# Dimension drop for diagonalizable
flows on homogeneous spaces

Dmitry Kleinbock  Department of Mathematics, Brandeis University, Waltham MA
<EMAIL_ADDRESS>and  Shahriar Mirzadeh  Department of Mathematics,
University of Cincinnati, Cincinnati OH<EMAIL_ADDRESS>
(Date: July 2022)

###### Abstract.

Let $X=G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$,
let $O$ be an open subset of $X$, and let $F=\\{g_{t}:t\geq 0\\}$ be a one-
parameter subsemigroup of $G$. Consider the set of points in $X$ whose
$F$-orbit misses $O$; it has measure zero if the flow is ergodic. It has been
conjectured that this set has Hausdorff dimension strictly smaller than the
dimension of $X$. This conjecture is proved when $X$ is compact or when $G$ is
a simple Lie group of real rank $1$, or, most recently, for certain flows on
the space of lattices. In this paper we prove this conjecture for arbitrary
${\operatorname{Ad}}$-diagonalizable flows on irreducible quotients of
semisimple Lie groups. The proof uses exponential mixing of the flow together
with the method of integral inequalities for height functions on $G/\Gamma$.
We also derive an application to jointly Dirichlet-Improvable systems of
linear forms.

###### 2010 Mathematics Subject Classification:

Primary: 37A17, 37A25; Secondary: 11J13.

The first-named author was supported by NSF grant DMS-1900560.

##  1\. Introduction

###  1.1. The set-up

Let $G$ be a connected Lie group, and let $\Gamma$ be a lattice in $G$. Denote
by $X$ the homogeneous space $G/\Gamma$ and by $\mu$ the $G$-invariant
probability measure on $X$. For an unbounded subset $F$ of $G$ and a non-empty
open subset $O$ of $X$ define the sets ${E(F,O)}$ and $\widetilde{E}(F,O)$ as
follows:

$\displaystyle E(F,O)$ $\displaystyle:=\\{x\in X:gx\notin O\ \forall\,g\in
F\\}$ (1.1) $\displaystyle\subset\ \widetilde{E}(F,O)$ $\displaystyle:=\\{x\in
X:\exists\text{ compact }Q\subset G\text{ such that }gx\notin O\ \forall\,g\in
F\smallsetminus Q\\}$ $\displaystyle=\bigcup_{\text{compact }Q\subset
G}E(F\smallsetminus Q,O)$

of points in $X$ whose $F$-trajectory always (resp., eventually) stays away
from $O$. If $F$ is a subgroup or a subsemigroup of $G$ acting ergodically on
$(X,{\mu})$, then the set $\\{gx:g\in F\\}$ is dense for $\mu$-almost all
$x\in X$, in particular $\mu\big{(}\widetilde{E}({F},O)\big{)}=0$.

The present paper studies the following natural question, asked several years
ago by Mirzakhani: for a subgroup or sub-semigroup $F\subset G$, if the set
$E(F,O)$ has measure zero, does it necessarily have less than full Hausdorff
dimension? it is reasonable to conjecture that the answer is always
affirmative; in other words, that the following ‘Dimension Drop Conjecture’
(DDC) holds: if $F\subset G$ is a subsemigroup and $O$ is an open subset of
$X$, then either $E(F,O)$ has positive measure, or its dimension is less than
the dimension of $X$. When $X$ is compact it follows from the variational
principle for measure-theoretic entropy, as outlined in [KW2, §7]; an
effective argument using exponential mixing was developed in [KMi1]. See also
[AAEKMU, Theorem 1.1 and Corollary 1.3] which explores the dimension drop
phenomenon in a different setting.

In the non-compact case a weaker statement that for any non-quasiunipotent
flow on a finite volume homogeneous space, the set of points that lie on
divergent trajectories has positive codimension is a conjecture made by Cheung
in [C]. A standard approach to this circle of problems is to use the
phenomenon of non-escape of mass on homogeneous spaces, going back to the work
of Eskin–Margulis–Mozes and Eskin–Margulis, see [EMM, EM] and also [KKLM].
This is precisely how Cheung’s conjecture has been recently verified by Guan
and Shi [GS]; see also [AGMS, RHW] for some related work. However combining
the non-escape of mass argument with an additional construction taking care of
the compact part of the space is more involved. Previously this was done in
the case when $G$ is a simple Lie group of real rank $1$ [EKP], and then, in
the most recent work of the authors [KMi2], when

$\displaystyle
X=\operatorname{SL}_{m+n}({\mathbb{R}})/\operatorname{SL}_{m+n}({\mathbb{Z}})\text{
and }$ (1.2) $\displaystyle
F=\big{\\{}\operatorname{diag}(e^{nt},\dots,e^{nt},e^{-mt},\dots,e^{-mt}):t\geq
0\big{\\}}.$

In this paper we generalize the approach of [KMi2] by exhibiting two abstract
assumptions sufficient for the validity of DDC. One takes care of the compact
part of the space, while the other deals with the non-escape of mass.

Let $G$ be a Lie group, $\Gamma$ a lattice in $G$ and $X=G/\Gamma$. We shall
start by introducing some notation. Fix a right-invariant Riemannian structure
on $G$, and denote by ‘${\operatorname{dist}}$’ the corresponding Riemannian
metric, using the same notation for the induced metric on homogeneous spaces
of $G$. In what follows, if $P$ is a subgroup of $G$, we will denote by
$B^{P}(r)$ the open ball in $P$ of radius $r$ centered at the identity element
with respect to the metric on $P$ corresponding to the Riemannian structure
induced from $G$. We will let $\nu$ stand for the Haar measure on $P$
normalized so that $\nu\big{(}B^{P}(r)\big{)}=1$. For simplicity, we use
$B(r)$ instead of $B^{G}(r)$ to denote a ball of radius $r$ in $G$ centered at
the identity element. Also, $B(x,\rho)$ will stand for the open ball in $X$
centered at $x\in X$ of radius $\rho$.

For $x\in X$ denote by $\pi_{x}$ the map $G\to X$ given by $\pi_{x}(g):=gx$,
and by $r_{0}(x)$ the injectivity radius of $x$:

${r_{0}(x):=}\,\sup\\{r>0:\pi_{x}\text{ is injective on }B(r)\\}.$

If $K$ is a subset of $X$, let us denote by $r_{0}(K)$ the injectivity radius
of $K$:

$r_{0}(K):=\inf_{x\in K}r_{0}(x)=\sup\\{r>0:\pi_{x}\text{ is injective on
}B(r)\ \ \forall\,x\in K\\};$

it is known that $r_{0}(K)>0$ if and only if $K$ is bounded.

The notation ${A\gg B}$ where $A$ and $B$ are quantities depending on certain
parameters, will mean ${A\geq CB}$, where $C$ is a constant independent on
those parameters.

Let now $F=\\{g_{t}:t\geq 0\\}$ be an ${\operatorname{Ad}}$-diagonalizable
one-parameter subsemigroup of $G$. A key role in our method will be played by
the unstable horospherical subgroup with respect to $F$, defined as

${H:=\\{g\in G:{\operatorname{dist}}({g_{t}}g{g_{-t}},e)\to
0\,\,\,as\,\,\,t\to-\infty\\}.}$ (1.3)

Equivalently, $H$ is the Lie group whose Lie algebra is a direct sum of
eigenspaces of ${\operatorname{Ad}}\,g_{1}$ corresponding to eigenvalues with
absolute value $>1$. More generally, we will work with connected subgroups $P$
of $H$ normalized by $F$ and will give conditions sufficient for ‘dimension
drop along $P$-orbits’; that is, ensuring a nontrivial upper estimate for

$\dim\big{(}\\{h\in P:hx\in\widetilde{E}({F},O)\\}\big{)},$

where $x\in X$ is arbitrary, and $O$ is a non-empty open subset of $X$.

Throughout the proof we will pay close attention to translates of $P$-orbits
in $X$ by $g_{t}$. it will be convenient to use the following notation: if $f$
a function on $P$ and $t\geq 0$, we will define the integral operator
$I_{f,t}$ acting on functions $\psi$ on $X$ via

${(I_{f,t}\psi)(x):=\int_{P}f({{h}})\psi(g_{t}{{h}}x)\,d{{\nu}}({{h}})\,.}$
(1.4)

In other words, $(I_{f,t}\psi)(x)$ is the integral of $\psi$ with respect to
the $g_{t}$-translate of the $\pi_{x}$-pushforward of the signed measure
$f\,d{{\nu}}$. When $f=1_{B}$ for a subset $B$ of $P$, we will write

$I_{B,t}=\int_{B}\psi(g_{t}{{h}}x)\,d{{\nu}}({{h}})$

in place of $I_{1_{B},t}$.

###  1.2. Exponential mixing and effective equidistribution

The first ingredient of our proof is the effective equidistribution of
$g_{t}$-translates of $P$-orbits on $X$. To introduce this property we will
work with Sobolev spaces of functions on $X$. Let us define

$C^{\infty}_{2}(X)=\big{\\{}h\in C^{\infty}(X):\|h\|_{\ell{,2}}<\infty\text{
for any }\ell\in{\mathbb{Z}}_{+}\big{\\}},$

where $\|\cdot\|_{\ell{,2}}$ is the “$L^{{2}}$, order $\ell$” Sobolev norm
(see §3 for more detail). Now let us introduce the following

######  Definition 1.1.

[KMi1] Say that a subgroup $P$ of $G$ has Effective Equidistribution Property
(EEP) with respect to the flow $(X,F)$ if there exists constants
$a,b,\lambda>0$ and $\ell\in{{\mathbb{N}}}$ such that for any $x\in X$ and
$t>0$ with

${t\ {\geq a+b\log\frac{1}{r_{0}({x})},}}$ (1.5)

any $f\in C^{\infty}(P)$ with $\operatorname{supp}f\subset B^{P}(1)$ and any
$\psi\in C^{\infty}_{2}(X)$ it holds that

${\left|{(I_{f,t}\psi)(x)-\int_{P}f\,d\nu{\mkern
1.0mu}\int_{X}\psi\,d\mu{\mkern 1.0mu}}\right|{\ll}\
{\max\big{(}{\left\|\psi\right\|_{C^{1}},\left\|\psi\right\|_{\ell{,2}}}\big{)}}\cdot{\left\|f\right\|_{{C^{\ell}}}}\cdot{e^{-\lambda
t}}{\mkern 1.0mu}.}$ (1.6)

Note that the constants $a,b$ in (1.5) and an implicit constant in (1.6) are
allowed to depend on $P$ and $F$, but not on $x,t,f$ and $\psi$.

A general principle that mixing implies equidistribution of unstable
leaves—that is, orbits of $H$ as in (1.3)—has been widely used in homogeneous
dynamics, starting perhaps with the Ph.D. thesis of Margulis [M]. Its
effective versions have been exploited in [KM1, KM2]. Specifically, let us say
that a flow $(X,F)$ is exponentially mixing if there exist $\gamma>0$ and
$\ell\in{\mathbb{Z}}_{+}$ such that for any $\varphi,\psi\in
C^{\infty}_{2}(X)$ and for any $t\geq 0$ one has

${\left|{({g_{t}}\varphi,\psi)-\int_{X}\varphi\,d\mu\int_{X}\psi\,d\mu}\right|\ll{e^{-\gamma
t}}{\left\|\varphi\right\|_{\ell{,2}}}{\left\|\psi\right\|_{\ell{,2}}}.}$

This property for non-quasiunipotent flows follows from the strong spectral
gap of the regular representation of $G$, see [KM1]; the latter is known to
hold for quotients of semisimple Lie groups without compact factors by
irreducible lattices, see [KS, p. 285].

The fact that property (EEP) for expanding horospherical subgroups follows
from exponential mixing was established in [KMi1] by a variation of the method
developed in [KM1]:

######  Theorem 1.2.

[KMi1, Theorem 2.5] Let $G$ be a Lie group, $\Gamma$ a lattice in $G$, and let
${F}$ be a one-parameter subsemigroup of $G$ whose action on $X=G/\Gamma$ is
exponentially mixing. Then $H$ as in (1.3) satisfies property (EEP) with
respect to the flow $(X,F)$.

###  1.3. Height functions and integral inequalities

The second ingredient of our proof is studying $g_{t}$-translates of
$P$-orbits in $X$ at infinity. For that it is helpful to have a family of
positive functions on $X$ which grow at infinity and behave nicely with
respect to integral operators of type (1.4) This is done via the method of
integral inequalities which goes back to [EMM] and [EM], and was recently
applied in [GS] for upper estimates for the Hausdorff dimension of the set of
points of $X$ with divergent $g_{t}$-trajectories. To state their result, it
will be convenient to introduce certain terminology, which will be used
throughout the paper. Namely, let us say that a non-negatve continuous
function $u$ on $X$ is a height function if it is proper, that is
$u(x)\to\infty$ if and only if $x\to\infty$ in $X$, and regular, that is there
exists a non-empty neighborhood $B$ of identity in $G$ and $C>0$ such that

${u(hx)\leq Cu(x)\text{ for every $h\in B$ and all }x\in X;}$ (1.7)

equivalently, if for any bounded $B\subset G$ there exists $C>0$ such that
(1.7) holds. Also let us say that $u$ satisfies the $(c,d)$-Margulis
inequality with respect to an operator $I:C(X)\to C(X)$ if for all $x\in X$
one has

$(Iu)(x)\leq cu(x)+d.$

See [EMo, SS], where functions $u$ satisfying the $(c,d)$-Margulis inequality
for some $c<1$ and $d\in{\mathbb{R}}$ are called Margulis functions. With this
terminology, let us introduce the following definition.

######  Definition 1.3.

Say that a subgroup $P$ of $G$ has Effective Non-Divergence Property  (ENDP)
with respect to the flow $(X,F)$ if there exists $0<c_{0}<1$ and $t_{0}>0$
such that for any $t\geq t_{0}$ one can find $d_{t}>0$ and a height function
$u_{t}$ such that $u_{t}$ satisfies the $(c_{0},d_{t})$-Margulis inequality
with respect to $I_{B^{P}(1),t}$.

In the course of proving the main result of [KMi2] the above property was
shown in the case (1.2), see [KMi2, Proposition 3.4.]. The proof followed a
construction fron [KKLM] and used functions on the space of lattices coming
from the work of Eskin, Margulis and Mozes [EMM]. To get more examples, we
will quote the following result from [GS]:

######  Theorem 1.4.

[GS, Lemma 4.3] Let $X=\prod_{i=1}^{n}G_{i}/\Gamma_{i}$, where each
$G_{i}/\Gamma_{i}$ is a non-uniform irreducible quotient of a semisimple Lie
group without compact factors, and let $F$ be be a one-parameter
${\operatorname{Ad}}$-diagonalizable subsemigroup of $G=\prod_{i=1}^{n}G_{i}$
such that the projection of $F$ to each $G_{i}$ is unbounded. Also let $H$ be
as in (1.3) and let $B=B^{H}(1)$. Then there exist $\alpha,t_{0}>0$ and, for
any $t\geq t_{0}$, a height function $u_{t}$ on $X$ and $d_{t}\in{\mathbb{R}}$
such that for any $t\geq t_{0}$ the function $u_{t}$ satisfies an $(e^{-\alpha
t},d_{t})$-Margulis inequality with respect to $I_{B,t}$. Consequently, $H$
has property (ENDP) with respect to $(X,F)$.

We remark that, since the flow $(X,F)$ as in the above theorem is
exponentially mixing, the expanding horospherical subgroup $H$ has property
(EEP) with respect to $(X,F)$ as well.

###  1.4. The main results

We are now ready to state our main theorem.

######  Theorem 1.5.

Let $G$ be a Lie group, $\Gamma$ a lattice in $G$, $X=G/\Gamma$, $F$ a one-
parameter ${\operatorname{Ad}}$-diagonalizable subsemigroup of $G$, $H$ the
unstable horospherical subgroup relative to $F$, and $P$ a subgroup of $H$
which is normalized by $F$ and has properties (EEP) and (ENDP) with respect to
the flow $(X,F)$. Then for any non-empty open subset $O$ of $X$ one has

${\inf_{x\in X}\operatorname{codim}\left(\big{\\{}h\in
P:hx\in\widetilde{E}({F},O)\big{\\}}\right)>0.}$

Applying the above theorem with $P=H$ and using Theorems 1.2 and 1.4 together
with the standard slicing technique, we get

######  Corollary 1.6.

Let $G,\Gamma,X$ and ${F}$ be as in Theorem 1.4. Then for any non-empty open
subset $O$ of $X$ one has $\dim\widetilde{E}({F},O)<\dim X$; that is, DDC
holds in this generality.

We remark that the main result of [KMi2] in the case (1.2) actually contains
an effective upper bound for the dimension of $E({F},O)$. In the more general
set-up of this paper it is also possible to make our estimates effective. For
that one needs to find a lower bound for the injectivity radius of compact
sets $\\{x:u_{t}(x)\leq M\\}$. Such bounds can be obtained for a large class
of height functions including $u_{t}$ as in Theorem 1.4 using a similar
procedure as in [SS, Proposition 26] and [BQ, Lemma 6.3]. We have decided not
to overcomplicate the exposition with the proof of the stronger result;
however see §10 for some indications of the proof.

Another remark is that, similarly to [KMi2], we could have considered cyclic
semigroups $F$ of the form $\\{g^{t}:t\in{\mathbb{Z}}_{+}\\}$, where $g$ is an
${\operatorname{Ad}}$-diagonalizable element of $G$. Then, after replacing
$g_{t}$ with $g^{t}$ in (1.4), the conclusions of Theorem 1.5 and Corollary
1.6 can be established for discrete-time actions, with minor modifications of
the proofs.

The structure of the paper is as follows. In the next section we state a
technical theorem (Theorem 2.1) and show how it implies Theorem 1.5 and
Corollary 1.6. The proof of Theorem 2.1 occupies the bulk of the paper. It has
two main ingredients: one deals with orbits staying inside a fixed compact
subset of $X$, which are handled in §§4–5 with the help of the effective
equidistribution assumption. The other one (§§6–7) takes care of orbits
venturing far away into the cusp of $X$; there we use property (ENDP) via the
method of integral inequalities for height functions on $X$. The two
ingredients are combined in §§8–9. Some concluding remarks, including an
application to joint Dirichlet improvement in Diophantine approximation, are
presented in §10.

Acknowledgements. The authors are grateful to Alex Eskin for bringing
Mirzakhani’s question to their attention, and to Victor Beresnevich for useful
remarks, in particular for asking a question that led to Theorem 10.1.

##  2\. Theorem 2.1 $\Rightarrow$ Theorem 1.5 $\Rightarrow$ Corollary 1.6

Let us introduce the following notation: for a non-empty open subset $O$ of
$X$ and $r>0$ denote by $\sigma_{r}O$ the inner $r$-core of $O$, defined as

${\sigma_{r}O:=\\{x\in X:{\operatorname{dist}}(x,O^{c})>r\\}.}$ (2.1)

This is an open subset of $O$, whose measure is close to $\mu(O)$ for small
enough values of $r$.

Furthermore, for a closed subset $S$ of $X$ denote by $\partial_{r}S$ the
$r$-neighborhood of $S$, that is,

$\partial_{r}S:=\\{x\in X:{\operatorname{dist}}(x,S)<r\\}.$

In particular, for $z\in X$ we have $\partial_{r}\\{z\\}=B(z,r)$, the open
ball of radius $r$ centered at $z$. Note that we always have

${\partial_{r}S\subset\big{(}\sigma_{r}(S^{c})\big{)}^{c}\text{ for all
}S\subset X,\ r>0.}$ (2.2)

Also, given $G$, $H$, and $F=\\{g_{t}:t\geq 0\\}$ as in Theorem 1.5, for any
subgroup $P$ of $H$ that is normalized by $F$ define:

${\lambda_{\min}:=\min\big{\\{}\lambda:\,\lambda\text{ is an eigenvalue of
}{\operatorname{ad}}_{{g_{1}}}|_{\mathfrak{p}}\big{\\}}}$ (2.3)

and

${\lambda_{\max}:=\max\big{\\{}\lambda:\,\lambda\text{ is an eigenvalue of
}{\operatorname{ad}}_{{g_{1}}}|_{\mathfrak{p}}\big{\\}}.}$ (2.4)

Note that all eigenvalues of the restriction of
${\operatorname{ad}}_{{g_{1}}}$ to $\mathfrak{p}$, including $\lambda_{\min}$
and $\lambda_{\max}$, are positive.

In this section we derive Theorem 1.5 from the following crucial but technical
theorem.

######  Theorem 2.1.

Let $G,\Gamma,X,F=\\{g_{t}:t\geq 0\\}$ and $P$ be as in Theorem 1.5, and let
$p=\dim P$. Then there exist

$r_{*},C_{1},C_{2},a^{\prime},b^{\prime},\lambda>0$

such that the following holds:
For any $0<c<1$ there exist $t>0$ and a compact subset $Q$ of $X$ such that:

  1. (1)

For all $x\in X$, and for all $2\leq k\in{\mathbb{N}}$, the set

${{S(k,t,x):=\left\\{h\in P:{{g_{Nkt}}}hx\notin Q\,\,\,\forall
N\in{\mathbb{N}}\right\\}}}$ (2.5)

satisfies

${{\operatorname{codim}{S(k,t,x)}}\geq{\frac{1}{{\lambda_{\max}}kt}\log{\frac{1-c}{4c}}}.}$
(2.6)

  2. (2)

For all $2\leq k\in{\mathbb{N}}$, all $r$ satisfying

${{{e^{\frac{a^{\prime}-kt}{b^{\prime}}}}\leq
r<\frac{1}{4}\min\big{(}r_{0}\left(\partial_{1}Q\right),{r_{*}}\big{)}},}$
(2.7)

all $\theta\in{\left[r,\frac{r_{*}}{2}\right]}$, all $x\in X$, and for all
open subsets $O$ of $X$ we have

${{{\operatorname{codim}\left({\left\\{h\in P\smallsetminus
S(k,t,x):hx\in\widetilde{E}({F},O)\right\\}}\right)}\geq{{\frac{\mu\big{(}{\sigma_{4\theta}}O\big{)}-{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}-\frac{C_{2}}{r^{p}}e^{-\lambda
kt}}{{\lambda_{\max}}kt}}}}.}$ (2.8)

We now show how the two estimates are put together.

###### Proof of Theorem 1.5 assuming Theorem 2.1.

Recall that we are given the constants
$r_{*},C_{1},C_{2},a^{\prime},b^{\prime},\lambda>0$ such that statements (1),
(2) of Theorem 2.1 hold. Let $O$ be an open subset of $X$. Define

${\theta_{O}:=\sup\left\\{0<\theta\leq
1:\mu({\sigma_{4\theta}}O)\geq\frac{1}{2}\mu(O)\right\\},}$ (2.9)

then put $\theta:=\min(\theta_{O},\frac{r_{*}}{2})$ and

${c:=\min\left({\frac{1}{4e^{1/2}+1}},{{\left(\frac{\mu(O)}{128C_{1}}\cdot\theta^{p}\right)^{2}}}\right).}$
(2.10)

Now choose $t$ and $Q$ as in the assumption of Theorem 2.1. Then in view of
(2.10), statement (1) of Theorem 2.1 readily implies that for any $2\leq
k\in{\mathbb{N}}$ one has

${{\operatorname{codim}S(k,t,x)}\geq\frac{1}{2{\lambda_{\max}}kt}.}$ (2.11)

Next, let

${r:=\frac{1}{{{4}}}\min\big{(}r_{0}\left(\partial_{1}Q\right),{r_{*}},\theta_{O}\big{)}.}$
(2.12)

Clearly the second inequality in (2.7) is then satisfied. Now take $2\leq
k\in{\mathbb{N}}$ sufficiently large so that

${{e^{\frac{a^{\prime}-kt}{b^{\prime}}}}\leq
r\quad\text{and}\quad\frac{C_{2}}{r^{p}}e^{-\lambda kt}\leq\frac{\mu(O)}{8};}$
(2.13)

this will imply the first inequality in (2.7). Also it is easy to see from
(2.12) that $\theta\in{\left[r,\frac{r_{*}}{2}\right]}$; hence (2.8) holds.

Observe that since $\theta\leq\theta_{O}$, by definition of $\theta_{O}$, we
have

${\mu\big{(}{{{\sigma_{{4{\theta}}}}{O}}}\big{)}\leq\frac{\mu(O)}{2}.}$ (2.14)

Definition of $c$ implies

${{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}\underset{c<1/2}{<}{\frac{8C_{1}}{\theta^{p}}}\cdot
2\sqrt{c}\underset{\eqref{eq:c}}{\leq}}\frac{\mu(O)}{8}.}$ (2.15)

Hence, by combining (2.13), (2.14), and (2.15), we conclude that the numerator
in the right hand side of (2.8) is not less than $\mu(O)/4$. Thus (2.8)
implies

${\operatorname{codim}\big{(}{\\{h\in P\smallsetminus
S(k,t,x):hx\in\widetilde{E}({F},O)\\}}\big{)}}\geq{\frac{\mu(O)}{4{\lambda_{\max}}kt}}.$

Combining it with (2.11), we obtain

$\operatorname{codim}\big{(}{\widetilde{E}(F,O)\cap
Px}\big{)}\geq\frac{1}{4{\lambda_{\max}}kt}{\min\big{(}2,{{\mu(O)}}\big{)}}=\frac{\mu(O)}{4{\lambda_{\max}}kt},$

which is a positive number independent of $x$. This finishes the proof. ∎

###### Proof of Corollary 1.6.

Let $G,\Gamma,X$ and ${F}$ be as in Theorem 1.4 and let $H$ be as in (1.3).
Let $\mathfrak{g}$ be the Lie algebra of $G$, $\mathfrak{g}_{\mathbb{C}}$ its
complexification, and for $\lambda\in\mathbb{C}$, let $E_{\lambda}$ be the
eigenspace of ${\operatorname{Ad}}\,g_{1}$ corresponding to $\lambda$. Let
$\mathfrak{h}$, $\mathfrak{{h^{0}}}$, $\mathfrak{{h^{-}}}$ be the subalgebras
of $\mathfrak{g}$ with complexifications:

${\mathfrak{h}_{\mathbb{C}}}=\operatorname{span}({E_{\lambda}}:\left|\lambda\right|>1),\
\mathfrak{h}_{\mathbb{C}}^{0}=\operatorname{span}({E_{\lambda}}:\left|\lambda\right|=1),\
\mathfrak{h}_{\mathbb{C}}^{-}=\operatorname{span}({E_{\lambda}}:\left|\lambda\right|<1).$

Note that $\mathfrak{h}$ is the Lie algebra of $H$. Moreover,
$\mathfrak{h}^{-}$ is the Lie algebra of the stable horospherical subgroup
defined by

${H^{-}}:=\\{h\in G:{g_{t}}h{g_{-t}}\to e\,\,\,as\,\,t\to+\infty\\}.$

Since ${\operatorname{Ad}}\,g_{1}$ is assumed to be diagonalizable,
$\mathfrak{g}$ is the direct sum of $\mathfrak{h}$, $\mathfrak{{h^{0}}}$ and
$\mathfrak{{h^{-}}}$. Hence, if we denote the group ${H^{-}}{H^{0}}$ by
$\widetilde{H}$, $G$ is locally (at a neighborhood of identity) a direct
product of $H$ and $\widetilde{H}$.

Now let $O$ be a non-empty open subset of $X$, and fix ${0<\rho<1}$ such that
the following properties are satisfied:

  1. the multiplication map $\widetilde{H}\times H\to G$ is one to one on $B^{{\widetilde{H}}}(\rho)\times{B^{H}}(\rho),$

  2. ${g_{t}B^{\widetilde{H}}(\rho)g_{-t}\subset B^{\widetilde{H}}(2\rho)\text{ for any }t\geq 0,}$ (2.16)

  3. and

${\sigma_{2\rho}O\neq\varnothing.}$ (2.17)

Note that (2.16) can be satisfied since $F$ is
${\operatorname{Ad}}$-diagonalizable and the restriction of the map $g\to
g_{t}gg_{-t}$, $t\geq 0$, to $\widetilde{H}$ is non-expanding. Also (2.17) can
be achieved, since in view of (2.1) $\sigma_{r}O$ is non-empty when $r>0$ is
sufficiently small.

Now in view of (2.17), we can apply Theorem 1.5 with $O$ replaced with
$\sigma_{2\rho}O$ and conclude that there exists $\varepsilon>0$ such that

${\dim\left(\big{\\{}h\in
H:hx\in\widetilde{E}({F},\sigma_{2\rho}O)\big{\\}}\right)=\dim
H-\varepsilon<\dim H.}$ (2.18)

Choose $s>0$ such that ${B}(s)$ is contained in the product
$B^{{\widetilde{H}}}(\rho){B^{H}}(\rho)$, and for $x\in X$ denote

${{E_{x}}:=}\,\big{\\{}g\in{B}(s):gx\in\widetilde{E}({F},O)\big{\\}}.$

In view of the countable stability of Hausdorff dimension, in order to prove
the corollary it suffices to prove that for any $x\in X$,

$\dim E_{x}\leq\dim X-\varepsilon,$

where $\varepsilon$ is as in (2.18); note that $\widetilde{E}({F},O)$ can be
covered by countably many sets $\\{gx:g\in E_{x}\\}$, with the maps
$\pi_{x}:E_{x}\to X$ being Lipschitz and at most finite-to-one. Since every
${g\in B(s)}$ can be written as $g=h^{\prime}h$, where
$h^{\prime}\in{B^{\widetilde{H}}}(\rho)$ and $h\in{B^{H}}(\rho)$, for any
$y\in X$ we can write

$\displaystyle{\operatorname{dist}}({g_{t}}gx,y)$
$\displaystyle\leq{\operatorname{dist}}({g_{t}}h^{\prime}hx,{g_{t}}hx)+{\operatorname{dist}}({g_{t}}hx,y)$
$\displaystyle={\operatorname{dist}}\big{(}g_{t}h^{\prime}g_{-t}{g_{t}}hx,{g_{t}}hx\big{)}+{\operatorname{dist}}({g_{t}}hx,y).$

Hence in view of (2.16), $g\in{E_{x}}$ implies that $h{x}$ belongs to
$E({F},{\sigma_{{2\rho}}}U)$, and by using Wegmann’s Product Theorem [We] we
have:

$\displaystyle\dim{E_{x}}$
$\displaystyle\leq{\dim}\left(\\{h\in{B^{H}}(\rho):hx\in
E({F},{\sigma_{2\rho}}O)\\}\times{B^{\widetilde{H}}(\rho)}\right)$
$\displaystyle\leq{\dim}\big{(}\\{h\in{B^{H}}(\rho):hx\in\widetilde{E}({F},{\sigma_{2\rho}}O)\\}\big{)}+\dim{\widetilde{H}}$
$\displaystyle\underset{\eqref{eq:sliceH}}{\leq}\dim
H-\varepsilon+\dim\widetilde{H}=\dim X-\varepsilon.$

This ends the proof of the corollary. ∎

##  3\. Tessellations and Bowen boxes

Let $P$ be a connected subgroup of $H$ normalized by $F$. Following [KM1], say
that an open subset $V$ of $P$ is a tessellation domain relative to a
countable subset $\Lambda$ of $P$ if

  * •

$\nu(\partial V)=0$;

  * •

$V\gamma_{1}\cap V\gamma_{2}=\varnothing$ for different
$\gamma_{1},\gamma_{2}\in\Lambda$;

  * •

$P=\bigcup\limits_{\gamma\in\Lambda}{\overline{V}\gamma}.$

Note that $P$ is a connected simply connected nilpotent Lie group. Denote
$\mathfrak{p}:=\operatorname{Lie}(P)$ and $p:=\dim P$. As shown in [KM1,
Proposition 3.3], one can choose a basis of $\mathfrak{p}$ such that for any
$r>0$, $\exp\left(rI_{\mathfrak{p}}\right)$, where
$I_{\mathfrak{p}}\subset\mathfrak{p}$ is the cube centered at $0$ with side
length $1$ with respect to that basis, is a tessellation domain. Let us denote

${V_{r}:=\exp\left({{\frac{r}{4\sqrt{p}}}}{I_{\mathfrak{p}}}\right)}$ (3.1)

and choose a countable $\Lambda_{r}\subset P$ such that $V_{r}$ is a
tessellation domain relative to $\Lambda_{r}$.

Take $0<r_{*}<1/4$ such that the exponential map from $\mathfrak{p}$ to $P$ is
$2$-bi-Lipschitz on $B^{\mathfrak{p}}(r_{*})$. The latter implies that

${{B^{P}}\Big{(}\frac{r}{{16\sqrt{p}}}\Big{)}\subset{V_{r}}\subset{B^{P}}\left(\frac{r}{4}\right)\quad\text{for
any }0<r\leq r_{*}.}$ (3.2)

Also, the measure $\nu$ and the pushforward of the Lebesgue measure
${\operatorname{Leb}}$ on $\mathfrak{p}$ are absolutely continuous with
respect to each other with locally bounded Radon–Nikodym derivative. This
implies that there exists $0<c_{1}<c_{2}$ such that

${c_{1}{\operatorname{Leb}}(A)\leq\nu\big{(}\exp(A)\big{)}\leq
c_{2}{\operatorname{Leb}}(A)\quad\forall\,\text{measurable }A\subset
B^{\mathfrak{p}}(1).}$ (3.3)

In what follows we will be taking $\theta\geq r$ and approximating
${V_{\theta}}$ by the union of $\Lambda_{r}$-translates of ${V_{r}}$. The
following estimate will be helpful:

######  Lemma 3.1.

For any $0<r\leq\theta\leq r_{*}/2$

$\\#\\{\gamma\in\Lambda_{r}:{V_{r}}\gamma\cap{V_{\theta}}\neq\varnothing\\}\leq\frac{c_{2}}{c_{1}}\left(\frac{\theta}{r}+8\sqrt{p}\right)^{p}.$

###### Proof.

Note that if ${V_{r}}\gamma$ intersects ${V_{\theta}}$, then in view of (3.2)
we must have ${V_{r}}\gamma\subset\partial_{r/2}{V_{\theta}}$. Hence,

$\displaystyle\\#\\{\gamma\in\Lambda_{r}:{V_{r}}\gamma\cap{V_{\theta}}\neq\varnothing\\}$
$\displaystyle\leq\frac{\nu\left(\partial_{r/2}{V_{\theta}}\right)}{\nu\left({V_{r}}\right)}\underset{\eqref{eq:lb1}}{\leq}\frac{c_{2}}{c_{1}}\cdot\frac{{\operatorname{Leb}}\left(\partial_{r}\big{(}{\frac{\theta}{4\sqrt{p}}{I_{\mathfrak{p}}}}\big{)}\right)}{{\operatorname{Leb}}\left({\frac{r}{4\sqrt{p}}{I_{\mathfrak{p}}}}\right)}$
$\displaystyle=\frac{c_{2}}{c_{1}}\cdot\frac{\left(\frac{\theta}{4\sqrt{p}}+2r\right)^{p}}{\left(\frac{r}{4\sqrt{p}}\right)^{p}}=\frac{c_{2}}{c_{1}}\left(\frac{\theta}{r}+8\sqrt{p}\right)^{p},$

where in the second inequality above we were able to use the bi-Lipschitz
property of exp since

$\partial_{r}\big{(}{\frac{\theta}{4\sqrt{p}}{I_{\mathfrak{p}}}}\big{)}\subset{B^{\mathfrak{p}}\left({\frac{\theta}{{8}}+r}\right)}\subset
B^{\mathfrak{p}}(r_{*}).$

This finishes the proof. ∎

Recall that all eigenvalues of the restriction of
${\operatorname{ad}}_{{g_{1}}}$ to $\mathfrak{p}$ are positive. Using the bi-
Lipschitz property of $\exp$, one can conclude that

$\displaystyle\operatorname{diam}(g_{-t}{V_{r}}g_{t})$ $\displaystyle\leq
2\cdot\operatorname{diam}\left(\exp{\Big{(}{{{\frac{re^{-\lambda_{\min}t}}{4\sqrt{p}}}}{I_{\mathfrak{p}}}}\Big{)}}\right)$
(3.4) $\displaystyle\leq\frac{re^{-\lambda_{\min}t}}{2}\quad\text{for any
$0<r\leq r_{*}$ and any }t\geq 0,$

where $\lambda_{\min}$ is as in (2.3). Also let
$\delta:=\operatorname{Tr}{\operatorname{ad}}_{{g_{1}}}|_{\mathfrak{p}}$;
clearly one then has

${{\nu(g_{-t}Ag_{t})=e^{-\delta t}\nu(A)\text{ for any measurable }A\subset
P.}}$ (3.5)

Let us now define a Bowen $(t,r)$-box in $P$ to be a set of the form
$g_{-t}\overline{V_{r}}\gamma g_{t}$ for some $\gamma\in P$ and $t\geq 0$. The
following lemma, analogous to [KM1, Proposition 3.4] and [KMi1, Lemma 6.1],
gives an upper bound for the number of $\gamma\in\Lambda_{r}$ such that the
Bowen box $g_{-t}\overline{V_{r}}\gamma g_{t}$ has non-empty intersection with
$\overline{V_{r}}$:

######  Lemma 3.2.

For any $0<r\leq r_{*}/2$ and

${t\geq\frac{\log(8\sqrt{p})}{\lambda_{\min}},}$ (3.6)

one has

$\\#\\{\gamma\in\Lambda_{r}:{g_{-t}}{\overline{V_{r}}\gamma}{g_{t}}\cap\overline{V_{r}}\neq\varnothing\\}\leq
e^{\delta t}\left(1+{C_{0}}e^{-\lambda_{\min}t}\right),$

where

${{C_{0}}:={\frac{2^{p+3}{p}^{3/2}c_{2}}{c_{1}}}.}$ (3.7)

###### Proof.

Let $0<r\leq r_{*}/2$. One has:

$\displaystyle\\#$
$\displaystyle\\{\gamma\in\Lambda_{r}:{g_{-t}}{\overline{V_{r}}\gamma}{g_{t}}\cap\overline{V_{r}}\neq\varnothing\\}$
$\displaystyle=\\#$
$\displaystyle\\{\gamma\in\Lambda_{r}:{g_{-t}}{{V_{r}}\gamma}{g_{t}}\subset{V_{r}}\\}+\\#\\{\gamma\in\Lambda_{r}:{g_{-t}}{\overline{V_{r}}\gamma}{g_{t}}\cap\partial{V_{r}}\neq\varnothing\\}.$

Since $V_{r}$ is a tessellation domain of $P$ relative to $\Lambda_{r}$, the
first term in the above sum is not greater than
$\frac{\nu({V_{r}})}{\nu(g_{-t}{V_{r}}g_{t})}=e^{\delta t}$, while, in view of
(3.4), the second term is not greater than

${{\frac{{\nu\Big{(}\partial_{{\frac{re^{-\lambda_{\min}t}}{2}}}(\partial{{V_{r}}})\Big{)}}}{\nu({g_{-t}}{V_{r}}{g_{t}})}}\underset{\eqref{eq:lb1},\,\eqref{eq:delta}}{\leq}c_{2}e^{\delta
t}\
\frac{{{\operatorname{Leb}}\left(\partial_{{re^{-\lambda_{\min}t}}}\big{(}\partial({{\frac{r}{{4}\sqrt{{p}}}{I_{\mathfrak{p}}}}})\big{)}\right)}}{\nu({V_{r}})}.}$
(3.8)

(Here we used the fact that

${\partial_{re^{-\lambda_{\min}t}}\left({\frac{r}{{4}\sqrt{p}}{I_{\mathfrak{p}}}}\right)\subset{B^{\mathfrak{p}}\left(\frac{r}{{8}}+re^{-\lambda_{\min}t}\right)}\subset{B^{\mathfrak{p}}\left(\frac{9r}{8}\right)}\subset
B^{\mathfrak{p}}\left(\frac{9r_{*}}{16}\right)}\subset B^{\mathfrak{p}}(1),$

hence we can use the $2$-bi-Lipschitz property of $\exp$ to conclude that

$\exp\left(\partial_{{re^{-\lambda_{\min}t}}}\big{(}\partial({{\tfrac{r}{{4}\sqrt{{p}}}{I_{\mathfrak{p}}}}})\big{)}\right)\supset\partial_{{\frac{re^{-\lambda_{\min}t}}{2}}}(\partial{V_{r}}),$

and the estimate (3.3) is applicable.) It is easy to see that the numerator in
the right hand side of (3.8) is not greater that

$\displaystyle\qquad\left(\frac{r}{4\sqrt{p}}+2re^{-\lambda_{\min}t}\right)^{p}-\left(\frac{r}{2\sqrt{p}}-2re^{-\lambda_{\min}t}\right)^{p}$
$\displaystyle\underset{\text{(Mean Value
Theorem)}}{\leq}4re^{-\lambda_{\min}t}p\left(\frac{r}{4\sqrt{p}}+2re^{-\lambda_{\min}t}\right)^{p-1}$
$\displaystyle\underset{\eqref{eq:bigt}}{\leq}4pre^{-\lambda_{\min}t}\left(\frac{r}{2\sqrt{p}}\right)^{p-1}=2^{p+3}{p}^{3/2}\left(\frac{r}{4\sqrt{p}}\right)^{p}{e^{-\lambda_{\min}t}}\underset{\eqref{eq:defvr},\,\eqref{eq:lb1}}{\leq}\frac{2^{p+3}{p}^{3/2}}{c_{1}}\nu({V_{r}})e^{-\lambda_{\min}t},\qquad$

which finishes the proof. ∎

We conclude the section with a lemma, which is a slight modification of [KMi1,
Lemma 6.4], to be used at the last stage of the proof for switching from
coverings by Bowen boxes to coverings by balls.

######  Lemma 3.3.

For any $t>0$ and any $0<r\leq r_{*}$, any Bowen $(t,r)$-box in $P$ can be
covered with at most $e^{(p\lambda_{\max}-\delta)t}$ balls of radius
$re^{-\lambda_{\max}t}$, where $\lambda_{\max}$ is as in (2.4).

###### Proof.

Using the $2$-bi-Lipschitz property of $\exp$ again, one can cover
$g_{-t}\overline{V_{r}}g_{t}$ by at most as many balls of radius
$re^{-\lambda_{\max}t}$, as the number of translates of
$\frac{re^{-\lambda_{\max}t}}{\sqrt{p}}I_{\mathfrak{p}}$ needed to cover
${\operatorname{Ad}}(g_{-t})\left(\frac{r}{4\sqrt{p}}\overline{I_{\mathfrak{p}}}\right)$.
The latter can be written as the direct product of intervals
$I_{1},\dots,I_{p}$, where
$\min_{i}{\operatorname{Leb}}(I_{i})=\frac{re^{-\lambda_{\max}t}}{4\sqrt{p}}$.
Clearly each $I_{i}$ can be covered by the union of intervals of length
$\frac{re^{-\lambda_{\max}t}}{\sqrt{p}}$ whose total measure is at most
$4{\operatorname{Leb}}(I_{i})$. Hence
${\operatorname{Ad}}(g_{-t})\left(\frac{r}{4\sqrt{p}}\overline{I_{\mathfrak{p}}}\right)$
can be covered by at most

$\frac{4^{p}{\operatorname{Leb}}\left({\operatorname{Ad}}(g_{-t})\big{(}\frac{r}{4\sqrt{p}}{I_{\mathfrak{p}}}\big{)}\right)}{{\operatorname{Leb}}\left(\frac{re^{-\lambda_{\max}t}}{\sqrt{p}}{I_{\mathfrak{p}}}\right)}=\frac{4^{p}e^{-\delta
t}\Big{(}\frac{r}{4\sqrt{p}}\Big{)}^{p}}{\left(\frac{re^{-\lambda_{\max}t}}{\sqrt{p}}\right)^{p}}=e^{(p\lambda_{\max}-\delta)t}$

translates of $\frac{re^{-\lambda_{\max}t}}{\sqrt{p}}I_{\mathfrak{p}}$, which
finished the proof of the lemma. ∎

##  4\. Property (EEP) and a measure estimate

Our goal in this section is to use property (EEP) of $P$ to find a lower bound
for the measure of the sets of the type

${\\{h\in{V_{r}}:{g_{t}}hx\in O\\},}$ (4.1)

where $x\in X$, $O$ is a subset of $X$, $r>0$ is small enough, and $t>0$ is
large enough. This step is similar to [KMi1, Theorem 4.1], where balls in $P$
were used in place of tessellation domains $V_{r}$. For our new proof the use
of tessellations is crucial; to make the paper self-contained we present a
complete argument.

We start with the definition of Sobolev spaces. Let $G$ be a Lie group and
$\Gamma$ a discrete subgroup of $G$ such that $X=G/\Gamma$ admits a
$G$-invariant measure $\mu_{X}$. Fix a basis $\\{Y_{1},\dots,Y_{N}\\}$ for the
Lie algebra $\mathfrak{g}$ of $G$, and, given $h\in C^{\infty}(X)$,
$k\in{\mathbb{N}}$ and $\ell\in{{\mathbb{Z}}_{+}}$, define the “$L^{{k}}$,
order $\ell$” Sobolev norm $\|h\|_{\ell{,k}}$ of $h$ by

$\|h\|_{\ell{,k}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\sum_{|\alpha|\leq\ell}\|D^{\alpha}h\|_{{k}},$

where $\|\cdot\|_{{k}}$ stands for the $L^{k}$ norm,
$\alpha=(\alpha_{1},\dots,\alpha_{N})$ is a multiindex,
$|\alpha|=\sum_{i=1}^{N}\alpha_{i}$, and $D^{\alpha}$ is a differential
operator of order $|\alpha|$ which is a monomial in $Y_{1},\dots,Y_{N}$,
namely $D^{\alpha}=Y_{1}^{\alpha_{1}}\cdots Y_{N}^{\alpha_{N}}$. This
definition depends on the basis, however, a change of basis would only distort
$\|\cdot\|_{\ell{,k}}$ by a bounded factor. We will also use the operators
$D^{\alpha}$ to define $C^{\ell}$ norms of smooth functions $f$ on $X$:

$\|f\|_{C^{\ell}}:=\sup_{x\in X,\ |\alpha|\leq\ell}|D^{\alpha}f(x)|.$

The next lemmas provide a way to approximate subsets of $X$ and $P$
respectively by smooth functions. We start with a basic lemma constructing
test functions supported inside small neighborhoods of identity in $G$. It is
an immediate corollary of [Ka, Lemma 2.6], see also [KM1, Lemma 2.4.7(b)].

######  Lemma 4.1.

For each $\ell\in{\mathbb{Z}}_{+}$ there exists ${M_{G,\ell}}\geq 1$ with the
following property: for any $0<{{\varepsilon}}<1$ there exists a nonnegative
smooth function $\varphi_{{\varepsilon}}$ on $G$ such that

  1. (1)

the support of $\varphi_{{\varepsilon}}$ is inside $B(\varepsilon)$;

  2. (2)

$\|\varphi_{{\varepsilon}}\|_{1}=1$;

  3. (3)

$\|\varphi_{{\varepsilon}}\|_{\ell{,1}}\leq{M_{G,\ell}}\cdot{{\varepsilon}}^{-\ell}$.

The next lemma is a slightly easier version of [KMi1, Lemma 5.2]; we provide
the proof for the sake of completeness.

######  Lemma 4.2.

For any $\ell\in{\mathbb{Z}}_{+}$ there exist a constant $M_{\ell}>0$
(depending only on $\ell$ and $G$) such that for any nonempty open subset $O$
of $X$ and any $0<\varepsilon<1$ one can find a nonnegative function
$\psi_{\varepsilon}\in C^{\infty}(X)$ such that

  1. (1)

${1_{\sigma_{\varepsilon}O}}\leq{\psi_{\varepsilon}}\leq{1_{O}}$;

  2. (2)

$\max\big{(}{\left\|{{\psi_{\varepsilon}}}\right\|_{\ell{,2}}},{\left\|{{\psi_{\varepsilon}}}\right\|_{C^{\ell}}}\big{)}\leq{M_{\ell}}{\varepsilon^{-\ell}}$.

###### Proof.

Let $O$ be a nonempty open subset of $X$, and let $0<\varepsilon<1$. Now take
${\psi_{\varepsilon}}=\varphi_{{\varepsilon/2}}*{1_{\sigma_{{\varepsilon/2}}O}}$,
where $\varphi_{{\varepsilon/2}}$ is as in Lemma 4.1. It follows from the
definition of $\psi_{\varepsilon}$ and the normalization
$\|\varphi_{{\varepsilon}}\|_{1}=1$ that $\psi_{\varepsilon}(x)\leq 1$ for all
$x$. Also, since $\varphi_{{\varepsilon/2}}$ is supported on
$B(\varepsilon/2)$, the support of the function $\psi_{\varepsilon}$ is
contained in $\partial_{\varepsilon/2}\sigma_{{\varepsilon/2}}O\subset O$,
which implies ${\psi_{\varepsilon}}\leq{1_{O}}$. Furthermore, if
$x\in\sigma_{\varepsilon}O$ and $g\in B(\varepsilon/2)$, then
$gx\in\partial_{\varepsilon/2}\sigma_{\varepsilon}O\subset\sigma_{{\varepsilon/2}}O$,
i.e. $1_{\sigma_{{\varepsilon/2}}O}(gx)=1$. Therefore

${\psi_{\varepsilon}}(x)=\int_{G}\varphi_{{\varepsilon/2}}(g){1_{\sigma_{{\varepsilon/2}}O}}(gx)\,d{\mu_{G}}=\int_{G}\varphi_{{\varepsilon/2}}(g)\,{d\mu_{G}}=1.$

So property $(1)$ holds.

Let $\alpha=(\alpha_{1},\dots,\alpha_{N})$ be such that
$\left|\alpha\right|\leq\ell$. For any $x\in X$ we have

$\displaystyle\left|{{D^{\alpha}}{\psi_{\varepsilon}}(x)}\right|$
$\displaystyle=\left|{{D^{\alpha}}(\varphi_{\varepsilon/2}*1_{\sigma_{{\varepsilon/2}}O})(x)}\right|=\left|{{D^{\alpha}}\varphi_{\varepsilon/2}*1_{\sigma_{{\varepsilon/2}}O}(x)}\right|$
$\displaystyle\leq{\left\|D^{\alpha}{{\varphi_{{\varepsilon/2}}}}\right\|}_{1}\leq{\left\|{{\varphi_{{\varepsilon/2}}}}\right\|}_{\ell,1}\leq
M_{G,\ell}{(\tfrac{\varepsilon}{2})^{-\ell}},$

where $M_{G,\ell}$ is as in Lemma 4.1. Likewise, by Young’s inequality,

${{\left\|D^{\alpha}{{\psi_{\varepsilon}}}\right\|}_{2}\leq\|{D^{\alpha}}\varphi_{\varepsilon/2}*1_{\sigma_{{\varepsilon/2}}O}\|_{2}\leq{\left\|D^{\alpha}{{\varphi_{{\varepsilon/2}}}}\right\|}_{1}\cdot{\left\|1_{\sigma_{{\varepsilon/2}}O}\right\|}_{2}\leq{\left\|D^{\alpha}{{\varphi_{{\varepsilon/2}}}}\right\|}_{1}\leq
M_{G,\ell}{(\tfrac{\varepsilon}{{2}})^{-\ell}},}$

which implies (2) with $M_{\ell}={2}^{\ell}M_{G,\ell}$. ∎

The next lemma is a modification of [KMi1, Lemma 5.3] where we replace balls
of radius $r$ in $P$ with $V_{r}$; we omit the proof.

######  Lemma 4.3.

For any $\ell\in{\mathbb{Z}}_{+}$ there exist constants $M^{\prime}_{\ell}\geq
1$ (depending only on $\ell$ and $P$) such that the following holds: for any
$0<\varepsilon,r\leq r_{*}/2$, there exist functions
${f_{\varepsilon}}:P\to[0,1]$ such that

  1. (1)

${f_{\varepsilon}}=1\,\,on\,\,\,V_{r}$;

  2. (2)

${f_{\varepsilon}}=0\,\,on\,\,\,{\big{(}V_{r+\varepsilon}\big{)}^{c}}$;

  3. (3)

$\max\big{(}{\left\|{{f_{\varepsilon}}}\right\|_{\ell{,2}}}\,,{\left\|{{f_{\varepsilon}}}\right\|_{C^{\ell}}}\big{)}\leq{M^{\prime}_{\ell}}{\varepsilon^{-\ell}}$.

Here is the main result of the section, which is a modified and improved
version of [KMi1, Proposition 5.1].

######  Proposition 4.4.

Let $P$ be a subgroup of $G$ with property (EEP) with respect to the flow
$(X,F)$. Then for any open $O\subset X$, any $x\in X$, any

${0<r<\frac{1}{2}\min\big{(}r_{0}(x),r_{*}\big{)},}$ (4.2)

and any $t$ satisfying

${t\geq a^{\prime}+b\log\frac{1}{r_{0}(x)}}$ (4.3)

one has

${{\mathop{\nu}\big{(}{\\{h\in{V_{r}}:{g_{t}}hx\in
O\\}}\big{)}\geq\nu\left({V_{r}}\right)\mu({\sigma_{e^{-\lambda^{\prime}t}}}O)-{e^{-{\lambda^{\prime}t}}}.}}$

Here

${\lambda^{\prime}:=\frac{\lambda}{4\ell+2}}$ (4.4)

and

${a^{\prime}:=\max\left(a,\frac{1}{\lambda^{\prime}}\log\left({M_{\ell}{M^{\prime}_{\ell}}{{E}}}+{p}{c_{2}}\right),\frac{\log\frac{2}{r_{*}}}{2\lambda^{\prime}}\right),}$
(4.5)

where $\ell,\lambda,a,b$ are as in Definition 1.1, $E$ is an implicit constant
from (1.6), $c_{2}$ is as in (3.3), and $M_{\ell},M^{\prime}_{\ell}$ are as in
Lemmas 4.2 and 4.3.

###### Proof of Proposition 4.4.

Let $O\subset X$ be an open subset of $X$, and take $x\in X$ and $r$ as in
(4.2). Now set $f={1_{{V_{r}}}}$, take $t$ as in (4.3) and put
$\varepsilon:=e^{-2\lambda^{\prime}t}.$ Note that (4.3) and (4.5) give

${\varepsilon\leq\frac{r_{*}}{2}.}$ (4.6)

Now let ${\psi_{\varepsilon}}$ and ${f_{\varepsilon}}$ be the functions
constructed in Lemmas 4.2 and 4.3 respectively. Then we have

$\displaystyle\max(\|\psi_{{\varepsilon}}\|_{C^{1}},{\left\|\psi_{{\varepsilon}}\right\|_{\ell{,2}}})\cdot{\left\|f_{{\varepsilon}}\right\|_{{C^{\ell}}}}\cdot{e^{-\lambda
t}}$
$\displaystyle\leq\max(\|\psi_{{\varepsilon}}\|_{C^{\ell}},{\left\|\psi_{{\varepsilon}}\right\|_{\ell{,2}}})\cdot{\left\|f_{{\varepsilon}}\right\|_{{C^{\ell}}}}\cdot{e^{-\lambda
t}}$ (4.7) $\displaystyle\leq
M_{\ell}{\varepsilon^{-\ell}}{M^{\prime}_{\ell}}{\varepsilon^{-\ell}}e^{-\lambda
t}$
$\displaystyle=M_{\ell}{M^{\prime}_{\ell}}{e^{4\ell\lambda^{\prime}t-\lambda
t}}\underset{\eqref{eq:exponent}}{\leq}M_{\ell}{M^{\prime}_{\ell}}{e^{-2\lambda^{\prime}t}}.$

Furthermore, by (3.2) and (4.6),

${\operatorname{supp}f_{\varepsilon}\subset{V_{r+{e^{-2\lambda^{\prime}t}}}}\subset
V_{r+\frac{r_{*}}{2}}\subset V_{r_{*}}\subset B^{P}(1).}$ (4.8)

Also, in view of (4.5) we have $a^{\prime}\geq a$; hence, inequality (1.5) is
satisfied for any $x,t$ satisfying (4.3). Hence the estimate (1.6) can be
applied to ${\psi_{\varepsilon}}$, ${f_{\varepsilon}}$, $x$ and $t$, and, in
view of (4.7), yields

$\displaystyle\int_{P}{{f_{\varepsilon}}({h})}\psi_{\varepsilon}({g_{t}}{h}x)\,d\nu({h})$
$\displaystyle\geq\int_{P}{{f_{\varepsilon}}\,d}\nu\int_{X}{{\psi_{\varepsilon}}{\mkern
1.0mu}\,d\mu}-{M_{\ell}{M^{\prime}_{\ell}}{{E}}}{e^{-2\lambda^{\prime}t}}.$

Thus we have:

$\displaystyle\nu\left({\\{h\in{V_{r}}:{g_{t}}hx\in
O\\}}\right)=\int_{P}{f({h}){1_{O}}({g_{t}}}{h}x)\,d\nu({h})$
$\displaystyle\geq\int_{P}{f({h})\psi_{\varepsilon}({g_{t}}}{h}x)d\nu({h})\geq\int_{P}{{f_{\varepsilon}}({h})\psi_{\varepsilon}({g_{t}}}{h}x)\,d\nu({h})-\int_{P}{|{f_{\varepsilon}}-f|\,d\nu}$
$\displaystyle\geq\int_{P}{{f_{\varepsilon}}({h})\psi_{\varepsilon}({g_{t}}}{h}x)\,d\nu({h})-\nu\left(V_{r+{e^{-2\lambda^{\prime}t}}}\smallsetminus{V_{r}}\right).$

Since by (4.2) and (4.6) we have $r+e^{-2\lambda^{\prime}t}\leq r_{*}$, it
follows that
$\frac{r+{e^{-2\lambda^{\prime}t}}}{{4}\sqrt{p}}{I_{\mathfrak{p}}}\subset
B^{\mathfrak{p}}(1)$. So in view of (3.3)

$\displaystyle\nu\left(V_{r+{e^{-2\lambda^{\prime}t}}}\smallsetminus{V_{r}}\right)\leq
c_{2}{\operatorname{Leb}}\left(\frac{r+{e^{-2\lambda^{\prime}t}}}{{4}\sqrt{p}}{I_{\mathfrak{p}}}\smallsetminus{\frac{r}{{4}\sqrt{p}}{I_{\mathfrak{p}}}}\right)$
$\displaystyle{\underset{\text{(Mean Value
Theorem)}}{\leq}c_{2}\left(\frac{1}{\sqrt{4p}}\right)^{p}e^{-2\lambda^{\prime}t}p\big{(}r+e^{-2\lambda^{\prime}t}\big{)}^{{p-1}}\leq
c_{2}pe^{2\lambda^{\prime}t}.}$

Combining the above computations, we obtain

$\displaystyle\nu\left({\\{h\in{V_{r}}:{g_{t}}hx\in O\\}}\right)$
$\displaystyle\geq\int_{P}{{f_{\varepsilon}}({h})\psi_{\varepsilon}({g_{t}}}{h}x)\,d\nu({h})-{c_{2}}{p}{e^{-2\lambda^{\prime}t}}$
$\displaystyle\geq\int_{P}{f_{\varepsilon}}\,d\nu\int_{X}{\psi_{\varepsilon}\,d\mu-{M_{\ell}{M^{\prime}_{\ell}}{{E}}}{e^{-2\lambda^{\prime}t}}}-{c_{2}}{p}{e^{-2\lambda^{\prime}t}}$
$\displaystyle\geq\nu\left({V_{r}}\right){{\mu({\sigma_{\varepsilon}}O)}}-({M_{\ell}{M^{\prime}_{\ell}}{{E}}}+{c_{2}}{p}){e^{-2\lambda^{\prime}t}}$
$\displaystyle=\nu\left({V_{r}}\right)\mu({\sigma_{e^{-2\lambda^{\prime}t}}}O)-({M_{\ell}{M^{\prime}_{\ell}}{{E}}}+{c_{2}}{p}){e^{-2\lambda^{\prime}t}}$
$\displaystyle\underset{\eqref{eq:a'}}{\geq}\nu\left({V_{r}}\right)\mu({\sigma_{e^{-\lambda^{\prime}t}}}O)-{e^{-\lambda^{\prime}t}}.$

∎

##  5\. Coverings by Bowen boxes

For $x\in X$, $t>0,{N\in{\mathbb{N}}}$ and a subset $S$ of $X$ let us define

${{{A}^{N}_{x}(t,r,S):=\big{\\{}h\in\overline{V_{r}}:{g_{{\ell}t}}hx\in
S\,\,\,{\forall\,\ell\in\\{1,\dots,N\\}\mkern 1.0mu}\big{\\}}}.}$ (5.1)

Clearly the set (4.1) studied in the previous section has the same measure as
${A}^{1}_{x}(t,r,O)$. Our goal in this section will be to inductively use
Proposition 4.4 to find an effective covering result for the set
${A}^{N}_{x}(t,r,O^{c})$. We start with the following theorem, which is a
modified and improved version of [KMi1, Proposition 5.1]:

######  Theorem 5.1.

Let ${F}$ be a one-parameter ${\operatorname{Ad}}$-diagonalizable subsemigroup
of $G,$ and $P$ a subgroup of $G$ that has property (EEP) with respect to the
flow $(X,F)$. Then there exist positive constants
$a^{\prime},b^{\prime}\geq{\frac{\log({8}\sqrt{p})}{\lambda_{\min}}},C_{2},\lambda$
such that for any open $O\subset X$, any

${0<r<\frac{1}{2}\min\Big{(}r_{0}\big{(}\partial_{1/2}(O^{c})\big{)},r_{*}\Big{)},}$
(5.2)

any $x\in\partial_{r}{(O^{c})},$ any $t$ satisfying

${t\geq a^{\prime}+b^{\prime}\log\frac{1}{r},}$ (5.3)

and any $N\in{\mathbb{N}}$, the set ${A}^{N}_{x}(t,r,O^{c})$ can be covered
with at most

${e^{\delta Nt}}\left({1-\mu({\sigma_{r}}O)}+\frac{C_{2}}{r^{p}}e^{-\lambda
t}\right)^{N}$

Bowen $(Nt,r)$-boxes in $P$.

We remark that the above theorem, as well as its corollary proved later in
this section, is applicable only to the situations when the complement of $O$
is compact: indeed, otherwise $r_{0}\big{(}\partial_{1/2}(O^{c})\big{)}=0$ and
(5.2) is never satisfied.

Before we prove the theorem, we need the following lemma:

######  Lemma 5.2.

For any $x\in X$, any $O\subset X$, any $0<r\leq r_{*}$ and any $t>0$ we have

${\\#\\{\gamma\in{\Lambda_{r}}:\overline{V_{r}}\gamma g_{t}x\subset
O\\}\geq\frac{\nu\left({A}^{1}_{x}(t,r,{\sigma_{r/2}}{O})\right)}{\nu({g_{-t}}{V_{r}}{g_{t}})}.}$
(5.4)

###### Proof.

For any $\gamma\in{P}$ and any ${h}_{1},{h}_{2}\in\overline{V_{r}}$ we have:

$\displaystyle{\operatorname{dist}}\big{(}{h}_{1}\gamma g_{t}x,{h}_{2}\gamma
g_{t}x\big{)}\leq{\operatorname{dist}}({h}_{1},{h}_{2})\leq\operatorname{diam}(\overline{V_{r}})\leq
r/2.$ (5.5)

Hence, if

${A}^{1}_{x}(t,r,{\sigma_{r/2}}{O})\cap g_{-t}\overline{V_{r}}\gamma
g_{t}\neq\varnothing$

for $\gamma\in\Lambda_{r}$, then for some ${h}\in\overline{V_{r}}$ one has
$g_{t}{h}x\in\sigma_{r/2}{O}\cap\overline{V_{r}}\gamma g_{t}x$, and, in view
of (5.5) and ${\partial_{r/2}}({\sigma_{r/2}}{O})\subset O$, we can conclude
that $\overline{V_{r}}\gamma g_{t}x\subset O$. Thus

${A}^{1}_{x}(t,r,{\sigma_{r/2}}{O})\subset\bigcup_{\begin{subarray}{c}\gamma\in{\Lambda_{r}}\\\
\overline{V_{r}}\gamma g_{t}x\subset
O\end{subarray}}{{g_{-t}}\overline{V_{r}}\gamma{g_{t}}},$

and (5.4) follows from the definition of $V_{r}$ being a tessellation domain
relative to $\Lambda_{r}$. ∎

###### Proof of Theorem 5.1.

Take $a^{\prime},b,\lambda^{\prime}$ be as in Proposition 4.4, and
$\lambda_{\min}$ as in (2.3). Also set

${b^{\prime}:=\max\left(b,\frac{1}{\lambda^{\prime}},{\frac{\log({16}\sqrt{p})}{\lambda_{\min}}}\right).}$
(5.6)

Fix an open $O\subset X$, and take $r$ as in (5.2). Also take
$x\in\partial_{r}(O^{c})$ and $t$ as in (5.3).

First let us show how to derive the desired result for $N=1$ from Proposition
4.4. Observe that

$t\underset{\eqref{eq:t estimate
2}}{\geq}a^{\prime}+b^{\prime}\log\frac{1}{r}\underset{\eqref{eq:r estimate
2}}{\geq}b^{\prime}\log\frac{2}{r_{*}}\underset{(r_{*}<\frac{1}{4})}{>}b^{\prime}\underset{\eqref{eq:b'}}{{\geq}}{\frac{\log(8\sqrt{p})}{\lambda_{\min}}}.$

So, by combining Lemma 3.2 with Lemma 5.2, we conclude that
${A}^{1}_{x}(t,r,O^{c})$ can be covered with at most

$\displaystyle\\#\\{\gamma\in\Lambda_{r}:{g_{-t}}{\overline{V_{r}}\gamma}{g_{t}}\cap\overline{V_{r}}\neq\varnothing\\}-\\#\\{\gamma\in{\Lambda_{r}}:\overline{V_{r}}\gamma
g_{t}x\subset O\\}$ $\displaystyle\leq{e^{\delta
t}}\left(1+{C_{0}}e^{-\lambda_{\min}t}\right)-\frac{\nu\left({A}^{1}_{x}(t,r,{\sigma_{r/2}}{O})\right)}{\nu({g_{-t}}{V_{r}}{g_{t}})}$

Bowen $(t,r)$-boxes in $P$, where $C_{0}$ is as in (3.7). Note that whenever
$x\in\partial_{r}(O^{c})$, (4.2) and (4.3) follow from (5.2), (5.3) and (5.6).
Moreover, we have

${\lambda^{\prime}t\underset{\eqref{eq:t estimate
2}}{\geq}\lambda^{\prime}a^{\prime}+\lambda^{\prime}b^{\prime}\log\frac{1}{r}\underset{\eqref{eq:a'},\,\eqref{eq:b'}}{\geq}\frac{\log\frac{2}{r_{*}}}{2}+\log\frac{1}{r}\underset{(r_{*}<\frac{1}{4})}{\geq}\log\frac{2}{r}.}$
(5.7)

Hence one can apply Proposition 4.4 and conclude that ${A}^{1}_{x}(t,r,O^{c})$
can be covered with at most

$\displaystyle{e^{\delta
t}}\left({1+{C_{0}}e^{-\lambda_{\min}t}-\mu({\sigma_{e^{-\lambda^{\prime}t}}\sigma_{r/2}}O)}+\frac{e^{-\lambda^{\prime}t}}{\nu(V_{r})}\right)$
$\displaystyle\underset{\eqref{eq:lambdat}}{\leq}{e^{\delta
t}}\left({1+{C_{0}}e^{-\lambda_{\min}t}-\mu({\sigma_{r}}O)}+\frac{e^{-\lambda^{\prime}t}}{\nu(V_{r})}\right)$
$\displaystyle\leq{e^{\delta
t}}\left(1-\mu({\sigma_{r}}O)+\frac{C_{2}}{r^{p}}e^{-\lambda
t}\right)=:N(r,t)$

Bowen $(t,r)$-boxes in $P,$ where
$\lambda:=\min(\lambda_{\min},\lambda^{\prime})$ and

${C_{2}:={{C_{0}}+\frac{(4\sqrt{p})^{p}}{c_{1}}}.}$ (5.8)

Now let $g_{-t}\overline{V_{r}}\gamma g_{t}$ be one of the Bowen $(t,r)$-boxes
in the above cover which has non-empty intersection with
${A}^{1}_{x}(t,r,O^{c})$. Take any $q={g_{-t}}{h}\gamma{g_{t}}\in
g_{-t}\overline{V_{r}}\gamma g_{t}$; then ${{g_{t}}qx={h}\gamma g_{t}x}$,
hence ${\left\\{{{g_{t}}qx:q\in g_{-t}\overline{V_{r}}\gamma
g_{t}\,}\right\\}=\left\\{{h}\gamma g_{t}x:{h}\in\overline{V_{r}}\right\\}.}$
Consequently,

${\\{q\in g_{-t}\overline{V_{r}}\gamma g_{t}:g_{2t}qx\notin
O\\}=g_{-t}{{A}^{1}_{x}(t,r,O^{c})}\gamma g_{t}.}$ (5.9)

Note that since $\operatorname{diam}(\overline{V_{r}}\gamma)<r$ and
$g_{-t}\overline{V_{r}}\gamma g_{t}\cap{{A}^{1}_{x}(t,r,O^{c})}$ is non-empty,
we have $\gamma g_{t}x\in\partial_{r}(O^{c})$. Hence, by going through the
same procedure, this time using $\gamma g_{t}x$ in place of $x$, we can cover
the set in the left hand side of (5.9) with at most $N(r,t)$ Bowen
$(2t,r)$-boxes in $P$. Therefore, we conclude that the set
${A}^{2}_{x}(t,r,O^{c})$ can be covered with at most $N(r,t)^{2}$ Bowen
$(2t,r)$-boxes in $P$. By doing this procedure inductively, we can see that
for any $N\in{\mathbb{N}}$, the set ${A}^{N}_{x}(t,r,O^{c})$ can be covered
with at most

${N(r,t)^{N}=e^{\delta
Nt}\left({1-\mu({\sigma_{r}}O)}+\frac{C_{2}}{r^{p}}e^{-\lambda t}\right)^{N}}$

Bowen $(Nt,r)$-boxes in $P$. This finishes the proof. ∎

Next we are going to apply Theorem 5.1 to cover ${{A}^{N}_{x}(t,r,O^{c})}$
with Bowen $(Nt,\theta)$-boxes, where $r\leq\theta\leq\frac{r_{*}}{2}$.

######  Theorem 5.3.

Let ${F}$ be a one-parameter ${\operatorname{Ad}}$-diagonalizable subsemigroup
of $G,$ and $P$ a subgroup of $G$ that has property (EEP). Then, with
$a^{\prime},b^{\prime},C_{2},\lambda$ as in Theorem 5.1, for any open
$O\subset X$, any $t$ as in (5.3), any $r$ such that

${0<r<\frac{1}{4}\min\Big{(}r_{0}\big{(}\partial_{1}(O^{c})\big{)},r_{*}\Big{)},}$
(5.10)

any $x\in\partial_{r}{(O^{c})},$ any $N\in{\mathbb{N}}$, and any
$\theta\in{\left[{r},\frac{r_{*}}{2}\right]}$, the set
${{A}^{N}_{x}(t,r,O^{c})}$ can be covered with at most

${\frac{c_{2}}{c_{1}}}{\left(\frac{2r}{\theta}\right)^{p}}{e^{\delta
Nt}}\left(1-\mu\big{(}{\sigma_{{{4}{\theta}}}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
t}\right)^{N}$

Bowen $(Nt,\theta)$-boxes in $P$.

###### Proof.

Consider the covering of ${{A}^{N}_{x}(t,r,O^{c})}$ by Bowen boxes
$\left\\{g_{-Nt}\overline{V_{\theta}}\gamma
g_{Nt}:\gamma\in\Lambda_{\theta}\right\\}$. Let $R$ be one of those boxes, so
that

${R\cap{{A}_{x}^{N}\left(t,r,O^{c}\right)}\neq\varnothing.}$ (5.11)

Since $\theta<r_{*}$, in view of (3.4) we have
$\operatorname{diam}(R)\leq\frac{\theta}{2}e^{-\lambda_{\min}Nt}$;
furthermore,

${\theta e^{-\lambda_{\min}t}\underset{{\eqref{eq:t estimate 2}}}{\leq}\theta
e^{-\lambda_{\min}b^{\prime}\log\frac{1}{r}}\leq\theta
e^{-{\log(8\sqrt{p})}\log\frac{1}{r}}\underset{(\theta\leq
1)}{\leq}{\frac{r}{8\sqrt{p}}}.}$ (5.12)

Since $R\cap V_{r}\neq\varnothing$, it follows that

$R\subset\partial_{\frac{\theta}{2}e^{-\lambda_{\min}Nt}}\overline{V_{r}}\underset{\eqref{eq:theta-r}}{\subset}\partial_{{\frac{r}{16\sqrt{p}}}}\overline{V_{r}}\subset{V_{2r}},$

where in the last inclusion we again use the $2$-bi-Lipschitz property of
$\exp$.

We now claim that $R$ is contained in
${{A}^{N}_{x}\big{(}t,{2r},{{\partial_{{{2\theta}}}(O^{c})}}\big{)}}$. Indeed,
in view of (5.11) we can find $h_{1}^{\prime}:=g_{-Nt}h_{1}\gamma g_{Nt}\in R$
such that $g_{it}{h_{1}^{\prime}}x\in O^{c}$ for all $i\in\\{1,\dots,N\\}$
(here $h_{1}\in\overline{V_{\theta}}$). Then take any
$h_{2}^{\prime}:=g_{-Nt}h_{2}\gamma g_{Nt}\in R$, where again
$h_{2}\in\overline{V_{\theta}}$, and for any $i\in\\{1,\dots,N\\}$ write

$\displaystyle g_{it}{h_{2}^{\prime}}x$
$\displaystyle=(g_{-(N-i)t}{h_{2}h_{1}^{-1}}g_{(N-i)t})g_{it}{h_{1}^{\prime}}x$
$\displaystyle\in(g_{-(N-i)t}{h_{2}h_{1}^{-1}}g_{(N-i)t})O^{c}\underset{\eqref{eq:diam}}{\subset}\partial_{2\theta
e^{-\lambda_{\min}(N-i)t}}(O^{c})\subset\partial_{2\theta}(O^{c}).$

Note that since $\theta\leq r_{*}/2<1/8$, we have
$\partial_{1/2}\big{(}\partial_{2\theta}(O^{c})\big{)}\subset\partial_{1}O^{c}$,
which implies
$r_{0}\Big{(}\partial_{1/2}\big{(}\partial_{2\theta}(O^{c})\big{)}\Big{)}\geq
r_{0}(\partial_{1}O^{c})$. Thus, since (5.10) is satisfied, the following is
satisfied as well:

$0<2r<\frac{1}{2}\min\Big{(}r_{0}\big{(}\partial_{1/2}\big{(}\partial_{2\theta}(O^{c})\big{)},r_{*}\big{)}\Big{)}.$

Consequently, Theorem 5.1, applied to $O$ replaced with
$\sigma_{{{2}\theta}}O$ and $r$ replaced with $2r$, implies that

$\displaystyle\nu\Big{(}{{A}^{N}_{x}\big{(}t,{2r},{{\partial_{{{2\theta}}}(O^{c})}}\big{)}}\Big{)}\
$
$\displaystyle\,\underset{\eqref{eq:boundarysigma}}{\leq}\nu\Big{(}{{A}^{N}_{x}\big{(}t,{2r},{{(\sigma_{{{2\theta}}}O)^{c}\big{)}}}}\Big{)}$
$\displaystyle\ \ \leq\ \nu(g_{-Nt}{V_{2r}}g_{Nt})\cdot{e^{\delta
Nt}}\left(1-\mu\big{(}{\sigma_{2r}(\sigma_{{{{2}\theta}}}}O)\big{)}+\frac{C_{2}}{(2r)^{p}}e^{-\lambda
t}\right)^{N}$ $\displaystyle\underset{{(\theta\geq
r)}}{\leq}\nu({V_{2r}})\left(1-\mu\big{(}{\sigma_{{{4}{\theta}}}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
t}\right)^{N}$

for any
$x\in\partial_{r}(O^{c})\subset\partial_{2r}\big{(}(\sigma_{{{2}\theta}}O)^{c}\big{)}$.
This forces the number of $\gamma\in\Lambda_{\theta}$ such that

$g_{-Nt}\overline{V_{\theta}}\gamma
g_{Nt}\cap{{A}_{x}^{N}\left(t,r,O^{c}\right)}\neq\varnothing$

to be not greater than
$\left(1-\mu\big{(}{\sigma_{{{4}{\theta}}}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
t}\right)^{N}$ multiplied by

$\frac{\nu({V_{2r}})}{\nu(g_{-Nt}{V_{\theta}}g_{Nt})}\underset{\eqref{eq:lb1}}{\leq}\frac{c_{2}\big{(}\frac{2r}{4\sqrt{p}}\big{)}^{p}}{e^{-\delta
Nt}c_{1}\big{(}\frac{\theta}{4\sqrt{p}}\big{)}^{p}}={\frac{c_{2}}{c_{1}}}{e^{\delta
Nt}}{\left(\frac{2r}{\theta}\right)^{p}}.$

This finishes the proof of the theorem. ∎

##  6\. ENDP and iterations of Margulis inequality

Suppose $P$ is a subgroup of $G$ which satisfies (ENDP). Then by the
definition one can find $0<c_{0}<{1}$ and $t_{0}>0$ such that the following
holds: for any $t\geq t_{0}$ one can find a height function $u_{t}$ and
$d_{t}>0$ such that $u_{t}$ satisfies the $(c_{0},d_{t})$-Margulis inequality
with respect to $I_{{B^{P}(1),t}}u_{t}$; that is,

${(I_{{B^{P}(1),t}}u_{t})(x)\leq c_{0}u_{t}(x)+d_{t}.}$ (6.1)

Let $t_{1}>0$ be sufficiently large so that

${g_{-{t}}B^{P}(r)g_{t}\subset B^{P}(r/4)\,\,\,\text{for all}\,\,\,0<r\leq
1,\,t\geq t_{1},}$ (6.2)

and set

${t_{*}:=\max(t_{0},t_{1})}$ (6.3)

In the following proposition, by using inequality (6.1) $N$ times for $t$
sufficiently large, we prove that $u_{t}$ satisfies the
$\left(c_{0}^{N},{\frac{d_{t}}{1-c_{0}}}\right)$-Margulis inequality with
respect to $I_{{B^{P}(1/2),Nt}}$. The argument is similar to the proof of [SS,
Theorem 15].

######  Proposition 6.1.

Let $P$ be a subgroup of $G$ that has property (ENDP), and let
$\\{u_{t}\\}_{t>0}$ be the family of height functions in the definition of
(ENDP). Also let $t_{*}$ be as in (6.3). Then for any $t\geq t_{*}$ and any
$N\in{\mathbb{N}}$, $u_{t}$ satisfies the
$\left(c_{0}^{N},{\frac{d_{t}}{1-c_{0}}}\right)$ Margulis inequality with
respect to $I_{B^{P}(1/2),\,Nt}$. In other words, for any $t\geq t_{*}$, any
$N\in{\mathbb{N}}$ and any $x\in X$ one has

${(I_{B^{P}(1/2),\,Nt}u_{t})(x)\leq
c_{0}^{N}u_{t}(x)+{\frac{d_{t}}{1-c_{0}}}.}$ (6.4)

As a corollary, we get the following crucial statement which will be useful in
later sections:

######  Corollary 6.2.

Let $P$ be a subgroup of $G$ that has property (ENDP), and let $t_{1}$ be as
in (6.2). Then there exists a height function $u$ and $d>0$ such that for any
$0<c<1$ one can find positive $t_{c}\geq t_{1}$ such that for any
$t\in{\mathbb{N}}t_{c}$, $u$ satisfies the $(c,d)$-Margulis inequality with
respect to $I_{B^{P}(1/2),t}$. In other words, for all $x\in X$ we have

${(I_{B^{P}(1/2),\,t}u)(x)\leq cu(x)+d.}$ (6.5)

###### Proof.

Let $0<c<1$, and take $c_{0}$ as in Proposition 6.1. Choose $N$ sufficiently
large so that $c_{0}^{N}\leq c$, and set

$u:=u_{t_{1}},\,t_{c}:={Nt_{*}{\underset{\eqref{eq:ts}}{\geq}t_{1}}},\,d:={\frac{d_{t_{1}}}{1-c_{0}}}.$

Now let $t=nt_{c}=nN{t_{*}}$ be an element in ${\mathbb{N}}t_{c}$. Then, by
Proposition 6.1 applied with $N$ replaced by $nN$, we have

$(I_{B^{P}(1/2),\,t}u)(x)=(I_{B^{P}(1/2),\,nNt_{1}}u)(x)\leq
c_{0}^{nN}{u(x)}+d\leq c_{0}^{N}{u(x)}+d\leq cu(x)+d.$

This finishes the proof. ∎

###### Proof of Proposition 6.1.

Given $n\in{\mathbb{N}}$ and $t>0$, define $\eta_{n,t}:B^{P}(1)^{n}\rightarrow
P$ by

$\displaystyle\eta_{n,t}(h_{1},\dots,h_{n})$
$\displaystyle:=g_{-(n-1)t}h_{n}g_{t}\cdots h_{2}g_{t}h_{1}$ (6.6)
$\displaystyle\,\,=\tilde{h}_{n}\cdots\tilde{h}_{1},\,\,\,\text{where}\,\,\tilde{h}_{i}=g_{-(i-1)t}h_{i}g_{(i-1)t}.$

For any $n\in{\mathbb{N}}$ and $t>0$, let $\tilde{\nu}_{n,t}$ be the
pushforward of $\nu|_{B^{P}(1)}$ via the conjugation by $g_{nt}$, that is,
defined by

${\int_{P}\phi(h)\,d\tilde{\nu}_{n,t}(h)=\int_{B^{P}(1)}\phi(g_{-nt}hg_{nt})\,d\nu(h)}$
(6.7)

for all $\phi\in C_{c}(P).$ For any positive integer $n$ let

$\nu_{n,t}:=\tilde{\nu}_{n-1,t}\ast\cdots\ast\tilde{\nu}_{1,t}\ast\tilde{\nu}_{0,t}$

be the measure on $P$ defined by the $n$ convolutions. It is easy to see that
$\nu_{n,t}$ is absolutely continuous with respect to $\nu$, and $\nu_{n,t}$ is
the pushforward of $(\nu|_{B^{P}(1)})^{\otimes n}$ by the map $\eta_{n,t}$.
These measures were considered in [GS], and the following was shown:

######  Lemma 6.3.

[GS, Lemma 5.5] For all $t\geq t_{1}$ as in (6.2), all $h\in B^{P}(1/2)$, and
for all $n\in{\mathbb{N}}$ we have $\frac{d\nu_{n,t}}{d\nu}(h)\geq 1$.

Using Lemma 6.3, we have for all $N\in{\mathbb{N}}$ and all $t\geq t_{1}$:

$\displaystyle(I_{B^{P}(1/2),Nt}u)(x)$
$\displaystyle=\int_{B^{P}(1/2)}u(g_{Nt}hx)\,d\nu(h)\leq\int_{B^{P}(1/2)}u(g_{Nt}hx)\,d{\nu_{N,t}(h)}$
(6.8) $\displaystyle\leq{\int_{B^{P}(1/2)^{N}}u({g_{t}h_{{N}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes n}(h_{1},\dots,h_{N})}$
$\displaystyle\leq\int_{B^{P}(1)^{N}}u({g_{t}h_{{N}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes n}(h_{1},\dots,h_{N})$

Take $0<c_{0}<1$ and $t_{0}>0$ as in the definition of (ENDP), and let $t\geq
t_{*}=\max(t_{0},t_{1})$. Recall that $\nu\left(B^{P}(1)\right)=1$. Since
$t\geq t_{0}$, we can apply (6.1) and for any $i=2,\dots$ conclude that

$\displaystyle\int_{B^{P}(1)^{i}}u({g_{t}h_{{i}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes i}(h_{1},\dots,h_{i})$ (6.9)
$\displaystyle\leq\int_{B^{P}(1)^{i-1}}\left(c_{0}\cdot
u({g_{t}h_{{i-1}}\cdots
g_{t}h_{1}x})+d_{t}\right)\,d\nu^{\otimes{i-1}}(h_{1},\dots,h_{i-1})$
$\displaystyle=c_{0}\int_{B^{P}(1)^{i-1}}u({g_{t}h_{{i-1}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes{i-1}}(h_{1},\dots,h_{i-1})+d_{t}\cdot\nu\left(B^{P}(1)^{i-1}\right)$
$\displaystyle=c_{0}\int_{B^{P}(1)^{i-1}}u({g_{t}h_{{i-1}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes{i-1}}(h_{1},\dots,h_{i-1})+d_{t}.$

Let $N\in{\mathbb{N}}$. If $N=1$, then (6.4) follows immediately from the
combination of (6.1) and (6.8). If $N\geq 2$, then by using (6.9) repeatedly
and combining with (6.8) we obtain

$\displaystyle(I_{B^{P}(1/2),Nt}$ $\displaystyle
u)(x)\leq\int_{B^{P}(1)^{N}}u({g_{t}h_{{N}}\cdots g_{t}h_{1}x})\,d\nu^{\otimes
n}(h_{1},\dots,h_{N})$ (6.10) $\displaystyle\leq
c_{0}^{N-1}\int_{B^{P}(1)}u(g_{t}h_{1}x)\,d\nu(h_{1})+c_{0}^{N-2}d_{t}+\cdots+c_{0}d_{t}+d_{t}$
$\displaystyle\underset{\eqref{eq:mi1}}{\leq}c_{0}^{N}u(x)+c_{0}^{N-1}d_{t}+c_{0}^{N-2}d_{t}+\cdots+c_{0}d_{t}+d_{t}$
$\displaystyle<c_{0}^{N}u(x)+d_{t}(1+c_{0}+c_{0}^{2}+\cdots)=c_{0}^{N}u(x)+\frac{d_{t}}{1-c_{0}}.$

This finishes the proof. ∎

##  7\. (ENDP) and escape of mass

Let $P$ be a subgroup of $G$ that has property (ENDP). Take a height function
$u$, and for $M>0$ define the following sets:

$X_{>M}:=\\{x\in X:u(x)>M\\},\,\,\,\,\,X_{\leq M}:=\\{x\in X:u(x)\leq M\\}.$

Since $u$ is proper, the sets $X_{\leq M}$ are compact.

Since $u$ is regular, by definition there exists $C\geq 1$ such that

${C^{-1}u(x)\leq u(gx)\leq Cu(x)\text{ for all $g\in B(2)$ and }x\in X.}$
(7.1)

Moreover, it is easy to see from (7.1) that there exists $\alpha>0$ such that
for any $t>0$ we have

${e^{-\alpha t}u(x)\leq u(g_{t}x)\leq e^{\alpha t}u(x)}$ (7.2)

Now let $0<c<1$, take $d$ and $t_{c}\geq t_{1}$ as in Corollary 6.2, and let
$t\in{\mathbb{N}}t_{c}$. Note that (6.5) immediately implies that if
$u(x)\geq\frac{d}{c},$ then

${(I_{B^{P}(1/2),t}u)(x)\leq 2c\cdot u(x)}$ (7.3)

Now define

${\ell_{c,t}:=\max\left(\frac{d}{c},e^{\alpha t}\right)}$ (7.4)

In the following key proposition, we obtain an upper bound for the measure of
the sets of type $A_{x}^{N}\left(kt,\theta,X_{>C^{2}\ell_{c,t}^{2}}\right)$,
where $2\leq k\in{\mathbb{N}}$, $\theta\in(0,{r_{*}]}$, and $C$ is as in
(7.1). We will use this measure estimate to derive a covering result for the
sets of type $A_{x}^{N}\left(kt,\theta,X_{>C^{3}\ell_{c,t}^{2}}\right)$ in
Corollary 7.3.

######  Proposition 7.1.

For any $2\leq k\in{\mathbb{N}}$, any $\theta\in(0,{r_{*}]}$, any
$N\in{\mathbb{N}}$, and for any $x\in X$ we have

${\nu\left(A_{x}^{N}\big{(}kt,\theta,X_{>C^{2}\ell_{c,t}^{2}}\big{)}\right)\leq\left({\frac{4c}{1-c}}\right)^{N}\frac{\max\big{(}u(x),d\big{)}}{\ell_{c,t}^{2}}}$
(7.5)

###### Proof of Proposition 7.1.

Let ${2\leq k\in{\mathbb{N}}}$, $N\in{\mathbb{N}}$ and $x\in X$. Define

$Z_{x}(k,N):=\left\\{(h_{1},\dots,h_{Nk})\in
B^{P}(1/2)^{Nk}:{{u}}(g_{t}h_{nk}\cdots
g_{t}h_{1}x)>C\ell_{c,t}^{2}\,\,\,\forall n\in\\{1,\dots,N\\}\right\\}.$

We need the following lemma:

######  Lemma 7.2.

For all $\theta\in(0,{r_{*}]}$ and for all $h\in
A_{x}^{N}\big{(}kt,\theta,X_{>C^{2}\ell_{c,t}^{2}}\big{)}$ one has
$\eta^{-1}_{Nk,t}(h)\subset Z_{x}(k,N)$, where $\eta_{Nk,t}$ is defined as in
(6.6).

###### Proof.

Let $\theta\in(0,{r_{*}]}$ and let $h\in
A_{x}^{N}\big{(}kt,\theta,X_{>C^{2}\ell_{c,t}^{2}}\big{)}$. Suppose that

$\eta_{Nk,t}(h_{1},\dots,h_{Nk})=h.$

Then for any $1\leq i\leq N$ we have

$\displaystyle\operatorname{dist}(g_{ikt}h,g_{t}h_{ik}\cdots g_{t}h_{1})$
$\displaystyle\underset{\eqref{eq:eta}}{=}\operatorname{dist}({g_{ikt}\tilde{h}_{Nk}\cdots\tilde{h}_{1}},g_{ikt}\tilde{h}_{ik}\cdots\tilde{h}_{1})$
$\displaystyle\ \ {=}\
\,\begin{cases}\operatorname{dist}({g_{ikt}\tilde{h}_{Nk}\cdots\tilde{h}_{ik+1}g_{-ikt}},e)&\text{
if }i<N,\\\ 0&\text{ if }i=N.\end{cases}$

Moreover, if $i<N$ one has

$\displaystyle\operatorname{dist}({g_{ikt}\tilde{h}_{Nk}\cdots\tilde{h}_{ik+1}g_{-ikt}},e)$
$\displaystyle\ \ \leq\
\,\operatorname{dist}(g_{ikt}\tilde{h}_{ik+1}g_{-ikt},e)+\cdots+\operatorname{dist}(g_{ikt}\tilde{h}_{Nk}g_{-ikt},e)$
$\displaystyle\underset{\eqref{eq:eta}}{=}\operatorname{dist}(h_{ik+1},e)+\operatorname{dist}(g_{-t}h_{ik+2}g_{t},e)+\operatorname{dist}(g_{-2t}h_{ik+3}g_{2t},e)$
$\displaystyle\ \ {+}\
\,\cdots+\,\operatorname{dist}(g_{-((N-i)k-1)t}h_{Nk}g_{((N-i)k-1)t},e)$
$\displaystyle\underset{\eqref{eq:conj1}}{\leq}1+\frac{1}{4}+\frac{1}{4^{2}}+\cdots\frac{1}{4^{(N-i)k-1}}<2.$

Hence, in view of (7.1), for any $1\leq i\leq N$, $g_{ikt}h{x}\in
X_{>C^{2}\ell_{c,t}^{2}}$ implies that $g_{t}h_{ik}\cdots g_{t}h_{1}{x}\in
X_{>C\ell_{c,t}^{2}}$. This finishes the proof. ∎

Now let $\theta\in(0,{r_{*}]}$. Note that in view of (3.2) we have
$\overline{V_{\theta}}\subset{B^{P}(r_{*}/2)}\subset B^{P}(1/2)$; moreover,
$kt\geq kt_{c}\geq t_{1}$. Thus, by Lemma 7.2 and Lemma 6.3 we have:

$\displaystyle\nu\left(A_{x}^{N}\big{(}kt,\theta,X_{>C^{2}\ell_{c,t}^{2}}\big{)}\right)$
$\displaystyle\leq\nu_{Nk,\,t}\left(A_{x}^{N}\big{(}kt,\theta,X_{>C^{2}\ell_{c,t}^{2}}\big{)}\right)$
(7.6) $\displaystyle\leq\nu^{\otimes Nk}\big{(}Z_{x}(k,N)\big{)},$

where $\nu_{Nk,\,t}$ is defined as in (6.7). So it suffices to estimate
$\nu^{\otimes Nk}\big{(}Z_{x}(k,N)\big{)}$. Define

$s(k,N,x):=\int_{Z_{x}(k,N)}u(g_{t}h_{Nk}\cdots g_{t}h_{1}x)\,d\nu^{\otimes
Nk}(h_{1},\dots,h_{Nk}).$

Since $\ell_{c,t}\geq e^{\alpha t}$, in view of (7.1) and (7.2) we have
$u(g_{t}h_{k-1}\cdots g_{t}h_{{1}}x)>\ell_{c,t}$ whenever
$(h_{1},\dots,h_{k})\in Z_{x}(k,1)$. Hence,

$\displaystyle s(k,1,x)$
$\displaystyle\leq\int_{B^{P}(1/2)^{k-1}}1_{{X_{>\ell_{c,t}}}}(g_{t}h_{{k-1}}\cdots
g_{t}h_{1}x)u(g_{t}h_{k}\cdots g_{t}h_{{1}}x)\,d\nu^{\otimes
k-1}(h_{1},\dots,h_{k-1})$ (7.7) $\displaystyle\leq
2c\int_{B^{P}(1/2)^{k-1}}u({g_{t}h_{{k-1}}\cdots g_{t}h_{1}x})\,d\nu^{\otimes
k-1}(h_{1},\dots,h_{k-1}),$

where the second inequality follows from (7.3) applied with $x$ replaced by
$g_{t}h_{k-1}\cdots g_{t}h_{1}x$, and from the fact that
$\ell_{c,t}\geq{\frac{d}{c}}$.

Again recall that $\nu\left(B^{P}(1)\right)=1$. By applying (6.5) we get:

$\displaystyle\int_{B^{P}(1/2)^{k-1}}u({g_{t}h_{{k-1}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes k-1}(h_{1},\cdots,h_{k-1})$ $\displaystyle\leq
c\int_{B^{P}(1/2)^{k-2}}u({g_{t}h_{{k-2}}\cdots g_{t}h_{1}x})\,d\nu^{\otimes
k-2}(h_{1},\cdots,h_{k-2})+d\cdot\nu\left({B^{P}(1/2)}^{k-2}\right)$
$\displaystyle\leq c\int_{B^{P}(1/2)^{k-2}}u({g_{t}h_{{k-2}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes k-2}(h_{1},\cdots,h_{k-2})+d.$

Therefore, if we apply (6.5) repeatedly, similarly to (6.10) we get

$\displaystyle\int_{B^{P}(1/2)^{k-1}}u({g_{t}h_{{k-1}}\cdots
g_{t}h_{1}x})\,d\nu^{\otimes k-1}(h_{1},\cdots,h_{k-1})$ (7.8)
$\displaystyle\leq
c^{k-1}u(x)+\frac{d}{1-c}\leq\frac{2}{1-c}\cdot\max\big{(}u(x),d\big{)}.$

So by combining (7.7) and (7.8) we have:

${s(k,1,x)\leq{\frac{4c}{1-c}}\cdot\max\big{(}u(x),d\big{)}\,\,\,\text{for all
}x\in X.}$ (7.9)

Note that since $\ell_{c,t}\geq e^{\alpha t},$ in view of (7.1), (7.2) and
(7.4)

${(h_{1},\dots,h_{ik})\in Z_{x}(k,i)\Rightarrow u(g_{t}h_{(i-1)k}\cdots
g_{t}h_{{1}}x)\geq\ell_{c,t}\geq\frac{d}{c}\geq d.}$ (7.10)

Now for any $2\leq i\in{\mathbb{N}}$ we can write

$\displaystyle s(k,i,x)=\int_{Z_{x}(k,i)}u(g_{t}h_{ik}\cdots
g_{t}h_{{1}}x)\,d\nu^{\otimes ik}(h_{1},\dots,h_{ik})$
$\displaystyle=\int_{Z_{x}(k,i-1)}\int_{Z_{g_{t}h_{(i-1)k}\cdots
g_{t}h_{{1}}x}(k,1)}u(g_{t}h_{ik}\cdots g_{t}h_{{1}}x)\,d\nu^{\otimes
k}(h_{(i-1)k+1},\dots,h_{ik})\,d\nu^{\otimes(i-1)k}(h_{1},\dots,h_{(i-1)k})$
$\displaystyle=\int_{Z_{x}(k,i-1)}s(k,1,g_{t}h_{(i-1)k}\cdots
g_{t}h_{{1}}x)\,d\nu^{\otimes(i-1)k}(h_{1},\dots,h_{(i-1)k})$
$\displaystyle\underset{\eqref{eq:mainind},\,\eqref{eq:implic}}{\leq}\int_{Z_{x}(k,i-1)}{\frac{4c}{1-c}}\cdot
u(g_{t}h_{(i-1)k}\cdots
g_{t}h_{{1}}x)\,d\nu^{\otimes(i-1)k}(h_{1},\dots,h_{(i-1)k})$
$\displaystyle={\frac{4c}{1-c}}\cdot s(k,i-1,x)$

Thus, by repeatedly using the above computation, for any $N\in{\mathbb{N}}$ we
conclude that

$s(k,N,x)\leq\left({\frac{4c}{1-c}}\right)^{N-1}s(k,1,x)\underset{\eqref{eq:mainind}}{\leq}\left({\frac{4c}{1-c}}\right)^{N}\max\big{(}u(x),d\big{)}.$

Note that $s(k,N,x)\geq\ell_{c,t}^{2}\cdot\nu^{\otimes
Nk}\big{(}Z_{x}(k,N)\big{)}$. Hence (7.5) follows from the above inequality
and (7.6). ∎

As a corollary, we get the following crucial covering result:

######  Corollary 7.3.

Let $P$ be a subgroup of $G$ with property (ENDP). Then for any $0<c<1$ there
exists $t_{c}>0$ such that for all $t\in{\mathbb{N}}t_{c}$ and $2\leq
k\in{\mathbb{N}}$ satisfying $kt\geq\frac{\log(8\sqrt{p})}{\lambda_{\min}}$,
all $\theta\in{(0,r_{*}/2]}$, all $N\in{\mathbb{N}}$, and for all $x\in X$,
the set

${A_{x}^{N}\big{(}kt,\theta,X_{>C^{3}\ell_{c,t}^{2}}\big{)}=\left\\{h\in\overline{V_{\theta}}:u(g_{ikt}hx)>C^{3}\ell_{c,t}^{2}\,\,\,\forall\,i\in\\{1,\dots,N\\}\right\\}}$

can be covered with at most

$\frac{{e^{\delta
Nkt}}}{{\nu({V_{\theta}})}}{\left({\frac{4c}{1-c}}\right)^{N}}\frac{\max\big{(}u(x),d\big{)}}{\ell_{c,t}^{2}}$

Bowen $(Nkt,\theta)$-balls in $P$.

###### Proof.

Let $0<c<1$, take $t_{c}$ as in Corollary 6.2, and let $t\in{\mathbb{N}}t_{c}$
and $2\leq k\in{\mathbb{N}}$ be such that
$kt\geq\frac{\log(8\sqrt{p})}{\lambda_{\min}}$. Also let
$\theta\in(0,{r_{*}/2]}$, $N\in{\mathbb{N}}$, and $x\in X$. Take a covering of
$\overline{V_{\theta}}$ with Bowen $(Nkt,\theta)$-boxes in $P$. Now let $R$ be
one of the Bowen boxes in this cover which has non-empty intersection with
${A_{x}^{N}\big{(}kt,\theta,X_{>C^{3}\ell_{c,t}^{2}}\big{)}}$. Note that in
view of (3.4), we have

$\operatorname{diam}(R)\leq\frac{\theta}{2}e^{-\lambda_{\min}Nkt}\leq\frac{\theta}{2}e^{-\lambda_{\min}kt}\leq\frac{\theta}{16\sqrt{p}}.$

So, since $R\cap\overline{V_{\theta}}\neq\varnothing$, we must have

${R\subset\partial_{\frac{\theta}{16\sqrt{p}}}\overline{V_{\theta}}\subset{V_{2\theta}},}$
(7.11)

where in the last inclusion we use the $2$-bi-Lipschitz property of $\exp$.

Now let ${h}\in{{R}\cap
A_{x}^{N}\big{(}kt,\theta,X_{>C^{3}\ell_{c,t}^{2}}\big{)}}$. Then

$u(g_{ikt}{h}x)>C^{3}\ell_{c,t}^{2}\text{ for all }1\leq i\leq N.$

On the other hand, if we denote the center of $R$ by $h_{0}$, then for all
$1\leq i\leq N$ we have for all $h^{\prime}\in R$:

$\displaystyle g_{ikt}h^{\prime}x$
$\displaystyle=\left(g_{ikt}h^{\prime}h_{0}^{-1}g_{-ikt}\right)g_{ikt}h_{0}x$
$\displaystyle\in\left(g_{-(N-i)kt}\overline{V_{\theta}}g_{(N-i)kt}\right)g_{ikt}h_{0}{x}$
$\displaystyle\underset{\eqref{eq:diam}}{\in}B\left({\frac{\theta}{2}\cdot
e^{-\lambda_{\min}(N-i)kt}}\right)g_{ikt}h_{0}x$ $\displaystyle\subset
B({\theta}/{2})g_{ikt}h_{0}x\subset B(1/2)g_{ikt}h_{0}x$

This implies that

${g_{ikt}Rx\subset B(1)g_{ikt}hx\text{ for all }1\leq i\leq N}$ (7.12)

Now in view of (7.11) and (7.12) we can conclude that

${{R}\subset{A_{x}^{N}\big{(}kt,{2\theta},X_{>C^{2}\ell_{c,t}^{2}}\big{)}.}}$
(7.13)

Therefore, by (7.5) and (7.13) applied with $\theta$ replaced with $2\theta$,
the set ${A_{x}^{N}\big{(}kt,{\theta},X_{>C^{3}\ell_{c,t}^{2}}\big{)}}$ can be
covered with at most

$\displaystyle\frac{{\nu}{\left(A_{x}^{N}\big{(}kt,{2\theta},X_{>{C^{2}}\ell_{c,t}^{2}}\big{)}\right)}}{\nu(g_{-Nkt}{V_{{\theta}}}g_{Nkt})}$
$\displaystyle\leq\frac{({\frac{4c}{1-c}})^{N}\max\big{(}u(x),d\big{)}}{\nu(g_{-Nkt}{V_{\theta}}g_{Nkt})\cdot\ell_{c,t}^{2}}$
$\displaystyle={\frac{{e^{\delta
Nkt}}}{{\nu({V_{\theta}})}}{\left({\frac{4c}{1-c}}\right)^{N}}\frac{\max\big{(}u(x),d\big{)}}{\ell_{c,t}^{2}}}$

Bowen $(Nkt,\theta)$-boxes in $P$. This finishes the proof. ∎

##  8\. Combining the estimates of §5 and §7

The goal of this section is to describe a method making it possible to put
together properties (EEP) and (ENDP). In the next proposition nether (EEP) nor
(ENDP) are assumed to hold. Instead we will assume certain covering estimates
(similar to those we derived from (EEP) and (ENDP) in the previous sections)
and then combine them to derive an estimate on which our dimension bound is
based. This formalizes an argument which first appeared in [KKLM] and then was
used in [KMi2] to solve DDC in the case (1.2).

######  Proposition 8.1.

Let $P$ be a connected subgroup of $H$ normalized by $F$. Let $S,Q\subset X$,
$t$ satisfying (3.6), $r>0$, ${\theta\in[r,r_{*}/2]}$, and let
$k_{1},k_{2},a_{1},a_{2}\geq 1$ be given. Suppose that for any
$N\in{\mathbb{N}}$ the following two conditions hold:

  * (a)

For all $x\in\partial_{r}(S\cap Q)$ the set $A_{x}^{N}(t,r,S\cap Q)$ can be
covered with at most $k_{1}{e^{\delta Nt}}a_{1}^{N}$ Bowen $(Nt,\theta)$-boxes
in $P$.

  * (b)

For all $x\in\partial_{\theta}(S\cap Q)$ the set $A_{x}^{N}(t,\theta,Q^{c})$
can be covered with at most $k_{2}{e^{\delta Nt}}a_{2}^{N}$ Bowen
$(Nt,\theta)$-boxes in $P$.

Then for all $x\in\partial_{r}(S\cap Q)$ the set ${A}^{N}_{x}(t,r,S)$ can be
covered with at most $k_{3}{e^{\delta Nt}}a_{3}^{N}$ Bowen $(Nt,\theta)$-boxes
in $P$, where

${k_{3}={(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}k_{2}^{2},\quad
a_{3}=a_{1}+a_{2}+\sqrt{k_{3}a_{2}}.}$ (8.1)

###### Proof.

For any $h\in{A_{x}^{N}\left(t,r,S\right)}$, let us define:

${J_{h}}:=\big{\\{}j\in\\{1,\dots,N\\}:g_{jt}hx\in Q^{c}\big{\\}},$

and for any $J\subset\\{1,\dots,N\\}$, set:

$Z(J):=\left\\{h\in{A_{x}^{N}\left(t,r,S\right)}:{J_{h}}=J\right\\}.$

Note that

${{A_{x}^{N}\left(t,r,S\right)}=\bigcup_{J\subset\\{1,\dots,N\\}}Z(J)}$ (8.2)

Let $J$ be a subset of $\\{1,\dots,N\\}$. We can decompose $J$ and
$I:=\\{1,\dots,N\\}\smallsetminus J$ into sub-intervals of maximal size
$J_{1},\dots,J_{q}$ and $I_{1},\dots,I_{q^{\prime}}$ so that

$J=\bigsqcup_{j=1}^{q}J_{j}\text{ and }I=\bigsqcup_{i=1}^{q^{\prime}}I_{i}.$

Hence, we get a partition of the set $\\{1,\dots,N\\}$ as follows:

$\\{1,\dots,N\\}=\bigsqcup_{j=1}^{q}J_{j}\sqcup\bigsqcup_{i=1}^{q^{\prime}}I_{i}.$

Now we inductively prove the following

######  Claim 8.2.

For any integer $L\leq N$, if

${\\{1,\dots,L\\}=\bigsqcup_{j=1}^{\ell}J_{j}\sqcup\bigsqcup_{i=1}^{\ell^{\prime}}I_{i},}$
(8.3)

then the set $Z(J)$ can be covered with at most

${k_{2}^{d^{\prime}_{J,L}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,L}+1}{e^{\delta
Lt}}a_{1}^{\sum_{i=1}^{\ell^{\prime}}|I_{i}|-d_{J,L}}a_{2}^{\sum_{j=1}^{\ell}|J_{j}|}}$
(8.4)

Bowen $(Lt,\theta)$-boxes in $P,$ where $d_{J,L}$, $d^{\prime}_{J,L}$ are
defined as follows:

$d_{J,L}:=\\#\\{i\in\\{1,\dots,L\\}:\ i<L,\,i\in J\ and\ i+1\in I\\},$
$d^{\prime}_{J,L}:=\\#\\{i\in\\{1,\dots,L\\}:\ i<L,\,i\in I\ and\ i+1\in
J\\}.$

###### Proof of Claim 8.2.

We argue by induction on $\ell+\ell^{\prime}$. When $\ell+\ell^{\prime}=1$, we
have $d_{J,L}=d^{\prime}_{J,L}=0$, and there are two cases: either $\ell=1$
and $\\{1,\dots,L\\}=J_{1}$, or $\ell^{\prime}=1$ and $\\{1,\dots,L\\}=I_{1}$.
In the first case

$\displaystyle Z(J)\subset A_{x}^{L}\left(t,{r},Q^{c}\right)\subset
A_{x}^{L}\left(t,\theta,Q^{c}\right),$

Therefore, condition (b) applied with $N=L$ implies that this set can be
covered with at most

$\displaystyle k_{2}e^{\delta
Lt}a_{2}^{L}<{k_{1}}k_{2}{(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}}e^{\delta
Lt}a_{2}^{L}$

Bowen $(Lt,\theta)$-boxes in $P$. This finishes the proof of the first case.

In the second case, note that

$Z(J)\subset A_{x}^{L}\left(t,{r},S\cap Q\right).$

Moreover, by condition (a) applied with $N=L$, $A_{x}^{N}\left(t,{r},S\cap
Q\right)$ can be covered by at most

$k_{1}e^{\delta
Lt}a_{1}^{L}<{k_{1}}k_{2}{(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}}e^{\delta
Lt}a_{1}^{L}$

Bowen $(Lt,\theta)$-balls in $P$. This ends the proof of the base of the
induction.

In the inductive step, let $L^{\prime}>L$ be the next integer for which an
equation similar to (8.3) is satisfied. We have two cases. Either

${\\{1,\dots,L^{\prime}\\}=\\{1,\dots,L\\}\sqcup I_{\ell^{\prime}+1}}$ (8.5)

or

${\\{1,\dots,L^{\prime}\\}=\\{1,\dots,L\\}\sqcup J_{\ell+1}.}$ (8.6)

We start with the case (8.5). Note that in this case we have

${d_{J,L^{\prime}}=d_{J,L}+1\text{ and
}d^{\prime}_{J,L^{\prime}}=d^{\prime}_{J,L}.}$ (8.7)

By the induction hypothesis, an upper bound for the number of Bowen
$(Lt,\theta)$-boxes needed to cover $Z(J)$ is given by (8.4). Then observe
that:

  * •

In view of (3.6) and Lemma 3.2,

${e^{\delta t}(1+C_{0}e^{-\lambda_{\min}kt})\leq e^{\delta t}(1+C_{0})}$ (8.8)

is an upper bound for the number of Bowen $\big{(}(L+1)t,\theta\big{)}$-boxes
needed to cover an arbitrary Bowen $(Lt,\theta)$-box;

  * •

In view of Lemma 3.1,

${\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}$ (8.9)

is an upper bound for the number of Bowen $\big{(}(L+1)t,r\big{)}$-boxes
needed to cover an arbitrary Bowen $(Lt,{\theta})$-box.

Now let $B_{r}$ be a Bowen $\big{(}(L+1)t,r\big{)}$-box that has non-empty
intersection with $Z(J)$, and let $h\in B_{r}\cap Z(J)$. Since $h\in Z(J)$, it
follows that $g_{(L+1)t}hx\in S\cap Q$. Therefore, if we denote the center of
$B_{r}$ by $h_{0}$, we have

${{g_{(L+1)t}h_{0}x\in\overline{V_{r}}(S\cap Q)\subset\partial_{r}(S\cap
Q).}}$ (8.10)

Moreover, for any $h\in B_{r}$ and any positive integer $1\leq i\leq
L^{\prime}-(L+1)$ we have:

$\displaystyle g_{(L+1+i)t}hx$
$\displaystyle=g_{it}(g_{(L+1)t}hh_{0}^{-1}g_{-(L+1)t})(g_{(L+1)t}h_{0}x).$

Since the map $h\rightarrow g_{(L+1)t}hh_{0}^{-1}g_{-(L+1)t}$ sends $B_{r}$
into $\overline{V_{r}}$, the preceding equality implies that

$\displaystyle\left\\{h^{\prime}\in B_{r}:g_{(L+1+i)t}h^{\prime}x\in S\cap
Q\,\,\,\forall\,i\in\\{1,\dots,L^{\prime}-(L+1)\\}\right\\}$
$\displaystyle\subset
g_{-(L+1)t}A_{g_{(L+1)t}h_{0}x}^{L^{\prime}-(L+1)}\left(t,r,S\cap
Q\right)g_{(L+1)t}h_{0}.$

So, in view of the above inclusion and (8.10), we can go through the same
procedure and apply condition (a) with $N$ replaced with
$|I_{\ell^{\prime}+1}|-1=L^{\prime}-(L+1)$ and $x$ replaced with
$g_{(L+1)t}h_{0}x$, and conclude that $B_{r}\cap Z(J)$ can be covered with at
most

${k_{1}e^{\delta(|I_{\ell^{\prime}+1}|-1)t}a_{1}^{|I_{\ell^{\prime}+1}|-1}}$
(8.11)

Bowen $(L^{\prime}t,\theta)$-boxes in $P$. Multiplying the bounds (8.4),
(8.8), (8.9) and (8.11), we conclude that $Z(J)$ can be covered with at most

$\displaystyle\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}{e^{\delta(|I_{{\ell^{\prime}}+1}|-1)t}}a_{1}^{|I_{{\ell^{\prime}}+1}|-1}{(1+C_{0})}$
$\displaystyle\cdot
k_{2}^{d^{\prime}_{J,L}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left(\frac{2\theta}{r}\right)^{p}}k_{1}}\right)^{d_{J,L}+1}{e^{\delta(L+1)t}}a_{1}^{\sum_{i=1}^{\ell^{\prime}}|I_{i}|-d_{J,L}}a_{2}^{\sum_{j=1}^{\ell}|J_{j}|}$
$\displaystyle\underset{\eqref{eq:d11}}{=}k_{2}^{d^{\prime}_{J,L^{\prime}}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,L^{\prime}}+1}{e^{\delta
L^{\prime}t}}a_{1}^{\sum_{i=1}^{\ell^{\prime}}|I_{i}|-d_{J,L^{\prime}}}a_{2}^{\sum_{j=1}^{\ell}|J_{j}|}$

Bowen $(L^{\prime}t,\theta)$-boxes in $P$. This ends the proof of the claim in
this case.

Next assume (8.6). Note that in this case

${d_{J,L^{\prime}}=d_{J,L}\text{ and
}d^{\prime}_{J,L^{\prime}}=d^{\prime}_{J,L}+1.}$ (8.12)

Take a covering of $Z(J)$ with Bowen $(Lt,\theta)$-boxes in $P$. Suppose
$B^{\prime}$ is one of the Bowen $(Lt,\theta)$-boxes in the cover such that
$B^{\prime}\cap Z(J)\neq\varnothing$, and let $h_{1}$ be the center of
$B^{\prime}$. It is easy to see that $B^{\prime}\cap Z(J)\neq\varnothing$
implies:

${g_{Lt}h_{1}x\in\overline{V_{\theta}}(S\cap Q)\subset\partial_{\theta}(S\cap
Q).}$ (8.13)

On the other hand, for any $s\in B^{\prime}$ and any positive integer $1\leq
i\leq L^{\prime}-L$ we have:

$\displaystyle g_{(L+i)t}h_{1}x$
$\displaystyle=g_{it}(g_{Lt}hh_{1}^{-1}g_{-Lt})(g_{Lt}h_{1}x).$

Hence, since the map $h\rightarrow g_{Lt}{hh_{1}^{-1}}g_{-Lt}$ maps
$B^{\prime}$ into $\overline{V_{\theta}}$, the above equality implies

$\displaystyle\left\\{h\in B^{\prime}:g_{(L+i)t}hx\in
Q^{c}\,\,\,\forall\,i\in\\{1,\cdots,L^{\prime}-L\\}\right\\}\subset
g_{-Lt}A_{g_{Lt}h_{1}x}^{L^{\prime}-L}\left(t,\theta,Q^{c}\right)g_{Lt}h_{1}$

So in view of the above inclusion and (8.13), we can apply condition (b) with
$g_{Lt}h_{1}x$ in place of $x$, and $|J_{\ell+1}|=L^{\prime}-L$ in place of
$N$. This way, we get that the set $B^{\prime}\cap Z(J)$ can be covered with
at most $k_{2}a_{2}^{|J_{\ell+1}|}{e^{\delta|J_{\ell+1}|t}}$ Bowen
$(L^{\prime}t,\theta)$-boxes in $P$. From this, combined with the induction
hypothesis, we conclude that $Z(J)$ can be covered with at most

$\displaystyle k_{2}a_{2}^{|J_{\ell+1}|}{e^{\delta|J_{\ell+1}|t}}\cdot
k_{2}^{d^{\prime}_{J,L}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,L}+1}{e^{\delta
Lt}}a_{1}^{\sum_{i=1}^{\ell^{\prime}}|I_{i}|-d_{J,L}}a_{2}^{\sum_{j=1}^{\ell}|J_{j}|}$
$\displaystyle\underset{\eqref{eq:d21}}{=}k_{2}^{d^{\prime}_{J,L^{\prime}}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,L^{\prime}}+1}{e^{\delta
L^{\prime}t}}a_{1}^{\sum_{i=1}^{\ell^{\prime}}|I_{i}|-d_{J,L}}a_{2}^{\sum_{j=1}^{\ell+1}|J_{j}|}$

Bowen $(L^{\prime}t,\theta)$-boxes in $P$, finishing the proof of the claim. ∎

Now by letting $L=N$, we conclude that $Z(J)$ can be covered with at most

${k_{2}^{d^{\prime}_{J,N}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,N}+1}{e^{\delta
Nt}}a_{1}^{|I|-d_{J,N}}a_{2}^{|J|}}$ (8.14)

Bowen $(Nt,\theta)$-boxes in $P$.

Clearly

${d^{\prime}_{J,N}\leq d_{J,N}+1.}$ (8.15)

Also, note that since $d_{J,N}\leq\max(|I|,|J|)$, the exponents
$|I|-d_{J,N},|J|-d_{J,N}$ in (8.14) are non-negative integers. So, in view of
(8.2) and (8.14), the set ${A_{x}^{N}\left(t,{r},S\right)}$ can be covered
with at most

$\displaystyle\sum_{J\subset\\{1,\dots,N\\}}k_{2}^{d^{\prime}_{J,N}+1}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,N}+1}{e^{\delta
Nt}}a_{1}^{|I|-d_{J,N}}a_{2}^{|J|}$
$\displaystyle\underset{\eqref{eq:lastre1}}{\leq}{e^{\delta
Nt}}\sum_{J\subset\\{1,\dots,N\\}}k_{2}^{d_{J,N}+2}\left({(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}\right)^{d_{J,N}+1}a_{1}^{|I|-d_{J,N}}a_{2}^{|J|}$
$\displaystyle\leq k_{3}{e^{\delta
Nt}}\sum_{J\subset\\{1,\dots,N\\}}a_{1}^{|I|-d_{J,N}}a_{2}^{|J|}k_{3}^{d_{J,N}}$
$\displaystyle=k_{3}{e^{\delta
Nt}}\sum_{J\subset\\{1,\dots,N\\}}a_{1}^{N-|J|-d_{J,N}}a_{2}^{|J|-d_{J,N}}{({k_{3}a_{2}})^{d_{J,N}}}$

Bowen $(Nt,\theta)$-boxes in $P$, where
$k_{3}:={(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}k_{2}^{2}$.

To simplify the last expression we will use the following

######  Lemma 8.3.

[KMi2, Lemma 5.4] For any $n_{1},n_{2},n_{3}>0$ it holds that

$\sum_{J\subset\\{1,\dots,N\\}}n_{1}^{N-|J|-d_{J,N}}n_{2}^{|J|-d_{J,N}}n_{3}^{2d_{J,N}}\leq\left(n_{1}+n_{2}+n_{3}\right)^{N}.$

Applying the above lemma with $n_{1}=a_{1}$, $n_{2}=a_{2}$ and
$n_{3}=\sqrt{k_{3}a_{3}}$, we conclude that ${A_{x}^{N}\left(t,{r},S\right)}$
can be covered with at most

$k_{3}{e^{\delta Nt}}\left(a_{1}+a_{2}+\sqrt{k_{3}a_{3}}\right)^{N}$

Bowen $(Nt,\theta)$-boxes in $P$. The proof of Proposition 8.1 is now
complete. ∎

##  9\. Proof of Theorem 2.1

Given $P\subset G$ satisfying (ENDP), $0<c<1$, and $t>0$, let us define the
compact subset $Q_{c,t}$ of $X$ as follows:

${Q_{c,t}:=X_{\leq C^{3}\ell_{c,t}^{2}},}$ (9.1)

where $\ell_{c,t}$ is as in (7.4) and $C$ is as in (7.1).

######  Lemma 9.1.

Let $P$ be a subgroup of $G$ that has properties (EEP) and (ENDP). Then there
exist constants

$a^{\prime},b^{\prime},C_{1},C_{2},\lambda>0$

such that for any open subset $O$ of $X$ and all $N\in{\mathbb{N}}$ the
following holds: For all $0<c<1$ there exists $t_{c}>0$ such that for all
$t\in{\mathbb{N}}t_{c}$, $0<r<1$, and $2\leq k\in{\mathbb{N}}$ satisfying

${{{e^{\frac{a^{\prime}-kt}{b^{\prime}}}}\leq
r<\frac{1}{4}\min\big{(}r_{0}\left({\partial_{1}}Q_{c,t}\right),{r_{*}}\big{)}},}$
(9.2)

all $\theta\in{\left[r,\frac{r_{*}}{2}\right]}$, and for all
$x\in\partial_{r}\left(Q_{c,t}\cap O^{c}\right)$, the set
${A_{x}^{N}\left(kt,r,O^{c}\right)}$ can be covered with at most

$\frac{C_{1}}{\theta^{2p}}{e^{\delta
Nkt}}\left(1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
kt}+{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}\right)^{N}$

Bowen $(Nkt,\theta)$-boxes in $P$.

###### Proof.

Let $0<c<1$, take $t_{c}$ as in Corollary 7.3, and let $0<r<1$, $2\leq
k\in{\mathbb{N}}$, $t\in{\mathbb{N}}t_{c}$ be such that (9.2) is satisfied,
where $a^{\prime},b^{\prime}$ are as in Theorem 5.3. Also let
$\theta\in{\left[r,\frac{r_{*}}{2}\right]}$. Note that the second inequality
in (9.2), together with the fact that $r_{0}\big{(}\partial_{1}(O^{c}\cap
Q_{c,t})\big{)}\geq r_{0}(\partial_{1}Q_{c,t})$, implies condition (5.10) with
$O$ replaced by $O\cup Q_{c,t}^{c}$. Moreover, condition (5.3) with $t$
replaced by $kt$ follows from the first inequality in (9.2) . Hence, by
applying Theorem 5.3 with $O$ replaced with $O\cup Q_{c,t}^{c}$ and $t$
replaced with $kt$, we get that for all $x\in\partial_{r}(O^{c}\cap Q_{c,t})$
and for all $N\in{\mathbb{N}}$, the set $A_{x}^{N}\left(kt,r,O^{c}\cap
Q_{c,t}\right)$ can be covered with at most $k_{1}e^{\delta Nkt}a_{1}^{N}$
Bowen $(Nkt,\theta)$-boxes in $P$, where

${k_{1}=\frac{c_{2}}{c_{1}}\left(\frac{2r}{\theta}\right)^{p},\quad
a_{1}=1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
kt}}$ (9.3)

and $C_{2},\lambda$ are as in Theorem 5.3.

Moreover, in view of (9.1) and (7.4), for any $x\in\partial_{\theta}(O^{c}\cap
Q_{c,t})\subset\partial_{2}Q_{c,t}$ we have

$\frac{\max(u(x),d)}{\ell_{c,t}^{2}}\leq\frac{\max(C^{4}\ell_{c,t}^{2},d)}{\ell_{c,t}^{2}}\underset{\ell_{c,t}^{2}\geq\ell_{c,t}>d}{=}C^{4}.$

Also, note that

${kt\underset{\eqref{eq:r estimate
4}}{\geq}a^{\prime}+b^{\prime}\log\frac{1}{r}>b^{\prime}\log\frac{1}{r}>b^{\prime}\geq\frac{\log(8\sqrt{p})}{\lambda_{\min}}.}$
(9.4)

Thus, by applying Corollary 7.3 we get that for all
$x\in\partial_{\theta}(O^{c}\cap Q_{c,t})$ and for all $N\in{\mathbb{N}}$, the
set $A_{x}^{N}\left(kt,\theta,Q_{c,t}^{c}\right)$ can be covered with at most
$k_{2}e^{\delta Nkt}a_{2}^{N}$ Bowen $(Nkt,\theta)$-boxes in $P$, where

${k_{2}=\frac{C^{4}}{\nu\left({V_{\theta}}\right)},\,\quad
a_{2}=\frac{4c}{1-c}.}$ (9.5)

Now we put together the estimates we found to get an estimate for the number
of Bowen $(Nkt,\theta)$-boxes needed to cover the set
$A_{x}^{N}\left(kt,r,O^{c}\right)$. Observe that in view of (9.4), we have
$kt\geq\frac{\log(8\sqrt{p})}{\lambda_{\min}}$. So, we can apply Proposition
8.1 with $S=O^{c},\,Q=Q_{c,t}$ and $kt$ in place of $t$, and conclude that the
set $A_{x}^{N}\left(kt,r,O^{c}\right)$ can be covered with at most
$k_{3}e^{\delta Nkt}a_{3}^{N}$ Bowen $(Nkt,\theta)$-boxes in $P$, where
$k_{3},a_{3}$ are as in (8.1), $k_{1},a_{1}$ are as in (9.3), and
$k_{2},a_{2}$ are as in (9.5).

Finally, we need to estimate $k_{3}e^{\delta Nkt}a_{3}^{N}$ from above. We
have

$\displaystyle\,k_{3}\underset{\eqref{eq:a3}}{=}{(1+C_{0}){\frac{c_{2}}{c_{1}}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}k_{1}}k_{2}^{2}$
(9.6)
$\displaystyle\,\underset{\eqref{eq:a1},\,\eqref{eq:a2}}{=}{(1+C_{0}){\left(\frac{c_{2}}{c_{1}}\right)^{2}\left({\frac{\theta}{r}+8\sqrt{p}}\right)^{p}}\left(\frac{2r}{\theta}\right)^{p}}\left(\frac{C^{4}}{\nu\left({V_{\theta}}\right)}\right)^{2}$
$\displaystyle\quad\underset{(\theta\geq
r)}{\leq}{(1+C_{0}){\left(\frac{c_{2}}{c_{1}}\right)^{2}\left({2+16\sqrt{p}}\right)^{p}}}\left(\frac{C^{4}}{\nu\left({V_{\theta}}\right)}\right)^{2}$
$\displaystyle\quad\,\underset{\eqref{eq:lb1}}{\leq}(1+C_{0})\left(\frac{c_{2}}{c_{1}}\right)^{2}\left({2+16\sqrt{p}}\right)^{p}\left(\frac{(4\sqrt{p})^{p}}{c_{1}\theta^{p}}C^{4}\right)^{2}=\frac{C_{1}^{2}}{\theta^{2p}},$

where
$C_{1}:=\sqrt{1+C_{0}}\frac{c_{2}}{c_{1}}\left({2+16\sqrt{p}}\right)^{p/2}\frac{(4\sqrt{p})^{p}}{c_{1}}C^{4}\geq
1$. Furthermore, we have

$\displaystyle
a_{3}\underset{\eqref{eq:a3}}{=}a_{1}+a_{2}+\sqrt{k_{3}a_{2}}\underset{\eqref{eq:k3est}}{\leq}a_{1}+a_{2}+\sqrt{\frac{C_{1}^{2}}{\theta^{2p}}\cdot
a_{2}}$ (9.7)
$\displaystyle\underset{\eqref{eq:a1},\,\eqref{eq:a2}}{=}1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
kt}+\frac{4c}{1-c}+\sqrt{\frac{C_{1}^{2}}{\theta^{2p}}\cdot\frac{4c}{1-c}}$
$\displaystyle\,\quad\leq
1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
kt}+\frac{8C_{1}}{\theta^{p}}\cdot\frac{\sqrt{c}}{1-c}.$

Therefore, by combining (9.6) and (9.7) we obtain

$k_{3}e^{\delta Nkt}a_{3}^{N}\leq\frac{C_{1}}{\theta^{2p}}{e^{\delta
Nkt}}\left(1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
kt}+{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}\right)^{N}$

This ends the proof of the lemma. ∎

###### Proof of Theorem 2.1.

Let $0<c<1$. Take $t=t_{c}$ as in Lemma 9.1, and let $Q=Q_{c,t_{c}}$ be as in
(9.1). Also let $O$ be an open subset of $X$.

Proof of (1): Take $2\leq k\in{\mathbb{N}}$ and $x\in X$. Our goal is to find
an upper bound for the Hausdorff dimension of the set $S(k,t,x)$ defined in
(2.5). In view of (2.5) and the countable stability of Hausdorff dimension it
suffices to estimate the dimension of

$\left\\{h\in\overline{V_{{r_{*}}/{2}}}:{{g_{Nkt}}}hx\notin Q\,\,\,\forall
N\in{\mathbb{N}}\right\\},$

which, due to (9.1), coincides with
$\bigcap_{N\in{\mathbb{N}}}{A_{x}^{N}\big{(}kt,\frac{r_{*}}{2},X_{>C^{3}\ell_{c,t}^{2}}\big{)}}$.

From Corollary 7.3 applied with $\theta=\frac{r_{*}}{2}$, combined with Lemma
3.3 applied with $t$ replaced by $Nkt$ and ${r}=\frac{r_{*}}{2}$, we get that
for any $N\in{\mathbb{N}}$ the set
$A_{x}^{N}{\big{(}kt,\frac{r_{*}}{2},X_{>C^{3}\ell_{c,t}^{2}}\big{)}}$ can be
covered with at most

${{\frac{e^{p{\lambda_{\max}}Nkt}}{{\nu({V_{{r_{*}}/{2}}})}}}}\left({\frac{4c}{1-c}}\right)^{N}\frac{\max\big{(}u(x),d\big{)}}{\ell_{c,t}^{2}}$

balls of radius $\frac{r_{*}}{2}e^{-{\lambda_{\max}}Nkt}$ in $P$. Hence,

$\displaystyle\dim\bigcap_{N\in{\mathbb{N}}}{A_{x}^{N}\left(kt,\frac{r_{*}}{2},X_{>C^{3}\ell_{c,t}^{2}}\right)}$
$\displaystyle\leq\lim_{N\rightarrow\infty}\frac{\log\left({{\frac{e^{p{\lambda_{\max}}Nkt}}{{\nu({V_{{r_{*}}/{2}}})}}}}({\frac{4c}{1-c}})^{N}\frac{\max\big{(}u(x),d\big{)}}{\ell_{c,t}^{2}}\right)}{-\log{\frac{r_{*}}{2}}{e}^{-{\lambda_{\max}}Nk{t}}}$
$\displaystyle=\frac{\log{\frac{4c}{1-c}}e^{p{\lambda_{\max}}kt}}{{\lambda_{\max}}kt}={p-{\frac{1}{{\lambda_{\max}}kt}\log{\frac{1-c}{4c}}}.}$

Proof of (2): Let $2\leq k\in{\mathbb{N}}$, and $x\in X$. Our goal is to find
an upper bound for the Hausdorff dimension of the set

${\left\\{h\in P\smallsetminus
S(k,t,x):hx\in\widetilde{E}({F^{+}},O)\right\\}}$

Recall that

${S(k,t,x)^{c}=\left\\{h\in P:{{g_{Nkt}}}hx\in Q\text{ for some
}N\in{\mathbb{N}}\right\\}}.$

Therefore

$\displaystyle\left\\{h\in P\smallsetminus
S(k,t,x):hx\in\widetilde{E}({F^{+}},O)\right\\}=$ $\displaystyle\left\\{h\in
P:hx\in\widetilde{E}({F^{+}},O)\bigcap\left(\bigcup_{N\in{\mathbb{N}}}g_{-Nkt}Q\right)\right\\}$
$\displaystyle\subset$ $\displaystyle\left\\{h\in
P:hx\in\bigcup_{N\in{\mathbb{N}}}g_{-Nkt}\big{(}Q\cap\widetilde{E}({F^{+}},O)\big{)}\right\\}$
$\displaystyle=$ $\displaystyle\bigcup_{N\in{\mathbb{N}}}\left\\{h\in P:hx\in
g_{-Nkt}\big{(}Q\cap\widetilde{E}({F^{+}},O)\big{)}\right\\}.$

Hence, since Hausdorff dimension is countably stable, to complete the proof of
this part, it suffices to show that for any $N\in{\mathbb{N}}$ we have

${\dim\left\\{h\in P:hx\in
g_{-Nkt}\big{(}Q\cap\widetilde{E}({F^{+}},O)\big{)}\right\\}\leq
p-{\frac{\mu\big{(}{{\sigma_{4\theta}}}O\big{)}-\frac{C_{2}}{r^{p}}e^{-\lambda
kt}-{{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}}}{{\lambda_{\max}}kt}}}$
(9.8)

where $C_{1},C_{2},\lambda$ as in Lemma 9.1.

Now let $N\in{\mathbb{N}}$ and suppose $r>0$ is such that (2.7) is satisfied,
where $a^{\prime},b^{\prime}$ are as in Lemma 9.1. Note that, since $P$ is
normalized by $F^{+},$ we have $P=g_{-Nkt}Pg_{Nkt}$. Moreover, $V_{r}$ is a
tessellation domain. Hence, by countable stability of Hausdorff dimension, in
order to prove (9.8), it suffices to show that the Hausdorff dimension of the
set

$E^{\prime}_{N,x,r}:=\left\\{h\in g_{-Nkt}\,{\overline{V_{r}}}\,g_{Nkt}:hx\in
g_{-Nkt}\big{(}Q\cap\widetilde{E}({F^{+}},O)\big{)}\right\\}$

is not greater than the right-hand side of (9.8). For any $h\in
E^{\prime}_{N,x,r}$ we have

$\displaystyle g_{ikt}g_{Nkt}hx$
$\displaystyle=g_{ikt}(g_{Nkt}hg_{-Nkt})g_{Nkt}x\in
O^{c}\quad\forall\,i\in{\mathbb{N}},$

and at the same time $g_{Nkt}hg_{-Nkt}\in{\overline{V_{r}}}.$ Hence,

${E^{\prime}_{N,x,r}\subset
g_{-Nkt}\left(\bigcap_{i\in{\mathbb{N}}}{A_{g_{Nkt}x}^{i}\left(kt,r,O^{c}\right)}\right)g_{Nkt}.}$
(9.9)

Also, it is easy to see that if $E^{\prime}_{N,x,r}$ is non-empty, then

$g_{Nkt}x\in{\overline{V_{r}}}\left(Q\cap
O^{c}\right)\subset\partial_{r}\left(Q\cap O^{c}\right).$

So by applying Lemma 3.3 with $r$ replaced by $\theta$ and $t$ replaced with
$ikt$, and Lemma 9.1 with $t$ replaced by $kt$, we get that for any
$i\in{\mathbb{N}}$ and any $\theta\in{\left[r,\frac{r_{*}}{2}\right]}$, the
set $A_{g_{Nkt}x}^{i}\left(kt,r,O^{c}\right)$ can be covered with at most

$\displaystyle{\frac{C_{1}}{\theta^{2p}}e^{p{\lambda_{\max}}ikt}\left(1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
ikt}+{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}\right)^{i}}$

balls of radius $\theta e^{-{\lambda_{\max}}ikt}$ in $P$. Also, note that the
Hausdorff dimension is preserved by conjugation. So, we have for any
$\theta\in{\left[r,\frac{r_{*}}{2}\right]}$:

$\displaystyle\dim E^{\prime}_{N,x,r}$
$\displaystyle\underset{\eqref{eq:fininc}}{\leq}\dim\left(g_{-Nkt}\left(\bigcap_{i\in{\mathbb{N}}}{A_{g_{Nkt}x}^{i}\left(kt,r,O^{c}\right)}\right)g_{Nkt}\right)$
$\displaystyle{=}\dim\bigcap_{i\in{\mathbb{N}}}{A_{g_{Nkt}x}^{i}\left(kt,{r},O^{c}\right)}$
$\displaystyle\leq\lim_{i\rightarrow\infty}\frac{\log\left({\frac{C_{1}}{\theta^{2p}}\cdot
e^{p{\lambda_{\max}}ikt}\cdot\left(1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
ikt}+{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}\right)^{i}}\right)}{{-\log{\theta}{e}^{-{\lambda_{\max}}ikt}}}$
$\displaystyle=p-\frac{-\log\left(1-\mu\big{(}{\sigma_{4\theta}}O\big{)}+\frac{C_{2}}{r^{p}}e^{-\lambda
kt}+{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}\right)}{{\lambda_{\max}}kt}$
$\displaystyle\leq{p-\frac{\mu\big{(}{\sigma_{4\theta}}O\big{)}-\frac{C_{2}}{r^{p}}e^{-\lambda
kt}-{{{\frac{8C_{1}}{\theta^{p}}}{{\frac{\sqrt{c}}{1-c}}}}}}{{\lambda_{\max}}kt}}.$

This finishes the proof. ∎

##  10\. Concluding Remarks

###  10.1. Effective estimates

It is a natural problem to effectivize the estimates showing up in the
Dimension Drop Conjecture. Previous work of the authors on the subject [KMi1,
KMi2] contained explicit estimates, although with no claims of optimality.
Namely, this has been done under the assumption that the complement of $O$ is
compact (in particular, when $X$ is compact), and also in the special case
(1.2).

In the more general set-up of this paper it is also possible to make the
estimates effective. This however would require an additional ingredient:
finding a lower bound for the injectivity radii of compact sets
$\\{x:u_{t}(x)\leq M\\}$ arising from condition (ENDP). Such lower bounds can
be obtained immediately whenever the following condition is satisfied: Let $P$
be a subgroup of $G$ that has property (ENDP), and let $\\{u_{t}\\}_{t\geq
t_{0}}$ be the family of height functions as in Definition 1.3; then there
exist positive constants ${m}_{0},m$ such that

${{r_{0}(x)}^{-1}\geq m_{0}{u_{t}(x)}^{-m}\,\,\,\text{ for every }x\in
X,\,t\geq t_{0}.}$ (10.1)

This condition can be verified in many special cases. For example, in [SS, BQ]
certain height functions are constructed on homogeneous spaces of semisimple
Lie groups without compact factors, and for these height functions (10.1) is
verified in [SS, Proposition 26] and [BQ, Lemma 6.3] respectively. By using
the same method one can easily show that (10.1) holds for height functions
$u_{t}$ as in Theorem 1.4, and also for the family of height functions
constructed in [KMi2] in the case (1.2). A variation of our argument shows
that in the presence of (10.1) one has

${\inf_{x\in X}\operatorname{codim}\left(\big{\\{}h\in
P:hx\in\widetilde{E}({F},O)\big{\\}}\right)\gg\frac{\mu(O)}{\log\frac{1}{{\min\left(\theta_{O},\mu(O),r_{1}\right)}}},}$

where $\theta_{O}$ is as in (2.9), and $0<r_{1}<\frac{1}{2}$ is a uniform
constant independent of $O$.

###  10.2. Removing the ${\operatorname{Ad}}$-diagonalizability condition

We expect that by a slight modification of the proof of Theorem 1.5 one can
show that this theorem holds when $F$ is an arbitrary one-parameter unbounded
subsemigroup of a connected semisimple Lie group $G$; namely, the condition
that $F$ is ${\operatorname{Ad}}$-diagonalizable is not necessary. Indeed,
recall the Jordan decomposition of $F=\\{g_{t}\\}$: one can write
$g_{t}=k_{t}a_{t}u_{t}$, where $K_{F}=\\{k_{t}\\}$ is bounded,
$A_{F}=\\{a_{t}\\}$ is ${\operatorname{Ad}}$-diagonalizable, and
$U_{F}=\\{u_{t}\\}$ is ${\operatorname{Ad}}$-unipotent. These subgroups are
uniquely determined and commute with each other. If $A_{F}$ is trivial (in
other words, if $F$ is ${\operatorname{Ad}}$-quasiunipotent) and $U_{F}$ is
not, then Ratner’s Measure Classification Theorem and the work of Dani and
Margulis (see [St, Lemma 21.2] and [DM, Proposition 2.1]) imply that whenever
$O$ is non-empty, the set $\widetilde{E}({F},O)$ is contained in a countable
union of proper submanifolds of $X$; hence dimension drop takes place in a
stronger form. On the other hand, if $A_{F}$ is non-trivial, one can modify
our argument following the lines of [GS, §4], where an analog of (ENDP) was
considered with $(I_{f,t}\psi)(x)$ as in (1.4) replaced by a family of
operators

$\psi(\cdot)\mapsto\int_{P}f({{h}})\psi(a_{t}gu_{t}g^{-1}{{h}}\,\cdot)\,d{{\nu}}({{h}}),$

and with $g$ running through the centralizer of $A_{F}$ in $G$.

###  10.3. Jointly Dirichlet-improvable systems of linear forms: a dimension
bound

Fix $m,n\in{\mathbb{N}}$ and, given $c\leq 1$, say that $Y\in M_{m,n}$ is
$c$-Dirichlet improvable if for all sufficiently large $N$

there exists ${\bf p}\in{\mathbb{Z}}^{m}$ and ${\bf
q}\in{\mathbb{Z}}^{n}\smallsetminus\\{0\\}$ such that (10.2)
$\displaystyle\|Y{\bf q}-{\bf p}\|<cN^{-n/m}\text{ and }0<\|{\bf q}\|<N.$

(In this subsection $\|\cdot\|$ stands for the supremum norm on
${\mathbb{R}}^{m}$, ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{m+n}$.) We let
$\mathbf{DI}_{m,n}(c)$ be the set of $c$-Dirichlet improvable $Y\in M_{m,n}$.
Dirichlet’s theorem (see e.g. [Sc]) implies that
$\mathbf{DI}_{m,n}(1)=M_{m,n}$. Davenport and Schmidt [DS] proved that the
Lebesgue measure of $\mathbf{DI}_{m,n}(c)$ is zero for any $c<1$, and also
that $\bigcup_{c<1}\mathbf{DI}_{m,n}(c)$ contains the set of badly
approximable $m\times n$ matrices, which is known [Sc1] to have full Hausdorff
dimension; in other words, $\dim\mathbf{DI}(c)\to mn$ as $c\to 1$.

Recently in [KMi2] a solution of DDC for the case (1.2), that is for the space
$X$ of unimodular lattices in ${\mathbb{R}}^{m+n}$, was used to derive a
dimension drop result for the family $\\{\mathbf{DI}_{m,n}(c)\\}$: namely,
that $\dim\mathbf{DI}_{m,n}(c)<mn$ whenever $c<1$. Moreover, as explained in
[KSY, Remark 6], a combination of the methods from [KMi2] with measure
estimates obtained in [KSY] can produce an effective estimate for the
codimension of $\mathbf{DI}_{m,n}(c)$. The reduction to dynamics goes back to
Davenport, Schmidt and Dani [Da]. It proceeds by assigning an element
$h_{Y}:=\begin{bmatrix}I_{m}&Y\\\ 0&I_{n}\end{bmatrix}$ of
$G=\operatorname{SL}_{m+n}({\mathbb{R}})$ to $Y$. Arguing as in [KW1,
Proposition 2.1] or [KMi2, Proof of Theorem 1.5], one can see that
$Y\in\mathbf{DI}_{m,n}(c)$ if and only if
$h_{Y}{\mathbb{Z}}^{m+n}\in\widetilde{E}(F,O)$, where

${O=\left\\{\Lambda\in X:\|{\bf{v}}\|\geq c^{\frac{m}{m+n}}\text{ for all
}{\bf{v}}\in\Lambda\smallsetminus\\{0\\}\right\\}}$ (10.3)

(a subset of $X$ with non-empty interior), and $X$, $F$ are as in (1.2).

Our new Diophantine appplication is motivated by [BV, §2.7], where Beresnevich
and Velani introduced the notion of jointly singular $k$-tuples of matrices.
Namely, say that $(Y_{1},\dots,Y_{k})\in M_{n,m}^{k}$ is $c$-Dirichlet
improvable if for all sufficiently large $N$

there exist ${\bf p}\in{\mathbb{Z}}^{m},\,{\bf
q}\in{\mathbb{Z}}^{n}\smallsetminus\\{0\\}$ and $i\in\\{1,\dots,k\\}$ such
that (10.4) $\displaystyle\qquad\qquad\qquad|Y_{i}{\bf q}-{\bf
p}\|<cN^{-n/m}\text{ and }0<\|{\bf q}\|<N.$

Denote the set of $c$-Dirichlet improvable $(Y_{1},\dots,Y_{k})$ by
$\mathbf{DI}_{m,n}^{(k)}(c)$. Applying Dirichlet’s theorem for each $k$, it is
easy to see that $\mathbf{DI}_{m,n}^{(k)}(1)=M_{n,m}^{k}$. When $c<1$ one
wants for each large $N$ to improve the conclusion of Dirichlet’s theorem for
at least one of the matrices, and for different $N$ it does not have to be the
same matrix. It is clearly if one of the matrices is itself $c$-Dirichlet
improvable, then so is the whole $k$-tuple; however in general
$\mathbf{DI}_{m,n}^{(k)}(c)$ could be much larger than the set

$\big{\\{}(Y_{1},\dots,Y_{k})\in
M_{n,m}^{k}:Y_{i}\in\mathbf{DI}_{m,n}(c)\text{ for some
}i=1,\dots,k\big{\\}}.$

This raises a problem of showing some sort of dimension drop, which is
achieved by reducing the problem to a flow on the product of $k$ copies of $X$
as in (1.2). Indeed, it is not hard to see that the validity of (10.4) for all
sufficiently large $N$ is equivalent to the statement that for all
sufficiently large $t$

${\exists\,{\bf{v}}\in{\mathbb{Z}}^{m+n}\smallsetminus\\{0\\}\text{ and
}i\in\\{1,\dots,k\\}\text{ with
}\|g_{t}h_{Y_{i}}{\bf{v}}\|<c^{\frac{m}{m+n}}.}$ (10.5)

In its turn, (10.5) is equivalent to

$(g_{t}h_{Y_{1}}{\mathbb{Z}}^{m+n},\dots,g_{t}h_{Y_{k}}{\mathbb{Z}}^{m+n})\notin
O\times\cdots\times O,$

where $O$ is as in (10.3). We conclude that
$(Y_{1},\dots,Y_{k})\in\mathbf{DI}_{m,n}^{(k)}(c)$ if and only if
$(h_{Y_{1}}{\mathbb{Z}}^{m+n},\dots,h_{Y_{k}}{\mathbb{Z}}^{m+n})\in\widetilde{E}(F^{(k)},O\times\cdots\times
O)$, where

$F^{(k)}:=\\{(g_{t},\dots,g_{t}):t\geq 0\\}\subset\prod_{i=1}^{k}G$

is acting on $X^{(k)}:=\prod_{i=1}^{k}X$.

Since $F^{(k)}$ is a diagonalizable subsemigroup of $\prod_{i=1}^{k}G$ whose
expanding horospherical subgroup is precisely

$H^{(k)}:=\prod_{i=1}^{k}\\{h_{Y}:Y\in M_{m,n}\\},$

it follows from Theorem 1.4 that $H^{(k)}$ has property (ENDP) with respect to
$(X^{(k)},F^{(k)})$. Moreover, since the action of $F$ on $X$ is exponentially
mixing, by using Fubini’s Theorem it is straightforward to check that the
action of $F^{(k)}$ on $X^{(k)}$ is exponentially mixing as well; hence, by
Theorem 1.2, $H^{(k)}$ has property (ENDP) with respect to
$(X^{(k)},F^{(k)})$. Therefore, we can apply Theorem 1.5 with $P=H^{(k)}$ and
arrive at:

######  Theorem 10.1.

The Hausdorff dimension of $\mathbf{DI}_{m,n}^{(k)}(c)$ is strictly less than
$kmn$ for any $c<1$ and $k\in{\mathbb{N}}$.

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