem 1.3] and,
again, Kleinbock-Shi-Weiss [25, Theorem 1.2] and Björklund-Gorodnik [2,
Corollary 2.4] (see Theorem 6.3 below). In fact in [2] a uniform treatment was
given proving effective mixing of arbitrary order for the $g_{s}$-translates
of ${\mathcal{Y}}$; however we will not make use of this.

### Notation and conventions

Throughout the paper, the notation $\|\cdot\|$ denotes the supremum norm on
various Euclidean spaces or matrix spaces (which can be viewed as Euclidean
spaces on the matrix entries). Let $I\subset{\mathbb{R}}$ be an interval (not
necessarily bounded). A function $f:I\to{\mathbb{R}}$ is called increasing
(resp. decreasing) if $f(t_{1})\leq f(t_{2})$ (resp. $f(t_{1})\geq f(t_{2})$)
whenever $t_{1}<t_{2}$. All the vectors in this paper are column vectors. For
two positive quantities $A$ and $B$, we will use the notation $A\ll B$ or
$A=O(B)$ to mean that there is a constant $c>0$ such that $A\leq cB$, and we
will use subscripts to indicate the dependence of the constant on parameters.
We will write $A\asymp B$ for $A\ll B\ll A$.

Acknowledgements We are grateful to the anonymous referee for useful comments.

##  2\. Some preliminaries for Theorem 1.3

Fix an integer $d\geq 2$. In what follows we always denote
$G=\text{SL}_{d}({\mathbb{R}})$, $\Gamma=\text{SL}_{d}({\mathbb{Z}})$ and
$X_{d}=G/\Gamma$ the space of unimodular lattices in ${\mathbb{R}}^{d}$. Let
$\mu_{d}$ be the unique $G$-invariant probability measure on $X_{d}$. Let
$\Delta:X_{d}\to[0,\infty)$ be the function on $X_{d}$ defined as in (1.4). In
this section, we collect some preliminary results for our proof of Theorem
1.3. In fact, for simplicity of presentation we will prove an equivalent
measure estimate result. For any $r\in[0,1)$ let
$\mathcal{C}_{r}\subset{\mathbb{R}}^{d}$ be the open “${(1-r)}$-cube”, i.e.

$\mathcal{C}_{r}:=(r-1,1-r)^{d}.$

Let $K_{r}\subset X_{d}$ be the set of unimodular lattices having no nonzero
points in $\mathcal{C}_{r}$, i.e.

$K_{r}:=\big{\\{}\Lambda\in
X_{d}\>:\>\Lambda\cap\mathcal{C}_{r}=\\{\bm{0}\\}\big{\\}}.$

We note that by definition of $\Delta$,
$K_{r}=\Delta^{-1}[0,-\log\left(1-r\right)]$, or equivalently,
$\Delta^{-1}[0,r]=K_{1-e^{-r}}$. Since $1-e^{-r}=r+O(r^{2})\asymp r$ for all
$r\in(0,1)$, Theorem 1.3 can be equivalently restated as follows.

######  Theorem 2.1.

Let $\varkappa_{d}=\frac{d^{2}+d-4}{2}$ and $\lambda_{d}=\frac{d(d-1)}{2}$ be
as in Theorem 1.2. Then

$\displaystyle\mu_{d}\left(K_{r}\right)\asymp_{d}r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)},\qquad\textrm{as
$r\to 0^{+}$}.$

We will prove Theorem 2.1 by proving a lower bound and an upper bound
separately.

###  2.1. Haar measure and coordinates

Let $P<G$ be the maximal parabolic subgroup fixing the line spanned by
$\bm{e}_{d}\in{\mathbb{R}}^{d}$, and let $N<G$ be the transpose of the
unipotent radical of $P$. Here and hereafter,
$\\{{\text{\boldmath$e$}}_{i}:1\leq i\leq d\\}$ denotes the standard
orthonormal basis of ${\mathbb{R}}^{d}$. Explicitly,

$\displaystyle P=\\{p\in G\>:\>p\bm{e}_{d}=t\bm{e}_{d}\ \textrm{for some
$t\neq 0$}\\},$

and

$\displaystyle
N=\left\\{u_{\bm{x}}:=\left(\begin{smallmatrix}I_{d-1}&\bm{x}\\\
\bm{0}^{t}&1\end{smallmatrix}\right)\>:\>\bm{x}\in{\mathbb{R}}^{d-1}\right\\}.$

For any $p\in P$, let
${\text{\boldmath$b$}}_{1}:=p\bm{e}_{1},\ldots,\bm{b}_{d-1}:=p\bm{e}_{d-1}$ be
the first $d-1$ column vectors of $p$. We note that $p$ is uniquely determined
by $\bm{b}_{1},\ldots,\bm{b}_{d-1}$; we will sometimes denote $p\in P$ by
$p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}$ to indicate this dependence. For any
$g\in G$, let us denote by $\tilde{g}\in M_{d-1,d-1}({\mathbb{R}})$ the top
left $(d-1)\times(d-1)$ block of $g$. If $\det\tilde{g}\neq 0$, then $g$ can
be written uniquely as a product

(2.1) $\displaystyle
g=p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}u_{\bm{x}}\quad{\text{for some
$p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}\in P$ and }u_{\bm{x}}\in N}.$

Let $\nu$ be the (left and right) Haar measure on $G$, normalized so that it
agrees locally with $\mu_{d}$. In terms of the coordinates in (2.1), $\nu$ is
given by

(2.2)
$\operatorname{d}\\!{\nu}(g)=\frac{1}{\zeta(2)\cdots\zeta(d)}\,\operatorname{d}\\!{{\text{\boldmath$x$}}}\prod_{1\leq
i\leq d-1}\operatorname{d}\\!{{\text{\boldmath$b$}}}_{i},$

where $\zeta(\cdot)$ is the Riemann zeta function, and where
$\operatorname{d}\\!{{\text{\boldmath$x$}}}$ and
$\operatorname{d}\\!{{\text{\boldmath$b$}}}_{i}$ denote Lebesgue measure on
${\mathbb{R}}^{d-1}$ and ${\mathbb{R}}^{d}$, respectively. For later purpose,
we also note that the lattice $\Lambda$ represented by
$p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}u_{\bm{x}}$, i.e.
$\Lambda=p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}u_{\bm{x}}{\mathbb{Z}}^{d}$, has a
basis

$\Lambda={\mathbb{Z}}\bm{b}_{1}\oplus\cdots\oplus{\mathbb{Z}}\bm{b}_{d-1}\oplus{\mathbb{Z}}\bm{b}_{d},$

where
$\bm{b}_{d}:=\sum_{j=1}^{d-1}x_{j}\bm{b}_{j}+(\det\tilde{p})^{-1}\bm{e}_{d}$
is the $d$-th column vector of the matrix
$p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}u_{\bm{x}}$. Here $\tilde{p}$ is the top
left $(d-1)\times(d-1)$ block of $p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}$.

For our computation of the upper bounds, it will be more convenient to use
another set of coordinates: For any $g=(g_{ij})_{1\leq i,j\leq d}\in G$ with
$\det\tilde{g}\neq 0$, as mentioned above, we can write $g$ uniquely as in
(2.1). It is clear from this relation that $g$ and
$p_{\bm{b}_{1},\ldots,\bm{b}_{d-1}}$ share the same first $d-1$ column
vectors, i.e. $g\bm{e}_{j}=\bm{b}_{j}$ for all $1\leq j\leq d-1$. Moreover, as
noted above, for the $d$-th column vector we have

$\displaystyle
g\bm{e}_{d}=\sum_{j=1}^{d-1}x_{j}\bm{b}_{j}+(\det\tilde{p})^{-1}\bm{e}_{d}=\sum_{j=1}^{d-1}x_{j}\left(g\bm{e}_{j}\right)+(\det\tilde{g})^{-1}\bm{e}_{d}.$

In particular, we have $(g_{1d},\ldots,g_{d-1,d})^{t}=\tilde{g}\bm{x}$, which
further implies

$\operatorname{d}\\!{{\text{\boldmath$x$}}}=(\det\tilde{g})^{-1}\prod_{1\leq
i\leq d-1}\operatorname{d}\\!{g}_{id}.$

This relation, together with the relations $g\bm{e}_{j}=\bm{b}_{j},1\leq j\leq
d-1$ and the Haar measure description (2.2), immediately implies the
following:

######  Lemma 2.2.

For any (Borel) subset $\mathcal{K}$ of $\bigl{\\{}g\in
G\>:\>|\det\tilde{g}-1|<\frac{1}{2}\bigr{\\}}$, we have

(2.3)
$\displaystyle\nu(\mathcal{K})\asymp_{d}\int_{\mathcal{K}}\prod_{\begin{subarray}{c}1\leq
i,j\leq d\\\ (i,j)\neq(d,d)\end{subarray}}\operatorname{d}\\!{g}_{ij}.$

###  2.2. A small parameter for the lower bound

To prove the lower bound, we will construct a subset of $K_{r}$ whose measure
is of the same magnitude as $K_{r}$. For a lattice
$\Lambda=g{\mathbb{Z}}^{d}\in X_{d}$, to show $\Lambda\in K_{r}$, by
definition one needs to show $g\bm{m}\notin\mathcal{C}_{r}$ for all nonzero
$\bm{m}\in{\mathbb{Z}}^{d}$. If $g\in G$ is sufficiently close to the identity
element $I_{d}\in G$, so that $\Lambda$ has a basis close to the standard
basis $\\{\bm{e}_{i}:1\leq i\leq d\\}$, then one only needs to consider
vectors $\bm{m}\in{\mathbb{Z}}^{d}$ with small supremum norms. For this
reason, we will only focus on lattices that are close to ${\mathbb{Z}}^{d}$.
Recall that the set $K_{r}$ certainly does not get concentrated near the
lattice ${\mathbb{Z}}^{d}$ as $r\to 0^{+}$; indeed, we have
$\cap_{r>0}K_{r}=K_{0}=\Delta^{-1}\\{0\\}$, which as we have mentioned is a
finite union of compact submanifolds of positive codimension $\varkappa_{d}+1$
in $X_{d}$ (see also Section 2.3). The fact that it still suffices to consider
a small neighborhood of ${\mathbb{Z}}^{d}$ when proving the lower bound in
Theorem 2.1 is related to the fact that the mass of $K_{r}$ (with respect to
$\mu_{d}$) becomes concentrated near the lattice ${\mathbb{Z}}^{d}$ as $r\to
0^{+}$; see Remark 10.

Explicitly, we fix a small norm ball in $X_{d}$ around ${\mathbb{Z}}^{d}$ as
follows: For any $c>0$, let

(2.4) $\displaystyle G_{c}:=\left\\{g\in G\>:\>\|g-I_{d}\|<c\right\\}$

be the open ball in $G$ of radius $c$, centered at $I_{d}$ with respect to the
supremum norm on $M_{d,d}({\mathbb{R}})$. Let $\pi:G\to X_{d}$ be the natural
projection from $G$ to $X_{d}$. We fix a parameter $c_{0}\in(0,1)$ (which only
depends on $d$) so small that $\pi|_{G_{c_{0}}}$ is injective and, for any
vectors
${\text{\boldmath$b$}}_{1},\ldots,{\text{\boldmath$b$}}_{d}\in{\mathbb{R}}^{d}$
satisfying $\|{\text{\boldmath$b$}}_{i}-{\text{\boldmath$e$}}_{i}\|<c_{0}$ for
all $1\leq i\leq d$, every hyperplane of the form

$\displaystyle
m{\text{\boldmath$b$}}_{j}+\sum_{\begin{subarray}{c}1\leq\ell\leq d\\\
\ell\neq j\end{subarray}}{\mathbb{R}}{\text{\boldmath$b$}}_{\ell}$

with $m\in{\mathbb{R}}$, $|m|\geq 2$ and $1\leq j\leq d$, is disjoint from the
cube $[-1,1]^{d}$. In particular, if $\Lambda=g{\mathbb{Z}}^{d}$ for some
$g\in G_{c_{0}}$, then $\Lambda$ has a basis $\\{g\bm{e}_{i}:1\leq i\leq d\\}$
satisfying $\|g\bm{e}_{i}-\bm{e}_{i}\|\leq c_{0}$ for all $1\leq i\leq d$. It
follows that in order to prove that $\Lambda\in K_{r}$ for a given
$r\in(0,1)$, it suffices to verify that $\sum_{1\leq i\leq
d}m_{i}\left(g\bm{e}_{i}\right)\notin\mathcal{C}_{r}$ for all
$\bm{m}=(m_{1},\ldots,m_{d})\in\\{-1,0,1\\}^{d}\smallsetminus\\{\bm{0}\\}$.

###  2.3. Hajós’s Theorem and its consequences

Recall that

$K_{0}=\left\\{\Lambda\in
X_{d}\>:\>\Lambda\cap(-1,1)^{d}=\\{\bm{0}\\}\right\\}.$

As we mentioned in the introduction, the explicit description of $K_{0}$ was
conjectured (and proved in two and three dimensions) by Minkowski, and proved
in full generality by Hajós in 1941 [12]:

######  Theorem 2.3 (Hajós).

Let $U$ be the subgroup of upper triangular unipotent matrices in $G$. Let $W$
be the subgroup of permutation matrices in $\text{GL}_{d}({\mathbb{Z}})$. Then

$\displaystyle K_{0}=\bigcup_{w\in W}\left(wUw^{-1}\right){\mathbb{Z}}^{d}.$

If we set

(2.5) $\displaystyle U_{0}=\left\\{(u_{ij})\in
U\>:\>-\tfrac{1}{2}<u_{ij}\leq\tfrac{1}{2}\text{ for all }1\leq i<j\leq
d\right\\}$

so that $U_{0}$ is a fundamental domain for $U/(\Gamma\cap U)$, then we have
the following immediate corollary of Hajós’s Theorem.

######  Corollary 2.4.

Given any $\Lambda\in K_{0}$, there exist $w\in W$ and $u\in U_{0}$ such that
$\Lambda=wu{\mathbb{Z}}^{d}=wuw^{-1}{\mathbb{Z}}^{d}$.

###### Proof.

Since $\Lambda\in K_{0}$, by Theorem 2.3 we can find $u^{\prime}\in U$ and
$w\in W$ such that $\Lambda=wu^{\prime}w^{-1}{\mathbb{Z}}^{d}$; but since
$w^{-1}{\mathbb{Z}}^{d}={\mathbb{Z}}^{d}$, we have
$\Lambda=wu^{\prime}{\mathbb{Z}}^{d}$. Now using the fact that $U_{0}$ is a
fundamental domain for $U/(\Gamma\cap U)$, we can then find $u\in U_{0}$ such
that $u{\mathbb{Z}}^{d}=u^{\prime}{\mathbb{Z}}^{d}$. Thus
$\Lambda=wu^{\prime}{\mathbb{Z}}^{d}=wu{\mathbb{Z}}^{d}=wuw^{-1}{\mathbb{Z}}^{d}$,
finishing the proof. ∎

There is a geometric interpretation of $K_{0}$ in terms of lattice tilings by
unit cubes [36, Ch. 1.4]: Let us write
$\tfrac{1}{2}\mathcal{C}_{0}=(-\tfrac{1}{2},\tfrac{1}{2})^{d}$ for the unit
cube obtained by dilating $\mathcal{C}_{0}$ by a factor $\tfrac{1}{2}$. Then
for any $\Lambda\in X_{d}$, the family of cubes
$\bm{v}+\tfrac{1}{2}\mathcal{C}_{0}$, with $\bm{v}$ running through the
lattice $\Lambda$, forms a tiling of ${\mathbb{R}}^{d}$ (modulo a null set) if
and only if $\Lambda\in K_{0}$. More generally, for any $r\in[0,1)$ we write
$\frac{1}{2}{\mathcal{C}}_{r}=\bigl{(}\frac{1}{2}(r-1),\frac{1}{2}(1-r)\bigr{)}^{d}$.
Then for any $\Lambda\in X_{d}$, the condition $\Lambda\in K_{r}$, i.e.
$\Lambda\cap\mathcal{C}_{r}=\\{\mathbf{0}\\}$, is equivalent to the condition
that the cubes ${\text{\boldmath$v$}}+\tfrac{1}{2}{\mathcal{C}}_{r}$
($\bm{v}\in\Lambda$) are pairwise disjoint. When this holds, we write

$\displaystyle{\mathfrak{C}}_{\Lambda,r}:=\Lambda+\tfrac{1}{2}{\mathcal{C}}_{r}$

for the union of these disjoint cubes. This set is used in the statement of
the following simple bound, which is of crucial importance in our proof of the
upper bound in Theorem 2.1.

######  Lemma 2.5.

Let $\Lambda\in X_{d}$ and $r\in(0,\frac{1}{4})$ be such that
$\Lambda\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$, and let ${\mathcal{U}}$ be a
Borel subset of ${\mathbb{R}}^{d}$ which is disjoint from
${\mathfrak{C}}_{\Lambda,r}$ and which is contained in some translate of the
cube $(0,\frac{3}{4})^{d}$. Then $\operatorname{vol}({\mathcal{U}})<dr$.

###### Proof.

The set ${\mathfrak{C}}_{\Lambda,r}$ is invariant under translation by any
vector in $\Lambda$, and if ${\mathcal{F}}\subset{\mathbb{R}}^{d}$ is any
fundamental domain for ${\mathbb{R}}^{d}/\Lambda$, then

$\displaystyle\operatorname{vol}({\mathcal{F}}\cap{\mathfrak{C}}_{\Lambda,r})=\sum_{{\text{\boldmath$v$}}\in\Lambda}\operatorname{vol}\left({\mathcal{F}}\cap({\text{\boldmath$v$}}+\tfrac{1}{2}{\mathcal{C}}_{r})\right)=\sum_{{\text{\boldmath$v$}}\in\Lambda}\operatorname{vol}\left(({\mathcal{F}}-{\text{\boldmath$v$}})+\tfrac{1}{2}{\mathcal{C}}_{r}\right)=\operatorname{vol}(\tfrac{1}{2}{\mathcal{C}}_{r})=(1-r)^{d},$

where the first equality holds since the cubes
${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$
(${\text{\boldmath$v$}}\in\Lambda$) are pairwise disjoint, and the third
equality holds since the sets ${\mathcal{F}}-{\text{\boldmath$v$}}$
(${\text{\boldmath$v$}}\in\Lambda$) form a partition of ${\mathbb{R}}^{d}$.
Hence

$\displaystyle\operatorname{vol}({\mathcal{F}}\smallsetminus{\mathfrak{C}}_{\Lambda,r})=1-(1-r)^{d}<dr.$

Next, since ${\mathcal{U}}$ is contained in a translate of
$(0,\frac{3}{4})^{d}$, the difference between any two vectors in
${\mathcal{U}}$ lies in
$(-\frac{3}{4},\frac{3}{4})^{d}={\mathcal{C}}_{1/4}\subset{\mathcal{C}}_{r}$,
and since $\Lambda\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$, this implies that
the points in ${\mathcal{U}}$ are pairwise inequivalent modulo $\Lambda$.
Hence the set
$\big{(}{\mathcal{F}}\smallsetminus(\Lambda+{\mathcal{U}})\big{)}\cup{\mathcal{U}}$
is another fundamental domain for ${\mathbb{R}}^{d}/\Lambda$, and it contains
${\mathcal{U}}$. After replacing ${\mathcal{F}}$ by this set, we have
${\mathcal{U}}\subset{\mathcal{F}}$; thus
${\mathcal{U}}\subset{\mathcal{F}}\smallsetminus{\mathfrak{C}}_{\Lambda,r}$,
and hence
$\operatorname{vol}({\mathcal{U}})\leq\operatorname{vol}({\mathcal{F}}\smallsetminus{\mathfrak{C}}_{\Lambda,r})<dr$.
∎

##  3\. Proof of the lower bound

We keep the notation introduced in Section 2. In this section we prove the
lower bound in Theorem 2.1. We will do this by constructing, for every
sufficiently small $r$, an explicit subset $\underline{K}_{r}\subset X_{d}$
which we will show satisfies

$\displaystyle\underline{K}_{r}\subset
K_{r}\qquad\textrm{and}\qquad\mu_{d}(\underline{K}_{r})\gg_{d}r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)}.$

We start by giving a family of conditions which ensures that a given lattice
is contained in $K_{r}$. Recall from Section 2.2 that $c_{0}\in(0,1)$ is a
small fixed parameter with the property that for any $g\in G_{c_{0}}$ and
$0<r<1$, we have $g{\mathbb{Z}}^{d}\in K_{r}$ if and only if $\sum_{1\leq
j\leq d}m_{j}\left(g\bm{e}_{j}\right)\notin\mathcal{C}_{r}$ for all
$\bm{m}=(m_{1},\ldots,m_{d})^{t}\in\\{-1,0,1\\}^{d}\smallsetminus\\{\bm{0}\\}$.

######  Lemma 3.1.

Let $c_{0}\in(0,1)$ be as above, let $r\in(0,c_{0}/d)$ and let
$\Lambda=p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}u_{\bm{x}}{\mathbb{Z}}^{d}\in X_{d}$
with $\bm{b}_{j}=(b_{1j},\ldots,b_{dj})^{t}\in{\mathbb{R}}^{d}$
($j=1,\ldots,d-1$) and $\bm{x}\in(0,c_{0}/d)^{d-1}$. Let
$\tilde{p}=(b_{ij})_{1\leq i,j\leq d-1}$ be the top left $(d-1)\times(d-1)$
block of $p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}$ as before. Suppose
$\bm{b}_{1},\ldots,\bm{b}_{d-1},\bm{x}$ further satisfies

(3.1) $\displaystyle b_{ij},-b_{ji}\in(-c_{0},0),\ \forall\ 1\leq j<i\leq
d-1;\ b_{d\ell}\in(-c_{0},0),b_{\ell\ell}\in(1-r,1),\ \forall\ 1\leq\ell\leq
d-1,$ (3.2) $\displaystyle b_{ij}<\sum_{k=1}^{j-1}b_{ik},\qquad\forall\ 2\leq
j<i\leq d,$ (3.3) $\displaystyle
1-r<\det\tilde{p}<(1-r)^{-1}\qquad\textrm{and}\qquad\sum_{j=1}^{d-1}|b_{dj}|x_{j}<(\det\tilde{p})^{-1}-(1-r),$

and

(3.4) $\displaystyle b_{ii}+\sum_{j=1}^{d-1}b_{ij}x_{j}>1-r,\qquad\forall\
1\leq i\leq d-1.$

Then $\Lambda\in K_{r}$.

###### Proof.

Let us set $g:=p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}u_{\bm{x}}$ and
${\text{\boldmath$b$}}_{d}=(b_{1d},\ldots,b_{dd})^{t}:=g{\text{\boldmath$e$}}_{d}$;
then ${\text{\boldmath$b$}}_{j}=g{\text{\boldmath$e$}}_{j}$ for all $1\leq
j\leq d$, and
$\bm{b}_{d}:=\sum_{j=1}^{d-1}x_{j}\bm{b}_{j}+(\det\tilde{p})^{-1}\bm{e}_{d}$;
in particular $b_{id}=\sum_{j=1}^{d-1}b_{ij}x_{j}$ for $1\leq i\leq d-1$. We
will start by proving that $g\in G_{c_{0}}$, i.e.
$\|\bm{b}_{j}-\bm{e}_{j}\|<c_{0}$ for all $1\leq j\leq d$. In fact, if $1\leq
j\leq d-1$ then $\|\bm{b}_{j}-\bm{e}_{j}\|<c_{0}$ is an immediate consequence
of (3.1) and $0<r<c_{0}/d<c_{0}$; thus it remains to show
$\|\bm{b}_{d}-\bm{e}_{d}\|<c_{0}$. For each $1\leq i\leq d-1$ we have

$\displaystyle\left|b_{id}\right|=\Biggl{|}\sum_{j=1}^{d-1}b_{ij}x_{j}\Biggr{|}\leq\sum_{j=1}^{d-1}\left|b_{ij}\right|\left|x_{j}\right|<\sum_{j=1}^{d-1}\frac{c_{0}}{d}<c_{0},$

where for the second inequality we used the assumption that
$\bm{x}\in(0,c_{0}/d)^{d-1}$ and the fact that $|b_{ij}|<1$ for all $1\leq
i,j\leq d-1$, which is immediate from (3.1). It remains to prove that
$1-c_{0}<b_{dd}<1+c_{0}$. In fact, we have the following stronger bound:

(3.5) $\displaystyle 1-r<b_{dd}<1+c_{0}.$

Indeed, using $b_{dj}<0$ and $x_{j}>0$ ($1\leq j\leq d-1$) in combination with
(3.3), we get

$\displaystyle
b_{dd}=\sum_{j=1}^{d-1}b_{dj}x_{j}+(\det\tilde{p})^{-1}=(\det\tilde{p})^{-1}-\sum_{j=1}^{d-1}\left|b_{dj}\right|x_{j}>1-r$

as well as

$\displaystyle
b_{dd}=\sum_{j=1}^{d-1}b_{dj}x_{j}+(\det\tilde{p})^{-1}<(\det\tilde{p})^{-1}<(1-r)^{-1}<1+2r\leq
1+c_{0}.$

(For the second to last inequality we used the fact that $0<r<c_{0}/d<1/2$.)
This finishes the proof of (3.5), and hence $g\in G_{c_{0}}$.

Because of $g\in G_{c_{0}}$, in order to show $\Lambda\in K_{r}$, it suffices
to prove that $g\bm{m}=\sum_{1\leq j\leq
d}m_{j}\bm{b}_{j}\notin\mathcal{C}_{r}$ for all
$\bm{m}\in\\{-1,0,1\\}^{d}\smallsetminus\\{\bm{0}\\}$. Thus we now let the
vector
$\bm{m}=(m_{1},\ldots,m_{d})^{t}\in\\{-1,0,1\\}^{d}\smallsetminus\\{\bm{0}\\}$
be given, and show that $g\bm{m}\notin\mathcal{C}_{r}$.

First assume that all the nonzero entries of $\bm{m}$ are of the same sign.
After replacing $\bm{m}$ by $-\bm{m}$ if necessary, we may assume $m_{j}\geq
0$ for all $1\leq j\leq d$. Let $k\in\\{1,\ldots,d\\}$ be the smallest integer
such that $m_{k}=1$. If $k=d$ then $\bm{m}=\bm{e}_{d}$, and thus
$g\bm{m}=\bm{b}_{d}\notin\mathcal{C}_{r}$ by (3.5). In the remaining case when
$k<d$, the $k$-th coordinate of $g\bm{m}$ satisfies

(3.6) $\displaystyle m_{k}b_{kk}+\sum_{j=k+1}^{d}m_{j}b_{kj}\geq
b_{kk}+m_{d}b_{kd}\geq\min\\{b_{kk},b_{kk}+b_{kd}\\},$

where we used the fact that, by (3.1), $b_{kj}>0$ for each $k<j\leq d-1$.
Furthermore,

(3.7) $\displaystyle\min\\{b_{kk},b_{kk}+b_{kd}\\}>1-r,$

since $b_{kk}>1-r$ by (3.1) and
$b_{kk}+b_{kd}=b_{kk}+\sum_{j=1}^{d-1}b_{kj}x_{j}>1-r$ by (3.4). It follows
from (3.6) and (3.7) that the $k$-th coordinate of $g\bm{m}$ is larger than
$1-r$, and so $g\bm{m}\notin{\mathcal{C}}_{r}$. This completes the proof in
the case when all the nonzero entries of $\bm{m}$ are of the same sign.

It remains to treat the case when $\\{-1,1\\}\subset\\{m_{j}:1\leq j\leq
d\\}$. Then let $1\leq i_{1}<i_{2}\leq d$ be the indices which record the
latest instance when the signs of the entries of $\bm{m}$ change, i.e. the
unique indices such that $m_{i_{1}}m_{i_{2}}=-1$, $m_{j}=0$ for
$i_{1}<j<i_{2}$ and $m_{j}\in\\{m_{i_{2}},0\\}$ for $i_{2}<j\leq d$ (the last
two conditions are void if $i_{1}+1=i_{2}$ or $i_{2}=d$, respectively). Again
after replacing $\bm{m}$ by $-\bm{m}$ if necessary, we may assume
$m_{i_{1}}=-1$, $m_{i_{2}}=1$, and thus $m_{j}\geq 0$ for all $i_{2}<j\leq d$.
Now consider the $i_{2}$-th coordinate of $g\bm{m}$ which is

$\sum_{1\leq
j<i_{1}}m_{j}b_{i_{2},j}-b_{i_{2},i_{1}}+b_{i_{2},i_{2}}+\sum_{i_{2}<j\leq
d}m_{j}b_{i_{2},j}.$

Here we have

(3.8) $\displaystyle\sum_{1\leq
j<i_{1}}m_{j}b_{i_{2},j}-b_{i_{2},i_{1}}\geq\sum_{1\leq
j<i_{1}}b_{i_{2},j}-b_{i_{2},i_{1}}>0,$

where the first inequality holds since $|m_{j}|\leq 1$ and $b_{i_{2},j}<0$ for
all $j<i_{1}$, and the second inequality holds by (3.2) (except if $i_{1}=1$;
in that case (3.8) simply says that $b_{i_{2},i_{1}}<0$, which holds by
(3.1)). Furthermore,

(3.9) $\displaystyle b_{i_{2},i_{2}}+\sum_{i_{2}<j\leq
d}m_{j}b_{i_{2},j}\quad\begin{cases}=b_{dd}>1-r&\text{if
}\>i_{2}=d,\\\\[2.0pt] \geq b_{i_{2},i_{2}}+m_{d}b_{i_{2},d}>1-r&\text{if
}\>i_{2}<d,\end{cases}$

where we used (3.5) in the case $i_{2}=d$, and in the case $i_{2}<d$ we used
the fact that $m_{j}\geq 0$ and $b_{i_{2},j}>0$ for all $i_{2}<j\leq d-1$ (if
any), and then applied (3.7) with $k=i_{2}$. Adding the inequalities in (3.8)
and (3.9), we conclude that the $i_{2}$-th coordinate of $g\bm{m}$ is larger
than $1-r$; hence $g\bm{m}\notin\mathcal{C}_{r}$. This concludes the proof of
the lemma. ∎

We next give another family of conditions, which implies the conditions in
Lemma 3.1, and which is more suitable for the measure computations which we
are going to carry out.

######  Proposition 3.2.

Let $c_{0}\in(0,1)$ be as above, let $r\in(0,c_{0}/d)$ and let
$\Lambda=p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}u_{\bm{x}}{\mathbb{Z}}^{d}\in X_{d}$
with $\bm{b}_{j}=(b_{1j},\ldots,b_{dj})^{t}\in{\mathbb{R}}^{d}$
($j=1,\ldots,d-1$) and $\bm{x}\in(0,c_{0}/d)^{d-1}$. Assume that

(3.10) $\displaystyle b_{ij},-b_{ji}\in(-c_{0},0),\ \forall\ 1\leq j<i\leq
d-1;\ b_{d\ell}\in(-c_{0},0),b_{\ell\ell}\in(1-\tfrac{r}{2d},1),\ \forall\
1\leq\ell\leq d-1,$ (3.11) $\displaystyle b_{ij}<db_{i,j-1}\
(\Leftrightarrow|b_{ij}|>d|b_{i,j-1}|),\qquad\forall\ 2\leq j<i\leq d,$ (3.12)
$\displaystyle|b_{ij}b_{ji}|<\frac{r}{d!},\quad\forall 1\leq j<i\leq
d-1,\qquad\textrm{and}\qquad\sum_{j=1}^{d-1}|b_{dj}|x_{j}<\frac{r}{2},$

and

(3.13) $\displaystyle b_{kj}>b_{ij}\
(\Leftrightarrow|b_{kj}|<|b_{ij}|),\qquad\forall 1\leq j<k<i\leq d.$

Then $\Lambda\in K_{r}$.

###### Proof.

In view of Lemma 3.1, it suffices to show that the conditions (3.1), (3.2),
(3.3) and (3.4) are fulfilled. Among these, (3.1) is an immediate consequence
of (3.10). Furthermore, (3.2) follows from (3.10) and (3.11) by the following
computation, valid for any $2\leq j<i\leq d$:

$\displaystyle
b_{ij}<db_{i,j-1}<(j-1)b_{i,j-1}\leq\sum_{k=1}^{j-1}b_{ik}\quad(<0),$

where the last relation is an equality when $j=2$, while for $j\geq 3$ it is a
strict inequality which holds since (3.11) forces
$b_{i,j-1}<b_{i,j-2}<\cdots<b_{i,1}$. Also (3.4) is easily proved: Let $1\leq
i\leq d-1$. Then for every $1\leq j\leq d-1$ we have $x_{j}>0$ and
$b_{ij}>b_{dj}$ (the latter holds by (3.13) if $j<i$ and by (3.10) if $j\geq
i$). Hence

$b_{ii}+\sum_{j=1}^{d-1}b_{ij}x_{j}>b_{ii}+\sum_{j=1}^{d-1}b_{dj}x_{j}>1-\frac{r}{2d}-\frac{r}{2}>1-r,$

where for the second last inequality we used (3.10) and the second part of
(3.12).

It remains to prove (3.3). We first note that the ordering assumptions in
(3.11) and (3.13) imply

(3.14)
$\displaystyle|b_{i^{\prime},j^{\prime}}|\leq|b_{ij}|\qquad\textrm{whenever
$1\leq j^{\prime}<i^{\prime}\leq d$, $1\leq j<i\leq d$ and $i^{\prime}\leq i$
and $j^{\prime}\leq j$.}$

Now let $\tilde{p}=(b_{ij})_{1\leq i,j\leq d-1}$ be as in Lemma 3.1. Define

$\varphi_{\tilde{p}}:=\sum_{\begin{subarray}{c}\sigma\in S_{d-1}\\\
\sigma\neq\operatorname{id}\end{subarray}}\prod_{1\leq i\leq
d-1}\left|b_{\sigma(i)i}\right|,$

where $S_{d-1}$ is the symmetric group over the finite set
$\\{1,2,\ldots,d-1\\}$ and $\operatorname{id}\in S_{d-1}$ is its identity
element. Then

(3.15)
$\displaystyle\prod_{i=1}^{d-1}b_{ii}-\varphi_{\tilde{p}}\leq\det\tilde{p}\leq\prod_{i=1}^{d-1}b_{ii}+\varphi_{\tilde{p}}.$

If $d=2$ then $S_{d-1}=\\{\operatorname{id}\\}$ and $\varphi_{\tilde{p}}=0$.
Now assume $d\geq 3$, and consider an arbitrary permutation $\sigma\in
S_{d-1}\smallsetminus\\{\operatorname{id}\\}$. Let $(i_{1}\ldots i_{\ell})$ be
a cycle of $\sigma$ of length $\ell\geq 2$, meaning that
$\sigma(i_{j})=i_{j+1}$ for all $1\leq j\leq\ell-1$ and
$\sigma(i_{\ell})=i_{1}$. Without loss of generality, we may assume
$i_{\ell}=\max\\{i_{j}:1\leq j\leq\ell\\}$. Let
$i_{j_{0}}:=\min\left\\{i_{j}:1\leq j\leq\ell\right\\}$. Then (3.14) applies
for the pairs $(i^{\prime},j^{\prime})=(\sigma(i_{j_{0}}),i_{j_{0}})$ and
$(i,j)=(i_{\ell},i_{1})$, so that
$\bigl{|}b_{\sigma(i_{j_{0}}),i_{j_{0}}}\bigr{|}\leq\bigl{|}b_{i_{\ell},i_{1}}\bigr{|}$.
We also have $|b_{ij}|<1$ for all $1\leq i,j\leq d-1$, by (3.10). Hence

$\prod_{1\leq i\leq
d-1}\left|b_{\sigma(i)i}\right|\leq\left|b_{\sigma(i_{j_{0}}),i_{j_{0}}}b_{i_{1},i_{\ell}}\right|\leq\left|b_{i_{\ell},i_{1}}b_{i_{1},i_{\ell}}\right|<\frac{r}{d!},$

where the last inequality holds by (3.12). The above holds for every
$\sigma\in S_{d-1}\smallsetminus\\{\operatorname{id}\\}$; hence

(3.16) $\displaystyle\varphi_{\tilde{p}}<(d-1)!\frac{r}{d!}=\frac{r}{d}.$

Note that (3.16) also holds when $d=2$, trivially.

Using (3.15), (3.16), and the fact that $1-\frac{r}{2d}<b_{ii}<1$ for all
$1\leq i\leq d-1$ (cf. (3.10)), we get

(3.17)
$\displaystyle\det\tilde{p}<\prod_{i=1}^{d-1}b_{ii}+\frac{r}{d}<1+\frac{r}{d}<(1-r)^{-1},$

and

$\displaystyle\det\tilde{p}>\prod_{i=1}^{d-1}b_{ii}-\frac{r}{d}>\left(1-\frac{r}{2d}\right)^{d-1}-\frac{r}{d}>1-\frac{r}{2}-\frac{r}{d}\geq
1-r.$

Hence we have proved the first condition in (3.3). For the second condition in
(3.3), in view of the second condition in (3.12) it suffices to show
$\tfrac{r}{2}<(\det\tilde{p})^{-1}-(1-r)$, or equivalently, that
$\det\tilde{p}$ is smaller than $(1-\tfrac{r}{2})^{-1}$. But this is true
since by (3.17), $\det\tilde{p}<1+\tfrac{r}{d}<(1-\tfrac{r}{2})^{-1}$. This
finishes the proof. ∎

We can now give the

###### Proof of the lower bound in Theorem 2.1.

Keep $0<r<c_{0}/d$, and define

(3.20)
$\displaystyle\underline{K}_{r}:=\left\\{p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}u_{\bm{x}}{\mathbb{Z}}^{d}\in
X_{d}\>:\>\begin{array}[]{ll}(\bm{b}_{1},\ldots,\bm{b}_{d-1},\bm{x})\in({\mathbb{R}}^{d})^{d-1}\times(0,c_{0}/d)^{d-1}\\\
\textrm{\ \ satisfies}\
\eqref{equ:con1},\eqref{equ:con2},\eqref{equ:con3},\eqref{equ:con4}\end{array}\right\\}.$

By Proposition 3.2 we have $\underline{K}_{r}\subset K_{r}$. It remains to
bound $\mu_{d}\left(\underline{K}_{r}\right)$ from below. Set

$\displaystyle{\underline{\mathcal{K}}_{r}}:=\left\\{p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}u_{\bm{x}}\in
G\>:\>\begin{array}[]{ll}(\bm{b}_{1},\ldots,\bm{b}_{d-1},\bm{x})\in({\mathbb{R}}^{d})^{d-1}\times(0,c_{0}/d)^{d-1}\\\
\textrm{\ \ satisfies}\
\eqref{equ:con1},\eqref{equ:con2},\eqref{equ:con3},\eqref{equ:con4}\end{array}\right\\},$

so that $\pi({\underline{\mathcal{K}}_{r}})=\underline{K}_{r}$. Here $\pi:G\to
X_{d}$ is the natural projection as before. By immediate inspection of the
proofs of Proposition 3.2 and Lemma 3.1, we have
${\underline{\mathcal{K}}_{r}}\subset G_{c_{0}}$, and by our choice of
$c_{0}$, $\pi|_{G_{c_{0}}}$ is injective (see Section 2.2). Hence
$\mu_{d}\left(\underline{K}_{r}\right)=\nu\left({\underline{\mathcal{K}}_{r}}\right)$,
and by (2.2) we have

$\displaystyle\nu\left({\underline{\mathcal{K}}_{r}}\right)$
$\displaystyle\asymp_{d}\prod_{1\leq k\leq
d-1}\left(\int_{1-\frac{r}{2d}}^{1}\operatorname{d}\\!{b}_{kk}\right)\int_{\mathcal{R}}\delta\bigl{(}\,(b_{ij})_{1\leq
j<i\leq d}\,\bigr{)}\prod_{1\leq j<i\leq d}\operatorname{d}\\!{b}_{ij},$

where

$\mathcal{R}:=\left\\{(b_{ij})_{1\leq j<i\leq
d}\in(-c_{0},0)^{d(d-1)/2}\>:\>\textrm{$(b_{ij})$ satisfies \eqref{equ:con2}
and \eqref{equ:con4}}\right\\},$

and

$\displaystyle\delta\bigl{(}\,(b_{ij})_{1\leq j<i\leq d}\,\bigr{)}:$
$\displaystyle=\prod_{1\leq j<i\leq
d-1}\int_{0}^{\min\left\\{c_{0},\frac{r}{d!|b_{ij}|}\right\\}}\operatorname{d}\\!{b}_{ji}\times\int_{\left\\{\bm{x}\in\left(0,\tfrac{c_{0}}{d}\right)^{d-1}\>:\>\sum_{j=1}^{d-1}|b_{dj}|x_{j}<\frac{r}{2}\right\\}}\prod_{1\leq
j\leq d-1}\operatorname{d}\\!{x}_{j}$
$\displaystyle\asymp_{d,c_{0}}\prod_{1\leq j<i\leq
d-1}\min\left\\{1,\frac{r}{|b_{ij}|}\right\\}\times\prod_{1\leq j\leq
d-1}\min\left\\{1,\frac{r}{|b_{dj}|}\right\\}$ $\displaystyle=\prod_{1\leq
j<i\leq d}\min\left\\{1,\frac{r}{|b_{ij}|}\right\\}.$

Hence

(3.21)
$\displaystyle\mu_{d}\left(\underline{K}_{r}\right)=\nu\left({\underline{\mathcal{K}}_{r}}\right)\asymp_{d,c_{0}}r^{d-1}\int_{\mathcal{R}}\left(\prod_{1\leq
j<i\leq
d}\min\left\\{1,\frac{r}{|b_{ij}|}\right\\}\operatorname{d}\\!{b}_{ij}\right).$

Now for each $1\leq j<i\leq d$, we make a change of variable,
$b_{ij}=-d^{j-1}z_{ij}$, to simplify the ordering condition (3.11). Then all
the $z_{ij}$’s are positive, and the conditions (3.11) and (3.13) become

(3.22) $\displaystyle z_{i^{\prime}j^{\prime}}<z_{ij}\quad\textrm{whenever
$1\leq j^{\prime}<i^{\prime}\leq d$, $\>1\leq j<i\leq d$, $\>i^{\prime}\leq
i$, $\>j^{\prime}\leq j$ and $(i^{\prime},j^{\prime})\neq(i,j)$}.$

Moreover, for any $j<i$, the condition $b_{ij}\in(-c_{0},0)$ corresponds to
$z_{ij}\in(0,c_{0}/d^{j-1})$, and we note that each of these intervals
contains the fixed interval $(0,c_{0}/d^{d-1})$. In fact, let us restrict each
$z_{ij}$ to the even smaller interval $(r,c_{0}/d^{d-1})$, and assume
$r<c_{0}/d^{d-1}$ so that this interval is non-empty; then
$r/|b_{ij}|=rd^{1-j}/z_{ij}<d^{1-j}\leq 1$, so that
$\min\left\\{1,r/|b_{ij}|\right\\}=rd^{1-j}/z_{ij}\asymp_{d}r/z_{ij}$. Note
also that
$|\operatorname{d}\\!{b}_{ij}|=d^{j-1}\,\operatorname{d}\\!{z}_{ij}\asymp_{d}\operatorname{d}\\!{z}_{ij}$;
hence we get from (3.21):

$\displaystyle\mu_{d}(\underline{K}_{r})$
$\displaystyle\gg_{d,c_{0}}r^{d-1+\frac{d(d-1)}{2}}\int_{\mathcal{R}^{\prime}}\prod_{1\leq
j<i\leq
d}\frac{\operatorname{d}\\!{z}_{ij}}{z_{ij}}=r^{\frac{(d-1)(d+2)}{2}}\int_{\mathcal{R}^{\prime}}\prod_{1\leq
j<i\leq d}\frac{\operatorname{d}\\!{z}_{ij}}{z_{ij}},$

where

$\displaystyle\mathcal{R}^{\prime}:=\left\\{(z_{ij})_{1\leq j<i\leq
d}\in(r,c_{0}/d^{d-1})^{d(d-1)/2}\>:\>(z_{ij})\ \textrm{satisfies
\eqref{equ:ordering}}\right\\}.$

The last integrand is invariant under every permutation of the variables
$(z_{ij})_{1\leq j<i\leq d}$, and the integration regions
$\sigma(\mathcal{R}^{\prime})$ with $\sigma$ running through all these
permutations cover (modulo a null set) the cube
$(r,c_{0}/d^{d-1})^{d(d-1)/2}$; hence we obtain

(3.23) $\displaystyle\mu_{d}\left(\underline{K}_{r}\right)$
$\displaystyle\gg_{d,c_{0}}r^{\frac{(d-1)(d+2)}{2}}\prod_{1\leq j<i\leq
d}\left(\int_{r}^{c_{0}/d^{d-1}}\frac{\operatorname{d}\\!{z}_{ij}}{z_{ij}}\right)\asymp_{d,c_{0}}r^{\frac{(d-1)(d+2)}{2}}\log^{\frac{d(d-1)}{2}}\Bigl{(}\frac{1}{r}\Bigr{)},$

where the last estimate is valid e.g. for all $0<r<c_{0}/(2d^{d-1})$.
Recalling that $c_{0}$ only depends on $d$, we see that the lower bound in
Theorem 2.1 is now proved. ∎

##  4\. Proof of the upper bound

We keep the notation introduced in Section 2. In this section we prove the
upper bound in Theorem 2.1. The main step in the proof is to show that for $r$
small, $K_{r}$ is contained in a certain set of more explicit nature; see
Proposition 4.2 below. To prepare for the statement of this result, we start
by introducing the following set, for any $r\in(0,1)$ and $C>0$:

$\displaystyle{\overline{\mathcal{K}}_{r,C}}:=\left\\{g=(g_{ij})\in
G\>:\>\begin{array}[]{ll}1-r\leq g_{ii}\leq 1+Cr,\ \forall\ 1\leq i\leq d\\\
|g_{ij}|<1,\;|g_{ij}g_{ji}|\leq Cr,\ \forall\ 1\leq i\neq j\leq d\\\
|\det\tilde{g}-1|<\tfrac{1}{2}\end{array}\right\\}.$

Here $\tilde{g}$ denotes the top left $(d-1)\times(d-1)$ block of $g$ as
before. In view of the Haar measure description (2.3), it is not difficult to
compute the measure of ${\overline{\mathcal{K}}_{r,C}}$:

######  Lemma 4.1.

For any $r\in(0,\frac{1}{2})$ and $C>0$ we have

$\displaystyle\nu\bigl{(}{\overline{\mathcal{K}}_{r,C}}\bigr{)}\ll_{d,C}r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)}.$

###### Proof.

Let us define

$\displaystyle{\overline{\mathcal{K}}^{\prime}_{r,C}}:=\left\\{g=(g_{ij})\in
G\>:\>\begin{array}[]{ll}1-r\leq g_{ii}\leq 1+Cr,\ \forall\ 1\leq i\leq d-1\\\
|g_{ij}|<1,\;|g_{ij}g_{ji}|\leq Cr,\ \forall\ 1\leq i\neq j\leq d\\\
|\det\tilde{g}-1|<\tfrac{1}{2}\end{array}\right\\}$

by disregarding the restriction on $g_{dd}$. Then clearly
${\overline{\mathcal{K}}_{r,C}}\subset{\overline{\mathcal{K}}^{\prime}_{r,C}}$.
Moreover, in view of the Haar measure description (2.3) we have (noting also
that $\varkappa_{d}+1=\lambda_{d}+d-1$)

$\displaystyle\nu\left({\overline{\mathcal{K}}^{\prime}_{r,C}}\right)$
$\displaystyle\ll_{d}\prod_{1\leq i\leq
d-1}\int_{1-r}^{1+Cr}\operatorname{d}\\!{g}_{ii}\prod_{1\leq j<i\leq
d}\int_{-1}^{1}\int_{-1}^{1}{\chi_{\left\\{(g_{ij},g_{ji})\>:\>|g_{ij}g_{ji}|<Cr\right\\}}}\,\operatorname{d}\\!{g}_{ij}\operatorname{d}\\!{g}_{ji}$
$\displaystyle\asymp_{C}r^{d-1}\left(r\log\Bigl{(}\frac{1}{r}\Bigr{)}\right)^{\lambda_{d}}=r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)}.$

Thus

$\displaystyle\nu\left({\overline{\mathcal{K}}_{r,C}}\right)\leq\nu\left({\overline{\mathcal{K}}^{\prime}_{r,C}}\right)\ll_{d,C}r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)},$

finishing the proof. ∎

Recall that $\pi:G\to X_{d}$ is the natural projection from $G$ to $X_{d}$.

######  Proposition 4.2.

There exist $r_{0}>0$ and $C>0$ (depending only on $d$) such that for all
$r\in(0,r_{0})$

(4.1) $\displaystyle K_{r}\subset\bigcup_{w\in
W}\pi\bigl{(}w\,{\overline{\mathcal{K}}_{r,C}}\,w^{-1}\bigr{)}.$

Let us first give a quick

###### Proof of the upper bound in Theorem 2.1 assuming Proposition 4.2.

The Haar measure $\nu$ on $G$ is preserved by conjugation by any element $w\in
W$ (even though $w$ may be outside $G$). Hence it follows from (4.1) that, for
any $r\in(0,r_{0})$,

$\displaystyle\mu_{d}(K_{r})$ $\displaystyle\leq\sum_{w\in
W}\mu_{d}\bigl{(}\pi(w\,{\overline{\mathcal{K}}_{r,C}}\,w^{-1})\bigr{)}\leq\sum_{w\in
W}\nu(w\,{\overline{\mathcal{K}}_{r,C}}\,w^{-1})=d!\,\nu({\overline{\mathcal{K}}_{r,C}}).$

Using this inequality, the upper bound in Theorem 2.1 now follows from Lemma
4.1. ∎

######  Remark 10.

Proposition 4.2, in combination with the lower bound in Theorem 2.1, also
implies that as $r\to 0^{+}$, the mass of $K_{r}$ with respect to $\mu_{d}$
becomes concentrated near the lattice ${\mathbb{Z}}^{d}$. In precise terms, if
${\mathcal{O}}$ is any fixed neighborhood of ${\mathbb{Z}}^{d}$ in $X_{d}$,
then

(4.2)
$\displaystyle\mu_{d}\left(K_{r}\smallsetminus{\mathcal{O}}\right)\ll_{d}r^{\varkappa_{d}+1}\log^{\lambda_{d}-1}\Bigl{(}\frac{1}{r}\Bigr{)},\quad\text{and
thus }\quad\frac{\mu_{d}(K_{r}\cap{\mathcal{O}})}{\mu_{d}(K_{r})}\to
1\qquad\textrm{as $r\to 0^{+}$}.$

Indeed, we can fix $\varepsilon>0$ so that ${\mathcal{O}}$ contains the set
$\pi(G_{\varepsilon})$ with $G_{\varepsilon}\subset G$ the norm ball defined
as in (2.4); then by arguing along the same lines as above, the first relation
in (4.2) will follow from the following bound:

$\displaystyle\nu\left({\overline{\mathcal{K}}_{r,C}}\smallsetminus
G_{\varepsilon}\right)\ll_{d,C}r^{\varkappa_{d}+1}\log^{\lambda_{d}-1}\Bigl{(}\frac{1}{r}\Bigr{)},$

for $C>0$ and $r$ small. However, for $r<\varepsilon/C$,
$g=(g_{ij})\in{\overline{\mathcal{K}}_{r,C}}$ forces $|g_{ii}-1|<\varepsilon$
for all $i$, and so the set ${\overline{\mathcal{K}}_{r,C}}\smallsetminus
G_{\varepsilon}$ is contained in the union $\cup_{i^{\prime}\neq
j^{\prime}}\bigl{\\{}g=(g_{ij})\in{\overline{\mathcal{K}}_{r,C}}\>:\>|g_{i^{\prime}j^{\prime}}|\geq\varepsilon\bigr{\\}}$.
Therefore, it suffices to prove that for any given $1\leq i^{\prime}\neq
j^{\prime}\leq d$ we have

$\displaystyle\nu\bigl{(}\bigl{\\{}g=(g_{ij})\in{\overline{\mathcal{K}}_{r,C}}\>:\>|g_{i^{\prime}j^{\prime}}|\geq\varepsilon\bigr{\\}}\bigr{)}\ll_{d,C}r^{\varkappa_{d}+1}\log^{\lambda_{d}-1}\Bigl{(}\frac{1}{r}\Bigr{)}.$

This is shown by following the proof of Lemma 4.1 and using

$\int_{\varepsilon\leq|x|\leq
1}\int_{-1}^{1}{\chi_{\\{(x,y)\>:\>|xy|<Cr\\}}}\,\operatorname{d}\\!{y}\,\operatorname{d}\\!{x}\ll_{\varepsilon,C}r.$

Finally, the second relation in (4.2) follows from the first relation combined
with the lower bound in Theorem 2.1.

The remainder of this section is devoted to the proof of Proposition 4.2.

###  4.1. Bounds on diagonal entries

Recall that $U_{0}$ is the fixed fundamental domain for ${U/(\Gamma\cap U)}$
given in (2.5). The next lemma shows that if a lattice $\Lambda\in K_{r}$ has
a representative sufficiently close to some element in $U_{0}$, then the
diagonal entries of such a representative satisfy the desired bounds.

######  Lemma 4.3.

Let $g=(g_{ij})\in G$ and $r\in(0,\frac{1}{8})$, and assume that
$g{\mathbb{Z}}^{d}\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$ and
$\|g-u\|<\frac{1}{8}$ for some $u\in U_{0}$. Then

(4.3) $\displaystyle 0\leq g_{ii}-(1-r)\ll_{d}r,\qquad\forall\>1\leq i\leq d.$

###### Proof.

Note that it follows from $\|g-u\|<\frac{1}{8}$ and $u\in U_{0}$ that
$|g_{ij}|<\frac{5}{8}$ for all $1\leq i<j\leq d$, $|g_{ii}-1|<\frac{1}{8}$ for
all $1\leq i\leq d$, and $|g_{ij}|<\frac{1}{8}$ for all $1\leq j<i\leq d$. For
each $i$ we have $g{\text{\boldmath$e$}}_{i}\notin{\mathcal{C}}_{r}$, since
$g{\mathbb{Z}}^{d}\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$. Combining this with
the fact that $|g_{ji}|<\frac{5}{8}<1-r$ for all $j\neq i$, we conclude that
$|g_{ii}|\geq 1-r$. Since also $|g_{ii}-1|<\frac{1}{8}$, we must in fact have
$g_{ii}\geq 1-r$, i.e. we have proved the left inequality in (4.3).

Next, let $i\in\\{1,\ldots,d\\}$ be given, and let ${\mathcal{U}}$ be the open
box

$\displaystyle{\mathcal{U}}={\mathcal{I}}_{1}\times\cdots\times{\mathcal{I}}_{d},\quad\text{with
}\>\left\\{\begin{array}[]{ll}{\mathcal{I}}_{i}=\bigl{(}\frac{1}{2}(1-r),g_{ii}-\frac{1}{2}(1-r)\bigr{)},\\\\[5.0pt]
{\mathcal{I}}_{j}=\bigl{(}g_{ji}^{+}-\frac{3}{8},g_{ji}^{-}+\frac{3}{8}\bigr{)}\
(j\neq i),\end{array}\right.$

where $g_{ji}^{+}:=\max\\{g_{ji},0\\}$ and $g_{ji}^{-}:=\min\\{g_{ji},0\\}$.
Note that each interval ${\mathcal{I}}_{j}$ ($j\neq i$) has length
$|{\mathcal{I}}_{j}|>\frac{1}{8}$, since $|g_{ji}|<\frac{5}{8}$; furthermore
$|{\mathcal{I}}_{i}|=g_{ii}-(1-r)\geq 0$ (thus ${\mathcal{I}}_{i}$ and
${\mathcal{U}}$ are empty if $g_{ii}=1-r$, but otherwise non-empty).

We claim that ${\mathcal{U}}$ is disjoint from
${\mathfrak{C}}_{g{\mathbb{Z}}^{d},r}:=g{\mathbb{Z}}^{d}+\tfrac{1}{2}{\mathcal{C}}_{r}$.
Indeed, assume the opposite; then there is some ${\text{\boldmath$v$}}\in
g{\mathbb{Z}}^{d}$ such that
${\mathcal{U}}\cap({\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r})\neq\varnothing$.
We must have ${\text{\boldmath$v$}}\neq\mathbf{0}$ and
${\text{\boldmath$v$}}\neq g{\text{\boldmath$e$}}_{i}$, since ${\mathcal{U}}$
is, by construction, disjoint both from $\frac{1}{2}{\mathcal{C}}_{r}$ and
from $g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$. Pick a point
${\text{\boldmath$x$}}\in{\mathcal{U}}\cap({\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r})$.
It follows from
${\text{\boldmath$x$}}\in{\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$
and $\frac{1}{2}(1-r)>\frac{1}{4}$ that at least one of the points
${\text{\boldmath$x$}}-\frac{1}{4}{\text{\boldmath$e$}}_{i}$ or
${\text{\boldmath$x$}}+\frac{1}{4}{\text{\boldmath$e$}}_{i}$ also lies in
${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$. But we have

$\displaystyle{\text{\boldmath$x$}}-\tfrac{1}{4}{\text{\boldmath$e$}}_{i}\in\tfrac{1}{2}{\mathcal{C}}_{r},$

since
$x_{j}\in{\mathcal{I}}_{j}\subset\big{(}-\frac{1}{2}(1-r),\frac{1}{2}(1-r)\big{)}$
for all $j\neq i$ and
$x_{i}-\frac{1}{4}<g_{ii}-\frac{1}{2}(1-r)-\frac{1}{4}<\frac{1}{2}(1-r)$. We
also have

$\displaystyle{\text{\boldmath$x$}}+\tfrac{1}{4}{\text{\boldmath$e$}}_{i}\in
g{\text{\boldmath$e$}}_{i}+\tfrac{1}{2}{\mathcal{C}}_{r},$

since $x_{j}\in{\mathcal{I}}_{j}\subset
g_{ji}+\big{(}-\frac{1}{2}(1-r),\frac{1}{2}(1-r)\big{)}$ for all $j\neq i$ and
$x_{i}+\frac{1}{4}>\frac{1}{2}(1-r)+\frac{1}{4}>g_{ii}-\frac{1}{2}(1-r)$.
Hence we have arrived at a contradiction against the fact that
${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$ is disjoint from both
$\frac{1}{2}{\mathcal{C}}_{r}$ and
$g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$. This completes the
proof of the fact that ${\mathcal{U}}$ is disjoint from
${\mathfrak{C}}_{g{\mathbb{Z}}^{d},r}$.

Note also that ${\mathcal{U}}$ is contained in a translate of the cube
$(0,\frac{3}{4})^{d}$, since each interval ${\mathcal{I}}_{j}$ has length at
most $\frac{3}{4}$. Hence Lemma 2.5 applies, and yields that
$\operatorname{vol}({\mathcal{U}})<dr$. But we have noted that
$|{\mathcal{I}}_{j}|>\frac{1}{8}$ for each $j\neq i$, and
$|{\mathcal{I}}_{i}|=g_{ii}-(1-r)$; hence
$\operatorname{vol}({\mathcal{U}})\geq 8^{1-d}(g_{ii}-(1-r))$. Combining these
facts, we obtain the right bound in (4.3) with the implied constant
$8^{d-1}d$. ∎

###  4.2. A technical choice of lattice representatives

For any $\varepsilon>0$ let us write

$\displaystyle U_{\varepsilon}=\bigl{\\{}(u_{ij})\in
U\>:\>-\tfrac{1}{2}+4\varepsilon<u_{ij}\leq\tfrac{1}{2}+\varepsilon\text{ for
all }1\leq i<j\leq d\bigr{\\}}.$

######  Lemma 4.4.

There exist constants $0<a<1$ and $A>1$, which only depend on $d$, such that
the following holds: given any $u\in U$ and $\varepsilon\in(0,a)$, there exist
$\gamma\in\Gamma\cap U$ and $B\in[1,A]$ such that $u\gamma\in
U_{B\varepsilon}$.

###### Proof.

We will show that the statement of this lemma holds with $A:=4^{2^{d}}$ and
$a:=2^{-(3+2^{d+1})}$. Let us set

$\displaystyle U[t]:=\bigl{\\{}(u_{ij})\in
U\>:\>-\tfrac{1}{2}+t<u_{ij}\leq\tfrac{1}{2}+t\text{ for all }1\leq i<j\leq
d\bigr{\\}}\qquad(t\in{\mathbb{R}}).$

Note that $U_{0}=U[0]$, and for each $t$, $U[t]$ is a fundamental domain for
$U/(\Gamma\cap U)$.

For the given $u\in U$ and $\varepsilon\in(0,a)$, and for each
$k\in{\mathbb{Z}}_{\geq 0}$, we let $u^{(k)}$ be the unique element in
$u\Gamma\cap U[4^{k}\varepsilon]$, and set

$\displaystyle P^{(k)}=\sum_{j=2}^{d}P^{(k)}_{j},\quad\text{where}\quad
P^{(k)}_{j}:=\sum_{i=1}^{j-1}2^{i-1}\cdot{\chi_{(0,\infty)}(u_{ij}^{(k)})}.$

We now claim:

(4.4) $\displaystyle\forall\,k\in{\mathbb{Z}}_{\geq 0}:\quad
k<\log_{4}(\tfrac{1}{8\varepsilon})\text{ and }u^{(k)}\notin
U_{4^{k}\varepsilon}\quad\Longrightarrow\quad P^{(k)}<P^{(k+1)}.$

To prove (4.4), assume $0\leq k<\log_{4}(\frac{1}{8\varepsilon})$ and
$u^{(k)}\notin U_{4^{k}\varepsilon}$. Then $u^{(k+1)}\neq u^{(k)}$, since
otherwise $u^{(k)}$ would lie in the intersection $U[4^{k}\varepsilon]\cap
U[4^{k+1}\varepsilon]=U_{4^{k}\varepsilon}$. Let us denote by
${\text{\boldmath$u$}}_{j}^{(k)}$ the $j$th column vector of $u^{(k)}$. It
follows that
${\text{\boldmath$u$}}_{j}^{(k+1)}\neq{\text{\boldmath$u$}}_{j}^{(k)}$ for at
least one $j\in\\{2,\ldots,d\\}$, and for each such index $j$ we can argue as
follows: Since $u^{(k)},u^{(k+1)}\in u\Gamma\cap U=u(\Gamma\cap U)$, we have
$\bm{u}_{j}^{(k+1)}=\bm{u}_{j}^{(k)}+\sum_{1\leq i<j}m_{i}\bm{u}_{i}^{(k)}$
for some $m_{1},\ldots,m_{j-1}\in{\mathbb{Z}}$. Because of
${\text{\boldmath$u$}}_{j}^{(k+1)}\neq{\text{\boldmath$u$}}_{j}^{(k)}$, there
exists $i\in\\{1,\ldots,j\\}$ such that $m_{i}\neq 0$; let us fix $i$ to be
the largest such index. Thus $u_{ij}^{(k+1)}=u_{ij}^{(k)}+m_{i}$; in
particular $\left|u_{ij}^{(k+1)}-u_{ij}^{(k)}\right|\geq 1$. On the other
hand, note that
$u_{ij}^{(k)}\in\left(-\frac{1}{2}+4^{k}\varepsilon,\frac{1}{2}+4^{k}\varepsilon\right]$,
$u_{ij}^{(k+1)}\in\left(-\frac{1}{2}+4^{k+1}\varepsilon,\frac{1}{2}+4^{k+1}\varepsilon\right]$
and
$\left(\frac{1}{2}+4^{k+1}\varepsilon)-(-\frac{1}{2}+4^{k}\varepsilon\right)=1+3\cdot
4^{k}\varepsilon<2$ (since $k<\log_{4}(\frac{1}{8\varepsilon})$). Hence we
must have $m_{i}=1$, implying that

${\text{\boldmath$u$}}_{j}^{(k+1)}\in{\text{\boldmath$u$}}_{j}^{(k)}+{\text{\boldmath$u$}}_{i}^{(k)}+{\mathbb{Z}}{\text{\boldmath$u$}}_{i-1}^{(k)}+\cdots+{\mathbb{Z}}{\text{\boldmath$u$}}_{1}^{(k)}.$

Thus $u_{i^{\prime}j}^{(k+1)}=u_{i^{\prime}j}^{(k)}$ for all $i^{\prime}>i$,
and $u_{ij}^{(k+1)}=u_{ij}^{(k)}+1>0$ while
$u_{ij}^{(k)}\leq-\frac{1}{2}+4^{k+1}\varepsilon<0$ (again since
$k<\log_{4}(\frac{1}{8\varepsilon})$). It follows that
$P_{j}^{(k+1)}-P_{j}^{(k)}\geq 2^{i-1}-\sum_{i^{\prime}<i}2^{i^{\prime}-1}=1$.
On the other hand we clearly have $P_{j}^{(k+1)}=P_{j}^{(k)}$ for each
$j\in\\{2,\ldots,d\\}$ such that
${\text{\boldmath$u$}}_{j}^{(k+1)}={\text{\boldmath$u$}}_{j}^{(k)}$. Hence
$\sum_{j=2}^{d}P^{(k+1)}_{j}>\sum_{j=2}^{d}P^{(k)}_{j}$, i.e.
$P^{(k+1)}>P^{(k)}$. This finishes the proof of (4.4).

Next note that by definition, for each $k$, $P^{(k)}$ is a non-negative
integer satisfying

$P^{(k)}\leq\sum_{j=2}^{d}\sum_{i=0}^{j-2}2^{i}<2^{d}.$

This implies that we cannot have $P^{(k)}<P^{(k+1)}$ for all
$k\in\\{0,1,\ldots,2^{d}-1\\}$. But our assumption on $\varepsilon$ implies
that $\log_{4}(\frac{1}{8\varepsilon})>2^{d}$ (recall that
$0<\varepsilon<a=2^{-(3+2^{d+1})}$); hence it now follows from (4.4) that
$u^{(k)}\in U_{4^{k}\varepsilon}$ for at least one
$k\in\\{0,1,\ldots,2^{d}-1\\}$. Furthermore, for such $k$ we have
$4^{k}<4^{2^{d}}=A$ and $u^{(k)}\in u(\Gamma\cap U)$ by construction. Hence
the lemma is proved. ∎

Using Lemma 4.4, Corollary 2.4 and a compactness argument we have the
following technical lemma, which gives us a good choice of lattice
representatives for lattices in $K_{r_{0}}$ for some small $r_{0}>0$.

######  Lemma 4.5.

There exist constants $0<a<1$ and $A>1$, which only depend on $d$, such that
the following holds: for any ${\varepsilon_{0}}\in(0,a)$ there exists
$r_{0}>0$ such that for every $\Lambda\in X_{d}$ satisfying
$\Lambda\cap{\mathcal{C}}_{r_{0}}=\\{\mathbf{0}\\}$, there exist $g\in G$,
$w\in W$, $B\in[1,A]$ and $u\in U_{B{\varepsilon_{0}}}$ such that
$\Lambda=g{\mathbb{Z}}^{d}$ and $\|g-wuw^{-1}\|<{\varepsilon_{0}}$.

###### Proof.

Let $a$ and $A$ be as in Lemma 4.4; we will prove that the statement of the
lemma holds with these $a,A$. The proof is by contradiction; thus we assume
that the statement of the lemma is false, i.e. we assume that there exist some
${\varepsilon_{0}}\in(0,a)$ and a sequence $r_{1}>r_{2}>\cdots$ in $(0,1)$
with $r_{j}\to 0$, and a corresponding sequence
$\Lambda_{1},\Lambda_{2},\ldots$ in $X_{d}$, such that
$\Lambda_{j}\cap{\mathcal{C}}_{r_{j}}=\\{\mathbf{0}\\}$ for each $j$, and
furthermore, for each $j$ we have that there do not exist any $g\in G$, $w\in
W$, $B\in[1,A]$ and $u\in U_{B{\varepsilon_{0}}}$ satisfying
$\Lambda_{j}=g{\mathbb{Z}}^{d}$ and $\|g-wuw^{-1}\|<{\varepsilon_{0}}$. Now
for every $j$ we have ${\mathcal{C}}_{r_{1}}\subset{\mathcal{C}}_{r_{j}}$, and
thus $\Lambda_{j}\cap{\mathcal{C}}_{r_{1}}=\\{\mathbf{0}\\}$. Hence by
Mahler’s Compactness Theorem, after passing to a subsequence we may assume
that $\Lambda_{j}$ tends to a limit point in $X_{d}$. Let us call this limit
point $\Lambda_{0}$; thus $\Lambda_{j}\to\Lambda_{0}$ in $X_{d}$ as
$j\to\infty$. Let us also fix a representative $g_{0}\in G$ such that
$\Lambda_{0}=g_{0}{\mathbb{Z}}^{d}$.

Recall that the standard topology on $X_{d}=G/\Gamma$ is given by the metric

$\displaystyle\operatorname{dist}(g\Gamma,g^{\prime}\Gamma):=\inf\\{d(g\gamma,g^{\prime})\>:\>\gamma\in\Gamma\\}\qquad(g,g^{\prime}\in
G),$

where $d(\cdot,\cdot)$ is any fixed right $G$-invariant Riemannian metric on
$G$. Hence the fact that $\Lambda_{j}$ converges to
$\Lambda_{0}=g_{0}{\mathbb{Z}}^{d}$ implies that there exist
$g_{1},g_{2},\ldots\in G$ such that $\Lambda_{j}=g_{j}{\mathbb{Z}}^{d}$ for
each $j$ and $d(g_{j},g_{0})\to 0$ as $j\to\infty$.

Using $\Lambda_{j}\cap{\mathcal{C}}_{r_{j}}=\\{\mathbf{0}\\}$ for each $j$,
and $r_{j}\to 0$, we claim that

(4.5) $\displaystyle\Lambda_{0}\cap(-1,1)^{d}=\\{\mathbf{0}\\}.$

Indeed, assume the opposite; this means that there exists some
${\text{\boldmath$m$}}\in{\mathbb{Z}}^{d}\smallsetminus\\{\bm{0}\\}$ such that
$g_{0}{\text{\boldmath$m$}}$ belongs to $(-1,1)^{d}$, i.e.
$\|g_{0}{\text{\boldmath$m$}}\|<1$. Set
$r:=\frac{1}{2}(1-\|g_{0}{\text{\boldmath$m$}}\|)$; then
$g_{0}{\text{\boldmath$m$}}\in{\mathcal{C}}_{r}$. We have
$g_{j}{\text{\boldmath$m$}}\to g_{0}{\text{\boldmath$m$}}$ as $j\to\infty$,
since $d(g_{j},g_{0})\to 0$; hence for all sufficiently large $j$ we have
$g_{j}{\text{\boldmath$m$}}\in{\mathcal{C}}_{r}$ (since ${\mathcal{C}}_{r}$ is
open). Also for all sufficiently large $j$ we have $r_{j}<r$. Hence there
exists some $j$ for which $r_{j}<r$ and
$g_{j}{\text{\boldmath$m$}}\in{\mathcal{C}}_{r}\subset{\mathcal{C}}_{r_{j}}$.
This contradicts the fact that
$\Lambda_{j}\cap{\mathcal{C}}_{r_{j}}=\\{\mathbf{0}\\}$ for all $j$. Hence
(4.5) is proved.

It follows from (4.5) and Corollary 2.4 that
$\Lambda_{0}=wu^{\prime}w^{-1}{\mathbb{Z}}^{d}$ for some $w\in W$ and
$u^{\prime}\in U_{0}$. Next, by Lemma 4.4 (and since ${\varepsilon_{0}}<a$),
there exist $\gamma\in\Gamma\cap U$ and $B\in[1,A]$ such that
$u:=u^{\prime}\gamma\in U_{B{\varepsilon_{0}}}$. Using
$w^{-1}{\mathbb{Z}}^{d}={\mathbb{Z}}^{d}$ and
$\gamma^{-1}{\mathbb{Z}}^{d}={\mathbb{Z}}^{d}$, we then have
$\Lambda_{0}=wuw^{-1}{\mathbb{Z}}^{d}$. Hence $wuw^{-1}=g_{0}\gamma_{0}$ for
some $\gamma_{0}\in\Gamma$. Now
$d(g_{j}\gamma_{0},wuw^{-1})=d(g_{j}\gamma_{0},g_{0}\gamma_{0})=d(g_{j},g_{0})\to
0$ as $j\to\infty$. This implies that every matrix entry of $g_{j}\gamma_{0}$
tends to the corresponding entry of $wuw^{-1}$, i.e. we have
$\|g_{j}\gamma_{0}-wuw^{-1}\|\to 0$ as $j\to\infty$. In particular there
exists some $j$ such that $\|g_{j}\gamma_{0}-wuw^{-1}\|<{\varepsilon_{0}}$.
Now we have a contradiction against our previous assumption; namely for our
chosen $j$, if we set $g:=g_{j}\gamma_{0}$ then
$\Lambda_{j}=g{\mathbb{Z}}^{d}$ and $\|g-wuw^{-1}\|<{\varepsilon_{0}}$. This
completes the proof of Lemma 4.5. ∎

######  Remark 11.

By a similar compactness argument as in the proof of Lemma 4.5, one can also
prove a more basic statement: for any $\varepsilon_{0}>0$ there exists
$r_{0}>0$ such that every $\Lambda\in K_{r_{0}}$ has a representative $g\in G$
(i.e. $\Lambda=g{\mathbb{Z}}^{d}$) satisfying $\|g-wuw^{-1}\|<\varepsilon_{0}$
for some $w\in W$ and $u\in U_{0}$. The purpose of the choice of the more
technical lattice representatives in Lemma 4.5 (with $U_{B\varepsilon_{0}}$ in
place of $U_{0}$) is to ensure the following property, which is a key
ingredient in the proof of the important Lemma 4.6 below: for any
$g=(g_{ij})\in G$ satisfying $\|g-u\|<\varepsilon_{0}$ for some $u\in
U_{B\varepsilon_{0}}$, and any $1\leq i<j\leq d$, we have (cf. (4.8))

$\displaystyle|g_{ki}-g_{kj}|<1-B\varepsilon_{0}\quad\textrm{for all
$k\notin\\{i,j\\}$}.$

A crucial consequence of this is that if
$g{\text{\boldmath$e$}}_{i}-g{\text{\boldmath$e$}}_{j}\notin\mathcal{C}_{r}$
for some $r<B\varepsilon_{0}$, then either $|g_{ii}-g_{ij}|>1-r$ or
$|g_{ji}-g_{jj}|>1-r$ must hold.

###  4.3. Bounds on off-diagonal symmetric pairs

The next lemma shows that if a lattice $\Lambda$ in $K_{r}$ has a
representative as in Lemma 4.5, then its entries satisfy the desired bounds
for proving Proposition 4.2.

######  Lemma 4.6.

Let $A>1$, $0<\varepsilon_{0}<(16A)^{-1}$ and
$0<r_{0}<\tfrac{1}{2}\varepsilon_{0}$. Let $g=(g_{ij})\in G$ and
$r\in(0,r_{0})$, and assume that
$g{\mathbb{Z}}^{d}\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$ and that there exist
$B\in[1,A]$ and $u\in U_{B\varepsilon_{0}}$ such that
$\|g-u\|<\varepsilon_{0}$. Then

(4.6) $\displaystyle 0\leq g_{ii}-(1-r)\ll_{d}r\quad\text{for all }\>1\leq
i\leq d,$

and

(4.7) $\displaystyle|g_{ij}g_{ji}|\ll_{d,\varepsilon_{0}}r\quad\text{for all
}\>1\leq i<j\leq d.$

###### Proof.

Let us write $\varepsilon=B\varepsilon_{0}$. Note that
$\|g-u\|<\varepsilon_{0}\leq\varepsilon$ and $u\in U_{\varepsilon}$ together
imply that

(4.8)
$\displaystyle-\tfrac{1}{2}+3\varepsilon<g_{ij}<\tfrac{1}{2}+2\varepsilon\quad\text{
and }\quad|g_{ji}|<\varepsilon,\quad\forall 1\leq i<j\leq d.$

It also follows that $|g_{ii}-1|<\varepsilon$ for all $i$; however these
inequalities may be sharpened using Lemma 4.3. Indeed, we have
$\varepsilon<\frac{1}{16}$ since $A\varepsilon_{0}<\frac{1}{16}$; hence the
above inequalities imply that $\|g-u^{\prime}\|<\frac{1}{8}$ for some
$u^{\prime}\in U_{0}$. Hence Lemma 4.3 applies, yielding that (4.6) holds.

Now let $1\leq i<j\leq d$ be given. We separate the proof of (4.7) into three
cases. (Note that (4.7) holds trivially if $g_{ij}=0$ or $g_{ji}=0$; hence we
may without loss of generality assume $g_{ij}g_{ji}\neq 0$.)

Case I: $g_{ij}>0$ and $g_{ji}>0$. In this case we will build the proof on the
fact that
$g{\text{\boldmath$e$}}_{i}-g{\text{\boldmath$e$}}_{j}\notin{\mathcal{C}}_{r}$,
which holds since $g{\mathbb{Z}}^{d}\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$.
Using (4.8) and $r<r_{0}<2r_{0}\leq\varepsilon<\frac{1}{16}$, it follows that
$|g_{ki}-g_{kj}|<1-\varepsilon<1-r$ for all $k\notin\\{i,j\\}$. Hence we must
have either $|g_{ii}-g_{ij}|\geq 1-r$ or $|g_{ji}-g_{jj}|\geq 1-r$. If
$|g_{ii}-g_{ij}|\geq 1-r$, then because of $g_{ii}\geq 1-r$ and
$0<g_{ij}<\frac{1}{2}+2\varepsilon$ it follows that $g_{ii}-g_{ij}\geq 1-r$,
and so by (4.6), $0<g_{ij}\ll r$. Similarly if $|g_{ji}-g_{jj}|\geq 1-r$ then
$0<g_{ji}\ll r$. In both cases, it follows that (4.7) holds for our $i,j$.

Case II: $g_{ij}<0$ and $g_{ji}<0$. In this case we will prove the desired
bound by proving the stronger assertion that either $|g_{ij}|\leq Cr$ or
$|g_{ji}|\leq Cr$, with $C:=12d(\varepsilon_{0}/8)^{2-d}$. Assume the
opposite, i.e. assume that

(4.9) $\displaystyle g_{ij}<-Cr\qquad\text{and}\qquad
g_{ji}<-Cr\qquad(C:=12d(\varepsilon_{0}/8)^{2-d}).$

We will prove that this leads to a contradiction.

Set ${\mathcal{J}}_{r}:=\bigl{(}-\tfrac{1}{2}(1-r),\tfrac{1}{2}(1-r)\bigr{)}$,
so that $\frac{1}{2}{\mathcal{C}}_{r}={\mathcal{J}}_{r}^{d}$. For each
$k\notin\\{i,j\\}$ we introduce the following open interval:

(4.10)
$\displaystyle\widetilde{{\mathcal{I}}}_{k}:={\mathcal{J}}_{r}\cap(g_{ki}+{\mathcal{J}}_{r})\cap(g_{kj}+{\mathcal{J}}_{r}).$

Using $|g_{ki}|,|g_{kj}|,|g_{ki}-g_{kj}|<1-\varepsilon$ (see (4.8)) it follows
that $\widetilde{{\mathcal{I}}}_{k}$ has length
$|\widetilde{{\mathcal{I}}}_{k}|>(1-r)-(1-\varepsilon)>\tfrac{1}{2}\varepsilon_{0}$.
Define ${\mathcal{I}}_{k}\subset\widetilde{{\mathcal{I}}}_{k}$ to be the open
interval of length $\frac{1}{4}\varepsilon_{0}$ with the same center as
$\widetilde{{\mathcal{I}}}_{k}$. Let us also set
${\mathcal{I}}_{i}=\bigl{(}\frac{1}{2}(1-r)+g_{ij},\frac{1}{2}(1-r)\bigr{)}$
and ${\mathcal{I}}_{j}=\bigl{(}g_{jj}-\frac{1}{2}(1-r),\frac{3}{4}\bigr{)}$.
Then by construction,

(4.11)
$\displaystyle{\mathcal{I}}_{i}\subset{\mathcal{J}}_{r},\qquad{\mathcal{I}}_{i}\cap(g_{ii}+{\mathcal{J}}_{r})=\varnothing,\qquad{\mathcal{I}}_{i}\cap(g_{ij}+{\mathcal{J}}_{r})=\varnothing,$

and

(4.12) $\displaystyle{\mathcal{I}}_{j}\subset
g_{jj}+{\mathcal{J}}_{r},\qquad{\mathcal{I}}_{j}\cap{\mathcal{J}}_{r}=\varnothing.$

Furthermore, $|{\mathcal{I}}_{i}|=|g_{ij}|$ and
$\frac{1}{8}<|{\mathcal{I}}_{j}|<\frac{1}{2}$. Now let ${\mathcal{U}}$ be the
open box ${\mathcal{U}}={\mathcal{I}}_{1}\times\cdots\times{\mathcal{I}}_{d}$.
Then
$\operatorname{vol}({\mathcal{U}})>\frac{1}{8}|g_{ij}|\cdot(\frac{1}{4}\varepsilon_{0})^{d-2}>dr$,
where we used the first part of our assumption (4.9). Note also that
$|{\mathcal{I}}_{k}|<\frac{3}{4}$ for all $k$. Hence by Lemma 2.5,
${\mathcal{U}}\cap{\mathfrak{C}}_{g{\mathbb{Z}}^{d},r}\neq\varnothing$, i.e.
there exists some ${\text{\boldmath$v$}}\in g{\mathbb{Z}}^{d}$ such that

$\displaystyle{\mathcal{U}}\cap({\text{\boldmath$v$}}+\tfrac{1}{2}{\mathcal{C}}_{r})\neq\varnothing.$

It follows from the disjointness relations in (4.11) and (4.12) that
${\mathcal{U}}$ is disjoint from the three cubes
$g{\text{\boldmath$e$}}_{i}+\tfrac{1}{2}{\mathcal{C}}_{r}$,
$g{\text{\boldmath$e$}}_{j}+\tfrac{1}{2}{\mathcal{C}}_{r}$ and
$\tfrac{1}{2}{\mathcal{C}}_{r}$; hence
${\text{\boldmath$v$}}\notin\\{\mathbf{0},g{\text{\boldmath$e$}}_{j},g{\text{\boldmath$e$}}_{i}\\}$.
Let us fix a point
${\text{\boldmath$y$}}\in{\mathcal{U}}\cap({\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r})$.
Then the line ${\text{\boldmath$y$}}+{\mathbb{R}}{\text{\boldmath$e$}}_{i}$
goes through both the cubes
${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$ and
$g{\text{\boldmath$e$}}_{j}+\frac{1}{2}{\mathcal{C}}_{r}$ (the latter holds
since ${\mathcal{I}}_{k}\subset g_{kj}+{\mathcal{J}}_{r}$ for all $k\neq i$;
see (4.10) and (4.12)); hence since these two cubes are disjoint, we must have
$v_{i}\geq g_{ij}+1-r$. It also follows from $y_{i}\in
v_{i}+{\mathcal{J}}_{r}$ and $y_{i}\in{\mathcal{I}}_{i}$ that
$v_{i}<y_{i}+\frac{1}{2}(1-r)<1-r$. In summary:

(4.13) $\displaystyle g_{ij}+1-r\leq v_{i}<1-r.$

Similarly, using the fact that the line
${\text{\boldmath$y$}}+{\mathbb{R}}{\text{\boldmath$e$}}_{j}$ goes through the
two disjoint cubes ${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$ and
$\frac{1}{2}{\mathcal{C}}_{r}$, and also using $y_{j}\in
v_{j}+{\mathcal{J}}_{r}$ and $y_{j}\in{\mathcal{I}}_{j}$, it follows that

(4.14) $\displaystyle 1-r\leq v_{j}<y_{j}+\tfrac{1}{2}(1-r)<\tfrac{5}{4}.$

Next, for each $k\notin\\{i,j\\}$, since $y_{k}\in{\mathcal{I}}_{k}$ and
$y_{k}\in v_{k}+{\mathcal{J}}_{r}$, we have that both of the intervals
$(y_{k}-\tfrac{1}{8}\varepsilon_{0},y_{k}]$ and
$[y_{k},y_{k}+\tfrac{1}{8}\varepsilon_{0})$ are contained in
$\widetilde{{\mathcal{I}}}_{k}$, and at least one of them is contained in
$v_{k}+{\mathcal{J}}_{r}$; hence there exists an open subinterval
${\mathcal{I}}_{k}^{\prime}$ of
$\widetilde{{\mathcal{I}}}_{k}\cap(v_{k}+{\mathcal{J}}_{r})$ of length
$\frac{1}{8}\varepsilon_{0}$. Let us also set
${\mathcal{I}}_{i}^{\prime}=(\tfrac{2}{3},\tfrac{3}{4})$ and
${\mathcal{I}}_{j}^{\prime}=\bigl{(}g_{ji}+\tfrac{1}{2}(1-r),v_{j}-\tfrac{1}{2}(1-r)\bigr{)}$,
and then let ${\mathcal{U}}^{\prime}$ be the open box
${\mathcal{U}}^{\prime}={\mathcal{I}}_{1}^{\prime}\times\cdots\times{\mathcal{I}}_{d}^{\prime}$.
Using (4.14) we have $|{\mathcal{I}}_{j}^{\prime}|\geq-g_{ji}$; hence
$\operatorname{vol}({\mathcal{U}}^{\prime})\geq(\frac{1}{8}\varepsilon_{0})^{d-2}\cdot\frac{1}{12}\cdot|g_{ji}|>dr$,
where we used the second part of our assumption (4.9). Note also that
$|{\mathcal{I}}_{k}^{\prime}|<\frac{3}{4}$ for all $k$; for $k=j$ this uses
$v_{j}<\frac{5}{4}$ (see (4.14)) and $g_{ji}>-\varepsilon$ (see (4.8)). Hence,
by Lemma 2.5,
${\mathcal{U}}^{\prime}\cap{\mathfrak{C}}_{g{\mathbb{Z}}^{d},r}\neq\varnothing$,
i.e. there exists some ${\text{\boldmath$v$}}^{\prime}\in g{\mathbb{Z}}^{d}$
such that

$\displaystyle{\mathcal{U}}^{\prime}\cap({\text{\boldmath$v$}}^{\prime}+\tfrac{1}{2}{\mathcal{C}}_{r})\neq\varnothing.$

Choose a point
${\text{\boldmath$z$}}\in{\mathcal{U}}^{\prime}\cap({\text{\boldmath$v$}}^{\prime}+\tfrac{1}{2}{\mathcal{C}}_{r})$.
Note that for every $k\notin\\{i,j\\}$ we have
${\mathcal{I}}_{k}^{\prime}\subset(g_{ki}+{\mathcal{J}}_{r})\cap(v_{k}+{\mathcal{J}}_{r})$
by construction. Furthermore, using $1-r\leq g_{ii}<1+\varepsilon$ we have
${\mathcal{I}}_{i}^{\prime}\subset g_{ii}+{\mathcal{J}}_{r}$, and using (4.13)
we have ${\mathcal{I}}_{i}^{\prime}\subset v_{i}+{\mathcal{J}}_{r}$. Hence the
line ${\text{\boldmath$z$}}+{\mathbb{R}}{\text{\boldmath$e$}}_{j}$ goes
through of the cubes $g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$
and ${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$. Of course this line
also goes through the cube
${\text{\boldmath$v$}}^{\prime}+\tfrac{1}{2}{\mathcal{C}}_{r}$. Note also that
$z_{j}\in{\mathcal{I}}_{j}^{\prime}$, and by construction,
${\mathcal{I}}_{j}^{\prime}$ is disjoint from and lies between the two
intervals $g_{ji}+{\mathcal{J}}_{r}$ and $v_{j}+{\mathcal{J}}_{r}$. Hence we
must have
${\text{\boldmath$v$}}^{\prime}\notin\\{g{\text{\boldmath$e$}}_{i},{\text{\boldmath$v$}}\\}$;
thus the three cubes
$g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$,
${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$ and
${\text{\boldmath$v$}}^{\prime}+\frac{1}{2}{\mathcal{C}}_{r}$ are pairwise
disjoint, and the two intervals $g_{ji}+{\mathcal{J}}_{r}$ and
$v_{j}+{\mathcal{J}}_{r}$ must lie at a distance $\geq 1-r$ from each other.
However, this is impossible, since
$v_{j}-g_{ji}<\frac{5}{4}+\varepsilon<2(1-r)$. This completes the proof in
Case II.

Case III: $g_{ij}g_{ji}<0$. If $g_{ji}<0$ then let us swap the values of $i$
and $j$; thus from now on we have $g_{ji}>0$ and $g_{ij}<0$, but either $i<j$
or $i>j$. If $g_{ji}\leq g_{jj}-(1-r)$, then $g_{ji}\ll r$ by (4.6), and so
(4.7) holds for our $i,j$. Hence from now on we may assume
$g_{ji}>g_{jj}-(1-r)$. Now set:

$\displaystyle{\mathcal{I}}_{i}=\bigl{(}g_{ij}+\tfrac{1}{2}(1-r),g_{ii}-\tfrac{1}{2}(1-r)\bigr{)};$
$\displaystyle{\mathcal{I}}_{j}=\bigl{(}g_{jj}-\tfrac{1}{2}(1-r),g_{ji}+\tfrac{1}{2}(1-r)\bigr{)};$
$\displaystyle\text{and
}\>\widetilde{{\mathcal{I}}}_{k}=(g_{ki}+{\mathcal{J}}_{r})\cap(g_{kj}+{\mathcal{J}}_{r})\quad\text{for
$k\notin\\{i,j\\}$}.$

These are non-empty intervals. Indeed, ${\mathcal{I}}_{i}$ is non-empty since
$g_{ii}\geq 1-r$ and $g_{ij}<0$; ${\mathcal{I}}_{j}$ is non-empty because of
our assumption $g_{ji}>g_{jj}-(1-r)$, and for each $k\notin\\{i,j\\}$, it
follows from $|g_{ki}-g_{kj}|<1-\varepsilon$ (see (4.8)) that
$\widetilde{{\mathcal{I}}}_{k}$ is non-empty with
$|\widetilde{{\mathcal{I}}}_{k}|>(1-r)-(1-\varepsilon)\geq\tfrac{1}{2}\varepsilon_{0}$.
Now for each $k\notin\\{i,j\\}$ we choose an open subinterval
${\mathcal{I}}_{k}$ of $\widetilde{{\mathcal{I}}}_{k}$ of length
$\min\big{\\{}\frac{3}{4},|\widetilde{{\mathcal{I}}}_{k}|\big{\\}}$, and then
define ${\mathcal{U}}$ to be the open box
${\mathcal{U}}={\mathcal{I}}_{1}\times\cdots\times{\mathcal{I}}_{d}$. We claim
that

(4.15)
$\displaystyle{\mathcal{U}}\cap{\mathfrak{C}}_{g{\mathbb{Z}}^{d},r}=\varnothing.$

Indeed, assume the opposite; then there is some ${\text{\boldmath$v$}}\in
g{\mathbb{Z}}^{d}$ with
${\mathcal{U}}\cap\left({\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}\right)\neq\varnothing$.
By construction, ${\mathcal{I}}_{i}$ is disjoint from both the intervals
$g_{ij}+{\mathcal{J}}_{r}$ and $g_{ii}+{\mathcal{J}}_{r}$; hence
${\mathcal{U}}$ is disjoint from the two cubes
$g{\text{\boldmath$e$}}_{j}+\frac{1}{2}{\mathcal{C}}_{r}$,
$g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$, and thus
${\text{\boldmath$v$}}\notin\\{g{\text{\boldmath$e$}}_{j},g{\text{\boldmath$e$}}_{i}\\}$.
Let $x$ be a point in
${\mathcal{U}}\cap\left({\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}\right)$.
Then $x_{i}\in{\mathcal{I}}_{i}\cap(v_{i}+{\mathcal{J}}_{r})$. Note also that
the three intervals $g_{ij}+{\mathcal{J}}_{r}$, ${\mathcal{I}}_{i}$,
$g_{ii}+{\mathcal{J}}_{r}$ are adjacent to each other in this order along the
real line, with the length of ${\mathcal{I}}_{i}$ being

$\displaystyle|{\mathcal{I}}_{i}|=g_{ii}-g_{ij}-(1-r)<1+\varepsilon+(\tfrac{1}{2}-3\varepsilon)-(1-r)=\tfrac{1}{2}-2\varepsilon+r<\tfrac{1}{2}<1-r.$

But $v_{i}+{\mathcal{J}}_{r}$ has length $1-r$; hence there exists a number
$x_{i}^{\prime}\in v_{i}+{\mathcal{J}}_{r}$ lying either in
$g_{ii}+{\mathcal{J}}_{r}$ or $g_{ij}+{\mathcal{J}}_{r}$. Noticing also that
$x_{k}\in{\mathcal{I}}_{k}\subset(g_{ki}+{\mathcal{J}}_{r})\cap(g_{kj}+{\mathcal{J}}_{r})$
for all $k\neq i$, it now follows that the point
${\text{\boldmath$x$}}+(x_{i}^{\prime}-x_{i}){\text{\boldmath$e$}}_{i}$ lies
in the cube ${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$ and also in
one of the two cubes $g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$
or $g{\text{\boldmath$e$}}_{j}+\frac{1}{2}{\mathcal{C}}_{r}$. This is a
contradiction against the fact that
${\text{\boldmath$v$}}+\frac{1}{2}{\mathcal{C}}_{r}$ is disjoint from both
$g{\text{\boldmath$e$}}_{i}+\frac{1}{2}{\mathcal{C}}_{r}$ and
$g{\text{\boldmath$e$}}_{j}+\frac{1}{2}{\mathcal{C}}_{r}$; hence we have
completed the proof of (4.15).

We have $|{\mathcal{I}}_{k}|<\frac{3}{4}$ for all $1\leq k\leq d$; hence Lemma
2.5 applies, giving $\operatorname{vol}({\mathcal{U}})<dr$. But
$|{\mathcal{I}}_{k}|\geq\frac{1}{2}\varepsilon_{0}$ for all
$k\notin\\{i,j\\}$, and $|{\mathcal{I}}_{i}|\geq|g_{ij}|$; thus

$\displaystyle|g_{ij}|\bigl{(}g_{ji}-g_{jj}+1-r\bigr{)}=|g_{ij}||{\mathcal{I}}_{j}|\ll\operatorname{vol}({\mathcal{U}})\ll
r.$

Furthermore, $|g_{ij}|\big{(}g_{jj}-(1-r)\big{)}\ll r$ by (4.6). Adding the
last two bounds, we conclude that (4.7) holds for our $i,j$. ∎

###  4.4. Proof of Proposition 4.2

Finally, we can give the

###### Proof of Proposition 4.2.

Choose $0<a<1$ and $A>1$ as in Lemma 4.5. Fix a number $0<\beta<\frac{1}{16}$
so small that for every matrix $g\in G$ which has distance (w.r.t.
$\|\cdot\|$) less than $\beta$ to a matrix in $U_{0}$, we have
$|\det\tilde{g}-1|<\frac{1}{2}$, where $\tilde{g}$ is the top left
$(d-1)\times(d-1)$ block of $g$. Fix a number
$0<\varepsilon_{0}<\min\bigl{\\{}a,\beta/(2A)\bigr{\\}}$, and for this
$\varepsilon_{0}$, take $r_{0}>0$ as in Lemma 4.5. Note that the defining
property of $r_{0}$ trivially remains valid if we decrease $r_{0}$; hence we
may assume that $0<r_{0}<\frac{1}{2}\varepsilon_{0}$. Let $C>0$ be the maximum
of the implied constants in the two “$\ll$” bounds in Lemma 4.6, for our fixed
$d$ and $\varepsilon_{0}$.

Now let $r\in(0,r_{0})$ and $\Lambda\in K_{r}$ be given. This means that
$\Lambda\cap{\mathcal{C}}_{r}=\\{\mathbf{0}\\}$, and a fortiori,
$\Lambda\cap{\mathcal{C}}_{r_{0}}=\\{\mathbf{0}\\}$. Hence by our choice of
$A,a,\varepsilon_{0},r_{0}$ (see the statement of Lemma 4.5), there exist
$g^{\prime}\in G$, $w\in W$, $B\in[1,A]$ and $u\in U_{B{\varepsilon_{0}}}$
such that $\Lambda=g^{\prime}{\mathbb{Z}}^{d}$ and
$\|g^{\prime}-wuw^{-1}\|<{\varepsilon_{0}}$. Note that the norm $\|\cdot\|$ is
preserved by left and right multiplication by elements from $W$; hence letting
$g=(g_{ij}):=w^{-1}g^{\prime}w$ we have $\|g-u\|<\varepsilon_{0}$, and also
$g{\mathbb{Z}}^{d}\cap{\mathcal{C}}_{r}=w^{-1}(g^{\prime}{\mathbb{Z}}^{d}\cap{\mathcal{C}}_{r})=\\{\mathbf{0}\\}$
(this is true since $w{\mathbb{Z}}^{d}={\mathbb{Z}}^{d}$ and
${\mathcal{C}}_{r}=w^{-1}{\mathcal{C}}_{r}$). Hence by Lemma 4.6, and by our
choice of $C$, we have $0\leq g_{ii}-(1-r)\leq Cr$ for all $i$ and
$|g_{ij}g_{ji}|\leq Cr$ for all $i\neq j$. Furthermore, it follows from $u\in
U_{B{\varepsilon_{0}}}$ and $\|g-u\|<\varepsilon_{0}$ that
$\|g-u^{\prime}\|<(B+1)\varepsilon_{0}$ for some $u^{\prime}\in U_{0}$; hence
a fortiori $\|g-u^{\prime}\|<2A\varepsilon_{0}<\beta$, which implies that
$|\det\tilde{g}-1|<\frac{1}{2}$ by our choice of $\beta$. It also follows that
$|g_{ij}|<\frac{1}{2}+2A\varepsilon_{0}<1$ for all $i\neq j$. Hence
$g\in{\overline{\mathcal{K}}_{r,C}}$, and thus $g^{\prime}=wgw^{-1}\in
w\,{\overline{\mathcal{K}}_{r,C}}\,w^{-1}$ and
$\Lambda=g^{\prime}{\mathbb{Z}}^{d}\in\pi(w\,{\overline{\mathcal{K}}_{r,C}}\,w^{-1})$,
finishing the proof. ∎

##  5\. Measure estimates of the thickenings

Fix $m,n\in{\mathbb{N}}$ and let $d=m+n$. Let $\bm{\alpha}\in{\mathbb{R}}^{m}$
and $\bm{\beta}\in{\mathbb{R}}^{n}$ be two fixed weight vectors as in Theorem
1.2. As mentioned in Remark 9, in order to incorporate the case of general
weights, we need to consider a more general one-parameter subgroup of $G$
associated to $\bm{\alpha}$ and $\bm{\beta}$. Explicitly, for any
$s\in{\mathbb{R}}$ let us define

(5.1) $\displaystyle
g_{s}=g_{s}^{\bm{\alpha},\bm{\beta}}:=\operatorname{diag}(e^{\alpha_{1}s},\ldots,e^{\alpha_{m}s},e^{-\beta_{1}s},\ldots,e^{-\beta_{n}s})\in
G.$

Let $\Delta:X_{d}\to[0,\infty)$ be the function defined in (1.4). The main
result of this section is an asymptotic estimate for the measure of the
thickened set

$\displaystyle\widetilde{\Delta}_{r}:=\bigcup_{0\leq
s<1}g_{-s}\,\Delta^{-1}[0,r],$

when $r>0$ is small.

######  Theorem 5.1.

Let $\varkappa_{d}=\frac{d^{2}+d-4}{2}$ and $\lambda_{d}=\frac{d(d-1)}{2}$ be
as in Theorem 1.2. Then

$\displaystyle\mu_{d}\bigl{(}\widetilde{\Delta}_{r}\bigr{)}\asymp_{d}r^{\varkappa_{d}}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)},\qquad\textrm{as
$r\to 0^{+}$},$

where the implicit constant is independent of $r$ and the two weight vectors
$\bm{\alpha}$ and $\bm{\beta}$.

Just as for Theorem 2.1, we prove Theorem 5.1 by proving the upper and lower
bounds separately.

###  5.1. Proof of the upper bound

Note that for any $(\bm{x},\bm{y})\in{\mathbb{R}}^{m}\times{\mathbb{R}}^{n}$,

$g_{-s}\left(\begin{smallmatrix}\bm{x}\\\
\bm{y}\end{smallmatrix}\right)=(e^{-\alpha_{1}s}x_{1},\ldots,e^{-\alpha_{m}s}x_{m},e^{\beta_{1}s}y_{1},\ldots,e^{\beta_{n}s}y_{n})^{t}.$

This implies that

$\displaystyle|\Delta(g_{-s}\Lambda)-\Delta(\Lambda)|\leq\max\bigl{\\{}\alpha_{1},\ldots,\alpha_{m},\beta_{1},\ldots,\beta_{n}\bigr{\\}}|s|<|s|,\qquad\forall\
s\in{\mathbb{R}},\ \Lambda\in X_{d}.$

Hence

(5.2) $\displaystyle
g_{-s}\Delta^{-1}\big{[}0,r]\subset\Delta^{-1}[0,r+|s|\big{]},\qquad\forall\
s\in{\mathbb{R}}.$

Given any $r\in(0,1)$, let $q=\left\lceil{1/r}\right\rceil$. Using (5.2) and
the fact that $1/q\leq r$, we have

$\displaystyle\widetilde{\Delta}_{r}=\bigcup_{0\leq
s<1}g_{-s}\Delta^{-1}[0,r]=\bigcup_{k=0}^{q-1}\,\bigcup_{0\leq
s<1/q}g_{-k/q}\,g_{-s}\,\Delta^{-1}[0,r]\subset\bigcup_{k=0}^{q-1}g_{-k/q}\,\Delta^{-1}[0,2r],$

implying that (using the $G$-invariance of $\mu_{d}$ and $q\asymp r^{-1}$)

$\displaystyle\mu_{d}\bigl{(}\widetilde{\Delta}_{r}\bigr{)}\leq\sum_{k=0}^{q-1}\mu_{d}\bigl{(}g_{-k/q}\,\Delta^{-1}[0,2r]\bigr{)}\asymp
r^{-1}\mu_{d}\big{(}\Delta^{-1}[0,2r]\big{)}.$

Finally, by Theorem 1.3 we get

$\displaystyle\mu_{d}\bigl{(}\widetilde{\Delta}_{r}\bigr{)}\ll_{d}r^{-1}r^{\varkappa_{d}+1}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)}=r^{\varkappa_{d}}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)},\qquad\textrm{as
$r\to 0^{+}$}.$

This finishes the proof of the upper bound in Theorem 5.1.

###  5.2. Proof of the lower bound

In this subsection we prove the lower bound in Theorem 5.1. By the discussion
in the beginning of Section 2, we may replace the set $\Delta^{-1}[0,r]$ by
$K_{r}$, that is, it suffices to prove the following lower bound

(5.3) $\displaystyle\mu_{d}\Biggl{(}\,\bigcup_{0\leq
s<1}g_{-s}\,K_{r}\Biggr{)}\gg_{d}r^{\varkappa_{d}}\log^{\lambda_{d}}\Bigl{(}\frac{1}{r}\Bigr{)},\qquad\text{as
$r\to 0^{+}$}.$

The following lemma is the crucial ingredient in our proof of (5.3). Let
$c_{0}$ be the small parameter which we fixed in Section 2.2; after possibly
shrinking $c_{0}$, we may without loss of generality assume that
$0<c_{0}<(3e)^{-1}$. For $r\in(0,c_{0}/d)$, let $\underline{K}_{r}\subset
K_{r}$ be as in (3.20).

######  Lemma 5.2.

For any $r\in(0,c_{0}/d)$ and $s\in[r,1)$, the two sets $\underline{K}_{r}$
and $g_{-s}\,\underline{K}_{r}$ are disjoint.

###### Proof.

Assume the opposite; then there exist $r\in(0,c_{0}/d)$, $s\in[r,1)$ such that
$\underline{K}_{r}\cap g_{-s}\underline{K}_{r}\neq\varnothing$, or
equivalently $g_{s}\underline{K}_{r}\cap\underline{K}_{r}\neq\varnothing$.
Pick a lattice $\Lambda\in g_{s}\underline{K}_{r}\cap\underline{K}_{r}$; thus
$\Lambda\in\underline{K}_{r}$ and $g_{-s}\Lambda\in\underline{K}_{r}$. By the
definition of $\underline{K}_{r}$ in (3.20), we now have
$\Lambda=p_{\bm{b}_{1},\dots,\bm{b}_{d-1}}u_{\bm{x}}{\mathbb{Z}}^{d}$ for some
vectors $\bm{b}_{j}=(b_{1j},\ldots,b_{dj})^{t}\in{\mathbb{R}}^{d}$
($j=1,\ldots,d-1$) and $\bm{x}\in(0,c_{0}/d)^{d-1}$ satisfying (3.10), (3.11),
(3.12) and (3.13); moreover, we also have
$g_{-s}\Lambda=p_{\bm{b}^{\prime}_{1},\dots,\bm{b}^{\prime}_{d-1}}u_{\bm{x}^{\prime}}{\mathbb{Z}}^{d}$
for some vectors
$\bm{b}^{\prime}_{j}=(b_{1j}^{\prime},\ldots,b_{dj}^{\prime})^{t}\in{\mathbb{R}}^{d}$
($j=1,\ldots,d-1$) and $\bm{x}^{\prime}\in(0,c_{0}/d)^{d-1}$ which again
satisfy (3.10), (3.11), (3.12) and (3.13).

Because of ${\text{\boldmath$\alpha$}}\in({\mathbb{R}}_{>0})^{m}$ and
$\sum_{i=1}^{m}\alpha_{i}=1$, there exists an index $1\leq i\leq m$ such that
$\alpha_{i}\geq\tfrac{1}{m}$. Fixing such an $i$, we consider the vector

$\displaystyle{\text{\boldmath$y$}}=(y_{1},\ldots,y_{d})^{t}:=g_{-s}{\text{\boldmath$b$}}_{i}=\bigl{(}e^{-\alpha_{1}s}b_{1i},\ldots,e^{-\alpha_{m}s}b_{mi},e^{\beta_{1}s}b_{m+1,i},\ldots,e^{\beta_{n}s}b_{d,i}\bigr{)}^{t}.$

By (3.10) we have $|b_{ji}|<c_{0}$ for all $j\neq i$, and
$1-\frac{r}{2d}<b_{ii}<1$. Hence $|y_{j}|<ec_{0}$ for all $j\neq i$ (since
$0<s<1$ and $\beta_{\ell}\leq 1$ for all $1\leq\ell\leq n$), and
$0<y_{i}<e^{-\alpha_{i}s}<e^{-s/m}<1-\frac{s}{2m}<1-\frac{r}{2d}$. We have
${\text{\boldmath$b$}}_{i}\in\Lambda\smallsetminus\\{\mathbf{0}\\}$ and thus
${\text{\boldmath$y$}}=g_{-s}{\text{\boldmath$b$}}_{i}\in
g_{-s}\Lambda\smallsetminus\\{\mathbf{0}\\}$; also
$g_{-s}\Lambda\in\underline{K}_{r}\subset K_{r}$, and hence
${\text{\boldmath$y$}}\notin{\mathcal{C}}_{r}$. But for all $j\neq i$ we have
$|y_{j}|<ec_{0}<\frac{1}{3}<1-r$ (indeed, recall that $0<c_{0}<(3e)^{-1}$);
also $y_{i}>0$; hence ${\text{\boldmath$y$}}\notin{\mathcal{C}}_{r}$ implies
$y_{i}\geq 1-r$. In summary:

$\displaystyle|y_{j}|<ec_{0}\quad(\forall\ j\neq i)\qquad\text{and}\quad
1-r\leq y_{i}<1-\tfrac{r}{2d}.$

Furthermore, ${\text{\boldmath$b$}}_{i}^{\prime}\in g_{-s}\Lambda$, since
$g_{-s}\Lambda=p_{\bm{b}^{\prime}_{1},\dots,\bm{b}^{\prime}_{d-1}}u_{\bm{x}^{\prime}}{\mathbb{Z}}^{d}$;
and by (3.10) we have $|b^{\prime}_{ji}|<c_{0}$ for all $j\neq i$ and
$1-\frac{r}{2d}<b^{\prime}_{ii}<1$. It follows that
$|b^{\prime}_{ji}-y_{j}|<(e+1)c_{0}<(e+1)(3e)^{-1}<\frac{1}{2}<1-r$ for all
$j\neq i$, and $|b^{\prime}_{ii}-y_{i}|<r<1-r$; hence
${\text{\boldmath$b$}}^{\prime}_{i}-{\text{\boldmath$y$}}\in{\mathcal{C}}_{r}$.
Note also that ${\text{\boldmath$b$}}_{i}^{\prime}-{\text{\boldmath$y$}}\in
g_{-s}\Lambda$, and
${\text{\boldmath$b$}}_{i}^{\prime}-{\text{\boldmath$y$}}\neq\mathbf{0}$,
since $y_{i}<1-\frac{r}{2d}<b^{\prime}_{ii}$; hence we have obtained a
contradiction against $g_{-s}\Lambda\in\underline{K}_{r}\subset K_{r}$. This
completes the proof of the lemma. ∎

It follows from Lemma 5.2 that for any $r\in(0,c_{0}/d)$, the sets
$g_{-kr}\,\underline{K}_{r}$, for $k$ running through the integers in the
interval $0\leq k<1/r$, are pairwise disjoint. (Indeed, if
$g_{-kr}\,\underline{K}_{r}\cap
g_{-k^{\prime}r}\,\underline{K}_{r}\neq\varnothing$ for some $0\leq
k<k^{\prime}<1/r$ then $\underline{K}_{r}\cap
g_{(k-k^{\prime})r}\,\underline{K}_{r}\neq\varnothing$, contradicting Lemma
5.2.) Hence, using also $K_{r}\supset\underline{K}_{r}$, we have

$\displaystyle\mu_{d}\Biggl{(}\,\bigcup_{0\leq
s<1}g_{-s}\,K_{r}\Biggr{)}\geq\mu_{d}\Biggl{(}\,\bigcup_{0\leq
k<1/r}g_{-kr}\,\underline{K}_{r}\Biggr{)}=\sum_{0\leq
k<1/r}\mu_{d}\bigl{(}g_{-kr}\,\underline{K}_{r}\bigr{)}=\\#\bigl{(}{\mathbb{Z}}\cap[0,1/r)\bigr{)}\cdot\mu_{d}\bigl{(}\underline{K}_{r}\bigr{)},$

Here $\\#\big{(}{\mathbb{Z}}\cap[0,1/r)\big{)}\gg r^{-1}$, and for $r$
sufficiently small we have $\mu_{d}\bigl{(}\underline{K}_{r}\bigr{)}\gg
r^{\varkappa_{d}+1}\log^{\lambda_{d}}\bigl{(}\frac{1}{r}\bigr{)}$ by (3.23).
Hence we obtain the lower bound (5.3), and the proof of Theorem 5.1 is
complete. $\square$

##  6\. Some preliminaries for Theorem 1.2

In this section we collect some preliminary results for our proof of Theorem
1.2.

###  6.1. Dynamical interpretation of weighted $\psi$-Dirichlet matrices

Let $m,n\in{\mathbb{N}}$ and let $\bm{\alpha}\in{\mathbb{R}}^{m}$ and
$\bm{\beta}\in{\mathbb{R}}^{n}$ be two fixed weight vectors as before. Let
$t_{0}>0$ and let $\psi:[t_{0},\infty)\to(0,\infty)$ be a continuous
decreasing function which tends to zero at infinity. In this subsection we
give a dynamical interpretation of $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet
matrices which generalizes [26, Proposition 4.5]; see Proposition 6.2. Let us
first introduce the following modified Dani Correspondence which is a special
case of [20, Lemma 8.3].

######  Lemma 6.1.

Fix $m,n\in{\mathbb{N}}$ and let $d=m+n$. Let $t_{0}>0$, and let
$\psi:[t_{0},\infty)\to(0,\infty)$ be a continuous, decreasing function
satisfying (1.8) and (1.9). Then there exists a unique continuous, decreasing
function

$r=r_{\psi}:[s_{0},\infty)\to(0,\infty),\quad\textrm{where
$s_{0}=\frac{m}{d}\log t_{0}-\frac{n}{d}\log\psi(t_{0})$},$

such that

(6.1) the function $s\mapsto s+mr(s)$ is increasing,

and

(6.2) $\psi\left(e^{s-nr(s)}\right)=e^{-s-mr(s)}\quad\textrm{for all $s\geq
s_{0}$}.$

Conversely, given $s_{0}\in{\mathbb{R}}$ and a continuous, decreasing function
$r:[s_{0},\infty)\to(0,\infty)$ satisfying (6.1), there exists a unique
continuous, decreasing function $\psi=\psi_{r}:[t_{0},\infty)\to(0,\infty)$
with $t_{0}=e^{s_{0}-nr(s_{0})}$ satisfying (1.8), (1.9) and (6.2).
Furthermore, for any fixed $\alpha,\beta>0$ the series

(6.3) $\displaystyle\sum_{k\geq
t_{0}}\frac{\big{(}1-k\psi(k)\big{)}^{\alpha}\big{(}-\log\left(1-k\psi(k)\right)\big{)}^{\beta}}{k}$

diverges if and only if the series

(6.4) $\displaystyle\sum_{k\geq
s_{0}}r(k)^{\alpha}\log^{\beta}\Bigl{(}1+\frac{1}{r(k)}\Bigr{)}$

diverges.

###### Proof.

The $\psi$ and $r$-functions determine each other uniquely via the relation

(6.5) $\displaystyle\psi(t)^{1/m}e^{s/m}=t^{1/n}e^{-s/n}=e^{-r(s)},$

which captures the moment when the $a_{s}$-flow transforms the long and thin
‘rectangle’

$\big{\\{}(\bm{x},\bm{y})\in{\mathbb{R}}^{m}\times{\mathbb{R}}^{n}\>:\>\|\bm{x}\|^{m}<\psi(t),\
\|\bm{y}\|^{n}<t\big{\\}}$

determined by (1.2) into a cube (with side length $2e^{-r(s)}$). Here
$a_{s}=\operatorname{diag}(e^{s/m}I_{m},e^{-s/n}I_{n})$, as defined in the
introduction. This correspondence between $\psi(\cdot)$ and $r(\cdot)$ is a
special case of [20, Lemma 8.3], as here we assume that $\psi(\cdot)$
additionally satisfies (1.8) and (1.9), which on the $r$-function side
corresponds respectively to the assumptions that $r(\cdot)$ is decreasing and
$r(s)>0$ for all $s\geq s_{0}$. The equivalence of these additional
assumptions is easily checked using the following three relations, which
follow from (6.5):

(6.6) $\displaystyle e^{-dr(s)}=t\psi(t),\qquad s=\frac{m}{d}\log
t-\frac{n}{d}\log\psi(t),\qquad\textrm{and}\qquad t=e^{s-nr(s)}.$

Finally we prove the equivalence of the divergence of the two series. If
$\lim_{t\to\infty}t\psi(t)<1$ then both the functions $1-t\psi(t)$ and $r(s)$
are bounded away from zero (and positive), which implies that the two series
in (6.3) and (6.4) are divergent. Hence from now on we may assume that
$\lim_{t\to\infty}t\psi(t)=1$. Then $\lim_{s\to\infty}r(s)=0$ (by (6.6)), and
$F_{\psi}(t):=1-t\psi(t)$ is a decreasing function taking values in the
interval $(0,1)$ and satisfying $\lim_{t\to\infty}F_{\psi}(t)=0$. After
enlarging $s_{0}$ (thus also enlarging $t_{0}$) we may assume that
$0<r(s)<1/d$ for all $s\geq s_{0}$. Then by (6.6) we have, with
$t=t(s)=e^{s-nr(s)}$:

(6.7) $\displaystyle\frac{d}{2}r(s)<F_{\psi}(t)<d\,r(s)\quad\text{and }\quad
e^{s-1}<t<e^{s},\qquad\forall\ s\geq s_{0}.$

It follows that
$r(s)^{\alpha}\log^{\beta}\bigl{(}1+\frac{1}{r(s)}\bigr{)}\asymp_{d,\alpha,\beta}F_{\psi}(t)^{\alpha}\log^{\beta}\bigl{(}\frac{1}{F_{\psi}(t)}\bigr{)}$
for all $s\geq s_{0}$. Hence, using also $e^{k-1}<t(k)<e^{k}$ (see (6.7)), the
fact that $F_{\psi}(t)$ is decreasing, and $\sum_{e^{k}\leq
j<e^{k+1}}\frac{1}{j}\asymp 1$ ($\forall k\geq 1$), we have for all
sufficiently large integers $k$:

$\displaystyle
r(k)^{\alpha}\log^{\beta}\Bigl{(}1+\frac{1}{r(k)}\Bigr{)}\ll_{d,\alpha,\beta}F_{\psi}(e^{k-1})^{\alpha}\log^{\beta}\Bigl{(}\frac{1}{F_{\psi}(e^{k-1})}\Bigr{)}\ll_{\alpha,\beta}\sum_{e^{k-2}\leq
j<e^{k-1}}\frac{1}{j}F_{\psi}(j)^{\alpha}\log^{\beta}\Bigl{(}\frac{1}{F_{\psi}(j)}\Bigr{)},$

and similarly

$\displaystyle
r(k)^{\alpha}\log^{\beta}\Bigl{(}1+\frac{1}{r(k)}\Bigr{)}\gg_{d,\alpha,\beta}F_{\psi}(e^{k})^{\alpha}\log^{\beta}\Bigl{(}\frac{1}{F_{\psi}(e^{k})}\Bigr{)}\gg_{\alpha,\beta}\sum_{e^{k}\leq
j<e^{k+1}}\frac{1}{j}F_{\psi}(j)^{\alpha}\log^{\beta}\Bigl{(}\frac{1}{F_{\psi}(j)}\Bigr{)}.$

It follows that the series in (6.4) diverges if and only if
$\sum_{j}j^{-1}F_{\psi}(j)^{\alpha}\log^{\beta}\bigl{(}\frac{1}{F_{\psi}(j)}\bigr{)}$
diverges, that is, if and only if the series in (6.3) diverges. ∎

######  Remark 12.

Let $\psi$ and $r$ be as in Lemma 6.1 with $\lim_{t\to\infty}t\psi(t)=1$. Let
$F_{\psi}(t)=1-t\psi(t)$ be as above. Assume that the series (6.3) (and thus
also the series (6.4)) diverges for some $\alpha,\beta>0$. It is then not
difficult to see from the proof of Lemma 6.1 that for any
$\beta^{\prime}\geq\beta$ and for all large $s_{1}>s_{0}$,

$\displaystyle\sum_{s_{0}\leq k\leq
s_{1}}r(k)^{\alpha}\log^{\beta^{\prime}}\Bigl{(}1+\frac{1}{r(k)}\Bigr{)}\asymp_{d,\alpha,\beta^{\prime}}\sum_{t_{0}\leq
k\leq
e^{s_{1}}}k^{-1}F_{\psi}(k)^{\alpha}\log^{\beta^{\prime}}\Bigl{(}\frac{1}{F_{\psi}(k)}\Bigr{)}.$

In particular it follows that for any $\beta^{\prime}>\beta$ we have the
following equivalence:

$\displaystyle\liminf_{s_{1}\to\infty}\frac{\sum_{s_{0}<k\leq
s_{1}}r(k)^{\alpha}\log^{\beta^{\prime}}\bigl{(}1+\frac{1}{r(k)}\bigr{)}}{\Bigl{(}\sum_{s_{0}<k\leq
s_{1}}r(k)^{\alpha}\log^{\beta}\bigl{(}1+\frac{1}{r(k)}\bigr{)}\Bigr{)}^{2}}=0\hskip
180.0pt$
$\displaystyle\Longleftrightarrow\quad\liminf_{t_{1}\to\infty}\frac{\sum_{t_{0}\leq
k\leq
t_{1}}k^{-1}F_{\psi}(k)^{\alpha}\log^{\beta^{\prime}}\left(\frac{1}{F_{\psi}(k)}\right)}{\left(\sum_{t_{0}\leq
k\leq
t_{1}}k^{-1}F_{\psi}(k)^{\alpha}\log^{\beta}\left(\frac{1}{F_{\psi}(k)}\right)\right)^{2}}=0.$

Similarly, the above two limits inferior remain bounded simultaneously.

We now state the dynamical interpretation of
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet matrices.

######  Proposition 6.2.

Let $\psi$ be as in Theorem 1.2, and let $r=r_{\psi}$ be as in Lemma 6.1. Let
$\\{g_{s}\\}_{s\in{\mathbb{R}}}$ be the one-parameter subgroup associated to
the two fixed weight vectors $\bm{\alpha}$ and $\bm{\beta}$ as in (5.1). Set

$\omega_{1}:=\max\\{m\alpha_{i},n\beta_{j}\>:\>1\leq i\leq m,1\leq j\leq
n\\}\quad\text{and}\quad\omega_{2}:=\min\\{m\alpha_{i},n\beta_{j}\>:\>1\leq
i\leq m,1\leq j\leq n\\}.$

Then for any $A\in M_{m,n}({\mathbb{R}})$ we have, with $\Lambda_{A}$ as in
(1.5):

  * (1)

if $\Delta(g_{s}\Lambda_{A})>\omega_{1}r(s)$ for all sufficiently large $s$,
then $A$ is $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet;

  * (2)

if $\Delta(g_{s}\Lambda_{A})\leq\omega_{2}r(s)$ for an unbounded set of $s$,
then $A$ is not $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet.

######  Remark 13.

When $\bm{\alpha}=(\tfrac{1}{m},\ldots,\tfrac{1}{m})\in{\mathbb{R}}^{m}$ and
$\bm{\beta}=(\tfrac{1}{n},\ldots,\tfrac{1}{n})\in{\mathbb{R}}^{n}$, then
$\omega_{1}=\omega_{2}=1$ and Proposition 6.2 recovers [26, Proposition 4.5].

###### Proof of Proposition 6.2.

For any $t>\max\\{t_{0},1\\}$, define

$\displaystyle\mathcal{R}_{t}=\mathcal{R}^{\bm{\alpha},\bm{\beta}}_{t}:=\big{\\{}(\bm{x},\bm{y})\in{\mathbb{R}}^{m}\times{\mathbb{R}}^{n}\>:\>\|\bm{x}\|_{\bm{\alpha}}<\psi(t),\|\bm{y}\|_{\bm{\beta}}<t\big{\\}},$

so that
$(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times({\mathbb{Z}}^{n}\smallsetminus\\{\bm{0}\\})$
is a solution to (1.7) if and only if
$\left(A\bm{q}-\bm{p},\bm{q}\right)\in\mathcal{R}_{t}$. On the other hand, the
lattice $\Lambda_{A}$ consists exactly of the points

$\displaystyle\left(\begin{matrix}I_{m}&A\\\
0&I_{n}\end{matrix}\right)\left(\begin{matrix}-\bm{p}\\\
\bm{q}\end{matrix}\right)=\left(\begin{matrix}A\bm{q}-\bm{p}\\\
\bm{q}\end{matrix}\right)\qquad\textrm{for}\
(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times{\mathbb{Z}}^{n}.$

Moreover, if $(A\bm{q}-\bm{p},\bm{q})\in\Lambda_{A}\cap{\mathcal{R}}_{t}$ is
nonzero for some $(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times{\mathbb{Z}}^{n}$,
then we must have $\bm{q}\neq\bm{0}$. Indeed, otherwise we would have
$\|A\bm{q}-\bm{p}\|_{\bm{\alpha}}=\|\bm{p}\|_{\bm{\alpha}}\geq 1$, but
$(A\bm{q}-\bm{p},\bm{q})\in{\mathcal{R}}_{t}$ implies that
$\|A\bm{q}-\bm{p}\|_{\bm{\alpha}}<\psi(t)<1/t<1$ (since
$t>\max\\{t_{0},1\\}$), contradicting $\|A\bm{q}-\bm{p}\|_{\bm{\alpha}}\geq
1$. Thus there exists a solution
$(\bm{p},\bm{q})\in{\mathbb{Z}}^{m}\times({\mathbb{Z}}^{n}\smallsetminus\\{\bm{0}\\})$
to (1.7) if and only if $\Lambda_{A}\cap\mathcal{R}_{t}\neq\\{\bm{0}\\}$,
implying that $A\in M_{m,n}({\mathbb{R}})$ is
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet if and only if
$\Lambda_{A}\cap\mathcal{R}_{t}\neq\\{\bm{0}\\}$ for all sufficiently large
$t$. Now let $s=s(t)=\frac{m}{d}\log t-\frac{n}{d}\log\psi(t)$; then
$s\to\infty$ if and only if $t\to\infty$, and by (6.5) we have

$\displaystyle
g_{s}\mathcal{R}_{t}=\left\\{(\bm{x}^{\prime},\bm{y}^{\prime})\in{\mathbb{R}}^{m}\times{\mathbb{R}}^{n}\>:\>\|\bm{x}^{\prime}\|_{\bm{\alpha}}<e^{-mr(s)},\
\|\bm{y}^{\prime}\|_{\bm{\beta}}<e^{-nr(s)}\right\\}=:\mathcal{E}_{s}.$

It follows that $A$ is $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet if and only
if $g_{s}\Lambda_{A}\cap\mathcal{E}_{s}\neq\\{\bm{0}\\}$ for all sufficiently
large $s$. Next, note that we have the following simple relation:

(6.8)
$\displaystyle\bigl{(}-e^{-\omega_{1}r(s)},e^{-\omega_{1}r(s)}\bigr{)}^{d}\subset\mathcal{E}_{s}\subset\bigl{(}-e^{-\omega_{2}r(s)},e^{-\omega_{2}r(s)}\bigr{)}^{d},$

with $\omega_{1},\omega_{2}$ defined as in the statement of the proposition.
Note also that $\Delta(g_{s}\Lambda_{A})>\omega_{1}r(s)$ is equivalent with
$g_{s}\Lambda_{A}\cap\left(-e^{-\omega_{1}r(s)},e^{-\omega_{1}r(s)}\right)^{d}\neq\\{\bm{0}\\}$.
Hence, using the first inclusion relation in (6.8) we have

$\Delta(g_{s}\Lambda_{A})>\omega_{1}r(s)\ \textrm{for all sufficiently large
$s$}\ \Rightarrow g_{s}\Lambda_{A}\cap\mathcal{E}_{s}\neq\\{\bm{0}\\}\
\textrm{for all sufficiently large $s$},$

and the latter condition implies that $A$ is
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet. We have thus proved part (1) of the
proposition. Similarly, part (2) follows using the second inclusion relation
in (6.8). ∎

Let $\psi$ and $r=r_{\psi}$ be as above. For any integer $k>s_{0}$, let us
define

$\displaystyle\underline{B}_{k}:=\bigcup_{0\leq
s<1}g_{-s}\Delta^{-1}[0,\omega_{2}r(k+s)]\quad\textrm{and}\quad\overline{B}_{k}:=\bigcup_{0\leq
s<1}g_{-s}\Delta^{-1}[0,\omega_{1}r(k+s)].$

It follows that for any $\Lambda\in X_{d}$, we have
$g_{k}\Lambda\in\underline{B}_{k}$ (respectively
$g_{k}\Lambda\in\overline{B}_{k}$) if and only if there is some $k\leq s<k+1$
such that $g_{s}\Lambda\in\Delta^{-1}[0,\omega_{2}r(s)]$ (respectively
$g_{s}\Lambda\in\Delta^{-1}[0,\omega_{1}r(s)]$). In particular, in view of
Proposition 6.2, a given matrix $A\in M_{m,n}({\mathbb{R}})$ is
$\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet if
$g_{k}\Lambda_{A}\notin\overline{B}_{k}$ for all sufficiently large $k$, or
equivalently, if $g_{k}\Lambda_{A}\in\overline{B}_{k}$ holds only finitely
often. Similarly, $A$ is not $\psi_{\bm{\alpha},\bm{\beta}}$-Dirichlet if
$g_{k}\Lambda_{A}\in\underline{B}_{k}$ holds infinitely often.

###  6.2. Effective equidistribution and doubly mixing for certain
$g_{s}$-translates

Let $m,n\in{\mathbb{N}}$ and $d=m+n$ be as before. Let

$\mathcal{Y}=\left\\{\Lambda_{A}:A\in M_{m,n}({\mathbb{R}})\right\\}\cong
M_{m,n}({\mathbb{R}}/{\mathbb{Z}})$

be defined as in (1.5), and recall that ${\mathcal{Y}}$ is equipped with the
probability Lebesgue measure, Leb.

As mentioned in Remark 9, we will need an effective equidistribution and
doubly mixing result for the $g_{s}$-translates
$\\{g_{s}{\mathcal{Y}}\\}_{s>0}$ which is analogous to (1.13) and (1.14)
respectively. In fact, we will state a corollary of a more general effective
mixing result of arbitrary order proved by Björklund and Gorodnik [2, Theorem
2.2]. To state their result, let us first fix some notation.

Let $\mathfrak{g}=\mathfrak{sl}_{d}({\mathbb{R}})$ be the Lie algebra of $G$.
For each $Y\in\mathfrak{g}$, let us denote by $\mathcal{D}_{Y}$ the
corresponding Lie derivative (a first order differential operator) on
$C^{\infty}(G)$ defined by

$\displaystyle\mathcal{D}_{Y}(f)(g):=\frac{\text{d}}{\operatorname{d}\\!{t}}f(\exp(tY)g)\big{|}_{t=0},\qquad
f\in C^{\infty}(G).$

Here $\exp:\mathfrak{g}\to G$ denotes the usual exponential map from
$\mathfrak{g}$ to $G$. Note that this definition naturally extends to the
function space $C_{c}^{\infty}(X_{d})$ since we can view elements in
$C_{c}^{\infty}(X_{d})$ as right $\Gamma$-invariant smooth functions on $G$.
Fix an ordered basis $\\{Y_{1},\ldots,Y_{a}\\}$ of $\mathfrak{g}$. Then every
monomial $Z=Y_{1}^{\ell_{1}}\cdots Y_{a}^{\ell_{a}}$ defines a differential
operator of degree $\deg(Z):=\ell_{1}+\cdots+\ell_{a}$, via

$\displaystyle\mathcal{D}_{Z}:=\mathcal{D}_{Y_{1}}^{\ell_{1}}\circ\cdots\circ\mathcal{D}_{Y_{a}}^{\ell_{a}}.$

Now for each $\ell\in{\mathbb{N}}$ we define the “$L^{2}$, degree $\ell$”
Sobolev norm on $C_{c}^{\infty}(X_{d})$ by

$\displaystyle\|f\|_{{H^{\ell}}}:=\biggl{(}\sum_{\deg(Z)\leq\ell}\int_{X_{d}}|\mathcal{D}_{Z}(f)|^{2}\,\operatorname{d}\\!{\mu}_{d}\biggr{)}^{1/2},$

where the summation is over all the monomials $Z$ in
$\\{Y_{1},\ldots,Y_{a}\\}$ with degree no greater than $\ell$. Fix a metric
$\textrm{dist}(\cdot,\cdot)$ on $X_{d}=G/\Gamma$ which is induced from a right
$G$-invariant Riemannian metric on $G$. We also define the following Lipschitz
(semi-)norm on $C_{c}^{\infty}(X_{d})$ with respect to this metric:

$\displaystyle\|f\|_{\operatorname{Lip}}:=\sup\left\\{\frac{|f(x_{1})-f(x_{2})|}{\textrm{dist}(x_{1},x_{2})}\>:\>x_{1},x_{2}\in
X_{d},\ x_{1}\neq x_{2}\right\\},\qquad f\in C_{c}^{\infty}(X_{d}).$

Let us also write $\|\cdot\|_{C^{0}}$ for the uniform norm on $C_{c}(X_{d})$.
Finally, for any $f\in C_{c}^{\infty}(X_{d})$ we define

$\displaystyle\mathcal{N}_{\ell}(f):=\max\left\\{\|f\|_{C^{0}},\|f\|_{\operatorname{Lip}},\|f\|_{{H^{\ell}}}\right\\}.$

We can now state the result which we need from [2].

######  Theorem 6.3 ([2, Corollary 2.4]).

There exist $\ell\in{\mathbb{N}}$ and $\delta>0$ such that for every
$b\in{\mathbb{N}}$ and any $f_{0}\in C^{\infty}(\mathcal{Y})$,
$f_{1},\ldots,f_{b}\in C_{c}^{\infty}(X_{d})$ and $s_{1},\ldots,s_{b}>0$, we
have

$\displaystyle\int_{\mathcal{Y}}f_{0}(\Lambda_{A})\left(\prod_{i=1}^{b}f_{i}(g_{s_{i}}\Lambda_{A})\right)\,\operatorname{d}\\!{A}$
$\displaystyle=\textrm{Leb}(f_{0})\prod_{i=1}^{b}\mu_{d}(f_{i})+O_{b,f_{0}}\left(e^{-\delta
D(s_{1},\ldots,s_{b})}\prod_{i=1}^{b}\mathcal{N}_{\ell}(f_{i})\right),$

where $D(s_{1},\ldots,s_{b}):=\min\left\\{s_{i},|s_{i}-s_{j}|\>:\>1\leq i\neq
j\leq b\right\\}$.

(In fact, in [2, Corollary 2.4], the error term is also explicit in terms of
$f_{0}$.)

Taking $f_{0}\equiv 1$ on $\mathcal{Y}$ and $b=1,2$, we get the following
effective equidistribution and doubly mixing of the family of
$g_{s}$-translates $\\{g_{s}\mathcal{Y}\\}_{s>0}$ in $X_{d}$.

######  Corollary 6.4.

Let $\ell\in{\mathbb{N}}$ and $\delta>0$ be as in Theorem 6.3. Then for any
$f,f_{1},f_{2}\in C^{\infty}_{c}(X_{d})$ and $s,s_{1},s_{2}>0$,

(6.9)
$\displaystyle\int_{\mathcal{Y}}f(g_{s}\Lambda_{A})\,\operatorname{d}\\!{A}=\mu_{d}(f)+O\left(e^{-\delta
s}\mathcal{N}_{\ell}(f)\right),$

and

(6.10)
$\displaystyle\int_{\mathcal{Y}}f_{1}(g_{s_{1}}\Lambda_{A})f_{2}(g_{s_{2}}\Lambda_{A})\,\operatorname{d}\\!{A}=\mu_{d}(f_{1})\mu_{d}(f_{2})+O\left(e^{-\delta\min\\{s_{1},s_{2},|s_{1}-s_{2}|\\}}\mathcal{N}_{\ell}(f_{1})\mathcal{N}_{\ell}(f_{2})\right).$

###  6.3. Smooth approximations and estimates on norms

In this subsection we prove the existence of smooth functions $\phi\in
C_{c}^{\infty}(X_{d})$ bounding our shrinking targets from above and below in
an appropriate sense, with control on the norm ${\mathcal{N}}_{\ell}(\phi)$.
We follow the strategy of [22, Theorem 1.1] while allowing the small identity
neighborhoods of $G$ (against which we convolve) to shrink.

Recall that

$\displaystyle\widetilde{\Delta}_{r}:=\bigcup_{0\leq
s<1}g_{-s}\Delta^{-1}[0,r]\qquad{(0<r<1)}.$

######  Lemma 6.5.

Let $\varepsilon>0$. For any $0<r<1$, there exists $\phi_{r}\in
C^{\infty}_{c}(X_{d})$ satisfying
$\chi_{\widetilde{\Delta}_{r}}\leq\phi_{r}\leq\chi_{\widetilde{\Delta}_{2r}}$
and, for any $\ell\in{\mathbb{N}}$:

(6.11)
$\displaystyle\mathcal{N}_{\ell}(\phi_{r})\ll_{\ell,d,\varepsilon}r^{-L},\qquad\text{with
}\>L:=1+\max\left\\{0,\ell+\varepsilon-\tfrac{d^{2}}{4}-\tfrac{d}{4}\right\\}.$

(Note that the implied constant in the bound in (6.11) is independent of $r$.)

To prepare for the proof, let us define, for any $r>0$,

$\displaystyle\mathcal{O}_{r}:=\left\\{g\in
G\>:\>\max\\{\|g-I_{d}\|,\|g^{-1}-I_{d}\|\\}<\frac{r}{10d}\right\\}.$

Here the norm is the supremum norm on the matrix space
$M_{d,d}({\mathbb{R}})$. Clearly, $\mathcal{O}_{r}$ is an open neighborhood of
the identity element in $G$ and it is invariant under inversion. Let $\nu$ be
the normalized Haar measure of $G$ as in Section 2.1; recall that $\nu$
locally agrees with $\mu_{d}$.

We will need the following auxiliary lemma.

######  Lemma 6.6.

For every $0<r<1$, there exists a function ${\theta}_{r}\in C_{c}^{\infty}(G)$
satisfying ${\theta}_{r}\geq 0$,
$\operatorname{supp}({\theta}_{r})\subset\mathcal{O}_{r/2}$,
$\int_{G}{\theta}_{r}(g)\,d\nu(g)=1$ and
$\|\mathcal{D}_{Z}({\theta}_{r})\|_{L^{\infty}(G)}\ll_{\ell,d}r^{1-d^{2}-\ell}$
for every monomial $Z=Y_{1}^{\ell_{1}}\cdots Y_{a}^{\ell_{a}}$, where
$\ell=\deg(Z)=\ell_{1}+\cdots+\ell_{a}$.

(The implied constant in the bound on
$\|\mathcal{D}_{Z}({\theta}_{r})\|_{L^{\infty}(G)}$ depends only on $\ell$ and
$d$, and not on $r$.)

###### Proof (sketch).

Let $\varphi:\Omega\to{\mathbb{R}}^{a}$ ($a=d^{2}-1$) be an arbitrary
$C^{\infty}$ coordinate chart of an open neighbourhood $\Omega$ of $I_{d}$ in
$G$, with $\varphi(I_{d})=\mathbf{0}$. Let $\eta\in
C_{c}^{\infty}({\mathbb{R}}^{a})$ be a fixed bump function in
${\mathbb{R}}^{a}$, i.e. a function satisfying $\eta\geq 0$ and
$\int_{{\mathbb{R}}^{a}}\eta\,\operatorname{d}\\!{{\text{\boldmath$x$}}}=1$.
We may assume that the support of $\eta$ is contained in the unit ball
centered at the origin, ${\mathcal{B}}_{1}^{a}$. For each $t>0$ define
$\eta_{t}\in C_{c}^{\infty}({\mathbb{R}}^{a})$ through

$\displaystyle\eta_{t}({\text{\boldmath$x$}})=t^{-a}\eta(t^{-1}{\text{\boldmath$x$}})\qquad({\text{\boldmath$x$}}\in{\mathbb{R}}^{a}),$

and note that $\operatorname{supp}(\eta_{t})\subset{\mathcal{B}}_{t}^{a}$ and
$\int_{{\mathbb{R}}^{a}}\eta_{t}\,\operatorname{d}\\!{{\text{\boldmath$x$}}}=1$.
Let us choose the constant $c>0$ so small that
${\mathcal{B}}_{c}^{a}\subset\varphi(\Omega)$ and
$\varphi^{-1}({\mathcal{B}}_{cr}^{a})\subset{\mathcal{O}}_{r/2}$ for all
$0<r<1$. This is possible since the matrix entries of $g$ and $g^{-1}$ are
$C^{\infty}$ functions of $g\in G$. Now we may simply set, for each $0<r<1$,

$\displaystyle{\theta}_{r}:=v_{r}\cdot(\eta_{cr}\circ\varphi),$

where $v_{r}>0$ is chosen so as to make
$\int_{G}{\theta}_{r}\,\operatorname{d}\\!{\nu}=1$. One verifies that the
limit $\lim_{r\to 0^{+}}v_{r}$ exists and is a positive real number. Using
this fact, and recalling $a=d^{2}-1$, all the properties stated in the lemma
are straightforward to verify. ∎

###### Proof of Lemma 6.5.

We claim that for any $r_{1},r_{2}>0$,

(6.12)
$\displaystyle\mathcal{O}_{r_{1}}\widetilde{\Delta}_{r_{2}}\subset\widetilde{\Delta}_{r_{1}+r_{2}}.$

First we note that for any $h\in M_{d,d}({\mathbb{R}})$ and
$\bm{v}\in{\mathbb{R}}^{d}$, $\|h\bm{v}\|\leq d\|h\|\|\bm{v}\|$. This implies