2.1 Algebraic groups over number fields

We start by reviewing basic properties of semisimple algebraic groups over number fields (cf. [Reference Platonov and RapinchukPR94, Ch. 3–5]). Throughout the paper, $K$ denotes an algebraic number field (namely, a finite extension of the field of rationals $\mathbb {Q}$). Let $V_K$ be the set of normalized absolute values $|\cdot |_v$ of the field $K$. The set of non-trivial absolute values decomposes as

\[ V_K=V_K^\infty\sqcup V_K^f, \]

where $V_K^\infty$ is the finite set of Archimedean absolute values and $V_K^f$ the set of non-Archimedean absolute values. For $v\in V_K$, we write $K_v$ for the corresponding completion of $K$, and when $v\in V_K^f$, we denote by

\[ O_v:=\{x\in K_v:\, |x|_v\le 1\} \]

the ring of integers in $K_v$. For $S\subset V_K^f$, we write

\[ O_S:=\{x\in K:\, |x|_v\le 1\text{ for } v\in V_K^f\backslash S\}, \]

for the ring of $S$-integers,Footnote ¹ namely elements in $K$ that are integral with respect to every completion except possibly those in $S$. In particular, $O=O_\emptyset$ denotes the ring of integers in $K$.

We denote by

\[ {\mathbb{A}}_K:=\{(x_v)_{v\in V_K}:\, |x_v|_v\le 1\text{ for almost all } v\} \]

the ring of adéles of $K$, which is the restricted direct product of $K_v$, $v\in V_K$, with respect to compact open subrings $O_v\subset K_v$, $v\in V_K^f$.

Let ${\sf G}\subset \text {SL}_n$ be a connected semisimple linear algebraic group defined over the field $K$. The focus of our investigation is the distribution of the set of $S$-integral points

\[ \Gamma_S:={\sf G}(O_S). \]

We denote by

\[ G_v:={\sf G}(K_v)\quad \text{for } v\in V_K, \]

the locally compact groups of $K_v$-points of $\sf G$ equipped with the topology defined by the corresponding absolute values $|\cdot |_v$. We say that $\sf G$ is isotropic over $v$ if $G_v$ is non-compact, and $\sf G$ is isotropic over $S\subset V_K$ if $G_v$ is non-compact for at least one $v\in S$.

For $v\in V_K^f$, we consider

\[ {\sf G}(O_v):=\{g\in G_v:\, \|g\|_v\le 1\}, \]

which is a compact open subgroup of $G_v$. We introduce compact groups

\[ \mathcal{I}_S:=\prod_{v\in S}\; {\sf G}(O_v)\quad \text{and} \quad \mathcal{I}^S:=\prod_{v\in V_K^f\backslash S}\; {\sf G}(O_v). \]

For $S\subset V_K$, we write

\[ G_S:=\prod_{v\in S}'\, G_v=\{(g_v)_{v\in S}:\, g_v\in G_v,\, \|g_v\|_v\le 1\text{ for almost all } v\} \]

for the restricted direct product of the groups $G_v$ with respect to the compact open subgroups ${\sf G}(O_v)$, $v\in S\cap V_K^f$. Then $G_S$ is a locally compact group. For instance, ${\sf G}(\mathbb {A}_K\!)=G_{V_K}$ is the adéle group associated to ${\sf G}$. To simplify notation, we also write

\[ G_\infty:=G_{V_K^\infty}\quad\text{and}\quad G_f:=G_{V_K^f}. \]

We recall that when $\sf G$ is simply connected, the group $G_\infty$ is connected with respect to the Euclidean topology (cf. [Reference Platonov and RapinchukPR94, Ch. 7, Proposition 7.2]). In general, $G_\infty$ has finitely many connected components, and we denote by $G_\infty ^0$ the connected component of the identity in $G_\infty$.

For $v\in V_K$, we denote by $m_{v}$ the Haar measure on $G_v$, and when $v\in V_K^f$ we normalize it so that $m_v({\sf G}(O_v))=1$. Then for a subset $S\subset V_K^f$, the product measure $m_S:=\prod _{v\in S} m_v$ defines a Haar measure on $G_S$ such that $m_S(\mathcal {I}_S)=1$. We also denote by $m_\infty$ a Haar measure on $G_\infty$. Under the diagonal embedding $\Gamma _S\hookrightarrow G_\infty \times G_S$, the group $\Gamma _S$ is a discrete subgroup with finite covolume in $G_\infty \times G_S$ (cf. [Reference Platonov and RapinchukPR94, Ch. 5]). We normalize the measure $m_\infty$, so that $\Gamma _S$ has covolume one with respect $m_\infty \times m_S$.

We recall the strong approximation property [Reference Platonov and RapinchukPR94, § 7.4].

This result fails if the group $\sf G$ is not simply connected even for the embedding $\Gamma _S\hookrightarrow G_\infty$. Nonetheless, one can deduce the following result about the closure.

Proof. Let us consider the simply connected cover ${\sf p}:\tilde {\sf G}\to {\sf G}$. Then the groups

\[ U_v:={\sf p}^{-1}({\sf G}(O_v))\cap \tilde {\sf G}(O_v) \quad \text{with } v\in V_K^f, \]

are compact open subgroups of $\tilde G_v$. It follows from Theorem 2.1 that the image of the group

\[ \Gamma:=\tilde {\sf G}(O_S)\cap \bigg(\tilde G_\infty\times\prod_{v\in V_K^f\backslash S} U_v\bigg) \]

is dense in $\tilde G_\infty$. Since ${\sf p}(\Gamma )\subset \Gamma _S$, it follows that the closure $\overline {\Gamma _S}$ in $G_\infty$ contains ${\sf p}(\tilde G_\infty )$ which is an open and closed subgroup of finite index in $G_\infty$ (cf. [Reference Platonov and RapinchukPR94, § 3.2]).

2.2 Automorphic representations

Let $\sf G$ be a connected $K$-simple algebraic group defined over a number field $K$. We consider the Hilbert space

\[ \mathcal{H}_{\sf G}:=L^2({\sf G}(\mathbb{A}_K\!)/{\sf G}(K)) \]

consisting of square-integrable functions on the space ${\sf G}(\mathbb {A}_K\!)/{\sf G}(K)$ equipped with the invariant probability measure $\mu$. Let

\[ \mathcal{H}^0_{\sf G}:=L_0^2({\sf G}(\mathbb{A}_K\!)/{\sf G}(K))=\bigg\{ \phi\in \mathcal{H}_{\sf G}:\, \int_{{\sf G}(\mathbb{A}_K\!)/{\sf G}(K)} \phi\, d\mu=0 \bigg\} . \]

A continuous unitary character $\chi$ of ${\sf G}(\mathbb {A}_K\!)$ is called automorphic if $\chi ({\sf G}(K))=1$. Then $\chi$ can be considered as an element of $\mathcal {H}_{\sf G}$. We denote by $\mathcal {H}_{\sf G}^{00}$ the subspace of $\mathcal {H}_{\sf G}$ orthogonal to all automorphic characters. We note that when $\sf G$ is simply connected there no non-trivial automorphic characters and $\mathcal {H}_{\sf G}^{00}=\mathcal {H}_{\sf G}^{0}$.

Now we describe our choice of maximal compact subgroups $U_v$ of $G_v$. For all but finitely many $v\in V_K^f$:

1. (i) ${\sf G}(O_v)$ is a hyperspecial, good maximal compact subgroup of $G_v$;

2. (ii) the group $\sf G$ is unramified over $K_v$ (that is, $\sf G$ is quasi-split over $K_v$ and split over an unramified extension of $K_v$).

We say that $\sf G$ is unramified over such $v$. For those $v$, we set $U_v:={\sf G}(O_v)$. For the remaining (finite) set of finite places $v$, we fix a good special maximal compact subgroup $U_v$ of $G_v$. For any subset $S\subset V^f_K$, we set

\[ U_S:=\prod_{v\in S} U_v\quad \text{and} \quad U^S:=\prod_{v\in V_K^f\backslash S} U_v. \]

For places $v\in V_K^f$, we denote by $\pi _v^{\rm aut}=\pi _v$ the unitary representation of the group $G_v$ on the space $\mathcal {H}_{\sf G}$. The spherical integrability exponent of the representations $\pi _v$, with respect to the subgroup $U_v$, is defined by

(2.1) \begin{equation} \mathfrak{q}_v({\sf G}):=\inf\bigg\{ q \ge 2:\, \begin{array}{@{}l@{}} \forall\, U_v\text{-inv. } \phi\in \mathcal{H}^{00}_{\sf G}\\ \langle \pi_v(g)\phi,\phi\rangle\in L^q(G_v) \end{array}\bigg\}. \end{equation}

It is a fundamental result in the theory of automorphic representations that the integrability exponents $\mathfrak {q}_v({\rm G})$ are finite and, moreover, $\mathfrak {q}_v({\rm G})$ is uniformly bounded over $v$, see [Reference ClozelClo03]. These exponents can be estimated in terms of the Satake parameters of the corresponding spherical automorphic representations. We refer to [Reference ClozelClo07, Reference SarnakSar05] for surveys of some of these results.

More generally, for $S\subset V_K^f$, we denote by $\pi _S^{\rm aut}=\pi _S$ the unitary representation of the group $G_S$ on the space $\mathcal {H}_{\sf G}$. We define

(2.2)\begin{equation} \mathfrak{q}_S({\sf G}):=\inf\bigg\{ q \ge 2:\, \begin{array}{@{}l@{}} \forall \,U_S\text{-inv. } \phi\in \mathcal{H}^{00}_{\sf G}\\ \langle \pi_S(g)\phi,\phi\rangle\in L^q(G_S) \end{array}\bigg\}. \end{equation}

This integrability exponent is also finite and it can be estimated in terms of the exponents $\mathfrak {q}_v({\sf G})$, $v\in S$, see [Reference Ghosh, Gorodnik and NevoGGN13, Cor. 3.5].

Given a strongly continuous unitary representation $\pi :G_S\to \mathcal {U}(\mathcal {H})$ on a Hilbert space $\mathcal {H}$ and a finite Borel measure $\nu$ on $G_S$, we define the averaging operator

\[ \pi_S(\nu):\mathcal{H}\to \mathcal{H}:\, \phi\mapsto \int_{G_S} \pi(g)\phi\, d\nu(g). \]

We recall the following estimate on the norm of the averaging operators.

Theorem 2.3 [Reference Ghosh, Gorodnik and NevoGGN13, Proposition 3.8]

Let $\beta$ be a Haar-uniform probability measure supported on a $U_S$-bi-invariant bounded subset $B$ of $G_S$. Then

\[ \|\pi_S^{\rm aut}(\beta)|_{\mathcal{H}_{\sf G}^{00}}\|\ll_{S,\eta} m_{S}(B)^{-{1}/{\mathfrak{q}_S({\rm G})}+\eta}\quad\text{for all } \eta>0. \]

We note that although the measure $m_S$ in [Reference Ghosh, Gorodnik and NevoGGN13] was normalized differently (namely, so that $m_S(U_S)=1$), this gives the same bound up to a multiplicative constant.

An important fact, underlying our considerations below regarding ramified places, is that this result also holds for more general averaging operators. The crucial ingredient here is finiteness of a more general integrability exponent, which we define as follows:

(2.3)\begin{equation} \mathfrak{p}_S({\sf G}):= \inf \bigg\{p\ge 2:\, \begin{array}{@{}c@{}} \forall \,\phi_1,\phi_2 \text{ in a dense subspace of } \mathcal{H}_{\sf G}^{00}\\ \langle \pi_S^{\rm aut}(g)\phi_1,\phi_2\rangle \in L^p(G_S) \end{array}\bigg\}. \end{equation}

One says that a unitary representation $\pi :G_S\to \mathcal {U}(\mathcal {H})$ is $L^p$-integrable if for $\phi _1,\phi _2$ in a dense subset of $\mathcal {H}$, the functions $g\mapsto \langle \pi (g)\phi _1,\phi _2\rangle$ is in $L^p(G_S)$. Thus,

\[ \mathfrak{p}_S({\sf G})= \inf \big \{p\ge 2:\, \pi_S^{\rm aut}|_{\mathcal{H}_{\sf G}^{00}} \text{ is } L^p\text{-integrable} \big\}. \]

It was proved in [Reference Gorodnik, Maucourant and OhGMO08] that this exponent is finite provided that $\sf G$ is either simply connected or adjoint (see [Reference Gorodnik, Maucourant and OhGMO08, Theorem 3.20 and Theorem 3.7]). We shall show in Theorem 5.1 below that the exponent is finite for general $K$-simple groups.

We define

\[ \mathfrak{n}_S({\sf G}):=\bigg\{\begin{array}{@{}ll} \text{the least even integer } \ge \mathfrak{p}_S({\sf G})/2, & \text{if } \mathfrak{p}_S({\sf G})>2,\\ 1, & \text{if } \mathfrak{p}_S({\sf G})=2. \end{array} \]

With this notation, we have the following result.

Theorem 2.4 Let $W_S$ be a compact open subgroup of $G_S$, and $\beta$ a Haar-uniform probability measure supported on a $W_S$-bi-invariant bounded subset $B$ of $G_S$. Then

\[ \big\|\pi_S^{\rm aut}(\beta)|_{\mathcal{H}_{\sf G}^{00}}\big\| \ll_{W_S,\eta} m_S(B)^{-{1}/{4\mathfrak{n}_S({\sf G})}+\eta} \quad\text{for all } \eta>0. \]

Proof. The proof is a generalization of the proof of [Reference Gorodnik and NevoGN12, Cor. 6.7], and so we only provide an outline and refer to [Reference Gorodnik and NevoGN12, § 6] for further details. The main idea, which originated in [Reference Cowling, Haagerup and HoweCHH88] and [Reference NevoNev98], is to observe that a suitable tensor power of $\pi _S$ restricted to $\mathcal {H}_{\sf G}^{00}$ is weakly contained in (a multiple of) the regular representation $\lambda _S$ of $G_S$ on $L^2(G_S)$, and then use a generalization of the Kunze–Stein convolution inequality valid in $L^2(G_S)$. More precisely, the representation $(\pi _S|_{\mathcal {H}_{\sf G}^{00}})^{\otimes \mathfrak {n}_S({\sf G})}$ is $L^p$-integrable for all $p>2$, so that it is weakly contained in the regular representation $\lambda _G$ (by [Reference Cowling, Haagerup and HoweCHH88]), which allows us to deduce (by [Reference NevoNev98]) that for any probability density $\beta ^\prime$ on $G_S$

(2.4)\begin{equation} \big\|\pi_S^{\rm aut}(\beta^\prime)|_{\mathcal{H}_{\sf G}^{00}}\big\|\le \|\lambda_S(\beta^\prime)\|^{{1}/{\mathfrak{n}_S({\sf G})}}. \end{equation}

Now let

\[ B^\prime:=U_S B U_S \]

and denote by $\beta ^\prime$ the Haar-uniform probability measure supported on $B^\prime$. Since $U_S\cap W_S$ has finite index in both $U_S$ and $W_S$, it is clear that there exist $c_1,c_2>0$, depending only on $W_S$, such that

\[ c_1\, m_S(B^\prime)\le m_S(B)\le c_2\, m_S( B^\prime), \]

so that

(2.5)\begin{equation} \|\lambda_S(\beta)\|\le c\, \|\lambda_S(\beta^\prime)\| \end{equation}

for some $c>0$. Finally, using the decomposition of $m_S$ with respect to the Iwasawa decomposition on $G_S$, we deduce (cf. [Reference Gorodnik and NevoGN12, Theorem 6.6]) that $\|\lambda _S( \beta ^\prime )\|$ can be estimated in terms of the Harish-Chandra function on $G_S$. The Harish-Chandra function in this case is $L^{4+\eta }$-integrable for all $\eta >0$ (by [Reference Gorodnik and NevoGN12, Proposition 6.3]), and we deduce that

(2.6)\begin{equation} \|\lambda_S(\beta^\prime)\| \ll_{S,\eta} m(B^\prime)^{-{1}/{4}+\eta}\quad\text{for all } \eta>0. \end{equation}

We note that this argument applies to groups that possess Cartan and Iwasawa decompositions, and it does not require that the group be simply connected. Combining (2.4), (2.5) and (2.6), we deduce the theorem.

2.3 Mean ergodic theorem for simply connected groups

In our discussion below we aim to consider approximation by elements of $\Gamma _S$ subject to arbitrary additional congruence conditions. Such a condition is determined by a compact open subset $W\subset \mathcal {I}^S$, and to facilitate this discussion we will now reformulate Theorem 2.4 in a more general and explicit form.

Note that $U(W)$ depends on $S$ also, but we suppress this dependence in the notation. Let

\[ \Gamma_S(W):=\Gamma_S\cap (G_\infty \times G_S\times U(W)). \]

Since $U(W)$ is a finite-index subgroup of $\mathcal {I}^S$, it follows that $\Gamma _S(W)$ has finite index in $\Gamma _S$. In particular, $\Gamma _S(W)$ is a lattice subgroup of $G_\infty \times G_S$. We consider the homogeneous space

(2.7)\begin{equation} Y_{S,W}:=(G_\infty\times G_S)/\Gamma_S(W) \end{equation}

equipped with the Haar probability measure $\mu _{S,W}$.

The group $G_S$ naturally acts on the space $Y_{S,W}$ by left translations, and we introduce averaging operators

(2.8)\begin{equation} \pi_{S,W}(\beta):L^2(Y_{S,W})\to L^2(Y_{S,W}):\phi\mapsto \frac{1}{m_S(B)}\int_B \phi(g^{-1}y)\, dm_S(g) \end{equation}

defined for measurable subsets $B$ of $G_S$ with finite positive measures.

For simplicity, we skip the index $W$ in the above notation if $W=\mathcal {I}^S$.

Theorem 2.6 Assume that $\sf G$ is simply connected, and isotropic over $S\subset V_K^f$. Let $W \subset \mathcal {I}^S$ be a compact open subset, and let $W_S$ be a compact open subgroup of $G_S$. Let $\beta$ be the Haar-uniform probability measure supported on a $W_S$-bi-invariant bounded subset $B$ of $G_S$ with positive measure. Then there exists $\mathfrak {k}_S({\sf G})>0$ such that for every $\phi \in L^2(Y_{S,W})$, and any $\eta > 0$

\[ \bigg\| \pi_{S,W}(\beta)\phi-\int_{Y_{S,W}}\phi\,d\mu_{S,W}\bigg\|_{L^2(Y_{S,W})}\ll_{\eta,W_S} m_{S}(B)^{-\mathfrak{k}_{ S}({\sf G})+\eta}\, \|\phi\|_{L^2(Y_{S,W})}. \]

We note that the existence of the exponent $\mathfrak {k}_{ S}({\sf G})$ and its uniformity over $W\subset \mathcal {I}^S$, is a deep property of the automorphic representation. It is a consequence of the spectral gap property of automorphic representations $\pi _v$ (explained above) holding uniformly over congruence subgroups. In the case of SL$_2$ for example, this property is known as the Ramanujan–Petersson–Selberg eigenvalue bounds. We refer to [Reference Burger and SarnakBS91], [Reference Burger, Li and SarnakBLS92], [Reference ClozelClo03], [Reference SarnakSar05] [Reference Clozel, Oh and UllmoCOU01], [Reference Clozel and UllmoCU04] and [Reference ClozelClo07] for further discussion.

Proof of Theorem 2.6 To simplify notation, we set

\[ G:={\sf G}(\mathbb{A}_K\!)\quad\text{and}\quad\Gamma:={\sf G}(K). \]

We consider the action of the group $G_\infty \times G_S$ on the double-coset space $U(W)\backslash G/\Gamma$. The orbits of this action are open (and, consequently, also closed). Since $\sf G$ is assumed to be isotropic over $S$, it follows from the strong approximation property (Theorem 2.1) that the projection of $\Gamma$ to $G_{V_K^f\backslash S}$ is dense. Therefore, the above orbits are also dense. Hence, we conclude that this action is transitive, and

\[ Y_{S,W}\simeq U(W)\backslash G/\Gamma \]

as $(G_\infty \times G_S)$-spaces. In particular, we also deduce equivalence of unitary representations

(2.9)\begin{equation} L^2(Y_{S,W}) \simeq L^2(U(W)\backslash G/\Gamma) \end{equation}

of $G_S$. Furthermore, the space $L^2(U(W)\backslash G/\Gamma )$ can be identified with the subspace $L^2(G/\Gamma )^{U(W)}$ consisting of $U(W)$-invariant functions in $L^2(G/\Gamma )$. Since $\sf G$ is assumed to be simply connected, there are no non-trivial automorphic characters (see, for instance, [Reference Ghosh, Gorodnik and NevoGGN13, Lemma 4.1]). Hence, it follows from Theorem 2.4 (or from Theorem 2.3 if $B$ is $U_S$-bi-invariant) that there exists $\mathfrak {k}_S({\sf G})>0$ such that for every $\psi \in L^2(G/\Gamma )$, and any $\eta > 0$

\[ \| \pi^{\rm aut}_{S}(\beta)\psi-\mathcal{P}(\psi)\|_{L^2(G/\Gamma)}\ll_{W(S),\eta} m_{S}(B)^{-\mathfrak{k}_{ S}({\sf G})+\eta}\, \|\psi\|_{L^2(G/\Gamma)}, \]

where $\mathcal {P}$ denotes the orthogonal projection on the space of constant functions. Hence, the statement of the theorem follows from (2.9).

We shall also prove and utilize a version of Theorem 2.6 for general $K$-simple groups, but we will postpone this until § 5.2 below.

2.4 Riemannian local volume and distance estimates

We now note two local properties of distance and volume in any almost connected semisimple Lie group with finite center, denoted by $G_\infty$, which will be used in our arguments below. We fix a right-invariant Riemannian metric $\rho$ on $G_\infty$ and consider the corresponding balls

\[ B(g,r):=\{x\in G_\infty:\, \rho(x,g)\le r \}. \]

We first establish the following local estimates for the volume $m_\infty (B(g,r))$. By right invariance, it clearly suffices to consider only the case where the center is the identity element $e$. Recall that we use the notation $d=\dim _\mathbb {R} G_\infty$.

Proof. According to the volume formula for Riemannian balls [Reference SakaiSak96, p. 66], for sufficiently small $r$,

\[ m_\infty(B(e,r))=\int_0^r \omega(s)\, ds, \]

where $\omega$ is a continuous function satisfying

\[ c_1'\, s^{d-1}\le \omega(s)\le c_2'\, s^{d-1} \]

for some $c_1',c_2'>0$. This implies the first estimate. In addition,

\[ m_\infty(B(e,r+\epsilon))-m_\infty(B(e,r))=\int_r^{r+\epsilon}\omega(s)\, ds \ll (r+\epsilon)^{d-1}\, \epsilon, \]

which gives the second bound.

Now let us define two metrics $\rho$ and $\rho ^\prime$ on $G_\infty$ to be locally equivalent if for every compact neighborhood $Q\subset G_\infty$ of $I$, there exists a constant $C_Q$ such that for any two points $x\neq y\in G$ satisfying $xy^{-1}\in Q$, we have $C_Q^{-1}\le {\rho (x,y)}/{\rho ^\prime (x,y)}\le C_Q$.

Assuming that $G_\infty \subset GL_d(\mathbb {R})$ (namely fixing a faithful linear representation of $G_\infty$, but suppressing it from the notation), fix a (vector space) norm on $\text {Mat}_d(\mathbb {R})$ and consider the distance on $G_\infty$ given by $\rho ^\prime (x,y)=\|x-y\|$, $x,y \in G_\infty$.

3.1 The parameters of effective Diophantine approximation

Fix $S\subset V_K^f$. We recall that the group $\Gamma _S$ embeds diagonally in $G_\infty \times G_S$ as a lattice subgroup, and the Haar measures $m_\infty$ and $m_S$ on the factors are normalized so that $\Gamma _S$ has covolume one with respect to $m_\infty \times m_S$. We expect that $N_T(x)$ is approximated by the volume sum

(3.2)\begin{equation} V_T:=\sum_{1\le h\le T} m_\infty(\mathcal{B}_h)m_S(\Sigma_S(h)), \end{equation}

where

\[ \mathcal{B}_h:=\{x\in G_\infty:\, \rho(x,e)\le h^{-b}\}\quad\text{and}\quad \Sigma_S(h):=\{g\in G_S:\, {{\rm H}_f}(g)=h\}. \]

We recall that $\mathcal {I}_S=\prod _{v\in S} {\sf G}(O_v)$. Clearly, the set $\Sigma _S(h)$ is compact and $\mathcal {I}_S$-bi-invariant, but it might be empty for some choices of $h\in {\mathbb {N}}$, so that we introduce

\[ \mathcal{L}_S:=\{h\in{\mathbb{N}}:\, \Sigma_S(h)\ne \emptyset\}. \]

The estimates in this section will depend on the following three parameters:

We record some basic properties of the sets $\Sigma _S(h)$ and $\mathcal {L}_S$.

We note that in order to obtain the lower bound in terms of $h$ in part (c), it is essential to assume that the group $\sf G$ is isotropic for all $v\in S$. This is the only place where we use this condition. When this condition does not hold, it is still possible to formulate a weaker lower bound.

For $h\in \mathcal {L}_S$, we denote by $\sigma _h$ the Haar-uniform probability measure supported on the subset $\Sigma _S(h)^{-1}$ of $G_S$ and consider the corresponding averaging operator

\[ \pi_S(\sigma_h):L^2(Y_S)\to L^2(Y_S) \]

on the space $Y_S:=(G_\infty \times G_S)/\Gamma _S$ defined in (2.8). We note that $\Sigma _S(h)^{-1}\ne \Sigma _S(h)$ in general, but both sets have the same Haar measure. By Theorem 2.6, there exists $\mathfrak {k}_S({\sf G})=\mathfrak {k}_S\in (0,1/2)$ such that for all $\phi \in L^2(Y_S)$, and any $\eta > 0$

(SP)\begin{equation} \|\pi_S(\sigma_h)\phi -\int_{Y_S} \phi\, d\mu_S\|_{L^2(Y_S)}\ll_{S,\eta} \, m_{S}(\Sigma_S(h))^{-\mathfrak{k}_S+\eta}\quad\text{for all } h\in \mathcal{L}_S. \end{equation}

We also recall that by Lemma 2.8(a),

(V2)\begin{equation} h^{-b\mathfrak{d}}\ll m_{\infty}(\mathcal{B}_h)\ll h^{-b\mathfrak{d}}\quad\text{for all } h\ge h_0, \end{equation}

where $\mathfrak {d}:=\dim _{\mathbb {R}}(G_\infty )$.

We set

\[ b_0:=2\mathfrak{a}\mathfrak{k}/\mathfrak{d} \]

and for a positive scale $b< b_0$, define

\[ \theta_0(b):=\frac{(1-\mathfrak{k})\mathfrak{a}-b\mathfrak{d}/2}{\mathfrak{a}-b\mathfrak{d}} =\frac{1}{2}+\frac{(1/2-\mathfrak{k})\mathfrak{a}}{\mathfrak{a}-b\mathfrak{d}}. \]

Note that the condition on $b< b_0$ insures that $\theta _0(b)\in (0,1)$. With this notation, we prove the following result.

Theorem 3.2 Let $\sf G$ be a connected simply connected $K$-simple algebraic group defined over a number field $K$ and let $S$ be a finite set of finite places such that $\sf G$ is isotropic for all $v\in S$. Then for every $b\in (0,b_0)$ and $\theta \in (\theta _0(b), 1)$,

\[ \|N_T-V_T\|_{L^2(Q)}\ll_{S, Q,\theta} V_T^\theta, \]

where $Q$ is an arbitrary bounded measurable subset of $G_\infty$.

The proof of Theorem 3.2 will be based on the estimates (V1), (V2) and (SP).

Using a Borel–Cantelli argument we will also derive a pointwise bound for $N_T$.

Corollary 3.3 With the notation of Theorem 3.2, for every $\theta '>\theta$

\[ N_T(x)=V_T+O_{S, x,\theta'}(V_T^{\theta'})\quad\text{for a.e. } x\in G_\infty. \]

3.2 Towards a best-possible estimate

Let us note that it is often the case that the lower bound $m_S(\Sigma _S(h))\ll _S h^{\mathfrak {a}}$ holds, as well as the upper bound (V1). Then the sum (3.2) is bounded by

\[ V_T\ll \sum_{1\le h\le T,h\in\mathcal{L}_S} h^{-b\mathfrak{d}} \cdot h^{\mathfrak{a}}, \]

and it follows from Lemma 3.1(a) that $V_T$ is uniformly bounded when $b>\mathfrak {a}/\mathfrak {d}$. Therefore, the estimate in Theorem 3.2 is only interesting in the range $b\le \mathfrak {a}/\mathfrak {d}$. We highlight that Corollary 3.3 gives the following estimate.

Corollary 3.4 Suppose that additionally in Corollary 3.3 the automorphic representation of $G_S$ is tempered and that $S$ consists of unramified places. Then for every $b<\mathfrak {a}/\mathfrak {d}$,

\[ N_T(x)=V_T+O_{S, x,\eta}\big(V_T^{{1}/{2}+\eta}\big)\quad\text{for all } \eta>0 \text{ and a.e. } x\in G_\infty. \]

Indeed, under the temperedness condition the estimate (SP) holds with $\mathfrak {k}_S=1/2$ (see Theorem 2.3), so that in this case Theorem 3.2 covers all the relevant range of parameters $b<\mathfrak {a}/\mathfrak {d}$ and gives the exponent $\theta = {1}/{2}+\eta$ for every $\eta >0$. Hence, Corollary 3.4 is a direct consequence of Corollary 3.3.

Noting the fact that error term in the foregoing estimate is bounded (in essence) by the square root of the main term, it is natural to expect that the exponent it establishes is, in fact, the best possible. A full proof of this fact requires establishing the expected lower bound for $N_T$. We will establish lower bounds for discrepancy estimates in a separate paper, but the question of optimality of Corollary 3.4 remains open.

3.3 Proof of Theorem 3.2

Consider the function

\[ D_T(x):=N_T(x)-V_T,\quad \text{with } x\in G_\infty, \]

and recall that the metric $\rho$ is right-invariant, and so $B(x,h^{-b})= \mathcal {B}_{h}x= \mathcal {B}_{h}^{-1}x$. Therefore,

\begin{align*} D_T(x)&:=\sum_{1\le h\le T} (|\{\gamma \in\Gamma_S\cap B(x,h^{-b}):\, {{\rm H}_f}(\gamma)=h\}|-m_{\infty}(\mathcal{B}_{h}) m_{S}(\Sigma_S(h)))\\ &= \sum_{1\le h\le T} (|\Gamma_S\cap (\mathcal{B}_{h}x\times \Sigma_S(h))|-m_{\infty}(\mathcal{B}_{h}) m_{S}(\Sigma_S(h))). \end{align*}

The crucial ingredient of our proof is that $D_T(x)$ can be represented in terms of the averaging operators $\pi _S(\sigma _h)$. Let

\begin{align*} \chi_h(g_\infty,g_S):=\chi_{\mathcal{B}_h}(g_\infty)\chi_{\mathcal{I}_S}(g_S) \end{align*}

denote the characteristic function of the subset $\mathcal {B}_h \times \mathcal {I}_S$ of $G_\infty \times G_S$. The sum

\[ \phi_h (g):=\sum_{\gamma\in \Gamma_S} \chi_h(g\gamma)=\sum_{\gamma\in \Gamma_S} \chi_h(g\gamma^{-1}) \]

defines a measurable function with compact support on the homogeneous space $Y_S:=(G_\infty \times G_S)/\Gamma _S$. We observe that for $x\in G_\infty$ and $u\in \mathcal {I}_S$,

\begin{align*} \int_{a\in \Sigma_S(h)} \phi_h(a (x,u))\,dm_{S}(a) &= \sum_{\gamma\in \Gamma_S} \int_{\Sigma_S(h)} \chi_h(x\gamma^{-1}, au\gamma^{-1})\, d m_{S}(a)\\ &= \sum_{\gamma\in \Gamma_S} \int_{\Sigma_S(h)} \chi_{\mathcal{B}_h}( x\gamma^{-1}) \chi_{\mathcal{I}_S}(au\gamma^{-1})\, dm_{S}(a)\\ &= \sum_{\gamma\in \Gamma_S\cap B(x,h^{-b})} m_{S}(\mathcal{I}_S\gamma u^{-1}\cap \Sigma_S(h)). \end{align*}

Note that in the above computation, since $\Gamma _S$ is a discrete subgroup of $G_\infty \times G_S$, the non-zero summands in the sum above constitute a finite subset of $\gamma$. In order to evaluate the last expression, we use that the set $\Sigma _S(h)$ is $\mathcal {I}_S$-bi-invariant. Therefore, if $H_f(\gamma )=h$, then $\mathcal {I}_S \gamma u^{-1}\subset \Sigma _S(h)$ and $m_{S}( \mathcal {I}_S\gamma u^{-1}\cap \Sigma _S(h))$=1. On the other hand, if $H_f(\gamma )\neq h$, we have $\mathcal {I}_S\gamma u^{-1}\cap \Sigma _S(h)=\emptyset$. Hence, for every $u\in \mathcal {I}_S$,

\[ |\{\gamma \in\Gamma_S\cap B(x,h^{-b}):\, {{\rm H}_f}(\gamma)=h\}|=\int_{a\in \Sigma_S(h)} \phi_h(a (x,u))\,dm_{S}(a). \]

Furthermore,

\begin{align*} \int_{Y_S} \phi_h \, d\mu_S &=\int_{G_\infty\times G_S} \chi_{\mathcal{B}_h}(g_\infty)\, \chi_{\mathcal{I}_S}(g_S)\, dm_\infty(g_\infty) \, d m_S(g_S)\\ &= m_{\infty}(\mathcal{B}_h)m_S(\mathcal{I}_S)= m_{\infty}(\mathcal{B}_h). \end{align*}

Hence, we deduce that for every $u\in \mathcal {I}_S$,

\[ D_T(x)=\sum_{1\le h\le T} \bigg(\int_{a\in \Sigma_S(h)} \phi_{h}(a(x,u))\,d m_{S}(a) - m_{S}(\Sigma_S(h))\int_{Y_S} \phi_h \, d\mu_S\bigg). \]

Let ${Q}$ be a bounded measurable subset in $G_\infty$. Since the first integral in the previous line was just shown to be independent of $u\in \mathcal {I}_S$, we obtain

\begin{align*} &\int_{Q} \bigg|\int_{a\in \Sigma_S(h)} \phi_h(a (x,e))\,d m_{S}(a) - m_S(\Sigma_S(h))\int_{Y_S} \phi_h \, d\mu_S\bigg|^2\, d m_{\infty}(x)\\ &\quad =\int_{{Q}\times \mathcal{I}_S} \bigg|\int_{a\in \Sigma_S(h)} \phi_h(a (x,u)\,d m_{S}(a) - m_{S}(\Sigma_S(h))\int_{Y_S} \phi_h \, d\mu_S\bigg|^2\, d m_{\infty}(x)\,d m_{S}(u). \end{align*}

The set $Q\times \mathcal {I}_S$ projects onto a measurable subset $(Q\times \mathcal {I}_S)\Gamma _S$ of $Y_S$. Since $Q\times \mathcal {I}_S$ is bounded, there exists $N_Q$ such that every point in the image has preimage of cardinality at most $N_Q$. This implies that the last integral is bounded by

\begin{align*} & N_Q \int_{({Q}\times \mathcal{I}_S)\Gamma_S} \bigg|\int_{a\in \Sigma_S(h)} \phi_h(a y)\,d m_{S}(a) - m_{S}(\Sigma_S(h))\int_{Y_S} \phi_h \, d\mu_S\bigg|^2\, d\mu_S(y)\\ &\quad\le N_{Q}\int_{Y_S} \bigg|\int_{\Sigma_S(h)} \phi_h(a y)\,d m_{S}(a) - m_{S}(\Sigma_S(h))\int_{Y_S} \phi_h \, d\mu_S\bigg|^2\, d\mu_S(y)\\ &\quad= N_{Q}\, m_{S}(\Sigma_S(h))^2\bigg\|\pi_S(\sigma_h)\phi_h -\int_{Y_S} \phi_h \, d\mu_S \bigg\|_{L^2(Y_S)}^2. \end{align*}

Hence, using the bound (SP), we deduce that for any $\eta > 0$

\begin{align*} \|D_T\|_{L^2(Q)} &\le \sum_{1\le h\le T} \bigg\|\int_{a\in \Sigma_S(h)} \phi_{h}(a (\cdot,e))\,dm_{S}(a)-m_{S}(\Sigma_S(h))\int_{Y_S} \phi_h \, d\mu_S\bigg\|_{L^2(Q)}\\ & \ll_{S,Q,\eta} \sum_{1\le h\le T} m_{S}(\Sigma_S(h))^{1-\mathfrak{k}_S+\eta}\|\phi_h\|_{L^2(Y_S)}. \end{align*}

Furthermore, the $L^2$-norm of $\phi _h$ can be estimated as follows:

\begin{align*} \|\phi_h\|^2_{L^2(Y_S)} &=\int_{(G_\infty\times G_S)/\Gamma_S} \bigg(\sum_{\gamma\in \Gamma_S} \chi_h(g\gamma)\bigg)^2\,d(m_\infty\times m_S)(g)\\ &=\int_{(G_\infty\times G_S)/\Gamma_S} \sum_{\gamma_1,\gamma_2\in \Gamma_S} \chi_h(g\gamma_1)\chi_h(g\gamma_2)\,d(m_\infty\times m_S)(g)\\ &=\int_{(G_\infty\times G_S)/\Gamma_S} \sum_{\gamma,\delta\in \Gamma_S} \chi_h(g\delta)\chi_h(g\delta\gamma)\,d(m_\infty\times m_S)(g)\\ &= \int_{G_\infty\times G_S} \sum_{\gamma\in \Gamma_S} \chi_h(g)\chi_h(g\gamma)\,d(m_\infty\times m_S)(g)\\ & =\sum_{\gamma\in \Gamma_S} (m_\infty\times m_S)\big((\mathcal{B}_h\times \mathcal{I}_S)\cap (\mathcal{B}_h \times \mathcal{I}_S)\gamma^{-1}\big)\le N_h\, m_{\infty}(\mathcal{B}_h), \end{align*}

where $N_h$ denotes the number of $\gamma \in \Gamma _S$ such that $(\mathcal {B}_h\times \mathcal {I}_S)\cap (\mathcal {B}_h \times \mathcal {I}_S)\gamma ^{-1}\ne \emptyset$. Since the family of sets $\mathcal {B}_h\times \mathcal {I}_S$ is uniformly bounded, this number is uniformly bounded, and we conclude using (V2) that for any $\eta > 0$

\[ \|D_T\|_{L^2(Q)}\ll_{S,Q,\eta} \sum_{1\le h\le T} m_{S}(\Sigma_S(h))^{1-\mathfrak{k}_S+\eta} m_{\infty}(\mathcal{B}_h)^{1/2}\ll \sum_{1\le h\le T} m_{S}(\Sigma_S(h))^{1-\mathfrak{k}_S+\eta} h^{-b\mathfrak{d}/2}. \]

To complete the proof of Theorem 3.2 we employ the following computation, where we use parameters $A>0$ and $B>1$ that will be specified later. Using (V1), and writing $\mathfrak {k}_S-\eta =\mathfrak {k}$ for brevity, we obtain

\[ \sum_{1\le h\le T} m_{S}(\Sigma_S(h))^{1-\mathfrak{k}} h^{-b\mathfrak{d}/2} \ll_S \sum_{1\le h\le T} m_{S}(\Sigma_S(h))^{1-\mathfrak{k}+A} h^{-(\mathfrak{a}A+b\mathfrak{d}/2)}, \]

and applying Hölder's inequality, we conclude that this sum is bounded by

\[ \bigg(\sum_{1\le h\le T} m_{S}(\Sigma_S(h))^{(1-\mathfrak{k}+A)B} h^{-(\mathfrak{a}A+b\mathfrak{d}/2)B}\bigg)^{1/B}|\mathcal{L}_S\cap [1,T]|^{1-1/B}. \]

We choose $A$ and $B$ so that

\[ (1-\mathfrak{k}+A)B=1\quad\text{and}\quad (\mathfrak{a}A+b\mathfrak{d}/2)B=b\mathfrak{d}, \]

namely,

\[ A=\frac{(1-\mathfrak{k})b\mathfrak{d}-b\mathfrak{d}/2}{\mathfrak{a}-b\mathfrak{d}}\quad\text{and}\quad B=\frac{\mathfrak{a}-b\mathfrak{d}}{(1-\mathfrak{k})\mathfrak{a}-b\mathfrak{d}/2}. \]

Taking into account that $\mathfrak {k}\in (0,1/2)$ and $b<2\mathfrak {k}\mathfrak {a}/\mathfrak {d}$, a direct computation verifies that $A>0$ and $B>1$. Hence, using Lemma 3.1(a) and estimate (V2), we conclude that for every $\eta >0$,

\[ \|D_T\|_{L^2(Q)}\ll_{S,Q,\eta} \bigg(\sum_{1\le h\le T} m_{S}(\Sigma_S(h)) h^{-b\mathfrak{d}}\bigg)^{1/B} T^{\eta(1-1/B)}\ll_\eta V_T^{1/B} T^{\eta(1-1/B)}. \]

Finally, we note that it follows from (V1)–(V2) and Lemma 3.1(b) that

(3.3)\begin{equation} V_T\gg_S \sum_{h\in\mathcal{L}_S\cap [1,T]} h^{-b\mathfrak{d}}\cdot h^{\mathfrak{a}}\gg_S T^{\mathfrak{a}-b\mathfrak{d}}. \end{equation}

Since $\mathfrak {a}-b\mathfrak {d}>0$, we deduce from the previous estimate that for every $\eta '>0$,

\[ \|D_T\|_{L^2(Q)}\ll_{Q,\eta'} V_T^{\theta_0+\eta'}, \]

where

\[ \theta_0:=1/B=\frac{(1-\mathfrak{k})\mathfrak{a}-b\mathfrak{d}/2}{\mathfrak{a}-b\mathfrak{d}}. \]

Since $B>1$, we have $\theta _0\in (0,1)$. This completes the proof of Theorem 3.2.

Proof of Corollary 3.3 We note that the function $D_T(x)=N_T(x)-V_T$ is piecewise constant in $T$ and is determined by its values for $T\in \mathcal {L}_S$. Hence, it is sufficient to analyze the values $D_T(x)$ for $T\in \mathcal {L}_S$. Given any bounded measurable subset $Q$ of $G_\infty$, we have established in Theorem 3.2 the bound

\[ \int_{Q} |D_T(x)|^2\, dm_{\infty}(x)\ll_{S,{Q},\theta} V_T^{2\theta}. \]

Therefore, for every $\eta >0$, the sets

\[ \Omega_T:=\{x\in {Q}:\, |D_T(x)|\ge V_T^{\theta}\,T^\eta \} \]

satisfy

\[ m_{\infty}(\Omega_T)\ll_{S,{Q},\theta} T^{-2\eta}. \]

Hence, it follows from Lemma 3.1(a) that

\[ \sum_{T\in\mathcal{L}_S} m_{\infty}(\Omega_T)<\infty, \]

and by the Borel–Cantelli lemma, the $\limsup$ of the sets $\Omega _T$ with $T\in \mathcal {L}_S$ has measure zero. This means that for almost every $x\in {Q}$ and $T\in \mathcal {L}_S$,

\[ |D_T(x)|\le V_T^{\theta}\, T^{\eta}\quad\text{when } T\ge T_0(x,\eta). \]

Then using the estimate (3.3), we conclude that for almost all $x\in Q$,

\[ |D_T(x)|\ll_S V_T^{\theta+\eta/(\mathfrak{a}-b\mathfrak{d})} \quad\text{when } T\ge T_0(x,\eta). \]

Since $G_\infty$ can be exhausted by a countable union of bounded measurable sets $Q$, it follows that for every $\theta '>\theta$ and almost all $x\in G_\infty$,

\[ |D_T(x)|\ll_S V_T^{\theta'}\quad\text{when } T\ge T_0(x,\theta'). \]

Thus, in particular,

\[ |D_T(x)|\ll_{S,x,\theta'} V_T^{\theta'}\quad \text{for all } T, \]

which implies the corollary.

4.1 Discrepancy of rational points

We fix a subset $S$ of finite places of $K$ such that $\sf G$ is isotropic over $S$ and consider the $S$-arithmetic group $\Gamma _S:={\sf G}(O_S)$ which is exhausted by the increasing family of subsets

\[ R_S(h):=\{\gamma\in \Gamma_S:\, {{\rm H}_f}(\gamma)\le h \}. \]

We recall that $\mathcal {I}_S:=\prod _{v\in S} {\sf G}(O_v)$, and $\mathcal {I}^S:=\prod _{v\in V_K^f\backslash S} {\sf G}(O_v)$, and the Haar measure $m_S$ on $G_S$ is normalized so that $m_S(\mathcal {I}_S)=1$. We also denote by $m^S$ the Haar measure on $\mathcal {I}^S$ such that $m^S(\mathcal {I}^S)=1$. Under the diagonal embedding

\[ \Gamma_S \hookrightarrow G_\infty\times G_S, \]

$\Gamma _S$ is a lattice subgroup, and so the sets $R_S(h)$, while infinite, are locally finite namely deposit in every bounded subset of $G_\infty$ only finitely many elements. The Haar measure $m_\infty$ in $G_\infty$ is normalized so that $\Gamma _S$ has covolume one with respect to $m_\infty \times m_S$.

By the strong approximation property (Theorem 2.1), the diagonal embedding

\[ \Gamma_S \hookrightarrow G_\infty\times \mathcal{I}^S \]

is dense. Our goal is to analyze the discrepancy of distribution of this dense set. One can show (in fact, it follows from our results here) that the number of points from $R_S(h)$ contained in a bounded subset $\Omega$ of $G_\infty \times \mathcal {I}^S$ grows as

\[ v_S(h):=m_S(B_S(h)), \]

where $B_S(h):=\{g\in G_S:\, {{\rm H}_f}(g)\le h \}$. We define the discrepancy function as

(4.1)\begin{equation} \mathcal{D}(R_S(h),\Omega):= \bigg|\frac{|R_S(h)\cap \Omega|}{v_S(h)}-(m_\infty\times m^S)(\Omega)\bigg|. \end{equation}

Our goal is to produce an explicit estimate for this quantity for a natural collection of subsets of $G_\infty \times \mathcal {I}^S$. Let $E$ be a subset of $G_\infty$ and $W$ a compact open subset of $\mathcal {I}^S$. For $x\in G_\infty$, we set

\[ E(x):=Ex. \]

We will focus on analyzing the discrepancy for $\Omega =E(x)\times W$, where we allow an arbitrary congruence condition $W\subset \mathcal {I}^S$.

A crucial ingredient of our analysis is the estimate on averaging operators established in § 2.3, which we now recall. We consider the spaces

\[ Y_{S,W}:=(G_\infty\times G_S)/\Gamma_S(W) \]

equipped with the invariant probability measures $\mu _{S,W}$. Let $\beta _h$ be the uniform probability measure supported on the set $B_S(h)^{-1}$ and

\[ \pi_{S,W}(\beta_h):L^2(Y_{S,W})\to L^2(Y_{S,W}) \]

the corresponding averaging operator defined in (2.8). We note that $B_S(h)^{-1}\ne B_S(h)$ in general. According to Theorem 2.6, there exists an exponent $\mathfrak {k}_S({\sf G})=\mathfrak {k}_S\in (0,1/2]$, which is uniform in $W$, such that for every $\phi \in L^2(Y_{S,W})$, and every $\eta > 0$

(SP)\begin{equation} \bigg\| \pi_{S,W}(\beta_h)\phi-\int_{Y_{S,W}}\phi\,d\mu_{S,W}\bigg\|_{L^2(Y_{S,W})}\ll_{W,S,\eta} m_{S}(B_S(h))^{-\mathfrak{k}_S+\eta}\, \|\phi\|_{L^2(Y_{S,W})}. \end{equation}

The estimates in this section will depend on the parameters:

4.2 Mean-square discrepancy estimates

We will begin our discussion by establishing an $L^2$-bound for the discrepancy function $\mathcal {D}(R_S(h),E(x)\times W)$. Remarkably, this bound holds in great generality for measurable subsets $E$, and as noted already it is uniform in $W$.

For a subset $E$ of $G_\infty$, we define

\[ \mathcal{N}(E):=|\{\gamma\in {\sf G}(O):\, m_\infty(E\cap \gamma E)>0 \}|<\infty. \]

Clearly, $\mathcal {N}(E)$ is uniformly bounded when $G_\infty$ is compact, and finite if $E$ is bounded, but this is not the case in general.

Theorem 4.1 (Mean square discrepancy bound)

Let $E$ be any measurable subset of $G_\infty$ of finite measure satisfying $\mathcal {N}(E)<\infty$ and $W$ a compact open subset of $\mathcal {I}^S$. Then for any $\eta > 0$

\[ \|\mathcal{D}(R_S(h), E(\cdot)\times W)\|_{L^2(Q)}\ll_{S,Q, \eta} \mathcal{N}(E)^{1/2} m_\infty(E)^{1/2}m^S(W)^{1/2} v_S(h)^{-\mathfrak{k}_S({\sf G})+\eta} \]

for every bounded measurable subset $Q$ of $G_\infty$.

Note that in foregoing formula, the discrepancy decreases as the size of $W$ decreases, and this fact reflects our choice of normalization in (4.1). If we choose to normalize the discrepancy of rational points in $E$ by dividing by the measure of $W$, the error estimate on the right-hand side would depend on $W$ via $m^S(W)^{-1/2}$. Then the discrepancy will increase as the congruence conditions imposed become more stringent, as expected.

Remark 4.2 We say a subset ${Q}$ of $G_\infty$ is $\Gamma _S(W)$-injective if the projection map

\[ Q\times \mathcal{I}_S \to (G_\infty\times G_S)/\Gamma_S(W) \]

is injective. If ${Q}$ is $\Gamma _S(W)$-injective, then the implicit constant in the above estimate in Theorem 4.1 is independent of $Q$.

Proof of Theorem 4.1 Let again

\[ \chi(g_\infty,g_S):=\chi_{E^{-1}}(g_\infty)\chi_{\mathcal{I}_S}(g_S),\quad \text{for } (g_\infty,g_S)\in G_\infty\times G_S, \]

denote the characteristic function of the subset $E^{-1}\times \mathcal {I}_S$ of $G_\infty \times G_S$. We introduce the subset

\[ \Delta(W):=\Gamma_S\cap (G_\infty\times W) \]

of $\Gamma _S$. Since $W$ is bi-invariant under the subgroup $U(W)$, it is clear that $\Delta (W)$ is bi-invariant under the subgroup $\Gamma _S(W):=\Gamma _S\cap (G_\infty \times G_S\times U(W))$. Moreover, since the set $W$ is union of finitely many right cosets of $U(W)$, the set $\Delta (W)$ is a finite union of right cosets of $\Gamma _S(W)$. For $g=(g_\infty,g_S)\in G_\infty \times G_S$, we define

\[ \phi(g):=\sum_{\delta\in \Delta(W)} \chi(g_\infty\delta^{-1},g_S\delta^{-1}). \]

Since $\Delta (W)$ is $\Gamma _S(W)$-bi-invariant, this defines a measurable function on the space $Y_{S,W}=(G_\infty \times G_S)/\Gamma _S(W)$. First, let us compute the integral of $\phi$:

\begin{align*} \int_{Y_{S,W}} \phi \, d\mu_{S,W} &= \int_{(G_\infty\times G_S)/\Gamma_S(W)} \bigg(\sum_{\delta\in \Delta(W)} \chi(g\delta^{-1})\bigg) \,d\mu_{S,W}(g)\\ &=\int_{(G_\infty \times G_S)/\Gamma_S(W)} \bigg(\sum_{\delta\in \Delta(W)/\Gamma_S(W)}\sum_{\gamma\in \Gamma_S(W)} \chi(g \gamma^{-1}\delta^{-1})\bigg)\, d\mu_{S,W}(g) \\ &= \sum_{\delta\in \Delta(W)/\Gamma_S(W)}\int_{G_\infty \times G_S} \chi(g\delta^{-1})\, \frac{d(m_{\infty}\times m_{S})(g)}{|\Gamma_S/\Gamma_S(W)|} \\ &= \frac{|\Delta(W)/\Gamma_S(W)|}{|\Gamma_S/\Gamma_S(W)|} \int_{G_\infty \times G_S} \chi\, d(m_{\infty}\times m_{S}) \\ &= \frac{|\Delta(W)/\Gamma_S(W)|}{|\Gamma_S/\Gamma_S(W)|} m_{\infty}(E^{-1})m_S(\mathcal{I}_S)= \frac{|\Delta(W)/\Gamma_S(W)|}{|\Gamma_S/\Gamma_S(W)|} m_{\infty}(E). \end{align*}

Since the projection of $\Gamma _S$ to $\mathcal {I}^S$ is dense, the map $\gamma \to \gamma U(W)$, $\gamma \in \Gamma _S$, defines a bijection between the cosets $\Gamma _S/\Gamma _S(W)$ and $\mathcal {I}^S/U(W)$. In particular,

(4.2)\begin{equation} |\Gamma_S/\Gamma_S(W)|=|\mathcal{I}^S/U(W)|. \end{equation}

In addition, since $W$ is open, the projection of $\Delta (W)=\Gamma _S\cap (G_\infty \times G_S\times W)$ is dense in $W$, and we obtain that

(4.3)\begin{equation} |\Delta(W)/\Gamma_S(W)|=|W/U(W)|. \end{equation}

We recall that the measure $m^S$ on $\mathcal {I}^S$ is normalized so that $m^S(\mathcal {I}^S)=1$. This implies that

(4.4)\begin{equation} m^S(W)=\frac{|W/U(W)|}{|\mathcal{I}^S/U(W)|}. \end{equation}

Hence, we conclude from the above computations that

(4.5)\begin{equation} \int_{Y_{S,W}} \phi \, d\mu_{S,W} =m_\infty(E)m^S(W). \end{equation}

In particular, this shows that $\phi \in L^1(Y_{S,W})$. In fact, we will show later in the proof that $\phi \in L^2(Y_{S,W})$.

Since $\phi$ in non-negative and integrable, it follows from the Fubini–Tonelli theorem that for almost every $y\in Y_{S,W}$,

\begin{align*} \int_{a\in B_S(h)}\phi(ay)\, dm_{S}(a)<\infty. \end{align*}

We shall show that the discrepancy $\mathcal {D}(R_S(h), E(\cdot )\times W)$ can be approximated by such integrals. By the Fubini–Tonelli theorem again, for $x\in G_\infty$ and $u\in \mathcal {I}_S$,

\begin{align*} \int_{a\in B_S(h)}\phi\big(a(x,u)\big)\, dm_{S}(a) &=\sum_{\delta\in \Delta(W)} \int_{a\in B_S(h)} \chi(x\delta^{-1}, au\delta^{-1})\, dm_{S}(a)\\ & = \sum_{\delta\in \Delta(W)} \int_{B_S(h)} \chi_{E^{-1}}(x\delta^{-1})\chi_{\mathcal{I}_S} (au\delta^{-1}) \, dm_{S}(a)\\ &=\sum_{\delta\in \Delta(W)\cap (Ex\times G_S)} m_{S}(\mathcal{I}_S \delta u^{-1}\cap B_S(h)). \end{align*}

We recall that the sets $B_S(h):=\{b\in G_S:\, {{\rm H}_f}(b)\le h\}$ are $\mathcal {I}_S$-bi-invariant. Therefore, if $\delta \in B_S(h)$, we have $\mathcal {I}_S\delta u^{-1}\subset B_S(h)$, and if $\delta \notin B_S(h)$, we have $\mathcal {I}_S\delta u^{-1}\cap B_S(h)=\emptyset$. Hence, since $m_S(\mathcal {I}_S)=1$, it follows that for every $u\in \mathcal {I}_S$,

(4.6)\begin{equation} |R_S(h)\cap (E(x)\times W)|=\int_{a\in B_S(h)}\phi(a(x,u))\, dm_{S}(a). \end{equation}

Combining (4.5) and (4.6), we derive that

\begin{align*} \mathcal{D}(R_S(h), E(x)\times W) &= \bigg| \frac{1}{m_S(B_S(h))}\int_{a\in B_S(h)}\phi(a(x,e))\, dm_{S}(a)-\int_{Y_{S,W}} \phi\, d\mu_{S,W} \bigg|\\ &=\bigg| \pi_{{S,W}}(\beta_h) \phi (x,e)-\int_{Y_{S,W}} \phi\, d\mu_{S,W} \bigg|. \end{align*}

This representation allows us to apply the estimate (SP). Let $Q$ be a bounded measurable subset of $G_\infty$. Since the integral in (4.6) is independent of $u\in \mathcal {I}_S$, we obtain that

\begin{align*} &\int_Q \bigg| \int_{a\in B_S(h)}\phi(a(x,e))\, dm_{S}(a)-m_S(B_S(h))\int_{Y_{S,W}} \phi\, d\mu_{S,W} \bigg|^2 \, dm_\infty(x)\\ &\quad = \int_{Q\times \mathcal{I}_S} \bigg| \int_{a\in B_S(h)}\phi(a(x,u))\, dm_{S}(a)-m_S(B_S(h))\int_{Y_{S,W}} \phi\, d\mu_{S,W} \bigg|^2 \, dm_\infty(x)\,dm_S(u). \end{align*}

The set $Q\times \mathcal {I}_S$ projects to a measurable subset $(Q\times \mathcal {I}_S)\Gamma _S(W)$ in $Y_{S,W}$. Since this is a bounded subset of $G_\infty \times G_S$, the fibers of this map have cardinalities bounded by a uniform constant $N_Q$, which is independent of $W$ because $\Gamma _S(W)\subset \Gamma _S$. Hence, the last integral is bounded by

\begin{align*} & N_Q \int_{(Q\times \mathcal{I}_S)/\Gamma_S(W)} \bigg| \int_{a\in B_S(h)}\phi(ay)\, dm_{S}(a)-m_S(B_S(h))\int_{Y_{S,W}} \phi\, d\mu_{S,W} \bigg|^2 \, d\mu_{S,W}(y)\\ &\quad \le N_Q \int_{Y_{S,W}} \bigg| \int_{a\in B_S(h)}\phi(ay)\, dm_{S}(a)-m_S(B_S(h))\int_{Y_{S,W}} \phi\, d\mu_{S,W} \bigg|^2 \, d\mu_{S,W}(y)\\ & \quad = N_Q\, m_S(B_S(h))^2 \bigg\| \pi_{{S,W}}(\beta_h) \phi -\int_{Y_{S,W}} \phi \, d\mu_{S,W} \bigg\|^2_{L^2(Y_{S,W})}. \end{align*}

Thus, it follows from (SP) that for every $\eta > 0$

(4.7)\begin{equation} \|\mathcal{D}(R_S(h), E(\cdot)\times W)\|_{L^2(Q)}\ll_{S,Q,\eta} m_S(B_S(h))^{-\mathfrak{k}_S+\eta}\, \|\phi\|_{L^2(Y_{S,W})}. \end{equation}

The function $\phi$ is indeed square-integrable, and the $L^2$-norm $\|\phi \|_{L^2(Y_{S,W})}^2$ is computed as follows:

\begin{align*} &\int_{(G_\infty\times G_S)/\Gamma_S(W)} \bigg(\sum_{\delta_1,\delta_2\in \Delta(W)} \chi(g\delta_1^{-1})\chi(g\delta_2^{-1}) \bigg)\, d\mu_{S,W}(g)\\ &\quad =\int_{(G_\infty\times G_S)/\Gamma_S(W)} \bigg( \sum_{\delta_1,\delta_2\in \Delta(W)/\Gamma_S(W)}\sum_{\gamma_1,\gamma_2\in \Gamma_S(W)} \chi(g\gamma_1^{-1}\delta_1^{-1})\chi(g\gamma_2^{-1}\delta_2^{-1}) \bigg)\, d\mu_{S,W}(g)\\ &\quad =\int_{(G_\infty\times G_S)/\Gamma_S(W)} \bigg( \sum_{\delta_1,\delta_2\in \Delta(W)/\Gamma_S(W)}\sum_{\sigma,\gamma\in \Gamma_S(W)} \chi(g\sigma\delta_1^{-1})\chi(g\sigma\gamma^{-1}\delta_2^{-1}) \bigg)\, d\mu_{S,W}(g)\\ &\quad =\int_{G_\infty\times G_S} \bigg( \sum_{\delta_1,\delta_2\in \Delta(W)/\Gamma_S(W)}\sum_{\gamma\in \Gamma_S(W)} \chi(g\delta_1^{-1})\chi(g\gamma^{-1}\delta_2^{-1}) \bigg)\, \frac{d(m_\infty\times m_S)(g)}{|\Gamma_S:\Gamma_S(W)|}\\ &\quad = \int_{G_\infty\times G_S} \bigg( \sum_{\gamma\in \Delta(W)/\Gamma_S(W)}\sum_{\delta\in \Delta(W)} \chi(g\gamma^{-1})\chi(g\delta^{-1}) \bigg) \frac{d(m_\infty\times m_S)(g)}{|\Gamma_S:\Gamma_S(W)|}. \end{align*}

We observe that for any $\gamma,\delta \in \Delta (W)$,

\begin{align*} \int_{G_\infty\times G_S} \chi(g\gamma^{-1})\chi(g\delta^{-1})\, d(m_\infty\times m_S)(g)&= m_\infty(E^{-1}\gamma\cap E^{-1} \delta)m_S(\mathcal{I}_S\gamma\cap \mathcal{I}_S \delta)\\ &\le m_\infty(E)m_S(\mathcal{I}_S)=m_\infty(E). \end{align*}

Moreover, this integral is zero unless $m_\infty (E^{-1}\cap E^{-1}\delta \gamma ^{-1})> 0$ and $\delta \gamma ^{-1}\in \mathcal {I}_S$. But since $\gamma, \delta \in \Delta (W)$, we also have $\delta \gamma ^{-1}\in \mathcal {I}^S$, and so in this case it follows that $\delta \gamma ^{-1}\in {\sf G}(O)$. Recalling the definition of $\mathcal {N}(E)$, this implies that for fixed $\gamma \in \Delta (W)$,

\[ \bigg|\bigg\{\delta\in \Delta(W): \int_{G_\infty\times G_S} \chi(g\gamma^{-1})\chi(g\delta^{-1})\, d(m_\infty\times m_S)(g)\ne 0\bigg\}\bigg|\le \mathcal{N}(E). \]

Hence, applying the Fubini–Tonelli theorem once more, we conclude that

\[ \|\phi\|_{L^2(Y_{S,W})}^2\le \mathcal{N}(E)m_\infty(E)\frac{|\Delta(W)/\Gamma_S(W)|}{|\Gamma_S:\Gamma_S(W)|}=\mathcal{N}(E)m_\infty(E)m^S(W). \]

Here we have used (4.2)–(4.4) in the last step. Combining this estimate with (4.7), this completes the proof of Theorem 4.1.

4.3 Almost sure discrepancy estimates

We now turn to establish an almost sure bound for the discrepancy. We will use the mean-square bound of Theorem 4.1 and the Borel–Cantelli lemma, and the main difficulty here will be to establish an estimate which holds for all $h$ on a fixed set of full measure. For this argument, we need the following elementary lemma.

Proof. The proof proceeds by induction on $n$. We consider the function

\[ f(x):=x^\theta+y_2^\theta + \cdots + y_n^\theta- (x+y_2+\cdots + y_n)^\theta. \]

By the inductive assumption, $f(0)\le 0$. Using that $\theta \ge 1$, one checks that $f'(x)\le 0$ for $x\ge 0$. This implies the claim.

Theorem 4.4 (Almost sure discrepancy bound)

With notation as in Theorem 4.1, for every $0< \mathfrak {k}< \mathfrak {k}_S({\sf G})$ for almost all $x\in G_\infty$, and for every $\eta > 0$

\[ \mathcal{D}(R_S(h), E(x)\times W)\ll_{S,E,W,x,\mathfrak{k}, \eta} (\log v_S(h))^{3/2+\eta} v(h)^{-\mathfrak{k}}. \]

Proof. For $a, b\in \mathbb {Z}_{\ge 0}$, we consider the intervals

\[ I_{a,b}:=\{h\ge 1:\, 2^b a < v_S(h)\le 2^b (a+1) \}. \]

Note that for fixed $b$, they define a partition of $[1,\infty )$.

For $s\in \mathbb {Z}_{\ge 0}$, we set

\[ \mathcal{M}_{s,b}:=\{I_{a,b}:\, v_S(I_{a,b})\subset (0,2^s] \}\quad\text{and}\quad \mathcal{M}_{s}:=\bigsqcup_{b\le s}\, \mathcal{M}_{s,b}. \]

We observe that

\[ \bigcup_{I\in \mathcal{M}_{s}} I= [1,h_s]\quad \text{with } 2^{s-1}< v_S(h_s)\le 2^s. \]

For $I\subset [1,\infty )$, we set

\[ R_S(I):=\{\gamma\in \Gamma_S:\, {{\rm H}_f}(\gamma)\in I \}\quad\text{and}\quad B_S(I):=\{g\in G_S:\, {{\rm H}_f}(g)\in I \}. \]

The argument of the proof of Theorem 4.1 gives the following mean bound: for bounded measurable subsets $Q\subset G_\infty$,

\[ \int_Q \bigl|\, |R_S(I)\cap (E(x)\times W)|-m_\infty(E)m^S(W)m_S(B_S(I)) \bigr |^2 \, dm_\infty(x) \ll m_S(B_S(I))^{2-2\mathfrak{k}}. \]

The implicit constant here and the following computations depends on $S,E,W,Q$ and the difference $\mathfrak {k}_S-\mathfrak {k}$. Using Lemma 4.3,

\[ \sum_{I\in\mathcal{M}_{s,b}} m_S(B_S(I))^{2-2\mathfrak{k}}\le \bigg(\sum_{I\in\mathcal{M}_{s,b}} m_S(B_S(I))\bigg)^{2-2\mathfrak{k}} \le m_S(B_S(h_s))^{2-2\mathfrak{k}} \le 2^{s(2-2\mathfrak{k})}. \]

Hence,

(4.8)\begin{equation} \sum_{I\in\mathcal{M}_{s}} \int_Q \bigl |\, |R_S(I)\cap (E(x)\times W)| -m_\infty(E)m^S(W)m_S(B_S(I)) \bigr |^2 \, dm_\infty(x) \ll s 2^{s(2-2\mathfrak{k})}.\end{equation}

Let $\eta >0$. We consider the sets $\Upsilon _s$ consisting of $x\in Q$ such that

\[ \sum_{I\in\mathcal{M}_{s}} \bigl |\, |R_S(I)\cap (E(x)\times W)|-m_\infty(E)m^S(W)m_S(B_S(I)) \bigr |^2 \ge s^{2+\eta} 2^{s(2-2\mathfrak{k})}. \]

We deduce from (4.8) that

\[ m_\infty(\Upsilon_s)\ll s^{-1-\eta}. \]

It follows from the Borel–Cantelli lemma that the $\limsup$ of the sets $\Upsilon _s$ has measure zero. Hence, for almost all $x\in Q$ and all $s\ge s_0(x)$, we have the bound

(4.9)\begin{equation} \sum_{I\in\mathcal{M}_{s}} \bigl |\,|R_S(I)\cap (E(x)\times W)|-m_\infty(E)m^S(W)m_S(B_S(I)) \bigr |^2 \ll s^{2+\eta} 2^{s(2-2\mathfrak{k})}.\end{equation}

We shall use this bound to prove the theorem.

We first consider the case when $h$ is an end point of one of the intervals $I_{a,b}$. We choose the parameter $s$ such that $2^{s-1}< v_S(h)\le 2^s$. Using the binary representation, the interval $[1,h]$ can be written as a disjoint interval of at most $s$ intervals $I_i$ from $\mathcal {M}_s$. Then using (4.9) and the Cauchy–Schwartz inequality, we deduce that for almost all $x\in Q$ and $h\ge h_0(x)$, and for every $\eta > 0$

\begin{align*} &|\,|R_S(h)\cap (E(x)\times W)|-m_\infty(E)m^S(W)m_S(B_S(h))|\\ &\quad \le \sum_i |\,|R_S(I_i)\cap (E(x)\times W)|-m_\infty(E)m^S(W)m_S(B_S(I_i))|\\ &\quad \ll_\eta s^{3/2+\eta/2} 2^{s(1-\mathfrak{k})}\ll_\eta (\log v_S(h))^{3/2+\eta/2} v_S(h)^{1-\mathfrak{k}}. \end{align*}

For a general $h$, we observe that there exist $h_1$ and $h_2$ as above such that

\[ h_1\le h\le h_2\quad\text{and}\quad v_S(h_2)-v_S(h_1)\le 1. \]

Then

\[ |R_S(h_1)\cap (E(x)\times W)|\le |R_S(h)\cap (E(x)\times W)|\le |R_S(h_2)\cap (E(x)\times W)|, \]

and

\[ v_S(h_2)\le v_S(h)+1\quad\text{and}\quad v_S(h_1)\ge v_S(h)-1. \]

This provides upper and lower bounds on $|R_S(h)\cap (E(x)\times W)|$ that imply that for almost all $x\in Q$ and all sufficiently large $h$,

\[ |R_S(h)\cap (E(x)\times W)|=m_\infty(E)m^S(W)v_S(h) + O( (\log v_S(h))^{3/2+\eta/2} v_S(h)^{1-\mathfrak{k}}). \]

Clearly, this estimate also holds for all $h$ with an implicit constant depending on $x$ and $\eta$. Finally, exhausting $G_\infty$ by a countable union of bounded measurable sets, we deduce that this estimate holds for almost all $x\in G_\infty$, which implies Theorem 4.4.

4.4 Uniform discrepancy estimates

We now turn to establish a uniform pointwise bound on the discrepancy of rational points in the collection of sets $E(x)\times W$ with $E(x):=Ex\subset G_\infty$ and $W\subset \mathcal {I}^S$. In our discussion the set $E$ is fixed, the congruence condition $W$ is arbitrary and we consider all points $x\in G_\infty$. This requires imposing a regularity condition on the set $E$, namely the right-stability property (RS). The bound will now involve the dimension $\mathfrak {d}:=\dim _\mathbb {R} (G_\infty )$, whereas the almost sure pointwise bound in Theorem 4.4 did not.

We recall that by Lemma 2.8(a),

(V)\begin{equation} \epsilon^{\mathfrak{d}}\ll m_\infty(B(e,\epsilon)) \ll \epsilon^{\mathfrak{d}}\quad\text{for all } \epsilon\in (0,\epsilon_0). \end{equation}

Theorem 4.5 (Uniform discrepancy bound)

With notation as in Theorem 4.1, let $E$ be a right-stable finite-measure subset of $G_\infty$ satisfying $\mathcal {N}(E_{\epsilon _0}^+)<\infty$ and $W$ compact open subset of $\mathcal {I}^S$. For every $0 < \mathfrak {k} < \mathfrak {k}_S({\sf G})$ and for $x\in G_\infty$ the following pointwise bound for the discrepancy holds:

\[ \mathcal{D}(R_S(h), E(x)\times W) \ll_{S,E,x,\mathfrak{k}} m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}\, v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}, \]

provided that $m^S(W)\gg _{S,\mathfrak {k}} \, v_S(h)^{-2\mathfrak {k}}$.

When $m_\infty (E)>0$, this condition is also equivalent to the condition that the error term in the asymptotic expansion of the solution counting function:

\begin{align*} |R_S(h)\cap (E(x)\times W)|&=m_\infty(E)m^S(W)v_S(h)\\ &\quad + O_{S,E,x,\mathfrak{k}}( m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}\, v_S(h)^{1-2\mathfrak{k}/(d+2)}), \end{align*}

is bounded by the main term.

Explicitly, if the volume growth satisfies $v_S(h)\gg _{S,\mathfrak {a}} h^{\mathfrak {a}}$, with some $\mathfrak {a}>0$, then the estimate holds provided the height $h$ satisfies $h\gg _{S,\mathfrak {a},\mathfrak {k}} m^S(W)^{-1/2\mathfrak {a}\mathfrak {k}}$. Moreover, the above estimate is uniform for $x$ ranging in compact subsets of $G_\infty$.

In the proof, we use the estimates (SP), (RS) and (V).

Proof of Theorem 4.5 The starting point of our argument is the $L^2$-bound established in Theorem 4.1. Let $Q$ be a bounded measurable subset of $G_\infty$, and define the open set $Q':=B(e,\epsilon _0)Q$. According to Theorem 4.1, the following bound holds: for any measurable subset $F$ of $G_\infty$ with finite measure, and any given $0 < \mathfrak {k} < \mathfrak {k}_S({\sf G})$,

\[ \int_{Q'}\mathcal{D}(R_S(h), F(y)\times W)^2\, dm_\infty(y)\le C\, \mathcal{N}(F) m_\infty(F) m^S(W) v_S(h)^{-2\mathfrak{k}} \]

for some $C=C_{Q, S, \epsilon _0, \mathfrak {k}}>0$. This implies that for every $\delta >0$,

(4.10)\begin{align} & m_\infty(\{y\in Q':\, \mathcal{D}(R_S(h), F(y)\times W) >\delta \}) \nonumber\\ &\quad \le C \delta^{-2}\mathcal{N}(F)m_\infty(F) m^S(W) v_S(h)^{-2\mathfrak{k}}. \end{align}

We introduce a parameter $\epsilon \in (0,\epsilon _0)$ which we assume to satisfy

(4.11)\begin{equation} m_\infty(B(e,\epsilon))>C\, \delta^{-2}\mathcal{N}(E^+_\epsilon)m_\infty(E_\epsilon^+) m^S(W) v_S(h)^{-2\mathfrak{k}}. \end{equation}

Here $\epsilon _0$ is determined by conditions (RS) and (V). We note that for $x\in Q$, we have $B(e,\epsilon )x\subset Q'$. Hence, it follows from (4.10) applied to $F=E_\epsilon ^+$ that for every $x\in Q$, there exists $g\in B(e,\epsilon )$ such that

(4.12)\begin{equation} \mathcal{D}(R_S(h), E_\epsilon^+(gx)\times W)\le\delta. \end{equation}

We observe when $g\in B(e,\epsilon )=B(e,\epsilon )^{-1}$ that it follows from the definition of $E_\epsilon ^-$ and $E_\epsilon ^+$ (cf. (RS)) that

\[ E_\epsilon^-\cdot g\subset E \subset E_\epsilon^+ \cdot g, \ \text{since}\ E_\epsilon^-\cdot B(e,\epsilon)\subset E\subset E\cdot B(e,\epsilon)^{-1} =E_\epsilon^+ \]

and upon right-multiplication by $x$:

(4.13)\begin{equation} E_\epsilon^-\cdot gx=E_\epsilon^-(gx)\subset E\cdot x=E(x)\subset E^+_\epsilon\cdot gx= E^+_\epsilon(gx). \end{equation}

Hence, taking $g\in B(e,\epsilon )$ as in (4.12), we obtain that

\begin{align*} |R_S(h)\cap ( E(x)\times W) |&\le |R_S(h)\cap (E_\epsilon^+(gx)\times W)|\\ &\le ( m_\infty(E_\epsilon^+(gx)) m^S(W)+\mathcal{D}(R_S(h), E_\epsilon^+(gx)\times W))v_S(h)\\ &\le (m_\infty(E_\epsilon^+) m^S(W)+\delta)v_S(h). \end{align*}

Since the set $E$ is assumed to be right stable, we deduce that

\[ |R_S(h)\cap ( E(x)\times W) |\le ( m_\infty(E) m^S(W)+O(\epsilon\, m^S(W))+\delta)v_S(h) \]

for all $\epsilon \in (0,\epsilon _0)$ satisfying (4.11).

We also apply a similar argument to deduce a lower bound. Since it follows from (4.11) that also

\[ m_\infty(B(e,\epsilon))>C\, \delta^{-2}\mathcal{N}(E^-_\epsilon)m_\infty(E_\epsilon^-) m^S(W) v(h)^{-2\mathfrak{k}}, \]

we deduce from (4.10) with $F=E_\epsilon ^-$ that for every $x\in Q$, there exists $g\in B(e,\epsilon )$ such that

\[ \mathcal{D}(R_S(h), E_\epsilon^-(gx)\times W)\le\delta. \]

Hence, we deduce as above that

\begin{align*} |R_S(h)\cap (E(x)\times W)| &\ge |R_S(h)\cap (E_\epsilon^-(gx)\times W)|\\ &\ge (m_\infty(E_\epsilon^-(gx)) m^S(W)-\mathcal{D}(R_S(h), E_\epsilon^-(gx)\times W) )v_S(h)\\ &\ge (m_\infty(E) m^S(W) -O(\epsilon\,m^S(W))+\delta)v_S(h). \end{align*}

Combining the above estimates on $|R_S(h)\cap (E(x)\times W)|$, we deduce that for every $x\in Q$,

\[ \mathcal{D}(R_S(h), E(x)\times W)\le \delta+O(\epsilon\, m^S(W)) \]

provided that (4.11) is satisfied. To arrange (4.11), it is sufficient to pick $\delta$ of the form

\[ \delta=c\,m_\infty(B(e,\epsilon))^{-1/2}\mathcal{N}(E_\epsilon^+)^{1/2}m_\infty(E_\epsilon^+)^{1/2} m^S(W)^{1/2} v_S(h)^{-\mathfrak{k}} \]

with sufficiently large $c=c_{Q,S, \epsilon _0, \mathfrak {k}}>0$. Then we deduce using (V) that for all $x\in Q$,

(4.14)\begin{equation} \mathcal{D}(R_S(h), E(x)\times W)\ll_{E,Q} \epsilon^{-\mathfrak{d}/2} m^S(W)^{1/2} v_S(h)^{-\mathfrak{k}}+ \epsilon\,m^S(W). \end{equation}

We balance the two summands in the estimate and choose the parameter $\epsilon$ as

\[ \epsilon=( m^S(W)^{-1/2} v_S(h)^{-\mathfrak{k}})^{1/(\mathfrak{d}/2+1)}. \]

Therefore, when $h$ is sufficiently large, we have $\epsilon \in (0,\epsilon _0)$ provided that

\[ v_S(h)^{2\mathfrak{k}}m^S(W) > \epsilon_0^{-(\mathfrak{d}+2)}, \quad \text{namely } v_S(h)^{2\mathfrak{k}}\gg m^S(W)^{-1}, \]

the latter condition is equivalent to

\[ m^S(W)\gg m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}\, v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}, \]

and substituting this expression in the bound just established for the discrepancy we deduce that for every $x\in Q$,

\[ \mathcal{D}(R_S(h), E(x)\times W) \ll_{S,E,Q,\mathfrak{k}} m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}\, v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}. \]

This completes the proof of Theorem 4.5.

4.5 Uniform discrepancy at arbitrarily small scales

We now turn to establish a bound on the discrepancy which is uniform over a family of balls $B(x, \ell )$ in $G_\infty$ of arbitrarily small radius, for every $x\in G_\infty$. Let $\ell >0$, and $W$ be a compact open subset of $\mathcal {I}^S$. We will consider the family of subsets $\Omega$ given by $B(x,\ell )\times W\subset G_\infty \times \mathcal {I}^S,$ and our goal is to prove an explicit pointwise bound for the number of $\Gamma _S$-points contained in them.

Theorem 4.6 (Pointwise discrepancy bound at small scales)

With notation as in Theorem 4.1, let $W$ be a compact open subset of $\mathcal {I}^S$. Fix $0 < \mathfrak {k} < \mathfrak {k}_S({\sf G})$, let $x\in G_\infty$ and $\ell \in (0,\ell _0)$ (for suitable $\ell _0 >0$ independent of $x$) and set

\[ \mathcal{E}_{\ell,W}(h):= m_\infty(B(e,\ell))^{\mathfrak{d}/(\mathfrak{d}+2)}m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}v_S(h)^{1-2\mathfrak{k}/(\mathfrak{d}+2)}. \]

Then

(4.15)\begin{equation} |R_S(h)\cap B(x,\ell)\times W|= m_\infty(B(e,\ell))m^S(W)v_S(h)+O_{S,\mathfrak{k},x}(\mathcal{E}_{\ell,W}(h)) \end{equation}

provided that

(4.16)\begin{equation} m_\infty(B(e,\ell))^2 m^S(W)\ge v_S(h)^{-2\mathfrak{k}}. \end{equation}

This condition is equivalent to the condition that the error term in the foregoing asymptotic formula is bounded by the main term.

Explicitly, if $v_S(h) \gg _{S,\mathfrak {a}} h^{\mathfrak {a}}$, with some $\mathfrak {a}>0$, then the result holds at any scale $0< \ell < \ell _0$, provided $h$ satisfies

\[ h\gg_{S,\mathfrak{a},\mathfrak{k}}\ell^{-\mathfrak{d}/\mathfrak{k}\mathfrak{a}} m^S(W)^{-(\mathfrak{d}+2)/2\mathfrak{k}\mathfrak{a}}. \]

Moreover, this estimate is uniform for $x$ ranging in compact subsets of $G_\infty$.

Proof. We adopt the argument from the proof of Theorem 4.5 with some modifications. We note that there exists $r_0 > 0$ such that when $r\in (0,r_0)$,

\[ B(e,r)\cap \gamma B(e,r)=\emptyset\quad \text{for } \gamma\in {\sf G}(O)\backslash \{e\}, \]

so that $\mathcal {N}(B(e,r))=1$.

We fix a bounded measurable subset $Q$ of $G_\infty$ and define the open set $Q':=B(e,r_0)Q$. Applying Theorem 4.1 to the balls $B(y,\ell )=B(e,\ell )y$, we deduce that for every $r\in (0,r_0)$ and $\delta >0$, and some $C=C_{S,Q, r_0}$

(4.17) \begin{align} & m_\infty(\{y\in Q':\, \mathcal{D}(R_S(h), B(x,\ell)\times W)>\delta \}) \nonumber\\ &\quad \le C\, \delta^{-2}m_\infty(B(e,r)) m^S(W) v_S(h)^{-2\mathfrak{k}}. \end{align}

We introduce a positive parameter $\epsilon$ (to be chosen later) satisfying

(4.18)\begin{equation} \epsilon<\ell, \end{equation}

as well as

(4.19)\begin{equation} m_\infty(B(e,\epsilon))>C\, \delta^{-2}m_\infty(B(e,\ell+\epsilon)) m^S(W) v_S(h)^{-2\mathfrak{k}}. \end{equation}

We choose $\ell _0$ so that $0 < 2\ell _0< r_0$ and also so that Lemma 2.8 is applicable for $r < r_0$. Then it follows from (4.17) that for every $x\in Q$, there exists $g\in B(e,\epsilon )$ such that

\[ \mathcal{D}(R_S(h), B(gx,\ell+\epsilon)\times W)\le\delta. \]

It follows from the triangle inequality that

\[ B(gx,\ell-\epsilon)\times W\subset B(x,\ell)\times W\subset B(gx,\ell+\epsilon)\times W, \]

and so,

\begin{align*} |R_S(h)\cap (B(x,\ell)\times W)|&\le | R_S(h)\cap (B(gx,\ell+\epsilon)\times W)|\\ &\le ( m_\infty(B(e,\ell+\epsilon)) m^S(W)+\mathcal{D}(R_S(h), B(gx,\ell+\epsilon)\times W))v_S(h)\\ &\le ( m_\infty(B(e,\ell+\epsilon)) m^S(W)+\delta)v_S(h). \end{align*}

Since it follows from Lemma 2.8(b) that

(4.20)\begin{equation} m_\infty(B(e,\ell+\epsilon))\le \bigg(1+c\,\frac{\epsilon}{\ell}\bigg)m_\infty(B(e,\ell)), \end{equation}

we deduce that

\[ |R_S(h)\cap (B(x,\ell)\times W)|\le \bigg( m_\infty(B(e,\ell)) m^S(W)+\delta+c\,\frac{\epsilon}{\ell} m_\infty(B(e,\ell))m^S(W)\bigg)v_S(h). \]

A similar argument also gives the lower bound

\[ |R_S(h)\cap (B(x,\ell)\times W)|\ge \bigg( m_\infty(B(e,\ell)) m^S(W)-\delta-c\,\frac{\epsilon}{\ell} m_\infty(B(e,\ell))m^S(W)\bigg)v_S(h). \]

It follows that the discrepancy

\[ \mathcal{D}(R_S(h), B(x,\ell)\times W)=\bigg| \frac{|R_S(h)\cap (B(x,\ell)\times W)|}{v_S(h)} - m_\infty(B(e,\ell)) m^S(W) \bigg| \]

satisfies

\[ \mathcal{D}(R_S(h), B(x,\ell)\times W)\le \delta+ c\,\frac{\epsilon}{\ell} m_\infty(B(e,\ell))m^S(W). \]

The parameter $\delta$ have to satisfy (4.19), and since $m_\infty (B(e,\ell +\epsilon ))\le (1+c)m_\infty (B(e,\ell ))$ (using (4.20) and $\epsilon < \ell$) it is sufficient to pick $\delta$ of the form

\[ \delta=b_0\,m_\infty(B(e,\epsilon))^{-1/2} m_\infty(B(e,\ell))^{1/2} m^S(W)^{1/2} v_S(h)^{-\mathfrak{k}} \]

with sufficiently large $b_0>0$. Then using Lemma 2.8(a), we deduce the bound

(4.21)\begin{equation} \mathcal{D}(R_S(h), B(x,\ell)\times W)\ll \epsilon^{-\mathfrak{d}/2} m_\infty(B(e,\ell))^{1/2} m^S(W)^{1/2} v_S(h)^{-\mathfrak{k}}+ \frac{\epsilon}{\ell} m_\infty(B(e,\ell))m^S(W). \end{equation}

The two summands in the foregoing bound are balanced precisely when $\epsilon$ is a constant multiple of

(4.22)\begin{equation} \ell^{2/(\mathfrak{d}+2)}m_\infty(B(e,l))^{-1/(\mathfrak{d}+2)}m^S(W)^{-1/(\mathfrak{d}+2)}v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}, \end{equation}

and then the estimate that arises in (4.21) is given by a constant multiple of

\[ \ell^{-\mathfrak{d}/(\mathfrak{d}+2)}m_\infty(B(e,l)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2) }v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}. \]

By Lemma 2.8(a) $\ell \ll m_\infty (B(e,l)^{1/\mathfrak {d}}\ll \ell$, and so let us define the error term in (4.15) by

(4.23)\begin{equation} \mathcal{E}_{\ell,W}(h):= m_\infty(B(e,\ell))^{\mathfrak{d}/(\mathfrak{d}+2)}m^S(W)^{(\mathfrak{d}+1)/(\mathfrak{d}+2)}v_S(h)^{1-2\mathfrak{k}/(\mathfrak{d}+2)}. \end{equation}

We now claim that under the condition (4.16) stated in Theorem 4.6 namely $m_\infty (B(e,\ell ))^2 m^S(W)\ge v_S(h)^{-2\mathfrak {k}}$, the conclusion of Theorem 4.6 holds:

(4.24)\begin{equation} |R_S(h)\cap (B(e,\ell)\times W)|= m_\infty(B(e,\ell))m^S(W)v_S(h)+O_Q (\mathcal{E}_{\ell,W}(h)). \end{equation}

Indeed, (4.16) is equivalent to the condition that the error term $\mathcal {E}_{\ell,W}(h)$ defined in (4.23) is bounded by the main term in (4.24), namely,

(4.25)\begin{equation} m_\infty(B(e,\ell))m^S(W)v_S(h)\ge \mathcal{E}_{\ell,W}(h). \end{equation}

Furthermore, these two conditions are equivalent to the condition

(4.26)\begin{equation} m_\infty(B(e,\ell))^{-2/(\mathfrak{d}+2)} m^S(W)^{-1/(\mathfrak{d}+2)}v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}\le 1. \end{equation}

To establish (4.24) we apply (4.21) with the choice of the parameter $\epsilon$ being given by (4.22):

\[ \epsilon=c_0'\, \ell^{2/(\mathfrak{d}+2)} m_\infty(B(e,\ell))^{-1/(\mathfrak{d}+2)} m^S(W)^{-1/)(\mathfrak{d}+2)} v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}, \]

where $c_0'>0$ is a sufficiently small positive constant so that Lemma 2.8(b) holds. It follows from Lemma 2.8(a) and (4.26) that

\[ \epsilon \ll c_0' \,\ell^{-(\mathfrak{d}-2)/(\mathfrak{d}+2)} m^S(W)^{-1/(\mathfrak{d}+2)} v_S(h)^{-2\mathfrak{k}/(\mathfrak{d}+2)}\ll c_0'\,\ell. \]

Hence, with sufficiently small $c_0'$, condition (4.18) holds, and so the above estimates apply. Now the bound (4.24) follows from (4.21) and this completes the proof of Theorem 4.6.

5.1 Finiteness of the integrability exponent

Our first task is to show that the integrability exponent defined in (2.3) is finite.

Proof. When ${\sf G}$ is simply connected, this was already established in [Reference Gorodnik, Maucourant and OhGMO08], and we will reduce the proof to this case. We first consider the case when $S$ is finite, and then deal with the general case.

We recall that when ${\sf G}$ is simply connected, there is no non-trivial automorphic characters (see, for instance, [Reference Ghosh, Gorodnik and NevoGGN13, Lemma 4.1]), and it follows from [Reference Gorodnik, Maucourant and OhGMO08, Theorem 3.20 and Theorem 3.7] that for all functions $\phi,\psi \in L_0^2({\sf G}(\mathbb {A}_K\!)/{\sf G}(K))$ that are $U_\infty$-finite for a maximal compact subgroup $U_\infty$ of $G_\infty$ and $\mathcal {W}$-invariant for a compact open subgroup $\mathcal {W}$ of $G_f$, the functions

(5.1)\begin{equation} g\mapsto \langle \pi^{\rm aut}_S(g)\phi,\psi\rangle =\int_{{\sf G}(\mathbb{A}_K\!)/{\sf G}(K)} \phi (g^{-1}x)\overline{\psi(x)}\, d\mu(x) \end{equation}

are in $L^p(G_S)$ for a uniform $p>1$.

Now suppose that ${\sf G}$ is not necessarily simply connected, and that $S$ is finite. We fix a maximal compact subgroup $U_\infty$ of $G_\infty$ and a compact open subgroup $\mathcal {W}$ of $G_f$. We shall show that for any compactly supported $\phi,\psi \in \mathcal {H}^{00}_{\sf G}$ that are $U_\infty$-finite and $\mathcal {W}$-invariant, the matrix coefficients (5.1) are in $L^p(G_S)$. Since the span of such functions is dense in $\mathcal {H}^{00}_{\sf G}$ as we vary $\mathcal {W}$, this will imply that the integrability exponent is finite.

We consider the simply connected cover ${\sf p}:\tilde {\sf G}\to {\sf G}$ that induces the map

\[ {\sf p}: \tilde Y:= \tilde {\sf G}(\mathbb{A}_K\!)/\tilde {\sf G}(K) \longrightarrow Y:= {\sf G}(\mathbb{A}_K\!)/{\sf G}(K). \]

We denote by $\tilde \mu$ and $\mu$ the invariant probability measures on the spaces $\tilde Y$ and $Y$, respectively. According to [Reference Platonov and RapinchukPR94, Ch. 8, Proposition 8.8], there is an exact sequence

\[ \tilde {\sf G}(\mathbb{A}_K\!) \stackrel{\sf p}{\longrightarrow} {\sf G}(\mathbb{A}_K\!)\longrightarrow \prod_{v\in V_K} H^1(K_v, Z({\sf G})), \]

so that ${\sf p}(\tilde {\sf G}(\mathbb {A}_K\!))$ is a normal co-Abelian subgroup of ${\sf G}(\mathbb {A}_K\!)$. Let us consider the group $L:=\mathcal {W}{\sf p}(\tilde {\sf G}(\mathbb {A}_K\!)){\sf G}(K)$. Clearly, it is also a normal co-Abelian subgroup of ${\sf G}(\mathbb {A}_K\!)$. Furthermore, $L$ is open in ${\sf G}(\mathbb {A}_K\!)$. We also consider a subset $Y_0:=L/{\sf G}(K)$ of $Y$. Then

\[ Y=\bigsqcup_{\gamma\in \Delta} \gamma Y_0, \]

where $\Delta$ is a set of coset representatives for $L$ in ${\sf G}(\mathbb {A}_K\!)$. Since $Y_0$ is open, $\mu (Y_0)>0$. Thus, using that $\mu (Y)<\infty$, we conclude that $\Delta$ is finite. (In fact, this can be also deduced from the finiteness of class number of $\sf G$.)

Let $m_{\mathcal {W}}$ be the Haar probability measure on $\mathcal {W}$. The unique $L$-invariant probability measure $\mu _0$ on $Y_0$ can be given as

(5.2)\begin{equation} \int_{Y_0} f\, d\mu_0=\int_{w\in \mathcal{W}}\int_{\tilde{y}\in \tilde Y} f(w{\sf p}(\tilde{y}))\, d\tilde \mu(\tilde{y})\,dm_{\mathcal{W}}(w) \quad \text{for } f\in C_c(Y_0). \end{equation}

Indeed, the invariance of this measure is easy to check using that ${\sf p}(\tilde {\sf G}(\mathbb {A}_K\!))$ is co-Abelian, ${\sf p}$ is equivariant, and $\tilde {\mu }$ is invariant. On the other hand, it also follows from invariance that

(5.3)\begin{equation} \mu_0=|{\sf G}(\mathbb{A}_K\!):L|\cdot \mu|_{Y_0}. \end{equation}

Similarly, considering the exact sequence

\[ \tilde {G}_S \stackrel{\sf p}{\longrightarrow} {G}_S\longrightarrow \prod_{v\in S} H^1(K_v, Z({\sf G})), \]

we deduce that $H:={\sf p}(\tilde G_S)$ is an open, normal, co-Abelian subgroup of $G_S$. Moreover, since $S$ is finite, $H$ has finite index in $G_S$. We have the decomposition

\[ G_S =\bigsqcup_{\delta\in \Omega} \delta H, \]

where $\Omega$ is a finite subset of coset representatives for $H$ in $G_S$. Let $\tilde m_S$ be a Haar measure on $\tilde G_S$. Since $H$ is open in $G_S$, a Haar measure on $H$ is given by the restriction of the Haar measure $m_S$ on $G_S$. It follows from the uniqueness of Haar measure that for some $c>0$,

(5.4)\begin{equation} \int_H f\, dm_S=c\int_{\tilde{g}\in \tilde G_S} f({\sf p}(\tilde{g}))\, d\tilde m_S(\tilde{g}) \quad\text{for } f\in C_c(H). \end{equation}

Indeed, since $S$ is finite, the kernel of the map ${\sf p}:\tilde G_S\to G_S$ is finite, so that this map is proper. Thus, the right-hand side of (5.4) defines a Haar measure on $H$.

For $\phi,\psi \in \mathcal {H}_{\sf G}^{00}$ satisfying the invariance properties prescribed above, and $g\in G_S$,

\[ \langle \pi^{\rm aut}_S(g)\phi,\psi\rangle =\int_{Y} \phi (g^{-1}y)\overline{\psi(y)}\, d\mu(y)=\sum_{\gamma\in\Delta} \int_{Y_0} \phi (g^{-1}\gamma y)\overline{\psi(\gamma y)}\, d\mu(y), \]

and by the triangle inequality for $L^p$-norms,

(5.5)\begin{align} &\bigg(\int_{G_S} |\langle \pi^{\rm aut}_S(g)\phi,\psi\rangle |^p\, dm_S(g) \bigg)^{1/p} \nonumber\\ &\quad \le \sum_{\gamma\in\Delta} \bigg(\int_{G_S}\bigg| \int_{Y_0} \phi (g^{-1} \gamma y)\overline{\psi(\gamma y)}\, d\mu(y)\bigg|^p\,dm_S(g)\bigg)^{1/p} \nonumber\\ &\quad \le \sum_{\gamma\in\Delta} \bigg(\sum_{\delta\in\Omega} \int_H\bigg| \int_{Y_0} \phi ( h^{-1} \delta^{-1} \gamma y)\overline{\psi(\gamma y)}\, d\mu(y)\bigg|^p \,dm_S(h)\bigg)^{1/p} \nonumber\\ &\quad = \sum_{\gamma\in\Delta} \bigg(\sum_{\delta\in\Omega} \int_H\bigg| \int_{Y_0} \phi (\delta^{-1} \gamma h^{-1} y)\overline{\psi(\gamma y)}\, d\mu(y)\bigg|^p \,dm_S(h)\big)^{1/p}. \end{align}

Here in the last step we have used the fact that $L$ is co-Abelian. Indeed, denoting $\delta ^{-1}\gamma =z$, this property implies $h^{-1}z=zh^{-1} l'$ with $l'\in L$. Now $Y_0=L/{\sf G}(K)$ and writing a coset $y\in Y_0$ as $y=l {\sf G}(K)$ with $l\in L$, we have $h^{-1}zl {\sf G}(K)=zh^{-1}l'l{\sf G}(K)$. Therefore, the integrals over $Y_0=L/{\sf G}(K)$ in the last step above gives the same function of $h\in H$ in both cases.

We claim that for every $r_1,r_2\in {\sf G}(\mathbb {A}_K\!)$, the functions

\[ c_{r_1,r_2}(h):= \int_{Y_0} \phi (r_1 h y)\overline{\psi(r_2 y)}\, d\mu(y) \]

are in $L^p(H)$. In view of (5.2) and (5.3),

\[ c_{r_1,r_2}(h)=|{\sf G}(\mathbb{A}_K\!):L|^{-1} \int_{\tilde{y}\in \tilde Y} \phi (r_1 h {\sf p}(\tilde{y})) \overline{\psi(r_2 {\sf p}(\tilde{y}))}\, d\tilde\mu(\tilde{y}), \]

For a function $f$ on $Y$ and $g\in {\sf G}(\mathbb {A}_K\!)$, we define a function on $\tilde Y$ by

\[ \tilde f_g(\tilde{y}):=f(g {\sf p}(\tilde{y}))\quad \text{for } \tilde{y}\in \tilde Y. \]

Then when $f$ is $\mathcal {W}$-invariant, using (5.2), (5.3) and that $L$ is co-Abelian, we deduce that

\begin{align*} \int_{\tilde Y} \tilde f_g\, d\tilde \mu&=\int_{\mathcal{W}}\int_{\tilde Y} f(g w{\sf p}(\tilde{y}))\, d\tilde \mu(\tilde{y})\,dm_{\mathcal{W}}(w) =\int_{\mathcal{W}}\int_{\tilde Y} f(wg {\sf p}(\tilde{y}))\, d\tilde \mu(\tilde{y})\,dm_{\mathcal{W}}(w)\\ &=\int_{Y_0} f(gz)\, d\mu_0(z) =|{\sf G}(\mathbb{A}_K\!):L|\int_{g Y_0} f\, d \mu. \end{align*}

Let $\mathcal {X}({\sf G},L)$ denote the set of continuous unitary characters $\chi$ of ${\sf G}(\mathbb {A}_K\!)$ such that $\chi (L)=1$. It follows from the properties of characters of finite Abelian groups that

\[ \chi_{Y_0} =|{\sf G}(\mathbb{A}_K\!):L|^{-1} \sum_{\chi\in\mathcal{X}({\sf G},L)}\chi, \]

so that if $f\in \mathcal {H}_{\sf G}^{00}$, using the previous two identities we deduce

\begin{align*} \int_{\tilde Y} \tilde f_g\, d\tilde \mu&= \int_Y (f\circ g)\bigg(\sum_{\chi\in\mathcal{X}({\sf G},L)}\chi \bigg)\, d \mu= \int_{Y} f\bigg(\sum_{\chi\in\mathcal{X}({\sf G},L)}\chi\circ g^{-1} \bigg)\, d \mu \\ &=\sum_{\chi\in\mathcal{X}({\sf G},L)} \chi(g)^{-1}\int_{Y} f\chi\, d \mu=0. \end{align*}

This shows that if $f\in \mathcal {H}_{\sf G}^{00}$, then $\tilde f_g\in L^2_0(\tilde Y)$. In particular, $\tilde \phi _{r_1}, \tilde \psi _{r_2}\in L^2_0(\tilde Y)$. Using Hölder's inequality, the fact that $L$ is co-Abelian, and (5.4), we conclude that

\begin{align*} \int_H |c_{r_1,r_2}|^p \, dm_S&= \int_H \bigg|\int_{\mathcal{W}}\int_{\tilde Y} \phi (r_1 h w{\sf p}(\tilde{y}))\overline{\psi(r_2 w{\sf p}(\tilde{y}))}\, d\tilde\mu(\tilde{y})\,dm_{\mathcal{W}}(w)\bigg|^p \,dm_S(h)\\ &\le \int_H \int_{\mathcal{W}} \bigg|\int_{\tilde Y} \phi (r_1 h w{\sf p}(\tilde{y}))\overline{\psi(r_2 w{\sf p}(\tilde{y}))}\, d\tilde\mu(\tilde{y})\bigg|^p \,dm_{\mathcal{W}}(w)\,dm_S(h)\\ &\ll \int_{\mathcal{W}}\int_H \bigg|\int_{\tilde Y} \phi (r_1 wh {\sf p}(\tilde{y}))\overline{\psi(r_2 w{\sf p}(\tilde{y}))}\, d\tilde\mu(\tilde{y})\bigg|^p \,dm_S(h)\,dm_{\mathcal{W}}(w)\\ &\ll \int_{\mathcal{W}}\int_{\tilde G_S} \bigg|\int_{\tilde Y} \phi (r_1 w{\sf p}(\tilde{g}) {\sf p}(\tilde{y}))\overline{\psi(r_2 w{\sf p}(\tilde{y}))}\, d\tilde\mu(\tilde{y})\bigg|^p \,d\tilde m_S(\tilde{g})\,dm_{\mathcal{W}}(w)\\ &= \int_{\mathcal{W}}\int_{\tilde G_S} \bigg|\langle \tilde{\pi}^{\rm aut}_S(\tilde{g}^{-1}) \tilde \phi_{r_1w}, \tilde\phi_{r_2w}\rangle \bigg|^p \,d\tilde m_S(\tilde{g})\,dm_{\mathcal{W}}(w). \end{align*}

It is easy to see that $\tilde \phi _{r_1w}$ and $\tilde \phi _{r_2w}$ are $\widetilde U_\infty$-finite for suitable maximal compact subgroup $\widetilde U_\infty$ of $\widetilde {G}_\infty$ and $\widetilde {\mathcal {W}}$-finite for suitable compact open subgroup $\widetilde {\mathcal {W}}$ of $\tilde G$. We observe that the integrand in the above formula is locally constant in $w\in \mathcal {W}$, and it follows from the simply connected case that the functions $g\mapsto \langle \tilde {\pi }^{\rm aut}_S(\tilde {g}^{-1}) \tilde \phi _{r_1w}, \tilde \phi _{r_2w}\rangle$ are in $L^p(\tilde G_S)$. Hence, we conclude that

\[ \int_{\mathcal{W}}\int_H |c_{r_1w,r_2w}|^p \, dm_S\,dm_{\mathcal{W}}(w)<\infty, \]

and by (5.5) also

\[ \int_{G_S} |\langle \pi^{\rm aut}_S(g)\phi,\psi\rangle |^p\, dm_S(g)<\infty. \]

This proves that the representation $\pi ^{\rm aut}_S|_{\mathcal {H}_{\sf G}^{00}}$ is $L^p$-integrable when $S$ is finite, and $p$ is uniform and independent of $S$.

To deal with the general case with $S$ arbitrary and possibly infinite, we use that the representation $\pi ^{\rm aut}_S$ has the direct integral decomposition

(5.6)\begin{equation} \pi^{\rm aut}_S|_{\mathcal{H}_{\sf G}^{00}} =\int_{\widehat G_S} \rho^{\oplus n(\rho)} \, d\Pi(\rho) \end{equation}

with respect to a measure $\Pi$ on the unitary dual $\widehat G_S$, with $n(\rho )\in \mathbb {N}\cup \{{\infty }\}$ denoting the multiplicity. The irreducible representations $\rho$ in this decomposition are restricted tensor products of the form $\rho =\otimes _{v\in S} \rho _v,$ where $\rho _v$ are irreducible representations of $G_v$, which are $U_v$-spherical for almost all $v\in S$ (see [Reference FlathFla79]). For each $v\in S$, we also have the decomposition

(5.7)\begin{equation} \pi^{\rm aut}_v|_{\mathcal{H}_{\sf G}^{00}} =\int_{\widehat G_v} \rho_v^{\oplus n_v(\rho_v)} \, d\Pi_v(\rho_v). \end{equation}

We can conclude from the discussion in the previous paragraph that the representation $\pi ^{\rm aut}_v|_{\mathcal {H}_{\sf G}^{00}}$ is $L^p$-integrable. This condition implies that $\Pi _v$-almost all $\rho _v$ is $L^{p'}$-integrable for some finite $p' \ge p$. Indeed, $L^p$-integrability implies that a suitable tensor power $\big (\pi ^{\rm aut}_v|_{\mathcal {H}_{\sf G}^{00}}\big )^{\otimes N}$ is weakly contained in the regular representation of $G_v$. Since $\rho _v$ is weakly contained in $\pi ^{\rm aut}_v|_{\mathcal {H}_{\sf G}^{00}}$, it follows that for $\Pi _v$-almost all $\rho _v$, the tensor power $\rho _v^{\otimes N}$ is weakly contained in the regular representation of $G_v$. It follows that the $N$th power of the matrix coefficients of $U_v$-finite vectors $\rho _v$ satisfy the pointwise bound given by [Reference Cowling, Haagerup and HoweCHH88, Theorem 2]. Therefore, $\rho _v$ is $p^\prime$-integrable for $\Pi _v$-almost all $\rho _v$, and the matrix coefficients of $U_v$-finite vectors $\phi _v,\psi _v$ of $\rho _v$ satisfy, for suitable $k\in {\mathbb {N}}$,

(5.8)\begin{equation} |\langle \rho_v(g_v)\phi_v,\psi_v\rangle| \le d_v(\phi_v)^{1/2} d_v(\psi_v)^{1/2}\|\phi_v\| \|\psi_v\|\, \Xi_v(g_v)^{1/k}\quad\text{for } g_v\in G_v, \end{equation}

where $d_v(\phi _v):=\dim \langle \rho _v(U_v) \phi _v \rangle$, and $\Xi _v$ denotes the Harish-Chandra function on $G_v$. Since the measure $\Pi _v$ in (5.7) is the image of the measure $\Pi$ from (5.7) under the restriction map, $\Pi$-almost every representation $\rho$ appearing in the decomposition (5.6) is of the form $\rho =\otimes _{v\in S} \rho _v$ where the representations $\rho _v$ satisfy the bound (5.8). It follows from the description of the space of $U_S$-finite vectors for tensor products (see [Reference FlathFla79]), that for all $U_S$-finite vectors $\phi$ and $\psi$, there exists $c(\phi,\psi )>0$ such that

(5.9)\begin{equation} |\langle \rho(g)\phi,\psi\rangle| \le c(\phi,\psi)\,\Xi_S(g)^{1/k} \quad \text{for } g\in G_S, \end{equation}

where $\Xi _S(g):=\prod _{v\in S} \Xi _v(g_v)$ is the Harish-Chandra function on $G_S$. We recall that the Harish-Chandra function $\Xi _S$ is $L^{4+\eta }$-integrable for all $\eta >0$ (see [Reference Gorodnik and NevoGN12, Proposition 6.3]). We note that the argument in [Reference Gorodnik and NevoGN12] utilizes only the Cartan and Iwasawa decompositions of the group, and does not require that the group be simply connected. Hence, the estimate (5.9) implies that $\Pi$-almost every representation $\rho$ appearing in the decomposition (5.6) is $L^q$-integrable with a uniform $q$, and with the $L^q$-norm uniformly bounded. We conclude that the representation $\pi ^{\rm aut}_S|_{\mathcal {H}_{\sf G}^{00}}$ is $L^q$-integrable.