# Poisson approximation and Weibull asymptotics in the geometry of numbers

Michael Björklund  Mathematical Sciences, Chalmers, Chalmers Tvärgata 3, 412
58 Gothenburg, Sweden<EMAIL_ADDRESS>and  Alexander Gorodnik  Department
of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057,
Switzerland<EMAIL_ADDRESS>
###### Abstract.

Minkowski’s First Theorem and Dirichlet’s Approximation Theorem provide upper
bounds on certain minima taken over lattice points contained in domains of
Euclidean spaces. We study the distribution of such minima and show, under
some technical conditions, that they exhibit Weibull asymptotics with respect
to different natural measures on the space of unimodular lattices in
$\mathbb{R}^{d}$. This follows from very general Poisson approximation results
for shrinking targets which should be of independent interest. Furthermore, we
show in the appendix that the logarithm laws of Kleinbock-Margulis, Khinchin
and Gallagher can be deduced from our distributional results.

###### Key words and phrases:

Quantitative equidistribution, multiple mixing, Poisson approximation, Weibull
asymptotics

###### 2010 Mathematics Subject Classification:

Primary: 37A50; Secondary: 37A25, 60G70

MB was supported by GoCas Young Excellence grant 11423310 and Swedish VR-grant
11253320, AG was supported by the SNF grant 200021–182089.

##  1\. Introduction

In this paper we consider several extremal problems in Geometry of Numbers
such as the existence of lattice points in convex bodies, minimal values of
polynomial maps and Diophantine approximation. Our first results deal with
generic unimodular lattices, for which we use the following notation:

$\displaystyle X_{d}$ $\displaystyle=\textrm{the space of lattices in
$\mathbb{R}^{d}$ with covolume one},$ $\displaystyle\mu_{d}$
$\displaystyle=\textrm{the unique
$\operatorname{SL}_{d}(\mathbb{R})$-invariant probability measure on $X_{d}$.
}$

We write $\operatorname{Vol}_{d}$ for the Lebesgue measure on
$\mathbb{R}^{d}$, normalized so that
$\operatorname{Vol}_{d}\big{(}[0,1]^{d}\big{)}=1$.

###  1.1. Application I: Minkowski’s Theorem

Let $B\subset\mathbb{R}^{d}$ be a compact, convex, and symmetric set with non-
empty interior and let $\Lambda$ be a lattice in $\mathbb{R}^{d}$ of covolume
one. The Minkowski’s First Theorem ([20, Lecture II, Theorem 11]) says that

$\operatorname{Vol}_{d}(B)\geq 2^{d}\implies\Lambda\cap B\neq 0.$

In order to discuss whether this result can be improved, it is natural to
introduce the function

$c_{B}(\Lambda):=\inf\\{c>0\,:\,\Lambda\cap cB\neq 0\big{\\}},\quad\textrm{for
$\Lambda\in X_{d}$}.$ (1.1)

Let

$\mathscr{T}_{d}:=\big{\\{}(T_{1},\ldots,T_{d})\in\mathbb{R}_{+}^{d}:\,T_{1}\cdots
T_{d}=1\big{\\}},$ (1.2)

and for $T\in\mathscr{T}_{d}$, we consider the boxes

$B_{T}:=\prod_{p=1}^{d}[-T_{p},T_{p}]\subset\mathbb{R}^{d}.$

Since $\hbox{vol}(B_{T})=2^{d}$, it follows from Minkowski’s First Theorem
that $c_{B_{T}}(\Lambda)\leq 1.$ Furthermore, it is easy to see that

$\sup\big{\\{}c_{B_{T}}(\Lambda)\,:\,\Lambda\in X_{d}\big{\\}}=1,$ (1.3)

whence Minkowski’s First Theorem is in some sense optimal. However, we can
significantly improve it for generic lattices:

######  Theorem 1.1.

For $\mu_{d}$-almost every $\Lambda\in X_{d}$,

$\limsup_{\|T\|_{\infty}\to\infty}\;-\frac{\log
c_{B_{T}}(\Lambda)}{\log\log\|T\|_{\infty}}\geq\frac{d-1}{d}.$ (1.4)

This result can be proved using, for instance, the method introduced in [16].

######  Remark 1.2.

Theorem 1.1 is an example of a _logarithm law_. Such laws have been
extensively studied in Number Theory and Dynamics. Perhaps the first
occurrence of a logarithm law is Khinchin’s Theorem (cf. (1.8)). It was
realized by Sullivan [24] that Khinchin’s Theorem is intimately connected with
geodesic excursions to shrinking cuspidal neighborhoods in finite volume
hyperbolic orbifolds. Later this idea was generalized and refined in the
influential work of Kleinbock and Margulis [16].

It turns out that the logarithm law (1.4), as well as other logarithm laws
that we will discuss later, are very much related to _distributional_
convergence of extremal values of arithmetic functions. We show in the
appendix (see Proposition A.1) that Theorem 1.1 is a direct corollary of the
following result:

######  Theorem 1.3.

For a sequence of finite subsets $\Delta_{n}$ of $\mathscr{T}_{d}$ such that

$|\Delta_{n}|\to\infty\quad\quad\hbox{and}\quad\quad\frac{\displaystyle\min_{T\neq
T^{\prime}\in\Delta_{n}}\max_{p}|\log T_{p}-\log
T^{\prime}_{p}|}{\log|\Delta_{n}|}\to\infty,$ (1.5)

the minima

$\mathfrak{C}_{\Delta_{n}}(\Lambda):=\min_{T\in\Delta_{n}}c_{B_{T}}(\Lambda)$

satisfy

$|\Delta_{n}|^{1/d}\cdot\mathfrak{C}_{\Delta_{n}}\xLongrightarrow[\mu_{d}]{}\operatorname{Wei}\big{(}2^{-(d-1)/d}\zeta(d)^{1/d},d\big{)},$

where $\operatorname{Wei}$ denotes the Weibull distribution.

######  Remark 1.4.

Here and throughout this paper, $\zeta$ denotes Riemann’s $\zeta$-function. We
also recall that the _Weibull distribution_ $\operatorname{Wei}(\lambda,a)$
with scale $\lambda$ and shape $a$ is defined by the distribution function
$1-\exp({-(\lambda t)^{a}})$.

###  1.2. Application II: Minima of polynomial maps

We now turn to the very classical study of minimal values of homogeneous forms
restricted to lattice points (see [12, Ch. 6] and [20, Chapter IX]). Unlike
Theorem 1.3, this involves finding lattice points in _non-convex_ sets. We
consider the product form

$\Pi(x):=\prod_{p=1}^{d}|x_{p}|,\quad\textrm{for $x\in\mathbb{R}^{d}$}.$

The following result was proved by Kleinbock and Margulis ([16, Subsection
1.11]):

######  Theorem 1.5 (Kleinbock–Margulis).

For $\mu_{d}$-almost every $\Lambda\in X_{d}$,

$\limsup_{v\in\Lambda:\,\|v\|_{\infty}\to\infty}\;-\frac{\log\Pi(v)}{\log\log\|v\|_{\infty}}\geq
d-1.$ (1.6)

We establish a distributional version of this result. For boxes $B_{T}$ with
$T\in\mathscr{T}_{d}$ and a lattice $\Lambda$, we define

$m_{B_{T}}(\Lambda):=\inf\big{\\{}\Pi(v):\,0\neq v\in\Lambda\cap
B_{T}\big{\\}}.$

With this notation, we show:

######  Theorem 1.6.

Let $\Delta_{n}$ be a sequence of finite subsets of $\mathscr{T}_{d}$
satisfying (1.5), and

$\mathfrak{M}_{\Delta_{n}}(\Lambda):=\min_{T\in\Delta_{n}}m_{B_{T}}(\Lambda).$

Then

$|\Delta_{n}|(\log|\Delta_{n}|)^{d-1}\cdot\mathfrak{M}_{\Delta_{n}}\xLongrightarrow[\mu_{d}]{}\operatorname{Wei}\big{(}2^{-(d-1)}\zeta(d)/c_{d},1\big{)},$

where $c_{d}$ is the volume of the simplex
$\\{u\in[0,1]^{d-1}\,:\,u_{1}+\cdots+u_{d-1}\leq 1\big{\\}}$.

######  Remark 1.7.

We stress that Theorem 1.5 is in fact a direct consequence of Theorem 1.6 (see
Proposition A.1 in the Appendix).

This result can be generalized as follows. Let $F$ be a nontrivial homogeneous
polynomial on $\mathbb{R}^{d}$. For a lattice $\Lambda\in X_{d}$, we consider
the family of lattices

$\Lambda_{T}:=\\{(T_{1}\lambda_{1},\ldots,T_{d}\lambda_{d}):\,\lambda\in\Lambda\\},\quad
T=(T_{1},\ldots,T_{d})\in\mathscr{T}_{d}.$

Extremal behaviour for this family of lattices was studied in [12, Ch. 3]. We
investigate the asymptotics of the minima

$m_{T}(F,\Lambda):=\min\left\\{|F(v)|:\,0\neq
v\in\Lambda_{T}\cap[-1,1]^{d}\right\\}.$

######  Theorem 1.8.

There exist $c>0$, $a\in\mathbb{Q}^{+}$, $b\in\mathbb{N}_{0}$ such that for
any sequence of finite subsets $\Delta_{n}$ of $\mathscr{T}_{d}$ satisfying
(1.5), and

$\mathfrak{M}_{\Delta_{n}}(F,\Lambda):=\min_{T\in\Delta_{n}}m_{T}(F,\Lambda),$

one has

$|\Delta_{n}|^{a}(\log|\Delta_{n}|)^{b}\cdot\mathfrak{M}_{\Delta_{n}}(F,\cdot)\xLongrightarrow[\mu_{d}]{}\operatorname{Wei}\big{(}c,a\big{)}.$

###  1.3. Application III: Khinchin’s Theorem

Given $\alpha\in\mathbb{R}^{d-1}$, we define

$k_{T}(\alpha):=\min\big{\\{}T^{1/(d-1)}\|q\alpha-p\|_{\infty}\,:\,p\in\mathbb{Z}^{d-1},\enskip
1\leq q\leq T\big{\\}},\quad\textrm{for $T\geq 1$}.$

By Dirichlet’s Approximation Theorem, we have $k_{T}(\alpha)\leq 1$ for all
$\alpha\in\mathbb{R}^{d-1}$. Furthermore, according to Davenport and Schmidt
[8, Theorem 1], for Lebesgue almost every $\alpha\in[0,1]^{d-1}$,

$\limsup_{T\to\infty}k_{T}(\alpha)=1.$ (1.7)

On the other hand,

######  Theorem 1.9 (Khinchin).

For Lebesgue almost every $\alpha\in[0,1]^{d-1}$,

$\limsup_{T\to\infty}\,-\frac{\log k_{T}(\alpha)}{\log\log
T}\geq-\frac{1}{d-1},$ (1.8)

The limits (1.7) and (1.8) indicate that the function $T\mapsto k_{T}(\alpha)$
fluctuates quite wildly, and we wish to study its extremal behavior. More
precisely, we show that minima of the function $T\mapsto k_{T}(\cdot)$ on
sufficiently lacunary subsets of $[1,\infty)$ exhibit non-trivial Weibull
asymptotics with respect to the Lebesgue measure on $[0,1]^{d-1}$.

More generally, we can deal with approximation of systems of linear forms. Let
$d\geq 3$ and write $d=d_{1}+d_{2}$ with $d_{1},d_{2}\geq 1$. We set

$\mathscr{T}_{\underline{d}}^{+}:=\big{\\{}T=(T_{1},\ldots,T_{d})\in\mathscr{T}_{d}\,:\,T_{1},\ldots,T_{d_{1}}\geq
1,\,T_{d_{1}+1},\ldots,T_{d}\leq 1\big{\\}}.$ (1.9)

For $T\in\mathscr{T}_{\underline{d}}^{+}$, we define

$\lfloor T\rfloor_{\underline{d}}:=\min\big{(}\log T_{1},\ldots,\log
T_{d_{1}},-\log T_{d_{1}+1},\ldots,-\log T_{d}\big{)}.$ (1.10)

Given $T\in\mathscr{T}_{\underline{d}}^{+}$,
$\alpha\in\hbox{Mat}_{d_{1},d_{2}}([0,1])$ and $k>0$, we consider the
inequalities

$\Big{|}p_{l}+\sum_{j=1}^{d_{2}}\alpha_{lj}q_{j}\Big{|}\leq
k\,T_{l}^{-1},\,l=1\ldots,d_{1},\quad|q_{j}|\leq
T_{d_{1}+j},\,j=1,\ldots,d_{2},$

with $p\in\mathbb{Z}^{d_{1}}$ and $q\in\mathbb{Z}^{d_{2}}$ and define

$k_{T}(\alpha):=\min\Big{\\{}{\max}_{l}\;T_{l}\Big{|}p_{l}+\sum_{j=1}^{d_{2}}\alpha_{lj}q_{j}\Big{|}\,:\,\,\,p\in\mathbb{Z}^{d_{1}},\;\;q\in\mathbb{Z}^{d_{2}},|q_{j}|\leq
T_{{d_{1}}+j}\Big{\\}}.$

######  Theorem 1.10.

For a sequence of finite subsets $\Delta_{n}$ of
$\mathscr{T}_{\underline{d}}^{+}$ such that

$|\Delta_{n}|\to\infty,\quad\frac{\displaystyle\min_{T,T^{\prime}\in\Delta_{n}}\max_{p}|\log
T_{p}-\log
T^{\prime}_{p}|}{\log|\Delta_{n}|}\to\infty,\quad\frac{\displaystyle\min_{T\in\Delta_{n}}\lfloor
T\rfloor_{\underline{d}}}{\log|\Delta_{n}|}\to\infty,$ (1.11)

the minima

$\mathfrak{K}_{\Delta_{n}}(\alpha):=\min_{T\in\Delta_{n}}k_{T}(\alpha),\quad\textrm{for
$\alpha\in\hbox{\rm Mat}_{d_{1},d_{2}}([0,1])$},$

satisfy

$|\Delta_{n}|^{1/d_{1}}\cdot\mathfrak{K}_{\Delta_{n}}\xLongrightarrow[\hbox{\tiny\rm
Leb}]{}\operatorname{Wei}\big{(}2^{-(d-1)/d_{1}}\zeta(d)^{1/d_{1}},d_{1}\big{)}.$

######  Remark 1.11.

We again stress that Theorem 1.9 follows directly from Theorem 1.10 (see
Proposition A.2 in the Appendix).

###  1.4. Application IV: Gallagher’s Theorem

Now we consider a multiplicative version of Khinchin’s Theorem. Given
$\alpha\in\mathbb{R}^{d-1}$, we define

$g_{T}(\alpha):=\min\left\\{T\cdot{\prod}_{l=1}^{d-1}|q\alpha_{l}-p_{l}|\,:\,p\in\mathbb{Z}^{d-1},\enskip
1\leq q\leq T\right\\},\quad\textrm{for $T\geq 1$}.$

The following logarithm law was proved by Gallagher [10]:

######  Theorem 1.12 (Gallagher).

For a.e. $\alpha\in\mathbb{R}^{d-1}$,

$\limsup_{T\to\infty}\,-\frac{\log g_{T}(\alpha)}{\log\log T}\geq d-1,$ (1.12)

More generally, given $T\in\mathscr{T}_{\underline{d}}^{+}$,
$\alpha\in\hbox{Mat}_{d_{1},d_{2}}([0,1])$ and $g>0$, we consider the
inequalities

$\prod_{l=1}^{d_{1}}\Big{|}p_{l}+\sum_{j=1}^{d_{2}}\alpha_{lj}q_{j}\Big{|}\leq
g\,\left({\prod}_{l=1}^{d_{1}}T_{l}\right)^{-1},\quad|q_{j}|\leq
T_{d_{1}+j},\,j=1,\ldots,d_{2},$

with $p\in\mathbb{Z}^{d_{1}}$ and $q\in\mathbb{Z}^{d_{2}}$ and define

$g_{T}(\alpha):=\min\left\\{\prod_{l=1}^{d_{1}}T_{l}\Big{|}p_{l}+\sum_{j=1}^{d_{2}}\alpha_{lj}q_{j}\Big{|}\,:\,\,\,\begin{tabular}[]{ll}$q\in\mathbb{Z}^{d_{2}},\;|q_{j}|\leq
T_{{d_{1}}+j}$,\\\
$p\in\mathbb{Z}^{d_{1}},\;\Big{|}p_{l}+\sum_{j=1}^{d_{2}}\alpha_{lj}q_{j}\Big{|}\leq
T_{l}^{-1}$\end{tabular}\right\\}.$

######  Theorem 1.13.

For a sequence of finite subsets $\Delta_{n}$ of
$\mathscr{T}_{\underline{d}}^{+}$ satisfying (1.11), the minima

$\mathfrak{G}_{\Delta_{n}}(\alpha):=\min_{T\in\Delta_{n}}g_{T}(\alpha),\quad\textrm{for
$\alpha\in\hbox{\rm Mat}_{d_{1},d_{2}}([0,1])$},$

satisfy

$|\Delta_{n}|(\log|\Delta_{n}|)^{d_{1}-1}\cdot\mathfrak{G}_{\Delta_{n}}\xLongrightarrow[\hbox{\tiny\rm
Leb}]{}\operatorname{Wei}\big{(}2^{-(d_{1}-1)}\zeta(d)/c_{d_{1}},1\big{)}.$

######  Remark 1.14.

Just as before, it can be shown (see Proposition A.2 in the appendix) that
Theorem 1.13 implies Gallagher’s logarithm law (Theorem 1.12).

###  1.5. Structure of the paper

In the next section we discuss the theorems above in a very general context.
Theorem 2.7 encompasses Theorem 1.3, Theorem 1.6 and Theorem 1.8 (see the
discussion after Remark 2.9 for more details), while Theorem 2.7 generalizes
both Theorem 1.10 and Theorem 1.13.

##  2\. General framework

Let $X_{d}$ denote the space of unimodular lattices in $\mathbb{R}^{d}$. Given
a pair $(\eta,C)$, where $\eta$ is a non-negative continuous function on
$\mathbb{R}^{d}$ and $C$ is a convex subset of $\mathbb{R}^{d}$, we define the
function $\widetilde{\eta}:X_{d}\rightarrow[0,\infty)$ by

$\widetilde{\eta}(\Lambda):=\inf\big{\\{}\eta(\lambda)\,:\,\lambda\in\Lambda\cap
C,\enskip\lambda\neq 0\big{\\}},\quad\textrm{for $\Lambda\in X_{d}$}.$ (2.1)

Motivated by the problems introduced in the previous section, we want to
investigate the asymptotic behaviour of _minima_ of $\widetilde{\eta}$ along
_sparse_ samples taken from group orbits in $X_{d}$. More precisely, let $H$
be a closed subgroup of $\operatorname{SL}_{d}(\mathbb{R})$. Given a finite
subset $F\subset H$, we define

$\widetilde{\eta}_{F}(\Lambda):=\min_{h\in
F}\,\widetilde{\eta}(h.\Lambda),\quad\textrm{for $\Lambda\in X_{d}$},$ (2.2)

and consider the following problem

######  Problem 2.1.

Let $\theta$ be a Borel probability measure on $X_{d}$ and let $(F_{n})$ be a
sequence of finite subsets of $H$ such that $|F_{n}|\rightarrow\infty$. Can we
find a sequence $(\delta_{n})$ of positive real numbers such that
$(\widetilde{\eta}_{F_{n}}/\delta_{n})$ has a non-trivial
$\theta$-distributional limit?

Let us make a few remarks before we state our main theorems concerning this
question (see Theorem 2.7 and Theorem 2.10 below). Note that for all $t\geq
0$,

$\big{\\{}\widetilde{\eta}_{F_{n}}<t\big{\\}}=\bigcup_{h\in
F_{n}}h^{-1}.\big{\\{}\Lambda\in X_{d}\,:\,\Lambda\cap
C(t)\neq\\{0\\}\big{\\}},\vspace{-0.2cm}$ (2.3)

where

$C(t):=\big{\\{}x\in C\,:\,\eta(x)<t\big{\\}}.$

Hence, to study the $\theta$-distributional asymptotics of the sequence
$(\eta_{F_{n}}/\delta_{n})$, we need to understand

$\theta\Big{(}\big{\\{}\Lambda\in
X_{d}\,:\,N_{n}(\Lambda;\delta_{n}t)>0\big{\\}}\Big{)},\quad\textrm{as
$n\rightarrow\infty$},$

where

$N_{n}(\Lambda;t):=\\#\big{\\{}h\in F_{n}\,:\,h.\Lambda\cap
C(t)\neq\\{0\\}\big{\\}}.$ (2.4)

In what follows, we want to isolate conditions on $(\eta,C)$ and $(F_{n})$
which ensure that we can find a positive sequence $(\delta_{n})$ such that,
for some constants $a,m_{o}>0$,

$N_{n}(\,\cdot\,;\delta_{n}t)\xLongrightarrow[\theta]{}\operatorname{Poi}\big{(}m_{o}t^{a}\big{)},\quad\textrm{for
all $t\geq 0$},\vspace{-0.2cm}$ (2.5)

where $\operatorname{Poi}(m)$ denotes the standard Poisson distribution with
mean $m$, and the arrow denotes convergence in distribution (with respect to
$\theta$). We recall that (2.5) just means that for all $t\geq 0$ and for all
non-negative integers $k$,

$\lim_{n\rightarrow\infty}\theta\Big{(}\\{\Lambda\in
X_{d}\,:\,N_{n}(\Lambda;\delta_{n}t)=k\\}\Big{)}=\frac{(m_{o}t^{a})^{k}}{k!}e^{-m_{o}t^{a}}.$

Let us suppose that (2.5) holds. Then, by summing over all $k>0$, we see that

$\theta\Big{(}\big{\\{}\widetilde{\eta}_{F_{n}}/\delta_{n}\leq
t\big{\\}}\Big{)}=\theta\Big{(}\\{\Lambda\in
X_{d}\,:\,N_{n}(\Lambda;\delta_{n}t)>0\\}\Big{)}\rightarrow
1-e^{-m_{o}t^{a}},$

for all $t\geq 0$. In other words, if (2.5) holds, then

$\widetilde{\eta}_{F_{n}}/\delta_{n}\xLongrightarrow[\theta]{}\operatorname{Wei}\big{(}m_{o}^{-1/a},a\big{)},\quad\textrm{as
$n\rightarrow\infty$}.\vspace{-0.1cm}$ (2.6)

This shows that the Weibull asymptotics of the form (2.6) can be deduced from
the Poisson approximation of the form (2.5).

###  2.1. Volume regularity

In order to establish Poisson approximation of the form (2.5) (and
consequently also Weibull asymptotics of the form (2.6)), we need to assume
that the growth of the volumes of the sets $C(t)$ is stable under small
perturbations. The following definition captures a convenient form of
stability which is sufficient for our purposes.

######  Definition 2.2.

Let $a,c>0$ and $b\geq 0$. The pair $(\eta,C)$ is _$(a,b,c)$ -regular_ if

  * (i)

$\eta$ is locally Lipschitz, and $\gamma$-homogeneous for some $\gamma\geq 0$.

  * (ii)

$C(t)$ is a bounded subset of $\mathbb{R}^{d}$ for every $t\geq 0$ and there
exists a semi-norm $\tau$ such that $C=\\{\tau\leq 1\\}$.

  * (iii)

The limit

$c=\lim_{t\rightarrow 0^{+}}\frac{\operatorname{Vol}_{d}(C(t))}{t^{a}(-\log
t)^{b}}$

exists.

We give several examples of $(a,b,c)$-regular pairs $(\eta,C)$.

######  Example 2.3 (Norm balls).

Let $\|\cdot\|$ be a norm on $\mathbb{R}^{d}$, and let

$\eta=\tau=\|\cdot\|\quad\textrm{and}\quad C=\\{\tau\leq 1\\}.$

Then $\eta$ is clearly Lipschitz and $1$-homogeneous, and (iii) holds with

$a=d,\enskip b=0\quad\textrm{and}\quad
c=\operatorname{Vol}_{d}\big{(}\\{\|\cdot\|\leq 1\\}\big{)}.$

######  Example 2.4 (Products of linear forms).

Let

$\eta(x)=|x_{1}\cdots
x_{d}|,\quad\textrm{and}\quad\tau(x)={\max}_{k}\,|x_{k}|,\quad\textrm{for
$x\in\mathbb{R}^{d}$},$

and $C=\\{\tau\leq 1\\}$. Then $\eta$ is locally Lipschitz and
$d$-homogeneous, and $\tau$ is a norm. To verify (iii), one can use the
coordinate system and the integration formula from [4, Lemma 4.1]. Using the
coordinate system from [4], the set

$\\{x\in\mathbb{R}^{d}:\,0<x_{1},\ldots,x_{d}\leq 1,\,x_{1}\cdots x_{d}\leq
t\\}$

corresponds to

$\\{(u,s)\in\mathbb{R}^{d-1}\times\mathbb{R}:\,\,u_{1},\ldots,u_{d-1}\leq
0,\,u_{1}+\cdots+u_{d-1}\geq s,\,s\leq\log t\\},$

so that

$\operatorname{Vol}_{d}(C(t))=2^{d}c_{d}\int_{-\infty}^{\log
t}e^{s}s^{d-1}\,ds,$

where $c_{d}$ is the volume of the simplex
$\\{u\in[0,1]^{d-1}\,:\,u_{1}+\ldots+u_{d-1}\leq 1\big{\\}}$. Hence, (iii)
holds with

$a=1\quad\textrm{and}\quad b=d-1\quad\textrm{and}\quad c=2^{d}c_{d}.$

######  Example 2.5 (Polynomial maps).

Let $F$ be a nontrivial homogeneous polynomial map on $\mathbb{R}^{d}$. Let

$\eta(x)=|F(x)|,\quad\textrm{and}\quad\tau(x)={\max}_{k}\,|x_{k}|,\quad\textrm{for
$x\in\mathbb{R}^{d}$},$

and $C=\\{\tau\leq 1\\}$. Then $\eta$ is locally Lipschitz and
$d$-homogeneous, and $\tau$ is a norm. It remains to verify that the volume of
the sets

$C(t)=\\{x\in\mathbb{R}^{d}:\,\,|F(x)|\leq t,\,|x_{1}|,\ldots,|x_{d}|\leq
1\\}$

has the stated asymptotics as $t\to 0^{+}$. This can be deduce using the
resolution of singularities that there exist $c>0$, $a\in\mathbb{Q}^{+}$,
$b\in\mathbb{N}_{0}$ such that

$\operatorname{Vol}_{d}(C(t))\sim c\,t^{a}(-\log t)^{b}\quad\hbox{as $t\to
0^{+}$.}$

We refer, for instance, to [11, Th. 1] for a self-contained treatment.

######  Example 2.6 (Normed strips).

Let $\underline{d}=(d_{1},d_{2})$ with $d=d_{1}+d_{2}$. Suppose that
$\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ are norms on $\mathbb{R}^{d_{1}}$ and
$\mathbb{R}^{d_{2}}$ respectively. Define

$\eta(x)=\|x^{(1)}\|_{1}\quad\textrm{and}\quad\tau(x)=\|x^{(2)}\|_{2},\quad\textrm{for
$x=(x^{(1)},x^{(2)})\in\mathbb{R}^{d_{1}}\times\mathbb{R}^{d_{2}}$},$

and $C=\\{\tau\leq 1\\}$. Then $\eta$ is $1$-homogeneous, $\tau$ is a semi-
norm on $\mathbb{R}^{d}$ and (iii) holds with

$a=d_{1}\quad\textrm{and}\quad b=0\quad\textrm{and}\quad
c=\operatorname{Vol}_{d_{1}}\big{(}\\{\|\cdot\|_{1}\leq
1\\}\big{)}\cdot\operatorname{Vol}_{d_{2}}\big{(}\\{\|\cdot\|_{2}\leq
1\\}\big{)}.$

###  2.2. Poisson asymptotics with respect to $\mu_{d}$

For $T\in\mathscr{T}_{d}$, we set

$h(T):=\operatorname{Diag}\big{(}T_{1},\ldots,T_{d}\big{)}\in\operatorname{SL}_{d}(\mathbb{R}).$

Then

$H_{d}:=\big{\\{}h(T)\,:\,T\in\mathscr{T}_{d}\big{\\}}$

is a closed subgroup of $\operatorname{SL}_{d}(\mathbb{R})$. We will be
interested in analyzing the minima $\widetilde{\eta}_{F}(\Lambda)=\min_{h\in
F}\,\widetilde{\eta}(h.\Lambda)$ as above, where $F$ runs over sparse finite
subset of $H_{d}$ and $\Lambda$ is generic with respect to $\mu_{d}$. To
formulate this sparseness condition precisely, we define the distance

$\operatorname{dist}_{d}(h(T),h(T^{\prime})):=\max\big{\\{}\big{|}\log
T_{k}-\log T_{k}^{\prime}\big{|}\,:\,k=1,\ldots,d\big{\\}},\quad
T,T^{\prime}\in\mathscr{T}_{d}.$

If $F$ is a finite subset of $H_{d}$, we define the _spread_ $v_{d}(F)$ by

$v_{d}(F):=\min\big{\\{}\operatorname{dist}_{d}(h,h^{\prime})\,:\,h,h^{\prime}\in
F,\enskip h\neq h^{\prime}\big{\\}}.$ (2.7)

With this notation, we state our first main theorem:

######  Theorem 2.7.

Let $d\geq 3$ and let $(F_{n})$ be a sequence of finite subsets of $H_{d}$
such that

$|F_{n}|\rightarrow\infty\quad\textrm{and}\quad\frac{v_{d}(F_{n})}{\log|F_{n}|}\rightarrow\infty,\quad\textrm{as
$n\rightarrow\infty$}.$

Suppose that the pair $(\eta,C)$ is $(a,b,c)$-regular and set
$\delta_{n}:=|F_{n}|^{-1/a}\cdot(\log|F_{n}|)^{-b/a}$. Define the sequences
$(\widetilde{\eta}_{F_{n}})$ and $(N_{n})$ as in (2.2) and (2.4) respectively.
Then,

$N_{n}(\,\cdot\,;\delta_{n}u)\xLongrightarrow[\mu_{d}]{}\operatorname{Poi}\big{(}m_{o}u^{a}\big{)},\quad\textrm{for
all $u\geq 0$},$

where $m_{o}:=ca^{b}/2\zeta(d)$. In particular,

$\widetilde{\eta}_{F_{n}}/\delta_{n}\xLongrightarrow[\mu_{d}]{}\operatorname{Wei}\big{(}(2\zeta(d))^{1/a}/(ca^{b})^{1/a},a\big{)},\quad
n\rightarrow\infty.$

######  Remark 2.8.

To get a better idea of what the condition
$v_{d}(F_{n})/\log|F_{n}|\rightarrow\infty$ means, it might be instructive to
consider the case when the sets $F_{n}$ are contained in a fixed one-parameter
family in $H_{d}$. More precisely, we define

$T_{s}=(s,\ldots,s,s^{-1/(d-1)}),\quad\textrm{for $s>0$}.$

If $0<s_{n,1}<\ldots<s_{n,n}$, let
$F_{n}=\\{h(T_{s_{n,1}}),\ldots,h(T_{s_{n,n}})\\}$. Then
$v_{d}(F_{n})/\log|F_{n}|\rightarrow\infty$ if and only if

$\lim_{n\rightarrow\infty}\frac{\log\left(\min\left\\{\frac{s_{n,k+1}}{s_{n,k}}\,:\,k=1,\ldots,n-1\right\\}\right)}{\log|F_{n}|}=\infty.$

In other words, $v_{d}(F_{n})/\log|F_{n}|\rightarrow\infty$ if the lacunarity
constants of the sets $\\{s_{n,1},\ldots,s_{n,n}\\}$ grow faster than $\log
n$.

######  Remark 2.9.

We do not know if Theorem 2.7 holds when $d=2$. Our proof relies on precise
small volume asymptotics in Rogers’ formula (see Lemma 5.3), which we
currently only know how to establish for $d\geq 3$ (see Remark 5.4).

Now we show that Theorems 1.3–1.8 are immediate consequences of Theorem 2.7.

###### Proof of Theorem 1.3.

Let

$\eta=\tau=\|\cdot\|_{\infty},\quad C=\\{\tau\leq 1\\}=[-1,1]^{d}.$

Using the notation of Theorem 1.3, we obtain

$\displaystyle\widetilde{\eta}\big{(}h(T)^{-1}\Lambda\big{)}$
$\displaystyle=\inf\big{\\{}\|\rho\|_{\infty}:\,\rho\in
h(T)^{-1}\big{(}\Lambda\backslash\\{0\\}\big{)}\cap C\big{\\}}$
$\displaystyle=\inf\big{\\{}\|h(T)^{-1}\lambda\|_{\infty}:\,\lambda\in\Lambda\backslash\\{0\\}\cap
h(T)C\big{\\}}$
$\displaystyle=\min\big{\\{}{\max}_{i}\,T_{i}^{-1}|\lambda_{i}|\,:\,\,\lambda\in\Lambda\backslash\\{0\\}\cap
B_{T}\big{\\}}$
$\displaystyle=\min\big{\\{}c>0\,:\,\,\exists\lambda\in\Lambda\backslash\\{0\\}\cap
cB_{T}\big{\\}}=c_{B_{T}}(\Lambda).$

Given a finite set $\Delta_{n}$, we take
$F_{n}=\\{h(T)^{-1}:\,T\in\Delta_{n}\\}$. Then

$\mathfrak{C}_{\Delta_{n}}(\Lambda)=\widetilde{\eta}_{F_{n}}(\Lambda),$

so that we can apply Theorem 2.7 to study the minima
$\mathfrak{C}_{\Delta_{n}}(\Lambda)$ in the Minkowski Theorem. Finally, we
note that $(\|\cdot\|_{\infty},C)$ is $(d,0,2^{d})$-regular (see Example 2.3).
∎

###### Proof of Theorem 1.6.

Let

$\eta(x)=|x_{1}\cdots x_{d}|,\quad\tau(x)=\|x\|_{\infty},\quad C=\\{\tau\leq
1\\}=[-1,1]^{d}.$

Using the notation of Theorem 1.6, we obtain

$\displaystyle\widetilde{\eta}\big{(}h(T)^{-1}\Lambda\big{)}$
$\displaystyle=\inf\big{\\{}\Pi(\rho):\,\rho\in
h(T)\big{(}\Lambda\backslash\\{0\\}\big{)}\cap C\big{\\}}$
$\displaystyle=\inf\big{\\{}\Pi(\lambda):\,\lambda\in\Lambda\backslash\\{0\\}\cap
h(T)C\big{\\}}$
$\displaystyle=\inf\big{\\{}\Pi(\lambda):\,\lambda\in\Lambda\backslash\\{0\\}\cap
B_{T}\big{\\}}=m_{B_{T}}(\Lambda).$

Hence, the minima $\mathfrak{M}_{\Delta_{n}}(F,\Lambda)$ can be studied with
the help of Theorem 2.7. Finally, we note that the pair $(\eta,C)$ is
$(1,d-1,2^{d}c_{d})$-regular (see Example 2.4). ∎

###### Proof of Theorem 1.8.

Let

$\eta(x)=|F(x)|,\quad\tau(x)=\|x\|_{\infty},\quad C=\\{\tau\leq
1\\}=[-1,1]^{d}.$

Using the notation of Theorem 1.8, we obtain

$\displaystyle\widetilde{\eta}(h(T)\Lambda)$
$\displaystyle=\inf\big{\\{}|F(\rho)|:\,\rho\in
h(T)\big{(}\Lambda\backslash\\{0\\}\big{)}\cap C\big{\\}}$
$\displaystyle=\inf\big{\\{}|F(\rho)|:\,\lambda\in\Lambda_{T}\backslash\\{0\\}\cap[-1,1]^{d}\big{\\}}=m_{T}(F,\Lambda).$

Hence, the minima $\mathfrak{M}_{\Delta_{n}}(\Lambda)$ can be studied with the
help of Theorem 2.7. Finally, we note that the pair $(\eta,C)$ is
$(a,b,c)$-regular (see Example 2.5). ∎

###  2.3. Poisson approximation with respect to the horospherical measure

Our second main result involves a measure supported on the sets of lattices
arising in Diophantine approximation. Let $d\geq 3$ and
$\underline{d}=(d_{1},d_{2})$ such that $d_{1},d_{2}\geq 1$ and
$d=d_{1}+d_{2}$. Given a matrix
$\alpha\in\operatorname{Mat}_{d_{1},d_{2}}([0,1])$, we define the lattice
$\Lambda_{\alpha}\in X_{d}$ by

$\Lambda_{\alpha}=\big{\\{}(p_{1}+(\alpha q)_{1},\ldots,p_{d_{1}}+(\alpha
q)_{d_{1}},q)\,:\,\,p\in\mathbb{Z}^{d_{1}},\enskip
q\in\mathbb{Z}^{d_{2}}\big{\\}},$ (2.8)

and the probability measure $\nu_{\underline{d}}$ on $X_{d}$ by

$\int_{X_{d}}\varphi\,d\nu_{\underline{d}}=\int_{\operatorname{Mat}_{d_{1},d_{2}}([0,1])}\varphi(\Lambda_{\alpha})\,d\alpha,\quad\textrm{for
$\varphi\in C_{b}(X_{d})$},$

where $d\alpha$ denotes the Lebesgue probability measure on
$\operatorname{Mat}_{d_{1},d_{2}}([0,1])\cong[0,1]^{d_{1}d_{2}}$.

Let

$H^{+}_{\underline{d}}:=\big{\\{}h(T)\,:\,T\in\mathscr{T}_{\underline{d}}^{+}\big{\\}}.$

For $h(T)\in H^{+}_{\underline{d}}$, we set $\lfloor
h(T)\rfloor_{\underline{d}}:=\lfloor T\rfloor_{\underline{d}}$ (see (1.10)).
If $F$ is a finite subset of $H_{\underline{d}}^{+}$, we define the _spread_
$v_{\underline{d}}^{+}(F)$ by

$v^{+}_{\underline{d}}(F):=\min\big{\\{}\min\big{(}\lfloor
h\rfloor,\operatorname{dist}_{d}(h,h^{\prime})\big{)}\,:\,h\neq h^{\prime}\in
F\big{\\}}.$ (2.9)

Our second main result is about distribution of the minima
$\widetilde{\eta}_{F}(\Lambda)=\min_{h\in F}\,\widetilde{\eta}(h.\Lambda)$,
where $F$ runs over sparse finite subset of $H^{+}_{\underline{d}}$, and the
lattice $\Lambda$ is generic with respect to the measure $\nu_{d}$.

######  Theorem 2.10.

Let $d\geq 3$, and let $(F_{n})$ be a sequence of finite subsets of
$H^{+}_{\underline{d}}$ such that

$|F_{n}|\rightarrow\infty\quad\textrm{and}\quad\frac{v^{+}_{\underline{d}}(F_{n})}{\log|F_{n}|}\rightarrow\infty,\quad\textrm{as
$n\rightarrow\infty$}.$

Suppose that the pair $(\eta,C)$ is $(a,b,c)$-regular and set
$\delta_{n}:=|F_{n}|^{-1/a}\cdot(\log|F_{n}|)^{-b/a}$. Then,

$N_{n}(\,\cdot\,;\delta_{n}u)\xLongrightarrow[\nu_{\underline{d}}]{}\operatorname{Poi}\big{(}m_{o}u^{a}\big{)},\quad\textrm{for
all $u\geq 0$},$

where $m_{o}:=ca^{b}/2\zeta(d)$. In particular,

$\widetilde{\eta}_{F_{n}}/\delta_{n}\xLongrightarrow[\nu_{\underline{d}}]{}\operatorname{Wei}\big{(}(2\zeta(d))^{1/a}/(ca^{b})^{1/a},a\big{)},\quad
n\rightarrow\infty.$

######  Remark 2.11.

We have the same issues when $d=d_{1}+d_{2}=2$ as in Remark 2.9.

We shall show that Theorems 1.10 and 1.13 are direct consequences of Theorem
2.10.

###### Proof of Theorem 1.10.

Let $\pi_{1}:\mathbb{R}^{d}\to\mathbb{R}^{d_{1}}$ and
$\pi_{2}:\mathbb{R}^{d}\to\mathbb{R}^{d_{2}}$ denote the projections on the
first $d_{1}$ coordinates and the last $d_{2}$ coordinates respectively. Let

$\eta(x)=\|\pi_{1}(x)\|_{\infty},\quad\tau(x)=\|\pi_{2}(x)\|_{\infty},\quad
C=\\{\tau\leq 1\\}.$

We note that the pair $(\eta,C)$ is $(d_{1},0,2^{d})$-regular (see Example
2.6). Then for $T\in\mathscr{T}_{\underline{d}}^{+}$,

$k_{T}(\alpha)=\widetilde{\eta}\big{(}h(T)\Lambda_{\alpha}\big{)},$

so that Theorem 1.10 is a consequence of Theorem 2.10. ∎

###### Proof of Theorem 1.13.

Let

$\eta(x)=\Pi(\pi_{1}(x)),\quad\tau(x)=\|x\|_{\infty},\quad C=\\{\tau\leq
1\\},$

where $\Pi(y)=|y_{1}\cdots y_{d_{1}}|$ for $y\in\mathbb{R}^{d_{1}}$. The pair
$(\eta,C)$ is $(1,d_{1}-1,2^{d}c_{d_{1}})$-regular, where $c_{d_{1}}$ is the
volume of the simplex $\\{u\in[0,1]^{d_{1}-1}\,:\,u_{1}+\ldots+u_{d_{1}-1}\leq
1\big{\\}}$. Indeed, this is a combination of Examples 2.5 and 2.6. Then for
$T\in\mathscr{T}_{\underline{d}}^{+}$,

$g_{T}(\alpha)=\widetilde{\eta}\big{(}h(T)\Lambda_{\alpha}\big{)},$

and Theorem 1.13 is a consequence of Theorem 2.10. ∎

###  2.4. Connections to earlier works

In Section 4 and Section 6, we explain how the results stated above can be
recast in the language of dynamical systems and shrinking targets (possibly
with respect to a non-invariant measure). There is a vast literature on
Poisson asymptotics (and thus on extremal laws) for shrinking targets, and we
will not attempt to make a comprehensive summary. The investigations in this
paper could be said to originate from some surprising observations concerning
Poisson approximation in the gap distribution of Laplacian eigenvalues by
Sinai [22, 23] (see also the subsequent work by Hirata [13] for a more
dynamical framework, closer to the spirit of this paper). In the setting of
homogeneous spaces, the distribution of extreme values of excursions for
partially hyperbolic one-parameter subgroups was studied by Pollicott [17] and
Kirsebom [15].

Recently, Dologopyat, Fayad and Liu [9] have conceptualized a lot of the early
developments by emphasizing the role of quantitative multiple mixing
($\mathcal{W}$-mixing in our paper). Their setup and arguments in [9, Section
3] are close to the ones presented in Section 4 below. In particular, their
notion of _simple admissible targets_ [9, Definition 3.2] captures a similar
form of volume growth stability under perturbations as our notion of
$(a,b,c)$-regularity, and their [9, Proposition 3.9] is close to our Theorem
4.4 (specialized to $\mathbb{R}$-flows).

###  2.5. An overview of the paper

In Section 3 we state a common generalization (Theorem 3.1) of Theorem 2.7 and
Theorem 2.10, and deduce these theorems from it, highlighting the role of
volume regularity. In Section 4 we discuss Poisson asymptotics for general
shrinking targets. We introduce the notion of $\mathcal{W}$-equidistribution
and show that $\mathcal{W}$-equidistribution ensures that we always have
Poisson asymptotics under some weak assumptions on the shrinking targets
(Theorem 4.4). In Section 6, we deduce Theorem 3.1 from Theorem 4.4. To do
this, we need small volume asymptotics for hitting sets in the space of
lattices. Such asymptotics are proved in Section 5 (Lemma 5.3), using
classical formulas of Siegel and Rogers.

##  3\. The role of volume regularity

In this section we extend Theorem 2.7 and Theorem 2.10 to a larger class of
shrinking targets (see Theorem 3.1 below). In Subsection 3.3 we explain how
$(a,b,c)$-regularity of the pair $(\eta,C)$ implies that the sequence
$(C(u\delta_{n}))$, which appear in both Theorem 2.7 and Theorem 2.10, fit
this framework, and thereby proving these theorems (assuming Theorem 3.1).

###  3.1. Definitions and notation

We denote by $\|\cdot\|_{\infty}$ the $\ell^{\infty}$-norm on
$\mathbb{R}^{d}$, and define

$D_{R}:=\big{\\{}x\in\mathbb{R}^{d}\,:\,\|x\|_{\infty}\leq
R\big{\\}},\quad\textrm{for $R\geq 0$}.$

If $t\in\mathbb{R}$ and $B\subset\mathbb{R}^{d}$, we write

$t\cdot B=\\{tx\,:\,x\in B\big{\\}}.$

We say that a subset $B\subset\mathbb{R}^{d}$ is _balanced_ if $tB\subset B$
for all $|t|\leq 1$. Given $\varepsilon>0$, let

$W_{\varepsilon}:=\big{\\{}g\in\operatorname{SL}_{d}(\mathbb{R})\,:\,\|\,g-\operatorname{id}\|_{\textrm{op}}<\varepsilon\big{\\}},$
(3.1)

where $\|\cdot\|_{\textrm{op}}$ denotes the operator norm on
$\operatorname{SL}_{d}(\mathbb{R})$ with respect to $\|\cdot\|_{\infty}$. If
$B$ is a subset of $\mathbb{R}^{d}$, we write

$W_{\varepsilon}.B=\\{g.x\,:\,g\in W_{\varepsilon},\enskip x\in B\big{\\}}.$

###  3.2. Poisson approximation

Let $(F_{n})$ be a sequence of finite subsets of $H_{d}$ such that
$|F_{n}|\rightarrow\infty$ and let $(B_{n})$ be a sequence of bounded and
balanced sets in $\mathbb{R}^{d}$. We define

$N_{n}(\Lambda):=\\#\big{\\{}h\in F_{n}\,:\,h.\Lambda\cap
B_{n}\neq\\{0\\}\big{\\}},\quad\textrm{for $\Lambda\in X_{d}$}.$ (3.2)

In the next subsection we show how one can prove Theorem 2.7 and Theorem 2.10
from the following theorem.

######  Theorem 3.1.

Let $d\geq 3$ and suppose that there exist

  * $(i)$

a sequence $(\varepsilon_{n})$ of positive integers such that
$\varepsilon_{n}\gg|F_{n}|^{-\sigma}$ for some $\sigma>0$.

  * $(ii)$

two sequences $(B_{n}^{-})$ and $(B_{n}^{+})$ of bounded and balanced subsets
of $\mathbb{R}^{d}$ such that

$W_{\varepsilon_{n}}.B_{n}^{-}\subset B_{n}\quad\textrm{and}\quad
W_{\varepsilon_{n}}.B_{n}\subset B_{n}^{+}$

and the limits

$\xi:=\lim_{n\rightarrow\infty}|F_{n}|\cdot\operatorname{Vol}_{d}(B_{n}^{+})=\lim_{n\rightarrow\infty}|F_{n}|\cdot\operatorname{Vol}_{d}(B_{n}^{-})$

exist.

Then,

  * (i)

If $v_{d}(F_{n})/\log|F_{n}|\rightarrow\infty$, then

$N_{n}\xLongrightarrow[\mu_{d}]{}\operatorname{Poi}\big{(}\xi/2\zeta(d)\big{)}.$

  * (ii)

If $F_{n}\subset H_{\underline{d}}^{+}$ for all $n$ and
$v_{\underline{d}}^{+}(F_{n})/\log|F_{n}|\rightarrow\infty$, then

$N_{n}\xLongrightarrow[\nu_{\underline{d}}]{}\operatorname{Poi}\big{(}\xi/2\zeta(d)\big{)}.$

###  3.3. Proofs of Theorem 2.7 and Theorem 2.10 assuming Theorem 3.1

Let $a,c>0$ and $b\geq 0$, and suppose that $(\eta,C)$ is a $(a,b,c)$-regular
pair. In particular,

$C(t)=\big{\\{}x\in\mathbb{R}^{d}\,:\,\eta(x)<t,\enskip\tau(x)\leq
1\big{\\}},\quad\textrm{for all $t\geq 0$},$

for some semi-norm $\tau$ on $\mathbb{R}^{d}$, and $C(t)$ is a balanced and
bounded subset of $\mathbb{R}^{d}$ for every $t\geq 0$.

Let $(F_{n})$ be a sequence of finite subsets of $H_{d}$ and set
$\delta_{n}=|F_{n}|^{-1/a}(\log|F_{n}|)^{-b/a}$. Fix $u>0$. We want to apply
Theorem 3.1 to the sets

$B_{n}=C(\delta_{n}u)\quad\textrm{and}\quad B_{n}^{\pm}=t_{n}^{\pm}\cdot
C(\delta_{n}(u\pm\beta_{n})),$ (3.3)

for some suitable sequences $(t^{\pm}_{n})$ and $(\beta_{n})$ of positive real
numbers. More precisely, we wish to choose four sequences
$(t_{n}^{\pm}),(\varepsilon_{n})$ and $(\beta_{n})$ such that

$\lim_{n\rightarrow\infty}t_{n}^{\pm}=1\quad\textrm{and}\quad\varepsilon_{n}\gg|F_{n}|^{-\sigma},\quad\textrm{for
some $\sigma>0$},$

and (for all sufficiently large $n$, depending on $u$),

  * (1)

$W_{\varepsilon_{n}}.(t_{n}^{-}\cdot C(\delta_{n}(u-\beta_{n})))\subset
C(\delta_{n}u)$.

  * (2)

$W_{\varepsilon_{n}}.C(\delta_{n}u)\subset t_{n}^{+}\cdot
C(\delta_{n}(u+\beta_{n}))$.

  * (3)

$\lim_{n}|F_{n}|\operatorname{Vol}_{d}\big{(}C(\delta_{n}(u\pm\beta_{n})\big{)}=c\cdot
u^{a}a^{b}$.

If we can do this, then Theorem 3.1, applied with $\xi=c\cdot u^{a}a^{b}$,
implies:

  * •

(Theorem 2.7) If $v_{d}(F_{n})/\log|F_{n}|\rightarrow\infty$, then

$N_{n}\xLongrightarrow[\mu_{d}]{}\operatorname{Poi}\big{(}cu^{a}a^{b}/2\zeta(d)\big{)},\quad\textrm{for
all $u>0$}.$

  * •

(Theorem 2.10) If $F_{n}\subset H_{\underline{d}}^{+}$ for all $n$ and
$v^{+}_{\underline{d}}(F_{n})/\log|F_{n}|\rightarrow\infty$, then

$N_{n}\xLongrightarrow[\nu_{\underline{d}}]{}\operatorname{Poi}\big{(}cu^{a}a^{b}/2\zeta(d)\big{)},\quad\textrm{for
all $u>0$}.$

Let us now address the choices of the four sequences
$(t_{n}^{\pm}),(\varepsilon_{n})$ and $(\beta_{n})$. The following lemma will
be useful.

######  Lemma 3.2.

There exists a constant $M$ such that

$W_{\varepsilon}.C(t)\subseteq(1+M\varepsilon)\cdot C(t+M\varepsilon),$

for all $\varepsilon,t\in[0,1]$.

###### Proof.

Fix $\varepsilon,t\in[0,1]$. Note that

$\displaystyle W_{\varepsilon}.C(t)$
$\displaystyle\subseteq\big{\\{}x\in\mathbb{R}^{d}\,:\,\exists\,y\in
C(t)\enskip\textrm{such
that}\enskip\|x-y\|_{\infty}<\varepsilon\|y\|_{\infty}\big{\\}}$
$\displaystyle\subseteq\big{\\{}x\in\mathbb{R}^{d}\,:\,\exists\,y\enskip\textrm{such
that}\enskip\|x-y\|_{\infty}<\varepsilon\|y\|_{\infty},\enskip\eta(y)<t,\enskip\tau(y)\leq
1\big{\\}}.$

Since $C(1)$ is a bounded subset of $\mathbb{R}^{d}$, there exists a constant
$M_{o}$ such that $C(1)\subset D_{M_{o}}$. Hence, for every $x\in
W_{\varepsilon}.C(t)$, we can find $y\in D_{M_{o}}$ such that

$\|x\|\leq(1+\varepsilon)\|y\|_{\infty},\enskip\eta(x)<t+|\eta(x)-\eta(y)|\quad\textrm{and}\quad\tau(x)\leq
1+|\tau(x)-\tau(y)|.$

Since both $\eta$ and $\tau$ are locally Lipschitz and $x,y\in D_{2M_{o}}$, we
can find a constant $L$ such that

$|\eta(x)-\eta(y)|\leq L\|x-y\|_{\infty}\leq
LM_{o}\,\varepsilon\quad\textrm{and}\quad|\tau(x)-\tau(y)|\leq
L\|x-y\|_{\infty}\leq LM_{o}\,\varepsilon$

We conclude that

$W_{\varepsilon}.C(t)\subseteq\big{\\{}x\in\mathbb{R}^{d}\,:\,\eta(x)<t+LM_{o}\,\varepsilon,\enskip\tau(x)\leq
1+LM_{o}\,\varepsilon\big{\\}}.$

Set $M:=LM_{o}$. Since $\eta$ is $\gamma$-homogeneous for some $\gamma\geq 0$
and $\tau$ is $1$-homogeneous, we thus see that

$W_{\varepsilon}.C(t)\subset(1+M\varepsilon)\cdot
C\Big{(}\frac{t+M\varepsilon}{(1+M\varepsilon)^{\gamma}}\Big{)}\subset(1+M\varepsilon)\cdot
C(t+M\varepsilon),$

which finishes the proof. ∎

Let $(s_{n})$ be a sequence of positive real numbers converging to zero (to be
specified later), and set

$\varepsilon_{n}=\delta_{n}s_{n}\quad\textrm{and}\quad
t_{n}^{-}=\frac{1}{1+M\varepsilon_{n}}\quad\textrm{and}\quad
t_{n}^{+}=1+M\varepsilon_{n}\quad\textrm{and}\quad\beta_{n}=Ms_{n},$

where $M$ is as in the lemma above. Note that $t_{n}^{\pm}\rightarrow 1$ as
$n\rightarrow\infty$. Since $(s_{n})$ and $(\delta_{n})$ converge to zero, we
may without loss of generality assume that $n$ is large enough so that
$\beta_{n}<u$ and $\delta_{n}u\leq 1$, in which case we can apply Lemma 3.2
with

$t=\delta_{n}u\quad\textrm{and}\quad\varepsilon=\varepsilon_{n}=\delta_{n}s_{n},$

and readily deduce the inclusions (1) and (2) above. Concerning the limit in
(3), we argue as follows. Firstly, since $\delta_{n}(u\pm\beta_{n})\rightarrow
0$ and $(\eta,C)$ is $(a,b,c)$-regular, we have

$c=\lim_{n\rightarrow\infty}\frac{\operatorname{Vol}_{d}\big{(}C(\delta_{n}(u\pm\beta_{n}))\big{)}}{\delta_{n}^{a}(u\pm\beta_{n})^{a}(-\log\delta_{n}(u\pm\beta_{n}))^{b}}.$

Hence, since $\delta_{n}=|F_{n}|^{-1/a}(\log|F_{n}|)^{-b/a}$,

$\displaystyle\lim_{n\rightarrow\infty}|F_{n}|\operatorname{Vol}_{d}\big{(}C(\delta_{n}(u\pm\beta_{n}))\big{)}$
$\displaystyle=c\cdot\lim_{n\rightarrow\infty}|F_{n}|\delta_{n}^{a}(u\pm\beta_{n})^{a}(-\log\delta_{n}(u\pm\beta_{n}))^{b}$
$\displaystyle=c\cdot\lim_{n\rightarrow\infty}\frac{(u\pm\beta_{n})^{a}(a\log|F_{n}|+r^{\pm}_{n}(u))^{b}}{(\log|F_{n}|)^{b}}$
$\displaystyle=c\cdot u^{a}a^{b},$

where $r^{\pm}_{n}(u):=\frac{b}{a}\log\log|F_{n}|+\log(u\pm\beta_{n})$. This
proves (3). It remains to choose the sequence $(s_{n})$ so that
$\varepsilon_{n}\gg|F_{n}|^{-\sigma}$ for some $\sigma>0$. Since
$\delta_{n}=|F_{n}|^{-1/a}(\log|F_{n}|)^{-b/a}$, we can for instance take
$s_{n}=(\log|F_{n}|)^{b/a}/|F_{n}|^{1/a}$, in which case $\sigma=2/a$.

##  4\. Poisson approximation for shrinking targets

In this section we show how equidistribution (or mixing) of all orders can be
used to establish Poisson asymptotics in some situations which involve
shrinking targets. We begin by proving a very general criterion for Poisson
approximation (Lemma 4.1). Then we introduce the notion of
$\mathcal{W}$-equidistribution which is a simultaneous generalization of both
(quantitative) equidistribution (of all orders) and (quantitative) mixing (of
all orders). Our main result in this section is Theorem 4.4, which provides
criteria for when the hitting times to shrinking targets in a
$\mathcal{W}$-equidistributing setting are asymptotically Poissonian.

###  4.1. A general criterion for Poisson approximation

Let $(Z,\nu)$ be a probability space and let $H$ be an infinite set. Let
$(F_{n})$ be a sequence of finite subsets of $H$ such that
$|F_{n}|\rightarrow\infty$. Suppose that for every $n$ and $h\in F_{n}$, we
are given a measurable subset $A_{n,h}\subset Z$. Define

$N_{n}(z):=\\#\big{\\{}h\in F_{n}\,:\,z\in A_{n,h}\big{\\}},\quad\textrm{for
$z\in Z$}.$ (4.1)

We will prove that the sequence $(N_{n})$ is $\nu$-asymptotically Poissonian
with mean $m$, provided that the sets $A_{n,h}$, for $h\in F_{n}$, are ”almost
independent” for large $n$. To make this more precise, we define

$\Delta_{n,r}(m):=\max\Big{\\{}\big{|}\,|F_{n}|^{r}\cdot\nu\Big{(}\bigcap_{h\in
F^{\prime}}A_{n,h}\Big{)}-m^{r}\big{|}\,:\,F^{\prime}\subset
F_{n},\enskip|F^{\prime}|=r\Big{\\}},$ (4.2)

for positive integers $r,n$ and a real number $m\geq 0$. If $|F_{n}|<r$, we
set $\Delta_{n,r}(m)=0$ for all $m$. Note that if the sets $A_{n,h}$ for $h\in
F_{n}$ are $\nu$-independent, and if $\nu(A_{n,h})=m$ for all $h$, then
$\Delta_{n,r}(m)=0$.

The following lemma will be useful in the next subsection.

######  Lemma 4.1.

Suppose that there exists $m\geq 0$ such that

$\lim_{n\rightarrow\infty}\Delta_{n,r}(m)=0,\quad\textrm{for all $r\geq 1$}.$

Then

$N_{n}\xLongrightarrow[\nu]{}\operatorname{Poi}(m)\quad\hbox{as
$n\rightarrow\infty$. }$

###### Proof.

By the classical method of moments (see e.g. [2, Theorem 30.1]), it suffices
to show that

$\lim_{n\rightarrow\infty}\int_{Z}\binom{N_{n}}{r}\,d\nu=\frac{m^{r}}{r!},\quad\textrm{for
all $r\geq 1$}.$ (4.3)

By [19, Section 7.3, Eq. (3.4)],

$\int_{Z}\binom{N_{n}}{r}\,d\nu=\sum_{F^{\prime}}\nu\Big{(}\bigcap_{h\in
F^{\prime}}A_{n,h}\Big{)},\quad\textrm{for all $r\geq 1$},$

where the sum is taken over all finite subsets $F^{\prime}\subset F_{n}$ with
$|F^{\prime}|=r$. Hence,

$\displaystyle\int_{Z}\binom{N_{n}}{r}\,d\nu$
$\displaystyle=\sum_{F^{\prime}}\Big{(}\nu\Big{(}\bigcap_{h\in
F^{\prime}}A_{n,h}\Big{)}-\frac{m^{r}}{|F_{n}|^{r}}\Big{)}+\binom{|F_{n}|}{r}\frac{m^{r}}{|F_{n}|^{r}}$
$\displaystyle=\frac{1}{|F_{n}|^{r}}\cdot\sum_{F^{\prime}}\Big{(}|F_{n}|^{r}\nu\Big{(}\bigcap_{h\in
F^{\prime}}A_{n,h}\Big{)}-m^{r}\Big{)}+\binom{|F_{n}|}{r}\frac{m^{r}}{|F_{n}|^{r}}.$

Since $|F_{n}|\rightarrow\infty$ as $n\rightarrow\infty$ and
$\lim_{n\rightarrow\infty}\Delta_{n,r}(m)=0$, we conclude that

$\displaystyle\varlimsup_{n\rightarrow\infty}\Big{|}\int_{Z}\binom{N_{n}}{r}\,d\nu-\frac{m^{r}}{r!}\Big{|}$
$\displaystyle\leq\varlimsup_{n\rightarrow\infty}\Big{(}\,\frac{1}{|F_{n}|^{r}}\binom{|F_{n}|}{r}\,\Delta_{n,r}(t)+\Big{(}\frac{1}{|F_{n}|^{r}}\binom{|F_{n}|}{r}-\frac{1}{r!}\Big{)}\cdot
m^{r}\,\Big{)}$
$\displaystyle=\varlimsup_{n\rightarrow\infty}\frac{\Delta_{n,r}(m)}{r!}=0.$

Since $r$ is arbitrary, this proves (4.3), and we are done. ∎

###  4.2. Quantitative multiple equidistribution

The main aim of this subsection is to define the notion of
$\mathcal{W}$-equidistribution for actions of real Lie groups on locally
compact spaces. To be able to do this, we first need to discuss $C^{q}$-norms
and good pairs.

####  4.2.1. $C^{q}$-norms

Let $G$ be a real Lie group with Lie algebra $\mathfrak{g}$. Given
$Y\in\mathfrak{g}$ we denote by $D_{Y}$ the differential operator defined by

$(D_{Y}\rho)(g)=\frac{d}{dt}\rho(e^{tY}g)\mid_{t=0},\quad\varphi\in
C_{c}^{\infty}(G).$

Let $\\{Y_{1},\ldots,Y_{s}\\}$ be a basis for $\mathfrak{g}$, where
$s=\dim\mathfrak{g}$. Given $\alpha\in\mathbb{N}^{s}$, we define the
differential operator $D_{\alpha}$ by

$D_{\alpha}:=D_{Y_{1}}^{\alpha_{1}}\cdots D_{Y_{s}}^{\alpha_{s}}.$

We refer to $|\alpha|:=\alpha_{1}+\ldots+\alpha_{s}$ as the _order_ of
$D_{\alpha}$, and define, for $q\geq 1$,

$\|\rho\|_{C^{q}}:=\max\\{\|D_{\alpha}\rho\|_{\infty}\,:\,|\alpha|\leq
q\big{\\}},\quad\textrm{for $\rho\in C_{c}^{\infty}(G)$}.$ (4.4)

####  4.2.2. Good pairs and $\mathcal{W}$-equidistribution

Let $Z$ be a locally compact and second countable metrizable space, endowed
with a jointly continuous action of $G$. We assume that there exists a
$G$-invariant Borel probability measure $\mu$ on $Z$. Let $\mathcal{A}$ be a
$G$-invariant sub-algebra of $\mathbb{C}\cdot 1+C_{c}(Z)$, and let
$\mathcal{S}=(\mathcal{S}_{q})$ be a family of semi-norms on $\mathcal{A}$.

######  Definition 4.2 (Good pairs).

We say that the pair $(\mathcal{A},\mathcal{S})$ is _good_ if for every Borel
set $A\subset Z$ such that either $A$ or $A^{c}$ is pre-compact, and for every
$\rho\in C_{c}^{\infty}(G)$, we have

$\rho*\chi_{A}\in\mathcal{A}\quad\textrm{and}\quad\mathcal{S}_{q}(\rho*\chi_{A})\ll_{q}\|\rho\|_{C^{q}},$

for all $q\geq 1$, where the implicit constants only depend on $q$, and not on
the set $A$.

Let $H$ be a closed subgroup of $G$. For every $r\geq 1$, let $w_{r}$ be a
non-negative function on the space of $r$-element subsets of $H$, and set
$\mathcal{W}=(w_{r})$.

######  Definition 4.3 ($\mathcal{W}$-equidistribution and
$\mathcal{W}$-mixing).

Let $\nu$ be a Borel probability measure on $Z$. We say that _$\nu$
$\mathcal{W}$-equidistributes to $\mu$ along $H$ with respect to the pair
$(\mathcal{A},\mathcal{S})$_ if for every $r\geq 1$, there exist
$\delta_{r}>0$ and $q_{r}\geq 1$ such that for every $\varphi\in\mathcal{A}$
and $F\subset H$ with $|F|=r$, we have

$\Big{|}\int_{Z}\prod_{h\in F}\varphi\circ
h\,d\nu-\Big{(}\int_{Z}\varphi\,d\mu\Big{)}^{r}\Big{|}\ll_{r}e^{-\delta_{r}w_{r}(F)}\,\mathcal{S}_{q_{r}}(\varphi)^{r},$

where the implicit constants only depend on $r$. In the case when $\nu=\mu$,
we say that _$\mu$ is $\mathcal{W}$-mixing with respect to
$(\mathcal{A},\mathcal{S})$_.

In Section 6 below, we consider the case $Z=X_{d}$ and
$G=\operatorname{SL}_{d}(\mathbb{R})$ and $\mu=\mu_{d}$, and provide examples
of good pairs $(\mathcal{A},\mathcal{S})$ and triples $(H,\nu,\mathcal{W})$
such that $\nu$ equidistributes to $\mu$ along $H$ with respect to the pair
$(\mathcal{A},\mathcal{S})$.

###  4.3. $\mathcal{W}$-equidistribution and Poisson approximation

Our aim is now to connect $\mathcal{W}$-equidistribution with Poisson
approximation. This is done in Theorem 4.4 below.

In what follows, let $H$ be a closed subgroup of $G$ and let
$(\mathcal{A},\mathcal{S})$ be a good pair. Let $\nu$ be a probability measure
on $Z$ and suppose that $\nu$ $\mathcal{W}$-equidistributes to $\mu$ along $H$
with respect to $(\mathcal{A},\mathcal{S})$, for some $\mathcal{W}=(w_{r})$.
For every $r\geq 1$, we extend $w_{r}$ to a function $\widetilde{w}_{r}$
defined on all finite subsets of $H$ by

$\widetilde{w}_{r}(F)=\min\\{w_{r}(F^{\prime})\,:\,F^{\prime}\subset
F,\enskip|F^{\prime}|=r\big{\\}}.$

If $|F|<r$, we set $\widetilde{w}_{r}(F)=0$.

Let $(F_{n})$ be a sequence of finite subsets of $H$ such that
$|F_{n}|\rightarrow\infty$, and let $(A_{n})$ be a sequence of Borel subsets
of $Z$ such that for every $n$, either $A_{n}$ or $A_{n}^{c}$ is pre-compact.
We define

$N_{n}(z)=\\#\big{\\{}h\in F_{n}\,:\,h.z\in A_{n}\big{\\}},\quad\textrm{for
$z\in Z$}.$ (4.5)

The next theorem is the main result of this section. It provides a criterion
for when the sequence $(N_{n})$ is $\nu$-asymptotically Poissonian in terms of
$\mathcal{W}$ and the behaviour of the sequence $\mu(A^{\pm}_{n})$ of real
numbers, where $A_{n}^{\pm}$ are Borel sets in $Z$ which approximate the set
$A_{n}$ from above and below. The spirit of the criterion is similar to [9,
Theorem 2.10 and Theorem 3.7], which deal with $\mathbb{R}$-flows.

######  Theorem 4.4.

Suppose that

$\lim_{n\rightarrow\infty}\frac{\widetilde{w}_{r}(F_{n})}{\log|F_{n}|}=\infty,\quad\textrm{for
all $r\geq 1$}.$

and that there exist

  * $(i)$

a sequence $(V_{n})$ of open identity neighbourhoods in $G$ and a sequence
$(\rho_{n})$ of non-negative smooth functions on $G$ such that
$\operatorname{supp}(\rho_{n})\subset V_{n}$ and $\int_{G}\rho_{n}\,dm_{G}=1$
for all $n$, with the property that for every $q\geq 1$, there is
$\sigma_{q}>0$, such that

$\|\rho_{n}\|_{\mathcal{C}^{q}(G)}\ll_{q}|F_{n}|^{\sigma_{q}},$

for all $n$, where the implicit constants only depend on $q$.

  * $(ii)$

two sequences $(A_{n}^{-})$ and $(A_{n}^{+})$ of Borel sets in $Z$ such that

$V_{n}.A_{n}^{-}\subset A_{n}\quad\textrm{and}\quad V_{n}.A_{n}\subset
A_{n}^{+},\quad\textrm{for all $n$},$

and such that

$m:=\lim_{n\rightarrow\infty}|F_{n}|\mu(A_{n}^{+})=\lim_{n\rightarrow\infty}|F_{n}|\mu(A_{n}^{-}).$

Then, $N_{n}\xLongrightarrow[\nu]{}\operatorname{Poi}(m)$, as
$n\rightarrow\infty$.

###  4.4. Proof of Theorem 4.4

We define $\varphi_{n}^{\pm}=\rho_{n}*\chi_{A_{n}^{\pm}}$. Since $\rho_{n}\geq
0$ and $\operatorname{supp}(\rho_{n})\subset V_{n}$, the inclusions in our
assumption (ii) imply

$\varphi_{n}^{-}\leq\chi_{A_{n}}\leq\varphi_{n}^{+},$

and thus, for every finite subset $F\subset H$

$\int_{Z}\prod_{h\in F}\varphi_{n}^{-}\circ h\,d\nu\leq\nu\Big{(}\bigcap_{h\in
F^{\prime}}h^{-1}.A_{n}\Big{)}\leq\int_{Z}\prod_{h\in
F^{\prime}}\varphi_{n}^{+}\circ h\,d\nu.$ (4.6)

Furthermore, since $\mu$ is $G$-invariant and $\int_{G}\rho_{n}\,dm_{G}=1$, we
have

$\int_{Z}\varphi^{+}_{n}\,d\mu=\mu(A_{n}^{+})\quad\textrm{and}\quad\int_{Z}\varphi_{n}^{-}\,d\mu=\mu(A_{n}^{-}).$
(4.7)

Since $(\mathcal{A},\mathcal{S})$ is a good pair, and since either $A_{n}$ of
$A_{n}^{c}$ is pre-compact, we have $\varphi_{n}^{\pm}\in\mathcal{A}$, and

$\mathcal{S}_{q}(\varphi_{n}^{\pm})\ll_{q}\|\rho_{n}\|_{C^{q}(G)}\ll_{q}|F_{n}|^{\sigma_{q}},\quad\textrm{for
all $q\geq 1$},$ (4.8)

where we in the last inequality used our assumption (i). We stress that the
implicit constants are independent of $n$.

By assumption, $\nu$ $\mathcal{W}$-equidistributes to $\mu$ along $H$ with
respect to $(\mathcal{A},\mathcal{S})$, so it follows from (4.7) and (4.8)
that

$\Big{|}\int_{X}\prod_{h\in F^{\prime}}\varphi_{n}^{\pm}\circ
h\,d\nu-\mu(A_{n}^{\pm})^{r}\Big{|}\ll_{r}|F_{n}|^{r\sigma_{q_{r}}}\cdot
e^{-\delta_{r}w_{r}(F^{\prime})}$ (4.9)

for every subset $F^{\prime}\subset F_{n}$ with $r$ elements. We stress that
the implicit constants only depend on $r$. From (4.6) and (4.9) we now
conclude that

$\displaystyle|F_{n}|^{r}\nu\Big{(}\bigcap_{h\in
F^{\prime}}h^{-1}.A_{n})-m^{r}$
$\displaystyle\leq|F_{n}|^{r}\Big{(}\int_{Z}\prod_{h\in
F^{\prime}}\varphi_{n}^{+}\circ
h-\mu(A_{n}^{+})^{r}\Big{)}+\big{(}|F_{n}|\mu(A_{n}^{+})\big{)}^{r}-m^{r}$
$\displaystyle\ll_{r}\big{(}|F_{n}|\mu(A_{n}^{+})\big{)}^{r}-m^{r}+|F_{n}|^{r(1+\sigma_{q_{r}})}\cdot
e^{-\delta_{r}w_{r}(F^{\prime})}$
$\displaystyle\ll_{r}\big{(}|F_{n}|\mu(A_{n}^{+})\big{)}^{r}-m^{r}+|F_{n}|^{r(1+\sigma_{q_{r}})}\cdot
e^{-\delta_{r}\widetilde{w}_{r}(F_{n})},$ (4.10)

for every subset $F^{\prime}$ of $F_{n}$ with $r$ elements. Similarly,

$|F_{n}|^{r}\nu\Big{(}\bigcap_{h\in
F^{\prime}}h^{-1}.A_{n})-m^{r}\gg_{r}\big{(}|F_{n}|\mu(A_{n}^{-})\big{)}^{r}-m^{r}-|F_{n}|^{r(1+\sigma_{q_{r}})}\cdot
e^{-\delta_{r}\widetilde{w}_{r}(F_{n})},$ (4.11)

for every subset $F^{\prime}$ of $F_{n}$ with $r$ elements.

Since we assume that

$\lim_{n\rightarrow\infty}\frac{\widetilde{w}_{r}(F_{n})}{\log|F_{n}|}=\infty\quad\textrm{and}\quad
m=\lim_{n\rightarrow\infty}|F_{n}|\mu(A_{n}^{+})=\lim_{n\rightarrow\infty}|F_{n}|\mu(A_{n}^{-}),$

we see from (4.10) and (4.11) that

$\lim_{n\rightarrow\infty}\max\Big{\\{}\Big{|}|F_{n}|^{r}\nu\Big{(}\bigcap_{h\in
F^{\prime}}h^{-1}.A_{n}\Big{)}-m^{r}\Big{|}\,:\,F^{\prime}\subset
F_{j},\enskip|F^{\prime}|=r\Big{\\}}=0,$

for every $r\geq 1$. By Lemma 4.1, this shows that
$N_{n}\xLongrightarrow[\nu]{}\operatorname{Poi}(m)$, as $n\rightarrow\infty$.

##  5\. Siegel transforms

We recall that $X_{d}$ denotes the space of unimodular lattices in
$\mathbb{R}^{d}$, equipped with the Chabauty topology. The standard
$\operatorname{SL}_{d}(\mathbb{R})$-action on $X_{d}$ is jointly continuous
and there is a unique $\operatorname{SL}_{d}(\mathbb{R})$-invariant Borel
probability measure on $X_{d}$, which we denote by $\mu_{d}$.

###  5.1. Hitting sets

Given a Borel set $B\subset\mathbb{R}^{d}$, we define the _hitting set_
$\Omega_{B}\subset X_{d}$ by

$\Omega_{B}=\big{\\{}\Lambda\in X_{d}\,:\,\Lambda\cap B\neq\\{0\\}\big{\\}}.$
(5.1)

We note that $\Omega_{B}$ is a Borel set in $X_{d}$. By Mahler’s Compactness
Criterion [7, Chapter V, Theorem IV], the complement $\Omega_{B}^{c}$ is
compact in $X_{d}$ if $B$ contains a neighborhood of $0$ in $\mathbb{R}^{d}$.

###  5.2. Siegel transforms

Given $\Lambda\in X_{d}$, we denote by $\Lambda_{\textrm{pr}}$ the set of
primitive vectors in $\Lambda$. If $f:\mathbb{R}^{d}\rightarrow\mathbb{C}$ is
a Borel function with bounded support, we define its _Siegel transform_
$\widehat{f}:X_{d}\rightarrow\mathbb{C}$ by

$\widehat{f}(\Lambda)=\sum_{\hskip
5.69046pt\lambda\in\Lambda_{\textrm{pr}}}f(\lambda),\quad\textrm{for
$\Lambda\in X_{d}$}.$ (5.2)

In particular, if $B$ is a bounded Borel set in $\mathbb{R}^{d}$, then
$\widehat{\chi}_{B}(\Lambda)=|B\cap\Lambda_{\textrm{pr}}|$.

The following theorem summarizes important special cases of two fundamental
formulas of Siegel and Rogers.

######  Theorem 5.1 (Siegel’s and Rogers’ formulas).

Let $B\subset\mathbb{R}^{d}$ be a bounded and symmetric Borel set.

  * $(i)$

Siegel, [21, Section 2] If $d\geq 2$, then $\widehat{\chi}_{B}\in
L^{1}(\mu_{d})$ and

$\int_{X_{d}}\widehat{\chi}_{B}(\Lambda)\,d\mu_{d}(\Lambda)=\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)},$

  * $(ii)$

Rogers, [18, Theorem 5] If $d\geq 3$, then $\widehat{\chi}_{B}\in
L^{2}(\mu_{d})$, and

$\int_{X_{d}}\widehat{\chi}_{B}(\Lambda)^{2}\,d\mu_{d}(\Lambda)=2\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)}+\Big{(}\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)}\Big{)}^{2}.$

###  5.3. Volume asymptotics

We recall that a subset $B\subset\mathbb{R}^{d}$ is _balanced_ if $tB\subset
B$ for all $|t|\leq 1$. Note that every balanced set is symmetric.

######  Lemma 5.2.

Let $d\geq 2$. For every bounded and balanced Borel set
$B\subset\mathbb{R}^{d}$, we have

$\Omega_{B}=\\{\widehat{\chi}_{B}>0\\}=\\{\widehat{\chi}_{B}\geq 2\\}.$

and

$\mu_{d}(\Omega_{B})\leq\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}.$

###### Proof.

Since $B$ is balanced, we have $\frac{1}{k}B\subseteq B$ for every integer
$k\geq 1$, and thus

$\displaystyle\Omega_{B}$ $\displaystyle=\Big{\\{}\Lambda\,:\,\Lambda\cap
B\neq\\{0\\}\Big{\\}}=\Big{\\{}\Lambda\,:\,\bigcup_{k\geq
1}\Big{(}k.\Lambda_{\textrm{pr}}\cap B\Big{)}\neq\\{0\\}\Big{\\}}$
$\displaystyle=\Big{\\{}\Lambda\,:\,\bigcup_{k\geq
1}k.\Big{(}\Lambda_{\textrm{pr}}\cap\frac{1}{k}B\Big{)}\neq\\{0\\}\Big{\\}}=\big{\\{}\Lambda\,:\,\Lambda_{\textrm{pr}}\cap
B\neq\\{0\\}\big{\\}}.$

This shows that $\Omega_{B}=\\{\widehat{\chi}_{B}>0\\}$. Furthermore, since
$B$ is symmetric, $|\Lambda_{\textrm{pr}}\cap B|$ must be an even integer for
every $\Lambda\in X_{d}$, which proves that we also have
$\Omega_{B}=\\{\widehat{\chi}_{B}\geq 2\\}$. Finally, we note that by Siegel’s
formula (Theorem 5.1 (i)),

$\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)}=\int_{X_{d}}\widehat{\chi}_{B}\,d\mu_{d}\geq\int_{\\{\widehat{\chi}_{B}\geq
2\\}}\widehat{\chi}_{B}\,d\mu_{d}\geq 2\cdot\mu_{d}(\Omega_{B}).$

∎

######  Lemma 5.3.

Let $d\geq 3$. For every bounded and balanced Borel set
$B\subset\mathbb{R}^{d}$, we have

$\Big{|}\,\mu_{d}(\Omega_{B})-\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}\,\Big{|}\leq\Big{(}\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}\Big{)}^{2}.$

###### Proof.

By Lemma 5.2, we have
$\mu_{d}(\Omega_{B})\leq\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}$, so we
only need to show that

$\mu_{d}(\Omega_{B})\geq\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}-\Big{(}\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}\Big{)}^{2}.$
(5.3)

We note that

$\int_{X_{d}}\widehat{\chi}_{B}\,d\mu_{d}=\int_{X_{d}}\chi_{\\{\widehat{\chi}_{B}>0\\}}\cdot\widehat{\chi}_{B}\,d\mu_{d}\leq\mu_{d}(\Omega_{B})^{1/2}\cdot\Big{(}\int_{X_{d}}\widehat{\chi}_{B}^{2}\,d\mu_{d}\Big{)}^{1/2},$

and thus, by Theorem 5.1 (i) and (ii).

$\Big{(}\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)}\Big{)}^{2}\leq\mu_{d}(\Omega_{B})\cdot\Big{(}2\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)}+\Big{(}\frac{\operatorname{Vol}_{d}(B)}{\zeta(d)}\Big{)}^{2}\Big{)}.$

From this we conclude that

$\mu_{d}(\Omega_{B})\geq\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}\cdot\frac{1}{1+\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}}\geq\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}-\Big{(}\frac{\operatorname{Vol}_{d}(B)}{2\zeta(d)}\Big{)}^{2},$

where we in the last inequality used the trivial inequality $\frac{1}{1+t}\geq
1-t$ for all $t\geq 0$. ∎

######  Remark 5.4.

Kleinbock and Margulis have proved a version of Lemma 5.3 for _convex_ sets in
[16, Proposition 7.1]. They claim that their lemma also holds for $d=2$, but
unfortunately their argument relies on [16, Theorem 7.3], which cannot be
applied in the case when $d=k=2$ (the relevant parameters in [16, Proposition
7.1] for $d=2$).

There are some versions of Roger’s formula for $d=2$ by Athreya-Margulis [1,
Subsection 4.2] and Kelmer-Yu [14, Theorem 1]. However, it seems to be
difficult to use these formulas to deduce small volume asymptotics as in Lemma
5.3.

##  6\. Proof of Theorem 3.1

###  6.1. A good pair

Let $\mathcal{A}=\mathbb{C}\cdot 1+C^{\infty}_{c}(X_{d})$. Note that if $A$ is
Borel subset of $X_{d}$ such that either $A$ or its complement $A^{c}$ is
compact, and $\rho$ is a complex-valued, smooth and compactly supported
function on $G$, then $\rho*\chi_{A}\in\mathcal{A}$. We will construct a
family $\mathcal{S}=(\mathcal{S}_{q})$ of norms on the algebra $\mathcal{A}$
such that the pair $(\mathcal{A},\mathcal{S})$ is good (see Definition 4.2).
Given a non-zero element $Y$ in the Lie algebra
$\mathfrak{s}\mathfrak{l}_{d}(\mathbb{R})$, we define the differential
operator $D_{Y}$ on $\mathcal{A}$ by

$D_{Y}\varphi(x)=\frac{d}{dt}\varphi(e^{tY}.x)\mid_{t=0},\quad\textrm{for
$x\in X_{d}$}.$

Let $\\{Y_{1},\ldots,Y_{d^{2}-1}\\}$ be a basis for
$\mathfrak{s}\mathfrak{l}_{d}(\mathbb{R})$. For every
$\alpha=(\alpha_{1},\ldots,\alpha_{d^{2}-1})\in\mathbb{N}^{d^{2}-1}$, we set
$D_{\alpha}:=D_{1}^{\alpha_{1}}\cdots D_{d^{2}-1}^{\alpha_{d^{2}-1}}$, and for
$q\geq 1$ and $\varphi\in\mathcal{A}$, we define

$\|\varphi\|_{C^{q}(X_{d})}=\max\\{\|D_{\alpha}\varphi\|_{\infty}\,:\,\alpha_{1}+\ldots+\alpha_{d^{2}-1}\leq
q\big{\\}}.$

Let us fix a proper right-invariant distance function on
$\operatorname{SL}_{d}(\mathbb{R})$. Since $\operatorname{SL}_{d}(\mathbb{R})$
acts transitively on $X_{d}$ with uniformly discrete stabilizers, we can
define a (non-invariant) quotient metric on $X_{d}$, and write
$\operatorname{Lip}$ for the corresponding Lipschitz norm. We now define

$S_{q}(\varphi)=\max(\|\varphi\|_{C^{q}(X_{d})},\operatorname{Lip}(\varphi)),\quad\textrm{for
$q\geq 1$}.$ (6.1)

It is not difficult to see that if $A\subset X_{d}$ is a Borel set such that
either $A$ or $A^{c}$ is pre-compact, and $\rho\in C_{c}^{\infty}(G)$, then

$S_{q}(\rho*\chi_{A})\ll_{q}\|\rho\|_{C^{q}},\quad\textrm{for all $q\geq 1$},$

where the implict constants are independent of the Borel set $A$, and
$\|\cdot\|_{C^{q}}$ is defined as in (4.4). Hence $(\mathcal{A},\mathcal{S})$
is a good pair.

We recall that

$W_{\varepsilon}=\big{\\{}g\in\operatorname{SL}_{d}(\mathbb{R})\,:\,\|g-\operatorname{id}\|_{\textrm{op}}<\varepsilon\big{\\}},\quad\textrm{for
$\varepsilon>0$},$

where $\|\cdot\|_{\textrm{op}}$ is the operator norm with respect to the
$\ell^{\infty}$-norm on $\mathbb{R}^{d}$. The proof of the following simple
lemma is left to the reader.

######  Lemma 6.1.

There is a family $(\widetilde{\rho}_{\varepsilon})$ of non-negative smooth
functions on $\operatorname{SL}_{d}(\mathbb{R})$ such that
$\operatorname{supp}\widetilde{\rho}_{\varepsilon}\subset W_{\varepsilon}$ and
$\int\widetilde{\rho}_{\varepsilon}\,dm_{\operatorname{SL}_{d}(R)}=1$ for
every $\varepsilon>0$, and with the property that for every $q\geq 1$, there
exists $\kappa_{q}>0$ such that

$\|\widetilde{\rho}_{\varepsilon}\|_{C^{q}}\ll\varepsilon^{-\kappa_{q}},\quad\textrm{for
all $\varepsilon>0$},$

where the implicit constants only depend on $q$.

###  6.2. $\mathcal{W}$-mixing for $\mu_{d}$

Let $H_{d}<\operatorname{SL}_{d}(\mathbb{R})$ be as in Subsection 2.2. If
$F\subset H_{d}$ is a finite set with $r$ elements, we define
$w_{r}(F):=v_{d}(F)$, where $v_{d}$ is defined as in (2.7), and set
$\mathcal{W}=(w_{r})$. The following theorem was proved by the authors and
Manfred Einsiedler [6, Theorem 1.1].

######  Theorem 6.2.

For every $d\geq 2$, $\mu_{d}$ is $\mathcal{W}$-mixing with respect to
$(\mathcal{A},\mathcal{S})$.

######  Remark 6.3.

Theorem 6.2 is not explicitly stated in [6] with the norms $(\mathcal{S}_{q})$
above. However, these norms satisfy the properties (N1-N4) listed in [6,
Subsection 2.2]; this is enough to guarantee $\mathcal{W}$-mixing.

###  6.3. $\mathcal{W}$-equidistribution of $\nu_{\underline{d}}$ to $\mu_{d}$
along $H^{+}_{\underline{d}}$

Let $\underline{d}=(d_{1},d_{2})$ such that $d=d_{1}+d_{2}$, and let
$H^{+}_{\underline{d}}<\operatorname{SL}_{d}(\mathbb{R})$ be as in Subsection
2.3. If $F\subset H_{d}^{+}$ is a finite set with $r$ elements, we define
$w_{r}(F):=v_{\underline{d}}^{+}(F)$, where $v_{\underline{d}}^{+}$ is defined
as in (2.9). The following theorem was proved by the authors [5, Theorem 1.1].

######  Theorem 6.4.

For every $\underline{d}=(d_{1},d_{2})$ such that $d=d_{1}+d_{2}\geq 2$,
$\nu_{\underline{d}}$ $\mathcal{W}$-equidistributes to $\mu_{d}$ along
$H_{\underline{d}}^{+}$.

######  Remark 6.5.

Theorem 6.4 for one-parameter sub-semigroups of $H_{d}^{+}$ was proved earlier
by the authors in [3, Theorem 2.2].

###  6.4. Proof of Theorem 3.1

Let $d\geq 3$, and let $(F_{n})$ be a sequence of finite subsets of $H_{d}$
such that $|F_{n}|\rightarrow\infty$ and _either_

  * (1)

$\displaystyle\lim_{n\rightarrow\infty}\frac{v_{d}(F_{n})}{\log|F_{n}|}=\infty$,
or

  * (2)

$F_{n}\subset H_{\underline{d}}^{+}$ and
$\displaystyle\lim_{n\rightarrow\infty}\frac{v^{+}_{\underline{d}}(F_{n})}{\log|F_{n}|}=\infty$.

Theorem 6.2 tells us that we are in the setting of Theorem 4.4, and (1)
implies that the first assumption of Theorem 4.4 (about $\widetilde{w}_{r}$)
is satisfied. Theorem 6.4, in combination with (2), says the same thing. We
will check that the other two conditions (i) and (ii) (with $m=\xi/2\zeta(d))$
in Theorem 4.4 are satisfied for

$V_{n}:=W_{\varepsilon_{n}}\quad\textrm{and}\quad\rho_{n}:=\widetilde{\rho}_{\varepsilon_{n}}\quad\textrm{and}\quad
A_{n}:=\Omega_{B_{n}}\quad\textrm{and}\quad
A_{n}^{\pm}:=\Omega_{B_{n}^{\pm}},$

where $(\varepsilon_{n})$, $(B_{n})$ and $(B_{n}^{\pm})$ are as in the
statement of Theorem 3.1. If these conditions hold, then Theorem 4.4 implies
that

  * •

$N_{n}\xLongrightarrow[\mu_{d}]{}\operatorname{Poi}\big{(}\xi/2\zeta(d)\big{)}$
\- in the case of (1).

  * •

$N_{n}\xLongrightarrow[\nu_{\underline{d}}]{}\operatorname{Poi}\big{(}\xi/2\zeta(d)\big{)}$
\- in the case of (2).

#### Proof of (i) in Theorem 4.4

Since $\varepsilon_{n}\gg|F_{n}|^{-\sigma}$ for some $\sigma>0$, we conclude
from Lemma 6.1 that

$\|\rho_{n}\|_{C^{q}}\ll\varepsilon_{n}^{-\kappa_{q}}\ll|F_{n}|^{\sigma\kappa_{q}},\quad\textrm{for
all $q\geq 1$},$

where the implicit constants only depend on $q$. This proves (i) with
$\sigma_{q}:=\sigma\kappa_{q}$.

#### Proof of (ii) in Theorem 4.4

The inclusions are obvious. To prove that

$\lim_{n\rightarrow\infty}|F_{n}|\mu_{d}(A_{n}^{\pm})=\frac{\xi}{2\zeta(d)},$

we argue as follows. First note that

$|F_{n}|\cdot\mu_{d}(A_{n}^{\pm})=|F_{n}|\cdot\Big{(}\mu_{d}(\Omega_{B_{n}^{\pm}})-\frac{\operatorname{Vol}_{d}(B_{n}^{\pm})}{2\zeta(d)}\Big{)}+|F_{n}|\cdot\frac{\operatorname{Vol}_{d}(B_{n}^{\pm})}{2\zeta(d)}.$
(6.2)

By our assumption in Theorem 3.1, the second term converges to $\xi/2\zeta(d)$
when $n\rightarrow\infty$. In particular,
$\operatorname{Vol}_{d}(B_{n}^{\pm})\rightarrow 0$. By Lemma 5.3, the absolute
value of the first term is bounded from above by

$|F_{n}|\cdot\Big{(}\frac{\operatorname{Vol}_{d}(B_{n}^{\pm})}{2\zeta(d)}\Big{)}^{2}=\Big{(}|F_{n}|\cdot\frac{\operatorname{Vol}_{d}(B_{n}^{\pm})}{2\zeta(d)}\Big{)}\cdot\frac{\operatorname{Vol}_{d}(B_{n}^{\pm})}{2\zeta(d)},$

Since the first factor on the right hand side stays bounded as
$n\rightarrow\infty$ while the second factor tends to zero, we conclude that
the first term in (6.2) tends to zero as $n\rightarrow\infty$. Hence,
$\lim_{n}|F_{n}|\,\mu(A_{n}^{\pm})=\xi/2\zeta(d)$, which finishes the proof of
(ii).

##  Appendix A Weibull asymptotics implies a logarithm Law

Let $(X,\mu)$ be a probability space and let $f_{T}$, with
$T\in\mathscr{T}_{d}$, be positive measurable functions on $X$. We _assume_
that the minima of these functions are Weibull distributed, i.e. we suppose
that there exist $a,c>0$, $b\geq 0$ such that given any sequence of finite
subsets $\Delta_{n}$ of $\mathscr{T}_{d}$ satisfying

$|\Delta_{n}|\to\infty\quad\hbox{and}\quad\frac{\displaystyle\min_{T\neq
T^{\prime}\in\Delta_{n}}\max_{p}|\log T_{p}-\log
T^{\prime}_{p}|}{\log|\Delta_{n}|}\to\infty,$ (A.1)

the minima

$\mathfrak{M}_{\Delta_{n}}:=\min\big{\\{}f_{T}:\,T\in\Delta_{n}\big{\\}}$

satisfy

$|\Delta_{n}|^{a}(\log|\Delta_{n}|)^{b}\cdot\mathfrak{M}_{\Delta_{n}}\xLongrightarrow[\mu]{}\operatorname{Wei}\big{(}c,a\big{)}.$

######  Proposition A.1.

Under the above assumption, for every $\delta<d-1$ and $\mu$-a.e. $x\in X$,

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{a\delta}f_{T}(x)=0.$

In particular, it follows from Theorem 1.3 that the Minkowski constants (1.1)
satisfy

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{\frac{d-1}{d}-\varepsilon}c_{B_{T}}(\Lambda)=0\quad\hbox{for
all $\varepsilon>0$ and $\mu_{d}$-a.e. $\Lambda\in X_{d}$}$

(cf. (1.4)). Also from Theorem 1.6 we obtain that

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{d-1-\varepsilon}m_{B_{T}}(\Lambda)=0\quad\hbox{for
all $\varepsilon>0$ and $\mu_{d}$-a.e. $\Lambda\in X_{d}$}.$

In particular, there exist $v_{i}\in\Lambda$ such that

$\lim_{i\to\infty}\,(\log\|v_{i}\|_{\infty})^{d-1-\varepsilon}\Pi(v_{i})=0$

(cf. (1.6)).

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

For the sake of a contradiction, we assume that there exist $c_{1},c_{2}>0$
such that

$\mu\big{(}\\{x\in
X:\,{\liminf}_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{a\delta}f_{T}(x)>c_{1}\\}\big{)}>c_{2}.$

For a finite subset $\Delta\subset\mathscr{T}_{d}$, we write

$\displaystyle\rho(\Delta):=\max\\{(\log\|T\|_{\infty})^{a\delta}:\,T\in\Delta\\}\quad\hbox{and}\quad\mathfrak{M}_{\Delta}(x):=\min\\{f_{T}(x):\,T\in\Delta\\}.$

We observe that

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{a\delta}f_{T}(x)>c_{1}$

means that there exists $T_{0}(x)$ such that

$(\log\|T\|_{\infty})^{a\delta}f_{T}(x)>c_{1}\quad\hbox{when
$\|T\|_{\infty}\geq T_{0}(x)$.}$

In particular, when $\|T\|_{\infty}\geq T_{0}(x)$ for all $T\in\Delta$,

$\rho(\Delta)\mathfrak{M}_{\Delta}(x)\geq\min\\{(\log\|T\|_{\infty})^{a\delta}f_{T}(x):\,T\in\Delta\\}>c_{1}.$

Let us pick a sequence of finite subsets $\Delta_{n}$ such that

$\min\\{\|T\|_{\infty}:\,T\in\Delta_{n}\\}\to\infty\quad\hbox{ as
$n\to\infty$.}$ (A.2)

Then the set

$\\{x\in X:\,\exists
T_{0}(x):\,(\log\|T\|_{\infty})^{a\delta}f_{T}(x)>c_{1}\hbox{ when
$\|T\|_{\infty}\geq T_{0}(x)$}\\}$

is contained in

$\\{x\in X:\,\exists
n_{0}(x):\,\rho(\Delta_{n})\mathfrak{M}_{\Delta_{n}}(x)>c_{1}\hbox{ for all
$n\geq n_{0}(x)$}\\}.$

In particular, we conclude that

$\mu\big{(}\\{x\in X:\,\exists
n_{0}(x):\,\rho(\Delta_{n})\mathfrak{M}_{\Delta_{n}}(x)>c_{1}\hbox{ for all
$n\geq n_{0}(x)$}\\}\big{)}>c_{2}.$

Setting

$\Omega_{n}:=\\{x\in
X:\,\rho(\Delta_{n})\mathfrak{M}_{\Delta_{n}}(x)>c_{1}\\},$

we conclude that $\mu(\liminf\Omega_{n})>c_{2}$. This implies that

$\mu(\Omega_{n})>c_{2}\quad\hbox{ for all sufficiently large $n$.}$ (A.3)

Let us for now additionally assume that the sets $\Delta_{n}$ satisfy (A.1)
and

$\frac{|\Delta_{n}|^{a}(\log|\Delta_{n}|)^{b}}{\rho(\Delta_{n})}\to\infty.$
(A.4)

Below we show that such sequences of sets really do exist.

We choose $u\in\mathbb{R}$ so that $e^{-(\lambda u)^{a}}<c_{2}$. Since
$c_{1}\frac{|\Delta_{n}|^{a}(\log|\Delta_{n}|)^{b}}{\rho(\Delta_{n})}>u$ for
sufficiently large $n$, we deduce that

$\mu(\Omega_{n})\leq\mu\big{(}\\{x\in
X:\,|\Delta_{n}|^{a}(\log|\Delta_{n}|)^{b}\,\mathfrak{M}_{\Delta_{n}}(x)>u\\}\big{)}\to
e^{-(\lambda u)^{a}}<c_{2}.$

However, this contradicts (A.3). Therefore, we conclude that for all
$c_{1}>0$,

$\mu\big{(}\\{x\in
X:\,{\liminf}_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{a\delta}f_{T}(x)>c_{1}\\}\big{)}=0.$

This implies the proposition, provided that we can show that we can construct
$(\Delta_{n})$ satisfying the conditions (A.1), (A.2), (A.4). For this we use
parameters $\ell_{n},\theta_{n}\to\infty$. Let

$\Delta_{n}:=\big{\\{}T(s_{1},\ldots,s_{d-1}):\,1\leq
s_{1},\ldots,s_{d-1}\leq\ell_{n}\big{\\}},$

where

$T(s_{1},\ldots,s_{d-1}):=\big{(}\theta_{n}^{s_{1}},\ldots,\theta_{n}^{s_{d-1}},\theta_{n}^{-\sum_{k}s_{k}}\big{)}.$

Then $|\Delta_{n}|=\ell_{n}^{d-1}\to\infty$, and the condition (A.1) is
equivalent to $\frac{\log\theta_{n}}{\log\ell_{n}}\to\infty$, so that
$\theta_{n}=\ell_{n}^{\omega_{n}}$ with $\omega_{n}\to\infty$. The condition
(A.2) is obviously satisfied, and the condition (A.4) holds when

$\frac{\ell_{n}^{a(d-1)}}{\log(\theta_{n}^{\ell_{n}})^{a\delta}}=\frac{\ell_{n}^{a(d-1)-a\delta}}{(\omega_{n}\log\ell_{n})^{a\delta}}\to\infty.$

Since $\delta<d-1$, this can be arranged by taking for instance
$\omega_{n}=\log\ell_{n}$. ∎

An analogue of Proposition A.1 also holds in the setting of Theorems 1.10 and
1.13. We fix a tuple $\underline{d}=(d_{1},d_{2})$ such that $d=d_{1}+d_{2}$,
$d_{1},d_{2}\geq 1$ and consider the sub-semigroup
$\mathscr{T}^{+}_{\underline{d}}$ of $\mathscr{T}_{d}$ defined in (1.9). Let
$\mathscr{C}$ be a cone in $\mathscr{T}_{\underline{d}}^{+}$ with finitely
many faces. We consider positive measurable functions $f_{T}$,
$T\in\mathscr{C}$, on $X$ and assume that the minima of these functions
satisfy the Weibull distribution. Namely, we suppose that there exist $a>0$,
$b\geq 0$ such that given any sequence of finite subsets $\Delta_{n}$ of
$\mathscr{C}$ satisfying $|\Delta_{n}|\to\infty$ and

$\frac{\displaystyle\min_{T\neq T^{\prime}\in\Delta_{n}}\max_{p}|\log
T_{p}-\log
T^{\prime}_{p}|}{\log|\Delta_{n}|}\to\infty\quad\hbox{and}\quad\frac{\displaystyle\min_{T\in\Delta_{n}}\lfloor
T\rfloor_{\underline{d}}}{\log|\Delta_{n}|}\to\infty$ (A.5)

the minima

$\mathfrak{M}_{\Delta_{n}}:=\min\big{\\{}f_{T}:\,T\in\Delta_{n}\big{\\}}$

satisfy

$|\Delta_{n}|^{a}(\log|\Delta_{n}|)^{b}\cdot\mathfrak{M}_{\Delta_{n}}\xLongrightarrow[\mu]{}\operatorname{Wei}\big{(}c,a\big{)}.$

######  Proposition A.2.

Under the above assumption, if $d_{0}=\dim(\mathscr{C})$, for every
$\delta<d_{0}$ and $\mu$-a.e. $x\in X$,

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{a\delta}f_{T}(x)=0.$

Let us apply this proposition in the setting of Theorem 1.10 with $d_{1}=d-1$
and $d_{2}=1$ with the cone

$\mathcal{C}=\\{(t^{1/(d-1)},\ldots,t^{1/(d-1)},t^{-1}):\,t\geq 1\\}.$

We conclude that

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{\frac{1}{d-1}-\epsilon}k_{T}(\alpha)=0.$

for every $\varepsilon>0$ and Lebesgue almost every $\alpha\in[0,1]^{d-1}$
(cf. (1.8)).

In the setting of Theorem 1.13 with $d_{1}=d-1$ and $d_{2}=1$ with the cone

$\mathcal{C}=\\{(t^{1/(d-1)}s_{1},\ldots,t^{1/(d-1)}s_{d-1},t^{-1}):\,\,s_{1},\ldots,s_{d-1}\geq
t^{-1/(d-1)},\,t\geq 1,\,s_{1}\cdots s_{d-1}=1\\},$

we deduce that

$\liminf_{\|T\|_{\infty}\to\infty}\,(\log\|T\|_{\infty})^{d-1-\epsilon}g_{T}(\alpha)=0.$

for every $\varepsilon>0$ and Lebesgue almost every $\alpha\in[0,1]^{d-1}$
(cf. (1.12)).

###### Proof of Proposition A.2.

The proof proceeds as in Proposition A.1. We just have to justify existence of
a sequence of finite subsets $\Delta_{n}$ of $\mathcal{C}$ satisfying the
required conditions. We use parameters $\ell_{n},\theta_{n}\to\infty$. Since
$\dim(\mathcal{C})=d_{0}$, there exist (multiplicatively) independent

$A_{n,k}=(\theta_{n}^{w_{1,k}},\ldots,\theta_{n}^{w_{d,k}}),\quad
k=1,\ldots,d_{0},$

elements in $\mathcal{C}$ (here $w_{l,k}>0$ for $l\leq d_{1}$ and $w_{l,k}>0$
for $l>d_{1}$). Let

$\Delta_{n}:=\big{\\{}T(s_{1},\ldots,s_{d_{0}}):\,1\leq
s_{1},\ldots,s_{d_{0}}\leq\ell_{n}\big{\\}},$

where

$T(s_{1},\ldots,s_{d_{0}}):=A_{n,1}^{s_{1}}\cdots A_{n,d_{0}}^{s_{d_{0}}}.$

One can check as in the proof of Proposition A.1 that when
$\theta_{n}=\log\ell_{n}$, the sets $\Delta_{n}$ satisfy the required
conditions. We note that in this case we also need an additional condition in
(A.5), but this condition also follows from
$\frac{\log\theta_{n}}{\log\ell_{n}}\to\infty$. ∎

## References

  * [1] Athreya, J.S.; Margulis, G.A. _Logarithm laws for unipotent flows, I_. J. Mod. Dyn. 3 (2009), no. 3, 359–378.
  * [2] Billingsley, P. _Probability and measure._ Third edition. A Wiley–Interscience Publication. John Wiley & Sons, Inc., New York, 1995.
  * [3] Björklund, M.; Gorodnik, A. _Central limit theorems for Diophantine approximants._ Math. Ann. 374 (2019), no. 3–4, 1371–1437.
  * [4] Björklund, M.; Gorodnik, A. _Counting in generic lattices and higher rank actions_. arXiv:2101.04931.
  * [5] Björklund, M.; Gorodnik, A. _Effective multiple equidistribution of translated measures._ To appear in IMRN, arXiv:2105.05468
  * [6] Björklund, M.; Einsiedler, M.; Gorodnik, A. _Quantitative multiple mixing._ J. Eur. Math. Soc. (JEMS) 22 (2020), no. 5, 1475–1529.
  * [7] Cassels, J. W. S. _An introduction to the geometry of numbers._ Corrected reprint of the 1971 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1997.
  * [8] Davenport, H.; Schmidt, W. M. _Dirichlet’s theorem on diophantine approximation. II._ Acta Arith. 16 (1969/70), 413–424.
  * [9] Dolgopyat, D.; Fayad, B.; Sixu L., _Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence_. arxiv2103.08382.
  * [10] Gallagher, P. _Metric simultaneous diophantine approximation._ J. London Math. Soc. 37 (1962), 387–390.
  * [11] Greenblatt, M. _Resolution of singularities, asymptotic expansions of integrals and related phenomena._ J. Anal. Math. 111 (2010), 221–245.
  * [12] Gruber, P. M.; Lekkerkerker, C. G. _Geometry of numbers._ Second edition. North-Holland Mathematical Library, 37. North-Holland Publishing Co., Amsterdam, 1987.
  * [13] Hirata, M.: _Poisson law for Axiom A diffeomorphisms._ Ergodic Theory Dynam. Systems 13 (1993), no. 3, 533–556.
  * [14] Kelmer, D.; Yu, S. _The second moment of the Siegel transform in the space of symplectic lattices._ IMRN 2021, no. 8, 5825–5859.
  * [15] Kirsebom, M. _Extreme value distributions for one-parameter actions on homogeneous spaces._ Nonlinearity 33 (2020), no. 3, 1218–1239.
  * [16] Kleinbock, D. Y.; Margulis, G. A. _Logarithm laws for flows on homogeneous spaces._ Invent. Math. 138 (1999), no. 3, 451–494.
  * [17] Pollicott, M. _Limiting distributions for geodesics excursions on the modular surface._ Spectral analysis in geometry and number theory, 177–185, Contemp. Math., 484, Amer. Math. Soc., Providence, RI, 2009.
  * [18] Rogers, C.A. _Mean values over the space of lattices._ Acta Math. 94 (1955), 249–287.
  * [19] Ross, S., _A first course in probability. Second edition._ Macmillan Co., New York; Collier Macmillan Ltd., London, 1984.
  * [20] Siegel, C.L., _Lectures on the geometry of numbers._ Springer-Verlag, Berlin, 1989.
  * [21] Siegel, C.L., _A mean value theorem in geometry of numbers._ Ann. of Math. 46 (1945), 340–347.
  * [22] Sinai Y. G., _Some mathematical problems in the theory of quantum chaos_. Physica A 163 (1990), 197–204.
  * [23] Sinai Y.G. _Mathematical problems in the theory of quantum chaos._ Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1469. Springer, Berlin, Heidelberg.
  * [24] Sullivan, D. _Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics._ Acta Math. 149 (1982), no. 3–4, 215–237.