# New lower bound on ball packing density
in high-dimensional hyperbolic spaces

Irene Gil Fernández Mathematics Institute and DIMAP, University of Warwick,
UK. Email<EMAIL_ADDRESS>IGF was supported by the Warwick
Mathematics Institute Centre for Doctoral Training, and gratefully
acknowledges funding from the University of Warwick and the UK Engineering and
Physical Sciences Research Council (Grant number: EP/TS1794X/1) Jaehoon Kim
Department of Mathematical Sciences, KAIST, South Korea. Email:
<EMAIL_ADDRESS>J.K. was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government(MSIT) No.
RS-2023-00210430 Hong Liu Extremal Combinatorics and Probability Group
(ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea. Email:
<EMAIL_ADDRESS>H.L. was supported by the Institute for Basic Science
(IBS-R029-C4). Oleg Pikhurko Mathematics Institute and DIMAP, University of
Warwick, UK. Email<EMAIL_ADDRESS>O.P. was supported by ERC
Advanced Grant 101020255.

###### Abstract

We present a new lower bound on the Bowen–Radin maximal density of radius-$R$
ball packings in the $m$-dimensional hyperbolic space, improving on the basic
covering bound by factor $\Omega(m(R+\ln m))$ as $m$ tends to infinity. This
is done by applying the recent theorem of Campos, Jenssen, Michelen and
Sahasrabudhe on independent sets in graphs with sparse neighbourhoods.

##  1 Introduction

Let $R>0$ be a positive real and let $(V,d)$ be a metric space endowed with a
Borel non-zero measure $\mu$. An _$R$ -packing_ in $V$ is a subset $X\subseteq
V$ such that any two distinct points of $X$ are at distance at least $2R$. If
the measure $\mu$ is _finite_ (that is, $\mu(V)<\infty$), then we define the
_density_ of an $R$-packing $X$ by

$D_{R}(X):=\frac{\mu\left(B_{R}(X)\right)}{\mu(V)},$

where we denote

$B_{R}(X):=\\{y\in V:\exists\,x\in X,\ d(x,y)\leqslant R\\}.$

In other words, $D_{R}(X)$ is the fraction of the measure space $V$ covered by
closed radius-$R$ balls around the points of $X$. The problem of determining
or estimating the _$R$ -packing density_ $D_{R}(V)$ of a given metric space
$V$ (that is, the supremum of the densities of $R$-packings in $V$) is the
archetypical problem of coding theory which also has a very large number of
applications to other fields.

One important case is when $V={\mathbbm{R}}^{m}$, endowed with the Euclidean
distance and the Lebesgue (uniform) measure. Since the measure is not finite
here, one defines the _$R$ -packing density_ $D({\mathbbm{R}}^{m})$ as the
limit of the packing densities of growing cubes that exhaust
${\mathbbm{R}}^{m}$:

$D({\mathbbm{R}}^{m}):=\lim_{n\to\infty}D_{R}\left(\,[-n,n]^{m}\,\right).$ (1)

It is easy to see that the limit exists and, by the scaling properties of the
Lebesgue measure, does not depend on the radius $R$.

It is clear that the packing density of the real line
$D({\mathbbm{R}}^{1})=1$. The packing density of ${\mathbbm{R}}^{m}$ for
$m=2,3,8,24$ was determined by respectively Thue [Thue92], Hales [Hales05],
Viazovska [Viazovska17], and Cohn, Kumar, Miller, Radchenko and Viazovska
[CohnKumarMillerRadchenkoViazovska17]. However, the value of
$D({\mathbbm{R}}^{m})$ is still unknown for any other $m$.

When $m\to\infty$, the best known upper and lower bounds on
$D({\mathbbm{R}}^{m})$ are exponentially far apart. The best known upper bound
$D({\mathbbm{R}}^{m})\leqslant 2^{-(0.5990...+o(1))\,m}$ is due to
Kabatjanskiĭ and Levenšteĭn [KabatjanskiiLevenstein78], see also
[CohnElkies03, CohnZhao14].

The trivial _covering lower bound_ $D({\mathbbm{R}}^{m})\geqslant 2^{-m}$
(take a maximal packing and observe that balls of doubled radius cover the
whole space) was improved by a factor of $\Omega(m)$ by Rogers [Rogers47].
After a sequence of improvements in the constant factor [DavenportRogers47,
Ball92, Vance11, Venkatesh13], Venkatesh [Venkatesh13] was the first to beat
the $\Omega(m\cdot 2^{-m})$ lower bound by showing that
$D({\mathbbm{R}}^{m})=\Omega(m\ln\ln m\cdot 2^{-m})$ for some infinite
(sparse) sequence of dimensions $m$. In a recent breakthrough, Campos,
Jenssen, Michelen and Sahasrabudhe [CamposJenssenMichelenSahasrabudhe] proved
that $D({\mathbbm{R}}^{m})\geqslant(\frac{1}{2}-o(1))m\ln m\cdot 2^{-m}$. This
was done by discretising the problem and then applying their new result (which
is stated as Theorem 12 here) on the existence of a large independent set in
graphs with sparse neighbourhoods.

The aim of this paper is to observe that the method from
[CamposJenssenMichelenSahasrabudhe] also applies to ball packing in hyperbolic
spaces. Let ${\mathbbm{H}}^{m}$ denote the $m$-dimensional hyperbolic space,
equipped with the standard metric $d_{m}$ and the standard (isometry
invariant) measure $\mu_{m}$. (See Section 2 for all formal definitions.)

The study of packings in ${\mathbbm{H}}^{m}$ was held back for a long while by
the absence of a good definition of the maximum packing density, see e.g. the
discussion in [FejestothKuperberg93, Pages 831–834]. Analogues of (1) lack
many desired properties because growing polygons or balls in
${\mathbbm{H}}^{m}$ have most of their mass near their boundary. So some
“local” approaches were considered; see [FejestothFejestothKuperberg23lapss,
Chapter 11] for a survey of known results from this point of view. However,
even here one has to be careful in view of the following striking example of
Böröczky [Boroczky74] (also described on Page 833 of [FejestothKuperberg93]).
Namely, Böröczky showed that there is an $R$-packing
$X\subseteq{\mathbbm{H}}^{2}$ in the hyperbolic plane and two tilings
${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ of ${\mathbbm{H}}^{2}$ by single
polygonal tiles $T_{1}$ and $T_{2}$ respectively such that each tile of
${\mathcal{T}}_{i}$ contains one radius-$R$ disk centred at a point of $X$ and
is disjoint from all other such disks; at the same time, $T_{1}$ and $T_{2}$
have different areas. Thus if we measure the “density” of this packing $X$
using the fraction occupied by the corresponding disks within each tile of
${\mathcal{T}}_{1}$ (resp. ${\mathcal{T}}_{2}$), then we get two different
numbers.

A solution was proposed by Bowen and Radin [BowenRadin03, Definition 5] who
presented a new parameter of $(m,R)$ where, roughly speaking, one considers
distributions on $R$-packings in ${\mathbbm{H}}^{m}$ that are invariant under
isometries and maximises the probability that a fixed point is covered by a
ball. This parameter has many nice properties and is the one that will be used
in this paper as the definition of the _$R$ -packing density_ of
${\mathbbm{H}}^{m}$ and thus denoted by $D_{R}({\mathbbm{H}}^{m})$. We refer
the reader to Section 3 for the formal definition of
$D_{R}({\mathbbm{H}}^{m})$ and to [Radin04, BowenRadin04] for further
discussion and motivation behind it.

For dimension $m=2$ and countably many radii $R$, Bowen and Radin
[BowenRadin03, Theorem 2] were able to determine $D_{R}({\mathbbm{H}}^{2})$.
Namely, for every integer $n>6$, if we tile ${\mathbbm{H}}^{2}$ by
(equilateral) triangles having all three angles $2\pi/n$ and take a “uniform
shift” of the set of the triangles’ vertices then this distribution attains
$D_{R}({\mathbbm{H}}^{2})$, where $R=R_{n}$ is the half of the side length of
the triangles. As far as we know, these are the only pairs $(m,R)$ with
$m\geqslant 2$ for which the exact value of $D_{R}({\mathbbm{H}}^{m})$ is
known.

Let us discuss the known upper bounds on $D_{R}({\mathbbm{H}}^{m})$ for
$m\to\infty$. Before Bowen and Radin’s paper [BowenRadin03], Fejes Tóth
[Fejestoth43] (for $m=2$), Böröczky and Florian [BoroczkyFlorian64] (for
$m=3$) and Böröczky [Boroczky78] (any $m$) proved that for every $R$-packing
$X\subseteq{\mathbbm{H}}^{m}$ and any $x\in X$ the fraction of volume occupied
by the _$R$ -ball_ $B_{R}(x):=B_{R}(\\{x\\})$ around $x$ inside the Dirichlet-
Voronoi cell of $x\in X$ is at most $\delta_{m}(2R)$, which is defined to be
the fraction of volume of a regular simplex of side length $2R$ occupied by
(touching) balls of radius $R$ at its vertices. It follows from these results
(see Lemma 3 here) that $D_{R}({\mathbbm{H}}^{m})\leqslant\delta_{m}(2R)$.
More recently, Cohn and Zhao [CohnZhao14, Theorem 4.1] showed that,
independently of $R>0$, the Bowen–Radin density satisfies

$D_{R}({\mathbbm{H}}^{m})\leqslant\inf_{\pi/3\leqslant\theta\leqslant\pi}\sin^{m-1}(\theta/2)A(m,\theta),$
(2)

where $A(m,\theta)$ be the maximum number of unit vectors in the Euclidean
space ${\mathbbm{R}}^{m}$ such that the scalar product of every two is at most
$\cos\theta$; in other words, $A(m,\theta)$ is the maximum size of a spherical
code with minimum angle $\theta$. In particular, the method of Kabatjanskiĭ
and Levenšteĭn [KabatjanskiiLevenstein78] (that also applies to spherical
codes) gives that

$D_{R}({\mathbbm{H}}^{m})\leqslant 2^{-(0.5990...+o(1))\,m},$ (3)

see [CohnZhao14, Corollary 4.2]. For large $m$, this bound is smaller than
$\delta_{m}(0):=\lim_{R\to 0}\delta_{m}(R)$, which can be shown to be the
volume ratio coming from $m+1$ touching unit balls in ${\mathbbm{R}}^{m}$.
Note that Marshall [Marshall99, Theorem 2] proved that, for all large $m$, the
function $\delta_{m}(R)$ is strictly increasing in $R$ (and thus
$\delta_{m}(0)\leqslant\delta_{m}(R)$ for any $R>0$ for such $m$).

The covering principle (that if $X$ is a maximal $R$-packing then $2R$-balls
around points of $X$ cover the whole space) translates with some work (see
Lemma 8) into the basic lower bound

$D_{R}({\mathbbm{H}}^{m})\geqslant
L_{R}({\mathbbm{H}}^{m}):=\frac{\mu_{m}(B_{R})}{\mu_{m}(B_{2R})},$ (4)

where $\mu_{m}(B_{r})$ denotes the volume of some (equivalently, any) $r$-ball
in ${\mathbbm{H}}^{m}$.

There are various constructions of packings in small dimensions (see e.g.
[FejestothFejestothKuperberg23lapss, Chapter 11] for references) and it is
plausible that their “locally” measured densities translate into lower bounds
on the Bowen-Radin density. However, we are not aware of any improvements to
(4) for large $m$ apart that, as far as we can see, the method of Jenssen,
Joos and Perkins [JenssenJoosPerkins18, JenssenJoosPerkins19], which is based
on the hard-sphere model of statistical physics, can be applied to improve the
bound in (4) by factor $\Omega(m)$. Anyway, this is superseded by the main
result of this paper as follows.

###### Theorem 1

For every $\varepsilon>0$ there is $m_{0}$ such that for any $m\geqslant
m_{0}$ it holds for every $R\in(0,\infty)$ that

$D_{R}({\mathbbm{H}}^{m})\geqslant(1-\varepsilon)\,m\ln\left(\sqrt{m}\cosh(2R)\right)\frac{\mu_{m}(B_{R})}{\mu_{m}(B_{2R})},$

where $\ln$ is the natural logarithm and $\cosh$ is the hyperbolic cosine.

Note that $\ln(\cosh(2R))$ is at least $2R-1$, so Theorem 1 improves the bound
in (4) by factor at least $\Omega(m(R+\ln m))$.

Organisation of the paper. We recall some notions related to
${\mathbbm{H}}^{m}$ in Section 2 and define the Bowen-Radin density in Section
3. We describe the method from [BowenRadin03] of lower bounding
$D_{R}({\mathbbm{H}}^{m})$ via $R$-packings in some finite-volume space $M$ in
Section 4. Various estimates are collected in Section 5. Finally, Theorem 1 is
proved in Section 6. Since $m$ and $R$ will be reserved, respectively, for the
dimension of the studied hyperbolic space and the packing radius, we may omit
them from our notation if the meaning is clear.

##  2 Hyperbolic space

This section contains just a bare minimum of material sufficient to formally
define the hyperbolic space ${\mathbbm{H}}^{m}$ and some needed related
notions. For a detailed introduction to hyperbolic spaces see e.g. Bridson and
Haefliger [BridsonHaefliger:msnpc, Section 2] or Ratcliffe [Ratcliffe19fhm].

One representation of ${\mathbbm{H}}^{m}$ (for more details see e.g.
[Ratcliffe19fhm, Chapter 3]) is to take the bilinear form

$\langle u,v\rangle_{m,1}:=-u_{m+1}v_{m+1}+\sum_{i=1}^{m}u_{i}v_{i},\quad
u,v\in{\mathbbm{R}}^{m+1},$ (5)

identify ${\mathbbm{H}}^{m}$ with upper sheet of the hyperboloid

${\mathcal{H}}:=\\{u\in{\mathbbm{R}}^{m+1}:\langle u,u\rangle_{m,1}=-1\\}$

(namely, the sheet where $u_{m+1}>0$), and define the metric $d_{m}$ by $\cosh
d_{m}(u,v)=-\langle u,v\rangle_{m,1}$ for $u,v\in{\mathbbm{H}}^{m}$. The group
of isometries of ${\mathbbm{H}}^{m}$ can be identified with $O(m,1)_{0}$, the
group of $(m+1)\times(m+1)$-matrices $A$ which leave the bilinear form in (5)
invariant and do not swap the two sheets of ${\mathcal{H}}$; see
[BridsonHaefliger:msnpc, Theorem 2.24] or [Ratcliffe19fhm, Theorem 3.2.3]. We
fix one point of ${\mathbbm{H}}^{m}$, say
$(0,\dots,0,1)\in{\mathbbm{R}}^{m+1}$ in the above representation, which we
will denote by ${\mathcal{O}}$ and refer to as the _origin_.

Let ${\mathcal{G}}$ be the group of orientation-preserving isometries of
${\mathbbm{H}}^{m}$; it corresponds to the subgroup $\mathrm{SO}(m,1)_{0}$ of
index 2 in $O(m,1)_{0}$, which consists of matrices with determinant 1. It is
a topological group with the topology inherited from
$O(m,1)_{0}\subseteq{\mathbbm{R}}^{(m+1)^{2}}$.

The space ${\mathbbm{H}}^{m}$ is equipped with a Borel isometry-invariant
measure $\mu_{m}$ whose push-forward under the projection on the first $m$
coordinates of ${\mathbbm{R}}^{m+1}$ has density

$\rho(x_{1},\hskip 0.9pt.\hskip 0.3pt.\hskip 0.3pt.\hskip
1.5pt,x_{m}):=(1+(x_{1}^{2}+\hskip 0.9pt.\hskip 0.3pt.\hskip 0.3pt.\hskip
1.5pt+x_{m}^{2}))^{-1/2}$ (6)

with respect to the Lebesgue measure on ${\mathbbm{R}}^{m}$. The group
${\mathcal{G}}$ is unimodular and locally compact, thus there is a Haar
measure $\mu_{{\mathcal{G}}}$ on ${\mathcal{G}}$ (which is both right- and
left-invariant). It is a standard result (see e.g. [Ratcliffe19fhm, Lemma 4 of
Section 11.6]) that, by scaling $\mu_{{\mathcal{G}}}$, we can assume that the
projection map
$\pi_{{\mathcal{O}}}:({\mathcal{G}},\mu_{{\mathcal{G}}})\to({\mathbbm{H}}^{m},\mu_{m})$,
where $\gamma\mapsto\gamma.{\mathcal{O}}$, is measure-preserving.

We consider the natural (left) action
${\mathcal{G}}{\curvearrowright}{\mathbbm{H}}^{m}$. For
$\gamma\in{\mathcal{G}}$ and $A\subseteq{\mathbbm{H}}^{m}$, we denote
$\gamma.A:=\\{\gamma.x:x\in A\\}$. For $x\in{\mathbbm{H}}^{m}$, let
$\Sigma_{x}:=\\{\gamma\in{\mathcal{G}}:\gamma.x=x\\}$ be the _stabiliser_ of
$x$. In particular, we have the natural homeomorphism of topological spaces
${\mathcal{G}}/\Sigma_{{\mathcal{O}}}\cong{\mathbbm{H}}^{m}$.

##  3 Definition of the Bowen–Radin density

In this section, we give the definition of the packing density
$D_{R}({\mathbbm{H}}^{m})$ introduced by Bowen and Radin [BowenRadin03]. We
also state some results from [BowenRadin03], occasionally providing more
details (usually when these were implicitly assumed but not stated in
[BowenRadin03]).

Let $R>0$ be a positive real. Let $S_{R}$ consist of $R$-packings
$X\subseteq{\mathbbm{H}}^{m}$ that are _relatively dense_ , that is, such
that, for every $x\in{\mathbbm{H}}^{m}$, the (closed radius-$2R$) ball
$B_{2R}(x)$ around $x$ contains at least one element of $X$. This is slightly
weaker than the notion of a maximal packing (namely, a relatively dense
packing can have the distance from some $x$ to $X$ exactly $2R$ and thus not
be a maximal one).

Bowen and Radin [BowenRadin03] considered the following metric $d_{R}$ on
$S_{R}$:

$d_{R}(X,Y):=\sup_{r\in[1,\infty)}\frac{1}{r}\,h(B_{r}({\mathcal{O}})\cap
X,B_{r}({\mathcal{O}})\cap Y),\quad X,Y\in S_{R},$ (7)

where

$h(A,B):=\max\left\\{\sup_{a\in A}\inf_{b\in B}d_{m}(a,b),\ \sup_{b\in
B}\inf_{a\in A}d_{m}(a,b)\right\\},\quad A,B\subseteq{\mathbbm{H}}^{m},$

is the usual _Hausdorff distance_ on the subsets of ${\mathbbm{H}}^{m}$. As
noted in [BowenRadin03, Page 25], the metric space $(S_{R},d_{R})$ is compact.
Let ${\mathcal{M}}(R)$ be the set of Borel probability measures on
$(S_{R},d_{R})$. This space is compact in weak-$*$ topology by Prokhorov’s
theorem.

The group ${\mathcal{G}}$ naturally acts on $S_{R}$ and the corresponding map
${\mathcal{G}}\times S_{R}\to S_{R}$ is continuous. A measure
$\mu\in{\mathcal{M}}(R)$ is called _${\mathcal{G}}$ -invariant_ if for every
$\gamma\in{\mathcal{G}}$ and every Borel set $E\subseteq S_{R}$ we have
$\mu(\gamma.E)=\mu(E)$. Let ${\mathcal{M}}_{I}(R)$ consist of all
${\mathcal{G}}$-invariant measures in ${\mathcal{M}}(R)$. Let
${\mathcal{M}}_{I}^{\mathbf{e}}(R)$ consist of those measures
$\mu\in{\mathcal{M}}_{I}(R)$ that are _ergodic_ , that is, are extreme points
of the convex set ${\mathcal{M}}_{I}(R)$.

For $p\in{\mathbbm{H}}^{m}$, define

$F_{p}(X):=\left\\{\begin{array}[]{ll}1,&\mbox{if $d_{m}(x,X)\leqslant R$,}\\\
0,&\mbox{otherwise},\end{array}\right.\quad\text{ for }X\in S_{R},$

to be the indicator function that $R$-balls around $X$ cover $p$.

###### Lemma 2

For every $p\in{\mathbbm{H}}^{m}$, the function $F_{p}$ is a Borel function on
$(S_{R},d_{R})$.

Proof. Let us show that $F_{p}^{-1}(1)=\\{X\in S_{R}:F_{p}(X)=1\\}$ is a
closed subset of $(S_{R},d_{R})$. Take any Cauchy sequence
$(X_{n})_{n=1}^{\infty}$ in $F_{p}^{-1}(1)$. Each $X_{n}$ has a point $x_{n}$
in $B_{R}(p)$. By the compactness of the closed ball $B_{R}(p)$, we can pass
to a subsequence of $n$ so that $x_{n}$ converges to some $x\in B_{R}(p)$ as
$n\to\infty$. Since $X$ has at most one point in $B_{R/2}(x)$, it follows from
the definition of the Hausdorff distance by taking any
$r>d_{m}(x,{\mathcal{O}})$ in (7) that $x\in X$. Thus $F_{p}(X)=1$ and the set
$F_{p}^{-1}(1)$ is closed (and thus Borel).

Since the function $F_{p}$ is $\\{0,1\\}$-valued and $F_{p}^{-1}(1)$ is
closed, the function $F_{p}$ is Borel.

Thus for any $\mu\in{\mathcal{M}}_{I}(R)$, we can define the _average density_

$D(\mu):=\int_{S_{R}}F_{{\mathcal{O}}}(X)\,\mathrm{d}\mu(X).$ (8)

By the ${\mathcal{G}}$-invariance of $\mu$ and the transitivity of
${\mathcal{G}}{\curvearrowright}{\mathbbm{H}}^{m}$, we could have taken any
other point of ${\mathbbm{H}}^{m}$ instead of the origin in this definition,
without changing the value.

Following Bowen and Radin [BowenRadin03, Definition 5], we define the _$R$
-packing density of ${\mathbbm{H}}^{m}$_ as

$D_{R}({\mathbbm{H}}^{m}):=\sup_{\mu\in{\mathcal{M}}_{I}^{\mathbf{e}}(R)}D(\mu).$
(9)

We refer the reader to [BowenRadin03] and also to [BowenRadin04, Radin04] for
discussion and further properties of this parameter. For example,
[BowenRadin03, Theorem 1] implies that the supremum in (9) is attained by some
measure $\mu\in{\mathcal{M}}_{I}^{\mathbf{e}}(R)$.

At this point, let us observe that the classical “local” upper bounds from
[Fejestoth43, BoroczkyFlorian64, Boroczky78] also apply to
$D_{R}({\mathbbm{H}}^{m})$.

###### Lemma 3

For any real $R\geqslant 0$ and integer $m$, we have
$D_{R}({\mathbbm{H}}^{m})\leqslant\delta_{m}(2R)$.

Proof. Take any $\mu\in{\mathcal{M}}_{I}^{\mathbf{e}}(R)$. Bowen and Radin
[BowenRadin03, Proposition 3] showed that $D(\mu)$ can be computed as the
expectation over a random packing $X$ distributed according to $\mu$ of the
ratio of volume of $B_{R}({\mathcal{O}})$ to the volume of the Dirichlet-
Voronoi cell of $X$ containing the origin ${\mathcal{O}}\in{\mathbbm{H}}^{m}$
in its interior. (Note that, by the invariance of $\mu$, the probability that
${\mathcal{O}}$ lies on the boundary of a Dirichlet-Voronoi cell of $X$ is
zero.)

By the results from [Fejestoth43, BoroczkyFlorian64, Boroczky78] mentioned in
the Introduction, the integrated function is always at most $\delta_{m}(2R)$,
so its average is at most $\delta_{m}(2R)$, as desired.

##  4 Lower bounds via a finite-volume space

Here we describe the method from [BowenRadin03] of proving lower bounds on
$D_{R}({\mathbbm{H}}^{m})$ coming from ball packings in some finite-volume
space. We find it more convenient to use the following definition of a lattice
which is stronger than the more standard one assumed in [BowenRadin03] (so all
the cited positive results from [BowenRadin03] involving lattices remain
true).

A subgroup ${\mathcal{L}}\subseteq{\mathcal{G}}$ is called a _(uniform
torsion-free) lattice_ if (i) there is $g>0$ such that for every
$x\in{\mathbbm{H}}^{m}$ and every non-identity $\gamma\in{\mathcal{L}}$ we
have $d_{m}(x,\gamma.x)\geqslant g$; (ii) the quotient space
$M:={\mathbbm{H}}^{m}/{\mathcal{L}}$ is compact. Denote the supremum of $g$
that work in (i) as $g({\mathcal{L}})$ and call it the _girth_ of
${\mathcal{L}}$. The topological space $M$ comes with the natural metric

$d_{M}(x,y):=d_{m}(\pi_{M}^{-1}(x),\pi_{M}^{-1}(y)),\quad x,y\in M,$

where $\pi_{M}:{\mathbbm{H}}^{m}\to M$ is the projection. Note that $\pi_{M}$
is a _local isometry_ (that is, there is $r>0$ such that for every
$x\in{\mathbbm{H}}^{m}$ the map $\pi_{M}$ gives an isometry between $B_{r}(x)$
and the ball of radius $r$ around $\pi_{M}(x)$ in $d_{M}$; in fact, we can let
$r$ be $\frac{1}{2}\,g({\mathcal{L}})$). Since ${\mathbbm{H}}^{m}$ has
constant curvature $-1$ the same applies to $M$; so $M$ is a hyperbolic
manifold (compact, without boundary). Since the measure $\mu_{m}$ on
${\mathbbm{H}}^{m}$ can be defined via the element of length, see e.g.
[Ratcliffe19fhm, Section 3.4], this definition carries over to $M$ giving that
there is $r$ (which we can take again to be $\frac{1}{2}\,g({\mathcal{L}})$)
such that for every $x\in{\mathbbm{H}}^{m}$ the restriction of $\pi_{M}$ to
$B_{r}(x)$ is measure-preserving. By the compactness of $M$, it can be covered
by finitely many balls of radius $\frac{1}{2}\,g({\mathcal{L}})>0$. Each of
these balls has finite volume (since finite-radius balls in
${\mathbbm{H}}^{m}$ have finite volume) so $\mu_{M}(M)<\infty$.

###### Lemma 4

Let a lattice ${\mathcal{L}}$ have girth $g\geqslant 4R$, define
$M:={\mathbbm{H}}^{m}/{\mathcal{L}}$, and let $\pi_{M}:{\mathbbm{H}}^{m}\to M$
be the projection. Let $R>0$ and let $Y\subseteq M$ be an $R$-packing. Define
$X:=\pi_{M}^{-1}(Y)\subseteq{\mathbbm{H}}^{m}$. Then the following statements
hold.

  1. 1.

The set $X$ is an $R$-packing in $({\mathbbm{H}}^{m},d_{m})$.

  2. 2.

If the packing $Y$ is relatively dense in $(M,d_{M})$ then $X$ is relatively
dense.

Proof. Let us show the first claim that $X$ is a packing. Take any distinct
$x,x^{\prime}\in X$ and let $y:=\pi_{M}(x)$ and
$y^{\prime}:=\pi_{M}(x^{\prime})$. If $y\not=y^{\prime}$, then
$d_{M}(y,y^{\prime})\geqslant 2R$ and, by the girth assumption,
$d_{m}(x,x^{\prime})\geqslant 2R$. If $y=y^{\prime}$ then, since $\pi_{M}$
maps $B_{g/2}(x)$ isometrically and bijectively to the ball of radius
$g/2\geqslant 2R$ around $y$ in $M$ and this ball contains $y^{\prime}$, we
have $d_{m}(x,x^{\prime})>2R$, giving the desired.

Let us show that the packing $X\subseteq{\mathbbm{H}}^{m}$ is relatively dense
if $Y$ is. Take any $x\in{\mathbbm{H}}^{m}$. Since the packing $Y$ is
relatively dense, we have that $d_{M}(y,Y)\leqslant 2R$, where
$y:=\pi_{M}(x)$. Pick $y^{\prime}\in Y$ with $d_{M}(y,y^{\prime})\leqslant
2R$. We have

$d_{m}(x,X)=d_{M}(y,Y)\leqslant d_{M}(y,y^{\prime})\leqslant 2R.$

Since $x\in{\mathbbm{H}}^{m}$ was arbitrary, the packing $X$ is indeed
relatively dense.

Suppose now that ${\mathcal{L}}$ is a lattice of girth at least $4R$,
$M={\mathbbm{H}}^{m}/{\mathcal{L}}$ and $Y\subseteq M$ is a relatively dense
$R$-packing in $(M,d_{M})$. Define
$X:=\pi_{M}^{-1}(Y)\subseteq{\mathbbm{H}}^{m}$. By Lemma 4, we have $X\in
S_{R}$.

Let us recall some definitions from [BowenRadin03]. Define

$O(X):=\\{\gamma.X:\gamma\in{\mathcal{G}}\\}\subseteq S_{R},$

to be the orbit of $X\in S_{R}$ under the action
${\mathcal{G}}{\curvearrowright}S_{R}$. The space $O(X)\subseteq S_{R}$
inherits its topology from $(S_{R},d_{R})$. Next, define
$\Gamma_{X}:=\\{\gamma\in{\mathcal{G}}:\gamma.X=X\\}$ to be the subgroup of
${\mathcal{G}}$ that preserves the packing $X$. Clearly,
${\mathcal{L}}\subseteq\Gamma_{X}$. Also, define the map
$q_{X}:O(X)\to{\mathcal{G}}/\Gamma_{X}$ by

$q_{X}(\gamma.X):=\gamma.\Gamma_{X},\quad\mbox{for $\gamma\in{\mathcal{G}}$}.$

(We omit trivial verifications that the stated maps are well-defined, this
meaning here that for any $\gamma,\gamma^{\prime}\in{\mathcal{G}}$ if
$\gamma.X=\gamma^{\prime}.X$ then
$\gamma.\Gamma_{X}=\gamma^{\prime}.\Gamma_{X}$.) The map $q_{X}$ gives an
isomorphism between the topological spaces $O(X)$ and
${\mathcal{G}}/\Gamma_{X}$. The group ${\mathcal{G}}/\Gamma_{X}$ is compact as
a quotient of a compact space
${\mathcal{G}}/{\mathcal{L}}\cong{\mathbbm{H}}^{m}/{\mathcal{L}}$, so it
admits the (unique) probability Haar measure $\eta$. Define the measure
$\widehat{\mu}_{X}$ on $O(X)$ by

$\widehat{\mu}_{X}(E):=\eta(q_{X}(E)),\quad\mbox{for a Borel subset
$E\subseteq O(X)$},$

that is, $\widehat{\mu}_{X}$ is the pushforward to $O(X)$ of the measure
$\eta$ on ${\mathcal{G}}/\Gamma_{X}$ by the map $q_{X}^{-1}$.

Let the measure $\mu_{X}$ be the extension of the measure $\widehat{\mu}_{X}$
to $S_{R}$, that is, we define $\mu_{X}(E):=\widehat{\mu}_{X}(E\cap O(X))$ for
a Borel subset $E\subseteq S_{R}$. Clearly, $\mu_{X}$ is a Borel probability
${\mathcal{G}}$-invariant measure on $S_{R}$, that is,
$\mu_{X}\in{\mathcal{M}}_{I}(R)$; moreover, its support
$\mathrm{supp}(\mu_{X})$ is a subset of $O(X)$.

###### Lemma 5

Under the above assumptions, any measure $\mu\in{\mathcal{M}}_{I}(R)$ that
satisfies $\mathrm{supp}(\mu)\subseteq O(X)$ must be equal to $\mu_{X}$.

Proof. Define a Borel probability measure $\nu$ on ${\mathcal{G}}/\Gamma_{X}$
by

$\nu(E):=\mu(q_{X}^{-1}(E))=\mu\left(\\{\gamma.X:\gamma\Gamma_{X}\in
E\\}\right),\quad\mbox{for Borel $E\subseteq{\mathcal{G}}/\Gamma_{X}$}.$

The measure $\nu$ is ${\mathcal{G}}$-invariant under the left action of
${\mathcal{G}}$ on ${\mathcal{G}}/\Gamma_{X}$. Indeed, for every
$\gamma^{\prime}\in{\mathcal{G}}$ and $E\subseteq{\mathcal{G}}/\Gamma_{X}$ we
have

$\displaystyle\nu(\gamma^{\prime}.E)$ $\displaystyle=$
$\displaystyle\mu\left(\\{\gamma.X:\gamma\Gamma_{X}\in\gamma^{\prime}.E\\}\right)$
$\displaystyle=$
$\displaystyle\mu\left(\\{\gamma^{\prime}\gamma^{\prime\prime}.X:\gamma^{\prime\prime}\Gamma_{X}\in
E\\}\right)$ $\displaystyle=$
$\displaystyle\mu\left(\\{\gamma^{\prime\prime}.X:\gamma^{\prime\prime}\Gamma_{X}\in
E\\}\right)\ =\ \nu(E),$

where the second equality uses the (invertible) substitution
$\gamma=\gamma^{\prime}\gamma^{\prime\prime}$ and the third equality follows
from the ${\mathcal{G}}$-invariance of $\mu$.

The above claims also hold for $\nu_{X}$, which is defined the same way as
$\nu$ except we replace $\mu$ by $\mu_{X}\in{\mathcal{M}}_{I}(R)$.

By the uniqueness of the Haar measure, we have $\nu=\nu_{X}$ (and both are
equal to the measure $\eta$). Since $q_{X}:O(X)\to{\mathcal{G}}/\Gamma_{X}$ is
a homeomorphism, we have that $\mu=\mu_{X}$, as required.

Now, it easily follows that the measure $\mu_{X}\in{\mathcal{M}}_{I}(R)$ is
ergodic, that is,

$\mu_{X}\in{\mathcal{M}}_{I}^{\mathbf{e}}(R).$ (10)

Indeed, if $\mu_{X}=c\mu^{\prime}+(1-c)\mu^{\prime\prime}$ for some
$c\in(0,1)$ and $\mu^{\prime},\mu^{\prime\prime}\in{\mathcal{M}}_{I}(R)$ then
the support of each of $\mu^{\prime}$ and $\mu^{\prime\prime}$ is a subset of
$\mathrm{supp}(\mu_{X})\subseteq O(X)$ so each of them is equal to $\mu_{X}$
by Lemma 5, proving the ergodicity of $\mu_{X}$.

###### Lemma 6

If ${\mathcal{L}}\subseteq{\mathcal{G}}$ is a lattice of girth at least $4R$
and $M={\mathbbm{H}}^{m}/{\mathcal{L}}$ then

$D_{R}({\mathbbm{H}}^{m})\geqslant D_{R}(M).$ (11)

Proof. Take any relatively dense $R$-packing $Y$ in $M$ and let
$X:=\pi_{M}^{-1}(Y)$. Then $X$ is a relatively dense $R$-packing in
${\mathbbm{H}}^{m}$ by Lemma 4. Moverover, it is invariant under the action of
the lattice ${\mathcal{L}}{\curvearrowright}{\mathbbm{H}}^{m}$.

Furthermore, [BowenRadin03, Proposition 1] gives that $D(\mu_{X})$ is equal to
the relative volume taken by the $R$-balls around $X$ inside any fundamental
domain of $\Gamma_{X}{\curvearrowright}{\mathbbm{H}}^{m}$. The lattice
${\mathcal{L}}$, as a subgroup of $\Gamma_{X}$, has finite index (which can be
bounded by $k!$ where $k$ is the maximum size of an $R$-packing in $M$, where
$k$ in turn can be bounded by $\mu_{M}(M)/\mu_{m}(B_{R})<\infty$). Thus a
fundamental domain ${\mathcal{F}}\subseteq{\mathbbm{H}}^{m}$ of
${\mathcal{L}}{\curvearrowright}{\mathbbm{H}}^{m}$ can be obtained by taking
the union of finitely many (pairwise disjoint) translates of a fundamental
domain of $\Gamma_{X}{\curvearrowright}{\mathbbm{H}}^{m}$ (one per each coset
of $\Gamma_{X}/{\mathcal{L}}$). Since each of the latter translates has the
same occupied ratio (by [BowenRadin03, Proposition 1]), this ratio is the same
as that for their union ${\mathcal{F}}$. Note that the restriction of
$\pi_{M}$ to the fundamental domain ${\mathcal{F}}$ is a measure-preserving
map between $({\mathcal{F}},\mu_{m})$ and $(M,\mu_{M})$. By the girth
assumption on ${\mathcal{L}}$, the $R$-balls around points of the
${\mathcal{L}}$-periodic tiling $X\subseteq{\mathbbm{H}}^{m}$ occupy the same
fraction of volume of ${\mathcal{F}}$ as the $R$-balls around points of $Y$ in
$M$. Since $\mu_{X}\in{\mathcal{M}}_{I}^{\mathbf{e}}(R)$, we have by
definition that $D_{R}({\mathbbm{H}}^{m})\geqslant D(\mu_{X})$. Since
$D(\mu_{X})$ is the radius-$R$ density of $Y$ in $M$ and any packing extends
to a relatively dense one, the lemma now follows.

###### Lemma 7

For every $m\in{\mathbbm{N}}$ and $r\in(0,\infty)$, there is a lattice
${\mathcal{L}}$ of isometries of ${\mathbbm{H}}^{m}$ of girth at least $r$.

Proof. This is Theorem 4.1 in [KuperbergKuperbergFejestoth98] that contains a
detailed proof. (The authors of [KuperbergKuperbergFejestoth98] wrote that
this fact had been known for a long time but they could not find a suitable
reference.)

Recall that we defined
$L_{R}({\mathbbm{H}}^{m}):=\mu_{m}(B_{R})/\mu_{m}(B_{2R})$ to be the ratio of
the volumes of balls of radius $R$ and $2R$ in ${\mathbbm{H}}^{m}$. We can now
argue that this gives a lower bound on the Bowen–Radin packing density.

###### Lemma 8

For every $m\in{\mathbbm{N}}$ and $R>0$, we have
$D_{R}({\mathbbm{H}}^{m})\geqslant L_{R}({\mathbbm{H}}^{m})$.

Proof. Take any lattice ${\mathcal{L}}\subseteq{\mathcal{G}}$ of girth at
least $4R$, which exists by Lemma 7. In $M:={\mathbbm{H}}^{m}/{\mathcal{L}}$
take a maximal packing $Y$. By the maximality of $Y$, the $2R$-balls around
points of $Y$ cover the whole space $M$. By the girth assumption, for any
$r\leqslant 2R$, the volume of any $r$-ball in $M$ is the same as the volume
of an $r$-ball in ${\mathbbm{H}}^{m}$. Thus $D_{R}(M)\geqslant
L_{R}({\mathbbm{H}}^{m})$. Now the lemma follows from (11).

##  5 Various estimates

We will use various facts about hyperbolic functions $\cosh
x:=({\mathrm{e}}^{x}+{\mathrm{e}}^{-x})/2$, $\sinh
x:=({\mathrm{e}}^{x}-{\mathrm{e}}^{-x})/2$, $\tanh x:=\frac{\sinh x}{\cosh x}$
and $\coth x:=1/\tanh x$. First, we have $\cosh^{2}x=1+\sinh^{2}x$. The
following formulas for angle doubling are easy to check directly:

$\cosh(2x)=2\cosh^{2}x-1\quad\mbox{ and }\quad\sinh(2x)=2\sinh x\cosh x.$

Also, we will use the monotonicity of the above hyperbolic functions on
$[0,\infty)$, approximations $\tanh x,\sinh x=(1+o(1))x$ for $x\to 0$ and the
following inequalities that are routine to check:

$\tanh x<1,\ \ 2\sinh x\leqslant\sinh(2x),\ \ \tanh(2x)\leqslant 2\tanh
x,\quad\mbox{for $x\in[0,\infty)$.}$ (12)

Furthermore, we will use the following formula (see e.g. [Ratcliffe19fhm,
Theorem 5.3.5] for a proof).

######  Lemma 9 (The First Law of Cosines)

If $\alpha,\beta,\gamma$ are the angles of a hyperbolic triangle and $a,b,c$
are the lengths of the opposite sides, then

$\cos\gamma=\frac{\cosh a\cosh b-\cosh c}{\sinh a\sinh b}.$ (13)

###### Lemma 10

Let $x,u\in{\mathbbm{H}}^{m}$ be two distinct points and let
$\tau:=d_{m}(x,u)>0$. Let $r>0$ satisfy $\tau<2r$. Then the intersection
$B_{r}(x)\cap B_{r}(u)$ is contained in (and thus has volume at most as that
of) a hyperbolic ball of radius $\sigma=\sigma(\tau,r)$, where $\sigma>0$ is
defined by

$\sinh^{2}(\sigma)=\sinh^{2}(r)-\cosh^{2}(r)\tanh^{2}\left(\tau/2\right).$
(14)

Proof. Let $w$ be the point on the geodesic line through $x$ and $u$ such that
$d_{m}(x,w)=d_{m}(w,u)$. Let $y\in B_{r}(x)\cap B_{r}(u)$ be the point at
distance $r_{1}$ from $x$ and $r_{2}$ from $u$ for some
$0<r_{1},r_{2}\leqslant r$. Consider the hyperbolic triangle with the vertices
$x,y,u$ and let $\alpha$ be the angle at the vertex $u$ in this triangle. Let
$\rho=d_{m}(y,w)$ be the distance between $y$ and $w$.

Applying Lemma 9 to the hyperbolic triangle $xyu$ and the hyperbolic triangle
$ywu$, we obtain

$\cos\alpha=\frac{\cosh\tau\cosh r_{2}-\cosh r_{1}}{\sinh\tau\sinh
r_{2}}\enspace\text{ and }\enspace\cos\alpha=\frac{\cosh(\tau/2)\cosh
r_{2}-\cosh\rho}{\sinh(\tau/2)\sinh r_{2}}.$

As these two expressions have the equal value, using the equalities
$\sinh\tau=2\sinh(\tau/2)\cosh(\tau/2)$ and $\cosh\tau=2\cosh^{2}(\tau/2)-1$,
one can obtain

$\displaystyle\cosh\rho$
$\displaystyle=\cosh(\tau/2)\cos(r_{2})-\frac{1}{2\cosh(\tau/2)}\left(\cosh\tau\cosh
r_{2}-\cosh r_{1}\right)$ $\displaystyle=\cosh(\tau/2)\cosh
r_{2}-\frac{(2\cosh^{2}(\tau/2)-1)\cosh r_{2}-\cosh r_{1}}{2\cosh(\tau/2)}$
$\displaystyle=\frac{\cosh r_{2}+\cosh
r_{1}}{2\cosh(\tau/2)}\leqslant\frac{\cosh r}{\cosh(\tau/2)}.$

The final inequality holds as $\cosh x$ is an increasing function of
$x\geqslant 0$ and $r_{1},r_{2}\leqslant r$. This yields

$\displaystyle\sinh^{2}(\rho)$
$\displaystyle=\cosh^{2}(\rho)-1\leqslant\frac{\cosh^{2}(r)}{\cosh^{2}(\tau/2)}-1=\cosh^{2}(r)-1-\frac{\cosh^{2}(r)(\cosh^{2}(\tau/2)-1)}{\cosh^{2}(\tau/2)}$
$\displaystyle=\sinh^{2}(r)-\cosh^{2}(R)\tanh^{2}(\tau/2).$

As $\sinh^{2}(x)$ is an increasing function, this shows that $\rho$ is at most
$\sigma$. As $r_{1},r_{2}$ are arbitrary numbers at most $r$, this proves that
any point $y\in B_{r}(x)\cap B_{r}(u)$ has distance at most $\sigma$ from $w$,
thus proving the lemma.

The volume of an $r$-ball in ${\mathbbm{H}}^{m}$ is given by the following
formula (see e.g. [Chavel06rgmi, Equation (III.4.1)]):

$\mu_{m}(B_{r})=\mathrm{vol}(S^{m-1})\int_{0}^{r}\sinh^{m-1}\eta\,\mathrm{d}\eta,$

where $\mathrm{vol}(S^{m-1})$ denotes the $(m-1)$-dimensional (surface)
measure of the unit sphere in the Euclidean space ${\mathbbm{R}}^{m}$. Recall
that

$\mathrm{vol}(S^{m-1})=\frac{2\pi^{m/2}}{\Gamma(m/2)}={\mathrm{e}}^{-(1/2+o_{m}(1))m\ln
m}.$

Thus, for $r\to\infty$ and any $m\geqslant 2$,

$\mu_{m}(B_{r})=(1/2^{m-1}+o_{r}(1))\mathrm{vol}(S^{m-1}){\mathrm{e}}^{(m-1)r}={\mathrm{e}}^{(m-1)r-(1/2+o_{r,m}(1))m\ln
m}.$

We will also need the following bound on the ratio between $\mu_{m}(B_{r})$
and $\mu_{m}(B_{R})$ from [CohnZhao14]*Lemma 4.6:

###### Lemma 11

For $0<r<R$ and the balls $B_{r},B_{R}$ of radius $r,R$ in
${\mathbbm{H}}^{m}$, we have

$\left(\frac{\sinh r}{\sinh
R}\right)^{m}\leqslant\frac{\mu_{m}(B_{r})}{\mu_{m}(B_{R})}\leqslant\left(\frac{\sinh
r}{\sinh R}\right)^{m-1}.$

Let us estimate the fraction
$L_{R}({\mathbbm{H}}^{m})=\mu_{m}(B_{R})/\mu_{m}(B_{2R})$:

$\displaystyle L_{R}({\mathbbm{H}}^{m})$ $\displaystyle=$
$\displaystyle\frac{\int_{0}^{R}({\mathrm{e}}^{(m-1)\eta}+O(m\,{\mathrm{e}}^{(m-3)\eta}))\,\mathrm{d}\eta}{\int_{0}^{2R}({\mathrm{e}}^{(m-1)\eta}+O(m\,{\mathrm{e}}^{(m-3)\eta}))\,\mathrm{d}\eta}\
=\
\frac{\frac{1}{m-1}{\mathrm{e}}^{(m-1)R}-O({\mathrm{e}}^{(m-3)R})}{\frac{1}{m-1}{\mathrm{e}}^{2(m-1)R}-O({\mathrm{e}}^{(m-3)2R})}$
(15) $\displaystyle=$
$\displaystyle(1+O(m\,{\mathrm{e}}^{-2R}))\,{\mathrm{e}}^{-(m-1)R}.$

The error term is $o(1)$ if $m{\mathrm{e}}^{-2R}\to 0$; otherwise we have to
be more careful with lower order terms. It is conceivable that this ratio is
monotone decreasing with $R$, so it is never larger than the lower bound
$2^{-m}$ from the Euclidean case (which is the limit as $R\to 0$ for any fixed
$m$).

Note that Lemmas 9–11 also hold in $M={\mathbbm{H}}^{m}/{\mathcal{L}}$ as long
as the distance between any two involved points is at most the girth of the
lattice ${\mathcal{L}}$.

##  6 Proof of Theorem 1

Recall that Theorem 1 states that for every $\varepsilon>0$ and large $m$, we
have

$D_{R}({\mathbbm{H}}^{m})\geqslant(1-\varepsilon)m\ln(\sqrt{m}\cosh(2R))\frac{\mu_{m}(B_{R})}{\mu_{m}(B_{2R})}.$
(16)

In order to prove this, we will use the following result of Campos, Jenssen,
Michelen and Sahasrabudhe [CamposJenssenMichelenSahasrabudhe] on independent
sets in graphs with small codegrees.

###### Theorem 12

Let $G$ be a graph on $n$ vertices such that $\Delta(G)\leqslant\Delta$ and
$\Delta_{2}(G)\leqslant 2^{-7}\Delta(\ln\Delta)^{-7}$. Then the independence
number $\alpha(G)$ satisfies

$\alpha(G)\geqslant(1-o(1))\frac{n\ln\Delta}{\Delta},$

where $o(1)$ tends to $0$ as $\Delta\rightarrow\infty$.

Our proof of Theorem 1 goes as follows. Take any small constant
$\varepsilon>0$ and assume that $m\geqslant m_{0}(\varepsilon)$ is
sufficiently large. Then take any number $R>0$. Take a lattice
$\mathcal{L}\subseteq{\mathcal{G}}$ of girth at least $8R$, which exists by
Lemma 7. Let $M:=\mathbb{H}^{m}/\mathcal{L}$ with measure $\mu=\mu_{M}$ and
distance $d_{M}$. Consider $X\sim\mathrm{Po}_{\lambda}(M)$, that is, the
Poisson process $X\subseteq M$ with intensity $\lambda\,\mu$ for some choice
of $\lambda$. Once we obtain $X$, we will delete some bad points from $X$ to
obtain $Y$, and consider $G_{Y}$, the graph with the vertex set $Y$ and the
edge set

$E(G_{Y}):=\\{\,\\{x,y\\}:x,y\in Y\text{ and }0<d_{M}(x,y)\leqslant 2R\\}.$
(17)

Based on our choice of $\lambda$ and the removal process, the graph $G_{Y}$
will satisfy the condition in Theorem 12. Thus Theorem 12 yields an
independent set in $G_{Y}$, which corresponds to an $R$-packing in $M$. This
gives a lower bound on $D_{R}(M)$ which is also a lower bound on
$D_{R}({\mathbbm{H}}^{m})$ by Lemma 6.

Let us give the details of the proof (for finding $Y$ after ${\mathcal{L}}$
and $M$ have been defined). We will use asymptotic notation, like $O(1)$, with
respect to $m\to\infty$ for constants that can be chosen independently of $R$.

Given $m$ and $R$, consider the following function of $x\in(0,R]$:

$\gamma(x):=\left\\{\begin{array}[]{ll}m\cdot\tanh^{2}{(x/2)}-50\tanh^{2}(2R)\cdot\left(\ln(m)+\ln\ln\left(\frac{\sinh(2R)}{\sinh
x}\right)\right),&\mbox{if $R<m$,}\\\
\cosh^{2}{(x/2)}-m\ln(\cosh(2R)),&\mbox{if $R\geqslant m$}.\end{array}\right.$
(18)

Note that, by the monotonicity of $\sinh$, the double logarithm in (18) is
well-defined. We define $\tau=\tau(m,R)$ to be the number between $0$ and $R$
satisfying $\gamma(\tau)=0$.

Let us show that such $\tau$ uniquely exists (if $m$ is sufficiently large).
Note that $\gamma(x)$ a strictly increasing and continuous function of
$x\in(0,R]$. So it is enough to show that it assumes both positive and
negative values. If $R\geqslant m$ then we have rather generously that
$\gamma(R)\geqslant{\mathrm{e}}^{R}/4-2Rm>0$ and $\gamma(\ln m)\leqslant
m-m^{2}<0$, as desired. Moreover, we have $\tau>\ln m$ in this case. So let us
consider the case $R<m$. Using $0<\tanh(2R)<4\tanh(R/2)$, we obtain that

$\gamma(R)\geqslant\tanh^{2}(R/2)\left(m-O(\ln m)\right)>0.$

On the other hand, if we let $x\to 0$ then $\gamma(x)$ tends to $-\infty$.
Thus $\tau\in(0,R]$ satisfying $\gamma(\tau)=0$ exists and is unique.

###### Claim 12.1

If $R<m$, then the following holds:

$\frac{\sinh(2R)}{\sinh\tau}=\Theta\left(\cosh(2R)\sqrt{\frac{m}{\ln(m)}}\right).$

Proof. Observe that $\tau=o(1)$ for otherwise
$R\geqslant\tau\geqslant\Omega(1)$ and $\gamma(\tau)=\Omega(m)$ cannot be 0.

Let $\kappa:=\ln m+\ln\ln\left(\frac{\sinh(2R)}{\sinh\tau}\right)$. By the
monotonicity of $\sinh$ and by $\sinh 2x\geqslant 2\sinh x$ it holds that, for
example,

$\ln\left(\frac{\sinh(2R)}{\sinh
x}\right)\geqslant\ln\left(\frac{\sinh(2R)}{\sinh R}\right)\geqslant\ln
2\geqslant\frac{1}{\sqrt{m}},\quad\mbox{for any $0<x\leqslant R$}.$ (19)

Thus $\kappa\geqslant\frac{1}{2}\ln m$.

It is enough to show that $\kappa=O(\ln m)$. Indeed, then
$c:=\frac{\tanh^{2}(\tau)}{\tanh^{2}(\tau/2)}$ satisfies $1\leqslant
c\leqslant 4$ and we have

$\displaystyle\frac{\sinh^{2}(2R)}{\sinh^{2}(\tau)}$
$\displaystyle=\frac{\cosh^{2}(2R)}{\cosh^{2}(\tau)}\cdot\frac{\tanh^{2}(2R)}{c\tanh^{2}(\tau/2)}=\frac{\cosh^{2}(2R)}{\cosh^{2}(\tau)}\cdot\frac{m}{50c\kappa}=\frac{\cosh^{2}(2R)}{\cosh^{2}(\tau)}\cdot\frac{m}{\Theta(\ln
m)}.$

from which the claim follows by $\cosh\tau=1+o(1)=\Theta(1)$.

The proof of $\kappa=O(\ln m)$ is routine except we have to be careful to rule
out the possibility that $\tau$ is extremely small relative to $R$. Suppose on
the contrary that $\kappa\not=O(\ln m)$. Then the identity $\gamma(\tau)=0$
implies by $\tau=o(1)$ that

$m\tau^{2}=\Theta\left(\frac{\sinh^{2}(2R)}{1+\sinh^{2}(2R)}\ln\ln\left(\frac{\sinh(2R)}{\tau}\right)\right)$
(20)

It follows that $R=o(1)$ as otherwise $\tau\geqslant m^{-1/2+o(1)}$ by (20),
contradicting $\kappa\not=O(\ln m)$. We obtain from (20) that
$m=\Theta\left((R/\tau)^{2}\ln\ln(R/\tau)\right)$. Since $m\to\infty$, it
follows that $R/\tau\to\infty$ and thus $R/\tau=\Theta\left(\sqrt{m/\ln\ln
m}\right)$, which again contradicts our assumption $\kappa\neq O(\ln m)$ .

Now we define the parameters $\Delta$ and $\lambda$ as follows:

$\Delta:=\frac{1}{m^{4}}\frac{\mu(B_{2R})}{\mu(B_{\tau})}\text{ and
}\lambda:=\frac{\Delta}{\mu(B_{2R})}.$

Recall that $\mu=\mu_{M}$ denotes the measure on $M$ and that, for every
$0\leqslant r\leqslant 4R$, it assigns the same measure to an $r$-ball in $M$
as the hyperbolic measure $\mu_{m}$ assigns to an $r$-ball in
${\mathbbm{H}}^{m}$ by the girth assumption on ${\mathcal{L}}$.

Let us show that

$\displaystyle\ln\Delta=(1+o(1))m\ln(\sqrt{m}\cosh(2R)).$ (21)

Indeed, if $R<m$ then (21) is a consequence of Lemma 11 and Claim 12.1, so
suppose that $R\geqslant m$. Then we have $\tau>\ln m$ and
$\sinh\tau\leqslant\cosh\tau\leqslant 2\cosh^{2}(\tau/2)$, thus (18) implies

$0<\ln(\sinh\tau)=O(\ln R+\ln m)=o(\ln(\sinh(2R))).$

Thus Lemma 11 implies that

$\displaystyle\ln\Delta$ $\displaystyle=$
$\displaystyle(m+O(1))(\ln(\sinh(2R))-\ln(\sinh\tau))+O(\ln m)$
$\displaystyle=$ $\displaystyle(1+o(1))m\ln(\sinh(2R))$ $\displaystyle=$
$\displaystyle(1+o(1))m\ln(\sqrt{m}\cosh(2R)),$

as $|\sinh(2R)-\cosh(2R)|\leqslant 1$ and $\ln(\sqrt{m})=o(\ln(\cosh(2R)))$,
proving (21).

In particular, we have that $\Delta\to\infty$ as $m\to\infty$.

With these choices, we have the following lemma. Recall that $G_{Y}$ is the
graph on $Y$ whose edge set is defined by (17).

###### Lemma 13

There exists $Y\subseteq M$ such that

$|Y|\geqslant\left(1-\frac{1}{m}\right)\frac{\Delta}{\mu(B_{2R})}\,\mu(M)$

and the graph $G_{Y}$ satisfies that

$\Delta(G_{Y})\leqslant\Delta+\Delta^{2/3}\text{ and
}\Delta_{2}(G_{Y})\leqslant\Delta\,(\ln\Delta)^{-10}.$

With this lemma and Theorem 12, we obtain that there is an $R$-packing in $M$
of size at least

$(1-o(1))\frac{|Y|\ln\Delta}{\Delta}\geqslant(1-1/m-o(1))\frac{\mu(M)\ln\Delta}{\mu(B_{2R})}.$

Thus (21) implies that, as $m\to\infty$,

$D_{R}({\mathbbm{H}}^{m})\geqslant
D_{R}(M)\geqslant(1-o(1))\ln\Delta\cdot\frac{\mu(B_{R})}{\mu(B_{2R})}\geqslant(1-\varepsilon)m\ln(\sqrt{m}\cosh(2R))\frac{\mu(B_{R})}{\mu(B_{2R})}.$

This proves (16) (that is, Theorem 1) in both cases. Thus it remains to prove
Lemma 13.

Proof of Lemma 13. We follow the argument in Section 2 of
[CamposJenssenMichelenSahasrabudhe]. In order to prove Lemma 13, we sample a
Poisson point process $X\subseteq M$ with the intensity $\lambda\,\mu$, and
obtain a desired set $Y\subseteq X$ by removing points $x\in X$ which satisfy
at least one of the following two conditions:

$\displaystyle|X\cap B_{2R}(x)|\geqslant\Delta+\Delta^{2/3}\text{,\ \ or\ \
$\exists\,y\in X$\ \ }|X\cap B_{2R}(x)\cap
B_{2R}(y)|\geqslant\Delta\,(\ln\Delta)^{-10}.$ (22)

We will show that $Y$ has almost the same size as $X$. The following _identity
of Mecke_ will be useful: for any bounded measurable set $\Lambda\subseteq M$
and events $(A_{x})_{x\in\Lambda}$ we have

$\displaystyle\mathbb{E}\left|\\{x\in X\cap\Lambda:A_{x}\text{ holds for
}X\\}\right|=\lambda\int_{\Lambda}\mathbb{P}[\,A_{x}\text{ holds for
}X\cup\\{x\\}\,]\,\mathrm{d}\mu(x).$ (23)

Also, the following tail bound for a Poisson random variable $Z$ will be used:

$\displaystyle\mathbb{P}[\,Z-\mathbb{E}Z\geqslant
t\,\mathbb{E}Z\,]\leqslant\exp\left(-\min\\{t,t^{2}\\}\frac{\mathbb{E}Z}{3}\right).$
(24)

First, we show the following claim stating that, on average, only a small
fraction of $x\in X$ satisfies the first bad condition in (22).

###### Claim 13.1

Let $X\sim\mathrm{Po}_{\lambda}(M)$. Then

$\mathbb{E}|\\{x\in X:|X\cap
B_{2R}(x)|\geqslant\Delta+\Delta^{2/3}\\}|\leqslant\frac{1}{m^{2}}\,\mathbb{E}|X|.$

Proof. Take any $x\in M$. Recall that $|X\cap B_{2R}(x)|$ is a Poisson random
variable of mean $\lambda\mu(B_{2R})=\Delta$. Thus using (24), we have

$\mathbb{P}[\,|X\cap
B_{2R}(x)|\geqslant\Delta+\Delta^{2/3}-1\,]\leqslant\exp\left(-\frac{1}{4}\Delta^{1/3}\right)\leqslant
m^{-2}.$

Thus we have by Mecke’s identity that

$\mathbb{E}\,\left|\\{x\in X:|X\cap
B_{2R}(x)|\geqslant\Delta^{2/3}\\}\right|=\lambda\int_{M}\mathbb{P}[\,|X\cap
B_{2R}(x)|\geqslant\Delta^{2/3}-1\,]\,\mathrm{d}\mu(x)\leqslant\frac{\lambda\,\mu(M)}{m^{2}},$

giving the desired by $\mathbb{E}|X|=\lambda\,\mu(M)$.

Before considering the second bad condition in (22), we prove the following
claim.

###### Claim 13.2

For any $x,y\in M$ at distance $d_{M}(x,y)\geqslant\tau$, it holds that

$\lambda\,\mu(B_{2R}(x)\cap B_{2R}(y))\leqslant\Delta\,(\ln\Delta)^{-15}.$

Proof. Note that we only have to consider the case where $\tau$ is at most
$4R$, as otherwise $B_{2R}(x)\cap B_{2R}(y)$ is empty. As the lattice
${\mathcal{L}}$ has girth at least $8R$, Lemma 10 with $r=2R$ applies to the
points $x,y$ in $M$ (instead of ${\mathbbm{H}}^{m}$). Hence we obtain that the
following where $\sigma=\sigma(2R,\tau)$ was defined in (14):

$\displaystyle\begin{split}\lambda\,\mu(B_{2R}(x)\cap
B_{2R}(y))&\leqslant\lambda\,\mu(B_{\sigma})\leqslant\Delta\,\frac{\mu(B_{\sigma})}{\mu(B_{2R})}\\\
&\leqslant\Delta\,\frac{\sinh^{m-1}(\sigma)}{\sinh^{m-1}(2R)}=\Delta\,\left(1-\coth^{2}(2R)\tanh^{2}\left(\tau/2\right)\right)^{(m-1)/2}.\end{split}$
(25)

Here, the penultimate inequality is from Lemma 11.

Recall that $\tau$ satisfies $\gamma(\tau)=0$ where $\gamma$ was defined in
(18). If $R<m$, then the final expression in (25) becomes

$\displaystyle\Delta\left(1-50\cdot\frac{\ln m+\ln\ln(\frac{\sinh
2R}{\sinh\tau})}{m}\right)^{(m-1)/2}\leqslant\Delta\,\frac{1}{m^{20}}\cdot\ln\left(\frac{\sinh
2R}{\sinh\tau}\right)^{-20}\leqslant\Delta\,(\ln\Delta)^{-15},$

as we want. Here, the final inequality holds as Lemma 11 implies
$\ln\Delta\leqslant m\ln(\frac{\sinh 2R}{\sinh\tau})$. If $R>m$, then as
$\coth^{2}(2R)\tanh^{2}(\tau/2)\geqslant\tanh^{2}(\tau/2)=1-\frac{1}{\cosh^{2}(\tau/2)}$,
the final term in (25) is bounded from above by

$\displaystyle\Delta\left(\frac{1}{\cosh^{2}{(\tau/2)}}\right)^{(m-1)/2}$
$\displaystyle=\Delta\left(\frac{1}{m\ln(\cosh(2R))}\right)^{(m-1)/2}$
$\displaystyle\leqslant\Delta\,(\ln\Delta)^{-(m-1)/4}\leqslant\Delta\,(\ln\Delta)^{-15},$

where the penultimate inequality follows from (21). This proves the claim.

Now we bound the number of points satisfying the second bad condition in (22).

###### Claim 13.3

Let $X\sim\mathrm{Po}_{\lambda}(M)$ and put $\eta:=(\ln\Delta)^{-10}$. Then we
have that

$\mathbb{E}|\\{x\in X:|X\cap B_{2R}(x)\cap B_{2R}(y)|\geqslant\eta\Delta\text{
for some }y\in X\\}|\leqslant\frac{1}{2m}\,\mathbb{E}|X|.$

Proof. Take any $x\in M$. For $y\in M$, let $I_{x,y}:=|X\cap B_{2R}(x)\cap
B_{2R}(y)|$. Using Markov’s inequality, we have

$\mathbb{P}[\,\exists y\in
X:I_{x,y}\geqslant\eta\Delta-1\,]\leqslant\mathbb{E}\,|B_{\tau}(x)\cap
X|+\mathbb{E}\,|\\{y\in X\setminus
B_{\tau}(x):I_{x,y}\geqslant\eta\Delta-1\\}|.$

We bound each of these two terms.

For the first term, using Lemma 11 and the definition of $\Delta$, we have

$\mathbb{E}\,|B_{\tau}(x)\cap
X|\leqslant\lambda\,\mu(B_{\tau}(x))\leqslant\frac{\Delta\,\mu(B_{\tau})}{\mu(B_{2R})}=\frac{1}{m^{4}}.$

For the second term, we only need to consider $y\in B_{4R}(x)$ as otherwise
$I_{x,y}=0$. (Note that $\eta\Delta\to\infty$.) Using (23) and Markov’s
inequality, the second term is

$\displaystyle\lambda\,$ $\displaystyle\int_{M\setminus
B_{\tau(x)}}\mathbb{P}[\,I_{x,y}\geqslant\eta\Delta-2\,]\,\mathrm{d}\mu(y)=\lambda\,\int_{B_{4R}(x)\setminus
B_{\tau(x)}}\mathbb{P}[\,I_{x,y}\geqslant\eta\Delta-2\,]\,\mathrm{d}\mu(y)$
$\displaystyle\leqslant\lambda\,\mu(B_{4R})\ \sup_{y\in B_{4R}(x)\setminus
B_{\tau}(x)}\mathbb{P}[\,I_{x,y}\geqslant\eta\Delta-2\,].$

Using Claim 13.2, we have for $y$ with $d_{M}(y,x)\geqslant\tau$ that

$\mathbb{E}I_{x,y}=\lambda\,\mu(B_{2R}(x)\cap
B_{2R}(y))\leqslant\Delta\,(\ln\Delta)^{-15}.$

As $I_{x,y}$ is a Poisson random variable, we can apply (24) to obtain that,
rather roughly,

$\displaystyle\lambda\,\mu(B_{4R})\
\mathbb{P}[\,I_{x,y}\geqslant\eta\Delta-2\,]$
$\displaystyle\leqslant\Delta\,\frac{\mu(B_{4R})}{\mu(B_{2R})}\exp\left(-\Delta(\ln\Delta)^{-15}\right)<\frac{1}{m^{2}}.$

The last inequality follow by observing by Lemma 11 and (21) that

$\frac{\mu(B_{4R})}{\mu(B_{2R})}\leqslant\left(\frac{\sinh(4R)}{\sinh(2R)}\right)^{m}\leqslant{\mathrm{e}}^{2Rm}\leqslant\frac{\exp\left(\Delta(\ln\Delta)^{-15}\right)}{\Delta\,m^{2}}.$

Thus we have shown that, for every $x\in X$,

$\mathbb{P}[\,\exists y\in
X:I_{x,y}\geqslant\eta\Delta-1\,]\leqslant\frac{1}{m^{4}}+\frac{1}{m^{2}}<\frac{1}{2m}.$

By Mecke’s idenity, we conclude that the expected number of $x\in X$
satisfying the second bad condition in (22) is at most
$\frac{1}{2m}\lambda\,\mu(M)$, proving the claim

Now we prove Lemma 13. Let $X$ be obtained from the Poisson point process in
$M$ with intensity $\lambda\,\mu$. Let $S_{1},S_{2}\subseteq X$ be the points
that satisfy the first and second bad properties in (22). Let
$Y:=X\setminus(S_{1}\cup S_{2})$. Then the previous claims imply that

$\mathbb{E}|Y|\geqslant\mathbb{E}|X|-\mathbb{E}|S_{1}|-\mathbb{E}|S_{2}|\geqslant\left(1-\frac{1}{m^{2}}-\frac{1}{2m}\right)\mathbb{E}|X|\geqslant\left(1-\frac{1}{m}\right)\frac{\Delta}{\mu(B_{2R})}\,\mu(M).$

There is an outcome $X$ such that $|Y|$ is at least its expected value,
finishing the proof of Lemma 13.

## Acknowledgements

The authors would like to thank Lewis Bowen for the helpful discussions.

## References