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Random walks, spectral gaps, and Khintchine’s theorem on fractals

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Abstract

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor’s middle 1/3 set. We obtain the first instances where a complete analogue of Khintchine’s Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of \({\mathbb {R}}^d\) (for any \(d\ge 1\)) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler’s problem is the Hausdorff measure on the “middle \(1/5\) Cantor set”; i.e. the set of numbers whose base \(5\) expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space \({\mathcal {L}}_{d+1}\) of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of S-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on \({\mathcal {L}}_{d+1}\).

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Notes

  1. Here and throughout, we refer to Theorem 6.1, which is the more precise form of Theorem B, as the equidistribution theorem.

  2. The extra factor of 2 in the bound on \(w_{d+1}\) ensures (9.4).

  3. Recall that \(W(\psi _0)\) was defined in (1.1) using the sup-norm on \(\mathbb {R}^d\).

  4. The sets \(I_n\) differ from \(A_n^*\) in removing the lower bound restriction on the denominators q.

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Acknowledgements

We would like to thank Jon Chaika and Samantha Fairchild for generously sharing their version of Proposition 11.1 and, in particular, for explaining how an estimate like (1.14) can be used for short range correlations. We further thank Manfred Einsiedler, Nimish Shah, Andreas Strömbergsson, and Barak Weiss for earlier discussions surrounding this project. Both authors thank the Hausdorff Research Institute for Mathematics at the Universität Bonn for its hospitality during the trimester program “Dynamics: Topology and Numbers”. M. L. thanks the Ohio State University for their hospitality during his visit where this project was started. M. L. acknowledges the financial support of the ISF through grant 1483/16. The authors would like to thank the referees for numerous corrections and suggestions that improved the exposition.

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Appendices

Appendix A. Spectral Gap for Missing Digit Cantor sets

The goal of this section is to prove Theorem C providing a stronger version of our Khintchine and equidistribution theorems in the special case of missing digit Cantor sets. This is done by weakening the hypothesis (6.5) in Theorem 6.1. A key input is a sharper estimate on the spectral gap of the operators \(\mathcal {P}_\lambda \), Proposition A.3. Additionally, we take advantage of the equal contraction ratios to show that, in fact, the Sobolev norm (not just the \(\textrm{L}^2\)-norm) of a suitable variant of the operators \(\mathcal {P}_\lambda ^n\) decays in n. Finally, we require a sharper form of Proposition 5.1 due to Strömbergsson as well as bounds towards Selberg’s eigenvalue conjecture by Kim–Sarnak.

First, we recall the definition of a missing digit Cantor set.

Definition A.1

A set \(\mathcal {K}\subset [0,1]\) is a missing digit Cantor set if there exists a prime number \(p\ge 3\) and \(\emptyset \ne \Lambda \subseteq \left\{ 0,\dots ,p-1\right\} \) such that \(\mathcal {K}\) consists of those \(x\in [0,1]\) whose digits in their base p expansion all belong to \(\Lambda \). A missing digit IFS (with attractor \(\mathcal {K}\)) is defined as follows:

$$\begin{aligned} \mathcal {F} = \left\{ f_i(x) = \frac{x+i}{p} : i\in \Lambda \right\} . \end{aligned}$$
(12.41)

Throughout the remainder of this section, we fix a missing digit Cantor set \(\mathcal {K}\) in base p and digit set \(\Lambda \) along with its associated missing digit IFS \(\mathcal {F}\).

In particular, in our notation, \(\rho =\rho _i = 1/p\) and \(b_i = i/p\). One checks that this IFS satisfies the open set condition. In particular, we have

$$\begin{aligned} s:=\dim _H(\mathcal {K}) = \log |\Lambda |/\log p. \end{aligned}$$

By [41], the s-dimensional Hausdorff measure of \(\mathcal {K}\) is positive and finite. We denote by \(\mu \) the restriction of this measure to \(\mathcal {K}\), normalized to be a probability measure. By [29], \(\mu \) is the self-similar measure associated to the probability vector \(\lambda _i=\rho ^s, i\in \Lambda \).

The following is the precise form of Theorem C.

Theorem A.2

The conclusions of Theorem A and Theorem 6.1 hold for \(\mu \) as above whenever

$$\begin{aligned} s > 0.839. \end{aligned}$$
(12.42)

Since we showed that Theorem 6.1 implies Theorem A, we only need to verify that in this special case the former holds under the condition (12.42).

A wasteful step in the proof of Theorem 6.1 is (6.29). To improve this estimate, we introduce slightly different operators than \(\alpha \cdot \mathcal {P}_\lambda \) which take advantage of the equal contraction ratios. For \(\omega \in \Lambda ^n\), we define

$$\begin{aligned} \tau _\omega = u({\textbf{0}},-b_\omega ) a(1,\rho _\omega ) = u({\textbf{0}},-b_\omega ) a(1,p^{-n}). \end{aligned}$$
(12.43)

Note that \(\tau _\omega \) has trivial Archimedean component. For \(\alpha \in \Lambda ^*\), let \(\alpha \cdot \mathcal {Q}_\lambda \) denote the averaging operator defined analogously to \(\alpha \cdot \mathcal {P}_\lambda \) in (4.7) with \(\tau _\omega \) in place of \(\gamma _\omega \). Note that \(a(\rho _\omega ,1)\tau _\omega =\gamma _\omega \); cf. (4.2). In particular, for any function \(\varphi \) on \(X_S\) and every \(n\in \mathbb {N}\), we have

$$\begin{aligned} (\alpha \cdot \mathcal {P}_\lambda )^n(\varphi )(x) = (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )\big (a(p^{-n},1)x\big ). \end{aligned}$$
(12.44)

1.1 A.1 Sharper version of Proposition 4.3

The following result provides a sharper rate of decay of the operator norm of \(\mathcal {P}_\lambda ^n\). It holds without restrictions on the dimension of the Cantor set.

Proposition A.3

Let \(\varepsilon >0\) and \(\delta _{\varepsilon }=\frac{25}{32}-2\varepsilon \). For all \(n\in \mathbb {N}\), \(\alpha \in \Lambda ^*\), and for every smooth \(K_{\textrm{f}}\)-invariant function \(\varphi \in \textrm{L}^2_{00}(X_S)\), we have

$$\begin{aligned} \big \Vert (\alpha \cdot \mathcal {P}_\lambda )^n (\varphi )\big \Vert ^2_{\textrm{L}^2} \ll _{\varepsilon ,p,s} \mathcal {S}_{2,1}(\varphi )^2 (p^{-(s-\varepsilon )n}+p^{-\delta _{\varepsilon } (n+|\alpha |)}). \end{aligned}$$

The same estimate holds for \(\mathcal {Q}_\lambda \) in place of \(\mathcal {P}_\lambda \).

Proof

Note that in view of bounds towards the Generalized Ramanujan Conjectures (GRC) for \(\textrm{SL}_2\) in [31, Proposition 2], for \(K_{\textrm{f}}\)-invariant functions, one can take the bound in Corollary 3.7 to be \(\xi _{\textbf{G}}^{25/32}\) instead of \(\xi _{\textbf{G}}^{1/2}\) (GRC predicts the exponent should be 1); cf. [56, Lem. 9.1]. In what follows, we let \(m:=|\alpha |\). Given \(\omega \in \Lambda ^{n}\), in analogy to the proof of Proposition 4.3, we denote \(\gamma _\omega ^\alpha =\gamma _\alpha \gamma _\omega \gamma _\alpha ^{-1}\) and similarly \(\tau _\omega ^\alpha =\gamma _\alpha \tau _\omega \gamma _\alpha ^{-1}\). Expanding \((\alpha \cdot \mathcal {P}_\lambda )^n\) according to (4.7), it follows from Corollary 3.7 and Proposition 4.5 that

$$\begin{aligned} \big \Vert (\alpha \cdot \mathcal {P}_\lambda )^n(\varphi )\big \Vert ^2&= \sum _{\eta ,\omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \langle \gamma _\eta ^\alpha \varphi ,\gamma _\omega ^\alpha \varphi \rangle \ll \mathcal {S}_{2,1}(\varphi )^2\\&\quad \times \sum _{\eta ,\omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \xi _\textbf{G}^{25/32}\left( \gamma _\omega ^\alpha (\gamma _\eta ^\alpha )^{-1}\right) \\&= \mathcal {S}_{2,1}(\varphi )^2 \sum _{\eta ,\omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \xi _\textbf{G}^{25/32}\big (u(p^{-m}(b_\eta -b_\omega ))\big )\\&\ll _\varepsilon \mathcal {S}_{2,1}(\varphi )^2 \sum _{\eta ,\omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \big \Vert u(p^{-m}(b_\eta -b_\omega ))\big \Vert _p^{-\frac{25}{64}+\varepsilon }, \end{aligned}$$

where \(\Vert {u(p^{-m}(b_\eta -b_\omega ))\Vert }_p\) denotes the norm of the adjoint action of \(u(p^{-m}(b_\eta -b_\omega ))\) on the Lie algebra of \(\textbf{G}(\mathbb {Q}_p)\). Note further that the above estimate holds for \(\mathcal {Q}_\lambda \) since \(\gamma _\omega ^\alpha (\gamma _\eta ^\alpha )^{-1}=\tau _\omega ^\alpha (\tau _\eta ^\alpha )^{-1}\). Hence, it suffices to bound the above average.

To calculate the adjoint norm, we find a polar decomposition of \(u(p^{-m}(b_\eta -b_\omega ))\). Note that for all \(\textbf{x}\in \mathbb {Q}_p\) with \(|\textbf{x}|_p> 1\), we have

$$\begin{aligned} \begin{pmatrix} 1 &{} 0 \\ \frac{-1}{\textbf{x}+1} &{}1 \end{pmatrix} u(\textbf{x}) \begin{pmatrix} 1 &{} 0 \\ 1 &{} 1 \end{pmatrix} \begin{pmatrix} 1 &{} \frac{-\textbf{x}}{\textbf{x}+1}\\ 0 &{} 1 \end{pmatrix} = \begin{pmatrix} 1+\textbf{x}&{} 0\\ 0&{} \frac{1}{\textbf{x}+1} \end{pmatrix}. \end{aligned}$$

Then, since \(|1/(\textbf{x}+1)|_p<1\) and \(|\textbf{x}/(\textbf{x}+1)|_p = 1\), we obtain

$$\begin{aligned} u(\textbf{x})=m_1 a\big ((1+\textbf{x})^2\big ) m_2 \end{aligned}$$

for some \(m_1,m_2\in \textbf{G}(\mathbb {Z}_p)\) (recall that \(\textbf{G}=\textrm{PGL}_2\)). Using (3.3), we get

$$\begin{aligned} \left\Vert {u(\textbf{x})} \right\Vert _p = \big \Vert a\big ((\textbf{x}+1)^2\big )\big \Vert _p = |\textbf{x}|_p^2. \end{aligned}$$

It follows that

$$\begin{aligned}{} & {} \big \Vert (\alpha \cdot \mathcal {P}_\lambda )^n(\varphi )\big \Vert ^2 \ll _{\varepsilon } \mathcal {S}_{2,1}(\varphi )^2 \\{} & {} \quad \times \left( \sum _{\eta =\omega \in \Lambda ^n} \lambda _\eta \lambda _\omega + p^{-(\frac{25}{32}-2\varepsilon )m}\sum _{\eta \ne \omega \in \Lambda ^n} \lambda _\eta \lambda _\omega |b_\eta -b_\omega |_p^{-\frac{25}{32}+2\varepsilon } \right) . \end{aligned}$$

Fix some \(\varepsilon >0\) and let \(\delta = 25/32-2\varepsilon \). For each \(\eta \ne \omega \in \Lambda ^n\), define

$$\begin{aligned} d(\eta ,\omega ) = \max \left\{ 1\le i\le n: \eta _i \ne \omega _i\right\} . \end{aligned}$$

Recall that \(b_j = j/p\) for all \(j\in \Lambda \). Let \(\eta \ne \omega \in \Lambda ^n\) and let \(d=d(\eta ,\omega )\). A simple calculation then shows that

$$\begin{aligned} b_{\eta }-b_\omega&= \sum _{i=1}^n (b_{\eta _i}-b_{\omega _i}) p^{-i+1} = \sum _{i=1}^{d} (\eta _i-\omega _i) p^{-i}\\&= p^{-d} \sum _{i=1}^{d} (\eta _i-\omega _i) p^{d-i}. \end{aligned}$$

By definition, we have \(\eta _d\ne \omega _d\). This implies that the integer \(\sum _{i=1}^{d} (\eta _i-\omega _i) p^{d-i}\) is coprime to p, i.e., a unit in \(\mathbb {Z}_{p}=\{\textbf{x}\in \mathbb {Q}_{p}:|\textbf{x}|_{p}\le 1\}\). Thus, it follows that

$$\begin{aligned} \sum _{\eta \ne \omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \left| b_\eta -b_\omega \right| _p^{-\delta }&= \sum _{\eta \ne \omega \in \Lambda ^n} \lambda _\eta \lambda _\omega p^{-\delta d(\eta ,\omega )}\\&= \sum _{\eta \in \Lambda ^n} \lambda _\eta \sum _{j=1}^n p^{-\delta j} \sum _{\begin{array}{c} \omega \in \Lambda ^n\\ d(\eta ,\omega )=j \end{array}} \lambda _\omega . \end{aligned}$$

We now specialize to the case where \(\lambda \) is the uniform probability vector with weight \(1/|\Lambda |\). Then, for each \(\eta \in \Lambda ^n\), we have

$$\begin{aligned}\lambda _\eta = |\Lambda |^{-n} = p^{-sn}, \end{aligned}$$

where \(s = \log |\Lambda |/\log p\). Hence, we obtain

$$\begin{aligned}&\sum _{\eta \ne \omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \left| b_\eta -b_\omega \right| _p^{-\delta }\\&\quad = p^{-2sn} \sum _{j=1}^n p^{-\delta j} \sum _{\eta \in \Lambda ^n} \underbrace{|\{\omega \in \Lambda ^n: d(\eta ,\omega )=j\}|}_{=|\Lambda |^{j-1}(|\Lambda |-1)}\\&\quad \leqslant p^{-2sn} \sum _{j=1}^n p^{-\delta j} \sum _{\eta \in \Lambda ^n} |\Lambda |^{j}= p^{-2sn} \sum _{j=1}^n p^{-\delta j} |\Lambda |^{n+j}. \end{aligned}$$

If \(s\ne \delta \), then, using that \(|\Lambda | = p^s\), we obtain

$$\begin{aligned} \sum _{\eta \ne \omega \in \Lambda ^n} \lambda _\eta \lambda _\omega \left| b_\eta -b_\omega \right| _p^{-\delta }&\leqslant p^{-sn+s-\delta } \sum _{j=0}^{n-1} p^{(s-\delta ) j}\\&= p^{-sn+s-\delta } \frac{p^{(s-\delta )n}- 1}{ p^{(s-\delta )} -1}\\&\leqslant \frac{p^{s-\delta }}{|p^{s-\delta }-1|}\big (p^{-\delta n}+p^{-sn}\big ). \end{aligned}$$

Otherwise, if \(s=\delta \), we get a bound of the form \(n p^{-s n}\). Finally, we note that

$$\begin{aligned} \sum _{\eta =\omega \in \Lambda ^n} \lambda _\eta \lambda _\omega = p^{-sn}. \end{aligned}$$

\(\square \)

In the proof of Theorem A.2, we will need an estimate on the decay of the \(\textrm{L}^4\)-norm of the operators \(\alpha \cdot \mathcal {Q}_\lambda \). We deduce this estimate in the following corollary.

Corollary A.4

For all \(q\ge 2\), \(n\in \mathbb {N}\), \(\alpha \in \Lambda ^*\), and for every bounded smooth \(K_{\textrm{f}}\)-invariant function \(\varphi \in \textrm{L}^2_{00}(X_S)\), we have

$$\begin{aligned} \big \Vert (\alpha \cdot \mathcal {Q}_\lambda )^n (\varphi )\big \Vert ^q_{\textrm{L}^q} \ll _{\varepsilon ,p,s,q} \mathcal {S}_{2,1}(\varphi )^2 \left\Vert {\varphi } \right\Vert _\infty ^{q-2}\cdot p^{-2o_\varepsilon n }, \end{aligned}$$

for every \(\varepsilon >0\), where \(2o_\varepsilon =\min \left\{ 25/32,s\right\} -\varepsilon \).

Proof

The case when \(q=2\) is exactly Proposition A.3. Hence, we may assume \(q>2\). Let \(\mu _S\) denote the \(\textbf{G}_S\)-invariant probability measure on \(X_S\). Using Fubini’s Theorem one checks that

$$\begin{aligned} \big \Vert (\alpha \cdot \mathcal {Q}_\lambda )^n (\varphi )\big \Vert ^q_{\textrm{L}^q}&=\int _0^\infty \mu _S(x: |(\alpha \cdot \mathcal {Q}_\lambda )^n (\varphi )(x)|^q>t)\;dt \\&= \int _0^{\left\Vert {\varphi } \right\Vert ^q_\infty } \mu _S(x: |(\alpha \cdot \mathcal {Q}_\lambda )^n (\varphi )(x)|^q>t)\;dt, \end{aligned}$$

where we used that \(\left\Vert {\alpha \cdot \mathcal {Q}_\lambda ^n(\varphi )} \right\Vert _\infty \le \left\Vert {\varphi } \right\Vert _\infty \). Hence, by Proposition A.3 and Chebychev’s inequality, we have for all \(t>0\),

$$\begin{aligned} \mu _S(x: |(\alpha \cdot \mathcal {Q}_\lambda )^n (\varphi )(x)|^q>t) \ll _{\varepsilon ,p,s} \mathcal {S}_{2,1}(\varphi )^2 p^{-2o_\varepsilon n } t^{-2/q}. \end{aligned}$$

Hence, since \(q>2\), we obtain

$$\begin{aligned} \big \Vert (\alpha \cdot \mathcal {Q}_\lambda )^n (\varphi )\big \Vert ^q_{\textrm{L}^q} \ll _{\varepsilon ,p,s} \mathcal {S}_{2,1}(\varphi )^2 p^{-2o_\varepsilon n } \left\Vert {\varphi } \right\Vert _\infty ^{q-2} \frac{q}{q-2}. \end{aligned}$$

\(\square \)

1.2 A.2 Sharper version of Proposition 5.1

The following result provides a sharper value of \(\kappa \) constituting the rate of equidistribution of horospherical measures on congruence covers.

Proposition A.5

(Prop. 3.1, [52])

Let \(\Delta \le \Gamma (1)\) be a congruence lattice and \(X_\Delta = \textbf{G}_\infty /\Delta \). Then, for every \(\varphi \in \textrm{B}_{2,3}^{\infty }(X_\Delta )\), \(x\in X_\Delta \) and \(t\ge 1\),

$$\begin{aligned} \int _0^1 \varphi (a(t) u(\textbf{x})x)\; \textrm{d} \textbf{x}= \int \varphi \; \textrm{d} m_{\textbf{G}_\infty ^+ \cdot x} + O\big (V_\Delta \cdot \mathcal {S}_{2,3}(\varphi ) \cdot t^{-\kappa }\cdot {\mathscr {Y}}^{1/2}_{\Delta }(x)\big ), \end{aligned}$$

where \(V_\Delta = \sqrt{[\Gamma (1):\Delta ]}\), \({\mathscr {Y}}_{\Delta }\) is a positive proper function on \(X_\Delta \) and if \(\lambda _1\in (0,1/4)\) is a uniform lower bound on the non-zero eigenvalues of the Laplacian on \(X_\Delta \) for all \(\Delta \), then

$$\begin{aligned} \kappa = \frac{1-\sqrt{1-4\lambda _1}}{2}. \end{aligned}$$

The implied constant is independent of \(\Delta \).

Proof

The statement in [52, Prop. 3.1] is stated in a slightly different form, we outline the needed modifications. First, the results in loc. cit. are stated for quotients of \(\textrm{SL}_2(\mathbb {R})\). Recall that \(\textbf{G}_\infty ^+\) is the image of \(\textrm{SL}_2(\mathbb {R})\) inside \(\textbf{G}_\infty \) and is a normal subgroup of index 2. In particular, for each \(\Delta \), \(X_\Delta \) consists of at most two connected components, each of which is isomorphic to \(\textrm{SL}_2(\mathbb {R})/\Delta '\), where \(\Delta '\le \textrm{SL}_{2}(\mathbb {Z})\) is a congruence lattice. We define \({\mathscr {Y}}_{\Delta }\) to be \({\mathscr {Y}}_{\Delta '}\) (in the notation of [52, Eq. (11)]) on each of the connected components of \(X_\Delta \).

The measure on \(X_\Delta \) defining the \(\textrm{L}^2\)-Sobolev norms \(\left\Vert {\cdot } \right\Vert _{W_k}\) in loc. cit. has total mass \(\asymp V_\Delta ^2\). In particular, this norm is equivalent to \(V_\Delta \cdot \mathcal {S}_{2,k}\) for all \(k\in \mathbb {N}\); cf. discussion following [52, Eq. (9)]. Note further that the statement is made for long horocycle orbits starting from a point p. The above statement is obtained from this result with \(T=t\) and with \(pa(T^{-1})\) in place of p in the notation in loc. cit using standard conjugation relations of a(t) and \(u(\textbf{x})\).

Next, we note that the implied constant in [52, Prop. 3.1] can be made independent of \(\Delta \). The dependence on the lattice comes from [52, Lem. 2.1]. Note that the bounds in [52, Lem. 2.2, 2.3] are not needed for our weaker error term \(t^{-\kappa }{\mathscr {Y}}_{\Delta }^{1/2}(x)\).

The dependence in [52, Lem. 2.1] arises from a choice of an injectivity radius to allow for a thick-thin decomposition of \({\mathbb {H}}^2/\Delta '\) in order to apply the Sobolev embedding theorem (cf. the choice of \(\epsilon \) in the proof of [52, Lem. 2.1] given in [22, Lem. 5.3]). As \(X_\Delta \) are all covers of \(X_\infty (1)\cong \textrm{SL}_2(\mathbb {R})/\textrm{SL}_2(\mathbb {Z})\), a choice of an injectivity radius in \(X_\infty (1)\) works for all of \(X_\Delta \).

Hence, the error term can be obtained by applying [52, Lem. 2.1] to Burger’s integral formula in [52, Eq. (23)] combined with the estimates on the height function in [52, pg 303] and the estimates on the intertwining operators given in [52, Eq. (22)] (or [12, pg. 791] with \(\alpha =\kappa \) in the notation of [12]) as is done in [52]. One uses [52, Lem. 2.2] to ensure pointwise convergence of the last integral in [52, Eq. (23)] to a bounded continuous function as is done towards the end of the proof so that the above bounds apply.

Finally, we note that the order \(3\) Sobolev norm in the statement (as opposed to \(\mathcal {S}_{2,4}\) in loc. cit.) arises from only applying the bounds of [52, Lem. 2.1] in the proof of [52, Prop. 3.1].

\(\square \)

1.3 A.3 Proof of Theorem A.2

We outline the needed modifications of the proof of Theorem 6.1 in this setting. We retain the notation in that proof, in particular the constants \(a,b_\varepsilon \) and c in the statement of Theorem 6.1. We begin by noting that the average contraction ratio r is \(p^{-1}\) in the case at hand.

Since missing digit Cantor sets satisfy the open set condition with the open set (0, 1), the proof of Theorem 6.1 shows that we can take the absolutely continuous measure \(\nu \) to be the Lebesgue measure on the unit interval. In this case, the Mass Term in (6.22) takes the form

$$\begin{aligned} \text {Mass Term} = p^{2\sigma m}, \qquad 2\sigma =1-s. \end{aligned}$$

By Lemma 4.2 and using (12.44), \((\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )\) can be regarded as a function on \(\textbf{G}_\infty /\Delta \), for some congruence lattice \(\Delta \). Hence, we may apply Proposition A.5 in place of Proposition 5.1 to obtain the following replacement of (6.27):

$$\begin{aligned}&\int _0^1 (\alpha \cdot \mathcal {P}_\lambda )^n(\varphi )^2\big ( a(t) u(\textbf{x})h_\alpha \Delta \big ) \;\textrm{d}\textbf{x}\nonumber \\&\quad = \int _0^1 (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )^2\big ( a(tp^{-n}) u(\textbf{x})h_\alpha \Delta \big ) \;\textrm{d}\textbf{x} \nonumber \\&\quad =\int (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )^2 \; \textrm{d} m_{\textbf{G}_\infty ^+/\Delta ^+} \nonumber \\&\qquad + O\left( V_{\Delta } \mathcal {S}_{2,\ell }\big ( (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )^2\big ) p^{\kappa n}t^{-\kappa } {\mathscr {Y}}^{1/2}_{\Delta }(h_\alpha \Delta )\right) , \end{aligned}$$
(12.45)

where \(\ell =3\), \(\Delta ^+=\textbf{G}_\infty ^+\cap \Delta \). Here, we use the fact that \(h_\alpha \in \textbf{G}_\infty ^+\) so that \(\textbf{G}_\infty ^+\cdot h_\alpha \Delta \cong \textbf{G}_\infty ^+/\Delta ^+\).

Note further that, by [52, Eq. (11)-(13)], \({\mathscr {Y}}_{\Delta }(x)\le {\mathscr {Y}}_{\Gamma (1)}(x) \ll \left\Vert {\textrm{Ad}_g} \right\Vert \), where \(g\in \textbf{G}_\infty \) is any representative of x and \(\left\Vert {\textrm{Ad}_g} \right\Vert \) denotes the norm of its adjoint action. In particular, the estimate \({\mathscr {Y}}^{1/2}_{\Delta }(x_\alpha )\ll \rho _\alpha ^{-C_0}\) holds for a suitable \(C_0\ge 1\) in place of the estimate (6.28).

The key point in introducing the operators \(\mathcal {Q}_\lambda \) is as follows. Since multiplication by elements of \(\textbf{G}_\textrm{f}\) commutes with differential operators on \(\textbf{G}_\infty \), one checks using Lemma 3.3 that

$$\begin{aligned} \mathcal {S}_{2,\ell }\big ( (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )^2\big ) \ll \mathcal {S}_{4,\ell }\big ( (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )\big )^2. \end{aligned}$$

Moreover, note that \(\mathcal {D}\varphi \) has mean 0 for any differential operator \(\mathcal {D}\). This can be checked by induction on the degree of the operator using the dominated convergence theorem, invariance of the Haar measure, and the limit definition of Lie derivatives; cf. proof of Lemma 10.4. In particular, Lemma 3.5 implies that the lift of \(\mathcal {D}\varphi \) to \(X_S\) belongs to \(\textrm{L}^2_{00}(X_S)\).

Hence, by Corollary A.4, applied with \(q=4\), for any \(\mathcal {D}\) of degree \(\le \ell \), we have

$$\begin{aligned} \left\Vert {(\alpha \cdot \mathcal {Q}_\lambda )^n(\mathcal {D}\varphi )} \right\Vert _{\mathrm {L^4}}^4 \ll _{\varepsilon ,p,s} \mathcal {S}_{\infty ,1}(\mathcal {D}\varphi )^4 p^{-2o_\varepsilon n} \leqslant \mathcal {S}_{\infty ,\ell +1}(\varphi )^4 p^{-2o_\varepsilon n}, \end{aligned}$$

where

$$\begin{aligned} 2o_\varepsilon := \min \left\{ 25/32,s \right\} -\varepsilon . \end{aligned}$$

It follows that

$$\begin{aligned} \mathcal {S}_{2,\ell }\big ( (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )^2\big ) \ll \mathcal {S}_{4,\ell }\big ( (\alpha \cdot \mathcal {Q}_\lambda )^n(\varphi )\big )^2 \ll _{\varepsilon ,p,s} \mathcal {S}_{\infty ,\ell +1}(\varphi )^2 p^{-o_\varepsilon n}. \end{aligned}$$

Additionally, by Lemma 4.2, the congruence lattice \(\Delta \) can be chosen so that \(V_\Delta \ll p^{3|\alpha |+3n/2}\). Finally, estimating the main term in (12.45) using Proposition A.3, we obtain the following sharper bound on the horospherical term:

$$\begin{aligned} \text {Horospherical term } \ll _{\varepsilon ,p} \mathcal {S}_{\infty ,\ell +1}(\varphi )^2 p^{(3+C_0)|\alpha |}\left( p^{-2o_\varepsilon n} + p^{2n\upsilon -\kappa \tau } \right) , \end{aligned}$$

where

$$\begin{aligned} 2\upsilon = 3/2+\kappa -o_\varepsilon , \qquad p^\tau = t. \end{aligned}$$

By known bounds towards Selberg’s eigenvalue conjecture due to [31, Proposition 2], we can take \(\lambda _1\ge 975/4096\) in Proposition A.5. In particular, we may take \(\kappa = 25/64\).

By combining the above estimates and balancing the rates as is done in the proof of Theorem 6.1 (cf. discussion following (6.33)), we see that the conclusion of that theorem holds in our setting if

$$\begin{aligned} 2\sigma (o_\varepsilon +\upsilon )< \kappa (\sigma +o_\varepsilon ) \Longleftrightarrow (1-s)(o_\varepsilon +3/2)<2\kappa o_\varepsilon , \end{aligned}$$

for some \(\varepsilon >0\) and with our choices of \(\sigma ,o_\varepsilon ,\upsilon \) and \(\kappa \) as above. This condition is in turn satisfied under our hypothesis (12.42) as can be shown by a direct calculation.

1.4 A.4 A version of Lebesgue density

In this subsection, we verify the version of Lebesgue density theorem for Bernoulli measures on symbolic spaces used in the proof of Lemma 2.7.

Let \(\Lambda \) be a finite set and \(\lambda \) be a probability vector on \(\Lambda \). For \(\alpha \in \Sigma := \Lambda ^\mathbb {N}\) and \(k\in \mathbb {N}\), denote by \(\Sigma (\alpha ,k)\) the cylinder set given by the prefix of \(\alpha \) of length k and denote this prefix by \(\alpha |_k\). We endow \(\Lambda \) with the discrete topology and \(\Sigma \) with the associated product topology.

Lemma A.6

Suppose \(B\subseteq \Sigma \) is a Borel set. Then, for \(\lambda ^\mathbb {N}\)-almost every \(x\in B\),

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{\lambda ^\mathbb {N}(B\cap \Sigma (x,k))}{\lambda ^\mathbb {N}(\Sigma (x,k))} = 1. \end{aligned}$$

Proof

We deduce this result from the corresponding well-known Lebesgue density theorem for Radon measures on the real line. Let \(p=2|\Lambda |\) and consider the auxiliary IFS given by

$$\begin{aligned} \mathcal {F}= \left\{ f_i(x) = (x+i)/p: 0\le i\le p-1, i \text { is even} \right\} . \end{aligned}$$

Let \(\mathcal {K}\) be its attractor and note that the images of \(\mathcal {K}\) under distinct maps in \(\mathcal {F}\) are disjoint. Let \(\pi :\Sigma \rightarrow \mathbb {R}\) be the coding map defined by \(\pi (\alpha ) = \lim _{k\rightarrow \infty } f_{\alpha |_k}(0)\) and \(\mu =\pi _*\lambda ^\mathbb {N}\) be the self-similar measure. Then, \(\pi \) is a homeomorphism onto its image \(\mathcal {K}\); cf. [29, Thm. 3.1.(3) and Thm. 4.4.(4)]. Hence, it suffices to show that

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{\mu (\pi (B)\cap \pi ( \Sigma (\alpha ,k)))}{\mu (\pi (\Sigma (\alpha ,k)))} = 1 \end{aligned}$$

for \(\lambda ^\mathbb {N}\)-almost every \(\alpha \in B\). Let \(\alpha \in B\). To relate the images of cylinder sets under \(\pi \) to intervals in \(\mathbb {R}\), one first checks that \(\pi (\Sigma (\alpha ,k))\) is contained in the image of [0, 1] under \(f_{\alpha |_k}\). Hence, by definition of \(\mathcal {F}\), given any \(\beta \in \Sigma \) such that \(\Sigma (\beta ,k)\ne \Sigma (\alpha ,k)\), the distance between \(\pi (\Sigma (\beta ,k))\) and \(\pi (\Sigma (\alpha ,k))\) is at least \(p^{-k}\). It follows that

$$\begin{aligned} \pi (\Sigma (\alpha ,k)) = B(\pi (\alpha ), p^{-k})\cap \mathcal {K}, \end{aligned}$$

where \(B(\pi (\alpha ), p^{-k})\) denotes the open interval around \(\pi (\alpha )\) of radius \(p^{-k}\). It follows by Lebesgue’s density theorem for Radon measures on \(\mathbb {R}\) that

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{\mu (\pi (B)\cap \pi ( \Sigma (\alpha ,k)))}{\mu (\pi (\Sigma (\alpha ,k)))} = \lim _{k\rightarrow \infty } \frac{\mu (\pi (B)\cap B(\pi (\alpha ),p^{-k}))}{\mu (B(\pi (\alpha ),p^{-k}))} = 1, \end{aligned}$$

for \(\lambda ^\mathbb {N}\)-almost every \(\alpha \in B\). Note that we are allowed to use open balls in this application of Lebesgue density since \(\mu \) is non-atomic. Indeed, it suffices to note that \(\lambda ^\mathbb {N}(\Sigma (\beta ,k))\le \lambda _{\max }^k\rightarrow 0\) for any \(\beta \in \Sigma \), where \(\lambda _{\max }\) denotes the largest component of \(\lambda \). \(\square \)

Appendix B. Congruence quotients

The goal of this appendix is to give proofs of several facts presented in Sect. 3.1 and used in the proof of Theorem 6.1. In Corollary B.11, we establish the correspondence between compact-open subgroups of \(\textbf{G}(\mathbb {A}_{\textrm{f}})\) and principal congruence subgroups of \(\textbf{G}(\mathbb {Z})\) which underlies the double coset decomposition (3.4). In Proposition B.13, we prove the uniform bound on the number of connected components of used in (6.31). At the end of this section, we will define general congruence subgroups; this extension is immediate but we include it for completess.

1.1 B.1 Integral structures

We begin by making an explicit choice of the integral structure on \(\textbf{G}\) used to define congruence groups. Given a ring R, we let

$$\begin{aligned} V_{R}=\textrm{Mat}_{d+1}(\mathbb {Z})\otimes _{\mathbb {Z}}R. \end{aligned}$$

Then \(V_{R}\) is an R-algebra which is a free R-module of rank \((d+1)^{2}\). The algebra \(V_{R}\) allows us to realize \(\textbf{G}(k)\) as a linear group whenever k is a field. More explicitly, we fix a faithful k-representation of \(\textbf{G}\) by choosing the standard basis \(\mathcal {E}_{d+1}\) of \(V_{k}\) which gives rise to an isomorphism \(\textrm{GL}(V_{k})\cong \textrm{GL}_{(d+1)^{2}}(k)\) and we define

$$\begin{aligned} \textbf{G}=\{g\in \textrm{GL}_{(d+1)^{2}}:\forall u,v\in \mathcal {E}_{d+1}\quad g(uv)=(gu)(gv)\}. \end{aligned}$$

In what follows, we let \(\Phi :\textrm{GL}_{d+1}\rightarrow \textbf{G}\) denote the k-representation given by

$$\begin{aligned} \Phi (x)(v)=xvx^{-1}\quad (x\in \textrm{GL}_{d+1},v\in \textrm{Mat}_{d+1}). \end{aligned}$$
(12.46)

By the Skolem–Noether theorem we have \(\Phi (\textrm{GL}_{d+1}(k))=\textbf{G}(k)\) for any field k and in particular \(\textbf{G}(k)\cong \textrm{GL}_{d+1}(k)/k^{\times }\), where we identify \(k^{\times }\) with the scalar diagonal matrices in \(\textrm{GL}_{d+1}(k)\). We record the following consequence of the above discussion which is used to apply the results of [24].

Lemma B.1

The group \(\textbf{G}\) is a connected group over k.

Proof

As \(\textbf{G}\) is an affine k-group, we only have to prove that it is connected. Recall that \(\textrm{GL}_{d+1}\) is an irreducible affine \(\mathbb {Q}\)-group. To this end, note that \(\textrm{GL}_{d+1}\) is the principal open set defined by the polynomial \(\det (x_{ij})\), i.e.,

$$\begin{aligned} \textrm{GL}_{d+1}(k)=\big \{(x_{ij})\in k^{(d+1)^{2}}:\det (x_{ij})\ne 0\big \}. \end{aligned}$$

This is a Zariski-open subset of affine space. As affine space is irreducible, every open subset of affine space is irreducible and hence \(\textrm{GL}_{d+1}(k)\) is irreducible. In particular, it follows that \(\textrm{GL}_{d+1}\) is a connected group over k; cf. [10, Prop. I.1.2]

By the Skolem–Noether theorem, \(\textbf{G}\) is therefore the image of a connected group under the morphism (12.46) and as morphisms map Zariski-connected sets to Zariski-connected sets, the claim follows. \(\square \)

In what follows, we let \(D=(d+1)^{2}\). We identify \(\textbf{G}(k)\) with its image in \(\textrm{GL}_{D}(k)\) given by the basis \(\mathcal {E}_{d+1}\).

Definition B.2

Let k be a field and let \(R\hookrightarrow k\) a subring. Then

$$\begin{aligned} \textbf{G}(R)=\textbf{G}(k)\cap \textrm{GL}_{D}(R). \end{aligned}$$

We denote by \(\mathcal {V}_{\textrm{f}}\subseteq \mathbb {N}\) the set of finite rational primes and we let \(\mathcal {V}=\mathcal {V}_{\textrm{f}}\cup \{\infty \}\). The following definition of adelic points and integral adelic points of a \(\mathbb {Q}\)-group is formulated for a general algebraic \(\mathbb {Q}\)-subgroup \(\textbf{H}\) of \(\textrm{GL}_{D}\). It encompasses in particular the cases \(\textbf{H}=\textbf{G}\) and \(\textbf{H}=\textrm{GL}_{D}\).

Definition B.3

Let \(S\subseteq \mathcal {V}\), \(S_{\textrm{f}}=S\setminus \{\infty \}\), and \(\textbf{H}\le \textrm{GL}_{D}\) be a \(\mathbb {Q}\)-subgroup. We set

$$\begin{aligned} \textbf{H}(\mathbb {Z}_{S_{\textrm{f}}})&=\prod _{p\in S_{\textrm{f}}}\textbf{H}(\mathbb {Z}_{p}),\\ \textbf{H}(\mathbb {Q}_{S_{\textrm{f}}})&=\left\{ (g_{p})_{p\in S_{\textrm{f}}}\in \prod _{p\in S_{\textrm{f}}}\textbf{H}(\mathbb {Q}_{p}):g_{p}\in \textbf{H}(\mathbb {Z}_{p})\right. \\&\qquad \left. \text { for all but finitely many }p\in S_{\textrm{f}}\right\} . \end{aligned}$$

If \(\infty \in S\), then \(\textbf{H}(\mathbb {Q}_{S})=\textbf{H}(\mathbb {R})\times \textbf{H}(\mathbb {Q}_{S_{\textrm{f}}})\). If \(S_{\textrm{f}}=\mathcal {V}_{\textrm{f}}\), we set \(\textbf{H}({\widehat{\mathbb {Z}}})=\textbf{H}(\mathbb {Z}_{S_{\textrm{f}}})\), \(\textbf{H}(\mathbb {A}_{\textrm{f}})=\textbf{H}(\mathbb {Q}_{S_{\textrm{f}}})\), \(\textbf{H}(\mathbb {A})=\textbf{H}(\mathbb {R})\times \textbf{H}(\mathbb {A}_{\textrm{f}})\), and \(\textbf{H}(\mathbb {R}\times {\widehat{\mathbb {Z}}})=\textbf{H}(\mathbb {R})\times \textbf{H}({\widehat{\mathbb {Z}}})\).

1.2 B.2 The fundamental compact-open subgroups

Given \(p\in \mathcal {V}_{\textrm{f}}\) and \(v\in \mathbb {N}\) we define a map

$$\begin{aligned} \pi _{p,v}:\textrm{Mat}_{D}(\mathbb {Z}_{p})\rightarrow \textrm{Mat}_{D}(\mathbb {Z}/p^{v}\mathbb {Z}) \end{aligned}$$

by coordinate-wise reduction mod \(p^{v}\). This map clearly defines a ring homomorphism and for any \(x\in \textrm{Mat}_{D}(\mathbb {Z}_{p})\) we have

$$\begin{aligned} (\det \circ \pi _{p,v})(x)\equiv \det (x)\,\textrm{mod}\,p^{v}. \end{aligned}$$

As \(\mathbb {Z}_{p}^{\times }\) and \((\mathbb {Z}/p^{v}\mathbb {Z})^{\times }\) consist precisely of the elements whose projections mod \(p^{v}\) do not vanish, this induces a group homomorphism \(\pi _{p,v}:\textrm{GL}_{D}(\mathbb {Z}_{p})\rightarrow \textrm{GL}_{D}(\mathbb {Z}/p^{v}\mathbb {Z})\). For the sake of completeness, we argue that it is surjective. To this end one notes that \(\mathbb {Z}/p^{v}\mathbb {Z}\) is a semi-local ring, so that \(\textrm{SL}_{D}(\mathbb {Z}/p^{v}\mathbb {Z})\) is generated by elementary matrices; cf. [26, Thm. 4.3.9]. Therefore \(\pi _{p,v}\) restricts to an epimorphism from \(\textrm{SL}_{D}(\mathbb {Z}_{p})\) to \(\textrm{SL}_{D}(\mathbb {Z}/p^{v}\mathbb {Z})\). Now one uses that

$$\begin{aligned} \textrm{GL}_{D}(\mathbb {Z}/p^{v}\mathbb {Z})\cong (\mathbb {Z}/p^{v}\mathbb {Z})^{\times }\ltimes \textrm{SL}_{D}(\mathbb {Z}/p^{v}\mathbb {Z}), \end{aligned}$$

where \((\mathbb {Z}/p^{v}\mathbb {Z})^{\times }\) identifies with the set of matrices of the form

$$\begin{aligned} \left\{ \begin{pmatrix} a &{} 0 \\ 0 &{} \textrm{Id}_{D-1} \end{pmatrix}: a\in (\mathbb {Z}/p^{v}\mathbb {Z})^{\times }\right\} . \end{aligned}$$

In what follows, we will denote \(L_{p}[p^{v}]=\ker \pi _{p,v}\) and \(L_{p}[1]=\textrm{GL}_{D}(\mathbb {Z}_{p})\). We set

$$\begin{aligned}K_{p}[p^{v}]=L_{p}[p^{v}]\cap \textbf{G}(\mathbb {Z}_{p}).\end{aligned}$$

Lemma B.4

The family \(\{L_{p}[p^{v}]\}\) is a basis of open neighbourhoods of the identity in \(\textrm{GL}_{D}(\mathbb {Z}_p)\). In particular, the family \(\{K_{p}[p^{v}]:v\in \mathbb {N}\}\) is a basis of open neighbourhoods of the identity in \(\textbf{G}(\mathbb {Z}_{p})\).

Proof

The group \(L_{p}[p^{v}]\le \textrm{GL}_{D}(\mathbb {Z}_{p})\) is closed and has finite index, therefore it is open. The topology on \(\textrm{GL}_{D}(\mathbb {Z}_{p})\) is induced by the metric \(\Vert \cdot \Vert _{p}\) on \(\textrm{Mat}_{D}(\mathbb {Z}_{p})\) given by

$$\begin{aligned} \Vert x\Vert _{p}=\max \{|x_{i,j}|_{p}:1\le i,j\le D\}\quad (x\in \textrm{Mat}_{D}(\mathbb {Z}_{p})). \end{aligned}$$

Let \(v\in \mathbb {N}\) and \(x\in \textrm{Mat}_{D}(\mathbb {Z}_{p})\), then

$$\begin{aligned} \Vert \textrm{Id}_{D}-x\Vert _{p}\le p^{-v}\iff x\in \textrm{Id}_{D}+p^{v}\textrm{Mat}_{D}(\mathbb {Z}_{p}). \end{aligned}$$

In particular, the collection

$$\begin{aligned} \{\textrm{Id}_{D}+p^{v}\textrm{Mat}_{D}(\mathbb {Z}_{p}):v\in \mathbb {N}\} \end{aligned}$$

is a basis of open neighbourhoods of the identity in \(\textrm{GL}_{D}(\mathbb {Z}_{p})\). \(\square \)

Given \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\), we denote by \(\mathcal {I}_{S_{\textrm{f}}}\) the set of natural numbers whose prime factorization involves only primes contained in \(S_{\textrm{f}}\). Given \(N\in \mathcal {I}_{S_{\textrm{f}}}\), we define \((v_{p}(N))_{p\in \mathcal {I}_{S_{\textrm{f}}}}\) by \(N=\prod _{p\in S_{\textrm{f}}}p^{v_{p}(N)}\). We set

$$\begin{aligned} L_{S_{\textrm{f}}}[N]=\prod _{p\in S_{\textrm{f}}}L_{p}[p^{v_{p}(N)}] \end{aligned}$$

and \(K_{S_{\textrm{f}}}[N]=L_{S_{\textrm{f}}}[N]\cap \textbf{G}(\mathbb {Z}_{S_{\textrm{f}}})\). If \(S_{\textrm{f}}=\mathcal {V}_{\textrm{f}}\), we write \(L_{\textrm{f}}[N]\) and \(K_{\textrm{f}}[N]\) for \(L_{S_{\textrm{f}}}[N]\) and \(K_{S_{\textrm{f}}}[N]\) respectively.

Corollary B.5

The family \(\{L_{S_{\textrm{f}}}[N]:N\in \mathcal {I}_{S_{\textrm{f}}}\}\) forms a basis of compact open neighbourhoods of the identity in \(\textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\). In particular, the family \(\{K_{S_{\textrm{f}}}[N]:N\in \mathcal {I}_{S_{\textrm{f}}}\}\) forms a basis of compact open neighbourhoods of the identity in \(\textbf{G}(\mathbb {Z}_{S_{\textrm{f}}})\).

Proof

The groups \(L_{S_{\textrm{f}}}[N]\) are open by definition of the product topology. Compactness follows from Tychonov’s theorem. In order to prove that they form a neighbourhood basis, let \(V\subseteq \textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\) be an open neighbourhood of the identity. Then there is a finite set \(T_{\textrm{f}}\subseteq S_{\textrm{f}}\) and for all \(p\in T_{\textrm{f}}\) an open neighbourhood \(V_{p}\) of the identity in \(\textrm{GL}_{D}(\mathbb {Z}_{p})\) such that

$$\begin{aligned} \prod _{p\in T_{\textrm{f}}}V_{p}\times \prod _{p\in S_{\textrm{f}}\setminus T_{\textrm{f}}}\textrm{GL}_{D}(\mathbb {Z}_{p})\subseteq V. \end{aligned}$$

Given \(p\in T_{\textrm{f}}\), let \(v_{p}\in \mathbb {N}\) be such that \(L_{p}[p^{v_{p}}]\subseteq V_{p}\) and define \(N=\prod _{p\in T_{\textrm{f}}}p^{v_{p}}\). Then, \(L_{S_{\textrm{f}}}[N]\subseteq V\) by definition. \(\square \)

1.3 B.3 Principal congruence subgroups

Similar to what was done in Sect. 1, we can define for any \(N\in \mathbb {N}\), with \(N\ge 2\), the group homomorphism \(\varpi _{N}:\textrm{GL}_{D}(\mathbb {Z})\rightarrow \textrm{GL}_{D}(\mathbb {Z}/N\mathbb {Z})\) given by projection mod N. Note that \(\varpi _{N}\) is not surjective. We let \(\Lambda (N)=\ker \varpi _{N}\) and \(\Gamma (N)=\Lambda (N)\cap \textbf{G}(\mathbb {Z})\). We also define \(\Lambda (1)=\textrm{GL}_{D}(\mathbb {Z})\) and \(\Gamma (1)=\textbf{G}(\mathbb {Z})\).

Definition B.6

Let \(N\in \mathbb {N}\). The family \(\{\Gamma (N):N\in \mathbb {N}\}\) is called the family of principal congruence subgroups. A subgroup \(\Delta \le \Gamma (1)\) is a congruence subgroup if it contains a principal congruence subgroup.

Let \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\). In what follows, we will view

$$\begin{aligned} \mathbb {Z}[S_{\textrm{f}}^{-1}]=\mathbb {Z}[\tfrac{1}{p}:p\in S_{\textrm{f}}] \end{aligned}$$

as a subring of \(\mathbb {Q}_{S_{\textrm{f}}}\) by embedding it diagonally. Similarly, \(\textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\) and \(\textbf{G}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\) become subgroups of \(\textrm{GL}_{D}(\mathbb {Q}_{S_{\textrm{f}}})\) and \(\textbf{G}(\mathbb {Q}_{S_{\textrm{f}}})\).

We are now ready to prove the first main result of this section.

Proposition B.7

Let \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\) and \(N\in \mathcal {I}_{S_{\textrm{f}}}\). Then, \(\Gamma (N)=K_{S_{\textrm{f}}}[N]\cap \textbf{G}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\). In particular, for all N, we have

$$\begin{aligned} \Gamma (N)=K_{\textrm{f}}[N]\cap \textbf{G}(\mathbb {Q}). \end{aligned}$$

Proof

It suffices to show that

$$\begin{aligned} \Lambda (N)=L_{S_{\textrm{f}}}[N]\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}]). \end{aligned}$$

If \(g\in \Lambda (N)\), i.e., \(g\in \textrm{GL}_{D}(\mathbb {Z})\) and \(g\equiv \textrm{Id}_{D}\,\textrm{mod}\,N\), then clearly for all p|N we have \(g\equiv \textrm{Id}_{D}\,\textrm{mod}\,p^{v_{p}(N)}\). As \(\det g\in \{\pm 1\}\subseteq \mathbb {Z}_{p}^{\times }\) for all \(p\in S_{\textrm{f}}\), we have \(g\in \textrm{GL}_{D}(\mathbb {Z}_{p})\) for all \(p\in S_{\textrm{f}}\). Combining these two facts, we obtain that \(\Lambda (N)\subseteq L_{S_{\textrm{f}}}[N]\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\).

Before we turn to the opposite inclusion, we note that

$$\begin{aligned} \textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])=\textrm{GL}_{D}(\mathbb {Z}). \end{aligned}$$

This, in particular, implies the result in the special case \(N=1\). The inclusion \(\textrm{GL}_{D}(\mathbb {Z})\subseteq \textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\) is clear. For the opposite inclusion, one first notes that \(\mathbb {Z}[S_{\textrm{f}}^{-1}]\cap \mathbb {Z}_{S_{\textrm{f}}}=\mathbb {Z}\) and hence \(\textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\subseteq \textrm{Mat}_{D}(\mathbb {Z})\). Let \(g\in \textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\). Then, \(\det g\in \mathbb {Z}\). On the other hand \(\det g\in \mathbb {Z}_{S_{\textrm{f}}}^{\times }\), i.e., we have \(\det g\in \mathbb {Z}_{p}^{\times }\) for all \(p\in S_{\textrm{f}}\). This means that \(\det g\) is coprime to p for all \(p\in S_{\textrm{f}}\). But, since \(\det g\in \mathbb {Z}[S_{\textrm{f}}^{-1}]^\times \), we get \(\det g\in \{\pm 1\}\). It follows that \(\textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\subseteq \textrm{GL}_{D}(\mathbb {Z})\).

Let now \(g\in L_{S_{\textrm{f}}}[N]\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])\). In particular

$$\begin{aligned} g\in \textrm{GL}_{D}(\mathbb {Z}_{S_{\textrm{f}}})\cap \textrm{GL}_{D}(\mathbb {Z}[S_{\textrm{f}}^{-1}])=\textrm{GL}_{D}(\mathbb {Z}). \end{aligned}$$

Therefore reduction mod N is just the standard reduction. By assumption we have for all \(p\in S_{\textrm{f}}\) that \(g\equiv \textrm{Id}_{D}\,\textrm{mod}\,p^{v_{p}(N)}\) and in particular \(g\in \Lambda (N)\). \(\square \)

1.4 B.3 Finiteness of class number and principal congruence subgroups

Similarly to what we did earlier, we will now regard \(\mathbb {Q}\) as a subfield of the ring \(\mathbb {A}=\mathbb {R}\times \mathbb {A}_{\textrm{f}}\) by diagonal embedding. Notice that this embedding differs from the composition of embeddings \(\mathbb {Q}\hookrightarrow \mathbb {A}_{\textrm{f}}\hookrightarrow \mathbb {A}\). Similarly, we can view \(\textbf{G}(\mathbb {Q})\) as a subgroup of \(\textbf{G}(\mathbb {A})\). It was proven by Borel and Harish-Chandra that \(\textbf{G}(\mathbb {Q})\) is a lattice in \(\textbf{G}(\mathbb {A})\).

Proposition B.8

Let

Then, \(\textbf{G}_{\infty }\) acts transitively on \(X_{\mathbb {A},1}\), i.e., \(\textbf{G}_{\infty }\backslash X_{\mathbb {A},1}\) is a singleton.

Proof

We first claim that \(\textbf{G}(\mathbb {A})=\Phi (\textrm{GL}_{d+1}(\mathbb {A}))\). To this end, let \(g\in \textbf{G}(\mathbb {A})\) and using Skolem–Noether choose \(x_{p}\in \textrm{GL}_{d+1}(\mathbb {Q}_{p})\), \(p\in \mathcal {V}_{\textrm{f}}\), such that \(g_{p}=\Phi (x_{p})\). By definition we have that \(g_{p}\in \textbf{G}(\mathbb {Z}_{p})\) for all but finitely many \(p\in \mathcal {V}_{\textrm{f}}\). Recall that \(g_{p}\in \textbf{G}(\mathbb {Z}_{p})\) implies that we can assume \(x_{p}\in \textrm{GL}_{d+1}(\mathbb {Z}_{p})\); cf. the proof of Lemma 3.1. It follows in particular that \(g=\Phi (x)\) for some \(x\in \textrm{GL}_{d+1}(\mathbb {A})\).

Let \(g\in \textbf{G}(\mathbb {A})\) arbitrary and choose \(x\in \textrm{GL}_{d+1}(\mathbb {A})\) such that \(g=\Phi (x)\). By [44, Prop. 8.1], we know that \(\textrm{GL}_{d+1}\) has class number one, i.e., \(x=k\gamma \), where \(k\in \textrm{GL}_{d+1}(\mathbb {R}\times {\widehat{\mathbb {Z}}})\) and \(\gamma \in \textrm{GL}_{d+1}(\mathbb {Q})\). In particular, we have \(g=\Phi (k)\Phi (\gamma )\). Note that \(\Phi (\gamma )\in \textrm{GL}_{D}(\mathbb {Q})\) by rationality of the representation \(\Phi \). Note that

$$\begin{aligned} k_{p}\textrm{Mat}_{d+1}(\mathbb {Z}_{p})k_{p}^{-1}=\textrm{Mat}_{d+1}(\mathbb {Z}_{p}) \end{aligned}$$

and hence \(\Phi (k_{p})\in \textrm{GL}_{D}(\mathbb {Z}_{p})\) for all \(p\in \mathcal {V}_{\textrm{f}}\), i.e., \(\Phi (k)\in \textrm{GL}_{D}(\mathbb {R}\times {\widehat{\mathbb {Z}}})\). Therefore

$$\begin{aligned} g\in \textbf{G}(\mathbb {R}\times {\widehat{\mathbb {Z}}})\textbf{G}(\mathbb {Q}). \end{aligned}$$

\(\square \)

Proposition B.9

Let \(N\in \mathbb {N}\). Then, the double quotient

is a finite union of \(\textbf{G}_{\infty }\)-orbits. Let \(x\in X_{\mathbb {A},N}\), then

as \(\textbf{G}_{\infty }\)-spaces, i.e., \(X_{\mathbb {A},N}\) is a disjoint union of finitely many copies of \(\textbf{G}_{\infty }/\Gamma (N)\).

Proof

Recall that \(K_{\textrm{f}}[1]=\textbf{G}({\widehat{\mathbb {Z}}})\) by definition. By Proposition B.8, we know that \(\textbf{G}_{\infty }\backslash X_{\mathbb {A},1}\) is a singleton. As \(K_{\textrm{f}}[N]\le K_{\textrm{f}}[1]\) is a finite index subgroup, the finiteness of \(\textbf{G}_{\infty }\backslash X_{\mathbb {A},N}\) follows immediately.

For the second part, using Proposition B.8, let \(\mathcal {R}_{D,N}\subseteq \textbf{G}({\widehat{\mathbb {Z}}})\) be a set of representatives for the double quotient \((\textbf{G}_{\infty }\times K_{\textrm{f}}[N])\backslash \textbf{G}(\mathbb {A})/\textbf{G}(\mathbb {Q})\). After possibly multiplying x by an element in \(\textbf{G}_{\infty }\), we can assume that \(x=K_{\textrm{f}}[N]\eta \textbf{G}(\mathbb {Q})\) for some \(\eta \in \mathcal {R}_{D,N}\). We will show that in this case

$$\begin{aligned} \textrm{Stab}_{\textbf{G}_{\infty }}(x)=\Gamma (N). \end{aligned}$$

Let \(g_{\infty }\in \textbf{G}_{\infty }\), then

$$\begin{aligned}&g_{\infty }\cdot K_{\textrm{f}}[N]\eta \textbf{G}(\mathbb {Q})=K_{\textrm{f}}[N]\eta \textbf{G}(\mathbb {Q})\\&\quad \iff \exists \gamma \in \textbf{G}(\mathbb {Q})\exists k\in K_{\textrm{f}}[N],\;(g_{\infty },\eta )=(\gamma ,k\eta \gamma )\\&\quad \iff g_{\infty }\in \textbf{G}(\mathbb {Q})\cap \eta ^{-1}K_{\textrm{f}}[N]\eta . \end{aligned}$$

Now note that \(K_{\textrm{f}}[N]\) is the kernel of the group homomorphism

$$\begin{aligned} \Psi :\textbf{G}({\widehat{\mathbb {Z}}})\rightarrow \prod _{p|N}\textbf{G}(\mathbb {Z}/p^{v_{p}(N)}\mathbb {Z}), \qquad \Psi ((k_{p})_{p\in \mathcal {V}_{\textrm{f}}})=(k_{p}\,\textrm{mod}\,p^{v_{p}(N)}\mathbb {Z})_{p|N}. \end{aligned}$$

In particular, \(K_{\textrm{f}}[N]\) is a normal subgroup and thus \(\eta ^{-1}K_{\textrm{f}}[N]\eta =K_{\textrm{f}}[N]\). Hence, if \(x\in X_{\mathbb {A},N}\) is arbitrary, letting \(\eta \in \mathcal {R}_{D,N}\) be such that \(x=K_{\textrm{f}}[N]\eta \textbf{G}(\mathbb {Q})\), Proposition B.7 implies that

$$\begin{aligned} g_{\infty }\in \textrm{Stab}_{\textbf{G}_{\infty }}(x)\iff g_{\infty }\in \textbf{G}(\mathbb {Q})\cap \eta ^{-1}K_{\textrm{f}}[N]\eta =\Gamma (N). \end{aligned}$$

\(\square \)

1.5 B.5 Correspondence in the S-arithmetic setup

We deduce analogous results to those obtained in the previous section for quotients of \(\textbf{G}_S\). In particular, the decomposition in (3.5) follows by Corollary B.11. First, we need the following.

Corollary B.10

Let \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\) finite and let \(S=\{\infty \}\cup S_{\textrm{f}}\). Then,

$$\begin{aligned} \textbf{G}(\mathbb {R}\times \mathbb {Z}_{S_{\textrm{f}}})\Gamma _{S}=\textbf{G}(\mathbb {Q}_{S}). \end{aligned}$$

Proof

Let

$$\begin{aligned} M_{S_{\textrm{f}}}=\prod _{p\in \mathcal {V}_{\textrm{f}}\setminus S_{\textrm{f}}}\textbf{G}(\mathbb {Z}_{p}). \end{aligned}$$

We first claim that

as \(\textbf{G}(\mathbb {Q}_{S})\)-spaces. To this end, we note that \(\textbf{G}(\mathbb {Q}_{S})\) acts transitively on the left hand side by Proposition B.8. Denote by \(x_{0}\) the identity coset in the left-hand side double quotient. In particular, it remains to show that

$$\begin{aligned} \textrm{Stab}_{\textbf{G}(\mathbb {Q}_S)}(x_{0})=\Gamma _{S}. \end{aligned}$$

Let e denote the identity in \(\prod _{p\not \in S}\textbf{G}(\mathbb {Z}_p)\) and \(g\in \textrm{Stab}_{\textbf{G}(\mathbb {Q}_S)}(x_{0})\). Arguing as in the proof of Proposition B.7, we get that \(g\in M_{S_{\textrm{f}}}\cap \textbf{G}(\mathbb {Q})=\Gamma _{S}\) as desired.

\(\square \)

The following is the analogue of Proposition B.9 and implies (3.4).

Corollary B.11

Let \(N\in \mathbb {N}\) and \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\) such that \(v_{p}(N)\ne 0\implies p\in S_{\textrm{f}}\). Let \(\mathcal {R}_{D,N}\) be as in the proof of Proposition B.9. Then, the projection \(\mathcal {R}_{D,N,S_{\textrm{f}}}\) of \(\mathcal {R}_{D,N}\) to \(K_{S_{\textrm{f}}}[1]\) is a set of representatives of the \(\textbf{G}_{\infty }\)-orbits in

Moreover, the map \(\mathcal {R}_{D,N}\rightarrow \mathcal {R}_{D,N,S_{\textrm{f}}}\) is a bijection and \(X_{\mathbb {Q}_{S},N}\) is a disjoint union of \(|\mathcal {R}_{D,N}|\)-many copies of \(\textbf{G}_{\infty }/\Gamma (N)\).

Proof

Note that \(K_{\textrm{f}}[N]=K_{S_{\textrm{f}}}[N]\times M_{S_{\textrm{f}}}\). Therefore, as in the proof of Corollary B.10, we obtain

Looking at these isomorphisms more explicitly, it is easy to check that \(\mathcal {R}_{D,N,S_{\textrm{f}}}\) is a set of representatives which is in one-to-one correspondence with \(\mathcal {R}_{D,N}\). We leave the rest of the proofs to the reader. \(\square \)

1.6 B.6 Counting connected components

In this section, we aim at finding a uniform bound on the number of connected components of

independently of N, where we assume that \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\) is finite, \(S=S_{\textrm{f}}\cup \{\infty \}\), and \(N\in \mathcal {I}_{S_{\textrm{f}}}\). The main result is Proposition B.13.

We first need a lemma about \(\textbf{G}_{S}^{+}\)-orbits. Given \(p\in \mathcal {V}_{\textrm{f}}\) and \(n\in \mathbb {N}\), let \(S_{p}[p^{n}]\le \textrm{SL}_{d+1}(\mathbb {Z}_{p})\) be the kernel of the homomorphism \(\textrm{SL}_{d+1}(\mathbb {Z}_{p})\rightarrow \textrm{SL}_{d+1}(\mathbb {Z}/p^{n}\mathbb {Z})\) given by reduction mod \(p^{n}\). For \(n=0\), we let \(S_{p}[p^{n}]=\textrm{SL}_{d+1}(\mathbb {Z}_{p})\). Recall the representation \(\Phi :\textrm{GL}_{d+1}\rightarrow \textbf{G}\) defined in (12.46). A calculation shows that \(\Phi (S_{p}[p^{n}])\subseteq K_{p}[p^{n}]\). In what follows, we let \(K_{p}[p^{n}]^{+}=\Phi (S_{p}[p^{n}])\) and, given \(N\in \mathcal {I}_{S_{\textrm{f}}}\), we define

$$\begin{aligned} K_{S_{\textrm{f}}}[N]^{+}=\prod _{p\in S_{\textrm{f}}}K_{p}[p^{v_{p}(N)}]^{+}\subseteq K_{S_{\textrm{f}}}[N]. \end{aligned}$$

In fact, \(K_{S_{\textrm{f}}}[N]^{+}\) is normal in \(\textbf{G}(\mathbb {Z}_{S_{\textrm{f}}})\). To this end, one checks that \(S_{p}[p^{n}]\lhd \textrm{GL}_{d+1}(\mathbb {Z}_{p})\) is a normal subgroup and then uses that

$$\begin{aligned} \textbf{G}(\mathbb {Z}_{p})=\Phi (\textrm{GL}_{d+1}(\mathbb {Z}_{p})) \end{aligned}$$

as already argued in the proof of Lemma 3.1.

Similarly, we note that \(\Phi (\textrm{SL}_{d+1}(\mathbb {Z}[S_{\textrm{f}}^{-1}]))\subseteq \Gamma _{S}\), and we will denote

$$\begin{aligned} \Gamma _{S}^{+}=\Phi (\textrm{SL}_{d+1}(\mathbb {Z}[S_{\textrm{f}}^{-1}])). \end{aligned}$$

As \(\textrm{SL}_{d+1}\) has the strong approximation property, we find

$$\begin{aligned} \textbf{G}_{S}^{+}=(\textbf{G}_{\infty }^{+}\times K_{S_{\textrm{f}}}[N]^{+})\Gamma _{S}^{+} \end{aligned}$$
(12.47)

for all \(N\in \mathcal {I}_{S_{\textrm{f}}}\); cf. [44, Thm. 7.12].

Lemma B.12

Let \(S_{\textrm{f}}\subseteq \mathcal {V}_{\textrm{f}}\), \(S=S_{\textrm{f}}\cup \{\infty \}\), \(N\in \mathcal {I}_{S_{\textrm{f}}}\), and \(x\in \textbf{G}_{S}/\Gamma _{S}\). Then,

$$\begin{aligned} \textbf{G}_{S}^{+}.x=(\textbf{G}_{\infty }^{+}\times K_{S_{\textrm{f}}}[N]^{+}).x. \end{aligned}$$

Proof

This follows immediately from Corollary B.10 and (12.47). To this end let \(g\in \textbf{G}_{\infty }\times K_{S_{\textrm{f}}}[N]\) and \(\eta \in \textbf{G}(\mathbb {Z}_{S_{\textrm{f}}})\) such that \(x=g\eta \Gamma _{S}\). As \(K_{S_{\textrm{f}}}[N]^{+}\) is normal in \(\textbf{G}(\mathbb {Z}_{S_{\textrm{f}}})\), it follows that

$$\begin{aligned}{} & {} \textbf{G}_{S}^{+}.x=g\eta \textbf{G}_{S}^{+}\Gamma _{S}=g\eta (\textbf{G}_{\infty }^{+}\times K_{S_{\textrm{f}}}[N]^{+})\Gamma _{S}=(\textbf{G}_{\infty }^{+}\times K_{S_{\textrm{f}}}[N]^{+}).x. \end{aligned}$$
(12.48)

\(\square \)

In what follows, we let

and we note that \(\textbf{G}_{\textrm{char}}\) is a finite set whose cardinality is bounded by the index \([\textbf{G}_S:\textbf{G}_S^+]\).

Proposition B.13

We have

In particular, given a finite set \( S\subseteq \mathcal {V}\) containing \(\infty \), the number of connected components of \(X_{\mathbb {Q}_{S},N}\) is at most \([\textbf{G}_S:\textbf{G}_S^+]\).

Proof

For the first bijection, we recall from the proof of Lemma B.12 that \(K_{S_{\textrm{f}}}[N]^{+}\le K_{S_{\textrm{f}}}[N]\) and therefore (12.48) yields that for all \(g\in \textbf{G}_{S}\)

$$\begin{aligned} (\textbf{G}_{\infty }^{+}\times K_{S_{\textrm{f}}}[N])g\Gamma _{S}&=K_{S_{\textrm{f}}}[N](\textbf{G}_{\infty }^{+}\times K_{S_{\textrm{f}}}[N]^{+})g\Gamma _{S}\\&=K_{S_{\textrm{f}}}[N]\textbf{G}_{S}^{+}g\Gamma _{S}. \end{aligned}$$

The second bijection follows by definition of \(\textbf{G}_{\textrm{char},S}\). The last part of the statement follows by definition of \(\textbf{G}_{\textrm{char},S}\). \(\square \)

The following corollary follows immediately.

Corollary B.14

Let \(W\subseteq K_{S_{\textrm{f}}}\) be an open subgroup. Then the quotient \(W\backslash \textbf{G}_{S}/\Gamma _{S}\) is a union of at most \([\textbf{G}_{S}:\textbf{G}_{S}^{+}]\)-many \(\textbf{G}_{\infty }^{+}\)-orbits.

Appendix C. KAK-decomposition and norms

In this section we will introduce a function on \(\textbf{G}(\mathbb {Q}_{v})\) which measures the size of an element. These functions are used in Sect. 4.3 to define norm-balls on \(\textbf{G}(\mathbb {Q}_{S})\). The main input is the KAK-decomposition, which is well-known for \(v=\infty \). We only discuss the case where v is a finite place of \(\mathbb {Q}\). In what follows, p is a natural prime.

Lemma C.1

Let \(D\in \mathbb {N}\) and denote by \(\Vert \cdot \Vert _{p}:\textrm{Mat}_{D}(\mathbb {Q}_{p})\rightarrow [0,\infty )\) the norm defined for \(x=(x_{ij})\) by

$$\begin{aligned} \Vert x\Vert _{p}= \max \{|x_{ij}|_{p}:1\le i,j\le D\}. \end{aligned}$$

Then, \(\Vert \cdot \Vert _{p}\) is an operator norm. Moreover, for all \(k\in \textrm{GL}_{D}(\mathbb {Z}_{p})\) we have \(\Vert k\Vert _{p}=1\).

Proof

The fact that \(\Vert \cdot \Vert _{p}\) is an operator norm follows from the ultrametric property of the p-adic absolute value. We leave the details to the reader. For the second part we note that for any \(x\in \textrm{GL}_{D}(\mathbb {Z}_{p})\) we have \(|\det x|_p=1\) and \(\Vert x\Vert _{p}\le 1\). On the other hand we have \(|\det x|_{p}\le \Vert x\Vert _{p}^{D}\) by the ultrametric property. Hence \(\Vert k\Vert _{p}=1\) for all \(k\in \textrm{GL}_{D}(\mathbb {Z}_{p})\). \(\square \)

In the following discussion, Lemma C.1 is used in the form of the following corollary.

Corollary C.2

Let \(D\in \mathbb {N}\), \(x\in \textrm{Mat}_{D}(\mathbb {Q}_{p})\) and \(k_{1},k_{2}\in \textrm{GL}_{D}(\mathbb {Z}_{p})\). Then

$$\begin{aligned} \Vert k_{1}xk_{2}\Vert _{p}=\Vert x\Vert _{p}. \end{aligned}$$

Proof

Since operator norms are submultiplicative, we have

$$\begin{aligned} \Vert x\Vert _{p}\le \Vert k_{1}^{-1}\Vert _{p}\Vert k_{1}xk_{2}\Vert _{p}\Vert k_{2}^{-1}\Vert _{p}=\Vert k_{1}xk_{2}\Vert _{p}\le \Vert k_{1}\Vert _{p}\Vert x\Vert _{p}\Vert k_{2}\Vert _{p}=\Vert x\Vert _{p}. \end{aligned}$$

\(\square \)

Lemma C.3

Let \(x\in \textrm{GL}_{2}(\mathbb {Q}_{p})\). Then there are \(m_{1},m_{2}\in \textrm{GL}_{2}(\mathbb {Z}_{p})\) and \(n,k\in \mathbb {Z}\) such that

$$\begin{aligned} x=m_{1}\begin{pmatrix} p^{n} &{} 0 \\ 0 &{} p^{k} \end{pmatrix}m_{2}. \end{aligned}$$

Proof

Let \(x=({\begin{matrix}a &{} b \\ c &{} d\end{matrix}})\). Assume that \(\Vert x\Vert _{p}=|a|_{p}\) and write \(a=p^{m}u\), \(b=p^{n}v\) and \(c=p^{\ell }w\) with \(u,v,w\in \mathbb {Z}_{p}^{\times }\). Then

$$\begin{aligned} \begin{pmatrix} 1 &{} 0 \\ -p^{\ell -m}\frac{w}{u} &{} 1 \end{pmatrix} \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \begin{pmatrix} 1 &{} -p^{n-m}\frac{v}{u} \\ 0 &{} 1 \end{pmatrix}= \begin{pmatrix} a &{} 0 \\ 0 &{} d^{\prime } \end{pmatrix}. \end{aligned}$$

As we assumed that a was maximal, it follows that the two unipotent matrices lie in \(\textrm{GL}_{2}(\mathbb {Z}_{p})\). Denote them by \(m_{1},m_{2}\) respectively. The resulting diagonal matrix is of the form

$$\begin{aligned} \begin{pmatrix} a &{} 0 \\ 0 &{} d^{\prime } \end{pmatrix}= \begin{pmatrix} u &{} 0 \\ 0 &{} z \end{pmatrix} \begin{pmatrix} p^{m} &{} 0 \\ 0 &{} p^{k} \end{pmatrix} \end{aligned}$$

with \(u,z\in \mathbb {Z}_{p}^{\times }\), and thus we have shown that

$$\begin{aligned} x=m_{1}^{-1}\begin{pmatrix} u &{} 0 \\ 0 &{} z \end{pmatrix} \begin{pmatrix} p^{n} &{} 0 \\ 0 &{} p^{k} \end{pmatrix} m_{2}^{-1}. \end{aligned}$$

As \(m_{1}\textrm{diag}(u,z)\in \textrm{GL}_{2}(\mathbb {Z}_p)\), we obtain the claim under the assumption that \(\Vert x\Vert _{p}=|a|_{p}\). If \(\Vert x\Vert _{p}\ne |a|_{p}\), then we distinguish two cases. If \(\Vert x\Vert _{p}=|d|_{p}\), then we conjugate x by the matrix \(m=({\begin{matrix} 0 &{} 1 \\ 1 &{} 0 \end{matrix}})\), so that the maximal entry comes to lie in the top left corner. If \(\Vert x\Vert _{p}=|b|_{p}\) or \(\Vert x\Vert _{p}=|c|_{p}\), then it is a property of non-archimedian absolute values that for either \(x({\begin{matrix} 1 &{} 0 \\ 1 &{} 1 \end{matrix}})\) or \(({\begin{matrix} 1 &{} 1 \\ 0 &{} 1 \end{matrix}})x\) the top left entry will be maximal. These operations all follow from multiplying x with matrices in \(\textrm{GL}_{2}(\mathbb {Z}_{p})\) and thus the claim is proven. \(\square \)

In what follows, given \(x\in \textrm{GL}_{d+1}(\mathbb {Q}_{p})\), we denote by \([x]\in \textbf{G}(\mathbb {Q}_{p})\) its image under \(\Phi \) as introduced in the proof of Proposition B.8.

Proposition C.4

Let p be a finite rational prime and assume that \(g\in \textbf{G}(\mathbb {Q}_{p})\). Then there exist uniquely determined nonnegative integers \(n_{1}\ge \cdots \ge n_{d}\) as well as elements \(k_{1},k_{2}\in \textbf{G}(\mathbb {Z}_{p})\) such that

$$\begin{aligned} g=k_{1}[\textrm{diag}(p^{-n_{1}},\ldots ,p^{-n_{d}},1)]k_{2}. \end{aligned}$$

Proof

As \([\textrm{GL}_{d+1}(\mathbb {Z}_{p})]=\textbf{G}(\mathbb {Z}_{p})\), cf. the proof of Proposition B.8, it suffices to prove the existence of a decomposition of any element \(x\in \textrm{GL}_{d+1}(\mathbb {Q}_{p})\) into a product of the form \(k_{1}Dk_{2}\), where D is a diagonal matrix whose non-zero entries are powers of p for a set of exponents uniquely determined (with multiplicity) by x. If this is the case, we can use elements in \(\textrm{GL}_{d+1}(\mathbb {Z}_{p})\) to arrange the diagonal entries in decreasing order with respect to the p-adic absolute value. Furthermore, there will be an element \(\omega \) in the center of \(\textrm{GL}_{d+1}(\mathbb {Q}_{p})\) such that \(\omega ^{-1}D\) is of the form required by the proposition and we note that \([\omega ^{-1}D]=[D]\).

So let \(x\in \textrm{GL}_{d+1}(\mathbb {Q}_{p})\) be arbitrary and in view of Lemma C.3 assume that \(d\ge 2\). For the existence of a decomposition of x, we apply elements in \(\textrm{GL}_{d+1}(\mathbb {Z}_{p})\) so that \(\Vert x\Vert _{p}=|x_{11}|_{p}\). Then one can use the copies of \(\textrm{SL}_{2}(\mathbb {Z}_{p})\) in \(\textrm{GL}_{d+1}(\mathbb {Q}_{p})\) associated with spans of pairs of the standard basis to reduce the matrix x to a matrix of the form

$$\begin{aligned} x^{\prime }=\begin{pmatrix} x_{11} &{} 0 \\ 0 &{} y \end{pmatrix}, \end{aligned}$$

where \(y\in \textrm{GL}_{d}(\mathbb {Q}_{p})\) and \(\Vert y\Vert _{p}\le |x_{11}|_{p}\). Using \(\mathbb {Q}_{p}=\bigsqcup _{n\in \mathbb {Z}}p^{n}\mathbb {Z}_{p}^{\times }\) and multiplication by a diagonal matrix with diagonal entries contained in \(\mathbb {Z}_{p}^{\times }\), we can assume without loss of generality that \(x_{11}=\Vert g\Vert _{p}^{-1}\) in the expression for \(x^{\prime }\) obtained above. Now we proceed by induction on d. \(\square \)

Let us give a more intrinsic interpretation of Proposition C.4. In what follows, we consider

$$\begin{aligned} \mathfrak {pgl}_{d+1}(\mathbb {Q}_{p})=\{v\in \textrm{Mat}_{d+1}(\mathbb {Q}_{p}):\textrm{tr}(v)=0\}. \end{aligned}$$

Then \(\textrm{Ad}:\textbf{G}(\mathbb {Q}_{p})\rightarrow \textrm{GL}(\mathfrak {pgl}_{d+1}(\mathbb {Q}_{p}))\) given by

$$\begin{aligned} \textrm{Ad}_{g}(v)=gvg^{-1}\qquad (v\in \mathfrak {pgl}_{d+1}(\mathbb {Q}_{p})) \end{aligned}$$

is a well-defined, faithful representation. We let \(\Vert \cdot \Vert _{p}\) be the norm on \(\textrm{GL}(\mathfrak {pgl}_{d+1}(\mathbb {Q}_{p}))\) induced by the operator norm \(\Vert \cdot \Vert _{p}\) on \(\textrm{GL}(\textrm{Mat}_{d+1}(\mathbb {Q}_{p}))\) via restriction of the isomorphism to \(\mathfrak {pgl}_{d+1}(\mathbb {Q}_{p})\). To this end we note that any isomorphism of \(\mathfrak {pgl}_{d+1}(\mathbb {Q}_{p})\) extends trivially to the center of \(\textrm{Mat}_{d+1}(\mathbb {Q}_{p})\).

Lemma C.5

Let \(k_{1},k_{2}\in \textbf{G}(\mathbb {Z}_{p})\) and let \(n_{1}\ge \cdots \ge n_{d}\) nonnegative integers. Set

$$\begin{aligned} g=k_{1}[\textrm{diag}(p^{-n_{1}},\ldots ,p^{-n_{d}},1)]k_{2}. \end{aligned}$$

Then \(\Vert \textrm{Ad}_{g}\Vert _{p}=\Vert \textrm{Ad}_{g^{-1}}\Vert _{p}=p^{n_{1}}\).

Proof

By Proposition C.4 and Corollary C.2, we can assume without loss of generality that \(g=[\textrm{diag}(p^{-n_{1}},\ldots ,p^{-n_{d}},1)]\). We let \(n_{d+1}=0\). An elementary calculation shows that the operator \(\textrm{Ad}_{g}\) acting on \(\textrm{Mat}_{d+1}(\mathbb {Q}_{p})\) is diagonalizable with eigenvalues \(p^{n_i-n_j}\) for \(:1\le i,j\le d+1\). Therefore \(\textrm{Ad}_{g}\) has maximal eigenvalue \(p^{-n_{1}}\) and \(\Vert \textrm{Ad}_{g}\Vert _{p}=p^{n_{1}}\). As \(\sigma (\textrm{Ad}_{g})\) is symmetric under multiplicative inversion, we also have \(\Vert \textrm{Ad}_{g^{-1}}\Vert _{p}=p^{n_{1}}\). \(\square \)

Appendix D. Modular character on the Borel subgroup

The goal of this section is to prove (4.28). Recall first that the p-adic value satisfies that for all \(x\in \mathbb {Q}_{p}\) and for all \(f\in \textrm{C}_{c}(\mathbb {Q}_{p})\)

$$\begin{aligned} |x|_{p}\int _{\mathbb {Q}_{p}}f(xt) \textrm{d} m_{\mathbb {Q}_{p}}(t)=\int _{\mathbb {Q}_{p}}f(t) \textrm{d} m_{\mathbb {Q}_{p}}(t), \end{aligned}$$

where \(m_{\mathbb {Q}_{p}}\) denotes any choice of a Haar measure on \(\mathbb {Q}_{p}\). This implies that up to normalization the Haar measure on \(\mathbb {Q}_{p}^{\times }\) is given by

$$\begin{aligned} \int _{\mathbb {Q}_{p}^{\times }}f(y) \textrm{d} m_{\mathbb {Q}_{p}^{\times }}(y)=\int _{\mathbb {Q}_{p}^{\times }}\frac{f(y)}{|y|_{p}} \textrm{d} m_{\mathbb {Q}_{p}}(y)\quad (f\in \textrm{C}_{c}(\mathbb {Q}_{p}^{\times })). \end{aligned}$$

We let \(B_{p}=\textbf{B}(\mathbb {Q}_{p})\) and note that \(B_{p}=A_{p}U_{p}\), where \(A_{p}\) denotes the image of the diagonal subgroup of \(\textrm{GL}_{d+1}(\mathbb {Q}_{p})\) and \(U_{p}\) is the (injective) image of the subgroup of upper triangular unipotent matrices. Note that \(U_{p}\) is homeomorphic to \(\mathbb {Q}_{p}^{d^{\prime }}\) with \(d^{\prime }=\frac{1}{2}d(d+1)\) and that the push forward \(m_{U_{p}}\) of the Haar measure on \(\mathbb {Q}_{p}^{d^{\prime }}\) to \(U_{p}\) defines a Haar measure on \(U_{p}\). Therefore one finds that for a matrix \(b=au\in B_{p}\) with \(u\in U_{p}\) and \(a=\textrm{diag}(a_{1},\ldots ,a_{d},1)\) we have

$$\begin{aligned} \textrm{d} m_{B_{p}}^{\textrm{left}}(b)&\propto \textrm{d} m_{U_{p}}(u)\prod _{i=1}^{d}\frac{ \textrm{d} m_{\mathbb {Q}_{p}}(a_{i})}{|a_{i}|_{p}^{d+2-i}},&\textrm{d} m_{B_{p}}^{\textrm{right}}(b)&\propto \textrm{d} m_{U_{p}}(u)\prod _{i=1}^{d}\frac{ \textrm{d} m_{\mathbb {Q}_{p}}(a_{i})}{|a_{i}|^{i}}. \end{aligned}$$

As the modular function satisfies \( \textrm{d} m_{B_{p}}^{\textrm{right}}(b)\propto \delta _{\textbf{B}}(b) \textrm{d} m_{B_{p}}^{\textrm{left}}(b) \), we find

$$\begin{aligned} \delta _{\textbf{B}}(b)=\prod _{i=1}^{d}|a_{i}|_{p}^{d+2-2i}. \end{aligned}$$

Equation (4.28) now follows by plugging in \(a_{{\textbf{n}}}\) for b.

Appendix E. Integrability of Matrix Coefficients

In this section, we show that \(\eta _{S}\in \textrm{L}^{v(d)+\varepsilon }(\textbf{G}_{S})\). Given \(v\in S\), let \(\eta _{v}:\textbf{G}(\mathbb {Q}_{v})\rightarrow (0,\infty )\) denote

$$\begin{aligned} \eta _{v}(g_{v})=\Vert g_{v}\Vert ^{-1}_{v} \quad (g_{v}\in \textbf{G}(\mathbb {Q}_{v})). \end{aligned}$$

As \(\eta _{\textbf{G}}\) is the product of the various \(\eta _{v}\), \(v\in S\), and as the Haar measure on \(\textbf{G}_{S}\) is the product measure, it suffices to show that \(\eta _{v}\in \textrm{L}^{v(d)+\varepsilon }(\textbf{G}_{v})\) for all places v of \(\mathbb {Q}\).

1.1 E.1 Integrability in the Archimedean place

We recall that the Haar measure on \(\textbf{G}_{\infty }\) is explicitly given by the formula

$$\begin{aligned} \int _{\textbf{G}_{\infty }}f(g) \textrm{d} g= & {} \int _{K}\int _{A_{\infty }^{+}}\int _{K}f(k_{1}ak_{2})\prod _{\alpha \in \Sigma _{\infty }^{+}}(\sinh \alpha (\log a))^{m_{\alpha }} \textrm{d} k_{1} \textrm{d} a \textrm{d} k_{2}\quad \\{} & {} \qquad (f\in \textrm{C}_{c}(\textbf{G}_{\infty })), \end{aligned}$$

where \(m_{\alpha }\) denotes the multiplicity of the positive root \(\alpha \in \Sigma _{\infty }^{+}\) and where the Haar measure on A is the push-forward of the Lebesgue measure on the Cartan subalgebra under the exponential map; cf. [37, Prop. 5.28]. Note that in our situation \(m_{\alpha }=1\) for all \(\alpha \in \Sigma _{\infty }^{+}\). By definition of the hyperbolic sine we have

$$\begin{aligned} \prod _{\alpha \in \Sigma _{\infty }^{+}}\sinh \alpha (\log a)\le e^{\beta (\log a)}\quad (a\in A_{\infty }^{+}), \end{aligned}$$

where \(\beta =\sum _{\alpha \in \Sigma _{\infty }^{+}}\alpha \). We recall that for any \(\alpha \in \Sigma _{\infty }^{+}\) there are \(1\le i<j\le d+1\) such that for all \(a=\textrm{diag}(a_{1},\ldots ,a_{d+1})\in A_{\infty }^{+}\), we have \( \alpha (\log a)=\log a_{i}-\log a_{j}\). Hence, we get

$$\begin{aligned} \beta (\log a)=\sum _{i=1}^{d+1}(d+2-2i)\log a_{i}. \end{aligned}$$

As argued in the proof of Lemma 4.7 and recalling that we parametrize \(A_{\infty }^{+}\) such that \(a_{d+1}=1\), we thus find that \(\beta (\log a)\le v(d)\log a_{1}\). Using the lower bound in (4.24), it follows that \( \eta _{\infty }(k_{1}ak_{2})\le a_{1}^{-1}\) and thus

$$\begin{aligned} \int _{\textbf{G}_{\infty }}\eta _{\infty }(g)^{q} \textrm{d} g&=\int _{K}\int _{A_{\infty }^{+}}\int _{K}\eta _{\infty }(k_{1}ak_{2})^{q}\prod _{\alpha \in \Sigma _{\infty }^{+}}\sinh \alpha (\log a) \textrm{d} k_{1} \textrm{d} a \textrm{d} k_{2}\\&\ll _{\vartheta }\int _{A_{\infty }^{+}}a_{1}^{-q+v(d)} \textrm{d} a. \end{aligned}$$

By definition of \(A_{\infty }^{+}\) and recalling that the Haar measure on the connected component of the diagonal subgroup of \(\textbf{G}_{\infty }\) is the push-forward of the Lebesgue measure on the Lie algebra

$$\begin{aligned} {\mathfrak {a}}\cong \{(t_{1},\ldots ,t_{d},t_{d+1}):t_{1}+\cdots +t_{d+1}=0\} \end{aligned}$$

under the exponential map, we have for any \(s>0\) that

$$\begin{aligned} \int _{A_{\infty }^{+}}a_{1}^{-s} \textrm{d} a=\int _{0}^{\infty }\int _{t_{d}}^{\infty }\cdots \int _{t_{2}}^{\infty }e^{-st_{1}} \textrm{d} t_{1}\cdots \textrm{d} t_{d}=\frac{1}{s^{d}}. \end{aligned}$$

Hence, whenever \(q>v(d)\), then

$$\begin{aligned} \int _{\textbf{G}_{\infty }}\eta _{\infty }(g)^{q} \textrm{d} g<\infty . \end{aligned}$$

1.2 E.2 Integrability in the finite places

In what follows, we note that

$$\begin{aligned} \textbf{G}(\mathbb {Q}_{p})=\bigsqcup _{n\in \mathbb {N}_{0}}S_{n}, \qquad S_{n}=\{g\in \textbf{G}(\mathbb {Q}_{p}):\Vert g\Vert _{p}=p^{n}\}. \end{aligned}$$

Moreover, Lemma 4.7 yields that for all \(n\in \mathbb {N}_{0}\) we have

$$\begin{aligned} \int _{\textbf{G}(\mathbb {Q}_{p})}\eta _{p}(g)^{q} \textrm{d} g&=\sum _{n\in \mathbb {N}_{0}}\int _{S_{n}}\eta _{p}(g)^{q} \textrm{d} g \ll _{d} \sum _{n\in \mathbb {N}_{0}} p^{-nq}\textrm{Vol}( {S_{n}} )\\&\ll _{\varepsilon ,d} \sum _{n\in \mathbb {N}_{0}}p^{n(-q+v(d)+\varepsilon )}. \end{aligned}$$

It follows that \(\eta _{p}\in \textrm{L}^{v(d)+\varepsilon }(\textbf{G}(\mathbb {Q}_{p}))\) whenever \(q>v(d)\).

1.3 E.3 Integrability of matrix coefficients

This argument was mentioned in the proof of Proposition 5.3 and it follows readily from Sect. 1. Recall from Proposition 4.5 that

$$\begin{aligned} \xi _{\infty }(g)^{2/(1-2\varepsilon )}\ll _{\varepsilon ,d}\eta _{\infty }(g)\quad (g\in \textbf{G}_{\infty }). \end{aligned}$$

Hence for all \(0<\varepsilon <\frac{1}{2}\) and for \(q>\frac{2v(d)}{1-2\varepsilon }\) we find

$$\begin{aligned} \int _{\textbf{G}_{\infty }}\xi _{\infty }(g)^{q} \textrm{d} g\ll _{\varepsilon ,d}\int _{\textbf{G}_{\infty }}\eta _{\infty }(g)^{(\frac{1}{2}-\varepsilon )q} \textrm{d} g<\infty \end{aligned}$$

as \((\frac{1}{2}-\varepsilon )q>v(d)\) and \(\eta _{\infty }\in \textrm{L}^{v(d)+\varepsilon }(\textbf{G}_{\infty })\).

Remark E.1

The same argument works for finite places.

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Khalil, O., Luethi, M. Random walks, spectral gaps, and Khintchine’s theorem on fractals. Invent. math. 232, 713–831 (2023). https://doi.org/10.1007/s00222-022-01171-4

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