Effective equidistribution of horospherical flows in infinite volume rank-one homogeneous spaces

Abstract

We prove effective equidistribution of horospherical flows in $\operatorname {SO}(n,1)^{\circ } / \Gamma $ when $\Gamma $ is geometrically finite and the frame flow is exponentially mixing for the Bowen–Margulis–Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson–Sullivan measure satisfies some friendly like properties when $\Gamma $ is geometrically finite.

1 Introduction

The group $G = \operatorname {SO}(n,1)^{\circ }$ with $n\ge 2$ can be considered as the group of orientation preserving isometries of the hyperbolic space ${\mathbb H}^n$ . Let $\Gamma \subseteq G$ be a geometrically finite and Zariski dense subgroup of G with infinite covolume. In this paper, we establish an effective rate of equidistribution of orbits under the action of a horospherical subgroup $U \subseteq G$ under a certain exponential mixing assumption (Assumption 1.1).

An early result on the equidistribution of horocyclic flows in $G/\Gamma $ for $G= \operatorname {SL}_2(\mathbb R)$ and $\Gamma $ a lattice was obtained by Dani and Smillie in [4]. They proved that if $U=\{(\begin {smallmatrix}1 & t\\ 0 & 1 \end {smallmatrix}): t\in \mathbb R\}$ and if x does not have a closed U-orbit in $G/\Gamma $ , then for every $f \in C_c(X)$ ,

(1) $$ \begin{align} \lim\limits_{T \to \infty} \frac{1}{T} \int_0^T f(u_t x)\,dt = m(f), \end{align} $$

where m denotes the normalized Haar probability measure on X. The lattice case is well understood in general, thanks to Ratner’s celebrated theorems on unipotent flows [28].

Results such as these are not considered to be effective, because they do not address the rate of convergence, and this is important in many applications. Burger proved effective equidistribution of horocyclic flows for $\operatorname {SL}_2(\mathbb R)/\Gamma $ when $\Gamma $ is a uniform lattice or convex cocompact with critical exponent at least $1/2$ in [3]. Sarnak proved effective equidistribution of translates of closed horocycles when $\Gamma $ is a non-uniform lattice in [31]. More general results were obtained for non-uniform lattices using representation theoretic methods by Flaminio and Forni in [9], and also by Strömbergsson in [36]. The case where $\Gamma =\operatorname {SL}_2(\mathbb Z)$ was also obtained independently by Sarnak and Ubis in [32]. The higher dimensional setting has recently been considered by Katz [15] and McAdam [21]. McAdam proved equidistribution of abelian horospherical flows in $\operatorname {SL}_n(\mathbb R)/\Gamma $ for $n \ge 3$ when $\Gamma $ is a cocompact lattice or $\operatorname {SL}_n(\mathbb Z)$ , and Katz proved equidistribution in greater generality when $\Gamma $ is a lattice in a semi-simple linear group without compact factors.

In infinite volume, we cannot hope for a result such as (1) for the Haar measure: by the Hopf ratio ergodic theorem, for almost every point,

$$ \begin{align*} \lim\limits_{T \to \infty} \frac{1}{T}\int_0^T f(u_t x)\,dt = 0. \end{align*} $$

This tells us that this is not the correct measure to consider. A key characteristic of the Haar probability measure in the lattice case is that it is the unique U-invariant ergodic Radon measure that is not supported on a closed U orbit, [4, 10]. By [3, 29, 40], the measure with this property in the infinite volume setting is the Burger–Roblin (BR) measure, which is defined fully in §3. The correct normalization will be given by the Patterson–Sullivan (PS) measure, which is a geometrically defined measure on U orbits. This is also defined in §3.

Maucourant and Schapira proved equidistribution of horocycle flows on geometrically finite quotients of $\operatorname {SL}_2(\mathbb R)$ in [20], and in [24], Mohammadi and Oh generalize these results to geometrically finite quotients of $\operatorname {SO}(n,1)^{\circ }$ for $n\ge 2$ , but these results are not effective. Oh and Shah also proved equidistribution on the unit tangent bundle of geometrically finite hyperbolic manifolds in [26]. In [8], Edwards proves effective results for geometrically finite quotients of $\operatorname {SL}_2(\mathbb R)$ .

In this paper, we extend these results to geometrically finite quotients of $\operatorname {SO}(n,1)^{\circ }$ , under the assumption of exponential mixing of the frame flow for the Bowen–Margulis– Sullivan (BMS) measure, which is defined in §3. More explicitly, in §§6–8 (but not §4 or §5), we will assume the following holds, where $\{a_s : s \in \mathbb R\}$ denotes the frame flow on $G/\Gamma $ .

Assumption 1.1. (Exponential mixing)

There exist $c,\kappa>0$ and $\ell \in \mathbb N$ which depend only on $\Gamma $ , such that for $\psi ,\varphi \in C_c^{\infty }(G/\Gamma )$ and $s>0$ ,

$$ \begin{align*} \bigg|\int_{X}\psi(a_{s}x)\varphi(x)\,dm^{\operatorname{BMS}}(x)-m^{\operatorname{BMS}}(\psi)m^{\operatorname{BMS}}(\varphi)\bigg|<cS_{\ell}(\psi)S_{\ell}(\varphi)e^{-\kappa s}. \end{align*} $$

Assumption 1.1 is known to hold when $\Gamma $ is convex cocompact by [30]. In [23], Mohammadi and Oh prove such a result for geometrically finite $\Gamma $ under a spectral gap assumption (see Definition 2.1), using decay of matrix coefficients. Edwards and Oh recently proved effective mixing for the geodesic flow on the unit tangent bundle of a geometrically finite hyperbolic manifold when the critical exponent is larger than $(n-1)/2$ in [7]. Further details about this assumption are discussed in §2.

We will need to restrict consideration to points satisfying the following geometric property, which means that the point does not travel into a cusp ‘too fast’. Here, d is a left-invariant Riemannian metric on $G/\Gamma $ that projects to the hyperbolic distance on ${\mathbb H}^n$ .

Definition 1.2. For $0<\varepsilon <1$ and $s_0\ge 1$ , we say that $x\in G/\Gamma $ with $x^- \in \Lambda (\Gamma )$ is $(\varepsilon ,s_0)$ -Diophantine if for all $s\ge s_0$ ,

$$ \begin{align*} d(\mathcal{C}_0,a_{-s}x)<(1-\varepsilon)s, \end{align*} $$

where $\mathcal {C}_0$ is a compact set arising from the thick-thin decomposition, and is fully defined in §3.2. We say that $x\in G/\Gamma $ with $x^- \in \Lambda (\Gamma )$ is Diophantine if x is $(\varepsilon ,s_0)$ -Diophantine for some $\varepsilon $ and $s_0$ .

Here, $\Lambda (\Gamma )$ denotes the set of limit points of $\Gamma $ , and is defined fully in §3, as is the notation $x^{\pm }$ . In the case that $\Gamma $ is a lattice, the condition $x^- \in \Lambda (\Gamma )$ is always satisfied. Also, if $\Gamma $ is convex cocompact, every point $x \in G/\Gamma $ with $x^-\in \Lambda (\Gamma )$ will be Diophantine, because all limit points are radial in this case (see §3).

Note that x is $(\varepsilon ,s_0)$ -Diophantine if $(1-\varepsilon )s$ is a bound on the asymptotic excursion rate of the geodesic $\{a_{-s}x\}$ , that is,

(2) $$ \begin{align} \limsup_{s\rightarrow\infty} \frac{d(\mathcal{C}_0,a_{-s}x)}{s}\leq 1-\varepsilon. \end{align} $$

Sullivan’s logarithm law for geodesics when $\Gamma $ is geometrically finite with $\delta _{\Gamma }>(n-1)/2$ was shown in [16, 35] (and is a strengthening of Sullivan’s logarithm law for non-compact lattices [38, §9]), and implies that for almost all $x \in G/\Gamma $ ,

(3) $$ \begin{align} \limsup\limits_{s\to\infty} \frac{d(\mathcal{C}_0,a_{-s}x)}{\log s} = \frac{1}{2\delta_{\Gamma}-k}, \end{align} $$

where k is the maximal cusp rank. In [16], Kelmer and Oh showed a strengthening of the above, considering excursion to individual cusps and obtaining a limit for the shrinking target problem of the geodesic flow. Note also that the result stated in [16] is for $x\in {\operatorname {T}^1}(G/\Gamma $ ), but since the distance function there is assumed to be K-invariant, where ${\mathbb H}^n=K\backslash G$ , and the set $\mathcal {C}_0$ is K-invariant as well (see §3.2), we can deduce the form above.

It follows from (3) that the limit on the left-hand side of (2) is zero for almost every point $x\in G/\Gamma $ (with respect to the invariant volume measure) in this case. Moreover, for any $\varepsilon $ , the Hausdorff dimension of the set of directions in ${\operatorname {T}^1}({\mathbb H}^n/\Gamma )$ around a fixed point in ${\mathbb H}^n/\Gamma $ that do not satisfy (2) is computed in [22, Theorem 1]. For geometrically finite $\Gamma $ , the Hausdorff dimension of the set of directions around a fixed point that do not satisfy (2) can be found in [11, 35].

The main goal of this paper is to establish the following two theorems. Here, $m^{\operatorname {BR}}$ denotes the BR measure, $m^{\operatorname {BMS}}$ denotes the BMS measure, and $\mu ^{\operatorname {PS}}$ denotes the PS measure. These measures are defined in §3. Throughout the paper, the notation

$$ \begin{align*} x \ll y \end{align*} $$

means there exists a constant c such that

$$ \begin{align*} x \le cy. \end{align*} $$

If a subscript is denoted, e.g. $\ll _{\Gamma }$ , this explicitly indicates that this constant depends on $\Gamma $ .

Let $U = \{u_{\textbf {t}} : \textbf {t} \in \mathbb R^{n-1}\}$ denote the expanding horospherical flow. Let $B_U(r)$ denote the ball in U of radius r with the max norm on $\mathbb R^{n-1}$ . See §3 for more details on notation.

Theorem 1.3. Assume that $\Gamma $ satisfies Assumption 1.1. For any $0<\varepsilon <1$ and $s_0\ge 1$ , there exist constants $\ell = \ell (\Gamma ) \in \mathbb N$ and $\kappa = \kappa (\Gamma , \varepsilon )>0$ satisfying for every $\psi \in C_c^{\infty }(G/\Gamma )$ , there exists $c = c(\Gamma ,\operatorname {supp}\psi )$ such that for every $x\in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, and for every $r \gg _{\Gamma ,\varepsilon } s_0$ ,

$$ \begin{align*} \bigg|\frac{1}{\mu^{\operatorname{PS}}_x(B_U(r))} \int_{B_U(r)}\psi(u_{\textbf{t}} x)\,d\mu^{\operatorname{PS}}_x(\textbf{t}) - m^{\operatorname{BMS}}(\psi)\bigg| \le c S_{\ell}(\psi) r^{-\kappa}, \end{align*} $$

where $S_{\ell }(\psi )$ is the $\ell $ -Sobolev norm.

For the Haar measure, we will prove the following equidistribution result.

Theorem 1.4. Assume that $\Gamma $ satisfies Assumption 1.1. For any $0<\varepsilon <1$ and $s_0\ge 1$ , there exist $\ell =\ell (\Gamma )\in \mathbb N$ and $\kappa =\kappa (\Gamma ,\varepsilon )>0$ satisfying for every $\psi \in C_c^{\infty }(G/\Gamma )$ , there exists $c = c(\Gamma ,\operatorname {supp}\psi )$ such that for every $x\in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, and for all $r \gg _{\Gamma ,\operatorname {supp}\psi ,\varepsilon } s_0$ ,

$$ \begin{align*} \bigg|\frac{1}{\mu^{\operatorname{PS}}_x(B_U(r))} \int_{B_U(r)}\psi(u_{\textbf{t}} x)\,d\textbf{t} - m^{\operatorname{BR}}(\psi)\bigg| \le c S_{\ell}(\psi)r^{-\kappa}, \end{align*} $$

where $S_{\ell }(\psi )$ is the $\ell $ -Sobolev norm.

Note that the assumption that x is Diophantine is required to obtain quantitative non-divergence results in §4, which is key in proving the above theorems. The dependence on a Diophantine condition is necessary, and is analogous to known effective equidistribution results for when $\Gamma $ is a non-cocompact lattice (see [21, 36]).

In [39], we apply the above result to obtain a quantitative ratio theorem for the distribution of orbits for $\Gamma $ acting on $U \backslash G$ . This improves upon the work of Maucourant and Schapira in [20].

A key step toward proving Theorem 1.3 is the following, which is proved in §6.

Theorem 1.5. Assume that $\Gamma $ satisfies Assumption 1.1. There exist $\kappa = \kappa (\Gamma )$ and $\ell = \ell (\Gamma )$ which satisfy the following: for any $\psi \in C_c^{\infty }(X)$ , there exists $c = c(\Gamma , \operatorname {supp}\psi )>0$ such that for any $f \in C_c^{\infty }(B_U(r))$ , $0<r<1$ , $x \in \operatorname {supp}m^{\operatorname {BMS}}$ , and $s\gg _{\Gamma } d(\mathcal {C}_0,x)$ , we have

$$ \begin{align*} \bigg|\int_U \psi(a_s u_{\textbf{t}} x)f(\textbf{t})\,d\mu^{\operatorname{PS}}_x(\textbf{t}) - \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BMS}}(\psi)\bigg| < cS_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s}. \end{align*} $$

In §6, we also prove an analogous statement for the Haar measure. Such a result is proven in [23] under a spectral gap assumption on $\Gamma $ , but we show in this paper how to prove it whenever the frame flow is exponentially mixing.

The proof will use similar techniques as in [24, 26]; in particular, we will rely on Margulis’ ‘thickening trick’ from his thesis [19].

In the proofs of our main theorems (Theorems 1.3 and 1.4), we use partition of unity arguments. In particular, the bounds we get are on slightly bigger sets. As a result, we need an effective bound on the PS measure of a small neighborhood of a boundary of a ball relative to the PS measure of that ball. The following theorem achieves this. It is shown using [5, Lemma 3.8] and [35, Theorem 2].

Theorem 1.6. There exists a constant $\alpha = \alpha (\Gamma )>0,$ such that for every $x \in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, for every $0<s\le T^{{\varepsilon }/({1-\varepsilon })}$ , every $0<\xi \ll _{\Gamma } 1,$ and every $T\gg _{\Gamma ,\varepsilon } s_0$ ,

$$ \begin{align*} \frac{\mu^{\operatorname{PS}}_{a_{-s}x}(B_U(\xi+T) )}{\mu^{\operatorname{PS}}_{a_{-s}x}(B_U(T))}-1\ll_{\Gamma} \xi^{\alpha}.\end{align*} $$

In the appendix, a stronger version is obtained under the assumption that all cusps of $G/\Gamma $ have maximal rank.

This paper is organized as follows. In §2, we discuss under what conditions Assumption 1.1 is known to hold. In §3, we set out notation used in the article, and define the measures we will be using, along with proving some important facts about them. In §4, we prove quantitative non-divergence of horospherical orbits of Diophantine points, which is needed in the following sections. In §5, we control the PS measure of the boundary of a set by proving Theorem 1.6. In §6, we use Margulis’ ‘thickening trick’ to prove Theorem 1.5 and an analogous result for the Haar measure, which are key in the proofs of Theorems 1.3 and 1.4. In §7, we use quantitative non-divergence and Theorem 1.5 to prove Theorem 1.3. In §8, we use Theorem 1.6 and the Haar measure analogue of Theorem 1.5 to prove Theorem 1.4. Several technical details of the proof of Theorem 1.6 are in the appendix, §A, and in §A.3, we prove stronger statements hold in the setting that all cusps have maximal rank, because the PS measure is absolutely friendly (see [5, Theorem 1.9]).

2 Known exponential mixing results

Throughout the paper, we assume the existence of an exponential mixing result (see Assumption 1.1). In this section, we elaborate on the conditions under which such a result is known. Here we assume that $\Gamma $ is a Zariski dense discrete subgroup of G.

There is a natural action of G on ${\mathbb H}^n$ and $\partial {\mathbb H}^n$ , the hyperbolic n-space and its boundary, respectively. Let $\Lambda (\Gamma )\subseteq \partial ({\mathbb H}^n)$ denote the limit set of X, that is, the set of all accumulation points of $\Gamma z$ for some $z\in {\mathbb H}^n\cup \partial ({\mathbb H}^n)$ . The convex core of X is the image in X of the minimal convex subset of ${\mathbb H}^n$ which contains all geodesics connecting any two points in $\Lambda (\Gamma )$ . We say that $\Gamma $ is convex cocompact if the convex core of ${\mathbb H}^n /\Gamma $ is compact, and geometrically finite if a unit neighborhood of the convex core of $\Gamma $ has finite volume.

For $\Gamma $ convex cocompact, Assumption 1.1 was proved by Sarkar and Winter in [30, Theorem 1.1].

Fix a point $w_o \in {\operatorname {T}^1}({\mathbb H}^n)$ and denote $M = \operatorname {Stab}_G(w_o)$ . Denote by $\hat {G}$ and $\hat {M}$ the unitary dual of G and M respectively. A representation $(\pi ,\mathcal {H})\in \hat {G}$ is called tempered if for any K-finite $v\in \mathcal {H}$ , the associated matrix coefficient function $g\mapsto \langle \pi (g)v,v\rangle $ belongs to $L^{2+\varepsilon }(G)$ for any $\varepsilon>0$ , and non-tempered otherwise. The non-tempered part of $\hat {G}$ consists of the trivial representation, and complementary series representations $\mathcal {U}(v,s-n+1)$ parameterized by $v\in \hat {M}$ and $s\in I_v$ , where $I_v\subseteq (({n-1})/{2},n-1)$ is an interval depending on v (see [12]).

Definition 2.1. The space $L^2(X)$ has a spectral gap if there exist $({n-1})/{2}<s_0=s_0(\Gamma )<\delta $ and $n_0=n_0(\Gamma )\in \mathbb {N}$ such that:

1. (1) the multiplicity of $\mathcal {U}(v,\delta _{\Gamma }-n+1)$ contained in $L^2(X)$ is at most $dim(v)^{n_0}$ for any $v\in \hat {M}$ ;

2. (2) $L^2(X)$ does not weakly contain any $\mathcal {U}(v,s-n+1)$ with $s\in (s_0,\delta )$ and $v\in \hat {M}$ .

According to [23, Theorem 3.27], if $\delta _{\Gamma }>({n-1})/{2}$ for $n=2,3$ , or if $\delta _{\Gamma }>n-2$ for $n\ge 4$ , then $L^2(X)$ has a spectral gap. If $\delta _{\Gamma }\leq ({n-1})/{2}$ , then there is no spectral gap, but it was conjectured that whenever $\delta _{\Gamma }>({n-1})/{2}$ , $L^2(X)$ has a spectral gap (see [23]). Note that if there are cusps of maximal rank $n-1$ , it follows that $\delta _{\Gamma }>({n-1})/{2}$ .

For $\Gamma $ geometrically finite such that $L^2(X)$ has a spectral gap and $\delta _{\Gamma }> ({n-1})/{2}$ , Mohammadi and Oh stated in [23, Theorem 1.6] an exponential mixing result similar to Assumption 1.1. In their statement, the constant c depends on $\Gamma $ and the support of the functions. The dependence on the support of the functions arises in the last part of the proof (see [23, §6.3]) and can be omitted by using the following lemma (the BR-measure is defined in §3.3), hence obtaining a result of the form needed in Assumption 1.1.

Lemma 2.2. If $\delta>(n-1)/2$ , then there exists $c=c(\Gamma )>0$ such that any $B\subset X$ of diameter smaller than $1$ satisfies

$$ \begin{align*}m^{\operatorname{BR}}(B)\le c.\end{align*} $$

Proof. For any $g\in G$ denote

$$ \begin{align*} \Phi_0(g)=|\nu_{g(o)}|, \end{align*} $$

where o is the projection of $w_o$ onto ${\mathbb H}^n$ and for any $x\in {\mathbb H}^n$ , $\nu _x$ is the Patterson–Sullivan density defined in §3.1. Since $\Phi _0$ is $\Gamma $ -invariant, it can be considered as a smooth function on X. Moreover, by assuming B contains $K=\operatorname {Stab}_G(o)$ and using the Cauchy–Schwartz inequality, we get

$$ \begin{align*} m^{\operatorname{BR}}(B)&=\int_{B}\Phi_0(g)\,dm^{\operatorname{Haar}}(g)\\ &\leq\sqrt{dm^{\operatorname{Haar}}(B)}\Vert \Phi_0\Vert _2\\ &\ll\Vert \Phi_0\Vert _2. \end{align*} $$

According to [37, §7] and by the assumption $\delta>(n-1)/2$ , we have that $\phi _0\in L^2(X)$ .

3 Notation and preliminaries

Recall from §1 that $G = \operatorname {SO}(n,1)^{\circ }$ and $\Gamma \subseteq G$ is a geometrically finite Kleinian subgroup of G. Denote

$$ \begin{align*}X:=G/\Gamma.\end{align*} $$

Here, G acts transitively on ${\mathbb H}^n$ , the hyperbolic n-space. Fix a reference point $o\in {\mathbb H}^n$ and let $K=\text {Stab}_G(o)$ , then $K\backslash G=\mathbb {H}^n$ . Let $\pi :G\rightarrow {\mathbb H}^n$ be the projection

(4) $$ \begin{align} \pi(g)=g(o).\end{align} $$

We will abuse notation and also write $\pi $ for the induced map from $G/\Gamma $ to ${\mathbb H}^n/\Gamma $ . For convenience, we will assume throughout the paper that we have chosen o so that $o\Gamma \in \pi (\mathcal {C}_0)$ , where $\mathcal {C}_0$ is defined in §3.2. This says that $o\Gamma $ is in the convex core of ${\mathbb H}^n/\Gamma $ .

Let d denote the left G-invariant metric on G which induces the hyperbolic metric on $K\backslash G=\mathbb {H}^n$ .

Recall from §2 that $\Lambda (\Gamma )\subseteq \partial ({\mathbb H}^n)$ denotes the limit set of X. We denote the Hausdorff dimension of $\Lambda (\Gamma )$ by $\delta _{\Gamma }$ . It is equal to the critical exponent of $\Gamma $ (see [27]).

We say that a limit point $\xi \in \Lambda (\Gamma )$ is radial if there exists a compact subset of X so that some (and hence every) geodesic ray toward $\xi $ has accumulation points in that set. An element $g\in G$ is called parabolic if the set of fixed points of g in $\partial (\mathbb {H}^n)$ is a singleton. We say that a limit point is parabolic if it is fixed by a parabolic element of $\Gamma $ . A parabolic limit point $\xi \in \Lambda (\Gamma )$ is called bounded if the stabilizer $\Gamma _{\xi }$ acts cocompactly on $\Lambda (\Gamma )-\{\xi \}$ .

We denote by $\Lambda _r (\Gamma )$ and $\Lambda _{bp} (\Gamma )$ the set of all radial limit points and the set of all bounded parabolic limit points respectively. Since $\Gamma $ is geometrically finite (see [2]),

$$ \begin{align*}\Lambda (\Gamma)=\Lambda_r (\Gamma)\cup\Lambda_{bp} (\Gamma).\end{align*} $$

Fix $w_o \in {\operatorname {T}^1}({\mathbb H}^n)$ and let $M = \operatorname {Stab}_G(w_o)$ so that ${\operatorname {T}^1}({\mathbb H}^n)$ may be identified with $M\backslash G$ . For $w \in {\operatorname {T}^1}({\mathbb H}^n)$ ,

$$ \begin{align*}w^{\pm} \in \partial{\mathbb H}^n\end{align*} $$

denotes the forward and backward endpoints of the geodesic w determined. For $g \in G$ , we define

$$ \begin{align*}g^{\pm} := w_o^{\pm} g.\end{align*} $$

Without loss of generality, we may assume that $w_o^{\pm }\in \Lambda (\Gamma )$ , and hence every $\gamma \in \Gamma $ will satisfy $\gamma ^{\pm }\in \Lambda (\Gamma ).$

Let $A=\{a_s : s\in \mathbb R\}$ be a one parameter diagonalizable subgroup such that M and A commute, and such that the right action on $M\backslash G={\operatorname {T}^1}({\mathbb H}^n)$ corresponds to unit speed geodesic flow. We parameterize A by $A=\{a_s : s\in \mathbb R\}$ , where

(5) $$ \begin{align} a_s = \begin{pmatrix} e^{s} & & \\ & I & \\ & & e^{-s} \end{pmatrix} \end{align} $$

and I denotes the $(n-1)\times (n-1)$ identity matrix.

Let U denote the expanding horospherical subgroup

$$ \begin{align*} U=\{g\in G : a_{-s}ga_{s}\rightarrow e\text{ as }s\rightarrow+\infty \},\end{align*} $$

let $\tilde U$ be the contracting horospherical subgroup

$$ \begin{align*} \tilde U=\{g\in G : a_{s}ga_{-s}\rightarrow e\text{ as }s\rightarrow+\infty \},\end{align*} $$

and let $P=MA\tilde U$ be the parabolic subgroup.

The group U is a connected abelian group, isomorphic to $\mathbb R^{n-1}$ . We may use the parameterization $\textbf {t}\mapsto u_{\textbf {t}}$ so that for any $s\in \mathbb R$ ,

(6) $$ \begin{align} a_{s}u_{\textbf{t}} a_{-s}=u_{e^s\textbf{t}}.\end{align} $$

Similarly, we parameterize $\tilde U$ by $\textbf {t} \mapsto v_{\textbf {t}} \in \tilde {U}$ so that for $s \in \mathbb R$ ,

(7) $$ \begin{align} a_{s}v_{\textbf{t}} a_{-s} = v_{e^{-s}\textbf{t}} .\end{align} $$

More explicitly, if $\textbf {t} \in \mathbb R^{n-1}$ is viewed as a row vector,

(8) $$ \begin{align} u_{\textbf{t}} = \begin{pmatrix} 1 & \textbf{t} & \frac{1}{2}\Vert \textbf{t}\Vert ^2 \\ & I & \textbf{t}^T \\ & & 1 \end{pmatrix} \end{align} $$

and

$$ \begin{align*}v_{\textbf{t}} = \begin{pmatrix} 1 & & \\ \textbf{t}^T & I & \\ \frac{1}{2}\|\textbf{t}\|^2 & \textbf{t} & 1 \end{pmatrix}.\end{align*} $$

For a subset H of G and $\eta>0$ , $H_{\eta }$ denotes the closed $\eta $ -neighborhood of e in H, that is,

$$ \begin{align*} H_{\eta}=\{h\in H : d(h,e)\le\eta\}.\end{align*} $$

For any $r>0$ , let

$$ \begin{align*} B_U (r)=\{u_{\textbf{t}} : \Vert \textbf{t}\Vert \le r \}\quad\mbox{and}\quad B_{\tilde{U}} (r)=\{v_{\textbf{t}} : \Vert \textbf{t}\Vert \le r \},\end{align*} $$

where $\Vert \textbf {t}\Vert $ is the sup-norm of $\textbf {t}\in \mathbb R^{n-1}$ .

Lemma 3.1. For $0<\eta <1/4$ and $p \in P_{\eta }$ , there exists $\rho _p : B_U(1) \to B_U(1+O(\eta ))$ that is a diffeomorphism onto its image and a constant $D=D(\eta )<3\eta $ such that

$$ \begin{align*}u_{\textbf{t}} p^{-1} \in P_D u_{\rho_p(\textbf{t})}.\end{align*} $$

Explicitly, if $p = a_s v_{\textbf {r}}$ , then $\rho _p(\textbf {t}) = ({\textbf {t} - \tfrac 12\|\textbf {t}\|^2\textbf {r}})/{e^s(1-(\textbf {t}\cdot \textbf {r})+\tfrac 14\|\textbf {r}\|^2\|\textbf {t}\|^2)}$ .

Proof. For $s \in \mathbb R$ and $\textbf {r} \in \mathbb R^{n-1}$ , let $p = a_sv_{\textbf {r}}$ . Then

$$\begin{align*}p^{-1} = \begin{pmatrix} e^{-s} & & \\[3pt] -e^{-s} \textbf{r}^T & I & \\[3pt] \tfrac12e^{-s}\|\textbf{r}\|^2 & -\textbf{r} & e^{s} \end{pmatrix},\end{align*}$$

so

$$ \begin{align*}u_{\textbf{t}} p^{-1} = \begin{pmatrix} e^{-s}(1-(\textbf{t}\cdot \textbf{r})+\frac{1}{4}\|\textbf{r}\|^2\|\textbf{t}\|^2) & \textbf{t} - \frac{1}{2} \|\textbf{t}\|^2 \textbf{r} & \frac{1}{2}e^{s}\|\textbf{t}\|^2 \\[3pt] -e^{-s}\textbf{r}^T + \frac{1}{2}e^{-s}\|\textbf{r}\|^2\textbf{t}^T & I - \textbf{t}^T \textbf{r} & e^{s}\textbf{t}^T \\[3pt] \frac{1}{2}e^{-s}\|\textbf{r}\|^2 & -\textbf{r} & e^{s}\end{pmatrix}.\end{align*} $$

Now, if $p' = a_{s'}v_{\textbf {r}'},$ we obtain that

$$ \begin{align*}p'u_{\textbf{t}'} = \begin{pmatrix} e^{s'} & e^{s'}\textbf{t}' & \frac{1}{2}e^{s'}\|\textbf{t}'\|^2 \\[3pt] \textbf{r}^{\prime T} & \textbf{r}^{\prime T}\textbf{t}' + I & \frac{1}{2}\|\textbf{t}'\|^2\textbf{r}^{\prime T} + \textbf{t}^{\prime T} \\[3pt] \frac{1}{2}e^{-s'}\|\textbf{r}'\|^2 & \frac{1}{2}e^{-s'}\|\textbf{r}'\|^2\textbf{t}' + e^{-s'}\textbf{r}' & e^{-s'}(\frac{1}{4}\|\textbf{r}'\|^2\|\textbf{t}'\|^2 +(\textbf{r}'\cdot \textbf{t}')+1)\end{pmatrix}.\end{align*} $$

We wish to solve for $\textbf {t}'$ .

Setting entries equal yields

$$ \begin{align*}\textbf{t} - \tfrac{1}{2}\|\textbf{t}\|^2\textbf{r} = e^{s'}\textbf{t}'\end{align*} $$

and

(9) $$ \begin{align} e^{s'}=e^{-s}\big(1-(\textbf{t}\cdot\textbf{r})+\tfrac{1}{4}\|\textbf{r}\|^2\|\textbf{t}\|^2\big).\end{align} $$

Combining these implies that

$$ \begin{align*}\textbf{t}' = \frac{\textbf{t} - {1}/{2}\|\textbf{t}\|^2\textbf{r}}{e^s(1-(\textbf{t}\cdot\textbf{r})+{1}/{4}\|\textbf{r}\|^2\Vert \textbf{t}\Vert ^2)}.\end{align*} $$

We define $\rho _p(\textbf {t})$ to be this quantity. One can directly check that it satisfies the claim.

3.1 Patterson–Sullivan and Lebesgue measures

For $x, y \in {\mathbb H}^n$ and $\xi \in \partial ({\mathbb H}^n)$ , the Busemann function is given by

$$ \begin{align*} \beta_{\xi}(x,y):=\lim\limits_{t\to \infty} d(x,\xi_t) - d(y,\xi_t),\end{align*} $$

where $\xi _t$ is a geodesic ray towards $\xi $ .

A family of finite measures $\{\mu _x : x\in {\mathbb H}^n\}$ on $\partial ({\mathbb H}^n)$ is called a $\Gamma $ -invariant conformal density of dimension $\delta _{\mu }>0$ if for every $x,y\in {\mathbb H}^n$ , $\xi \in \partial ({\mathbb H}^n)$ , and $\gamma \in \Gamma $ ,

(10) $$ \begin{align} \gamma_*\mu_x=\mu_{x\gamma}\quad\text{ and }\quad\frac{d\mu_y}{d\mu_x}(\xi)=e^{-\delta_{\mu} \,\beta_{\xi}(y,x)}, \end{align} $$

where $\gamma _*\mu _x(F)=\mu _x(F\gamma )$ for any Borel subset F of $\partial ({\mathbb H}^n)$ .

We let $\{\nu _x\}_{x \in {\mathbb H}^n}$ denote the Patterson–Sullivan density on $\partial {\mathbb H}^n$ , that is, the unique (up to scalar multiplication) conformal density of dimension $\delta _{\Gamma }$ .

For each $x\in {\mathbb H}^n$ , we denote by $m_x$ the unique probability measure on $\partial ({\mathbb H}^n)$ which is invariant under the compact subgroup $\text {Stab}_G (x)$ . Then $\{m_x : x\in {\mathbb H}^n\}$ forms a G-invariant conformal density of dimension $n-1$ , called the Lebesgue density. Fix $o \in {\mathbb H}^n$ .

For $g\in G$ , we can define measures on $Ug$ using the conformal densities defined previously. The Patterson–Sullivan measure (abbreviated as the PS measure)

(11) $$ \begin{align} d\mu^{\operatorname{PS}}_{Ug}(u_{\textbf{t}} g) := e^{\delta_{\Gamma} \beta_{(u_{\textbf{t}} g)^+}(o, u_{\textbf{t}} g(o))}d\nu_o((u_{\textbf{t}} g)^+), \end{align} $$

and the Lebesgue measure

$$ \begin{align*} \mu^{\operatorname{Leb}}_{Ug}(u_{\textbf{t}} g):=e^{(n-1) \beta_{(u_{\textbf{t}} g)^+}(o, u_{\textbf{t}} g(o))}dm_o((u_{\textbf{t}} g)^+).\end{align*} $$

We similarly define the opposite PS measure on $\tilde Ug$ :

(12) $$ \begin{align} d\mu^{\operatorname{PS}-}_{\tilde Ug}(v_{\textbf{t}} g) := e^{\delta_{\Gamma} \beta_{(v_{\textbf{t}} g)^-}(o,v_{\textbf{t}} g(o))}d\nu_o((v_{\textbf{t}} g)^-).\end{align} $$

The conformal properties of $m_x$ and $\nu _x$ imply that these definitions are independent of the choice of $o\in {\mathbb H}^n$ .

We often view $\mu ^{\operatorname {PS}}_{Ug}$ as a measure on U via

$$ \begin{align*}d\mu^{\operatorname{PS}}_{g}(\textbf{t}):= d\mu^{\operatorname{PS}}_{Ug}(u_{\textbf{t}} g),\end{align*} $$

and similarly for $\mu ^{\operatorname {PS}-}_{\tilde U g}$ on $\tilde U$ . For $g \in G, s \in \mathbb R$ and $E\subseteq U$ a Borel subset (or $E \subseteq \tilde U$ for $\mu ^{\operatorname {PS}-}$ ), these measures satisfy

(13) $$ \begin{align} & \mu^{\operatorname{Leb}}_g(E) = e^{(n-1)s}\mu^{\operatorname{Leb}}_{a_{-s}g}(a_{-s}E a_{s}), \end{align} $$

(14) $$ \begin{align} & \mu^{\operatorname{PS}}_g(E) = e^{\delta_{\Gamma} s}\mu^{\operatorname{PS}}_{a_{-s}g}(a_{-s} E a_{s}), \end{align} $$

(15) $$ \begin{align} & \mu^{\operatorname{PS}-}_g(E) = e^{\delta_{\Gamma} s}\mu^{\operatorname{PS}-}_{a_{s}g}(a_{s} E a_{-s}). \end{align} $$

In particular,

$$ \begin{align*}\mu^{\operatorname{PS}}_g(B_U(e^s)) = e^{\delta_{\Gamma} s} \mu^{\operatorname{PS}}_{a_{-s}g}(B_U(1))\quad\text{and}\quad\mu^{\operatorname{PS}-}_g(B_{U^-}(e^{s})) = e^{\delta_{\Gamma} s } \mu^{\operatorname{PS}-}_{a_s g} (B_U(1)).\end{align*} $$

The measure

$$ \begin{align*} d\mu^{\operatorname{Leb}}_{Ug}(u_{\textbf{t}} g)=d\mu^{\operatorname{Leb}}_U(u_{\textbf{t}})=d\textbf{t}\end{align*} $$

is independent of the orbit $Ug$ and is simply the Lebesgue measure on $U\equiv \mathbb R^{n-1}$ up to a scalar multiple.

We will need the following fundamental results, which are stated for $\mu ^{\operatorname {PS}}$ and U, but also hold if we replace them with $\mu ^{\operatorname {PS}-}$ and $\tilde U$ .

Lemma 3.2. The map $g \mapsto \mu ^{\operatorname {PS}}_g$ is continuous, where the topology on the space of regular Borel measures on U is given by $\mu _n \to \mu \iff \mu _n(f)\to \mu (f)$ for all $f\in C_c(U)$ .

Proof. This is clear from the definition of the PS measure, since it is defined using the Busemann function and stereographic projection.

Corollary 3.3. For any compact set $\Omega \subseteq G$ and any $r>0$ ,

$$ \begin{align*} 0<\inf\limits_{g \in \Omega, g^+ \in \Lambda(\Gamma)}\mu^{\operatorname{PS}}_g(B_U(r)g) \le \sup\limits_{g\in \Omega, g^+\in \Lambda(\Gamma)} \mu^{\operatorname{PS}}_g(B_U(r)g) < \infty. \end{align*} $$

To define the PS measure on $Ux$ for $x \in X,$ note that

(16) $$ \begin{align} \text{if }x^- \in \Lambda_r(\Gamma),\quad\text{then } u \mapsto ux \text{ is injective,}\end{align} $$

and we can define the PS measure on $Ux \subseteq X$ , denoted $\mu ^{\operatorname {PS}}_x$ , simply by pushforward of $\mu ^{\operatorname {PS}}_g$ , where $x = g\Gamma $ . In general, defining $\mu ^{\operatorname {PS}}_x$ requires more care, see e.g. [24, §2.3] for more details. As before, we can view $\mu ^{\operatorname {PS}}_x$ as a measure on U via

$$ \begin{align*} d\mu^{\operatorname{PS}}_x(\textbf{t})=d\mu^{\operatorname{PS}}_x(u_{\textbf{t}}x).\end{align*} $$

3.2 Thick–thin decomposition and the height function

There exists a finite set of $\Gamma $ -representatives $\xi _1,\ldots ,\xi _q\in \Lambda _{bp}(\Gamma )$ . For $i=1,\ldots ,q$ , fix $g_i\in G$ such that $g_i^- =\xi _i$ , and for any $R>0$ , set

(17) $$ \begin{align} \mathcal{H}_i(R):=\bigcup_{s>R} Ka_{-s} U g_i\quad\mbox{and}\quad\mathcal{X}_i(R):=\mathcal{H}_i(R)\Gamma \end{align} $$

(recall, $K=\text {Stab}_G (o)$ ). Each $\mathcal {H}_i(R)$ is a horoball of depth R.

The rank of $\mathcal {H}_i(R)$ is the rank of the finitely generated abelian subgroup $\Gamma _{\xi _i}=\operatorname {Stab}_{\Gamma }(\xi _i)$ . We say that the cusp has maximal rank if $\operatorname {rank}\Gamma _{\xi }=n-1$ . It is known that each rank is strictly smaller than $2\delta _{\Gamma }$ .

We denote

$$ \begin{align*}\operatorname{supp}m^{\operatorname{BMS}}:=\{g\Gamma\in X : g^{\pm}\in\Lambda(\Gamma)\}.\end{align*} $$

(For now, this is simply notation. The measure $m^{\operatorname {BMS}}$ will be defined in the next section, and this set is its support. It projects onto the convex core of ${\mathbb H}^n/\Gamma .$ ) Note that the condition $g^{\pm } \in \Lambda (\Gamma )$ is independent of the choice of representative of $x=g\Gamma $ in the above definition, because $\Lambda (\Gamma )$ is $\Gamma $ -invariant. Thus, the notation $x^{\pm } \in \Lambda (\Gamma )$ is well defined, even though $x^{\pm }$ itself is not.

According to [2], there exists $R_0\geq 1$ such that $\mathcal {X}_1(R_0),\ldots ,\mathcal {X}_q(R_0)$ are disjoint, and for some compact set $\mathcal {C}_0\subset G/\Gamma $ ,

$$ \begin{align*} \operatorname{supp}m^{\operatorname{BMS}}\subseteq\mathcal{C}_0\sqcup\mathcal{X}_1(R_0)\sqcup\cdots\sqcup\mathcal{X}_q(R_0).\end{align*} $$

For $1\le i\le q$ and $R\ge R_0$ , denote

$$ \begin{align*} {\mathcal{X}}(R):=\mathcal{X}_1(R)\sqcup\cdots\sqcup\mathcal{X}_q(R),\quad {\mathcal{C}}(R):=\operatorname{supp}m^{\operatorname{BMS}}-{\mathcal{X}}(R).\end{align*} $$

We will need a version of Sullivan’s shadow lemma, obtained by Schapira–Maucourant (see Proposition 5.1 and Remark 5.2 in [20]).

Proposition 3.4. There exists a constant $\unicode{x3bb} =\unicode{x3bb} (\Gamma )\ge 1$ such that for all $x \in \operatorname {supp}m^{\operatorname {BMS}}$ and all $T>0$ , we have

(18) $$ \begin{align} \kern-100pt \unicode{x3bb}^{-1} T^{\delta_{\Gamma}}e^{(k_1(x,T)-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(a_{-\log T}x))} &\le \mu^{\operatorname{PS}}_x(B_U(T)) \end{align} $$

(19) $$ \begin{align} & \kern140pt \le \unicode{x3bb} T^{\delta_{\Gamma}}e^{(k_1(x,T)-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(a_{-\log T}x))}\end{align} $$

and

(20) $$ \begin{align} \unicode{x3bb}^{-1} T^{\delta_{\Gamma}}e^{(k_2(x,T)-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(a_{\log T}x))} &\le \mu^{\operatorname{PS}-}_x(B_{\tilde U}(T)) \nonumber\\ &\le \unicode{x3bb} T^{\delta_{\Gamma}}e^{(k_2(x,T)-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(a_{\log T}x))}, \end{align} $$

where $k_1(x,T)$ is the rank of $\mathcal {X}_i(R_0)$ if $a_{-\log T}x\in \mathcal {X}_i(R_0)$ for some $1\le i\le \ell $ and equals 0 if $a_{-\log T}x\in \mathcal {C}_0$ , and $k_2(x,T)$ is defined analogously for $a_{\log T}x$ . Recall the definition of $\pi $ from (4) as the projection from G to ${\mathbb H}^n$ .

Definition 3.5. For $x \in G/\Gamma $ , we define the height of x by

(21) $$ \begin{align} \operatorname{height}(x)=d(\pi(\mathcal{C}_0),\pi(x)), \end{align} $$

where $\pi :G/\Gamma \to {\mathbb H}^n/\Gamma $ is the projection map as in (4), recalling that ${\mathbb H}^n/\Gamma \cong K\backslash G/\Gamma $ .

Lemma 3.6. For any $x\in \operatorname {supp}m^{\operatorname {BMS}}$ and $R\ge R_0$ , we have that

$$ \begin{align*}x\in\mathcal{C}(R)\iff\operatorname{height}(x)\le R-R_0.\end{align*} $$

Proof. The claim follows from the disjointness of $\mathcal {X}_i(R_0)$ , $1\le i\le q$ from $\mathcal {C}_0$ , and the fact that $\mathcal {X}_i(R)\subseteq \mathcal {X}_i(R_0)$ .

If $x\in \mathcal {C}(R)$ , then either $x\in \mathcal {C}_0$ , in which case $\operatorname {height}(x)=0$ and we are done, or $x\in \mathcal {X}_i(R_0)$ . Assume the latter, then the Busemann function between x and the boundary of $\mathcal {X}_i(R_0)$ (which intersects $\mathcal {C}_0)$ is at most $R-R_0$ . Thus, we may deduce the claim in this case.

Next, assume $x\in \mathcal {X}_i(R)$ for some i. The Busemann function between two points in different horoballs is at least $R-R_0$ . Since a point from $\mathcal {X}_i(R)$ cannot go into $\mathcal {C}_0$ without passing through $\mathcal {X}_i(R_0),$ this is a lower bound for the distance between the base points, that is, the height.

Corollary 3.7. Let $x \in G/\Gamma $ be $(\varepsilon ,s_0)$ -Diophantine. Then

$$ \begin{align*}\operatorname{height}(x) < (2-\varepsilon)s_0.\end{align*} $$

Proof. By Definition 1.2,

$$ \begin{align*}d(\mathcal{C}_0, a_{-s_0}x)<(1-\varepsilon)s_0.\end{align*} $$

Hence, we have that

$$ \begin{align*} \operatorname{height}(x)&\le d(\mathcal{C}_0,x)\\ &< d(\mathcal{C}_0, a_{-s_0}x)+d(a_{-s_0}x,x) \\ &< (1-\varepsilon)s_0+s_0.\\[-2.8pc] \end{align*} $$

The injectivity radius at $x \in X$ is defined to be the supremum over all $\varepsilon>0$ such that the map

$$ \begin{align*}h\mapsto hx \text{ is injective on }G_{\varepsilon}.\end{align*} $$

We denote the injectivity radius at x by

$$ \begin{align*}\operatorname{inj}(x).\end{align*} $$

The injectivity radius of a set $\Omega $ is defined to be

$$ \begin{align*}\inf_{x \in \Omega} \operatorname{inj}(x).\end{align*} $$

By the proof of [25, Proposition 6.7], there exists a constant $\sigma = \sigma (\Gamma )> 0$ such that for all $x \in \operatorname {supp}m^{\operatorname {BMS}}$ ,

(22) $$ \begin{align} \sigma^{-1}\operatorname{inj}(x) \le e^{-\operatorname{height}(x)} \le {\sigma} \operatorname{inj}(x). \end{align} $$

The following fact is well known, but we include a proof for completion.

Lemma 3.8. There exists $T_0=T_0(\Gamma )>0$ which satisfies the following. Let $x \in G/\Gamma $ with $x^- \in \Lambda (\Gamma )$ , and let $R>0$ be such that $d(\mathcal {C}_0,x)<R.$ Then there exists $\textbf {t} \in B_U(2(R+T))$ such that

$$ \begin{align*}(u_{\textbf{t}} x)^{\pm}\in\Lambda(\Gamma).\end{align*} $$

In particular, for every $0<\varepsilon <1$ , $s_0\ge 1$ , and $(\varepsilon ,s_0)$ -Diophantine point x, there exists $|\textbf {t}|\ll _{\Gamma } s_0$ such that

$$ \begin{align*}(u_{\textbf{t}} x)^{\pm}\in\Lambda(\Gamma).\end{align*} $$

Proof. Let $g,h' \in G$ be such that $x=g\Gamma $ , $h^{\prime -} =g^-$ , $h'\Gamma \in K\mathcal {C}_0$ , and

$$ \begin{align*}d(g,h')\le\operatorname{height}(x)<R.\end{align*} $$

Since $K\mathcal {C}_0$ is a compact set, by [24, Lemma 3.3], there exists a constant $T_0$ , which only depends on $\mathcal {C}_0$ (that is, on $\Gamma $ ) such that for some $\textbf {t}\in B_U(T_0)$ ,

$$ \begin{align*}(u_{\textbf{t}} h')^{\pm}\in\Lambda(\Gamma). \end{align*} $$

Fix $h:=u_{\textbf {t}} h'$ and observe that

(23) $$ \begin{align} d(g,h) < R+T_0. \end{align} $$

We must flow $h\Gamma $ with an element of A so that it lies on $Ux.$

Because $h^- = g^-,$ if $s = \beta _{g^-}(h,g)$ , then

$$ \begin{align*}a_s h \in Ug.\end{align*} $$

Since $\beta _{g^-}(h,g) \le d(h,g)$ , we arrive at

$$ \begin{align*} d(g,a_s h)&\le d(g,h)+d(h,a_s h)\\ &\le 2d(g,h)\\ &\le 2(R+T). \end{align*} $$

For $(\varepsilon ,s_0)$ -Diophantine x, observe that

$$ \begin{align*} d(\mathcal{C}_0,x)&\le d(\mathcal{C}_0,a_{-s_0}x) + d(a_{-s_0},x) \\ &< (1-\varepsilon)s_0+s_0 \\ &< 2s_0, \end{align*} $$

so we see that $R=2s_0$ works for all such points.

3.3 Bowen–Margulis–Sullivan and Burger–Roblin measures

Recall $\pi : G \to {\mathbb H}^n$ from (4). In this section, we will abuse notation and write $\pi $ for the restriction of $\pi $ to ${\operatorname {T}^1}({\mathbb H}^n)\cong M\backslash G.$ Recalling the fixed reference point $o \in {\mathbb H}^n$ as before, the map

$$ \begin{align*}w \mapsto(w^+,w^-, s:= \beta_{w^-}(o,\pi(w)))\end{align*} $$

is a homeomorphism between ${\operatorname {T}^1}({\mathbb H}^n)$ and

$$ \begin{align*}(\partial({\mathbb H}^n)\times\partial({\mathbb H}^n) - \{(\xi,\xi):\xi \in \partial({\mathbb H}^n)\})\times \mathbb R.\end{align*} $$

This homeomorphism allows us to define the Bowen–Margulis–Sullivan (BMS) and Burger–Roblin (BR) measures on ${\operatorname {T}^1}({\mathbb H}^n)$ , denoted by $\tilde {m}^{\operatorname {BMS}}$ and $\tilde {m}^{\operatorname {BR}}$ respectively:

$$ \begin{align*}d\tilde{m}^{\operatorname{BMS}}(w):=e^{\delta_{\Gamma} \beta_{w^+}(o,\pi(w))}e^{\delta_{\Gamma} \beta_{w^-}(o,\pi(w))}d\nu_o(w^+)d\nu_o(w^-)ds,\end{align*} $$

$$ \begin{align*}d\tilde{m}^{\operatorname{BR}}(w):= e^{(n-1)\beta_{w^+}(o,\pi(w))}e^{\delta_{\Gamma} \beta_{w^-}(o,\pi(w))} dm_o(w^+)d\nu_o(w^-)ds.\end{align*} $$

The conformal properties of $\{\nu _x\}$ and $\{m_x\}$ imply that these definitions are independent of the choice of $o\in {\mathbb H}^n$ . Using the identification of ${\operatorname {T}^1}({\mathbb H}^n)$ with $M\backslash G$ , we lift the above measures to G so that they are all invariant under M from the left. By abuse of notation, we use the same notation ( $\tilde {m}^{\operatorname {BMS}}$ and $\tilde {m}^{\operatorname {BR}}$ ). These measures are right $\Gamma $ -invariant, and hence induce locally finite Borel measures on X, which are the Bowen–Margulis–Sullivan measure $m^{\operatorname {BMS}}$ and the Burger–Roblin measure $m^{\operatorname {BR}}$ respectively.

Note that

$$ \begin{align*} \operatorname{supp}m^{\operatorname{BMS}}:=\{x\in X : x^{\pm}\in\Lambda(\Gamma)\} \end{align*} $$

and

$$ \begin{align*} \operatorname{supp}m^{\operatorname{BR}}=\{x\in X : x^{-}\in\Lambda(\Gamma)\}. \end{align*} $$

Recall $P = MA\tilde {U}$ , which is exactly the stabilizer of $w_o^+$ in G. We can define another measure $\nu $ on $Pg$ for $g \in G$ , which will give us a product structure for $\tilde {m}^{\operatorname {BMS}}$ and $\tilde {m}^{\operatorname {BR}}$ that will be useful in our approach. For any $g\in G$ , define

(24) $$ \begin{align} d\nu(pg) := e^{\delta_{\Gamma} \beta_{(pg)^-}(o, pg(o))} d\nu_o(w_o^-pg)dmds \end{align} $$

on $Pg$ , where $s = \beta _{(pg)^-}(o,pg(o))$ , $p = mav \in MA\tilde {U}$ , and $dm$ is the probability Haar measure on M.

Then for any $\psi \in C_c(G)$ and $g\in G$ , we have

(25) $$ \begin{align} \tilde{m}^{\operatorname{BMS}}(\psi)=\int_{Pg}\int_U \psi(u_{\textbf{t}} pg)\,d\mu^{\operatorname{PS}}_{pg}(\textbf{t})\,d\nu(pg) \end{align} $$

and

(26) $$ \begin{align} \tilde{m}^{\operatorname{BR}}(\psi)=\int_{Pg}\int_U \psi(u_{\textbf{t}} pg)\,d\textbf{t} \,d\nu(pg). \end{align} $$

Lemma 3.9. There exists a constant $\unicode{x3bb} =\unicode{x3bb} (\Gamma )> 1$ such that for all $g \in \operatorname {supp}\tilde {m}^{\operatorname {BMS}}$ and all $0<\varepsilon <\operatorname {inj}(g)$ , we have

$$ \begin{align*}&\unicode{x3bb}^{-1} \varepsilon^{\delta_{\Gamma} +{1}/{2}(n-1)(n-2)+1}e^{(k_2(x,\varepsilon)-\delta_{\Gamma})d(\pi(\mathcal{C}_0), \pi(a_{\log \varepsilon}x))} \\[3pt] &\le \nu(P_{\varepsilon} g) \\[3pt] &\le \unicode{x3bb} \varepsilon^{\delta_{\Gamma} +{1}/{2}(n-1)(n-2)+1}e^{(k_2(x,\varepsilon)-\delta_{\Gamma})d(\pi(\mathcal{C}_0), \pi(a_{\log \varepsilon}x))},\end{align*} $$

where $x = g\Gamma $ and $k_2(x,\varepsilon )$ is as defined in Proposition 3.4.

Proof. Let $x = g\Gamma $ . By Proposition 3.4, there exists $\tilde {\unicode{x3bb} }> 1$ such that for all such $\varepsilon $ ,

(27) $$ \begin{align} \tilde{\unicode{x3bb}}^{-1} \varepsilon^{\delta_{\Gamma}}e^{(k_2(x,\varepsilon)-\delta_{\Gamma})d(\pi(\mathcal{C}_0), \pi(a_{\log \varepsilon}x))} &\le \mu^{\operatorname{PS}-}_g(B_{\tilde U}(\varepsilon))\nonumber\\[4pt] &\le \tilde{\unicode{x3bb}} \varepsilon^{\delta_{\Gamma}}e^{(k_2(x,\varepsilon)-\delta_{\Gamma})d(\pi(\mathcal{C}_0), \pi(a_{\log \varepsilon}x))}. \end{align} $$

From (24), if m denotes the probability Haar measure on M, we then have

$$ \begin{align*} \nu(P_{\varepsilon} g) &\le \int_{A_{\varepsilon}}\int_{M_{\varepsilon}} \mu^{\operatorname{PS}-}_g(B_{\tilde U}(\varepsilon))\,dm\,ds\\[3pt] &\le C \tilde{\unicode{x3bb}} \varepsilon^{\delta_{\Gamma}+{1}/{2}(n-1)(n-2)+1}e^{(k_2(x,\varepsilon)-\delta_{\Gamma})d(\pi(\mathcal{C}_0), \pi(a_{\log \varepsilon}x))},\end{align*} $$

where C is determined by the scaling of the probability Haar measures on A and M. The lower bound follows similarly. Then, $\unicode{x3bb} =\max \{C \tilde {\unicode{x3bb} },\tilde {\unicode{x3bb} }\}$ satisfies the conclusion of the lemma.

3.4 Admissible boxes and smooth partitions of unity

Recall that for $\eta>0$ , we denoted by $G_{\eta }$ the closed $\eta $ -neighborhood of e in G.

Take $\varepsilon>0$ such that the map

$$ \begin{align*}g\mapsto gx \text{ is injective on }G_{\varepsilon} \text{ for all }x\in \Omega.\end{align*} $$

For $x\in X$ and $\eta _1>0$ , $\eta _2\geq 0$ less than $\operatorname {inj}(x)$ , we call

$$ \begin{align*}B = B_U(\eta_1)P_{\eta_2}x\end{align*} $$

an admissible box (with respect to the PS measure) if B is the injective image of $B_U(\eta _1)P_{\eta _2}$ in X under the map $h\mapsto h x$ and

$$ \begin{align*}\mu^{\operatorname{PS}}_{px}(B_U(\eta_1)px)\neq0\end{align*} $$

for all $p\in P_{\eta _2}$ . For $g\in G$ , we say that $B = B_U(\eta _1)P_{\eta _2}g$ is an admissible box if $B = B_U(\eta _1)P_{\eta _2}x$ is one.

Note that if $B_U(\eta _1)P_{\eta _2}g$ is an admissible box, then there exists $\varepsilon>0$ such that $B_U (\eta _1+\varepsilon )P_{\eta _2 + \varepsilon }g$ is also an admissible box. Moreover, every point has an admissible box around it by [26, Lemma 2.17].

The error terms in our main theorems are in terms of Sobolev norms, which we define here. For $\ell \in \mathbb N$ , $1\leq p\leq \infty $ , and $\psi \in C^{\infty }(X)\cap L^p(X)$ , we consider the following Sobolev norm:

$$ \begin{align*} S_{p,\ell}(\psi)=\sum\Vert U\psi\Vert _p, \end{align*} $$

where the sum is taken over all monomials U in a fixed basis of $\mathfrak {g}=\mbox {Lie}(G)$ of order at most $\ell $ , and $\Vert \cdot \Vert _p$ denotes the $L^p(X)$ -norm. Since we will be using $S_{2,\ell }$ most often, we set

$$ \begin{align*}S_{\ell}=S_{2,\ell}.\end{align*} $$

Our proofs will require constructing smooth indicator functions and partitions of unity with controlled Sobolev norms. We prove such lemmas below.

Lemma 3.10. Let H be a horospherical subgroup of G (that is, U or $\tilde {U}$ ). For every $\xi _1,\xi _2>0$ and $g \in G$ , there exists a non-negative smooth function $\chi _{\xi _1,\xi _2}$ defined on $H_{\xi _1+\xi _2}g$ such that $0 \le \chi _{\xi _1,\xi _2}\le 1 $ , $S_{\ell }(\chi _{\xi _1,\xi _2}) \ll _{n,\Gamma } \xi _1^{n-1}\xi _2^{-\ell -(n-1)/2},$ and

$$ \begin{align*}\chi_{\xi_1,\xi_2}(h) = \begin{cases} 0 & \text{ if } h \not\in H_{\xi_1+\xi_2 }g, \\ 1 &\text{ if } h \in H_{\xi_1-\xi_2}g.\end{cases}\end{align*} $$

Proof. According to [17, Lemma 2.4.7(b)], there exists $c_1=c_1(n)>0$ such that for every $\xi>0$ , there exists a non-negative smooth function $\sigma _{\xi }$ defined on $H_{\xi }$ such that

(28) $$ \begin{align} \int_{H}\sigma_{\xi}(h)\,dm^{\text{Haar}}(h)=1,\quad S_{\ell}(\sigma_{\xi})<c_1\xi^{-\ell-(n-1)/2}. \end{align} $$

For $g\in \Omega $ , let $\chi _{\xi _1,\xi _2}={\mathbf {1}}_{H_{\xi _1} g}*\sigma _{\xi _2}$ . Then for any $h\in H$ , we have $0\leq \chi _{\xi _1,\xi _2}(h)\leq 1$ and

$$ \begin{align*} \chi_{\xi_1,\xi_2}(h)=\begin{cases} 0 & \text{if }h\notin H_{\xi_1+\xi_2}g,\\ 1& \text{if }h\in H_{\xi_1-\xi_2}g. \end{cases} \end{align*} $$

Since for some $c_2=c_2(\Gamma )>0$ ,

$$ \begin{align*} S_{1,0}({\mathbf{1}}_{H_{\xi_1}g_0})=m^{\text{Haar}}(H_{\xi_1})<c_2{\xi_1}^{n-1},\end{align*} $$

by the properties of the Sobolev norm and (28), we arrive at

$$ \begin{align*} S_{\ell}(\chi_{\xi_1,\xi_2})\leq S_{1,0}({\mathbf{1}}_{H_{\xi_1}g_0})S_{\ell}(\sigma_{\xi_2})<c_1 c_2{\xi_1}^{n-1}{\xi_2}^{-\ell-(n-1)/2}.\\[-3pc] \end{align*} $$

Lemma 3.11. Let H be a horospherical subgroup of G, $r>0$ , $\ell \in \mathbb N$ , and let $E\subset H$ be bounded. Then, there exists a partition of unity $\sigma _1,\ldots ,\sigma _k$ of E in $H_r E$ , that is,

$$ \begin{align*} \sum_{i=1}^k \sigma_i(x)=\begin{cases} 0 & \text{if }x\notin H_r E,\\ 1 & \text{if }x\in E, \end{cases}\end{align*} $$

such that for some $u_1,\ldots ,u_k\in E$ and all $1\leq i\leq k$ ,

$$ \begin{align*} \sigma_i\in C_c^{\infty}(H_r u_i),\quad S_{\ell}(\sigma_i)\ll_n r^{-\ell+n-1}. \end{align*} $$

Moreover, if there exists $R>r$ such that $E=H_R$ , then $k\ll _n ({R}/{r})^{n-1}$ .

Proof. Let $\{u_1,\ldots ,u_k\}$ be a maximal ${r}/{4}$ -separated set in E. Then

(29) $$ \begin{align} E\subseteq\bigcup_{i=1}^k H_{r/2}u_i. \end{align} $$

Let $1\leq i\leq k$ . According to [13, Theorem 1.4.2], there exists $\chi _i\in C_c^{\infty }(H_{r} u_i)$ such that $0\leq \chi _i\leq 1$ , $\chi _i(u)=1$ for any $u\in H_{r/2}u_i$ , and for $1\leq m\leq \ell $ ,

(30) $$ \begin{align} |\chi_i^{(m)}|\ll r^{-m} \end{align} $$

(where the implied constant depends only on n). Let $\sigma _i$ be defined by

$$ \begin{align*} \sigma_i=\chi_i(1-\chi_{i-1})\cdots(1-\chi_1). \end{align*} $$

Then, each $\sigma _i\in C_c^{\infty }(H_{r} u_i)$ and

$$ \begin{align*} 1-\sum_{i=1}^k\sigma_i=\prod_{i=1}^k(1-\chi_i)=0 \quad\mbox{on }\bigcup_{i=1}^k H_{r}u_i\end{align*} $$

implies that $\sum _{i=1}^k\sigma _i=1$ on $\bigcup _{i=1}^k H_{r/2}u_i$ .

By the rules for differentiating a product and (30) for $1\leq m\leq \ell $ , we have

$$ \begin{align*} |\sigma_i^{(m)}|\le C r^{-m}, \end{align*} $$

where C is the multiplicity of the cover in (29). By Besicovitch covering theorem, C is bounded by a constant which depends only on n. Using the definition of the Sobolev norm, we arrive at

$$ \begin{align*} S_{\ell}(\sigma_i)\ll_{n} r^{-\ell+n-1}. \end{align*} $$

Now, assume there exists $R>r$ such that $E=H_R$ . Since the geometry of H is of an Euclidean space of dimension $\dim H$ , we then have

$$ \begin{align*} k\ll_n\bigg(\frac{R}{r}\bigg)^{n-1}.\\[-3.6pc] \end{align*} $$

Lemma 3.12. Let H be either U or G. There exists $\ell '=\ell '(H)>0$ such that for any integer $\ell>\ell '$ , $\eta>0$ , $H\in \{U,G\}$ , and $f\in C_c^{\infty }(H)$ , there exist functions $f_{\eta ,\pm }\in C_c^{\infty }(H)$ which are supported on an $2\eta $ neighborhood of $\operatorname {supp} f$ , and for any $h\in H$ , satisfy:

1. (1) $f_{\eta ,-}(h)\le \min _{w\in H_{\eta }}f(wh) \le \max _{w\in H_{\eta }}f(wh) \le f_{\eta ,+}(h)$ ;

2. (2) $|f_{\eta ,\pm }(h)-f(h)|\ll _{\operatorname {supp} f} \eta S_{\ell }(f)$ ;

3. (3) $S_{\ell }(f_{\eta ,\pm })\ll _{H,\operatorname {supp} f}\eta ^{-2\ell } S_{\ell }(f).$

Proof. First, according to [1], there exists $\ell '\in \mathbb N$ such that any $\ell>\ell '$ satisfies $S_{\infty ,1}(\psi )\ll _{\operatorname {supp} \psi } S_{\ell }(\psi )$ for any $\psi \in C_c^{\infty }(H)$ .

Let $f^{\prime }_{\eta ,\pm }$ be defined by

$$ \begin{align*} f^{\prime}_{\eta,+}(h):=\sup_{w\in H_{\eta}}f(wh)\quad\text{and}\quad f^{\prime}_{\eta,-}(h):=\inf_{w\in H_{\eta}}f(wh) \end{align*} $$

for any $h\in H$ .

As before, we use [17, Lemma 2.4.7(b)] to deduce that there exist $c_1=c_1(H)>0$ , $n_1=n_1(H)$ , and a non-negative smooth function $\sigma _{\eta }$ supported on $H_{\eta }$ such that

$$ \begin{align*} \int_{H}\sigma_{\eta}(h)\,dm^{\operatorname{Haar}}(h)=1,\quad S_{\ell}(\sigma_{\eta})<c_1\eta^{-\ell-n_1}. \end{align*} $$

Define $f_{\eta ,\pm }$ by

$$ \begin{align*} f_{\eta,\pm}:=f^{\prime}_{2\eta,\pm}*\sigma_{\eta}. \end{align*} $$

Then, $f_{\eta ,\pm }$ are smooth functions which are supported on a $2\eta $ neighborhood of $\operatorname {supp} f$ . Moreover, for any $h\in H$ ,

(31) $$ \begin{align} f^{\prime}_{\eta,+}(h) & = \int_{H_{\eta}}f^{\prime}_{\eta,+}(h)\sigma_{\eta}(u^{-1})\,dm^{\operatorname{Haar}}(u)\nonumber\\ & \le \int_{H_{\eta}}f^{\prime}_{2\eta,+}(uh)\sigma_{\eta}(u^{-1})\,dm^{\operatorname{Haar}}(u) \quad\text{by definition of } f^{\prime}_{2\eta,+}\\ & = f_{\eta,+}(h)\nonumber\\ & \le \int_{H_{\eta}}f^{\prime}_{3\eta,+}(h)\sigma_{\eta}(u^{-1})\,dm^{\operatorname{Haar}}(u) \quad\text{by (31) and definition of }f^{\prime}_{3\eta,+}\nonumber\\ & = f^{\prime}_{3\eta,+}(h).\nonumber \end{align} $$

In a similar way, one can show

$$ \begin{align*} f^{\prime}_{3\eta,-}\le f_{\eta,-} \le f^{\prime}_{\eta,-}, \end{align*} $$

proving the first inequality.

By the mean value theorem, for any $h\in H$ , $w\in H_{3\eta }$ ,

$$ \begin{align*} |f(wh)-f(h)|\ll \eta S_{\infty,1}(f)\ll_{\operatorname{supp} f}S_{\ell}(f).\end{align*} $$

Since $f^{\prime }_{3\eta ,-}\le f_{\eta ,-} \le f_{\eta ,+} \le f^{\prime }_{3\eta ,+},$ there exist some $w_{+},w_{-}\in H_{3\eta }$ such that

$$ \begin{align*} |f_{\eta,\pm}(h)-f(h)|\le |f(w_{\pm} h)-f(h)|,\end{align*} $$

and we have the second inequality.

Now, we have

$$ \begin{align*} S_{\ell}(f_{\eta,\pm})\le S_{\infty,1}(f^{\prime}_{2\eta,\pm})S_{\ell}(\sigma_{\eta})\ll_{H,\operatorname{supp} f}S_{\ell}(f)\eta^{-\ell-n_1+1}.\end{align*} $$

By choosing $\ell '>n_1$ , we may deduce the last inequality.

4 Quantitative non-divergence

In this section, we prove a quantitative non-divergence result that is crucial in the following sections. We use the notation established in §3.2. The results in this section hold for any $\Gamma $ that is geometrically finite, without need for Assumption 1.1.

Recall from §1 that for $0<\varepsilon <1$ and $s_0\ge 1$ , we say that $x\in X$ is $(\varepsilon ,s_0)$ -Diophantine if for all $\tau>s_0$ ,

(32) $$ \begin{align} d(\mathcal{C}_0,a_{-\tau}x)<(1-\varepsilon)\tau, \end{align} $$

where $\mathcal {C}_0$ is the compact set defined in §3.2. Let $R_0$ and q also be as defined in §3.2.

This section is dedicated to the proof of the following theorem, which says (in a quantitative way) that most of the U orbit of a Diophantine point is not in the cusp.

Theorem 4.1. There exists $\beta>0$ satisfying the following: for every $0<\varepsilon <1$ and $s_0\ge 1$ , and for every $(\varepsilon ,s_0)$ -Diophantine element $x\in X$ , every $R\ge R_0$ , every $T\gg _{\Gamma ,\varepsilon }s_0$ , and every $0<s\le T^{\varepsilon }$ , we have

$$ \begin{align*} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T/s)a_{-\log s}x\cap{\mathcal{X}}(R))\ll_{n,\Gamma} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T/s)a_{-\log s}x)e^{-\beta R}.\end{align*} $$

We now follow the notation of Mohammadi and Oh in [25, §6]. Equip $\mathbb {R}^{n+1}$ with the Euclidean norm. Recall from §3.2 that for $1\le i\le q$ , $g_i^-=\xi _i$ . Without loss of generality, we may further assume that $g_i$ satisfies $\Vert g_i^{-1}e_1\Vert =~1.$ Let

$$ \begin{align*}v_i=g_i^{-1}e_1.\end{align*} $$

Lemma 4.2. For any $i=1,\ldots ,q$ , $\Gamma v_i$ is a discrete subset of $\mathbb {R}^{n+1}$ .

Proof. Since $\xi _i$ is assumed to be a bounded parabolic limit point, by definition, $(\Lambda (\Gamma )\setminus \{\xi _i\})/ \Gamma _{\xi _i}=(\Lambda (\Gamma )\setminus \{\xi _i\})/ \Gamma _{v_i}$ is compact, where

$$ \begin{align*}G_{v_i}=g_i^{-1}MUg_i\quad\text{and}\quad\Gamma_{v_i}=\Gamma\cap G_{v_i}.\end{align*} $$

If $\gamma \in \Gamma _{v_i}$ , then $\mathcal {H}_i(R_0)\gamma =\mathcal {H}_i(R_0)$ . Therefore, the visual map induces a homeomorphism between $\mathcal {H}_i(R_0)/\Gamma _{v_i}$ and $(\partial {\mathbb H}^n\setminus \{\xi _i\})/\Gamma _{v_i}$ . It follows that the quotient of $\{g^+\in \Lambda : g\in \mathcal {H}_i(R_0)\}$ by the action of $\Gamma _{v_i}$ is compact. Using Iwasawa decomposition, it follows that there exists a compact set $U_0\subset U$ such that for any $g=kaug_i\in \mathcal {H}_i(R_0)$ such that $g^+\in \Lambda (\Gamma )$ , $k\in K$ , $a\in A$ , and $u\in U$ , there exist $\gamma \in \Gamma _{v_i}$ , $k'\in K$ , $u'\in U_0$ so that $g\gamma =k'au'g_i$ .

Since $\xi _i$ is assumed to be a parabolic limit point, there exists a parabolic element $\gamma _0\in \Gamma _{\xi _i}$ , that is, $\gamma _0=g_i^{-1}mug_i$ .

Assume for contradiction that there exists an infinite sequence $\{\gamma _j\}\in \Gamma $ such that $\{\gamma _j v_i\}$ converges. Note that translating by an element of $\gamma $ allows us to assume, without loss of generality, that the limit of this sequence is $0$ . Using the Iwasawa decomposition, we get that for all j, there exist $a_{t_j}\in A$ , $k_j\in K$ , and uniformly bounded $u_j\in U$ such that $\gamma _j=k_ja_{t_j}u_jg_i$ . Since

$$ \begin{align*} \Vert \gamma_jv_i\Vert =\Vert k_ja_{t_j}u_je_1\Vert =e^{t_j}, \end{align*} $$

we may deduce that $t_j\rightarrow -\infty $ . In particular, $\gamma _j\in H_i(R_0)$ for all large enough j.

We have

$$ \begin{align*} \gamma_j\gamma_0\gamma_j^{-1} &=(k_ja_{t_j}u_jg_i)(g_i^{-1}mug_i)(g_i^{-1} u_j^{-1} a_{t_j}^{-1} k_{j}^{-1})\\ &=k_ja_{t_j}u_jm uu_j^{-1} a_{t_j}^{-1} k_j^{-1}. \end{align*} $$

Since $u_jmu_j^{-1}=m_ju^{\prime }_j\in MU$ , with $u^{\prime }_j$ uniformly bounded, and since M centralizes A, we have

$$ \begin{align*} \gamma_j\gamma_0\gamma_j^{-1} &=k_jm_j a_{t_j} u^{\prime}_j u a_{t_j}^{-1} k_j^{-1}. \end{align*} $$

Since $u^{\prime }_j u$ is in a bounded subset of U, we get that $a_{t_j} u^{\prime }_j u a_{t_j}^{-1}\rightarrow e$ as $t_j\rightarrow -\infty $ . Since K and M are compact, it then follows that the sequence $\gamma _j\gamma _0\gamma _j^{-1}$ has a convergent subsequence. This contradicts the discreteness of $\Gamma $ , since the $\gamma _j$ terms were assumed to be distinct.

For any $g\in G$ , we have that $g\Gamma \in \mathcal {X}_i(R)$ if and only if there exists $\gamma \in \Gamma $ such that

(33) $$ \begin{align} \Vert g\gamma v_i\Vert \le e^{-R}. \end{align} $$

Indeed, by the Iwasawa decomposition and (17), if $g\Gamma \in \mathcal {X}_i(R)$ , then there exist $\gamma \in \Gamma $ , $k\in K$ , $s>R$ , and $u\in U$ , such that

$$ \begin{align*} \Vert g\gamma v_i\Vert =\Vert ka_{-s}ug_iv_i\Vert =\Vert a_{-s}e_i\Vert =e^{-s}. \end{align*} $$

Moreover, it follows from [25, Lemmas 6.4 and 6.5] that the $\gamma $ in (33) is unique. Note that both lemmas are proved under the additional assumption that $n=3$ , but the proofs also hold without it.

However, by [25, Lemma 6.5] and Lemma 4.2, there exists a constant $\eta _0 = \eta _0(\Gamma )>0$ such that if $g\Gamma \notin \mathcal {X}_i(R_0)$ , then for any $\gamma \in \Gamma $ ,

(34) $$ \begin{align} \Vert g\gamma v_i\Vert>\eta_0. \end{align} $$

Lemma 4.3. There exists $c=c(\Gamma )>0$ which satisfies the following. Let $\varepsilon , s_0>0$ and let $g \in G$ . If $x = g\Gamma $ is $(\varepsilon ,s_0)$ -Diophantine, then for any $T\gg _{\Gamma ,\varepsilon } s_0$ ,

(35) $$ \begin{align} \sup_{\Vert \textbf{t}\Vert \leq T}\inf_{\gamma\in\Gamma}\inf_{i=1,\ldots,q}\Vert u_{\textbf{t}} g\gamma v_i\Vert>cT^{\varepsilon}. \end{align} $$

Proof. Fix $T>T_0=\max \{s_0,\eta _0^{{1}/({\varepsilon -1})}\}$ . We will first show that

(36) $$ \begin{align} \inf_{\gamma\in\Gamma}\inf_{i=1,\ldots,q}\Vert a_{-\log T}g\gamma v_i\Vert>cT^{\varepsilon-1}, \end{align} $$

for some constant $1>c=c(\Gamma )>0$ .

There are two cases to consider. If $a_{-\log T}x\notin \mathcal {X}_i(R_0)$ , then (36) follows from (34) and the choice of T.

Otherwise, $a_{-\log T}x\in \mathcal {X}_i(R)$ for some maximal $R>R_0$ . According to Lemma 3.6, we have

$$ \begin{align*}d(x,\mathcal{C}_0)\ge R-R_0.\end{align*} $$

Then, because x is $(\varepsilon ,s_0)$ Diophantine and $T> s_0,$ by (32), we may deduce that

$$ \begin{align*}R-R_0<(1-\varepsilon)\log T.\end{align*} $$

Hence, $a_{-\log T}x \not \in \mathcal {X}_i((1-\varepsilon )\log T+R_0),$ so (33) implies (36).

Now, fix $\gamma \in \Gamma $ and $1\le i \le q$ , and let

$$ \begin{align*} \begin{pmatrix} x_1\\ \vdots\\ x_{n+1} \end{pmatrix}=a_{-\log T}g\gamma v_i.\end{align*} $$

According to (36), there exists $1\leq k\leq n$ such that $|x_k|>cT^{\varepsilon -1}$ . If $|x_1|>cT^{\varepsilon -1}$ , then it follows from the action of $a_{-\log T}$ on $\mathbb {R}^{n+1}$ that

$$ \begin{align*} \Vert g\gamma v_i\Vert \geq|cT x_1|>cT^{\varepsilon}. \end{align*} $$

Otherwise, there exists $2\leq k\leq n$ such that $|x_k|>cT^{\varepsilon -1}$ . Then, for any $\textbf {t}\in \mathbb {R}^{n-1}$ , the first coordinate of $u_{\textbf {t}}a_{-\log T}g\gamma v_i$ is

$$ \begin{align*} x_1+\textbf{t}\cdot \mathbf{x}'+\frac{1}{2}\Vert \textbf{t}\Vert ^2 x_{n+1}\quad\mbox{where } \mathbf{x}'=\begin{pmatrix} x_2\\ \vdots\\ x_n \end{pmatrix}.\end{align*} $$

In particular, by taking $t_k=\pm T$ (the kth entry in $\textbf {t}$ ) one can ensure that $\|a_{\log T}u_{\textbf {t}}a_{-\log T} g\gamma v_i\|>cT^{\varepsilon }$ .

A measure $\mu $ is called D-Federer if for all $v\in \operatorname {supp}(\mu )$ and $0 <\eta \leq 1$ ,

$$ \begin{align*} \mu(B(v,{3\eta}))\le D\mu(B(v,\eta)). \end{align*} $$

It is proved in §A (specifically Corollary A.9) that there exists $D=D(\Gamma )>0$ such that

(37) $$ \begin{align} \text{if }x\in X\text{ satisfies }x^-\in\Lambda(\Gamma)\text{, then }\mu^{\operatorname{PS}}_x\text{ is }D\text{-Federer}. \end{align} $$

Indeed, Corollary A.9 actually establishes that there exists a constant $\sigma = \sigma (\Gamma )>0$ such that for all $x \in \operatorname {supp}m^{\operatorname {BMS}}, c>1,$ and $T>0$ ,

(38) $$ \begin{align} \mu^{\operatorname{PS}}_x(B_U(cT)) \ll_{\Gamma} c^{\sigma} \mu^{\operatorname{PS}}_x(B_U(T)). \end{align} $$

For $f:\mathbb {R}^d\rightarrow \mathbb {R}$ and $B\subset \mathbb {R}^d$ , let

$$ \begin{align*} \Vert f\Vert _B:=\sup_{x\in B}|f(x)|.\end{align*} $$

Recall that $U \cong \mathbb R^{n-1}$ .

Lemma 4.4. Let $\sigma = \sigma (\Gamma )>0$ be as in (38). Let $y \in \operatorname {supp}m^{\operatorname {BMS}}$ and let $f: B_U(\eta )\to \mathbb R^{n-1}$ be such that there exists $b\ne 0$ so that for every coordinate function $f_i: B_U(\eta ) \to \mathbb R$ , there exist $a_i\in \mathbb R$ , such that

$$ \begin{align*}f_i(\textbf{t}) = a_i + b\textbf{t}_i.\end{align*} $$

Then for $0<\eta \le 1$ and $0<\varepsilon <1,$ we have

(39) $$ \begin{align} \mu^{\operatorname{PS}}_y(\{\textbf{t}\in B_U(\eta):\|f(\textbf{t})\|<\varepsilon\})\ll_{\Gamma}\bigg(\frac{\varepsilon}{\Vert f\Vert _{B_U(\eta)}}\bigg)^{\sigma}\mu^{\operatorname{PS}}_y(B_U(\eta)), \end{align} $$

where $\|f(x)\|$ denotes the max norm.

Proof. First, note that if $\|f\|_{B_U(\eta )} < 2\varepsilon $ , then the result holds by assuming that the implied coefficient in (39) is bigger than $2^{\sigma }$ : in this case, the right-hand side is greater or equal to

$$ \begin{align*}2^{\sigma}\bigg(\frac{\varepsilon}{\Vert f\Vert _{B_U(\eta)}}\bigg)^{\sigma}\mu^{\operatorname{PS}}_y(B_U(\eta)) &\ge2^{\sigma}\bigg(\frac{\varepsilon}{2\varepsilon}\bigg)^{\sigma}\mu^{\operatorname{PS}}_y(B_U(\eta))\\ & \ge\mu^{\operatorname{PS}}_y(\{\textbf{t}\in B_U(\eta):\|f(\textbf{t})\|<\varepsilon\}), \end{align*} $$

as desired. Thus, we now assume that

(40) $$ \begin{align}\Vert f\Vert _{B_U(\eta)} \ge 2\varepsilon.\end{align} $$

If $\|f(\textbf {t})\|\ge \varepsilon $ for all $\textbf {t} \in B_U(\eta )$ such that $(u_{\textbf {t}} y)^+\notin \Lambda (\Gamma )$ , then there is nothing to prove. So assume that $\|f(\textbf {t})\|<\varepsilon $ and $(u_{\textbf {t}} y)^+\in \Lambda (\Gamma )$ . Since each $f_i$ is linear, for all $\textbf {t}'\in B_U(\eta )$ with $\|f(\textbf {t}')\|<\varepsilon $ , we get that for all $1\le i\le n-1$ ,

$$ \begin{align*} |f_i(\textbf{t}')|= |a_i+b\textbf{t}'| <\varepsilon, \end{align*} $$

$$ \begin{align*}|b(t^{\prime}_i-t_i)|=|f_i(\textbf{t}')-f_i(\textbf{t})|<2\varepsilon.\end{align*} $$

Therefore,

$$ \begin{align*}\|f(x)\|<\varepsilon {\implies} x \in B_U(2\varepsilon/b)z.\end{align*} $$

Thus, by (38), we have that there exists $\sigma = \sigma (\Gamma )>0$ so that

$$ \begin{align*} \mu^{\operatorname{PS}}_y(\{x \in B_U(\eta)y : \|f(x)\|<\varepsilon\})&\le\mu^{\operatorname{PS}}_z( B_U(2\varepsilon/b))\\ & \ll_{\Gamma}\bigg(\frac{2\varepsilon}{b\eta}\bigg)^{\sigma}\mu^{\operatorname{PS}}_z( B_U(\eta))\\ & \ll_{\Gamma}\bigg(\frac{2\varepsilon}{b\eta}\bigg)^{\sigma}\mu^{\operatorname{PS}}_y( B_U(3\eta))\\ & \ll_{\Gamma}\bigg(\frac{6\varepsilon}{b\eta}\bigg)^{\sigma}\mu^{\operatorname{PS}}_y( B_U(\eta)). \end{align*} $$

Assuming $\|f(\textbf {t})\|<\varepsilon $ for some $\textbf {t}\in B_U(\eta )$ (otherwise, as before, there is nothing to prove), for any $\textbf {t}"\in B_U(\eta )$ and $1\le i\le n-1$ , we have

$$ \begin{align*} |f_i(\textbf{t}")|\le |f_i(\textbf{t}")-f_i(\textbf{t})|+|f_i(\textbf{t})|<2b\eta+\varepsilon.\end{align*} $$

Thus, $\Vert f\Vert _{B_U(\eta )y}-\varepsilon \le 2b\eta $ , so by (40),

$$ \begin{align*}\tfrac{1}{2}\Vert f\Vert _{B_U(\eta)} \le 2b \eta,\end{align*} $$

which completes the proof.

A function f which satisfies (39) with the implied constant C for any $\varepsilon>0$ and any ball $B\subset U\subset \mathbb {R}^m$ is called $(C, \sigma )$ -good on U with respect to $\mu $ . Observe that

(41) $$ \begin{align} \text{if }g\text{ is }(C,\sigma)\text{-good and if } |g(x)|\le |f(x)|\text{ for }\mu\text{-almost every }x\text{, then }f\text{ is }(C,\sigma)\text{-good.} \end{align} $$

In the proof of the following theorem, we use similar ideas to those which appear in the proof of [18, Lemma 5.2]. Note that the proof in this case is simplified by the third assumption, reflecting our rank-one setting.

Proposition 4.5. Given positive constants $C,\beta , D$ , and $0<\eta <1$ , there exists $C'=C'(C,\beta , D)>0$ with the following property. Suppose $\mu $ is a D-Federer measure on $\mathbb R^m$ , $f:\mathbb R^m\rightarrow \operatorname {SL}_k(\mathbb R)$ is a continuous map, $0\le \varrho \le \eta $ , $z\in \operatorname {supp}\mu $ , $\Lambda \subset \mathbb R^k$ , $B=B(z,r_0)\subset \mathbb R^m$ , and $\tilde B=B(z,3 r_0)$ satisfy the following.

1. (1) For any $v\in \Lambda $ , the function $\textbf {t}\mapsto \Vert f(\textbf {t})v\Vert $ is $(C,\beta )$ -good on $\tilde B$ with respect to $\mu $ .

2. (2) For any $v\in \Lambda $ , there exists $\textbf {t}\in B$ such that $\Vert f(\textbf {t})v\Vert \geq \varrho $ .

3. (3) For any $\textbf {t}\in B$ , there is at most one $v\in \Lambda $ which satisfies $\Vert f(\textbf {t})v\Vert <\eta $ .

Then, for any $0<\varepsilon <\varrho $ ,

$$ \begin{align*} \mu(\{\textbf{t}\in B : \text{there exists } v\in\Lambda\text{ such that }\Vert f(\textbf{t})v\Vert <\varepsilon\})\le C'\bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu(B). \end{align*} $$

Proof. For any $\textbf {t}\in B$ , denote

$$ \begin{align*} f_{\Lambda}(\textbf{t})=\min\{\Vert f(\textbf{t})v\Vert : v\in\Lambda\}. \end{align*} $$

Let

$$ \begin{align*} E=\{\textbf{t}\in B : f_{\Lambda}(\textbf{t})<\varrho\}\cap \operatorname{supp}\mu, \end{align*} $$

and for each $v \in \Lambda $ , define

$$ \begin{align*} E_v=\{\textbf{t} \in B : \|f(\textbf{t})v\|<\varrho\} \cap \operatorname{supp}\mu. \end{align*} $$

Observe that by assumption (3), the $E_v$ terms are a disjoint cover of E. For each $\textbf {t} \in E_v$ , define

$$ \begin{align*} r_{\textbf{t},v}=\sup\{r : \|f(\mathbf{s})v\|<\varrho \text{ for all } \textbf{s}\in B(\textbf{t},r)\}. \end{align*} $$

By assumption (2), we know that for every $\textbf {t}\in E$ , the set $B(\textbf {t},r_{\textbf {t},v})$ does not contain B. Thus, since $\textbf {t}\in B$ , we deduce that $r_{\textbf {t},v}<2r_0$ . For any fixed $r_{\textbf {t},v}<r^{\prime }_{\textbf {t},v}<2r_0$ , we have that

(42) $$ \begin{align} B(\textbf{t},r^{\prime}_{\textbf{t},v})\subset B(z,3r_0)=\tilde B, \end{align} $$

and by the definition of $r_{\textbf {t},v}$ , there exists $\textbf {s} \in B(\textbf {t},r^{\prime }_{\textbf {t},v})$ such that

$$ \begin{align*} \Vert f(\textbf{s})v\Vert \geq\varrho. \end{align*} $$

Note that $\{B(\textbf {t},r_{\textbf {t},v}) : \textbf {t}\in E, v\in \Lambda \}$ is a cover of E. According to the Besicovitch covering theorem, there exists a countable subset $I\subset E\times \Lambda $ such that $\{B(\textbf {t},r_{\textbf {t},v}) : (\textbf {t},v)\in ~I\}$ is a cover of E with a covering number bounded by a constant which only depends on m. Thus,

(43) $$ \begin{align} \sum_{(\textbf{t},v)\in I}\mu(B(\textbf{t},r_{\textbf{t},v})) \ll_{m} \mu\bigg(\bigcup_{(\textbf{t},v)\in I}B(\textbf{t},r_{\textbf{t},v})\bigg). \end{align} $$

By assumption (3) and the continuity of f, for any $(\textbf {t},v)\in I$ and $\mathbf{s}\in E\cap B(\textbf {t},r_{\textbf {t},v})$ ,

$$ \begin{align*} f_{\Lambda}(\textbf{s})=\Vert f(\textbf{s})v\Vert. \end{align*} $$

Thus,

$$ \begin{align*} \mu(\{\mathbf{s}\in B(\textbf{t},r_{\textbf{t},v}) : {f_{\Lambda}(\mathbf{s})}<\varepsilon\})&=\mu(\{\mathbf{s}\in B(\textbf{t},r_{\textbf{t},v}) : \Vert f(\mathbf{s})v\Vert <\varepsilon\})\\[3pt] &\le\mu(\{\mathbf{s}\in B(\textbf{t},r^{\prime}_{\textbf{t},v}) : \Vert f(\mathbf{s})v\Vert <\varepsilon\}).\end{align*} $$

Thus, assumption (1) and the assumption that $\mu $ is D-Federer together imply that

(44) $$ \begin{align} \mu(\{\mathbf{s}\in B(\textbf{t},r_{\textbf{t},v}) : {f_{\Lambda}(\mathbf{s})}<\varepsilon\})&\le\mu(\{\mathbf{s}\in B(\textbf{t},r^{\prime}_{\textbf{t},v}) : \Vert f(\mathbf{s})v\Vert <\varepsilon\})\nonumber\\[3pt] & \le C\bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu(B(\textbf{t},r^{\prime}_{\textbf{t},v}))\nonumber\\[3pt] & \le CD \bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu(B(\textbf{t},r_{\textbf{t},v})). \end{align} $$

Since E covers the set of points for which $f_{\Lambda }$ is less than $\varepsilon $ , we may now conclude

$$ \begin{align*} &\mu(\{\textbf{t}\in B : f_{\Lambda}(\textbf{t})<\varepsilon\})\\ &\quad\le \sum_{(\textbf{t},v)\in I}\mu(\{\mathbf{s}\in B(\textbf{t},r_{\textbf{t},v}) : f_{\Lambda}(\textbf{t})<\varepsilon\}) \\ & \quad\le CD \sum_{(\textbf{t},v)\in I}\bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu(B(\textbf{t},r_{\textbf{t},v})) \quad \text{by (44)}\\ & \quad\ll_{m} CD \bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu\bigg(\bigcup_{(\textbf{t},v)\in I}B(\textbf{t},r_{\textbf{t},v})\bigg)\quad \text{by (43)}\\ & \quad\ll_{m} CD \bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu(\tilde B ) \quad \text{by (42)} \\ & \quad\ll_{m} CD^2 \bigg(\frac{\varepsilon}{\varrho}\bigg)^{\beta}\mu(B ) \quad\mu\text{ is }D\text{-Federer.}\\[-3.6pc] \end{align*} $$

Remark 4.6. Fix $x\in X$ such that $x^-\in \Lambda (\Gamma )$ . Since the PS measure $\mu ^{\operatorname {PS}}_x$ is supported on $Ux\cap \operatorname {supp}m^{\operatorname {BMS}}$ , it follows from Lemma 4.4 and (37) that Proposition 4.5 holds for $\mu ^{\operatorname {PS}}_x$ and function f, which satisfies the assumption of Lemma 4.4.

We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1

Let $x_0=a_{-\log s}x$ and fix $g\in G$ such that $x=g\Gamma $ . By Lemma 4.3, for all $T \gg _{\Gamma ,\varepsilon } s_0,$ we have (35), that is, that

$$ \begin{align*}\sup_{\Vert \textbf{t}\Vert \leq T}\inf_{\gamma\in\Gamma}\inf_{i=1,\ldots,q}\Vert u_{\textbf{t}} g\gamma v_i\Vert>T^{\varepsilon}.\end{align*} $$

Let $f:\mathbb R^{n-1}\rightarrow \operatorname {SL}_{n+1}(\mathbb R)$ be defined by

$$ \begin{align*}f(\textbf{t})=u_{\textbf{t}} a_{-\log s}g.\end{align*} $$

We first show that parts (1), (2), and (3) of Proposition 4.5 for $\mu =\mu ^{\operatorname {PS}}_{x_0}$ , f, $\varrho =1$ , $z=x_0$ , $r=T/s$ , $\eta =e^{-R_0}$ , and

$$ \begin{align*}\Lambda=\Gamma \{v_1,\ldots,v_q\}.\end{align*} $$

Note that $0\notin \Lambda $ .

It follows from the action of $u_{\textbf {t}}$ on $\mathbb {R}^{n+1}$ that for any $v\in \mathbb R^{n+1}$ , there exists $v'=(v^{\prime }_1,\ldots ,v^{\prime }_{n+1})^T\in \mathbb R^{n+1}$ such that

(45) $$ \begin{align} f(\textbf{t})v=\big(v^{\prime}_1+\textbf{t}\cdot v"+\tfrac{1}{2}\Vert \textbf{t}\Vert ^2v^{\prime}_{n+1},v^{\prime}_2-t_1v^{\prime}_{n+1},\ldots,v^{\prime}_{n}-t_{n-1}v^{\prime}_{n+1},v^{\prime}_{n+1}\big)^T,\end{align} $$

where $v"=(v^{\prime }_2,\ldots ,v^{\prime }_{n})^T$ . Thus, if $v^{\prime }_{n+1}\ne 0$ , then $\textbf {t} \mapsto f(\textbf {t})v$ is bounded from below by a function which satisfies the assumption of Lemma 4.4. Therefore, by (41), for any $v\in \Lambda $ , the function $\textbf {t}\mapsto \Vert f(\textbf {t})v\Vert $ is $(C,\beta )$ -good on $\tilde B$ with respect to $\mu ^{\operatorname {PS}}_{x_0}$ for some $C=C(\Gamma )\ge 1$ , $\beta =\beta (\Gamma )>0$ , which proves (1) of Proposition 4.5. Note that these constants are uniform across all v so that $v_{n+1}'\ne 0.$

However, if $v\neq 0$ and

$$ \begin{align*}v_{n+1}' = 0,\end{align*} $$

then $\textbf {t} \mapsto f(\textbf {t})v$ is bounded below by some positive constant, and since positive constant functions are $(C,\beta )$ -good for any $C\ge 1,\beta>0,$ we conclude that so is this function by (41).

By (6), we have

$$ \begin{align*} u_{\textbf{t}}a_{-\log s}=a_{-\log s} u_{s\textbf{t}}.\end{align*} $$

Since multiplication by $a_{-\log s}$ only changes the matrix entries by scaling, using (35), for $i=1,\ldots ,q$ , we get

$$ \begin{align*} \sup_{\Vert \textbf{t}\Vert \leq T/s}\Vert a_{-\log s}u_{s\textbf{t}} g\gamma v_i\Vert>s^{-1}\sup_{\Vert \textbf{t}\Vert \leq T}\Vert u_{\textbf{t}} g\gamma v_i\Vert >s^{-1}T^{\varepsilon}.\end{align*} $$

Thus, for any $s\le T^{\varepsilon }$ , $\|\textbf {t}\|<T/s$ , and $v\in \Lambda $ ,

$$ \begin{align*} \Vert f(\textbf{t})v\Vert \ge 1, \end{align*} $$

which establishes (2) of Proposition 4.5.

Since $\eta =e^{-R_0}$ and $\mathcal {H}_i(R_0)$ terms are pairwise disjoint, part (3) of Proposition 4.5 follows from the uniqueness of $\gamma $ in (33) and (34).

According to 37, the measure $\mu ^{\operatorname {PS}}_{x_0}$ is D-Federer for any $D>0$ . Thus, we may now use (33) and Proposition 4.5 to deduce

$$ \begin{align*} &\mu^{\operatorname{PS}}_{x_0}(B_U(T/s)\cap\mathcal{H}_i(R))\\[3pt] &\quad=\mu^{\operatorname{PS}}_{x_0}(\{\textbf{t}\in B_U(T/s) : \text{ there exists } \gamma\in\Gamma,1\le i\le q\text{ such that }\Vert f(\textbf{t})\gamma v_i\Vert <e^{-R}\})\\[3pt] &\quad\ll e^{-R \beta} \mu^{\operatorname{PS}}_{x_0}(B_U(T/s)x_0), \end{align*} $$

where the implied constant depends on n and $\Gamma $ .

5 Friendliness properties of the PS measure

In this section, we prove several key properties of the PS measure, including that slightly enlarging a ball does not increase the measure too much and that scaling the size of the ball has a bounded multiplicative increase on the measure. Note that the results in this section hold for any $\Gamma $ that is geometrically finite; we do not require Assumption 1.1. In the setting that all cusps have maximal rank, stronger statements hold. See the appendix, specifically §A.3, for more details.

The main results in this section are the following, which both establish control over the measure of a slightly enlarged ball. Many technical details of the proofs are hidden in Proposition 5.4, which is proved in §A.

Theorem 5.1. There exists a constant $\alpha ' = \alpha '(\Gamma )>0$ such that for every $x \in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, for every $0<s\le T^{{\varepsilon }/({1-\varepsilon })}$ , every $0<\xi \ll _{\Gamma } 1,$ and every $T\gg _{\Gamma ,\varepsilon } s_0$ ,

$$ \begin{align*} \frac{\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(\xi+T) )}{\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T))}-1\ll_{\Gamma} \xi^{\alpha'}.\end{align*} $$

Theorem 5.2. There exist $\alpha '=\alpha '(\Gamma )>0$ , $\theta '=\theta '(\Gamma )\ge \alpha '$ , $\omega '=\omega '(\Gamma )\ge 2\delta _{\Gamma }$ , such that for any $g \in G$ with $g^- \in \Lambda (\Gamma )$ and $0<\xi <\eta \ll _{\Gamma } e^{-\operatorname {height}(g\Gamma )},$ we have that

$$ \begin{align*}\frac{\nu(P_{\xi+\eta}g)}{\nu(P_{\eta} g)}-1\ll_{\Gamma} e^{\omega'\operatorname{height}(g\Gamma)}\frac{\xi^{\alpha'}}{\eta^{\theta'}}.\end{align*} $$

Theorem 5.2 will be obtained as a corollary of the following.

Proposition 5.3. There exist constants $\alpha = \alpha (\Gamma )>0$ , $\theta =\theta (\Gamma )\ge \alpha $ , and $\omega =\omega (\Gamma )\ge 2\delta _{\Gamma }$ such that for $x \in G/\Gamma $ , which satisfies $x^+\in \Lambda (\Gamma )$ , and $0<\xi < \eta \ll _{\Gamma } e^{-\operatorname {height}(x)}$ , we have

$$ \begin{align*}\frac{\mu^{\operatorname{PS}}_x(B_U(\xi+\eta))}{\mu^{\operatorname{PS}}_x(B_U(\eta))}-1 \ll_{\Gamma} e^{\omega\operatorname{height}(x)}\frac{\xi^{\alpha}}{\eta^{\theta}}.\end{align*} $$

We first show how to obtain Theorem 5.2 from Proposition 5.3.

Proof of Theorem 5.2 assuming Proposition 5.3

Using the product structure of $\nu $ , we can write

$$ \begin{align*}\nu(P_{\eta} g) = \int_{A_{\eta}}\int_{M_{\eta}} \mu^{\operatorname{PS}-}_g(B_{\tilde{U}} (\eta))\,dm\,ds.\end{align*} $$

Then, by an analogous statement to Proposition 5.3 for $\mu ^{\operatorname {PS}-}$ , there exists a constant $c_0 = c_0(\Gamma )>0$ such that

$$ \begin{align*} \nu(P_{\eta+\xi}g) &= \int_{A_{\xi+\eta}}\int_{M_{\xi+\eta}}\mu^{\operatorname{PS}-}_g(B_{\tilde U}(\xi + \eta))\,dm\,ds \\[3pt] &\le\int_{A_{\xi+\eta}}\int_{M_{\xi+\eta}}\mu^{\operatorname{PS}-}_g(B_{\tilde U}(\eta))\bigg[1+c_0\frac{\xi^{\alpha}}{\eta^{\theta}} e^{\omega\operatorname{height}(g\Gamma)}\bigg]\, dm\, ds\\ &=\bigg[1+c_0\frac{\xi^{\alpha}}{\eta^{\theta}} e^{\omega\operatorname{height}(g\Gamma)}\bigg]\bigg[ \frac{(\xi+\eta)^{{1}/{2}(n-1)(n-2)+1}}{\eta^{{1}/{2}(n-1)(n-2)+1}}\nu(P_{\eta} g)\bigg]\\[4pt] &\le\bigg[1+c_0\frac{\xi^{\alpha}}{\eta^{\theta}} e^{\omega\operatorname{height}(g\Gamma)}\bigg]\bigg[1+c_1\frac{\xi}{\eta}\bigg]\nu(P_{\eta} g), \end{align*} $$

where $c_1>0$ is an absolute constant (which depends only on n) arising from the binomial theorem. Therefore,

(46) $$ \begin{align} \frac{\nu(P_{\xi+\eta}g)}{\nu(P_{\eta} g)}-1\ll_{\Gamma} e^{\omega\operatorname{height}(g\Gamma)}\frac{\xi^{\alpha}}{\eta^{\theta}}\cdot\frac{\xi}{\eta}+\frac{\xi}{\eta}+e^{\omega\operatorname{height}(g\Gamma)}\frac{\xi^{\alpha}}{\eta^{\theta}}. \end{align} $$

Since $\xi <\eta $ , the first term on the left-hand side of (46) is dominated by the last term, and so

$$ \begin{align*}\frac{\nu(P_{\xi+\eta}g)}{\nu(P_{\eta} g)}-1\ll_{\Gamma}\frac{\xi}{\eta}+e^{\omega\operatorname{height}(g\Gamma)}\frac{\xi^{\alpha}}{\eta^{\theta}}.\end{align*} $$

Since $e^{\omega \operatorname {height}(g\Gamma )}\ge 1$ , if we define

$$ \begin{align*}\alpha'=\min\{1,\alpha\},\quad \theta'=\max\{1,\theta\},\end{align*} $$

then both terms are dominated by

$$ \begin{align*}e^{\omega\operatorname{height}(g\Gamma)}\frac{\xi^{\alpha'}}{\eta^{\theta'}},\end{align*} $$

which completes the proof.

The following result, showing that the PS measure is not concentrated near hyperplanes, is proved in the appendix to improve the readability of this section. See Proposition A.11 for the proof. This result builds upon the work of Das et al. in [5], where it is shown that the PS density $\nu _o$ is friendly when $\Gamma $ is geometrically finite.

For a hyperplane $L \subset U \cong \mathbb R^{n-1}$ and $\xi>0$ , define

$$ \begin{align*}\mathcal{N}_U(L,\xi):=\{u_{\textbf{t}} y: y \in L, \textbf{t} \in B_U(\xi)\}.\end{align*} $$

Proposition 5.4. Let $\Gamma $ be geometrically finite and Zariski dense. There exist constants $\alpha = \alpha (\Gamma )>0, \omega =\omega (\Gamma ) \ge 0, $ and $\theta = \theta (\Gamma )>\alpha $ satisfying the following: for any $x \in G/\Gamma $ with $x^+\in \Lambda (\Gamma )$ , and for every $\xi>0$ and $0<\eta \ll _{\Gamma } e^{-\operatorname {height}(x)}$ , we have that for every hyperplane L,

$$ \begin{align*}\mu^{\operatorname{PS}}_x(\mathcal{N}_U(L,\xi)\cap B_U(\eta)) \ll_{\Gamma} e^{\omega \operatorname{height}(x)} \frac{\xi^{\alpha}}{\eta^{\theta}}\mu^{\operatorname{PS}}_x(B_U(\eta)).\end{align*} $$

We are now ready to prove Proposition 5.3.

Proof of Proposition 5.3

It follows from the geometry of $B_U(\xi +\eta )x-B_U(\eta )x$ that there exist hyperplanes $L_1,\ldots ,L_m$ , where m only depends on n, such that

$$ \begin{align*} B_U(\xi+\eta)x-B_U(\eta)x\subseteq\bigcup_{i=1}^{m} \mathcal{N}_U(L_i,2\xi).\end{align*} $$

For any $0<\xi <\eta \ll _{\Gamma } e^{-\operatorname {height}(x)},$ we have that

By (37), $\mu ^{\operatorname {PS}}_x$ is D-Federer (see Corollary A.8 for more detail), in particular,

$$ \begin{align*}\mu^{\operatorname{PS}}_x(B_U(2\eta)) \ll_{\Gamma} \mu^{\operatorname{PS}}_x(B_U(\eta)).\end{align*} $$

Thus, we obtain

$$ \begin{align*} \frac{\mu^{\operatorname{PS}}_x(B_U(\xi+\eta))}{\mu^{\operatorname{PS}}_x(B_U(\eta))} -1 \ll_{\Gamma}e^{{\omega}\operatorname{height}(x)}\frac{ \xi^{\alpha}}{\eta^{\theta}},\end{align*} $$

and relabeling the constants completes the proof.

In (37), we saw that $\mu ^{\operatorname {PS}}_x$ is Federer when $x \in \operatorname {supp}m^{\operatorname {BMS}}.$ Below, we show that $\mu ^{\operatorname {PS}}_x$ satisfies a similar condition for sufficiently large balls when x is Diophantine, but not necessarily a BMS point.

Corollary 5.5. There exists a constant $\sigma =\sigma (\Gamma )\ge \delta _{\Gamma }$ such that for every $c \ge 1$ and every $x \in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, if $T \gg _{\Gamma ,\varepsilon } s_0 $ , then

$$ \begin{align*}\mu^{\operatorname{PS}}_x(B_U(cT))\ll_{\Gamma} c^{\sigma} \mu^{\operatorname{PS}}_x(B_U(T)).\end{align*} $$

Proof. By Lemma 3.8, for some $T_0\gg _{\Gamma ,\varepsilon } s_0$ , there exists

$$ \begin{align*}y \in B_U(T_0)x\cap\operatorname{supp}m^{\operatorname{BMS}}.\end{align*} $$

Then for $T \ge T_0$ , we have

$$ \begin{align*}B_U(T-T_0)y\subseteq B_U(T)x\subseteq B_U(T+T_0)y.\end{align*} $$

Since $c \ge 1,$ we therefore have that for $T \ge 2T_0,$

Remark 5.6. Observe that if x is $(\varepsilon ,s_0)$ -Diophantine and $T \gg _{\Gamma ,\varepsilon } s_0$ , then T is sufficiently large to use Corollary 5.5 on $a_{-s}x$ for $s>0.$ To see this, observe that in the notation of the proof of Corollary 5.5, $T_0$ is such that for any $(\varepsilon ,s_0)$ -Diophantine point x, there exists $y \in \operatorname {supp}m^{\operatorname {BMS}}$ and $\textbf {t}\le T_0$ so that $x=u_{\textbf {t}} y$ . Then

$$ \begin{align*}a_{-s}x = a_{-s}u_{\textbf{t}} y=u_{e^{-s}\textbf{t}}a_{-s}y.\end{align*} $$

Thus, the distance to the nearest BMS point in the U orbit shrinks, and so T is still sufficiently large.

Proposition 5.7. Let $H_R = \{y \in G/\Gamma : \operatorname {height}(y)\le R\}$ . There exist constants $\alpha = \alpha (\Gamma )>0,$ and $\omega =\omega (\Gamma )\ge 0$ such that for every $x \in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine and for every $0<\xi <1/2,$ and $T\gg _{\Gamma ,\varepsilon } s_0$ ,

$$ \begin{align*} \frac{\mu^{\operatorname{PS}}_x((B_U(\xi+T) \cap H_R)-(B_U(T)\cap H_R) )}{\mu^{\operatorname{PS}}_x(B_U(T))}\ll_{\Gamma} e^{\omega R}{\xi^{\alpha}}.\end{align*} $$

Proof. Let $T_0\gg _{\Gamma ,\varepsilon } s_0$ satisfy the conclusion of Lemma 3.8. For $T \ge T_0$ , let

$$ \begin{align*}E_T:=\mathcal{N}(L,\xi)\cap B_U(T) \cap H_R\cap \operatorname{supp}m^{\operatorname{BMS}},\end{align*} $$

and observe that $\mu ^{\operatorname {PS}}_x(E_T)=\mu ^{\operatorname {PS}}_x(\mathcal {N}(L,\xi )\cap B_U(T)\cap H_R).$

Let $c_1=c_1(\Gamma )>0$ be the implied constant in Proposition 5.4. Fix $r=c_1e^{-R}$ and let $\{u_1,\ldots ,u_k\}$ be a maximal ${r}/{2}$ -separated set in $E_{T-{r}/{4}}$ . Then,

$$ \begin{align*}E_T\subseteq\bigcup_{i=1}^k B_U(r)u_i.\end{align*} $$

Note also that by (37), we have that there exists a constant $c_2=c_2(\Gamma )>0$ such that for all $u_i$ ,

(47) $$ \begin{align} \mu^{\operatorname{PS}}_{u_i}(B_U(r))=\mu^{\operatorname{PS}}_{u_i}(B_U(8(r/8)) \le c_2\mu^{\operatorname{PS}}_{u_i}(B_U(r/8)). \end{align} $$

Therefore,

By Corollary 5.5, there exists $\sigma =\sigma (\Gamma ) \ge \delta _{\Gamma }$ so that

$$ \begin{align*}\mu^{\operatorname{PS}}_x(B_U(T+1)) \subseteq \mu^{\operatorname{PS}}_x(B_U(2T)) \ll_{\Gamma} 2^{\sigma} \mu^{\operatorname{PS}}_x(B_U(T)).\end{align*} $$

Let

$$ \begin{align*}\omega' = \omega+\theta.\end{align*} $$

It follows from the geometry of $B_U(\xi +T)x-B_U(T)x$ that there exist $L_1,\ldots ,L_m$ , where m only depends on n, such that

$$ \begin{align*} B_U(\xi+T)x-B_U(T)x\subseteq\bigcup_{i=1}^{m} \mathcal{N}_U(L_i,2\xi).\end{align*} $$

Thus, we also have

$$ \begin{align*}(B_U(\xi + T)x - B_U(T)x) \cap H_R\subseteq\bigcup_{i=1}^{m} \mathcal{N}_U(L_i,2\xi).\end{align*} $$

We arrive at

$$ \begin{align*} \frac{\mu^{\operatorname{PS}}_x((B_U(\xi+T) \cap H_R)-(B_U(T)\cap H_R) )}{\mu^{\operatorname{PS}}_x(B_U(T))} & \le \sum_{i=1}^m\frac{\mu^{\operatorname{PS}}_x(\mathcal{N}(L_i,2\xi)\cap B_U(\xi+T))}{\mu^{\operatorname{PS}}_x(B_U(T))}\\[4pt] & \ll_{\Gamma}m e^{\omega'R}{\xi^{\alpha}} \frac{\mu^{\operatorname{PS}}_x( B_U(\xi+T))}{\mu^{\operatorname{PS}}_x(B_U(T))}. \end{align*} $$

By Corollary 5.5 again, we conclude that

$$ \begin{align*} \frac{\mu^{\operatorname{PS}}_x((B_U(\xi+T) \cap H_R)-(B_U(T)\cap H_R) )}{\mu^{\operatorname{PS}}_x(B_U(T))} \ll_{\Gamma}e^{\omega'R}\xi^{\alpha},\end{align*} $$

which completes the proof.

We are now ready to prove Theorem 5.1.

Proof of Theorem 5.1

Observe that by Lemma 3.6, for any $R>R_0$ ,

(48) $$ \begin{align} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T))&=\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T)\cap H_{R-R_0}) + \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T)\cap \mathcal{X}(R)),\nonumber\\[4pt] \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi))&=\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi)\cap H_{R-R_0}) + \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi)\cap \mathcal{X}(R)). \end{align} $$

By Theorem 4.1, for $T \gg _{\Gamma ,\varepsilon } s_0$ , $0<s\le T^{{\varepsilon }/({1-\varepsilon })}$ , and any $R \ge R_0,$

$$ \begin{align*} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi)\cap \mathcal{X}(R))&= \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U((s(T+\xi)/s) \cap \mathcal{X}(R)) \\[4pt] &\ll_{\Gamma} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi))e^{-\beta R}\\[4pt] &\ll_{\Gamma} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T))e^{-\beta R} \quad\text{by Corollary 5.5}. \end{align*} $$

Observe that use of Corollary 5.5 is justified if $T \gg _{\Gamma ,\varepsilon } s_0$ by Remark 5.6. Similarly, by Proposition 5.7 and the same reasoning as in Remark 5.6, for $T \gg _{\Gamma ,\varepsilon } s_0$ , we have

$$ \begin{align*} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi)\cap H_{R-R_0})-\mu^{\operatorname{PS}}_{a_{-s}x}(B_U(T)\cap H_{R-R_0})\ll_{\Gamma} e^{\omega R}\xi^{\alpha} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T)). \end{align*} $$

Putting this together with (48), we conclude

$$ \begin{align*} &\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T+\xi))-\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T))\\[4pt] &\quad\ll_{\Gamma} e^{\omega R}\xi^{\alpha} \mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T)) + 2e^{-\beta R}\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T))\\[4pt] &\quad\ll_{\Gamma} (e^{\omega R}\xi^{\alpha} +2e^{-\beta R})\mu^{\operatorname{PS}}_{a_{-\log s}x}(B_U(T)). \end{align*} $$

Taking $R=-{\alpha }/({\omega +\beta })\log \xi $ implies the result, provided that $\xi $ is sufficiently small so that this is larger than $R_0.$ Note that since $\alpha ,\omega ,\beta , R_0$ are all constants depending only on $\Gamma $ , this is equivalent to requiring $\xi \ll _{\Gamma } 1.$

6 Proof of Theorem 1.5

In this section, we keep the notation of §3.2. In particular, d denotes the hyperbolic distance, $\operatorname {height}$ is the height of a point in the convex core into the cusps, and $\mathcal {C}_0$ is the fixed compact set in $G/\Gamma $ which is defined in §3.2.

We will first prove the following proposition, which is a form of Theorem 1.5 for G. Theorem 1.5 will follow by a partition of unity argument.

Proposition 6.1. There exist $\kappa =\kappa (\Gamma )$ and $\ell =\ell (\Gamma )$ which satisfy the following: let $0<r<1$ , $\psi \in C_c^{\infty }(G)$ supported on an admissible box, and $f\in C_c^{\infty }(B_U (r))$ . Then, there exists $c=c(\Gamma ,\operatorname {supp}\psi )>0$ such that for any $g\in \operatorname {supp}\tilde {m}^{\operatorname {BMS}}$ , and $s\gg _{\Gamma } \operatorname {height}(g\Gamma ),$ we have

$$ \begin{align*} &\bigg|\sum\limits_{\gamma \in \Gamma}\int_{U}\psi(a_su_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t})- \mu^{\operatorname{PS}}_g(f)\tilde{m}^{\operatorname{BMS}}(\psi)\bigg|\\[3pt] &\quad<cS_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s}\mu^{\operatorname{PS}}_g(B_U(1)). \end{align*} $$

Proof. Without loss of generality, assume that f and $\psi $ are non-negative functions. Step 1: Setup and approximations.

Let $\kappa ',\ell '$ satisfy the conclusion of Assumption 1.1, and let $\ell>\ell '$ satisfy the conclusion of Lemma 3.12. Observe that $\ell $ can be increased if necessary while maintaining this property.

Because $\psi $ is supported on an admissible box, there exists $0<\eta _0<1/2$ (depending on $\operatorname {supp}\psi $ ) such that $G_{3\eta _0}\operatorname {supp}\psi $ is still an admissible box. For $0<\eta <\eta _0$ , let $\psi _{\eta ,\pm }$ satisfy the conclusion of Lemma 3.12 for $G,3\eta $ , and $\psi $ . In particular, for all small $\eta>0$ ,

(49) $$ \begin{align} S_{\ell'}(\psi_{\eta,\pm})\ll_{\operatorname{supp}\psi}\eta^{-2\ell}S_{\ell}(\psi). \end{align} $$

Since $\psi $ is uniformly continuous and the BMS measure is finite, we may deduce from Lemma 3.12(2) that

(50) $$ \begin{align} |\tilde{m}^{\operatorname{BMS}}(\psi_{\eta,\pm})-\tilde{m}^{\operatorname{BMS}}(\psi)|\ll_{\operatorname{supp}\psi,\Gamma}\eta S_{\ell}(\psi). \end{align} $$

According to Lemma 3.1, for any $p \in P_{\eta }$ , there exists $\rho _p : B_U(1) \to B_U(1+O(\eta ))$ that is a diffeomorphism onto its image and a constant $D=D(\eta )<3\eta $ such that

(51) $$ \begin{align}u_{\textbf{t}} p^{-1} \in P_D u_{\rho_p(\textbf{t})}.\end{align} $$

Step 2: Assuming that f is supported on a small ball.

We start by proving that there exists $\kappa> 0$ such that if $f\in C_c^{\infty }(B_U(r_1))$ , where $r_1\le \operatorname {inj}(g)$ , then for $s>0$ ,

(52) $$ \begin{align} & \sum_{\gamma\in\Gamma}\int_U\psi(a_s u_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g (\textbf{t})-\tilde{m}^{\operatorname{BMS}}(\psi)\mu^{\operatorname{PS}}_g(f)\nonumber\\[3pt] &\quad\ll_{\Gamma,\operatorname{supp}\psi}S_{\ell}(\psi)S_{\ell}(f)e^{-2\kappa s}\mu^{\operatorname{PS}}_g(B_U(1)). \end{align} $$

For any $s>0$ and $\gamma \in \Gamma $ , from (51), we have that

$$ \begin{align*} &\int_{B_U(r_1)} \psi(a_s u_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t}) \\[3pt] &\quad= \frac{1}{\nu(P_{\eta} g)}\int_{P_{\eta} g}\int_{B_U(r_1)}\psi(a_{s}u_{\textbf{t}} p^{-1} pg\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\,d\nu(pg)\\[3pt] &\quad\le \frac{1}{\nu(P_{\eta} g)} \int_{P_{\eta} g}\int_{B_U(r_1)} \psi_{\eta,+}(a_{s}u_{\rho_p(\textbf{t})}pg\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\,d\nu(pg), \end{align*} $$

where the last inequality follows since $a_s P_{3\eta }a_{-s}\subset P_{3\eta }$ for any positive s.

Step 2.1: Use the product structure of the BMS measure. For any $p\in P_{\eta }$ , $(u_{\textbf {t}} g)^+= (u_{\rho _p (\textbf {t})}pg)^+$ , the measures $d\mu ^{\operatorname {PS}}_{g}(\textbf {t})$ and $d({\rho _p}_{\ast }\mu ^{\operatorname {PS}}_{pg}(\textbf {t}))=d\mu ^{\operatorname {PS}}_{pg}(\rho _p (\textbf {t}))$ are absolutely continuous with each other, and the Radon–Nikodym derivative at $\textbf {t}$ is given by

(53) $$ \begin{align} \frac{d\mu^{\operatorname{PS}}_{g}(\textbf{t})}{d\mu^{\operatorname{PS}}_{pg}(\rho_p (\textbf{t}))}=e^{\delta_{\Gamma}\beta_{(u_{\textbf{t}} g)^+}(u_{\textbf{t}} g(o),u_{\rho_p(\textbf{t})}pg(o))}. \end{align} $$

Let $0<\xi <\eta $ . Let $\chi _{\eta ,\xi }$ satisfy the conclusion of Lemma 3.10 for $H=P$ , $\xi _1=\eta -\xi $ , $\xi _2=\xi $ , and g. Let $\varphi _{\eta ,g}$ be the function defined on $B_U (1) P_{\eta }g$ given by

$$ \begin{align*} \varphi_{\eta,g}(u_{\rho_p(\textbf{t})}pg):=\frac{f(\textbf{t})\chi_{\eta,\xi}(pg)}{\nu(P_{\eta} g)e^{\delta_{\Gamma}\beta_{(u_{\textbf{t}} g)^+}(u_{\textbf{t}} g(o),u_{\rho_p(\textbf{t})}pg(o))}}.\end{align*} $$

We will need a bound on $S_{\ell }(\varphi _{\eta ,g}).$ To that end, note that

$$ \begin{align*} |\beta_{(u_{\textbf{t}} g)^+}(u_{\textbf{t}} g(o), u_{\rho_p(\textbf{t})} pg(o))| &\le d(u_{\textbf{t}} g(o), u_{\rho_p(\textbf{t})}pg(o))\\[3pt] &= d(g(o), u_{-\textbf{t}} u_{\rho_p(\textbf{t})}pg(o)). \end{align*} $$

Since $u_{-\textbf {t}} u_{\rho _p(\textbf {t})}p\in G_{5\eta }$ , the above is bounded by some absolute constant (depending only on $\Gamma $ ) for all $\eta <\tfrac 12$ . Observe that this bound holds on the support of $f(\textbf {t})\chi _{\eta ,\xi }(pg).$ Moreover, because the Busemann function is Lipschitz, all of its derivatives are bounded.

It then follows immediately from the product rule and the definition of the Sobolev norm that

(54) $$ \begin{align}S_{\ell}(f(\textbf{t})\chi_{\eta,\xi}(pg)\exp(-\delta_{\Gamma} \beta_{(u_{\textbf{t}} g)^+}(u_{\textbf{t}} g(o),u_{\rho_p(\textbf{t})}pg(o)))) \ll_{\Gamma,\ell} S_{\ell}(f(\textbf{t})\chi_{\eta,\xi}(pg)).\end{align} $$

By [17, Lemma 2.4.7(a)], Lemma 3.10, (54), and Lemma 3.9, we have

(55) $$ \begin{align} S_{\ell}(\varphi_{\eta,g})&\ll_{\Gamma,\ell} \nu(P_{\eta} g)^{-1} S_{\ell}(f)S_{\ell}(\chi_{\eta,\xi})\nonumber\\[3pt] &\ll_{\Gamma,\ell} \eta^{-(\delta_{\Gamma} + {1}/{2}(n-1)(n-2)+1)}e^{(\delta_{\Gamma} - k_2(x,\eta))d(\pi(\mathcal{C}_0),\pi(a_{\log \eta}g))}S_{\ell}(\chi_{\eta,\xi})S_{\ell}(f)\nonumber \\[3pt] &\ll_{\Gamma,\ell} e^{\delta_{\Gamma}(|\log \eta|+\operatorname{height}(g\Gamma))} \eta^{-(\delta_{\Gamma} + {1}/{2}(n-1)(n-2)+1)}\eta^{n-1} {\xi}^{-\ell-(n-1)/2} S_{\ell}(f)\nonumber\\[3pt] &\ll_{\Gamma,\ell} e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)}\eta^{-(2\delta_{\Gamma} + {1}/{2}(n-1)(n-2)+1)}\eta^{n-1} {\xi}^{-\ell-(n-1)/2} S_{\ell}(f)\nonumber\\[3pt] &\ll_{\Gamma,\ell} e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)} \eta^{4n-({1}/{2})n^2-3-2\delta_{\Gamma}}\xi^{-\ell-(n-1)/2}S_{\ell}(f). \end{align} $$

Note that the dependence on $\ell $ arises from the exponential of the Busemann function in the denominator.

Also, using the product structure of $\tilde {m}^{\operatorname {BMS}}$ in (25), we get

$$ \begin{align*} & \frac{1}{\nu(P_{\eta}g)} \int_{P_{\eta} g}\int_{B_U(r_1)} \psi_{\eta,+}(a_{s}u_{\rho_p(\textbf{t})}pg\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\,d\nu(pg)\\[3pt] &\quad= \frac{1}{\nu(P_{\eta}g)}\int_{P_{\eta} g}\int_{B_U(r_1)} \psi_{\eta,+}(a_{s}u_{\rho_p(\textbf{t})}pg\gamma)f(\textbf{t})\frac{d\mu^{\operatorname{PS}}_{g}(\textbf{t})}{d\mu^{\operatorname{PS}}_{pg}(\rho_p (\textbf{t}))}\,d\mu^{\operatorname{PS}}_{pg}(\rho_p (\textbf{t}))\,d\nu(pg)\\[3pt] &\quad\leq\int_G \psi_{\eta,+}(a_{s}h\gamma)\varphi_{\eta,g}(h)\,d\tilde{m}^{\operatorname{BMS}}(h). \end{align*} $$

Step 2.2: Use the exponential mixing assumption.

By defining $\Psi _{\eta ,+}(h\Gamma ) = \sum \nolimits _{\gamma \in \Gamma } \psi _{\eta ,+}(h\gamma )$ and $\Phi _{\eta ,g}(h\Gamma ):=\sum \nolimits _{\gamma \in \Gamma }\varphi _{\eta ,g}(h\gamma ),$ we obtain

$$ \begin{align*} \sum_{\gamma\in\Gamma}\int_G\psi_{\eta,+}(a_sh\gamma)\varphi_{\eta,g}(h)\,d\tilde{m}^{\operatorname{BMS}}(h)\leq\int_{X} \Psi_{\eta,+}(a_s x)\Phi_{\eta,g}(x)\,dm^{\operatorname{BMS}}(x)\end{align*} $$

for any positive s. Note that

(56) $$ \begin{align} S_{\ell'}(\Psi_{\eta,+})=S_{\ell'}(\psi_{\eta,+})\quad\text{and}\quad S_{\ell'}(\Phi_{\eta,g})=S_{\ell'}(\varphi_{\eta,g}). \end{align} $$

In particular, (49) and (55) imply

(57) $$ \begin{align} &S_{\ell'}(\Psi_{\eta,+})\ll_{\operatorname{supp}\psi}\eta^{-2\ell}S_{\ell}(\psi)\quad\text{and}\nonumber\\[3pt] &S_{\ell'}(\Phi_{\eta,g}) \ll_{\Gamma} e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)} \eta^{4n-({1}/{2})n^2-3-2\delta_{\Gamma}}\xi^{-\ell-(n-1)/2}S_{\ell}(f). \end{align} $$

By Assumption 1.1,

$$ \begin{align*} &\int\Psi_{\eta,+}(a_{s}x)\Phi_{\eta,g}(x)\,dm^{\operatorname{BMS}}(x)-m^{\operatorname{BMS}}(\Psi_{\eta,+})m^{\operatorname{BMS}}(\Phi_{\eta,g})\\ &\quad\ll_{\Gamma} S_{\ell'}(\Psi_{\eta,+})S_{\ell'}(\Phi_{\eta,g})e^{-\kappa' s}. \end{align*} $$

Then, by (57), there exists $c_1=c_1(\Gamma ,\operatorname {supp}\psi )$ such that

$$ \begin{align*} &\sum_{\gamma\in\Gamma}\int_{B_U(r_1)} \psi(a_su_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t})\\[3pt] &\quad<m^{\operatorname{BMS}}(\Psi_{\eta,+})m^{\operatorname{BMS}}(\Phi_{\eta,g})\\[3pt] &\qquad+c_1e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)} \eta^{4n-({1}/{2})n^2-3-2\delta_{\Gamma}}\xi^{-\ell-(n-1)/2}S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa' s}. \end{align*} $$

Step 2.3: Rewrite in terms of $\psi $ and f.

Using Lemma 3.10 and (53), one can calculate

$$ \begin{align*} m^{\operatorname{BMS}}(\Phi_{\eta,g})&=\int_{G}\varphi_{\eta,g}(h)\,d\tilde{m}^{\operatorname{BMS}}(h)\\[3pt] &=\frac{1}{\nu(P_{\eta} g)}\int_{Pg}\int_U\frac{f(\textbf{t})\chi_{\eta,\xi}(p)}{e^{\delta_{\Gamma}\beta_{(u_{\textbf{t}} g)^+}(u_{\textbf{t}} g(o),u_{\rho_p(\textbf{t})}pg(o))}}\,d\mu^{\operatorname{PS}}_{pg}(\rho_p (\textbf{t}))\,d\nu(pg)\\[3pt] &=\frac{1}{\nu(P_{\eta} g)}\int_{Pg}\int_U f(\textbf{t})\chi_{\eta,\xi}(p)\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\,d\nu(pg)\\[3pt] &\leq\frac{\nu(P_{\eta+\xi} g)}{\nu(P_{\eta} g)}\int_{B_U(r_1)}f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t}). \end{align*} $$

Thus, by Theorem 5.2, there exist $\alpha ,\theta ,\omega ,c_0>0$ depending only on $\Gamma $ such that for any $0<\xi <\eta \ll _{\Gamma } e^{-\operatorname {height}(g\Gamma )}$ ,

$$ \begin{align*} m^{\operatorname{BMS}}(\Phi_{\eta,g})&\leq \bigg(1+c_2e^{\omega\operatorname{height}(g\Gamma)} \frac{\xi^{\alpha}}{\eta^{\theta}}\bigg)\int_{B_U(r_1)}f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\\[3pt] &=\bigg(1+c_2e^{\omega\operatorname{height}(g\Gamma)} \frac{\xi^{\alpha}}{\eta^{\theta}}\bigg)\int_{B_U(r_1)}f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\\[3pt] &=\bigg(1+c_2e^{\omega\operatorname{height}(g\Gamma)} \frac{\xi^{\alpha}}{\eta^{\theta}}\bigg)\mu^{\operatorname{PS}}_g(f). \end{align*} $$

Using (50), we get that there exists $c_3=c_3(\Gamma ,\operatorname {supp}\psi )$ such that

$$ \begin{align*} m^{\operatorname{BMS}}(\Psi_{\eta,+})&\leq\int_G\psi_{\eta,+}(g)\,d\tilde{m}^{\operatorname{BMS}}(g)\\[4pt] &<\tilde{m}^{\operatorname{BMS}}(\psi)+c_3\eta S_{\ell}(\psi). \end{align*} $$

To summarize, we have

$$ \begin{align*} &\sum_{\gamma\in\Gamma}\int_{B_U(r_1)}\psi(a_s u_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t})\\[3pt] &\quad\le \frac{1}{\nu(P_{\eta} g)}\sum_{\gamma\in\Gamma}\int_{P_{\eta} g}\int_{B_U(r_1)} \psi_{\eta,+}(a_{s}u_{\rho_p(\textbf{t})}pg\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_{g}(\textbf{t})\,d\nu(pg)\\[3pt] &\quad\leq\sum_{\gamma\in\Gamma}\int_G \psi_{\eta,+}(a_{s}h\gamma)\varphi_{\eta,g}(h)\,d\tilde{m}^{\operatorname{BMS}}(h)\\ &\quad\leq\int_{X} \Psi_{\eta,+}(a_s x)\Phi_{\eta,g}(x)\,dm^{\operatorname{BMS}}(x)\\[3pt] &\quad<m^{\operatorname{BMS}}(\Psi_{\eta,+})m^{\operatorname{BMS}}(\Phi_{\eta,g})+c_1{\eta}^{4n-({1}/{2})n^2-3-\delta_{\Gamma}-2\ell}{\xi}^{-\ell -(n-1)/2}S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa' s}\\[3pt] &\quad<(\tilde{m}^{\operatorname{BMS}}(\psi)+c_3\eta S_{\ell}(\psi))\bigg(\bigg(1+c_2e^{\omega\operatorname{height}(g\Gamma)} \frac{\xi^{\alpha}}{\eta^{\theta}}\bigg)\mu^{\operatorname{PS}}_g(f)\bigg)\\[3pt] & \qquad+c_1e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)} \eta^{4n-({1}/{2})n^2-3-2\delta_{\Gamma}}\xi^{-\ell-(n-1)/2}S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa' s}. \end{align*} $$

It follows from the proof of Lemma 3.12 that $\tilde {m}^{\operatorname {BMS}}(\psi )\ll _{\operatorname {supp}\psi }S_{\ell }(\psi )$ and $\mu ^{\operatorname {PS}}_g(f)\ll S_{\ell }(f)\mu ^{\operatorname {PS}}_g(B_U(1))$ . Then, using Proposition 3.4, we arrive at

$$ \begin{align*} &\sum_{\gamma\in\Gamma}\int_{B_U(r_1)}\psi(a_s u_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t})- \mu^{\operatorname{PS}}_g(f)\tilde{m}^{\operatorname{BMS}}(\psi)\\ &\quad\ll_{\Gamma}\bigg(e^{\omega\operatorname{height}(g\Gamma)} \frac{\xi^{\alpha}}{\eta^{\theta}} +e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)} \eta^{4n-({1}/{2})n^2-3-2\delta_{\Gamma}}\xi^{-\ell-(n-1)/2}e^{-\kappa' s}\bigg)\\ &\qquad\cdot S_{\ell}(\psi)S_{\ell}(f)\mu^{\operatorname{PS}}_g(B_U(1)). \end{align*} $$

Define

$$ \begin{align*}\kappa = \frac{3\alpha\theta\kappa'}{2\theta(2\ell+n-1)+9\alpha(2\delta_{\Gamma}+3+n^2/2-4n)},\end{align*} $$

and note that by making $\ell $ larger if necessary, we guarantee $\kappa>0$ .

Recall from (22) that

$$ \begin{align*}e^{-\operatorname{height}(g\Gamma)}\ll_{\Gamma}\operatorname{inj}(g).\end{align*} $$

For $s \ge \max \{\theta ,\omega \}\operatorname {height}(g\Gamma ) /\kappa $ , choose

(58) $$ \begin{align} \eta=e^{-\kappa s/\theta},\quad\xi=e^{-{4\kappa s}/{\alpha} }. \end{align} $$

Note that $\eta < \operatorname {inj}(g\Gamma )$ by choice of s, $\omega \operatorname {height}(g\Gamma )\le \kappa s$ , and $\xi <\eta $ since by Proposition 5.2, $\alpha <\theta $ . By Proposition 5.2, we have $\omega>\delta _{\Gamma }$ ; therefore, $\delta _{\Gamma } \operatorname {height}(g\Gamma )\le \kappa s$ . Note also that $\max \{\theta ,\omega \}\operatorname {height}(g\Gamma ) /\kappa \ll _{\Gamma } \operatorname {height}(g\Gamma )$ .

With these choices, we obtain

(59) $$ \begin{align}e^{\omega\operatorname{height}(g\Gamma)}\bigg(\frac{\xi}{\eta^{\theta'}}\bigg)^{\alpha'} +e^{\delta_{\Gamma} \operatorname{height}(g\Gamma)} \eta^{4n-({1}/{2})n^2-3-2\delta_{\Gamma}}\xi^{-\ell-(n-1)/2}e^{-\kappa' s} \le 2e^{-2\kappa s}. \end{align} $$

In a similar way, using $\psi _{\eta ,-}$ , one can show a lower bound, proving (52).

Step 3: Covering argument for general f.

We now deduce the claim by decomposing f into a sum of functions, each defined on a ball of radius $r_1$ in U.

Let $u_1,\ldots ,u_k$ and $\sigma _1,\ldots ,\sigma _k\in C_c^{\infty }(B_U(r))$ satisfy the conclusion of Lemma 3.11 for $E=B_U(r)$ and $r_1$ . For $1\leq i\leq k$ , let

$$ \begin{align*} f_i:=f\sigma_i.\end{align*} $$

Then, $f\le \sum _{i-1}^k f_i$ , and by Lemma 3.11 and [17, Lemma 2.4.7(a)],

(60) $$ \begin{align} S_{\ell}(f_i)\ll_{\Gamma} S_{\ell}(f)S_{\ell}(\sigma_i)\ll_{\Gamma} r_1^{-\ell+n-1} S_{\ell}(f). \end{align} $$

Since each $f_i$ is supported on $B_U(r_1)u_i$ for some $u_i\in B_U(1)$ , by (52), we have

$$ \begin{align*} & \sum_{\gamma\in\Gamma}\int_{B_U(r_1)}\psi(a_s u_{\textbf{t}} g\gamma)f_i(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t})-\tilde{m}^{\operatorname{BMS}}(\psi)\mu^{\operatorname{PS}}_g(f_i)\\ &\quad \ll_{\Gamma,\operatorname{supp}\psi}\mu^{\operatorname{PS}}_g(B_U(1)) S_{\ell}(\psi)S_{\ell}(f_i)e^{-2\kappa s}. \end{align*} $$

Summing the above expressions for $i=1,\ldots ,k$ , we get

$$ \begin{align*} &\sum_{\gamma\in\Gamma}\int_{B_U(r)}\psi(a_s u_{\textbf{t}} g\gamma)f(\textbf{t})\,d\mu^{\operatorname{PS}}_g(\textbf{t})-\tilde{m}^{\operatorname{BMS}}(\psi)\mu^{\operatorname{PS}}_g(f)\\[3pt] &\quad \ll k r_1^{-\ell+n-1}S_{\ell}(\psi)S_{\ell}(f)e^{-2\kappa s}\mu^{\operatorname{PS}}_g(B_U(1))\\[3pt] &\quad \ll\bigg(\frac{r}{r_1}\bigg)^{n-1} r_1^{-\ell+n-1}S_{\ell}(\psi)S_{\ell}(f)e^{-2\kappa s}\mu^{\operatorname{PS}}_g(B_U(1))\\[3pt] &\quad \ll r_1^{-\ell}S_{\ell}(\psi)S_{\ell}(f)e^{-2\kappa s}\mu^{\operatorname{PS}}_g(B_U(1))\\[3pt] &\quad \ll S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s}\mu^{\operatorname{PS}}_g(B_U(1)), \end{align*} $$

where the first inequality is by Lemma 3.11, the second inequality follows from $r_1=\operatorname {inj}(g)>e^{-\kappa s/\ell }$ , the third is by (58) and because $r<1$ , and the implied constants depend on $\Gamma $ and $\operatorname {supp}\psi $ .

As before, using similar arguments, one can show a lower bound, proving the claim.

We will now use a partition of unity argument to prove Theorem 1.5. For the reader’s convenience, we restate it in the following.

Theorem 6.2. There exist $\kappa = \kappa (\Gamma )$ and $\ell = \ell (\Gamma )$ which satisfy the following: for any $\psi \in C_c^{\infty }(X)$ , there exists $c = c(\Gamma , \operatorname {supp}\psi )>0$ such that for any $f \in C_c^{\infty }(B_U(r))$ , $0<r<1$ , $x \in \operatorname {supp}m^{\operatorname {BMS}}$ , and $s\gg _{\Gamma } \operatorname {height}(x)$ , we have

$$ \begin{align*} \bigg|\int_U \psi(a_s u_{\textbf{t}} x)f(\textbf{t})\,d\mu^{\operatorname{PS}}_x(\textbf{t}) - \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BMS}}(\psi)\bigg| < cS_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s}. \end{align*} $$

Proof. According to [26, Lemma 2.17], there exists an admissible box $B_y$ around y for any $y\in X$ . Then, $\{B_y : y\in \operatorname {supp}\psi \}$ is an open cover of the compact set $\operatorname {supp}\psi $ . Hence, there exists a minimal sub-cover $B_{y_1},\ldots ,B_{y_k}$ . Using a similar construction to that in Lemma 3.11, there exist $\sigma _1,\ldots , \sigma _k$ , a partition of unity for $\operatorname {supp}\psi $ , such that for $i=1,\ldots ,k$ , we have $\sigma _i \in C_c^{\infty }(B_{y_i})$ and for $i=1,\ldots ,k$ and $m=1,\ldots ,\ell $ ,

(61) $$ \begin{align} |\sigma_i^{(m)}|\ll_{\operatorname{supp}\psi,\Gamma} 1 \end{align} $$

(the implied constant depends on the chosen sub-cover).

Define $\psi _i=\psi \sigma _i$ . Then

(62) $$ \begin{align} \psi=\sum_{i=1}^k\psi_i, \end{align} $$

and by (61) and the product rule, we have

(63) $$ \begin{align} S_{\ell}(\psi_i)\ll_{\operatorname{supp}\psi,\Gamma}S_{\ell}(\psi). \end{align} $$

According to Proposition 6.1 and Proposition 3.4, there exist $c=c(\Gamma ,\operatorname {supp}\psi )>0$ , $\unicode{x3bb} =\unicode{x3bb} (\Gamma )>1$ such that for $s\gg _{\Gamma } \operatorname {height}(x)$ ,

$$ \begin{align*} &\int_{B_U(r)} \psi(a_s u_{\textbf{t}} x)f(\textbf{t})d\textbf{t}\\ &\quad= \sum\limits_{i=1}^k \int_{B_U(r)} \psi_i(a_s u_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t} \\ &\quad \le \sum\limits_{i=1}^k m^{\operatorname{BMS}}(\psi_i) \mu^{\operatorname{PS}}_x(f) + c S_{\ell}(\psi_i)S_{\ell}(f)e^{-\kappa s}\mu^{\operatorname{PS}}_x(B_U(1))\\ &\quad \le \sum\limits_{i=1}^k m^{\operatorname{BMS}}(\psi_i) \mu^{\operatorname{PS}}_x(f) + c \unicode{x3bb} S_{\ell}(\psi_i)S_{\ell}(f)e^{-\kappa s+(n-1-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(x))}\\ &\quad\ll_{\Gamma,\operatorname{supp}\psi} m^{\operatorname{BMS}}(\psi) \mu^{\operatorname{PS}}_x(f) + c \unicode{x3bb} S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s+(n-1-\delta_{\Gamma})\operatorname{height}(x)}, \end{align*} $$

where the last line follows by the definition of $\operatorname {height}(x)$ , and (62) and (63). Moreover, we may assume that $s \ge {2(n-1-\delta _{\Gamma })}/{\kappa }\operatorname {height}(x)$ without changing the assumption $s\gg _{\Gamma } \operatorname {height}(x)$ . Then,

$$ \begin{align*}e^{-\kappa s+(n-1-\delta_{\Gamma})\operatorname{height}(x)} \ll_{\Gamma} e^{-\kappa s/2 },\end{align*} $$

as desired.

We will now use Theorem 1.5 to prove a similar result for the Haar measure. This will be necessary for the proof of Theorem 1.4. Note that such a result is proven in [23] under a spectral gap assumption on $\Gamma $ , but we show here how to prove it whenever the frame flow is exponentially mixing.

Theorem 6.3. There exists $\kappa = \kappa (\Gamma )<1$ and $\ell = \ell (\Gamma )$ that satisfy the following: let $0<r<1$ , let $f \in C_c^{\infty }(B_U(r))$ , and let $\psi \in C_c^{\infty }(X)$ be supported on an admissible box. Then there exists $c = c(\Gamma ,\operatorname {supp}\psi )>0$ such that for every $x \in \operatorname {supp}m^{\operatorname {BMS}}$ and $s\gg _{\Gamma ,\operatorname {supp}\psi } \operatorname {height}(x)$ ,

$$ \begin{align*} \bigg| e^{(n-1-\delta_{\Gamma})s} \int_{B_U(r)}\psi(a_su_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t} - \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BR}}(\psi)\bigg|< cS_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s}. \end{align*} $$

Proof. Step 1: Setup and approximations. Assume $s \gg _{\Gamma } \operatorname {height}(x)$ , and let $\kappa ,\ell '$ satisfy the conclusion of Theorem 1.5 and $\ell>\ell '$ satisfy the conclusion of Lemma 3.12.

Since $\psi $ is assumed to be supported on an admissible box, there exist $r_0,\eta ,\varepsilon _0,\varepsilon _1>0$ (depending only on $\operatorname {supp} \psi $ ) and $z\in X$ such that

$$ \begin{align*}\operatorname{supp}{\psi}=B_U(r_0) P_{\eta}z,\end{align*} $$

and

$$ \begin{align*}G_{\varepsilon_0}\operatorname{supp}\psi\subset B_U(r_0+\varepsilon_1) P_{\eta+\varepsilon_1}z,\end{align*} $$

where $B_U(r+\varepsilon _1) P_{\eta +\varepsilon _1}z$ is also an admissible box. Denote $\eta '=\eta +\varepsilon _1$ and $r^{\prime }_0=r_0+\varepsilon _1$ .

Without loss of generality, assume that f is a non-negative function. Continuously extend $\psi $ to $P_{\eta '}$ by defining $\psi =0$ on $P_{\eta '}\setminus P_{\eta }$ .

For $0<\varepsilon <\varepsilon _0$ , let $\psi _{\varepsilon ,\pm }$ and $f_{\varepsilon ,\pm }$ for Lemma 3.12 for $G,\varepsilon ,\psi $ and $U,\varepsilon ,f$ respectively. By Lemma 3.12,

(64) $$ \begin{align} S_{\ell'}(\psi_{\varepsilon,\pm})\ll_{\Gamma,\operatorname{supp}(\psi)} \varepsilon^{-2\ell}S_{\ell}(\psi)\quad\text{and}\quad S_{\ell'}(f_{\varepsilon,\pm})\ll_{\Gamma}\varepsilon^{-2\ell}S_{\ell}(f). \end{align} $$

Moreover, by Lemma 3.12(2),

(65) $$ \begin{align} \|f_{\varepsilon,\pm}-f\|_{\infty} \le \varepsilon S_{\ell}(f).\end{align} $$

For $p \in P_{\eta '}$ , define

(66) $$ \begin{align} \varphi(p):= \mu^{\operatorname{PS}}_{pz}(B_U(r^{\prime}_0)pz). \end{align} $$

Step 1.1: Construct a smooth approximation to $1/\varphi $ . Since the Busemann function is smooth and $\varphi $ is bounded below by a positive quantity on $P_{\eta '}$ by Corollary 3.3, the mean value theorem implies that for any $0<\varepsilon <\varepsilon _0$ and all $p, p' \in P_{\varepsilon },$ there exists a constant $d = d(\Gamma ,\operatorname {supp}\psi )$ such that

(67) $$ \begin{align} \bigg|\frac{1}{\varphi(p)} - \frac{1}{\varphi(p')}\bigg| \le \frac{d\varepsilon}{\varphi(p)}.\end{align} $$

By Lemma 3.10, for any $\xi>0$ , there exists a non-negative smooth function $\chi _{\xi }$ with

(68) $$ \begin{align} 1_{P_{\varepsilon}-\xi}\le \chi_{\xi} \le {\mathbf{1}}_{P_{\varepsilon}}\end{align} $$

and $S_{\ell '}(\chi _{\xi }) \ll _{\Gamma ,n} (\varepsilon -\xi /2)^{n-1}(\xi /2)^{-\ell '-(n-1)/2}.$ Define

(69) $$ \begin{align}\sigma(p):= \frac{1}{\varphi}\ast \frac{\chi_{\xi}}{m(P_{\varepsilon-\xi})},\end{align} $$

where m denotes the probability Haar measure on P. Then, assuming $\varepsilon _0<1/2$ and $\xi \le \varepsilon ^2$ , by (67), (68), and (69), we have that

(70) $$ \begin{align} \frac{1-d\varepsilon}{\varphi(p)} & \le\frac{1}{m(P_{\varepsilon-\xi})}\int_{pP_{\varepsilon - \xi}}\frac{1}{\varphi(p')}\,dp' \end{align} $$

(71) $$ \begin{align} & \kern30pt \le \sigma(p) \nonumber\\ & \kern30pt \le \frac{1}{m(P_{\varepsilon-\xi})}\int_{pP_{\varepsilon}}\frac{1+d\varepsilon}{\varphi(p')}\,dp' \nonumber \\ & \le \bigg(\frac{\varepsilon}{\varepsilon-\xi}\bigg)^n\frac{1+d\varepsilon}{\varphi(p)}\nonumber\\[3pt] & \le\frac{1+d'\varepsilon}{\varphi(p)}, \end{align} $$

for some absolute constant $d'>0$ .

For $upz \in B_U(r^{\prime }_0) P_{\eta '} z$ and $0<\varepsilon <\varepsilon _0$ , let

$$ \begin{align*} \Psi_{\varepsilon,\pm}(upz)=\sigma(p)\int_{Upz}\psi_{c_1\varepsilon,\pm}(u_{\textbf{t}}pz)\,d\textbf{t}.\end{align*} $$

Then, by (67),

(72) $$ \begin{align} \sup_{w\in G_{\varepsilon}}\Psi_{\varepsilon,\pm}(wupz) &=\sup_{w\in P_{\varepsilon}}\sigma(wp)\int_{Uwpz}\psi_{c_1\varepsilon,+}(u_{\textbf{t}}wpz)\,d\textbf{t}\nonumber\\[3pt] &\le(1+d'\varepsilon)\Psi_{2\varepsilon,\pm}. \end{align} $$

Step 2: Bounding with PS measure.

Let

$$ \begin{align*}P(f,\psi, x; s) = \{p \in P_{\eta} : a_s\operatorname{supp}(f)x \cap B_U(r_0)pz \ne \emptyset\}.\end{align*} $$

By [23, Lemma 6.2], there exists an absolute constant $c_1>0$ such that

(73) $$ \begin{align} &e^{(n-1)s}\int_{B_U(r)}\psi(a_s u_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t}\\[3pt] &\quad\le(1+c_1\varepsilon)\sum_{p\in P(f,\psi,x;s)}f_{c_1e^{-s}\eta}(a_{-s}pz)\int_{Upz}\psi_{c_1\varepsilon,+}(u_{\textbf{t}}pz)\,d\textbf{t}.\nonumber \end{align} $$

It now follows from [23, Lemma 6.5], (70), and (72) that there exists an absolute constant $c_2>0$ such that

$$ \begin{align*} &e^{-\delta_{\Gamma} s}\sum_{p\in P(f,\psi,x;s)}f_{c_1e^{-s}\eta}(a_{-s}pz)\int_{Upz}\psi_{c_1\varepsilon,+}(u_{\textbf{t}}pz)\,d\textbf{t}\\[3pt] &\quad\le\frac{(1+c_2\varepsilon)(1+d'\varepsilon)}{1-d\varepsilon}\int_U\Psi_{2c_2\varepsilon,+}(a_{s}u_{\textbf{t}}x)f_{(c_1+c_2)e^{-s}\varepsilon_0,+}(\textbf{t})\,d\mu^{\operatorname{PS}}_{x}(\textbf{t}). \end{align*} $$

Note that (70) is needed because our definition of $\Psi _{\varepsilon , +}$ is not identical to $\Psi $ as defined in [23, Lemma 6.5]. The latter is bounded above by ${1}/({1-d\varepsilon }) \Psi _{\varepsilon ,+}$ by (70).

Combining the above with (73), we get that there exist constants $c_3,c_4=c_4(\Gamma ,\operatorname {supp}\psi )>0$ such that

$$ \begin{align*} &e^{(n-1-\delta_{\Gamma})s}\int_{B_U(r)}\psi(a_s u_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t}\\[3pt] &\quad\leq(1+c_4\varepsilon)\int_U\Psi_{c_3\varepsilon,+}(a_{s}u_{\textbf{t}}x)f_{c_3e^{-s}\varepsilon_0,+}(\textbf{t})\,d\mu^{\operatorname{PS}}_{x}(\textbf{t}). \end{align*} $$

It follows from Theorem 1.5 that for some constant $c_5=c_5(\Gamma ,\operatorname {supp}\psi )>0$ ,

(74) $$ \begin{align} &e^{(n-1-\delta_{\Gamma})s}\int_{B_U(r)}\psi(a_s u_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t}\nonumber\\[3pt] &\quad\leq(1+c_4\varepsilon)(\mu^{\operatorname{PS}}_x(f_{c_3e^{-s}\varepsilon_0,+})m^{\operatorname{BMS}}(\Psi_{c_3\varepsilon,+})+c_5S_{\ell'}(\Psi_{c_3\varepsilon,+})S_{\ell'}(f_{c_3e^{-s}\varepsilon_0,+})e^{-\kappa s}). \end{align} $$

Step 3: Bounding the error terms. We now show how to bound the various error terms to obtain the desired conclusion.

To compute $m^{\operatorname {BMS}}(\Psi )$ , we use (25), (65), and (71) to deduce that for some $c_6=c_6(\Gamma ,\operatorname {supp}\psi )$ , if $\xi = \varepsilon ^2$ ,

(75) $$ \begin{align} &m^{\operatorname{BMS}}(\Psi_{c_3\varepsilon,+})\nonumber\\[3pt] &\quad\le (1+d'\varepsilon)\!\int_{P_{\eta'} z} \int_{B_U(r^{\prime}_0)} \frac{1}{\mu^{\operatorname{PS}}_{pz}(B_U(r^{\prime}_0)pz)} \int_{B_U(r^{\prime}_0)pz}\psi_{c_1\varepsilon,\pm}(u_{\textbf{t}}pz)\,d\textbf{t} \,d\mu^{\operatorname{PS}}_{pz}(\textbf{t})\,d\nu(pz)\nonumber\\[3pt] &\quad\le (1+d'\varepsilon)\int_{P_{\eta'} z} \int_{B_U(r^{\prime}_0)pz}\psi_{c_1\varepsilon,\pm}(u_{\textbf{t}}pz)\,d\textbf{t} \,d\nu(pz)\nonumber\\[3pt] &\quad\le (1+d'\varepsilon)(m^{\operatorname{BR}}(\psi)+c_6\varepsilon S_{\ell}(\psi)). \end{align} $$

By Proposition 3.4, if s is sufficiently large so that $r+c_3e^{-s}\varepsilon _0\le 1$ (note that this requirement on s depends only on $\Gamma $ and $\operatorname {supp}\psi $ ), we have that

(76) $$ \begin{align} \mu^{\operatorname{PS}}_x(B_U(r+c_3e^{-s}\varepsilon_0))\leq\mu^{\operatorname{PS}}_x(B_U(1))\ll_{\Gamma} e^{(n-1-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(x))}. \end{align} $$

Hence, by (65) and (76), we have

(77) $$ \begin{align} \mu^{\operatorname{PS}}_x(f_{c_3e^{-s}\varepsilon_0,+})-\mu^{\operatorname{PS}}_x(f)&\ll_{\Gamma}e^{- s}\varepsilon_0 S_{\ell}(f)\mu^{\operatorname{PS}}_x(B_U(r+c_3e^{-s}\varepsilon_0)) \nonumber \\[3pt] &\ll_{\Gamma} e^{-s}\varepsilon_0 S_{\ell}(f) e^{(n-1-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(x))}. \end{align} $$

According to [17, Lemma 2.4.7(a)] and (64), if $\xi = \varepsilon ^2$ and

(78) $$ \begin{align}\varepsilon = e^{-{\kappa s}/{2(n+4\ell)}},\end{align} $$

then

(79) $$ \begin{align} S_{\ell'}(\Psi_{c_3\varepsilon,+}) &\ll_{\Gamma} S_{\ell'}(\psi_{c_3\varepsilon,+})S_{\ell'}(\sigma)\nonumber\\[3pt] &\ll_{\Gamma} (m(P_{\varepsilon-\xi}))^{-1} (\varepsilon-\xi/2)^{n-1}\xi^{-{\ell'}-(n-1)/2}\varepsilon^{-2\ell}S_{\ell}(\psi)\nonumber\\[3pt] &\ll \varepsilon^{-1-2\ell}\xi^{-{\ell'}-(n-1)/2}S_{\ell}(\psi) \nonumber\\[3pt] &\le e^{\kappa s/2} S_{\ell}(\psi). \end{align} $$

Using (74), (75), (77), and (79), we obtain

(80) $$ \begin{align} &e^{(n-1-\delta_{\Gamma})s}\int_{B_U(r)}\psi(a_su_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t} - \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BR}}(\psi)\nonumber\\[3pt] &\quad\le (1+c_4\varepsilon) [d'\varepsilon \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BR}}(\psi) + (1+d'\varepsilon)\{ c_6 \varepsilon\mu^{\operatorname{PS}}_x(f)S_{\ell}(\psi)\nonumber\\[3pt] &\qquad+(e^{- s}\varepsilon_0 m^{\operatorname{BR}}(\psi)S_{\ell}(f) + c_6 e^{- s}\varepsilon_0 \varepsilon S_{\ell}(f) S_{\ell}(\psi))e^{(n-1-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(x))}\}\nonumber\\[3pt] &\qquad+ c_8 S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa s/2} ]. \end{align} $$

These remaining error terms can be controlled as follows. Using (76), we can deduce

(81) $$ \begin{align} \mu^{\operatorname{PS}}_x(f)\le\Vert f\Vert _{\infty}\mu^{\operatorname{PS}}_x(B_U(r))\ll_{\Gamma}S_{\ell}(f)e^{(n-1-\delta_{\Gamma})(\pi(\mathcal{C}_0),\pi(x))}. \end{align} $$

We also have that

(82) $$ \begin{align}m^{\operatorname{BR}}(\psi) \ll_{\Gamma, \operatorname{supp}\psi} S_{\ell}(\psi).\end{align} $$

Combining (80), (81), and (82) implies

(83) $$ \begin{align} &e^{(n-1-\delta_{\Gamma})s}\int_{B_U(r)}\psi(a_su_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t} - \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BR}}(\psi)\nonumber\\[3pt] &\quad\ll_{\Gamma, \operatorname{supp} \psi} S_{\ell}(\psi)S_{\ell}(f) \cdot [ d'\varepsilon + c_8e^{-\kappa s/2}+ \nonumber\\[3pt] &\qquad (1+d'\varepsilon)(c_6\varepsilon(1+e^{-s}\varepsilon_0) + e^{- s}\varepsilon_0)e^{(n-1-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(x))} ]. \end{align} $$

Finally, by the choice of $\varepsilon $ in (78) and because we may assume without loss of generality that $\kappa <1$ , we obtain from (83) that there exists $\kappa '<1$ such that

$$ \begin{align*} &e^{(n-1-\delta_{\Gamma})s}\int_{B_U(r)}\psi(a_su_{\textbf{t}} x)f(\textbf{t})\,d\textbf{t} - \mu^{\operatorname{PS}}_x(f)m^{\operatorname{BR}}(\psi)\\[3pt] &\quad\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)S_{\ell}(f)e^{-\kappa' s + (n-1-\delta_{\Gamma})d(\pi(\mathcal{C}_0),\pi(x))}. \end{align*} $$

Recall that $d(\pi (\mathcal {C}_0),\pi (x))=\operatorname {height}(x)$ . Thus, if we assume that $s \ge {2(n-1-\delta _{\Gamma })}/{\kappa '} \operatorname {height}(x)$ (which means $s\gg _{\Gamma } \operatorname {height}(x)$ ), then

$$ \begin{align*}e^{-\kappa' s+(n-1-\delta_{\Gamma})\operatorname{height}(x)} \ll_{\Gamma} e^{-\kappa'/2 s},\end{align*} $$

which completes the proof.

7 Proof of Theorem 1.3

In this section, we prove Theorem 1.3, which is restated in the following for the reader’s convenience. The proof relies on the quantitative non-divergence result in Theorems 4.1 and 1.5.

Theorem 7.1. For any $0<\varepsilon <1$ and $s_0\ge 1$ , there exist constants $\ell = \ell (\Gamma ) \in \mathbb N$ and $\kappa = \kappa (\Gamma , \varepsilon )>0$ satisfying: for every $\psi \in C_c^{\infty }(G/\Gamma )$ , there exists $c = c(\Gamma ,\operatorname {supp}\psi )$ such that every $x\in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, and for every T with $T^{1-\varepsilon /2} \gg _{\Gamma } s_0$ ,

$$ \begin{align*}&\bigg|\frac{1}{\mu^{\operatorname{PS}}_x(B_U(T))} \int_{B_U(T)}\psi(u_{\textbf{t}} x)\,d\mu^{\operatorname{PS}}_x(\textbf{t}) - m^{\operatorname{BMS}}(\psi)\bigg| \le c S_{\ell}(\psi) T^{-\kappa},\end{align*} $$

where $S_{\ell }(\psi )$ is the $\ell $ -Sobolev norm.

Proof. Let $\beta>0$ satisfy the conclusion of Theorem 4.1 for $\varepsilon $ and $s_0$ . Let $\kappa '>0$ , $\ell \in \mathbb {N}$ satisfy the conclusion of Theorem 1.5.

Since x is $(\varepsilon ,s_0)$ -Diophantine, by Theorem 4.1, for $T_0\gg _{\Gamma } s_0$ and $R \ge R_0$ ,

(84) $$ \begin{align} \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0 \cap {\mathcal{X}}(R)) \ll \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0)e^{-\beta R}, \end{align} $$

where

(85) $$ \begin{align} s_{\varepsilon} := \frac{\varepsilon}{2}\log T, \quad T_0 := Te^{-s_{\varepsilon}} = T^{1-\varepsilon/2},\quad x_0 := a_{-s_{\varepsilon}}x. \end{align} $$

By (6) and (14), we have

$$ \begin{align*} \frac{1}{\mu^{\operatorname{PS}}_x(B_U(T))} \int_{B_U(T)} \psi(u_{\textbf{t}} x)\,d\mu^{\operatorname{PS}}_x(\textbf{t}) = \frac{1}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))}\int_{B_U(T_0)}\psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}).\end{align*} $$

Fix $R>R_0$ and define

$$ \begin{align*}Q_0 = B_U(T_0)x_0 \cap \mathcal{C}(R).\end{align*} $$

By the definition of ${\mathcal C}(R)$ ,

$$ \begin{align*}Q_0\subseteq \operatorname{supp}m^{\operatorname{BMS}}.\end{align*} $$

Let $\rho>0$ be smaller than half of the injectivity radius of $Q_0$ .

First, by Lemma 3.11, there exist $\{y : y \in I_0\}\subseteq Q_0$ and $f_y \in C_c^{\infty }(B_U(2\rho )y)$ satisfying

(86) $$ \begin{align} S_{\ell}(f_y)\ll \rho^{-\ell+n-1} \end{align} $$

and

$$ \begin{align*}\sum\limits_{y} f_y = 1 \text{ on }E_1:= \bigcup_{y\in I_0} B_U(\rho)y\supseteq Q_0, \end{align*} $$

which are $0$ outside of

$$ \begin{align*}E_2=\bigcup_{y\in I_0} B_U(2\rho)y.\end{align*} $$

Observe that

(87) $$ \begin{align} Q_0\subseteq E_1\subseteq E_2 \subseteq B_U(T_0+2\rho)x_0. \end{align} $$

Thus,

$$ \begin{align*}\int_{u_{\textbf{t}} x_0 \in E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t})\le \sum\limits_{y\in I_0} \int_{u_{\textbf{t}} x_0 \in B_U(2\rho)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)f_y(u_{\textbf{t}} x_0) \,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}). \end{align*} $$

Because $Q_0 \subseteq \operatorname {supp}m^{\operatorname {BMS}}$ , we may use Proposition 3.4 to deduce that there exists $\unicode{x3bb} =\unicode{x3bb} (\Gamma )\ge 1$ such that for any $y\in I_{0}$ , we have

$$ \begin{align*} \mu^{\operatorname{PS}}_y(B_U(\rho))&\ge\unicode{x3bb}^{-1}\rho^{\delta_{\Gamma}}e^{(k(y,\rho)-\delta_{\Gamma})d(\pi(\mathcal{C}_0), \pi(a_{-\log \rho}y))}\\[3pt] &\ge \unicode{x3bb}^{-1} \rho^{\delta_{\Gamma}}e^{-\delta_{\Gamma} d(\pi(\mathcal{C}_0), \pi(a_{-\log \rho}y))} \\[3pt] &\ge \unicode{x3bb}^{-1} \rho^{\delta_{\Gamma}}e^{-\delta_{\Gamma} (-\log\rho)}e^{-\delta_{\Gamma} \operatorname{height}(y)} \quad\text{since } \rho<1\\[3pt] &\gg_{\Gamma} \unicode{x3bb}^{-1} \rho^{2\delta_{\Gamma}}e^{-\delta_{\Gamma} \operatorname{height}(y)}\\[3pt] &\ge \unicode{x3bb}^{-1} \rho^{2\delta_{\Gamma}}e^{-\delta_{\Gamma} R}, \end{align*} $$

where the last line follows by Lemma 3.6.

Since $e^{s_{\varepsilon }} = T^{\varepsilon /2}$ , it follows from (86) and the above, that if we choose $\rho $ and R such that

(88) $$ \begin{align} e^{\delta_{\Gamma} R}\rho^{n-1-\ell-2\delta_{\Gamma}}\ll_{\Gamma} T^{\varepsilon\kappa' /4}, \end{align} $$

then, by the choice of $f_y$ , we have

(89) $$ \begin{align} S_{\ell}(f_y)\ll\mu^{\operatorname{PS}}_y(B_U(\rho))e^{\kappa' s_{\varepsilon}/2} \ll\mu^{\operatorname{PS}}_y(f_y)e^{\kappa' s_{\varepsilon}/2}, \end{align} $$

where the implied constant is absolute.

If we further assume that

(90) $$ \begin{align} T \gg_{\Gamma} e^{2R/\varepsilon}\end{align} $$

(with the implied constant coming from Theorem 1.5), then $s_{\varepsilon } \gg _{\Gamma } R$ , and by (89), Theorem 1.5, and Lemma 3.6, there exist $c_1,c_2>0$ which depend only on $\Gamma $ and $\operatorname {supp}\psi $ such that

$$ \begin{align*} &\sum_{y\in I_0} \int_{u_{\textbf{t}} x_0 \in B_U(2\rho)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)f_y(u_{\textbf{t}} x_0) \,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t})\\[3pt] &\quad\le \sum_{y\in I_0} (m^{\operatorname{BMS}}(\psi)\mu^{\operatorname{PS}}_y(f_y)+c_1 S_{\ell}(\psi)S_{\ell}(f_y)e^{-\kappa' s_{\varepsilon}})\\[3pt] &\quad\le \sum_{y\in I_0} \mu^{\operatorname{PS}}_y(f_y)(m^{\operatorname{BMS}}(\psi)+c_2 S_{\ell}(\psi)e^{-\kappa' s_{\varepsilon}/2}). \end{align*} $$

By Theorem 5.1, there exists $c_3=c_3(\Gamma )>0$ such that if $T_0\gg s_0$ , then there exist $\alpha =\alpha (\Gamma )>0$ , $c_3=c_3(\Gamma )>0$ such that

$$ \begin{align*} \sum\limits_{y\in I_0} \mu^{\operatorname{PS}}_y(f_y) &\le \mu^{\operatorname{PS}}_{x_0}(B_U(T_0+2\rho))\\ &\ll_{\Gamma} (1+c_3(2\rho)^{\alpha}) \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)). \end{align*} $$

If $m^{\operatorname {BMS}}(\psi )+c_2 S_{\ell }(\psi )e^{-\kappa ' s_{\varepsilon }/2}\ge 0$ , we arrive at

(91) $$ \begin{align}&\sum_{y\in I_0} \int_{u_{\textbf{t}} x_0 \in B_U(2\rho)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)f_y(u_{\textbf{t}} x_0) \,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t})\nonumber\\[3pt] &\quad\le\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))(1+c_3(2\rho)^{\alpha})(m^{\operatorname{BMS}}(\psi) + c_2 S_{\ell}(\psi)e^{-\kappa' s_{\varepsilon}/2}). \end{align} $$

However, if $m^{\operatorname {BMS}}(\psi )+c_2 S_{\ell }(\psi )e^{-\kappa ' s_{\varepsilon }/2}<0$ , by (84), there exists $c_6 = c_6(\Gamma )>0$ so that

(92) $$ \begin{align} &\sum_{y\in I_0} \int_{u_{\textbf{t}} x_0 \in B_U(2\rho)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)f_y(u_{\textbf{t}} x_0) \,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}) \nonumber\\ &\quad\le \mu^{\operatorname{PS}}_{x_0}(Q_0)(m^{\operatorname{BMS}}(\psi)+c_2 S_{\ell}(\psi)e^{-\kappa' s_{\varepsilon}/2})\nonumber\\[3pt] &\quad\le \mu^{\operatorname{PS}}_{x_0}(B_U(T_0))(1-c_6e^{-\beta R})(m^{\operatorname{BMS}}(\psi)+c_2 S_{\ell}(\psi)e^{-\kappa' s_{\varepsilon}/2}). \end{align} $$

Fix

(93) $$ \begin{align} \kappa:=\kappa'\varepsilon/4, \quad R>\frac{\kappa}{\beta}\log T,\quad \rho<T^{-\frac{\kappa}{\alpha}}, \end{align} $$

such that $\rho $ also satisfies the assumption of Theorem 5.1, and R which satisfies (88) and (90). Thus, (91) and (93) imply in either case that

(94) $$ \begin{align} &\frac{1}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))} \int_{u_{\textbf{t}} x_0 \in E_1}\psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}) - m^{\operatorname{BMS}}(\psi) \nonumber\\[3pt] &\quad\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)T^{-\kappa}, \end{align} $$

where we have used that by [1], $\|\psi \|_{\infty } \ll _{\operatorname {supp} \psi } S_{\ell }(\psi )$ , so $m^{\operatorname {BMS}}(\psi )\ll _{\operatorname {supp}\psi } S_{\ell }(\psi ).$

By (84),

$$ \begin{align*}\int_{B_U(T_0)x_0\setminus E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}) &\le \|\psi\|_{\infty} \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)\setminus E_1)\\[3pt] &\ll_{\operatorname{supp}\psi} S_{\ell}(\psi)\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0) e^{-\beta R}\\[3pt] &\ll_{\operatorname{supp}\psi} S_{\ell}(\psi)\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0) T^{-\kappa},\end{align*} $$

where we have again used that by [1], $\|\psi \|_{\infty } \ll _{\operatorname {supp} \psi } S_{\ell }(\psi )$ . Combining the above with (94) implies that

$$ \begin{align*}&\frac{1}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))} \int_{B_U(T_0)} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}) - m^{\operatorname{BMS}}(\psi) \\[3pt] &\quad\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)T^{-\kappa}. \end{align*} $$

For the lower bound, define

$$ \begin{align*}Q_1:= B_U(T_0-2\rho)x_0 \cap {\mathcal{C}}(R).\end{align*} $$

As before, according to Lemma 3.11, there exist $\{y : y \in I_1\}\subseteq Q_1$ and $f_y \in C_c^{\infty }(B_U(2\rho )y)$ satisfying

$$ \begin{align*}S_{\ell}(f_y)\ll \rho^{-\ell+n-1}\end{align*} $$

and

$$ \begin{align*}\sum_{y\in I_1} f_y = 1 \text{ on }E_4:= \bigcup_{y\in I_1} B_U(\rho)y \supseteq Q_1,\end{align*} $$

and which are $0$ outside of

(95) $$ \begin{align} \bigcup_{y\in I_1} B_U(2\rho)y\subseteq B_U(T_0)x_0. \end{align} $$

Hence,

$$ \begin{align*}\int_{u_{\textbf{t}} x_0 \in B_U(T_0)x_0} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\textbf{t}\ge \sum_{y\in I_1} \int_{u_{\textbf{t}} x_0 \in B_U(2\rho)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)f_y(u_{\textbf{t}} x_0)\,d\textbf{t}. \end{align*} $$

Moreover, by the same argument as in (89), we deduce that

(96) $$ \begin{align} S_{\ell}(f_y) \ll\mu^{\operatorname{PS}}_y(f_y)e^{\kappa' s_{\varepsilon}/2}.\end{align} $$

By Theorem 1.5 and (96), we have that

$$ \begin{align*}&\sum_{y\in I_1} \int_{u_{\textbf{t}} x_0 \in B_U(2\rho)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)f_y(u_{\textbf{t}} x_0) \,d\textbf{t} \\ &\quad\ge \sum\limits_{y \in I_1} (\mu^{\operatorname{PS}}_{x_0}(f_y) m^{\operatorname{BMS}}(\psi) - c_4 S_{\ell}(\psi) S_{\ell}(f_y) e^{-\kappa' s_{\varepsilon}})\\ &\quad\ge \sum\limits_{y \in I_1} \mu^{\operatorname{PS}}_{x_0}(f_y)(m^{\operatorname{BMS}}(\psi) - c_5S_{\ell}(\psi)e^{-\kappa's_{\varepsilon}/2}),\end{align*} $$

where $c_5$ arises from $c_4$ and the implied constant in (96).

Note that by replacing $\psi $ with $-\psi $ if necessary, we may assume that $m^{\operatorname {BMS}}(\psi ) - c_5S_{\ell }(\psi )e^{-\kappa 's_{\varepsilon }/2}<0$ . Thus, by observing that

$$ \begin{align*}\sum\limits_{y \in I_1} \mu^{\operatorname{PS}}_y(f_y) \le \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)),\end{align*} $$

we immediately conclude that

$$ \begin{align*} &\frac{1}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))} \int_{u_{\textbf{t}} x_0 \in E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}) -m^{\operatorname{BMS}}(\psi)\\ &\quad\gg_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)e^{-\kappa's_{\varepsilon}/2}. \end{align*} $$

Hence, (85) and (93) imply that

$$ \begin{align*} \frac{1}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))} \int_{u_{\textbf{t}} x_0 \in E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\mu^{\operatorname{PS}}_{x_0}(\textbf{t}) -m^{\operatorname{BMS}}(\psi)\gg_{\Gamma,\operatorname{supp}\psi}- S_{\ell}(\psi)T^{-\kappa}. \end{align*} $$

8 Proof of Theorem 1.4

In this section, we will prove Theorem 1.4 using Theorem 6.3. We will use a partition of unity argument for a cover of the intersection of $B_U(r)x$ with a fixed compact set by small balls centered at PS points.

We will need the following lemma.

Lemma 8.1. There exists an absolute constant $c>0$ satisfying the following: for $x\in X$ , $y\in Ux$ , $\psi \in C_c^{\infty }(X)$ supported on an admissible box of diameter smaller than $1$ , $0<\rho <\operatorname {inj}(y)$ , $f\in C_c^{\infty }(B_U(\rho )y)$ such that for $0\le f\le 1$ and $s>c$ , we have

$$ \begin{align*} e^{(n-1-\delta_{\Gamma})s}\int_{Ux}\psi(a_s u_{\textbf{t}} y)f(u_{\textbf{t}} y)\,d\textbf{t}\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)\mu^{\operatorname{PS}}_y(B_U(2\rho)y), \end{align*} $$

where $\ell \in \mathbb N$ satisfies the conclusion of Lemma 3.12.

Proof. Assume that for $0<\varepsilon _0,\varepsilon _1<1$ , $\psi $ is supported on the admissible box $B_U(\varepsilon _0)P_{\varepsilon _1}z$ for $z\in X$ . Without loss of generality, we may assume that $\psi $ is non-negative. Fix $y\in Ux$ .

For small $\eta>0$ , $h\in G_{\eta }\operatorname {supp}(\psi )$ , and $p\in P$ , let

$$ \begin{align*} \psi_{\eta,+}(h):=\sup_{w\in G_{\eta}}\psi(wh),\quad\Psi_{\eta,+}(ph):=\int_{Uph}\psi_{\eta,+}(u_{\textbf{t}} ph)\,d\textbf{t}, \end{align*} $$

and for $upz\in B_U(\varepsilon _0)P_{\varepsilon _1}z$ , let

$$ \begin{align*}\tilde{\Psi}_{\eta,+}(upz):=\frac{1}{\mu^{\operatorname{PS}}_{pz}(B_U(\varepsilon_0)pz)}\Psi_{\eta,+}(pz). \end{align*} $$

By the choice of $\ell $ , for any $\eta>0$ and $h\in G_{\eta }\operatorname {supp}(\psi )$ ,

$$ \begin{align*} |\psi_{\eta,+}(h)-\psi(h)|\ll \eta S_{\ell}(\psi), \end{align*} $$

and

$$ \begin{align*} |\psi(z)|\le S_{\infty,0}(\psi)\ll S_{\ell}(\psi),\end{align*} $$

where the implied constants depend on $\operatorname {supp}\psi $ . Since the diameter of $\operatorname {supp}\psi $ is smaller than $1$ , we may assume that the implied constants in the above are absolute. Then, for any $u\in U$ such that $a_s uy=u'pz\in B_U(\varepsilon _0)P_{\varepsilon _1}z$ and $0<\eta <1$ , we have

(97) $$ \begin{align} |\tilde\Psi_{\eta,+}(a_s u y)| &=\bigg|\frac{1}{\mu^{\operatorname{PS}}_{pz}(B_U(\varepsilon_0)pz)}\int_{U pz}\psi_{\eta,+}(u_{\textbf{t}} pz)\,d\textbf{t}\bigg|\nonumber\\[4pt] &=\frac{\mu^{\operatorname{Leb}}_{pz}(B_U(\varepsilon_0)pz)}{\mu^{\operatorname{PS}}_{pz}(B_U(\varepsilon_0)pz)} S_{\ell}(\psi)\nonumber\\[4pt] & \ll S_{\ell}(\psi), \end{align} $$

where the implied constant depends only on $\operatorname {supp}\psi $ .

For small $\eta>0$ and $uy\in B_U(\eta +\varepsilon _0) y$ , let

$$ \begin{align*} f_{\eta,+}(uy):=\sup_{w\in B_U(\eta)}f(wu y). \end{align*} $$

Using Lemmas 6.2 and 6.5 in [23], we get that for some absolute constant $c'>0$ ,

$$ \begin{align*} e^{(n-1-\delta_{\Gamma})s}\int_{B_U(\rho)y }\psi(a_{s}u_{\textbf{t}} y)f(u_{\textbf{t}} y)\,d\textbf{t} &\ll\int_{U}\tilde{\Psi}_{c'\rho,+}(a_su_{\textbf{t}} y)f_{c'e^{-s}\rho,+}(u_{\textbf{t}} y)\,d\mu^{\operatorname{PS}}_y(\textbf{t})\\[3pt] & \le\int_{B_U(\rho+c'e^{-s}\rho)y}\tilde{\Psi}_{c'\rho,+}(a_su_{\textbf{t}} y)\,d\mu^{\operatorname{PS}}_y(\textbf{t}), \end{align*} $$

where the implied constant is absolute. Then, by (97), we get

$$ \begin{align*} e^{(n-1-\delta_{\Gamma})s}\int_{B_U(\rho)y }\psi(a_{s}u_{\textbf{t}} y)f(u_{\textbf{t}} y)\,d\textbf{t} &\ll_{\operatorname{supp}\psi} \mu^{\operatorname{PS}}_y(B_U(\rho+c'e^{-s}\rho)) S_{\ell}(\psi)\\ &\le \mu^{\operatorname{PS}}_y(B_U(2\rho))S_{\ell}(\psi). \end{align*} $$

Choosing $c:=\log c'$ , we may conclude the claim.

We are now ready to prove Theorem 1.4. For the reader’s convenience, we restate that theorem in the following.

Theorem 8.2. For any $0<\varepsilon <1$ and $s_0\ge 1$ , there exist $\ell =\ell (\Gamma )\in \mathbb N$ and $\kappa =\kappa (\Gamma ,\varepsilon )>0$ satisfying: for every $\psi \in C_c^{\infty }(G/\Gamma )$ , there exists $c = c(\Gamma ,\operatorname {supp}\psi )$ such that for every $x\in G/\Gamma $ that is $(\varepsilon ,s_0)$ -Diophantine, and for all T such that $T^{1-\varepsilon /2} \gg _{\Gamma ,\operatorname {supp}\psi } s_0$ ,

$$ \begin{align*}\bigg|\frac{1}{\mu^{\operatorname{PS}}_x(B_U(T))} \int_{B_U(T)}\psi(u_{\textbf{t}} x)\,d\textbf{t} - m^{\operatorname{BR}}(\psi)\bigg| \le c S_{\ell}(\psi)T^{-\kappa},\end{align*} $$

where $S_{\ell }(\psi )$ is the $\ell $ -Sobolev norm.

Proof. We keep the notation of §4. By an argument similar to the proof of Theorem 1.5, we may assume that $\psi $ is supported on an admissible box. Because $\psi $ is compactly supported, we may also assume $\psi \ge 0$ by using a translation.

Let $\beta>0$ satisfy the conclusion of Theorem 4.1 for $\varepsilon $ and $s_0$ . Let $\kappa '>0$ , $\ell \in \mathbb {N}$ satisfy the conclusion of Theorem 6.3.

Since x is $(\varepsilon ,s_0)$ -Diophantine, by Theorem 4.1, for $T_0\gg _{\Gamma } s_0$ and $R\ge R_0$ , we have

(98) $$ \begin{align} \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0\cap {\mathcal{X}}(R)) \ll \mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0)e^{-\beta R},\end{align} $$

where

(99) $$ \begin{align} s_{\varepsilon}:=\frac{\varepsilon}{2}\log T,\quad x_0 := a_{-s_{\varepsilon}}x,\quad\mbox{and}\quad T_0 = T^{1-{\varepsilon}/{2}}. \end{align} $$

Observe that by (6), (13), and (14),

$$ \begin{align*}\frac{1}{\mu^{\operatorname{PS}}_x(B_U(T))} \int_{B_U(T)}\psi(u_{\textbf{t}} x)\,d\textbf{t}=\frac{e^{(n-1-\delta)s_{\varepsilon}}}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0))} \int_{B_U(T_0)}\psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\textbf{t}.\end{align*} $$

Fix $R>R_0$ and define

$$ \begin{align*}Q_0:= B_U(T_0)x_0 \cap {\mathcal{C}}(R).\end{align*} $$

Since for any $R\geq R_0$ , the set ${\mathcal {C}}(R)$ is in the convex core of ${\mathbb H}^n /\Gamma $ ,

(100) $$ \begin{align}Q_0\subseteq\operatorname{supp}m^{\operatorname{BMS}}.\end{align} $$

Let $\rho>0$ be smaller than half of the injectivity radius of $Q_0$ .

First, by Lemma 3.11, there exist $\{y : y \in I_0\}\subseteq Q_0$ and $f_y \in C_c^{\infty }(B_U(2\rho )y)$ satisfying

(101) $$ \begin{align} S_{\ell}(f_y)\ll \rho^{-\ell+n-1} \end{align} $$

and

$$ \begin{align*}\sum\limits_{y} f_y = 1 \text{ on }E_1:= \bigcup_{y\in I_0} B_U(\rho)y\supseteq Q_0,\end{align*} $$

which are $0$ outside of

$$ \begin{align*}E_2=\bigcup_{y\in I_0} B_U(2\rho)y.\end{align*} $$

By replacing references to Theorem 1.5 with references to Theorem 6.3, the exact same argument as in the proof of Theorem 1.3 will establish that for $T \gg _{\Gamma } e^{2R/\varepsilon }$ and

(102) $$ \begin{align} \kappa=\frac{\beta\varepsilon}{2},\quad R=\frac{\kappa\log T}{\beta},\quad \rho \le T^{-\kappa/\alpha}. \end{align} $$

We get that if we assume without loss of generality that $\kappa '<2\beta $ and also that $T\gg _{\Gamma } 1$ ,

(103) $$ \begin{align} &\frac{e^{(n-1-\delta_{\Gamma}) s_{\varepsilon}}}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0)} \int_{u_{\textbf{t}} x_0 \in E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\textbf{t}-m^{\operatorname{BR}}(\psi) \ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)T^{-\kappa}. \end{align} $$

We now want to bound the integral over $B_U(T_0)x_0\setminus E_1$ . Using Lemma 3.11 again, we may deduce that there exist $\{y : y \in I_1\}\subseteq B_U(T_0)x_0\setminus E_1$ and $f_y \in C_c^{\infty }(B_U(\rho /4)y)$ satisfying $\sum \nolimits _{y\in I_1} f_y = 1$ on $ \bigcup \nolimits _{y\in I_1} B_U(\rho /8)y$ and $0$ outside of

$$ \begin{align*}\bigcup\limits_{y\in I_1} B_U(\rho/4)y.\end{align*} $$

In particular, by the definition of $E_1$ , we have

$$ \begin{align*} E_3:=\bigcup\limits_{y\in I_1} B_U(\rho/2)y\subseteq (B_U(T_0)x_0\setminus Q_0)\cup (B_U(T_0+\rho/2)x_0\setminus B_U(T_0))x_0. \end{align*} $$

Using Lemma 8.1, we arrive at

$$ \begin{align*} &e^{(n-1-\delta_{\Gamma})s_{\varepsilon}}\int_{B_U(T_0) \setminus E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\textbf{t}\\ &\quad\le e^{(n-1-\delta_{\Gamma})s_{\varepsilon}}\sum_{y\in I_1} \int_{B_U(\rho/2)y} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} y)f_y(u_{\textbf{t}} y)\,d\textbf{t}\\ &\quad\ll\sum_{y\in I_1} S_{\ell}(\psi)\mu^{\operatorname{PS}}_y(B_U(\rho/2)y)\\ &\quad\le S_{\ell}(\psi)(\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0\setminus Q_0)+\mu^{\operatorname{PS}}_{x_0} ((B_U(T_0+\rho/2)\setminus B_U(T_0))x_0)). \end{align*} $$

Thus, by Theorem 5.1, there exists $\alpha =\alpha (\Gamma )>0$ such that using (98) and (102), we arrive at

$$ \begin{align*} &e^{(n-1-\delta_{\Gamma})s_{\varepsilon}}\int_{B_U(T_0) \setminus E_1} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\textbf{t} \\ &\quad\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0)(e^{-\beta R}+\rho^{\alpha} )\\ &\quad\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0)T^{-\kappa}. \end{align*} $$

Using (103), we may now deduce

$$ \begin{align*} \frac{e^{(n-1-\delta_{\Gamma})s_{\varepsilon}}}{\mu^{\operatorname{PS}}_{x_0}(B_U(T_0)x_0)}\int_{B_U(T_0)} \psi(a_{s_{\varepsilon}}u_{\textbf{t}} x_0)\,d\textbf{t}-m^{\operatorname{BR}}(\psi)\ll_{\Gamma,\operatorname{supp}\psi} S_{\ell}(\psi)T^{-\kappa}.\end{align*} $$

The lower bound follows similarly, as in the proof of Theorem 1.3.

Remark 8.3. The dependence of T on $\operatorname {supp}\psi $ in the previous proof arises from Theorem 6.3, through the quantity $s_{\varepsilon }$ . Upon closer inspection, one can verify that this means T depends on $\operatorname {supp}\psi $ through the maximum height of elements in $\operatorname {supp}\psi $ . In particular, we may choose a larger compact set containing $\operatorname {supp}\psi $ and have T depend on that compact set, rather than $\operatorname {supp}\psi $ specifically.