HIGHER MOMENT FORMULAE AND LIMITING DISTRIBUTIONS OF LATTICE POINTS

Abstract

We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of $\mathrm {SL}_d({\mathbb {R}})$. As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.

1 Introduction

Let $X_d$ denote the space of unimodular lattices in ${\mathbb {R}}^d$ which can be naturally identified with $\mathrm {SL}_d({\mathbb {Z}})\backslash \mathrm {SL}_d({\mathbb {R}})$ and denote by $\mu $ the Haar measure on $X_d$ normalised to be a probability measure. Let $f:{\mathbb {R}}^d \rightarrow {\mathbb {R}}$ be a bounded function of compact support. The Siegel transform ${\mathcal S}_{1}(f)$ of f is defined by

$$\begin{align*}{\mathcal S}_{1}(f)(\Lambda)=\sum_{{\mathbf{m}} \in \Lambda} f({\mathbf{m}}),\; \Lambda\in \mathrm{SL}_d({\mathbb{Z}})\backslash \mathrm{SL}_d({\mathbb{R}}). \end{align*}$$

In [20], Siegel proved that

$$ \begin{align*} \int_{X_d}{\mathcal S}_{1}(f)d\mu = \int_{{\mathbb{R}}^d}f(x)dx+f({\mathbf{0}}). \end{align*} $$

This result, often referred to as Siegel’s mean value formula, is a fundamental result in the geometry of numbers and has proved to be indispensable in homogeneous dynamics, especially in applications to Diophantine problems. Following Siegel’s result, Rogers [13] established intricate formulae for the higher moments of Siegel transforms (see Theorem 2.2 in Section 2). These formulae have since become an important tool in a wide variety of Diophantine problems. It is of considerable interest to prove analogues of Siegel’s and Rogers’ formulae for other homogeneous spaces. In this paper, we will establish explicit higher moment formulae for analogues of the Siegel transform on the following two homogeneous spaces, which are equipped with natural invariant probability measures $\mu _Y$ and $\mu _{q}$ on Y and $Y_{{\mathbf {p}}/q}$ , respectively (see Section 2).

• • The space $Y := \mathrm {ASL}_d({\mathbb {Z}})\backslash \mathrm {ASL}_d({\mathbb {R}})$ .

• • The space $Y_{{{\mathbf {p}}}/q} := \left \{\left ({\mathbb {Z}}^d+\dfrac {{\mathbf {p}}} q\right )g : g\in \mathrm {SL}_d({\mathbb {R}}) \right \}$ , where ${\mathbf {p}}\in {\mathbb {Z}}^d \smallsetminus \{{\mathbf {0}}\}$ and $q\in {\mathbb {N}}_{\ge 2}$ with $\gcd ({\mathbf {p}}, q)=1$ .

There have been many developments since Rogers’ work; among those pertinent to the present paper is the recent work [8] of the third named author where S-arithmetic versions of Rogers’ theorems are established. Analogues of Siegel transforms for Y and $Y_{{{\mathbf {p}}}/q}$ have been considered, and, in fact, a second moment formula has been obtained in each case – in the affine case, by El-Baz, Marklof and Vinogradov [4], where they were used to study the distribution of gaps between lattice directions (see also [2]), and in the congruence case, by Ghosh, Kelmer and Yu [6], where they were used to study effective versions of an inhomogeneous version of Oppenheim’s conjecture on quadratic forms. In fact, they have other applications as well. We refer the reader to [1] for an application of the congruence second moment formula to Diophantine approximation and to [5] for an S-arithmetic version of the congruence second moment formula with applications to quadratic forms.

The main results in the present paper are formulas computing all the higher moments of Siegel transforms for both the affine and congruence cases. We also obtain analogues of a modification to Rogers’ formula, due to Strömbergsson and Södergren [22]. Our proof of the higher moment formulae owes a lot to the breakthrough work of Marklof and Strömbergsson [11]. As will become clear, we make significant use of the ideas in Section $7$ of their paper. Our formulas are explicit but, as is the case with Rogers’ original formula, are heavy on notation and need some buildup to state. We therefore postpone stating them to the next section. The reader will find the higher moment formula for Siegel transforms on Y in Theorem 2.12, and the formula for Siegel transforms on $Y_{{{\mathbf {p}}}/q}$ in Theorem 2.13. If history is a reliable guide, then our higher moment formulae will find good uses in counting problems. In the present paper, we provide applications to limiting distributions in lattice point counting problems. We devote the remainder of the introduction to discussing these applications.

1.1 Applications to counting results

Our counting results are inspired by the work of Strömbergsson and Södergren [22]. Given $d \geq 2$ , a lattice $L \in X_d$ and a real number $x \geq 0$ , set

$$ \begin{align*}N_{d, L}(x):= \#\left\{m \in L \backslash \{0\}~:~|m| \leq \left(\frac{x}{V_d}\right)^{1/d}\right\},\end{align*} $$

where $V_d$ denotes the volume of the unit ball in ${\mathbb {R}}^d$ . Further, let

$$ \begin{align*}R_{d, L}(x) := N_{d, L}(x) - x\end{align*} $$

be the error term in the Gauss circle problem. Strömbergsson and Södergren proved several interesting results regarding the behaviour of $R_{n, L}$ , including the following central limit theorem for a random lattice L.

Theorem (Strömbergsson and Södergren [22])

Let $\phi : {\mathbb {Z}}_{+} \to {\mathbb {R}}_{+}$ be any function satisfying $\lim _{n \to \infty } \phi (d) = \infty $ and $\phi (d) = O_{\varepsilon }(e^{\varepsilon d})$ for every $\varepsilon> 0$ . Let $Z^{(B)}_d$ be the random variable

$$\begin{align*}Z^{(B)}_d := \frac{1}{\sqrt{2\phi(d)}}R_{d, L}(\phi(d)) \end{align*}$$

with L picked at random in $(X_d, \mu )$ . Then

$$\begin{align*}Z^{(B)}_d \rightarrow {\mathcal{N}}(0,1) \text{ as } d \to \infty \end{align*}$$

in distribution.

Earlier, Södegren [23] studied the distribution of lengths of lattice vectors in a random lattice of large dimension. Strömbergsson and Södergren used the central limit theorem above in conjunction with Södegren’s theorem to establish the following theorem indicating Poissonian behaviour for sequences growing sub-exponentially with respect to the dimension.

Theorem (Strömbergsson and Södergren [22])

Let $\phi : {\mathbb {Z}}_{+} \to {\mathbb {R}}_{+}$ be any function satisfying $\lim _{n \to \infty } \phi (d) = \infty $ and $\phi (d) = O_{\varepsilon }(e^{\varepsilon d})$ for every $\varepsilon> 0$ . Let $\mathcal {N}(x)$ be a Poisson distributed random variable with expectation $x/2$ . Then

$$ \begin{align*}\operatorname{\mathrm{Prob}}_{\mu}(N_{d, L}(x) \leq 2N) - \operatorname{\mathrm{Prob}}(\mathcal{N}(x) \leq N) \to 0 \text{ as } d \to \infty,\end{align*} $$

uniformly with respect to all $N, x \geq 0$ , satisfying $\min (x,N) \leq \phi (d)$ .

More generally, they considered the case of several pairwise disjoint subsets and studied the joint distribution of the normalised counting variables and obtained a functional central limit theorem.

In this paper, we are concerned with two natural variations on this theme. Namely, we will consider the lattice point counting problem where the lattice is chosen at random from the spaces $(Y,\ \mu _Y)$ and $(Y_{{{\mathbf {p}}}/q},\ \mu _q)$ .

We refer to these as the affine lattice point counting problem and the congruence lattice point counting problem, respectively. We prove analogues of the results of Strömbergsson and Södergren in the affine and congruence setting, and also analogues of results of Rogers [16], Schmidt [18] and Södergren [23] on Poissonian behaviour of lengths of lattice vectors in a randomly chosen lattice; see also related work of Kim [9]. The main tool in [22] is a version of Rogers’ formula; in fact, one needs all moments, not just the second moment. In an analogous fashion, Theorems 2.12 and 2.13 will play a starring role in the proofs of the results stated below.

1.2 Counting results

Our first two results are analogues of Södergren’s results [23] in the affine and congruence setting, respectively. For each $d \geq 2$ , let $\mathcal {S}={\mathcal S}_d = \{S_t: t \geq 0\}$ be an increasing family of subsets of $\mathbb {R}^{d}$ with $\operatorname {\mathrm {vol}}(S_t) = t$ , and for $\Lambda \in Y=\mathrm {ASL}_d({\mathbb {Z}})\setminus \mathrm {ASL}_d({\mathbb {R}})$ , set

$$\begin{align*}N_t(\Lambda) := \#{\left(S_t \cap \Lambda\right)}. \end{align*}$$

Denote by $\{N^{\lambda }(t) : t \geq 0\}$ a Poisson process on the non-negative real line with intensity  $\lambda $ .

Theorem 1.1. The stochastic process $\{N_t(\Lambda ) : t \geq 0\}$ converges weakly to $\{N^1(t) : t \geq 0\}$ as d goes to infinity.

Let $q\in {\mathbb {N}}_{\ge 2}$ be given. For each $d \geq 2$ , consider $\mathcal {S} ={\mathcal S}_d= \{S_t: t> 0\}$ , an increasing family of subsets of $\mathbb {R}^{d}$ and $\mathbf {p}/q \in \mathbb {Q}^{d}$ for some ${\mathbf {p}}={\mathbf {p}}_d\in {\mathbb {Z}}^d$ coprime with q. By abuse of notation, set

$$\begin{align*}N_t(\Lambda) = \#(S_t \cap \Lambda), \end{align*}$$

for $\Lambda \in (Y_{\mathbf {p}/q},\ \mu _q)$ .

Theorem 1.2.

1. (i) For $q \geq 3$ , the stochastic process $\{N_t(\Lambda ) : t> 0\}$ converges weakly to $\{N^1(t) : t> 0\}$ as d goes to infinity.

2. (ii) For $q = 2$ , assume that $S_t$ ’s are symmetric about origin, and let $\widetilde {N}_t = \frac {1}{2} N_t$ . Then the stochastic process $\left \{\widetilde {N}_t(\Lambda ) : t> 0\right \}$ converges weakly to $\{N^{1/2}(t) : t> 0\}$ as d goes to infinity.

Next, we establish a central limit theorem for the normalised error term in the lattice point problem for a random affine lattice.

Theorem 1.3. Let $\phi :{\mathbb {N}}\rightarrow {\mathbb {R}}_{>0}$ be a function for which

(1.1) $$ \begin{align} \lim_{d\rightarrow \infty} \phi(d)=\infty \quad\text{and}\quad \phi(d)=O_{\varepsilon}(e^{\varepsilon d}),\; \forall \varepsilon>0. \end{align} $$

Consider a sequence $\{S_d\}_{d\in {\mathbb {N}}}$ of Borel sets $S_d \subseteq {\mathbb {R}}^d$ such that $\operatorname {\mathrm {vol}}(S_d)=\phi (d)$ . Let

$$\begin{align*}Z^1_d=\frac {\#\left(\Lambda\cap S_d\right)- \phi(d)} {\sqrt {\phi(d)}} \end{align*}$$

be the random variable with $\Lambda \in (Y, \mu _Y)$ . Then

$$\begin{align*}Z^1_d \rightarrow {\mathcal{N}}(0,1)\text{ as } d\rightarrow \infty \end{align*}$$

in distribution.

We now turn to the space $Y_{{{\mathbf {p}}}/q}$ which can be viewed as a finite volume homogeneous space of $\mathrm {SL}_{d}({\mathbb {R}})$ (see Section 2.2) and therefore inherits a natural finite Haar measure $\mu _q$ .

Theorem 1.4. Let a function $\phi :{\mathbb {N}}\rightarrow {\mathbb {R}}_{>0}$ and a sequence $\{S_d\}$ of Borel sets be given as in Theorem 1.3. When $q=2$ , we further assume that each $S_d$ is symmetric with respect to the origin. Let

$$\begin{align*}Z^{{\mathbf{p}}/q}_d=\left\{\begin{array}{cl} \dfrac {\#\left(\Lambda\cap S_d\right)- \phi(d)} {\sqrt {2\phi(d)}}, &\text{if } q=2;\\[0.2in] \dfrac {\#\left(\Lambda\cap S_d\right)- \phi(d)} {\sqrt {\phi(d)}}, &\text{otherwise} \end{array} \right. \end{align*}$$

be a random variable associated with $\Lambda \in (Y_{{\mathbf {p}}/q}, \mu _q)$ . Then

$$\begin{align*}Z^{{\mathbf{p}}/q}_d \rightarrow {\mathcal{N}}(0,1)\text{ as } d\rightarrow \infty \end{align*}$$

in distribution.

The next two theorems are functional central limit theorems in the affine and congruence case respectively.

Theorem 1.5. Let a function $\phi :{\mathbb {N}}\rightarrow {\mathbb {R}}_{>0}$ be given as in Theorem 1.3. Consider a sequence $\{S_d\}_{d\in {\mathbb {N}}}$ of star-shaped Borel sets $S_d\subseteq {\mathbb {R}}^d$ centered at the origin such that $\operatorname {\mathrm {vol}}(S_d)=\phi (d)$ . Let us define the random function

$$\begin{align*}t\in [0,1] \mapsto Z^1_d(t):= \frac {\#\left(\Lambda \cap t^{1/d}S_d\right) - t\phi(d)}{\sqrt{\phi(d)}}, \end{align*}$$

where $\Lambda $ is a random affine lattice in $(Y, \mu _Y)$ . Here, $tS=\{t\mathbf {v}\in {\mathbb {R}}^d : \mathbf {v}\in S\}$ for any $t\in {\mathbb {R}}_{\ge 0}$ and $S\subseteq {\mathbb {R}}^d$ . Then $Z^1_d(t)$ converges in distribution to one-dimensional Brownian motion as d goes to infinity.

Theorem 1.6. Let a function $\phi :{\mathbb {N}}\rightarrow {\mathbb {R}}_{>0}$ and a sequence $\{S_d\}_{d\in {\mathbb {N}}}$ of Borel sets be as in Theorem 1.5. When $q=2$ , we further assume that each $S_d$ is symmetric with respect to the origin. Define the random function

$$\begin{align*}t\in [0,1] \mapsto Z^{{\mathbf{p}}/q}_d(t):= \left\{\begin{array}{cl} \dfrac{\#\left(\Lambda \cap t^{1/d}S_d\right) - t\phi(d)}{\sqrt{2\phi(d)}}, &\text{if } q=2;\\[0.2in] \dfrac{\#\left(\Lambda \cap t^{1/d}S_d\right) - t\phi(d)}{\sqrt{\phi(d)}}, &\text{otherwise.}\end{array} \right. \end{align*}$$

Then $Z^{{\mathbf {p}}/q}_d(t)$ converges in distribution to one-dimensional Brownian motion.

Structure of the paper

In Section 2, we state and prove the moment formulae for the affine and congruence cases. In fact, we provide two approaches, one kindly suggested to us by the referee. Section 3 is devoted to the study of Poissonian behaviour. In particular, analogues of results of Södergren [23] and Rogers [14, 15] in the affine and congruence setting are established. These results might be of independent interest. Section 4 contains affine and congruence versions of the variation on Rogers’ formula developed by Strömbergsson and Södergren. Finally, Section 5 is devoted to the proofs of the counting results.

2 Higher Moment Formulae

We define

$$\begin{align*}\mathrm{ASL}_d({\mathbb{R}}) := \left\{\left(\begin{array}{cc} g & 0 \\ \xi & 1 \\ \end{array}\right) : g\in \mathrm{SL}_d({\mathbb{R}}),\; \xi \in {\mathbb{R}}^d \right\} \end{align*}$$

and denote by $(\xi ,g)$ an element of $\mathrm {ASL}_d({\mathbb {R}})$ . One can identify the space of affine unimodular lattices with

$$\begin{align*}Y_d = Y =\mathrm{ASL}_d({\mathbb{Z}})\backslash \mathrm{ASL}_d({\mathbb{R}})\end{align*}$$

via the map

$$ \begin{align*}\mathrm{ASL}_d({\mathbb{Z}})(\xi,g) \mapsto {\mathbb{Z}}^d g+\xi.\end{align*} $$

We denote by $\mu _Y$ the Haar measure on $\mathrm {ASL}_d(\mathbb {R})$ normalised so that

$$ \begin{align*} \mu_Y(\mathrm{ASL}_d(\mathbb{Z}) \backslash \mathrm{ASL}_d(\mathbb{R})) = 1. \end{align*} $$

Let $F:({\mathbb {R}}^d)^k \rightarrow {\mathbb {R}}$ be a bounded function of compact support. Define the transform ${\mathcal S}_{k}(F)$ of F by

$$\begin{align*}{\mathcal S}_{k}(F)(\Lambda)=\sum_{\scriptsize \begin{array}{c} {\mathbf{m}}_i \in \Lambda\\ 1\le i \le k\end{array}} F({\mathbf{m}}_1, \ldots, {\mathbf{m}}_k),\; \Lambda\in \mathrm{ASL}_d({\mathbb{Z}})\backslash \mathrm{ASL}_d({\mathbb{R}}). \end{align*}$$

By a mild abuse of notation, we will use ${\mathcal S}_{k}(F)$ to also denote the function induced by the natural inclusion

$$ \begin{align*}\mathrm{SL}_d({\mathbb{Z}})\backslash\mathrm{SL}_d({\mathbb{R}})\hookrightarrow \mathrm{ASL}_d({\mathbb{Z}})\backslash \mathrm{ASL}_d({\mathbb{R}}).\end{align*} $$

Notation 2.1. We follow Rogers [13] in setting some notation and recalling the definition of admissible matrices.

1. (1) We will identify the k-th power $({\mathbb {R}}^d)^k$ of ${\mathbb {R}}^d$ with $\mathrm {Mat}_{k,d}({\mathbb {R}})$ . For a matrix D, denote by $[D]^j$ the j-th column of D and $[D]_i$ the i-th row of D.

2. (2) For $u\in {\mathbb {N}}$ and $r\in \{1, \ldots , k\}$ , the collection ${\mathfrak {D}}_{r,u}^k$ is the set of integral matrices $D=(d_{ij}) \in \mathrm {Mat}_{k, r}({\mathbb {Z}})$ such that the greatest common divisor of all elements of D is one and there are $1\le i_1 < \ldots < i_r \le k$ with the following properties:

   1. (i) ${{}^{\mathrm {t}}{([D]_{i_1}, \ldots , [D]_{i_r})}}=u\mathrm {Id}_r$ ;

   2. (ii) $d_{ij}=0$ for $1\le j \le r$ and $1\le i < i_j$ .

   We say that D is admissible if D satisfies the above properties.

3. (3) For each $D\in {\mathfrak {D}}_{r,u}^k$ ,

   1. (a) set $I_D :=\{i_1<\ldots <i_r\}$ , where $i_1<\ldots <i_r$ are as above;

   2. (b) let

      $$\begin{align*}\Phi^{(d)}(D,u)=\left\{\!\!\left(\begin{array}{c} \mathbf{n}_1 \\ \vdots \\ \mathbf{n}_r \end{array}\right) \in ({\mathbb{Z}}^d)^r : \frac D u \left(\begin{array}{c} \mathbf{n}_1 \\ \vdots \\ \mathbf{n}_r \end{array}\right)\in ({\mathbb{Z}}^d)^k\quad\text{and}\quad \begin{array}{c} \mathbf{n}_1, \ldots, \mathbf{n}_r \text{ are}\\ \text{linearly independent}\end{array}\!\!\!\!\right\}; \end{align*}$$

   3. (c) define $N(D,u)$ to be the number of vectors $\mathbf {v}\in \{0, 1 \ldots , u-1\}^r$ for which

      $$ \begin{align*}\frac 1 u D \:{{}^{\mathrm{t}}{\mathbf{v}}}\in {\mathbb{Z}}^k.\end{align*} $$

We are now ready to state Rogers’ famous integral formula for ${\mathcal S}_{k}(F)$ on $\mathrm {SL}_d({\mathbb {Z}})\backslash \mathrm {SL}_d({\mathbb {R}})$ introduced in [13].

Theorem 2.2 (Rogers [13])

Let $F:({\mathbb {R}}^d)^k \rightarrow \mathbb {R}_{\geq 0}$ , where $1\le k \le d-1$ , be a bounded function of compact support. Then,

$$\begin{align*}\begin{aligned} \int_{X_d} {\mathcal S}_{k}(F)(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu(\Lambda) = F\left(\begin{array}{c} {\mathbf{0}}\\ \vdots\\ {\mathbf{0}}\end{array}\right) +\sum_{r=1}^k \sum_{u\in {\mathbb{N}}} \sum_{D\in {\mathfrak{D}}^k_{r,u}} \frac {N(D,u)^d} {u^{dr}} \int_{({\mathbb{R}}^d)^r} F \left( \frac D u \left(\begin{array}{c} \mathbf{v}_1 \\ \vdots \\ \mathbf{v}_r \end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}_r. \end{aligned} \end{align*}$$

We note that Rogers did not comment on the nature of convergence of the RHS of the above equation. He did, however, mention [13 , second paragraph of page 279] that results in another paper of his [14, §9] imply absolute convergence for $d \geq [\tfrac {1}{4}k^2] + 2$ ). Schmidt [17] showed that in the case of a bounded compactly supported function $F : (\mathbb {R}^{d})^k \to \mathbb {R}_{\geq 0}$ , the above sum is absolutely convergent; in other words, both sides of the above equation are finite (and equal). Thus, Rogers’ theorem holds also for a bounded compactly supported function $F : (\mathbb {R}^{d})^k \to \mathbb {R}$ , and both sides of the above equation are finite in this case (since Rogers’ theorem holds for $|F|$ , we have absolute convergence of the sum, and we can rearrange the terms in the sum).

Theorem 2.2 follows from the fact that

$$\begin{align*}({\mathbb{Z}}^d)^k= \left\{{}^{\mathrm{t}}{({\mathbf{0}},\ldots, {\mathbf{0}})}\right\} \sqcup \bigsqcup_{r=1}^k \bigsqcup_{u\in {\mathbb{N}}} \bigsqcup_{D\in {\mathfrak{D}}^k_{r,u}} \frac D u \Phi^{(d)}(D,u) \end{align*}$$

and the following proposition.

Proposition 2.3 (Rogers [13])

Let $F:({\mathbb {R}}^d)^k \rightarrow {\mathbb {R}}$ be a bounded function of compact support. For each $D\in {\mathfrak {D}}^k_{r,u}$ , we have

$$\begin{align*}\int_{X_d} \sum_{\scriptsize \begin{array}{c} ^{\mathrm{t}}{(\mathbf{n}_1, \ldots, \mathbf{n}_r)}\\ \in \Phi^{(d)}(D,u)\end{array}} F\!\left(\frac D u \left(\begin{array}{c} \mathbf{n}_1 \\ \vdots \\ \mathbf{n}_r\end{array}\right) g\right) { {\mathrm{d}}}\mu(g) = \frac {N(D,u)^d} {u^{dr}} \int_{({\mathbb{R}}^d)^r} F \!\left( \frac D u \left(\begin{array}{c} \mathbf{v}_1 \\ \vdots \\ \mathbf{v}_r \end{array}\right)\right)\!{ {\mathrm{d}}} \mathbf{v}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}_r. \end{align*}$$

2.1 Higher moment formulae for Y

In [4], El-Baz, Marklof and Vinogradov established a second moment formula for the Siegel transform on $ Y=\mathrm {ASL}_2({\mathbb {Z}})\backslash \mathrm {ASL}_2({\mathbb {R}})$ which easily extends to the case when $d\ge 3$ (see [4, Appendix B]). We will generalise their result to higher moment formulae for the transform ${\mathcal S}_{k}(\cdot )$ on Y. It is well-known that

$$\begin{align*}\bigcup_{g\in \mathcal F} \left\{(\xi,g) : \xi \in [0,1)^d g \right\} \end{align*}$$

is a fundamental domain for Y, where $\mathcal F$ is any fixed fundamental domain for $\mathrm {SL}_d({\mathbb {Z}})\backslash \mathrm {SL}_d({\mathbb {R}})$ . Thus, one can take the probability $\mathrm {ASL}_d({\mathbb {R}})$ -invariant measure $\mu _Y$ on Y as the measure inherited from the product of the Haar measure $\mu $ on $\mathrm {SL}_d({\mathbb {R}})$ and the Lebesgue measure on ${\mathbb {R}}^d$ .

Theorem 2.4. Let $F:({\mathbb {R}}^d)^k\rightarrow {\mathbb {R}}$ be a bounded compactly supported function, and $d\ge 2$ . We have the following:

1. (i) For $k = 1$ ,

   (2.1) $$ \begin{align} \begin{aligned} &\int_Y {\mathcal S}_{1}(F)(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_Y(\Lambda) = \int_{{\mathbb{R}}^d} F({\mathbf{y}}) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}. \end{aligned} \end{align} $$

2. (ii) For $2\le k\le d$ ,

   (2.2) $$ \begin{align} \begin{aligned} &\int_Y {\mathcal S}_{k}(F)(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_Y(\Lambda) = \int_{({\mathbb{R}}^d)^k} F\left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_2\\ \vdots\\ {\mathbf{y}}_k\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1{\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_2\cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_k +\int_{{\mathbb{R}}^d} F\left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_1\\ \vdots\\ {\mathbf{y}}_1\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 \\ &\hspace{0.4in}+\sum_{r=1}^{k-2} \sum_{u\in {\mathbb{N}}} \sum_{D\in {\mathfrak{D}}^{k-1}_{r,u}} \frac {N(D,u)^d} {u^{dr}}\int_{({\mathbb{R}}^d)^{r+1}} F\left(D' \left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_2\\ \vdots\\ {\mathbf{y}}_{r+1}\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1{\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_2\cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_{r+1}, \end{aligned} \end{align} $$

   where $D'$ for $D\in {\mathfrak {D}}^{k-1}_{r,u}$ is $k\times (r+1)$ matrix defined by

   (2.3) $$ \begin{align} D'=\left(\begin{array}{c|c} 1 & 0\;\cdots\;0 \\ \hline \begin{array}{c} 1 \\ \vdots\\ 1\end{array} & \dfrac D u \end{array}\right). \end{align} $$

   Here, as a convention, for $k=2$ , let us assume that $\sum _{r=1}^{0}$ is the empty summation.

Finally, both sides of the equation (2.2) are finite.

Proof. We first remark that the $k=1$ case is classical and can be proved using the folding and unfolding argument. When $k=2$ , the result can be deduced from [4, Appendix B], where the authors proved the second moment formula for $d=2$ . However, their proof can be seen to work in full generality. We will therefore focus on the case when $k\ge 3$ .

Fix any fundamental domain $\mathcal F$ for $\mathrm {SL}_d({\mathbb {Z}})\backslash \mathrm {SL}_d({\mathbb {R}})$ . For each $g\in \mathcal F$ , by the change of variables $\xi =\eta g$ , we have

$$\begin{align*}\begin{aligned} \int_Y {\mathcal S}_{k}(F)({\mathbb{Z}}^d g+\xi) {\hspace{0.5mm} {\mathrm{d}}}\mu(g){\hspace{0.5mm} {\mathrm{d}}}\xi &=\int_Y {\mathcal S}_{k}(F)(({\mathbb{Z}}^d +\eta)g) {\hspace{0.5mm} {\mathrm{d}}}\mu(g){\hspace{0.5mm} {\mathrm{d}}}\eta\\[4pt] &=\int_{\mathcal F}\int_{[0,1)^d}\sum_{\scriptsize \begin{array}{c} {\mathbf{m}}_i\in {\mathbb{Z}}^d\\ 1\le i\le k\end{array}} F\left(\begin{array}{c} ({\mathbf{m}}_1 +\eta)g\\ ({\mathbf{m}}_2+\eta)g\\ \vdots\\ ({\mathbf{m}}_k +\eta)g\end{array}\right){\hspace{0.5mm} {\mathrm{d}}}\eta {\hspace{0.5mm} {\mathrm{d}}}\mu(g). \end{aligned} \end{align*}$$

For each $g\in \mathcal F$ and ${\mathbf {m}}_1\in {\mathbb {Z}}^d$ , put ${\mathbf {y}}_1=(\eta +{\mathbf {m}}_1)g$ and ${\mathbf {m}}^{\prime }_j={\mathbf {m}}_j-{\mathbf {m}}_1$ for $2\le j\le k$ . Since $\bigcup _{{\mathbf {m}}_1\in {\mathbb {Z}}^d}\left ({\mathbf {m}}_1+[0,1)^d\right )= {\mathbb {R}}^d$ , the above expression is

$$\begin{align*}\begin{aligned} &=\int_{\mathcal F} \int_{{\mathbb{R}}^d} \sum_{\scriptsize \begin{array}{c} {\mathbf{m}}^{\prime}_j\in {\mathbb{Z}}^d\\ 2\le j\le k\end{array}} F\left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_1+{\mathbf{m}}^{\prime}_2 g\\ \vdots\\ {\mathbf{y}}_1+{\mathbf{m}}^{\prime}_k g\end{array}\right){\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 {\hspace{0.5mm} {\mathrm{d}}}\mu(g)\\ &=\int_{{\mathbb{R}}^d} F\left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_1\\ \vdots\\ {\mathbf{y}}_1\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 +\int_{({\mathbb{R}}^d)^k} F\left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_2\\ \vdots\\ {\mathbf{y}}_k\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1{\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_2\cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_k\\ &+\sum_{r=1}^{k-2} \sum_{u=1}^{\infty} \sum_{D\in {\mathfrak{D}}^{k-1}_{r,u}} \frac {N(D,u)^d} {u^{dr}}\int_{({\mathbb{R}}^d)^{r+1}} F\left(D' \left(\begin{array}{c} {\mathbf{y}}_1\\ {\mathbf{y}}_2\\ \vdots\\ {\mathbf{y}}_{r+1}\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1{\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_2\cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_{r+1}, \end{aligned} \end{align*}$$

where $D'$ is defined as in (2.3) In the last equality, we applied Theorem 2.2 to the function

$$\begin{align*}F' : ({\mathbf{y}}_2, \ldots, {\mathbf{y}}_k) \mapsto \int_{{\mathbb{R}}^d} F\left({\mathbf{y}}_1, {\mathbf{y}}_1+{\mathbf{y}}_2, \ldots, {\mathbf{y}}_1+{\mathbf{y}}_k\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1. \end{align*}$$

Observe that it is enough to prove finiteness for $F \geq 0$ . Indeed, for general F, finiteness for $|F|$ proves the absolute convergence of the sum in the RHS of (2.2). We note that (for $F \geq 0$ ) $F'$ is a compactly supported bounded positive function, and hence, invoking Schmidt [17, Theorem 2] for this function proves our claim.

2.2 Higher moment formulae for $Y_{{\mathbf {p}}/q}$

Recall that for ${\mathbf {p}}\in {\mathbb {Z}}^d \smallsetminus \{{\mathbf {0}}\}$ and $q\in {\mathbb {N}}_{\ge 2}$ such that $\gcd ({\mathbf {p}}, q)=1$ , we set

$$\begin{align*}Y_{{{\mathbf{p}}}/q}:=\left\{\left({\mathbb{Z}}^d+\frac {{\mathbf{p}}} q\right)g : g\in \mathrm{SL}_d({\mathbb{R}}) \right\}\subseteq Y. \end{align*}$$

We remark that the space $Y_{\mathbf {p}/q}$ does not depend on $\mathbf {p}$ because $Y_{\mathbf {p}/q}$ is the space of all affine grids $L + \mathbf {v}$ , where L is a unimodular lattice in $\mathbb {R}^{d}$ and $\mathbf {v} \in \mathbb {R}^{d}$ is a representative of a torsion point of order q in the torus $\mathbb {R}^{d}/L$ . Indeed, one can see that for such $L + \mathbf {v}$ , $\exists ~ g \in \mathrm {SL}_{d}(\mathbb {R})$ such that $L = \mathbb {Z}^{d}g$ , and since $q\mathbf {v} \in L$ , we have $\mathbf {v} = \frac {\mathbf {w}g}{q}$ , where $\mathbf {w} \in \mathbb {Z}^{d}$ and $\mathbf {w}$ is of order q in $(\mathbb {Z}/q\mathbb {Z})^d$ (since $\mathbf {v}$ is of order q). Therefore, $L + \mathbf {v} = {\left (\mathbb {Z}^{d} + \frac {\mathbf {w}}{q}\right )}g$ . Since $\mathbf {p}$ is also of order q in $(\mathbb {Z}/q\mathbb {Z})^d$ and $\mathrm {SL}_{d}(\mathbb {Z})$ acts transitively on elements of order q in $(\mathbb {Z}/q\mathbb {Z})^d$ , $\exists ~ \gamma \in \mathrm {SL}_{d}(\mathbb {Z})$ such that $\mathbf {w} = \mathbf {p}\gamma $ . Hence,

$$\begin{align*}L + \mathbf{v} = {\left(\mathbb{Z}^{d} + \frac{\mathbf{p}\gamma}{q}\right)}g = {\left(\mathbb{Z}^{d} + \frac{\mathbf{p}}{q}\right)} \gamma g. \end{align*}$$

Let $\{\mathbf {e}_j\}$ be the canonical basis of ${\mathbb {R}}^d$ . Define

$$\begin{align*}\begin{aligned} \Gamma(q)&=\left\{\gamma\in \mathrm{SL}_d({\mathbb{Z}}): \gamma \equiv \mathrm{Id}_d\quad\mod q\right\},\\ \Gamma_1(q)&=\left\{\gamma\in \mathrm{SL}_d({\mathbb{Z}}): \mathbf{e}_1\gamma\equiv \mathbf{e}_1\quad\mod q\right\}, \end{aligned} \end{align*}$$

and $X_q=\Gamma (q)\backslash \mathrm {SL}_d({\mathbb {R}})$ . If we choose any $\gamma _{{{\mathbf {p}}}}\in \mathrm {SL}_d({\mathbb {Z}})$ for which ${{\mathbf {p}}}=r\mathbf {e}_1 \gamma _{\mathbf {p}}$ , where $r=\gcd {{\mathbf {p}}}$ , then $Y_{{{\mathbf {p}}}/q}$ can be identified with $\gamma _{{{\mathbf {p}}}}^{-1} \Gamma _1(q)\gamma _{{\mathbf {p}}}\backslash \mathrm {SL}_d({\mathbb {R}})$ ([6, Lemma 3.1]). Denote by $\mu _q$ the Haar measure on $\mathrm {SL}_d(\mathbb {R})$ normalised so that $\mu _q(Y_{\mathbf {p}/q}) = 1$ . More precisely, let $J_q=[\mathrm {SL}_d({\mathbb {Z}}):\Gamma _1(q)]$ . We can see that $\mu _q=\frac 1 {J_q} \mu $ , which is independent of the choice of ${\mathbf {p}}$ .

Recall that we identify the k-tuple $({\mathbb {R}}^d)^k$ of ${\mathbb {R}}^d$ with $\mathrm {Mat}_{k,d}({\mathbb {R}})$ . Let $\{E_{ij} : 1\le i \le k, 1\le j\le d\}$ be the standard basis for $({\mathbb {R}}^d)^k$ ; that is, the $(k,\ell )$ -entry $[E_{ij}]_{k\ell }=0$ except that $[E_{ij}]_{ij}=1$ .

The Lemma below essentially follows from the definition. However, we provide a proof since it is vital in setting up and proving moment formulas for congruence quotients.

Lemma 2.5. For each $D\in {\mathfrak {D}}^k_{r,u}$ , where ${\mathfrak {D}}^k_{r,u}$ is as in Notation 2.1, define

$$\begin{align*}\Lambda_{D}=\left\{\left(\begin{array}{c} \ell_1\\ \vdots\\ \ell_r\end{array}\right)\in {\mathbb{Z}}^r : \frac D u \left(\begin{array}{c} \ell_1\\ \vdots\\ \ell_r\end{array}\right)\in {\mathbb{Z}}^k \right\}. \end{align*}$$

It follows that $\frac D u: \Lambda _D \rightarrow \frac D u {\mathbb {R}}^r$ is injective, and moreover,

$$\begin{align*}\frac D u \Lambda_D= \frac D u{\mathbb{R}}^r \cap {\mathbb{Z}}^k. \end{align*}$$

In other words, the set $\frac D u \Lambda _D$ is a primitive sublattice of ${\mathbb {Z}}^k$ of rank r, which is given by intersecting with the rational subspace $\frac D u{\mathbb {R}}^r\subseteq {\mathbb {R}}^k$ .

Proof. One direction as well as the injectivity is obvious. Let us show the other direction. Suppose that $\boldsymbol {\ell }\in {\mathbb {R}}^r$ satisfies that $\frac D u\boldsymbol {\ell }\in {\mathbb {Z}}^k$ . Considering indices $1\le i_1< \ldots < i_r\le k$ in Notation 2.1 (2), we have that $\boldsymbol {\ell }=([\frac D u \boldsymbol {\ell }]^{i_1}, \ldots , [\frac D u \boldsymbol {\ell }]^{i_r})\in {\mathbb {Z}}^r$ . This proves the lemma since $\Lambda _D={\mathbb {Z}}^r \cap \left (\frac D u\right )^{-1}{\mathbb {Z}}^k$ .

Notation 2.6. For each $D\in {\mathfrak {D}}^k_{r,u}$ , since $\Lambda _D$ defined as in Lemma 2.5 is primitive, one can find elements $\mathbf b_{1}, \ldots , \mathbf b_{k-r}$ in ${\mathbb {Z}}^k$ such that for any ${\mathbb {Z}}$ -basis $\{\mathbf b_{k-r+1}, \ldots , \mathbf b_{k}\}$ of $\frac D u \Lambda $ , it holds that

$$\begin{align*}{\mathbb{Z}}^k={\mathbb{Z}}\mathbf b_1 \oplus \cdots \oplus {\mathbb{Z}}\mathbf b_k. \end{align*}$$

Fix such a set $\{\mathbf b_{1}, \ldots , \mathbf b_{k-r}\}$ for each $D\in {\mathfrak {D}}^k_{r,u}$ and denote

$$\begin{align*}{\mathcal R}(D)= {\mathbb{Z}}\mathbf b_1 \oplus \cdots \oplus {\mathbb{Z}}\mathbf b_{k-r}\end{align*}$$

so that ${\mathbb {Z}}^k=\bigsqcup _{\boldsymbol {\ell }\in {\mathcal R}(D)} \left (\boldsymbol {\ell }+\frac D u \Lambda _D\right )$ . We also define the set $P_t({\mathcal R}(D))$ for every $t\in {\mathbb {N}}$ with $\gcd (t,q)=1$ as

$$\begin{align*}P_t({\mathcal R}(D))=\{\boldsymbol{\ell}\in {\mathcal R}(D): \gcd(\boldsymbol{\ell}, t)=1\}. \end{align*}$$

We are now ready to formulate the higher moment formula for $Y_{{\mathbf {p}}/q}$ , based on Notation 2.6. The formula in equation (2.4) below depends on a choice of ${\mathcal R}(D)$ for each ${\mathfrak {D}}^k_{r,u}$ . We are very grateful to the anonymous referee for providing an alternative formulation which does not involve any ad hoc choices. This formulation can be found in Theorem 2.13. We have chosen to include both formulations because we believe that (2.4) is more “intrinsic” in some sense (i.e., more indicative of the proof); see, for instance, the similarity with the second moment formula proven in [6] (see also [11, Proposition 7.6]).

Theorem 2.7. Let $d\ge 3$ and $1\le k \le d-1$ . Let $F:({\mathbb {R}}^d)^k\rightarrow {\mathbb {R}}$ be bounded and compactly supported. Then

1. (1) For $k = 1$ ,

   $$\begin{align*}\int_{Y_{{{\mathbf{p}}}/q}} {\mathcal S}_{1}(F) (\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_q(\Lambda) =\int_{{\mathbb{R}}^d} F\left({{\mathbf{y}}}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}. \end{align*}$$

2. (2) For $2 \leq k \leq d-1$ ,

   (2.4) $$ \begin{align} \begin{aligned} &\int_{Y_{{{\mathbf{p}}}/q}} {\mathcal S}_{k}(F) (\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_q(\Lambda) =\int_{({\mathbb{R}}^d)^k} F\left({{}^{\mathrm{t}}{({\mathbf{y}}_1, \ldots, {\mathbf{y}}_k)}}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_k\\ &+\int_{{\mathbb{R}}^d} F\left({{}^{\mathrm{t}}{({\mathbf{y}}, \ldots, {\mathbf{y}})}}\right) d{\mathbf{y}} +\hspace{-0.1in}\sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}} \\ (t,q)=1 \end{array}} \hspace{-0.1in}\sum_{\scriptsize \begin{array}{c} \boldsymbol{\ell}\neq {\mathbf{0}} \\ \in {\mathbb{Z}}^{k-1}\end{array}} \int_{{\mathbb{R}}^d}F\left(\left(\begin{array}{c} t{\mathbf{y}} \\ (t+\ell_1q){\mathbf{y}} \\ \vdots \\ (t+\ell_{k-1}q){\mathbf{y}}\end{array}\right)\right)d{\mathbf{y}}\\ &+\sum_{r=1}^{k-2} \sum_{u\in {\mathbb{N}}} \sum_{D\in {\mathfrak{D}}^{k-1}_{r,u}} \left[\frac {N(D,u)^d}{u^{dr}} \int_{({\mathbb{R}}^d)^{r+1}} F\left(D'\left(\begin{array}{c} {\mathbf{y}}_1 \\ \vdots \\ {\mathbf{y}}_{r+1}\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_{r+1} \right.\\ &\hspace{0.5in}\left. \sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}} \\ (t,q)=1 \end{array}} \sum_{\scriptsize \begin{array}{c} \boldsymbol{\ell}\in \\ P_t({\mathcal R}(D))\end{array}} \frac{N(D,u)^d}{t^d\cdot u^{dr}} \int_{({\mathbb{R}})^{r+1}} F\left(D^{\prime}_{t,\boldsymbol{\ell}} \left(\begin{array}{c} {\mathbf{y}}_1 \\ \vdots \\ {\mathbf{y}}_{r+1}\end{array}\right)\right)d{\mathbf{y}}_1 \cdots d{\mathbf{y}}_{r+1}\right],\\ \end{aligned} \end{align} $$
   where $D'$ and $D^{\prime }_{t,\boldsymbol {\ell }}$ for $D\in {\mathfrak {D}}^{k-1}_{r,u}$ and $\boldsymbol {\ell }={{}^{\mathrm {t}}{(\ell _1, \ldots , \ell _{k-1})}}\in P_t({\mathcal R}(D))$ are $k\times (r+1)$ matrices defined as follows:
   $$\begin{align*}D'=\left(\begin{array}{c|c} 1 & 0 \, \cdots \, 0 \\ \hline \begin{array}{c} 1 \\ \vdots \\ 1 \end{array} & \dfrac 1 u D \end{array}\right) \quad\text{and}\quad D^{\prime}_{t,\boldsymbol{\ell}} =\left(\begin{array}{c|c} t & 0 \, \cdots \, 0 \\ \hline \begin{array}{c} t+\ell_1q \\ \vdots \\ t+\ell_{k-1}q \end{array} & \dfrac 1 u D \end{array}\right). \end{align*}$$
   Here, if $k=2$ , we will consider $\sum _{m=1}^0$ as the empty summation.

Finally, both sides of the equation (2.4) are finite.

Notice that the right-hand side of the above expression does not depend on ${\mathbf {p}}\in {\mathbb {Z}}^d \smallsetminus \{{\mathbf {0}}\}$ , once $\gcd ({\mathbf {p}},q)=1$ .

We need several lemmas for the proof of Theorem 2.7. Let

$$\begin{align*}H=\left\{\left(\begin{array}{cc} 1 & 0 \\ ^{\mathrm{t}}{\mathbf{v}'} & g' \end{array}\right) : \mathbf{v}'\in {\mathbb{R}}^{d-1}\;\text{and}\; g'\in \mathrm{SL}_{d-1}({\mathbb{R}}) \right\} \end{align*}$$

and denote an element of H by $[\mathbf {v}', g']$ . Let us identify $\mathrm {SL}_{d-1}({\mathbb {R}})$ with the subgroup $\{[0, g']:g'\in \mathrm {SL}_{d-1}({\mathbb {R}})\}$ of H. One can define the Haar measure $\mu _H$ on H by the product of $\mu '$ and the Lebesgue measure on ${\mathbb {R}}^{d-1}$ , where $\mu '$ is the Haar measure such that $\mu '(X_{d-1})=1$ .

Notice the difference between H and $\mathrm {ASL}_{d-1}({\mathbb {R}})$ . For instance, a fundamental domain of $(\mathrm {SL}_d({\mathbb {Z}})\cap H)\backslash H$ is given by $[0,1)^{d-1} \times \mathcal F_{d-1}$ , where $\mathcal F_{d-1}$ is a fundamental domain of $\mathrm {SL}_{d-1}({\mathbb {Z}})\backslash \mathrm {SL}_{d-1}({\mathbb {R}})$ , whereas that of $\mathrm {ASL}_{d-1}({\mathbb {Z}})\backslash \mathrm {ASL}_{d-1}({\mathbb {R}})$ is given by

$$\begin{align*}\left\{[\xi'g', g']: g'\in \mathcal F_{d-1}\;\text{and}\;\xi' \in [0,1)^{d-1}\right\}.\end{align*}$$

Proposition 2.8. Let $F:({\mathbb {R}}^d)^k \rightarrow {\mathbb {R}}_{\ge 0}$ , where $d\ge 3$ and $1\le k \le d-2$ , be a bounded and compactly supported function. Suppose that $\xi =(z_1, \xi ')\in {\mathbb {R}}^d$ with $z_1\in {\mathbb {R}}$ and $\xi '\in {\mathbb {Z}}^{d-1}$ . Then,

$$\begin{align*}\begin{aligned} &\int_{\mathrm{SL}_d({\mathbb{Z}})\cap H\backslash H} {\mathcal S}_{k}(F)\left(({\mathbb{Z}}^d+\xi)g\right){\hspace{0.5mm} {\mathrm{d}}}\mu_H(g)\\ &=\sum_{\ell_1, \ldots, \ell_k\in {\mathbb{Z}}} F\left(\sum_{i=1}^k (z_1+\ell_i)E_{i1}\right)\\ &\hspace{0.2in}+\sum_{r=1}^k \sum_{u\in{\mathbb{N}}} \sum_{D\in {\mathfrak{D}}^k_{r,u}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_k)}} \\ \in {\mathcal R}(D)\end{array}}\\ &\hspace{0.5in}\frac {N(D,u)^{d}} {u^{dr}}\int_{({\mathbb{R}}^d)^r} F\left(\sum_{i=1}^k (z_1+\ell_i)E_{i1}+ \frac {D} {u} \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots\\ \mathbf{x}_r \end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_r. \end{aligned} \end{align*}$$

Note that H is the isotropy subgroup of $\mathbf {e}_1$ in $\mathrm {SL}_d({\mathbb {R}})$ . We will compute the integral $\int _{\mathrm {SL}_d({\mathbb {Z}})\cap H \backslash H} {\mathcal S}_{k}(F) {\hspace {0.5mm} {\mathrm {d}}}\mu _H$ in two steps: we first process the integrals associated to the first column in $({\mathbb {R}}^d)^k\simeq \mathrm {Mat}_{k,d}({\mathbb {R}})$ and then apply Theorem 2.2 to the integrals associated to the remaining columns. For this, we need the lemma below which describes the relation between the primitive sublattice $\frac D u \Lambda _D$ of ${\mathbb {Z}}^k$ for $D\in {\mathfrak {D}}^k_{r,u}$ and its sublattice $\frac C w \Lambda _C$ for some $C\in {\mathfrak {D}}^k_{r-1,w}$ .

Lemma 2.9. Recall Notation 2.6. Let $D\in {\mathfrak {D}}^k_{r,u}$ with $r\ge 2$ .

1. (a) For $1\le j_0\le r$ and $a_1, \ldots , a_{j_0-1}\in {\mathbb {Q}}$ , define $C_1\in \mathrm {Mat}_{k,r-1}({\mathbb {Q}})$ by

   $$\begin{align*}\left[C_1\right]^j =\left\{\begin{array}{cl} [D/u]^j +a_j[D/u]^{j_0} & \text{for } 1\le j<j_0; \\[0.05in] {[D/u]^{j+1}} & \text{for } j_0\le j \le r-1. \end{array} \right. \end{align*}$$
   Let $w\in {\mathbb {N}}$ be the least common denominator of $C_1$ and $C:=wC_1\in {\mathfrak {D}}^k_{r-1,w}$ . Let ${\mathfrak {D}}^{k}_{r-1,w}(D)$ be the collection of such matrices C. There is a one-to-one correspondence between
   $$\begin{align*}\bigcup_{w\in {\mathbb{N}}}{\mathfrak{D}}^k_{r-1,w}(D) \;\text{and}\;\left\{(r-1)\text{-dimensional rational subspaces in } D{\mathbb{R}}^r\right\}. \end{align*}$$

2. (b) For each $C\in {\mathfrak {D}}^k_{r-1,w}(D)$ , define

   $$\begin{align*}\Lambda_D(C)=\left\{{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_r)} \in \Lambda_D : a_1 \ell_1 + \cdots + a_{j_0-1}\ell_{j_0-1}=\ell_{j_0} \right\}. \end{align*}$$
   Then there is a natural isomorphism from $\Lambda _C$ to $\Lambda _D(C)$ so that
   $$ \begin{align*} \frac D u \Lambda_D(C)= \frac C w \Lambda_C \subseteq \frac D u \Lambda_D \subseteq {\mathbb{Z}}^k. \end{align*} $$
   For each such pair $(D,C)$ , one can choose (and fix from now on) an element $\mathbf b_D\in \Lambda _D-\Lambda _D(C)$ so that if we let ${\mathcal R}_D(C)={\mathbb {Z}} \mathbf b_D \smallsetminus \{{\mathbf {0}}\}$ , then it holds that
   $$\begin{align*}\Lambda_D-\Lambda_D(C)=\bigsqcup_{\boldsymbol{\ell}\in {\mathcal R}_D(C)} \left(\boldsymbol{\ell} + \Lambda_D(C)\right). \end{align*}$$

3. (c) For a given $C\in {\mathfrak {D}}^k_{r-1,w}$ , let $D_1\in {\mathfrak {D}}^k_{r,u_1}$ and $D_2\in {\mathfrak {D}}^k_{r,u_2}$ be such that $D_1\neq D_2$ and $C \in {\mathfrak {D}}^k_{r-1,w}(D_1) \cap {\mathfrak {D}}^k_{r-1,w}(D_2)$ . Then

   $$\begin{align*}\frac {D_1} {u_1} {\mathcal R}_{D_1}(C)\cap \frac {D_2} {u_2} {\mathcal R}_{D_2}(C)=\emptyset. \end{align*}$$
   Hence, for any $\boldsymbol {\ell }_1 \in \frac {D_1} {u_1} {\mathcal R}_{D_1}(C)$ and $\boldsymbol {\ell }_2\in \frac {D_2} {u_2} {\mathcal R}_{D_2}(C)$ , it follows that
   $$\begin{align*}\left(\boldsymbol{\ell}_1 + \frac C w \Lambda_C\right) \cap \left(\boldsymbol{\ell}_2 + \frac C w \Lambda_C\right) = \emptyset. \end{align*}$$

4. (d) For a given $C\in {\mathfrak {D}}^k_{r-1,w}$ , one can choose ${\mathcal R}(C)$ in Notation 2.6 to be

   $$\begin{align*}{{\mathcal R}}(C)=\left\{{{}^{\mathrm{t}}{(0,\ldots, 0)}}\right\} \sqcup \bigsqcup_{u\in {\mathbb{N}}} \bigsqcup_{\scriptsize \begin{array}{c} D\in {\mathfrak{D}}^k_{r,u}\;\text{s.t.}\\ C\in {\mathfrak{D}}^k_{r-1,w}(D)\end{array}}\frac D u {\mathcal R}_D(C) \end{align*}$$
   and vice versa.

Proof. (a) One way is obvious from its construction. Suppose that $W\subseteq D{\mathbb {R}}^r$ is a codimension-one rational subspace of $D{\mathbb {R}}^r$ . Then there is $C\in {\mathfrak {D}}^{k}_{r-1,w}$ so that $\frac C w {\mathbb {R}}^{r-1}=W$ . We want to show that $C\in {\mathfrak {D}}^k_{r-1,w}(D)$ . Pick any $^{\mathrm {t}}{({\mathbf {m}}_1, \ldots , {\mathbf {m}}_k)}\in ({\mathbb {R}}^d)^k\in \frac C w ({\mathbb {R}}^d)^k \cap ({\mathbb {Z}}^d)^k$ with rank $(r-1)$ and $^{\mathrm {t}}{({\mathbf {m}}^{\prime }_1, \ldots , {\mathbf {m}}^{\prime }_r)}$ and $^{\mathrm {t}}{({\mathbf {m}}^{\prime \prime }_1, \ldots , {\mathbf {m}}^{\prime \prime }_{r-1})}$ be such that

$$\begin{align*}\frac D u \left(\begin{array}{c} {\mathbf{m}}^{\prime}_1\\ \vdots \\ {\mathbf{m}}^{\prime}_r\end{array}\right)= \left(\begin{array}{c} {\mathbf{m}}_1\\ \vdots \\ {\mathbf{m}}_k \end{array}\right)= \frac C w \left(\begin{array}{c} {\mathbf{m}}^{\prime\prime}_1 \\ \vdots \\ {\mathbf{m}}^{\prime\prime}_{r-1}\end{array}\right). \end{align*}$$

Let $I_D=\{i_1<\ldots <i_r\}$ be as in Notation 2.1 (3). Since $\operatorname {\mathrm {rk}}{{}^{\mathrm {t}}{({\mathbf {m}}_1, \ldots , {\mathbf {m}}_k)}}=r-1$ , and by definition of D and C, there is $1\le j_0\le r$ for which

$$\begin{align*}\begin{gathered} {\mathbf{m}}^{\prime}_1={\mathbf{m}}_{i_1}={\mathbf{m}}^{\prime\prime}_1, \;\ldots,\;{\mathbf{m}}^{\prime}_{j_0-1}={\mathbf{m}}_{i_{j_0-1}}={\mathbf{m}}^{\prime\prime}_{j_0-1};\\ {\mathbf{m}}^{\prime}_{j_0}=a_1{\mathbf{m}}^{\prime}_1+\cdots+a_{j_0-1}{\mathbf{m}}^{\prime}_{j_0-1}\;\text{for some}\; a_1, \ldots, a_{j_0-1}\in {\mathbb{Q}};\\ {\mathbf{m}}^{\prime}_{j_0+1}={\mathbf{m}}_{i_{j_0+1}}={\mathbf{m}}^{\prime\prime}_{j_0}, \;\ldots,\; {\mathbf{m}}^{\prime}_{r}={\mathbf{m}}_{i_r}={\mathbf{m}}^{\prime\prime}_{r-1}. \end{gathered}\end{align*}$$

It is easily seen that $C_1$ constructed from D with $j_0$ and $a_1, \ldots , a_{j_0-1}\in {\mathbb {Q}}$ in Property (a) is equal to C.

(b) It is obvious from the definition that $C\in {\mathfrak {D}}^k_{r-1,w}$ . The map

$$\begin{align*}\begin{aligned} {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_{r-1})}} \mapsto {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_{j_0-1}, a_1 \ell_1 + \cdots + a_{j_0-1}\ell_{j_0-1}, \ell_{j_0}, \ldots, \ell_{r-1})}} \end{aligned} \end{align*}$$

gives an isomorphism from $\Lambda _C$ to $\Lambda _D(C)$ , and by definition,

$$\begin{align*}\frac C w {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_{r-1})}}= \frac D u {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_{j_0-1}, a_1 \ell_1 + \cdots + a_{j_0-1}\ell_{j_0-1}, \ell_{j_0}, \ldots, \ell_{r-1})}} \in {\mathbb{Z}}^k. \end{align*}$$

Recall Lemma 2.5. Since $\frac C w \Lambda _C$ is primitive, there is an element $\mathbf b\in \frac D u \Lambda _D$ for which $\frac D u \Lambda _D=\frac C w \Lambda _C\oplus {\mathbb {Z}}\mathbf b$ . Set $\mathbf b_D:=\left (\frac D u\right )^{-1}\mathbf b$ .

(c) Let ${\mathcal R}_{D_i}(C)$ be generated by $\mathbf b_{D_i}$ for $i=1,2$ . From the fact that $\frac {D_1} {u_1} {\mathbb {R}}^r \cap \frac {D_2} {u_2} {\mathbb {R}}^r = \frac {C} {w} {\mathbb {R}}^{r-1}$ , in other words,

$$\begin{align*}\frac {D_i}{u_i}{\mathbb{R}}^r=\frac C w {\mathbb{R}}^{r-1}\oplus {\mathbb{R}} \left(\frac {D_i}{u_i}\mathbf b_{D_i}\right)\;(i=1,2), \end{align*}$$

it is obvious that $\frac {D_1}{u_1}{\mathcal R}_{D_1}(C) \cap \frac {D_2}{u_2}{\mathcal R}_{D_2}(C)=\emptyset $ . Moreover, for any $\boldsymbol {\ell }_i\in \frac {D_i}{u_i}$ , where $i=1,2$ ,

$$\begin{align*}\boldsymbol{\ell}_i+\frac C w \Lambda_C\subseteq \boldsymbol{\ell}_i+\frac C w {\mathbb{R}}^{r-1}, \end{align*}$$

which are affine subspaces of $\frac C w {\mathbb {R}}^{r-1}$ lying on $\frac {D_i} {u_i} {\mathbb {R}}^r-\frac C w {\mathbb {R}}^{r-1}$ for $i=1,2$ , respectively. Hence, they are disjoint.

To deduce (d) from (c), it suffices to show that

$$\begin{align*}\frac C w \Lambda_C + \Bigg(\left\{{{}^{\mathrm{t}}{(0,\ldots, 0)}}\right\} \sqcup \bigsqcup_{u\in {\mathbb{N}}} \bigsqcup_{\scriptsize \begin{array}{c} D\in {\mathfrak{D}}^k_{r,u}\\ C\in {\mathfrak{D}}^k_{r-1,w}(D)\end{array}}\frac D u {\mathcal R}_D(C)\Bigg) =\frac C w \Lambda_C + {\mathcal R}(C). \end{align*}$$

Let $\boldsymbol {\ell } \in {\mathcal R}(C)$ be given. Since $\frac C w {\mathbb {R}}^{r-1} \oplus {\mathbb {R}}\boldsymbol {\ell }$ is a rational subspace of rank r, there are $u\in {\mathbb {N}}$ and $D\in {\mathfrak {D}}^k_{r,u}$ , which are uniquely determined, such that $\frac C w {\mathbb {R}}^{r-1} \oplus {\mathbb {R}} \boldsymbol {\ell }=\frac D u {\mathbb {R}}^r$ . It is obvious that $\boldsymbol {\ell } \in \frac D u \Lambda _D - \Lambda _D(C)$ ; hence, there is $\boldsymbol {\ell }'\in \frac D u{\mathcal R}_D(C)$ so that

$$ \begin{align*}\boldsymbol{\ell}\in \boldsymbol{\ell}'+\frac D u\Lambda_D(C)=\boldsymbol{\ell}'+ \frac C w \Lambda_C,\end{align*} $$

as asserted in the claim.

Proof of Proposition 2.8

Fix a fundamental domain $\mathcal F'$ of $\mathrm {SL}_{d-1}({\mathbb {Z}})\backslash \mathrm {SL}_{d-1}({\mathbb {R}})$ so that $\mathcal F'\times [0,1)^{d-1}$ is a fundamental domain of $(\mathrm {SL}_d({\mathbb {Z}})\cap H)\backslash H$ .

Recall that $({\mathbb {Z}}^d)^k \smallsetminus \{{{}^{\mathrm {t}}{({\mathbf {0}}, \ldots , {\mathbf {0}})}}\}$ is partitioned into $ \bigsqcup _{r=1}^k \bigsqcup _{u\in {\mathbb {N}}} \bigsqcup _{D\in {\mathfrak {D}}^k_{r,u}} \frac {D} u \Phi ^{(d)}(D,u), $ where ${\mathfrak {D}}^k_{r,u}$ and $\Phi ^{(d)}(D,u)$ are as in Notation 2.1.

By taking $g=[\mathbf {v}',g']$ and from Rogers’ formula, we have that

$$\begin{align*}\begin{aligned} &\int_{(\mathrm{SL}_d({\mathbb{Z}})\cap H)\backslash H} {\mathcal S}_{k}(F) \left(({\mathbb{Z}}^d+\xi)[\mathbf{v}',g']\right){\hspace{0.5mm} {\mathrm{d}}}\mu_{H}([\mathbf{v}',g']) \\ &=\int_{\mathcal F'\times [0,1)^{d-1}}\hspace{-0.18in} \sum_{\scriptsize \begin{array}{c} \ell_i\in {\mathbb{Z}}\\ {\mathbf{m}}^{\prime}_i\in {\mathbb{Z}}^{d-1}\\ 1\le i \le k \end{array}}\hspace{-0.18in} F\left(\begin{array}{c|c} (\ell_1+z_1)+{\mathbf{m}}^{\prime}_1{{}^{\mathrm{t}}{\mathbf{v}'}} & {\mathbf{m}}^{\prime}_1g' \\ \vdots & \vdots \\ (\ell_k+z_1)+{\mathbf{m}}^{\prime}_k{{}^{\mathrm{t}}{\mathbf{v}'}} & {\mathbf{m}}^{\prime}_kg' \end{array} \right){\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}'{\hspace{0.5mm} {\mathrm{d}}}\mu'(g')\\ &= F\left(\begin{array}{c|c} z_1 & 0, \ldots, 0 \\ \vdots & \vdots \\ z_1 & 0, \ldots, 0\end{array}\right) +\sum_{r=1}^k \sum_{u\in {\mathbb{N}}} \sum_{D\in {\mathfrak{D}}^k_{r,u}}\int_{\mathcal F'\times [0,1)^{d-1}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\mathbf{n}_1, \ldots, \mathbf{n}_r)}}\\ \in \Phi^{(d)}(D,u)\end{array}}\\ &\hspace{1in}\hspace{-12pt} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right)+ \dfrac D u \left(\begin{array}{c} \ell^{\prime}_1 \\ \vdots \\ \ell^{\prime}_r\end{array}\right)+ \dfrac D u \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_r\end{array}\right){{}^{\mathrm{t}}{\mathbf{v}'}} & \dfrac D u \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_r\end{array}\right)g' \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}'{\hspace{0.5mm} {\mathrm{d}}}\mu'(g), \end{aligned}\end{align*}$$

where $\mathbf {n}_j=(\ell ^{\prime }_j, \mathbf {n}^{\prime }_j)$ for $1\le j\le r$ .

Now, let $D\in {\mathfrak {D}}^k_{r,u}$ be given. For each ${{}^{\mathrm {t}}{((\ell ^{\prime }_1, \mathbf {n}^{\prime }_1), \ldots , (\ell ^{\prime }_r, \mathbf {n}^{\prime }_r))}}\in \Phi ^{(d)}(D,u)$ , the rank of ${{}^{\mathrm {t}}{(\mathbf {n}^{\prime }_1, \ldots , \mathbf {n}^{\prime }_r)}}$ is either r or $r-1$ .

Assume that $r\ge 2$ . It is easy to verify that

$$\begin{align*}\begin{aligned} &\Phi^{(d)}(D,u) =\left(\Lambda_D\times \Phi^{(d-1)}(D,u)\right) \sqcup \\ &\hspace{0.1in}\bigsqcup_{w\in {\mathbb{N}}} \bigsqcup_{C\in {\mathfrak{D}}^k_{r-1,w}(D)} \hspace{-0.1in}(\Lambda_D-\Lambda_D(C))\times \left\{\!{\small\left(\!\!\!\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \sum_{k=1}^{j_0-1} a_k\mathbf{n}^{\prime}_k \\ \vdots \\ \mathbf{n}^{\prime}_{r-1}\end{array}\!\!\!\right)_{r\times(d-1)}}: \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_{r-1}\end{array}\right)\in\Phi^{(d-1)}(C,w)\!\right\}, \end{aligned}\end{align*}$$

where ${\mathfrak {D}}^k_{r-1,w}(D)$ and $\Lambda _D(C)$ are as in Lemma 2.9.

Let us first compute the following integral

(2.5) $$ \begin{align} \begin{aligned} &\int_{\mathcal F'\times [0,1)^{d-1}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell^{\prime}_1, \ldots, \ell^{\prime}_r)}}\\ \in \Lambda_D\end{array}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\mathbf{n}^{\prime}_1, \ldots, \mathbf{n}^{\prime}_r)}}\\ \in \Phi^{(d-1)}(D,u)\end{array}}\\ &\hspace{0.4in} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right)+ \dfrac D u \left(\begin{array}{c} \ell^{\prime}_1 \\ \vdots \\ \ell^{\prime}_r\end{array}\right)+ \dfrac D u \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_r\end{array}\right){{}^{\mathrm{t}}{\mathbf{v}'}} & \dfrac D u \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_r\end{array}\right)g' \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}'{\hspace{0.5mm} {\mathrm{d}}}\mu'(g'). \end{aligned}\end{align} $$

Fix $g'\in \mathcal F'$ and $N:={{}^{\mathrm {t}}{(\mathbf {n}^{\prime }_1, \ldots , \mathbf {n}^{\prime }_r)}}\in \Phi ^{(d-1)}(D,u)$ . Set $J_N=\{1\le j_1<\ldots <j_r\le d-1\}$ such that $ N^{J_N}:=\left ([N]^{j_1}, \ldots , [N]^{j_r}\right ) $ has a nonzero determinant. Denote by

$$\begin{align*}N\:{{}^{\mathrm{t}}{\mathbf{v}'}}= N^{J_N}\:{{}^{\mathrm{t}}{\mathbf{v}^{\prime}_{J_N}}} + N^{J_N^c}\:{{}^{\mathrm{t}}{\mathbf{v}^{\prime}_{J_N^c}}}, \end{align*}$$

where $\mathbf {v}^{\prime }_{J_N}=(v_j)_{j\in J_N}\in {\mathbb {R}}^r$ and $\mathbf {v}^{\prime }_{J_N^c}=(v_i)_{i\in J_N^c}\in {\mathbb {R}}^{(d-1)-r}$ . Define

$$\begin{align*}G\left(\begin{array}{c} x_1 \\ \vdots \\ x_r\end{array}\right) := \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell^{\prime}_1, \ldots, \ell^{\prime}_r)}}\\ \in \Lambda_D\end{array}} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right) +\dfrac D u \left(\begin{array}{c} \ell^{\prime}_1 \\ \vdots \\ \ell^{\prime}_r\end{array}\right) +\dfrac D u \left(\begin{array}{c} x_1 \\ \vdots \\ x_r\end{array}\right) & \dfrac D u N g' \end{array}\right). \end{align*}$$

Obviously, G is $\Lambda _D$ -invariant so that it is $u{\mathbb {Z}}^r$ -invariant, and $G(N\:\cdot )$ is ${\mathbb {Z}}^{d-1}$ -invariant. We want to compute the integral

$$\begin{align*}&\int_{[0,1)^{d-1}} G\left(N\left(\begin{array}{c} v_1 \\ \vdots \\ v_{d-1} \end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}} v_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} v_{d-1} \\&\quad=\frac {1} {u^{d-1}} \int_{[0,u)^{d-1}} G\left(N^{J_N} \mathbf{v}^{J_N}+ N^{J_N^c} \mathbf{v}^{J_N^c}\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}^{J_N} {\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}^{J_N^c}. \end{align*}$$

By the change of variables

$$\begin{align*}\mathbf{v}^{J_N} \mapsto N^{J_N} \mathbf{v}^{J_N}+ N^{J_N^c} \mathbf{v}^{J_N^c}={{}^{\mathrm{t}}{(x_1, \ldots, x_r)}}, \end{align*}$$

the above integral is

$$\begin{align*}\begin{aligned} &=\frac {1}{u^{d-1}} \int_{[0,u)^{d-1-r}}\int_{N^{J_N}[0,u)^r+N^{J_N^c}\mathbf{v}^{J_N^c}} G\left(\begin{array}{c} x_1 \\ \vdots \\ x_r \end{array}\right) |\det N^{J_N}|^{-1} {\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r {\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}^{J_N^c}\\ &=\frac {1}{u^{d-1}} \int_{[0,u)^{d-1-r}}\int_{N^{J_N}[0,u)^r} G\left(\begin{array}{c} x_1 \\ \vdots \\ x_r \end{array}\right) |\det N^{J_N}|^{-1} {\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r {\hspace{0.5mm} {\mathrm{d}}}\mathbf{v}^{J_N^c}\\ &=\frac {1} {u^r} \int_{[0,u)^r} G\left(\begin{array}{c} x_1 \\ \vdots \\ x_r \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r. \end{aligned}\end{align*}$$

Let $\mathcal F_{\Lambda _D}$ be a fundamental domain for $\Lambda _D$ in $[0,u)^r$ . Since $[0,u)^r$ is an $N(D,u)$ -covering of $\mathcal F_{\Lambda _D}$ , it follows that

$$\begin{align*}\begin{aligned} &\int_{[0,1)^{d-1}} G\left(N\left(\begin{array}{c} v_1 \\ \vdots \\ v_{d-1} \end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}} v_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} v_{d-1}\\ &=\frac {N(D,u)} {u^r} \int_{\mathcal F_{\Lambda_D}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell^{\prime}_1, \ldots, \ell^{\prime}_r)}}\\ \in \Lambda_D\end{array}} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right) +\dfrac D u \left(\begin{array}{c} x_1+\ell^{\prime}_1 \\ \vdots \\ x_r+\ell^{\prime}_r\end{array}\right) & \dfrac D u N g' \end{array}\right){\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r\\ &= \frac {N(D,u)} {u^r} \int_{{\mathbb{R}}^r} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right) +\dfrac D u \left(\begin{array}{c} x_1\\ \vdots \\ x_r\end{array}\right) & \dfrac D u N g' \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r. \end{aligned}\end{align*}$$

Therefore, applying Proposition 2.3, the integral (2.5) is

$$\begin{align*}\begin{aligned} &\frac {N(D,u)} {u^r} \int_{\mathrm{SL}_{d-1}({\mathbb{Z}})\backslash \mathrm{SL}_{d-1}({\mathbb{R}})} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\mathbf{n}^{\prime}_1, \ldots, \mathbf{n}^{\prime}_r)}}\\ \in \Phi^{(d-1)}(D,u)\end{array}}\\ &\hspace{1in}\int_{{\mathbb{R}}^r} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right) +\dfrac D u \left(\begin{array}{c} x_1\\ \vdots \\ x_r\end{array}\right) & \dfrac D u \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_r\end{array}\right)g' \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r {\hspace{0.5mm} {\mathrm{d}}}\mu'(g')\\ &=\frac {N(D,u)} {u^r} \cdot \frac {N(D,u)^{d-1}} {u^{(d-1)r}}\\ &\hspace{0.5in}\int_{({\mathbb{R}}^{d-1})^r}\int_{{\mathbb{R}}^r} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right) +\dfrac D u \left(\begin{array}{c} x_1\\ \vdots \\ x_r\end{array}\right) & \dfrac D u \left(\begin{array}{c} \mathbf{x}^{\prime}_1 \\ \vdots \\ \mathbf{x}^{\prime}_r\end{array}\right) \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} x_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} x_r {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}^{\prime}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}^{\prime}_r\\ &= \frac {N(D,u)^d} {u^{dr}} \int_{({\mathbb{R}}^d)^r} F\left(\sum_{i=1}^k z_1E_{i1} +\dfrac D u\left(\begin{array}{c} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_r\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_r. \end{aligned}\end{align*}$$

Now, let us fix $C\in {\mathfrak {D}}^k_{r-1,w}(D)$ and let $\Lambda _D(C)$ and ${\mathcal R}_D(C)$ be as in Lemma 2.9. We want to compute

(2.6) $$ \begin{align}\begin{aligned} &\int_{\mathcal F'\times [0,1)^{d-1}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell^{\prime}_1, \ldots, \ell^{\prime}_r)}}\\ \in \Lambda_D-\Lambda_D(C)\end{array}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\mathbf{n}^{\prime}_1, \ldots, \mathbf{n}^{\prime}_{r-1})}}\\ \in \Phi^{(d-1)}(C,w)\end{array}}\\[5pt] &\hspace{0.1in} F\left(\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right)+ \dfrac D u \left(\begin{array}{c} \ell^{\prime}_1 \\ \vdots \\ \ell^{\prime}_{r}\end{array}\right)+ \dfrac C w \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_{r-1}\end{array}\right){{}^{\mathrm{t}}{\mathbf{v}'}} & \dfrac C w \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_{r-1}\end{array}\right)g' \end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}'{\hspace{0.5mm} {\mathrm{d}}} \mu'(g').\\ \end{aligned}\end{align} $$

Since $\frac D u(\Lambda _D-\Lambda _D(C))= \frac D u ({\mathcal R}_D(C)+\Lambda _D(C))=\frac D u {\mathcal R}_D(C) + \frac C w \Lambda _C$ from Lemma 2.9 (a), the integral (2.6) is

$$ \begin{align*}\begin{aligned} &\sum_{\boldsymbol{\ell}\in {\mathcal R}_D(C)} \int_{{\mathcal F}'\times [0,1)^{d-1}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell^{\prime}_1, \ldots, \ell^{\prime}_{r-1})}}\\ \in \Lambda_C\end{array}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\mathbf{n}^{\prime}_1, \ldots, \mathbf{n}^{\prime}_{r-1})}}\\ \in \Phi^{(d-1)}(C,w)\end{array}}\\ &\hspace{0.2in} F\left(\!\!\!\!\!\!\!\!\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right)+ \dfrac D u \boldsymbol{\ell} + \dfrac C w \left(\begin{array}{c} \ell^{\prime}_1 \\ \vdots \\ \ell^{\prime}_{r-1}\end{array}\right)+ \dfrac C w \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_{r-1}\end{array}\right){{}^{\mathrm{t}}{\mathbf{v}'}} & \dfrac C w \left(\begin{array}{c} \mathbf{n}^{\prime}_1 \\ \vdots \\ \mathbf{n}^{\prime}_{r-1}\end{array}\right)g' \!\!\!\!\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}'{\hspace{0.5mm} {\mathrm{d}}} \mu'(g'). \end{aligned}\end{align*} $$

Repeating the same argument with the above, where now we put $N={{}^{\mathrm {t}}{(\mathbf {n}^{\prime }_1, \ldots , \mathbf {n}^{\prime }_{r-1})}}$ with ${{}^{\mathrm {t}}{(\mathbf {n}^{\prime }_1, \ldots , \mathbf {n}^{\prime }_{r-1})}}\in \Phi ^{(d-1)}(C,w)$ and

$$\begin{align*}G\left(\begin{array}{c} x_1 \\ \vdots \\ x_{r-1} \end{array}\right) :=\hspace{-0.2in} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell^{\prime}_1, \ldots, \ell^{\prime}_{r-1})}}\\ \in \Lambda_C\end{array}}\hspace{-0.1in} F\left(\hspace{-0.1in}\begin{array}{c|c} \left(\begin{array}{c} z_1 \\ \vdots \\ z_1\end{array}\right) +\dfrac D u \boldsymbol{\ell} + \dfrac C w \left(\begin{array}{c} \ell^{\prime}_1 \\ \vdots \\ \ell^{\prime}_{r-1}\end{array}\right) +\dfrac C w \left(\begin{array}{c} x_1 \\ \vdots \\ x_{r-1}\end{array}\right) & \dfrac C w N g' \end{array}\right), \end{align*}$$

we have that the integral (2.6) is

$$\begin{align*}\begin{aligned} \frac {N(C,w)^d} {w^{d(r-1)}} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_k)}}\\ \in \frac D u {\mathcal R}_D(C)\end{array}} \int_{({\mathbb{R}}^d)^{r-1}} F\left(\sum_{i=1}^k (z_1+\ell_i)E_{i1} + \frac C w \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_{r-1}\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_{r-1}. \end{aligned}\end{align*}$$

If $r=1$ and $\operatorname {\mathrm {rk}}{{}^{\mathrm {t}}{(\mathbf {n}^{\prime }_1, \ldots , \mathbf {n}^{\prime }_r)}}=0$ , that is, ${{}^{\mathrm {t}}{(\mathbf {n}^{\prime }_1, \ldots , \mathbf {n}^{\prime }_r)}}={{}^{\mathrm {t}}{({\mathbf {0}}, \ldots , {\mathbf {0}})}}$ and the integral is

$$\begin{align*}F\left(\sum_{i=1}^k (z_1 +\ell_i)E_{i1}\right), \end{align*}$$

where ${{}^{\mathrm {t}}{(\ell _1, \ldots , \ell _k)}}\neq {{}^{\mathrm {t}}{(0, \ldots , 0)}}$ . Otherwise, they form $\Lambda _D\times \Phi ^{(d-1)}(D,u)$ , and one can proceed the same computation with the first case when $r\ge 2$ .

Now the proposition follows from Lemma 2.9 (c) after rearranging the summation with respect to $C\in {\mathfrak {D}}^{k}_{r-1, w}$ for $1\le r-1 \le k$ and $w\in {\mathbb {N}}$ .

For each ${\mathbf {y}}\in {\mathbb {R}}^d \smallsetminus \{{\mathbf {0}}\}$ , define

$$\begin{align*}X_q({\mathbf{y}})=\left\{\Gamma(q) g\in X_q : {\mathbf{y}} \in \left({\mathbb{Z}}^d+ \frac {{\mathbf{p}}} q \right)g\right\}. \end{align*}$$

It is known that for each $t\in {\mathbb {N}}$ with $\gcd (t,q)=1$ , there is ${\mathbf {k}}_t\in {\mathbb {Z}}^d+{\mathbf {p}}/q$ with $\gcd (q{\mathbf {k}}_t)=t$ so that we have the decomposition

$$\begin{align*}X_q({\mathbf{y}})= \bigsqcup_{\scriptsize \begin{array}{c} t\in {\mathbb{N}} \\ (t,q)=1\end{array}}X_q({\mathbf{k}}_t, {\mathbf{y}}), \end{align*}$$

where $X_q({\mathbf {k}}_t, {\mathbf {y}}) :=\{\Gamma (q)g\in X_q: {\mathbf {k}}_t g={\mathbf {y}}\}$ (see [11, Page 1993] for details). Note that the above decomposition holds for any such choice of $\mathbf {k}_t$ . Moreover, if we put $g_t\in \mathrm {SL}_d({\mathbb {R}})$ for each $t\in {\mathbb {N}}$ with $\gcd (t,q)=1$ and $g_{\mathbf {y}}\in \mathrm {SL}_d({\mathbb {R}})$ , respectively, such that $\mathbf {e}_1 g_t={\mathbf {k}}_t$ and $\mathbf {e}_1g_{\mathbf {y}}={\mathbf {y}}$ , it follows that

(2.7) $$ \begin{align} X_q({\mathbf{k}}_t, {\mathbf{y}})\simeq g_t^{-1}\left( (g_t\Gamma(q)g_t^{-1}\cap H)\backslash H\right) g_{,}y \end{align} $$

and one can define the probability measure $\nu _{\mathbf {y}}$ on $X_q({\mathbf {y}})$ for which $\nu _{\mathbf {y}}|_{X_q({\mathbf {k}}_t, {\mathbf {y}})}$ is the pull-back measure of $\frac 1 {I_q\zeta (d)} \mu _H$ , where $I_q:=[\mathrm {SL}_d({\mathbb {Z}}):\Gamma (q)]$ , with respect to the above identification (see [11], especially (7.10) $\sim $ (7.15) and Proposition 7.5).

Proposition 2.10. Let $d\ge 3$ and $1\le k \le d-1$ . Suppose that ${{\mathbf {p}}}\in {\mathbb {Z}}^d \smallsetminus \{{\mathbf {0}}\}$ and $q\in {\mathbb {N}}_{\ge 2}$ such that $\gcd (q, {{\mathbf {p}}})=1$ . Let $P_t({\mathcal R}(D))$ be as in Notation 2.6 after fixing ${\mathcal R}(D)$ for each $D\in {\mathfrak {D}}^k_{r,u}$ .

Let $F:({\mathbb {R}}^d)^k\rightarrow {\mathbb {R}}_{\ge 0}$ be a bounded and compactly supported function. For any ${\mathbf {y}}\in {\mathbb {R}}^d\smallsetminus \{{\mathbf {0}}\}$ , it follows that

$$\begin{align*}\begin{aligned}&\int_{X_q({\mathbf{y}})} {\mathcal S}_{k}(F) \left(\left({\mathbb{Z}}^d+ \frac {{\mathbf{p}}} q\right)g \right) {\hspace{0.5mm} {\mathrm{d}}} \nu_{{\mathbf{y}}}(g)\\[5pt]&=F\left({{}^{\mathrm{t}}{({\mathbf{y}}, \ldots, {\mathbf{y}})}}\right)+\sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}}\\ (t,q)=1\end{array}} \frac 1 {t^d} \sum_{\scriptsize \begin{array}{c} {{}^{\mathrm{t}}{(\ell_1, \ldots, \ell_k)}}\in {\mathbb{Z}}^k\\ (\ell_1, \ldots, \ell_k, t)=1\end{array}}F\left({{}^{\mathrm{t}}{\left(\frac {t+\ell_1 q} t {\mathbf{y}}, \ldots, \frac{t+\ell_k} t {\mathbf{y}}\right)}}\right)\\[5pt]&+\sum_{r=1}^{k-1}\sum_{u\in {\mathbb{N}}} \sum_{D\in {\mathfrak{D}}^k_{r,u}}\left[\frac {N(D,u)^d}{u^{dr}}\int_{({\mathbb{R}}^d)^r} F\left(\left(\begin{array}{c} {\mathbf{y}} \\ \vdots \\ {\mathbf{y}} \end{array}\right) + \frac D u \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_r\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_r\right.\\[5pt]&+\sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}}\\ (t,q)=1\end{array}} \sum_{\scriptsize \begin{array}{c} \boldsymbol{\ell}\in\\ P_t({\mathcal R}(D))\end{array}}\frac {N(D,u)^{d}} {t^d\cdot u^{dr}}\times\\[5pt]&\hspace{0.8in}\left.\int_{({\mathbb{R}}^d)^r} F\left( \left(\begin{array}{c} \dfrac{t+\ell_1q}{t} {\mathbf{y}} \\ \vdots \\ \dfrac{t+\ell_k q}{t} {\mathbf{y}}\end{array}\right) + \frac D u \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots\\\mathbf{x}_r\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_r\right]\\[5pt]&+\int_{({\mathbb{R}}^d)^k} F\left({{}^{\mathrm{t}}{(\mathbf{x}_1, \ldots, \mathbf{x}_k)}}\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_k. \end{aligned} \end{align*}$$

Proof. Recall the definitions of $g_t$ , $g_{\mathbf {y}}$ as in (2.7). If we let $a_{t/q}=\mathrm {diag}(t/q, q/t, 1, \ldots , 1),$ then one can further assume that $g_t=a_{t/q}\gamma _t$ for some $\gamma _t\in {\mathrm {SL}}_d({\mathbb {Z}})$ ([11, Page 1993]). By the definition of $\nu _{{\mathbf {y}}}$ on $X_q({\mathbf {y}})$ ,

$$\begin{align*}\begin{aligned} &\int_{X_q({\mathbf{y}})}{\mathcal S}_{k}(F) \left(\left({\mathbb{Z}}^d+ \frac {{\mathbf{p}}} q\right)g \right) {\hspace{0.5mm} {\mathrm{d}}} \nu_{{\mathbf{y}}}(g)\\[5pt]&=\frac 1 {I_q \zeta(d)} \sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}}\\(t,q)=1\end{array}}\int_{(g_t \Gamma(q) g_t^{-1} \cap H)\backslash H}\sum_{\scriptsize \begin{array}{c} {\mathbf{m}}_i \in {\mathbb{Z}}^d \\1\le i\le k\end{array}}F\left(\left(\begin{array}{c} {\mathbf{m}}_1+{{\mathbf{p}}}/q \\ \vdots \\{\mathbf{m}}_k+{{\mathbf{p}}}/q \end{array}\right) g_t^{-1}h g_{\mathbf{y}}\right) {\hspace{0.5mm} {\mathrm{d}}}\mu_H (h)\\[5pt]&=\frac {q^d} {I_q \zeta(d)} \sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}}\\(t,q)=1\end{array}} \frac 1 {t^d}\int_{(\Gamma(q)\cap H)\backslash H}\sum_{\scriptsize \begin{array}{c} {\mathbf{m}}_i \in {\mathbb{Z}}^d \\1\le i \le k\end{array}}F\left(\left(\begin{array}{c} {\mathbf{m}}_1+{{\mathbf{p}}}/q \\ \vdots \\{\mathbf{m}}_k+{{\mathbf{p}}}/q \end{array}\right) \gamma_t^{-1}h a_{t/q}^{-1}g_{\mathbf{y}}\right) {\hspace{0.5mm} {\mathrm{d}}}\mu_H (h). \end{aligned} \end{align*}$$

Note that $({\mathbb {Z}}^d+{{\mathbf {p}}}/q)\gamma _t^{-1}= ({\mathbb {Z}}^d+{\mathbf {k}}_t)\gamma _t^{-1}= {\mathbb {Z}}^d+ (t/ q) \mathbf {e}_1$ and $(\Gamma (q)\cap H)\setminus H$ is a $(q^{d-1}I_q^{(d-1)})$ -covering of $(\mathrm {SL}_d({\mathbb {Z}})\cap H)\setminus H$ , where $I_q^{(d-1)}:=[\mathrm {SL}_{d-1}({\mathbb {Z}}): \mathrm {SL}_{d-1}({\mathbb {Z}})\cap \Gamma (q)]$ , and one can apply Proposition 2.8. Since $E_{i1}a_{t/q}^{-1}=(q/t) E_{i1}$ , the above expression equals

$$\begin{align*}\begin{aligned} \frac {q^{2d-1}I_q^{(d-1)}} {I_q \zeta(d)}\sum_{\scriptsize \begin{array}{c} t\in {\mathbb{N}}\\(t,q)=1\end{array}}\frac 1 {t^d}&\left[\vphantom{\left(\begin{array}{c} \dfrac{t+\ell_1q}{t} {\mathbf{y}} \\ \vdots \\ \dfrac{t+\ell_k q}{t} {\mathbf{y}}\end{array}\right)}\sum_{\ell_1, \ldots, \ell_k\in {\mathbb{Z}}}F\left( {{}^{\mathrm{t}}{\left(\frac {t+\ell_1q} t {\mathbf{y}}, \ldots, \frac {t+\ell_k q} t {\mathbf{y}}\right)}}\right)\right.\\[5pt]&+\sum_{r=1}^k \sum_{u\in {\mathbb{N}}} \sum_{D\in {\mathfrak{D}}_{r,u}}\sum_{\scriptsize \begin{array}{c} \boldsymbol{\ell}={{}^{\mathrm{t}}{(\ell_1,\ldots, \ell_k)}}\\\in {\mathcal R}(D)\end{array}} \frac {N(D,u)^{d}} {u^{dr}}\times\\[5pt]&\hspace{-0.2in}\left.\int_{({\mathbb{R}}^d)^r} F\left( \left(\begin{array}{c} \dfrac{t+\ell_1q}{t} {\mathbf{y}} \\ \vdots \\ \dfrac{t+\ell_k q}{t} {\mathbf{y}}\end{array}\right) +\frac D u \left(\begin{array}{c} \mathbf{x}_1\\ \vdots\\\mathbf{x}_r\end{array}\right)a_{t/q}^{-1} g_{{\mathbf{y}}}\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_1\cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_r\right]. \end{aligned} \end{align*}$$

We will use the well-known fact that

$$\begin{align*}\frac {I_q \zeta(d)}{q^{2d-1}I_q^{(d-1)}} =\sum_{\scriptsize \begin{array}{c} t_1\in {\mathbb{N}}\\ (t_1, q)=1\end{array}} \frac 1 {t_1^d}. \end{align*}$$

For the first summation, which is the case when $r=0$ , put $t=t_1\cdot t_2$ , where $t_1=\gcd (\ell _1,\ldots , \ell _k,t)$ . By renaming $(\ell _1/t_1, \ldots , \ell _k/t_1)$ by $(\ell _1, \ldots , \ell _k)$ , it follows that

For the case when $r=k$ , we only have $u=1$ , $D=\mathrm {Id}_k$ and ${\mathcal R}(D)=\{{{}^{\mathrm {t}}{(0,\ldots ,0)}}\}$ . After a change of variables, the integral in this case is

$$\begin{align*}\begin{aligned}&\frac {q^{2d-1} I_q^{(d-1)}} {I_q \zeta(d)}\sum_{\scriptsize \begin{array}{c}t\in {\mathbb{N}} \\(t,q)=1 \end{array}} \frac 1 {t^d}\int_{({\mathbb{R}}^d)^k} F\left( \left(\begin{array}{c} {\mathbf{y}} \\ \vdots \\ {\mathbf{y}}\end{array}\right) + \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_k\end{array}\right) a^{-1}_{t/q} g_{\mathbf{y}} \right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_k\\&=\frac {q^{2d-1} I_q^{(d-1)}} {I_q \zeta(d)}\sum_{\scriptsize \begin{array}{c}t\in {\mathbb{N}} \\(t,q)=1 \end{array}} \frac 1 {t^d}\int_{({\mathbb{R}}^d)^k} F \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_k\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_k\\&=\int_{({\mathbb{R}}^d)^k} F \left(\begin{array}{c} \mathbf{x}_1 \\ \vdots \\ \mathbf{x}_k\end{array}\right) {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}} \mathbf{x}_k.\end{aligned} \end{align*}$$

Suppose that for $1\le r \le k-1$ and $u\in {\mathbb {N}}$ , $D\in {\mathfrak {D}}^k_{r,u}$ and ${\mathcal R}(D)$ are given. By rearranging the summation, it holds that

$$\begin{align*}\begin{aligned}&\frac {q^{2d-1} I_q^{(d-1)}} {I_q \zeta(d)} \sum_{\scriptsize \begin{array}{c}t\in {\mathbb{N}} \\(t,q)=1 \end{array}} \frac 1 {t^d}\sum_{\scriptsize \begin{array}{c} \boldsymbol{\ell}={{}^{\mathrm{t}}{(\ell_1,\ldots, \ell_k)}}\\\in {\mathcal R}(D)\end{array}} \frac {N(D,u)^{d}} {u^{dr}}\\&\hspace{0.8in}\int_{({\mathbb{R}}^d)^r} F\left( \left(\begin{array}{c} \dfrac{t+\ell_1q}{t} {\mathbf{y}} \\ \vdots \\ \dfrac{t+\ell_k q}{t} {\mathbf{y}}\end{array}\right) +\frac D u \left(\begin{array}{c} \mathbf{x}_1\\ \vdots\\\mathbf{x}_r\end{array}\right)a_{t/q}^{-1} g_{{\mathbf{y}}}\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_1\cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf{x}_r\\\end{aligned} \end{align*}$$

where for the first equality, as before, we put $t=t_1t_2$ with $t_1=\gcd (\boldsymbol {\ell },t)$ and rename $t_2$ and $\boldsymbol {\ell }/t_1$ by t and $\boldsymbol {\ell }$ , respectively. This completes the proof of the proposition.

To prove Theorem 2.13, we need one more lemma which has appeared in [6, (3.6)] and also in [11, (7.25)].

Lemma 2.11 [6, (3.6)]

For a Borel measurable function $\varphi : X_q\times {\mathbb {R}}^d\rightarrow {\mathbb {R}}_{\ge 0}$ , we have

$$\begin{align*}\frac 1 {I_q} \int_{X_q} \sum_{{\mathbf{m}}\in {\mathbb{Z}}^d} \varphi\left(\Gamma(q)g, \left({\mathbf{m}}+\frac {{\mathbf{p}}} q\right) g\right){\hspace{0.5mm} {\mathrm{d}}}\mu(g) =\int_{{\mathbb{R}}^d \smallsetminus \{{\mathbf{0}}\}}\int_{X_q({\mathbf{y}})} \varphi(\Gamma(q)g, {\mathbf{y}}) {\hspace{0.5mm} {\mathrm{d}}} \nu_{{\mathbf{y}}}(g){\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}. \end{align*}$$

Proof of Theorem 2.7

Let $F : (\mathbb {R}^{d})^k \to \mathbb {R}_{\geq 0}$ be compactly supported bounded and

$$\begin{align*}\begin{aligned} \varphi(\Gamma(q)g, {\mathbf{y}}) &=\sum_{\scriptsize \begin{array}{c} {\mathbf{m}}_i\in {\mathbb{Z}}^d\\ 1\le i \le k-1\end{array}} F\left({\mathbf{y}}, \left({\mathbf{m}}_1+\frac {{\mathbf{p}}} q\right)g, \ldots, \left({\mathbf{m}}_{k-1}+\frac {{\mathbf{p}}} q\right)g\right)\\ &={\mathcal S}_{k-1}(F_{\mathbf{y}})\left(\left({\mathbb{Z}}^d+\frac {{\mathbf{p}}} q\right)g\right), \end{aligned}\end{align*}$$

where $F_{\mathbf {y}}:({\mathbb {R}}^d)^{k-1}\rightarrow {\mathbb {R}}_{\ge 0}$ is defined by

$$\begin{align*}F_{\mathbf{y}}({\mathbf{y}}_2, \ldots, {\mathbf{y}}_k)=F({\mathbf{y}}, {\mathbf{y}}_2, \ldots, {\mathbf{y}}_k). \end{align*}$$

By Lemma 2.11, we have that

$$\begin{align*}\begin{aligned} \frac 1 {J_q} \int_{Y_{{{\mathbf{p}}}/q}} {\mathcal S}_{k}(F) (\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu(\Lambda) &=\frac 1 {I_q} \int_{X_q} \sum_{{\mathbf{m}}\in {\mathbb{Z}}^d} \varphi\left(\Gamma(q)g,\left({\mathbf{m}}+\frac {{\mathbf{p}}} q\right)g \right) {\hspace{0.5mm} {\mathrm{d}}}\mu(g)\\ &=\int_{{\mathbb{R}}^d \smallsetminus \{{\mathbf{0}}\}}\int_{X_q({\mathbf{y}})} \varphi(\Gamma(q)g, {\mathbf{y}}) {\hspace{0.5mm} {\mathrm{d}}} \nu_{{\mathbf{y}}}(g){\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}\\ &=\int_{{\mathbb{R}}^d \smallsetminus \{{\mathbf{0}}\}}\int_{X_q({\mathbf{y}}_1)} {\mathcal S}_{k-1}(F_{{\mathbf{y}}_1})(\Gamma(q)g) {\hspace{0.5mm} {\mathrm{d}}} \nu_{{\mathbf{y}}_1}(g) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1. \end{aligned}\end{align*}$$

For the first equality, let us recall that $X_q=\Gamma (q)\setminus \mathrm {SL}_d({\mathbb {R}})$ is a $I_q/J_q$ -covering of $Y_{{\mathbf {p}}/q}$ , where $I_q=[\mathrm {SL}_d({\mathbb {Z}}):\Gamma (q)]$ and $J_q=[\mathrm {SL}_d({\mathbb {Z}}):\Gamma _1(q)]$ .

Thus, for $F \geq 0$ , the equations in Theorem 2.7 immediately follow from Proposition 2.10, where we replace ${\mathbf {y}}_1$ by $\frac 1 t {\mathbf {y}}_1$ .

Let us deal with the finiteness claim for $F \geq 0$ : we will show that the LHS of (2.4) is finite. Define $\Phi : Y_{\mathbf {p}/q} \to X_d$ by $\Phi (\Lambda ) := \Lambda - \Lambda $ for every $\Lambda \in Y_{\mathbf {p}/q}$ . This map induces the measure $\Phi _*(\mu _q)$ on $X_d$ , which is easily seen to be $\mathrm {SL}_d(\mathbb {R})$ -invariant. Therefore, $\Phi _*(\mu _q)$ equals to $\mu $ up to a positive constant. In fact, $\Phi $ is the natural $J_q$ -to-1 covering map from $Y_{\mathbf {p}/q}$ to $X_d$ ; thus, $\Phi _*(\mu ) = J_q\mu $ . For $\Lambda \in Y_{\mathbf {p}/q}$ , we have $\Lambda \subseteq q^{-1}\Phi (\Lambda )$ . Therefore, $\mathcal {S}_k(F)(\Lambda ) \leq \mathcal {S}_k(F_q)(\Phi (\Lambda ))$ , where $F_q : (\mathbb {R}^{d})^k \to \mathbb {R}_{\geq 0}$ , $\mathbf {x} \mapsto F(q^{-1}\mathbf {x})$ is a compactly supported function. Hence,

$$ \begin{align*} \int_{Y_{\mathbf{p}/q}} \mathcal{S}_k(F)(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}{\mu_q(\Lambda)} &\leq \int_{Y_{\mathbf{p}/q}} \mathcal{S}_k(F_q)(\Phi(\Lambda)) {\hspace{0.5mm} {\mathrm{d}}}{\mu_q(\Lambda)} \\ &= J_q\int_{X_d} \mathcal{S}_k(F_q)(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}{\mu(\Lambda)} < \infty, \end{align*} $$

by Schmidt [17]. Thus, in the present case, the sum in the RHS of (2.4) is convergent.

We can now use classical techniques to prove Theorem 2.7 for a compactly supported bounded function $F : (\mathbb {R}^{d})^k \to \mathbb {R}$ . We first note that Theorem 2.7 holds for $F_+ := \max (F, 0)$ and $F_- := \max (-F, 0)$ . Finiteness for the function $|F|$ implies that the sum with F in the RHS of (2.4) is absolutely convergent. Furthermore, $\mathcal {S}_k(F)(\Lambda ) = \mathcal {S}_k(F_+)(\Lambda ) - \mathcal {S}_k(F_-)(\Lambda )$ (for a.e. $\Lambda $ ); we can integrate and rearrange to see that Theorem 2.7 holds.

2.3 Higher moment formulae revisited

For applications to Poisson distribution which are proved in the next section, we will need that the “admissible matrices” appearing in the higher moment formula for Y are contained in ${\mathfrak {D}}^{k'}_{r',u'}$ for some $k', r'$ and $u'$ , which does not hold in Theorem 2.4 and Theorem 2.7. In the process of proving the needed variations of the higher moment formulae, we will define canonical sets of admissible matrices for each cases. In particular, we will see that the set of “congruence-admissible matrices” can be defined without using any choice of ${\mathcal R}(D)$ in Notation 2.6.

Let us first refine the higher moment formula for the space Y of affine lattices in ${\mathbb {R}}^d$ .

Theorem 2.12. Let $F:({\mathbb {R}}^d)^k\rightarrow {\mathbb {R}}_{\ge 0}$ be bounded and compactly supported. For $d\ge 3$ and $3\le k\le d$ ,

(2.8) $$ \begin{align} \int_Y {\mathcal S}_{k}(F)(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_Y(\Lambda) =\sum_{m=1}^k \sum_{u\in {\mathbb{N}}} \sum_{\widetilde{D}\in {\mathfrak{A}}^k_{m,u}} \frac {N(\widetilde{D},u)^d} {u^{dm}} \int_{({\mathbb{R}}^d)^m} F\left(\frac {\widetilde{D}} u \left(\begin{array}{c} {\mathbf{y}}_1 \\ \vdots \\ {\mathbf{y}}_m \end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_m, \end{align} $$

where ${\mathfrak {A}}^k_{m,u}$ is a subset of ${\mathfrak {D}}^k_{m,u}$ given by

$$\begin{align*}{\mathfrak{A}}^k_{m,u}=\left\{C\in {\mathfrak{D}}^k_{m,u} : \sum_{i=1}^{m} [C]^i={{}^{\mathrm{t}}{(u, \ldots, u)}} \right\}. \end{align*}$$

Notice that when $m=1$ and k, the only possible u is $u=1$ and

$$\begin{align*}{\mathfrak{A}}^k_{1,1}=\left\{{{}^{\mathrm{t}}{(1, \ldots, 1)}}\right\} \quad\text{and}\quad {\mathfrak{A}}^k_{k,1}=\left\{\mathrm{Id}_k\right\}, \end{align*}$$

which corresponds to the first and second integrals of the RHS in (2.2), respectively.

Proof. Assume that $2\le m\le k-1$ so that $1\le r:=m-1\le k-2$ . Recall the $k\times m$ matrix $D'$ in Theorem 2.4 from $D\in {\mathfrak {D}}^{k-1}_{r,u}$ .

Take the map

(2.9) $$ \begin{align} D\in {\mathfrak{D}}^{k-1}_{r,u} \quad\mapsto\quad uD' \quad\mapsto\quad \widetilde{D} \in {\mathfrak{D}}^k_{m,u}, \end{align} $$

where we define $[\widetilde {D}]^1=[uD']^1 -\sum _{j=2}^m [uD']^j$ and $[\widetilde {D}]^j=[uD']^j$ for $2\le j\le m$ . Clearly, the map is injective and $\widetilde D\in {\mathfrak {A}}^k_{m,u}$ .

Conversely, for any $C\in {\mathfrak {A}}^k_{m,u}$ , denote by D the right-bottom minor of C of the size $(k-1)\times r$ . Then one can verify that $D\in {\mathfrak {D}}^{k-1}_{r,u}$ and $\widetilde D=C$ .

Moreover, it is easy to show from their definitions that

$$\begin{align*}\frac {N(\widetilde{D},u)^d} {u^{d(r+1)}} =\frac {N(D,u)^d} {u^{dr}}, \end{align*}$$

and the map $uD' \mapsto \widetilde D$ is the simple change of variables ${\mathbf {y}}_{j}+{\mathbf {y}}_1 \mapsto {\mathbf {y}}^{\prime }_{j}$ for $2\le j \le m$ :

$$\begin{align*}\int_{(\mathbb{R}^{d})^{m}} F {\left({D'} \begin{pmatrix} \mathbf{y}_1 \\ \vdots \\ \mathbf{y}_{m} \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_{m}} = \int_{(\mathbb{R}^{d})^{m}} F {\left(\frac{\widetilde D}{u} \begin{pmatrix} \mathbf{y}_1 \\ \vdots \\ \mathbf{y}_{m} \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_{m}}. \\[-54pt]\end{align*}$$

In contrast to the affine case, in the congruence case it is difficult and complicated to describe the subset of matrices $\widetilde {D}\in {\mathfrak {D}}^k_{m,u_0}$ , for given $1\le m \le k$ and $u_0\in {\mathbb {N}}$ , such that

$$\begin{align*}\widetilde{D}{\mathbb{R}}^m= D'{\mathbb{R}}^m \;\text{or}\; D^{\prime}_{t,\boldsymbol{\ell}}{\mathbb{R}}^m \end{align*}$$

for some $t\in {\mathbb {N}}$ with $(t,q)=1$ and $\boldsymbol {\ell }\in P_t({\mathcal R}(D))$ appearing in Theorem 2.7.

For each $u\in {\mathbb {N}}$ and $D\in {\mathfrak {D}}^{k-1}_{m-1,u}$ , once we fix ${\mathcal R}(D)$ in Notation 2.6, by the map

(2.10) $$ \begin{align}\begin{gathered} D \mapsto uD'\mapsto \widetilde{D}\;\text{as in}\ (2.9)\\ (D, t, \boldsymbol{\ell}) \mapsto u_0 D^{\prime}_{t,\boldsymbol{\ell}}=u_0\left(\begin{array}{c|c} 1 & 0 \, \cdots \, 0 \\ \hline \begin{array}{c} (t+\ell_1q)/t \\ \vdots \\ (t+\ell_{k-1}q)/t \end{array} & \dfrac 1 u D \end{array}\right) \mapsto \widetilde{D}, \end{gathered}\end{align} $$

where $\widetilde D$ is defined by

$$\begin{align*}[\widetilde{D}]^1=[u_0D^{\prime}_{t,\boldsymbol{\ell}}]^1-\sum_{j=1}^{m-1} \frac {t+\ell_{i_j}q} t[u_0D^{\prime}_{t,\boldsymbol{\ell}}]^{j+1} \quad\text{and}\quad [\widetilde{D}]^j=[u_0D^{\prime}_{t,\boldsymbol{\ell}}]^j\;(j=2,\ldots, m), \end{align*}$$

and $u_0\in {\mathbb {N}}$ is taken such that $\widetilde {D}\in \mathrm {Mat}_{k,m}({\mathbb {Z}})$ with $\gcd \widetilde {D}=1$ . Clearly, $\widetilde D\in {\mathfrak {D}}^k_{m, u_0}$ .

Hence, one can attempt to define such a subset ${\mathfrak {C}}^k_{m,u_0}$ of ${\mathfrak {D}}^k_{m,u_0}$ by

(2.11) $$ \begin{align} {\mathfrak{C}}^k_{m,u_0}:= \left\{C\in {\mathfrak{D}}^k_{m,u_0}: \begin{array}{c} C=\widetilde{D}\;\text{for some}\;D\in {\mathfrak{D}}^{k-1}_{m-1,u}\;\text{or}\\ (D,t,\boldsymbol{\ell})\in {\mathfrak{D}}^{k-1}_{m-1,u}\times {\mathbb{N}}\times {\mathcal R}(D)\;\text{in Notation}\ 2.6\\ \text{defined as in}\ (2.10) \end{array} \right\} \end{align} $$

and reformulate the higher moment formula using these ${\mathfrak {C}}^k_{m,u_0}$ .

As things stand, ${\mathfrak {C}}^k_{m,u_0}$ seems to depend on an ad hoc choice of a set of representatives $\mathcal {R}(D)$ . However, the anonymous referee has kindly provided us with an argument using the Riesz representation theorem which shows that the set ${\mathfrak {C}}^k_{m,u_0}$ is independent to the choice of ${\mathcal R}(D)$ regardless of its role in the construction. With this as background, we now provide a cleaner definition of the set ${\mathfrak {C}}^k_{m,u_0}$ , meaning that we do not need an ad hoc choice of ${\mathcal R}(D)$ for each $D\in {\mathfrak {D}}^{k-1}_{m-1,r}$ . This definition was also suggested by the referee.

Theorem 2.13. Let $d\ge 3$ and $1\le k \le d-1$ . Let $F:({\mathbb {R}}^d)^k\rightarrow {\mathbb {R}}_{\ge 0}$ be bounded and compactly supported. Then

(2.12) $$ \begin{align} \begin{aligned} &\int_{Y_{{{\mathbf{p}}}/q}} {\mathcal S}_{k}(F) (\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_q(\Lambda) =\int_{({\mathbb{R}}^d)^k} F\left({{}^{\mathrm{t}}{({\mathbf{y}}_1, \ldots, {\mathbf{y}}_k)}}\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_k\\ &\hspace{0.5in}+\sum_{m=1}^{k-1} \sum_{u\in {\mathbb{N}}} \sum_{\widetilde {D}\in {\mathfrak{C}}^k_{m,u_0}} \frac {N(\widetilde {D},u_0)^d} {u_0^{dm}} \int_{({\mathbb{R}}^d)^m} F\left(\frac {\widetilde D} {u_0} \left(\begin{array}{c} {\mathbf{y}}_1 \\ \vdots \\ {\mathbf{y}}_m\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}}_m, \end{aligned}\end{align} $$

where for $1\le m\le k-1$ and $u_0\in {\mathbb {N}}$ ,

(2.13) $$ \begin{align} {\mathfrak{C}}^k_{m,u_0} =\left\{C\in {\mathfrak{D}}^k_{m,u_0}: \exists \mathbf{v}=\left(\hspace{-0.05in}\begin{array}{c} v_1 \\ \vdots \\ v_k\end{array}\hspace{-0.05in}\right)\hspace{-0.05in} \in \frac C {u_0} \Lambda_C\;\;\text{s.t.}\begin{array}{c} \gcd(v_1,q)=1,\\ v_1\equiv \cdots \equiv v_k\quad\mod q,\;\text{and}\\ |v_1|=\min ({\mathbb{N}} \cap \{\mathbf{v}'\cdot \mathbf{e}_1: \mathbf{v}'\in \frac C u \Lambda_C \})\end{array} \right\}. \end{align} $$

Here, $\mathbf {e}_1={{}^{\mathrm {t}}{(1,0,\ldots , 0)}}\in {\mathbb {R}}^k$ and $\mathbf {v}_1\cdot \mathbf {v}_2={{}^{\mathrm {t}}{\mathbf {v}_1}}\mathbf {v}_2$ is the standard dot product of ${\mathbb {R}}^k$ .

Proof. We will consider the case when $m\ge 2$ ; then the case when $m=1$ would be easily seen. Let us first show that the sets defined as in (2.11) and (2.13) are identical.

Assume that $C=\widetilde {D}$ is an element of the set in (2.11). Note that

$$\begin{align*}\frac C {u_0} \Lambda_C = C{\mathbb{R}}^m \cap {\mathbb{Z}}^k = D'{\mathbb{R}}^m \cap {\mathbb{Z}}^k\;\text{or}\; D^{\prime}_{t,\boldsymbol{\ell}}{\mathbb{R}}^m \cap {\mathbb{Z}}^k, \end{align*}$$

where $D'$ or $D^{\prime }_{t,\boldsymbol {\ell }}$ is as in (2.10) for some $D\in {\mathfrak {D}}^{k-1}_{r,u}$ ( $r:=m-1$ ), or $D\in {\mathfrak {D}}^{k-1}_{r,u}$ , $t\in {\mathbb {N}}$ with $(t,q)=1$ , and $\boldsymbol {\ell }={{}^{\mathrm {t}}{(\ell _1, \ldots , \ell _{k-1})}} \in {\mathcal R}(D)\subset {\mathbb {Z}}^{k-1}$ settled in Notation 2.6, respectively. In particular,

$$\begin{align*}\mathbf{v}:={{}^{\mathrm{t}}{(1, \ldots, 1)}} \quad\text{or}\quad {{}^{\mathrm{t}}{\left(t, t+\ell_1 q, \ldots, t+\ell_{k-1}q\right)}} \in \frac C {u_0} \Lambda_C,\end{align*}$$

respectively.

It suffices to show that

(2.14) $$ \begin{align} \frac C {u_0} \Lambda_C= {\mathbb{Z}}\mathbf{v} \oplus {{}^{\mathrm{t}}{\left(0, \dfrac D u \Lambda_D \right)}}, \end{align} $$

where ${{}^{\mathrm {t}}{\left (0, \frac D u \Lambda _D \right )}}$ is the embedded image of $\frac D u \Lambda _D\subseteq {\mathbb {R}}^{k-1}$ into the last $(k-1)$ coordinates of ${\mathbb {R}}^k$ , since then it gives the fact that $v_1=\min ({\mathbb {N}} \cap \{\mathbf {v}'\cdot \mathbf {e}_1: \mathbf {v}'\in \frac C u \Lambda _C \})$ . The inclusion of the reverse direction is obvious.

Pick an arbitrary $\mathbf w\in \frac C {u_0} \Lambda _C$ . Since $C{\mathbb {R}}^m= {\mathbb {R}} \mathbf {v} \oplus {{}^{\mathrm {t}}{(0, D{\mathbb {R}}^{r-1})}}$ , one can take

$$\begin{align*}\mathbf w=\frac {c_1}{c_2}\mathbf{v}+ {{}^{\mathrm{t}}{(0,\mathbf{v}')}},\end{align*}$$

where $c_1\in {\mathbb {Z}}$ , $c_2\in {\mathbb {N}}$ with $\gcd (c_1,c_2)=1$ and $\mathbf {v}'\in D{\mathbb {R}}^{r-1}\subseteq {\mathbb {R}}^{k-1}$ . Since $\mathbf w\in {\mathbb {Z}}^k$ , it holds that

(2.15) $$ \begin{align} \frac {c_1}{c_2}v_1 \in {\mathbb{Z}} \;\Leftrightarrow\; c_2|v_1 \quad\text{and}\quad \frac {c_1}{c_2}q\boldsymbol{\ell} + \mathbf{v}' \in {\mathbb{Z}}^{k-1}. \end{align} $$

If $v_1=1$ , then automatically $c_2=1$ and $\mathbf {v}'\in {\mathbb {Z}}^{k-1}\cap D{\mathbb {R}}^r=\frac D u \Lambda _D$ , which implies that $\mathbf w\in {\mathbb {Z}}\mathbf {v} \oplus {{}^{\mathrm {t}}{\left (0, \frac D u \Lambda _D\right )}}$ .

Suppose that $v_1=t\ge 2$ so that $\boldsymbol {\ell }\neq {{}^{\mathrm {t}}{(0,\ldots ,0)}}$ . Denote $\ell =\gcd (\boldsymbol {\ell })$ and $\widehat {\boldsymbol {\ell }}=\frac 1 \ell \boldsymbol {\ell }$ , the primitive vector of the $\boldsymbol {\ell }$ -direction. Following Notation 2.6, let $\mathbf b_{k-m},\ldots , \mathbf b_{k-1}$ be the basis of $\frac D u \Lambda _D$ . Then it follows from the definition of ${\mathcal R}(D)$ that $\{\widehat {\boldsymbol {\ell }}, \mathbf b_{k-m}, \ldots , \mathbf b_{k-1}\}$ is a primitive set; that is,

$$\begin{align*}{\mathbb{Z}}\widehat{\boldsymbol{\ell}}\,{\oplus}\, {\mathbb{Z}} \mathbf b_{k-m} \oplus \cdots \oplus {\mathbb{Z}} \mathbf b_{k-1} =\left({\mathbb{R}}\widehat{\boldsymbol{\ell}}\oplus {\mathbb{R}} \mathbf b_{k-m} \oplus \cdots \oplus {\mathbb{R}} \mathbf b_{k-1}\right) \cap {\mathbb{Z}}^{k-1}. \end{align*}$$

Hence, the second condition in (2.15) implies that

$$\begin{align*}\frac {c_1}{c_2} q \ell \in {\mathbb{Z}} \;\Leftrightarrow\; c_2 | q\ell \quad\text{and}\quad \mathbf{v}'\in \frac D u \Lambda_D. \end{align*}$$

Since $c_2|t$ from (2.15) as well and $\gcd (t,q\ell )=1$ , we obtain the fact that $c_2=1$ and $\mathbf w\in {\mathbb {R}}\mathbf {v}\oplus {{}^{\mathrm {t}}{\left (0, \frac D u \Lambda _D\right )}}$ . And this shows one inclusion.

Conversely, let $C\in {\mathfrak {D}}^k_{m,u_0}$ be such that there is $\mathbf {v}\in \frac C {u_0} \Lambda _C$ satisfying three conditions in (2.13). One can easily extract $D\in {\mathfrak {D}}^k_{r,u}$ from the right-bottom $(k-1)\times r$ -minor of C by making a primitive matrix which will be D, and u is the unique nonzero entry of the first nonzero row of D. Fix any ${\mathcal R}(D)$ .

Notice that the third condition is equivalent to saying that

$$\begin{align*}\frac C {u_0}\Lambda_C= {\mathbb{Z}}\mathbf{v} \oplus \left(\frac C {u_0}{\mathbb{R}}^m \cap {{}^{\mathrm{t}}{\left(0, {\mathbb{Z}}^{k-1}\right)}}\right)={\mathbb{Z}}\mathbf{v}{{}^{\mathrm{t}}{\left(0, \frac D u \Lambda_D\right)}}. \end{align*}$$

Set $v_1=t$ (if $v_1<0$ , replace $\mathbf {v}$ by $-\mathbf {v}$ ). From the first and second conditions, $\gcd (t,q)=1$ and $\mathbf {v}={{}^{\mathrm {t}}{(t, t+\ell ^{\prime }_1q, \ldots , t+\ell ^{\prime }_{k-1}q)}}$ for some $\boldsymbol {\ell }'={{}^{\mathrm {t}}{(\ell ^{\prime }_1, \ldots , \ell ^{\prime }_{k-1})}}\in {\mathbb {Z}}^{k-1}$ . Since ${\mathcal R}(D)\simeq {\mathbb {Z}}^{k-1}/\frac D u \Lambda _D$ , there is the unique $\boldsymbol {\ell }={{}^{\mathrm {t}}{(\ell _1, \ldots , \ell _{k-1})}}\in {\mathcal R}(D)$ for which $\boldsymbol {\ell }+\frac D u \Lambda _D=\boldsymbol {\ell }'+\frac D u \Lambda _D$ and

$$\begin{align*}\frac C {u_0} \Lambda_C={\mathbb{Z}} \left(\begin{array}{c} t \\ t+\ell_1q \\ \vdots \\ t+\ell_{k-1}q \end{array}\right) \oplus \left(\begin{array}{c} 0 \\ \\ \dfrac D u \Lambda_D \\[0.1in] \end{array}\right). \end{align*}$$

This shows that ${{}^{\mathrm {t}}{(t, t+\ell _1q, \ldots , t+\ell _{k-1}q)}}$ is primitive. If $\boldsymbol {\ell }={{}^{\mathrm {t}}{(0,\ldots , 0)}}$ , then it holds that $C=\widetilde {D}$ of the first type described in (2.11). If $\boldsymbol {\ell }\neq {{}^{\mathrm {t}}{(0,\ldots , 0)}}$ , then $\gcd (\boldsymbol {\ell }, t)=1$ so that $\boldsymbol {\ell }\in P_t({\mathcal R}(D))$ and $C=\widetilde {D}$ defined from $(D,t,\boldsymbol {\ell })$ .

Now, to establish the theorem, considering the change of variables on ${\mathbf {y}}_1$ in Theorem 2.7, it is left to show that

$$\begin{align*}\frac {N(\widetilde D, u_0)^d}{u_0^{dm}} = \frac {N(D, u)^d}{t^d \cdot u^{dr}}, \end{align*}$$

where we put $t=1$ when $\widetilde {D}\in {\mathfrak {C}}^k_{m,u_0}$ is of the first type in (2.11). Recall that $N(\widetilde D, u_0)$ is the number of integral solutions ${\mathbf {z}}={{}^{\mathrm {t}}{(z_1, \ldots , z_m)}}\in {\mathbb {Z}}^m$ modulo $u_0$ for which $\frac {\widetilde {D}} {u_0} {\mathbf {z}} \in {\mathbb {Z}}^k$ . Equivalently, $N(\widetilde D, u_0)$ is the number of integral solutions ${\mathbf {z}}\in {\mathbb {Z}}^m$ modulo $u_0$ for which $D'{\mathbf {z}}\in {\mathbb {Z}}^k$ or $D^{\prime }_{t,\boldsymbol {\ell }}{\mathbf {z}}\in {\mathbb {Z}}^k$ , respectively.

Based on (2.14), it follows that $z_1\in t{\mathbb {Z}}$ and there are $(u_0/t)$ -number of such $z_1\in {\mathbb {Z}}$ modulo $u_0$ . Moreover, as long as $z_1\in t{\mathbb {Z}}$ , $t[D']^1\in {\mathbb {Z}}^{k}$ and we reduce that

$$\begin{align*}\frac D u {{}^{\mathrm{t}}{(z_2, \ldots, z_k)}}\in {\mathbb{Z}}^{k-1} \end{align*}$$

modulo $u_0$ , and the number of such ${{}^{\mathrm {t}}{(z_2, \ldots , z_k)}}$ is $(u_0/u)^r(N(D,u))$ . Therefore,

$$\begin{align*}\frac {N(\widetilde D, u_0)^d}{u_0^{dm}} =\frac {(u_0/t)^d \cdot (u_0/u)^{dr} N(D,u)^d} {u_0^{dm}} =\frac {N(D,u)^d}{t^d \cdot u^{dr}}.\\[-42pt] \end{align*}$$

3 Poissonian Behaviour

3.1 Affine case

In this section, we prove Theorem 1.1. Recall that for each $d \geq 2$ , we set $\mathcal {S} = \{S_t: t \geq 0\}$ to be an increasing family of subsets of $\mathbb {R}^{d}$ with $\operatorname {\mathrm {vol}}(S_t) = t$ , and for $\Lambda \in Y$ , set

$$\begin{align*}N_t(\Lambda) := \#{\left(S_t \cap \Lambda\right)}. \end{align*}$$

Denote by $\{N^{\lambda }(t) : t \geq 0\}$ a Poisson process on the non-negative real line with intensity  $\lambda $ .

For $\Lambda \in Y$ , we order the lengths of vectors in $\Lambda $ as $0\leq \ell _1 \leq \ell _2 \leq \ell _3 \leq \cdots $ , and let $\mathscr {V}_i$ denote the volume of the closed ball of radius $\ell _i$ centered at origin. If we take $\mathcal {S} = \{B_t: t \geq 0\}$ to be the family of closed balls with $\operatorname {\mathrm {vol}}(B_t) = t$ around origin, then

$$\begin{align*}N_t(\Lambda) = \#\{i : \mathscr{V}_i \leq t\}. \end{align*}$$

In this specific case, Theorem 1.1 is equivalent to the following:

Theorem 3.1. For any fixed n, the n-dimensional random variable $(\mathscr {V}_1, \ldots , \mathscr {V}_n)$ converges in distribution to the distribution of the first n points of a Poisson process on the non-negative real line with intensity $1$ as $d \to \infty $ .

In this form, the theorem determines the limit distribution of lengths of vectors in a random lattice as $d \to \infty $ .

We will now prove the above general Theorem 1.1 by proving a joint moment formula for $N_t(\cdot )$ . Let $k \geq 1$ and $0 \leq V_1 \leq \cdots \leq V_k$ . We use, by abuse of notation, $N_i( \cdot )$ to denote $N_{V_i}( \cdot )$ . Note that $N_i( \cdot ) = \widehat {\rho _i}( \cdot )$ , where $\rho _i$ is the characteristic function of $S_{V_i}$ . We calculate, following Södergren [23], the “main term” of the joint moment of $N_i$ ’s. In this regard, we apply Theorem 2.12 with $F = \prod _{i = 1}^{k}\rho _i$ defined as

$$\begin{align*}F \begin{pmatrix} \mathbf{y}_1 \\ \vdots \\ \mathbf{y}_k \end{pmatrix} = \prod_{i = 1}^{k} \rho_i(\mathbf{y}_i). \end{align*}$$

We consider the sub-collection of the RHS of (2.8) consisting of terms corresponding to $m = 1$ and k, and terms from the sum corresponding to $u = 1$ and $\widetilde {D} \in \mathfrak {A}_{m, 1}^{k}$ satisfying that $\widetilde {D}$ has exactly one nonzero entry in each row, with all nonzero entries of $\widetilde {D}$ being of modulus 1. The set of such matrices $\widetilde {D} \in \mathfrak {A}_{m, 1}^{k}$ is $\mathfrak {M}^k$ , where

$$ \begin{align*} \mathfrak{R}_1^k &= \bigcup_{2 \leq m \leq k-1} {\left({\left(\bigcup_{u \geq 2} \mathfrak{A}_{m, u}^k\right)} \cup {\left\{\widetilde{D} = {\left(\widetilde{D}_{ij}\right)} \in \mathfrak{A}_{m, 1}^k : \exists |\widetilde{D}_{ij}| \geq 2\right\}}\right)}, \\ \mathfrak{R}_{2}^k &= {\left\{\widetilde{D} \in {\left(\bigcup_{2 \leq m \leq k-1} \mathfrak{A}_{m, 1}^k\right)} \smallsetminus \mathfrak{R}_1^k: \substack{\displaystyle\exists~\text{row such that at least} \\ \\ \displaystyle\text{two entries are nonzero}}\right\}}, \\ \mathfrak{M}^k &= {\left(\bigcup_{2 \leq m \leq k-1} \mathfrak{A}_{m, 1}^k \smallsetminus {\left(\mathfrak{R}_1^k \cup \mathfrak{R}_{2}^k\right)}\right)} \cup \left\{\mathrm{Id}_k, \begin{pmatrix} 1 \\ \vdots \\ 1 \end{pmatrix}\right\}. \end{align*} $$

Here, we want to mention that we will use the same notations ${\mathfrak {R}}^k_1$ , ${\mathfrak {R}}^k_2$ and ${\mathfrak {M}}^k$ for the analogous (but different) sets in each subsection (see Subsection 3.2 and Section 5). This will hopefully cause no confusion.

We denote this sub-collection of the RHS of (2.8) as $M_{d, k}^{\text {affine}}$ and the rest of the terms as $R_{d, k}^{\text {affine}}$ . That is,

$$\begin{align*}\mathbb{E}{\left(\prod_{i = 1}^{k} N_i\right)} = M_{d, k}^{\text{affine}} + R_{d, k}^{\text{affine}}, \end{align*}$$

where

(3.1) $$ \begin{align} M_{d, k}^{\text{affine}} = \sum_{\widetilde{D} \in \mathfrak{M}^{k}} \int_{(\mathbb{R}^{d})^{m}} \prod_{i= 1}^{k}\rho_i{\left(\widetilde{D} \begin{pmatrix} {\mathbf{y}}_1 \\ {\mathbf{y}}_2 \\ \vdots \\ {\mathbf{y}}_{m} \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1}{\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_2} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} \end{align} $$

and

(3.2) $$ \begin{align} R_{d, k}^{\text{{affine}}} = \sum_{\widetilde{D} \in \mathfrak{R}_1^{k} \cup \mathfrak{R}_2^{k}} \frac{N(\widetilde{D}, u)^d}{u^{dm}} \int_{(\mathbb{R}^{d})^{m}} \prod_{i= 1}^{k}\rho_i{\left(\frac{1}{u}\widetilde{D} \begin{pmatrix} {\mathbf{y}}_1 \\ {\mathbf{y}}_2 \\ \vdots \\ {\mathbf{y}}_{m} \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1}{\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_2} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}}. \end{align} $$

Let $(\alpha , \beta )$ be a division of $\{1, \ldots , k\}$ ; that is, $\alpha = \{\alpha _1 < \cdots < \alpha _m\}$ and $\beta = \{\beta _1 < \cdots < \beta _{k - r}\}$ are complementary subsets of $\{1, \ldots , k\}$ with $\alpha \neq \varnothing $ . Define

$$\begin{align*}\mathfrak{M}_{\alpha, \beta}^{\text{affine}} := \left\{\widetilde{D} \in \mathfrak{M}^k: I_{\widetilde{D}} = \alpha\right\} \end{align*}$$

and let $M_{\alpha , \beta }^{\text {affine}}$ denote the cardinality of $\mathfrak {M}_{\alpha , \beta }^{\text {affine}}$ . We allow for the case $(\alpha , \beta ) = {\left (\{1, \ldots , k\}, \varnothing \right )}$ , in which case $\mathfrak {M}_{\alpha , \beta }^k = \{\mathrm {Id}_k\}$ . Thus, we can rewrite (3.1) as

(3.3) $$ \begin{align} M_{d, k}^{\text{{affine}}} = \sum_{(\alpha, \beta)} \sum_{\widetilde{D} \in \mathfrak{M}_{\alpha, \beta}^{\text{affine}}} \int_{(\mathbb{R}^{d})^{m}} \prod_{i= 1}^{k}\rho_i{\left(\widetilde{D} \begin{pmatrix} {\mathbf{y}}_1 \\ {\mathbf{y}}_2 \\ \vdots \\ {\mathbf{y}}_{m} \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1}{\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_2} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}}, \end{align} $$

where the outer sum is over all possible divisions $(\alpha , \beta )$ of $\{1, \ldots , k\}$ .

Remark 3.2. It follows from the definition of $\widetilde {D}$ that for $\widetilde {D} \in \mathfrak {M}^k$ , the nonzero entries of the matrix $\widetilde {D}$ can only be 1. Since $\widetilde {D} \notin {\mathfrak {R}}^k_1$ , we already know that entries of $\widetilde {D}\in \{0, \pm 1\}$ . The fact that $-1$ is not possible for entries of $\widetilde {D}$ comes from notations in Theorem 2.12. Suppose that there is a row having $-1$ in its entries. Let $(x_1, x_2, \ldots , x_{m})$ be such a row. If $x_1=-1$ , since $x_1=1 - \sum _{\ell =2}^{m} x_{\ell }$ , there should be at least one nonzero element in $(x_2, \ldots , x_{m})$ , which contradicts the fact that in each row, there is only one nonzero entry. One can also obtain a contradiction when one assumes that there is some $2\le i_0\le m$ for which $x_{i_0}=-1$ .

Lemma 3.3. With notations as above,

(3.4) $$ \begin{align} M_{d, k}^{\text{affine}} = \sum_{(\alpha, \beta)} M_{\alpha, \beta}^{\text{affine}} \prod_{i = 1}^{m} V_{\alpha_i}. \end{align} $$

Proof. Consider any matrix $\widetilde {D} = \left (\widetilde {D}_{ij}\right ) \in \mathfrak {M}_{\alpha , \beta }^{\text {{affine}}}$ and let $\lambda _{\ell }$ be such that $\widetilde {D}_{\beta _{\ell }, \lambda _{\ell }} = 1$ , $1 \leq \ell \leq k - m$ . Then, as $V_i$ ’s are increasing, the following calculation finishes the proof:

$$ \begin{align*} &\int_{(\mathbb{R}^{d})^{m}} \prod_{i = 1}^{k} \rho_i\left(\widetilde{D}\begin{pmatrix} {\mathbf{y}}_1 \\ {\mathbf{y}}_2 \\ \vdots \\ {\mathbf{y}}_{m} \end{pmatrix}\right) {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_2} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} = \int_{(\mathbb{R}^{d})^{m}} \prod_{i = 1}^k \rho_i{\left(\sum_{j = 1}^{m} \widetilde{D_{ij}} {\mathbf{y}}_j\right)} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} \\ =& \int_{(\mathbb{R}^{d})^{m}} \prod_{i = 1}^{m} \rho_{\alpha_i}({\mathbf{y}}_i) \prod_{\ell = 1}^{k - m} \rho_{\beta_{\ell}}(\mathbf{y}_{\lambda_{\ell}}) {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} = \int_{(\mathbb{R}^{d})^m} \prod_{i = 1}^{m} \rho_{\alpha_i}(\mathbf{y}_i) = \prod_{i = 1}^{m} V_{\alpha_i}.\\[-45pt] \end{align*} $$

We shall now mention some estimates regarding $R_{d, k}^{\text {affine}}$ . These estimates are originally due to Rogers [14, 16], and they were generalised to Lemma 3.4 (below) by Södergren [23]. For $D \in \mathfrak {D}_{r, u}^k$ set

$$\begin{align*}I(D, u) := \int_{(\mathbb{R}^{d})^r} \prod_{i = 1}^{k} \rho_i{\left(\frac{1}{u}D \begin{pmatrix} {\mathbf{y}}_{1} \\ \vdots \\ {\mathbf{y}}_{r} \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{r}}. \end{align*}$$

Lemma 3.4 (Estimates from [14], [16] and [23])

For $d> [k^2/4] + 2$ ,

1. (i) $\displaystyle \sum _{r = 1}^{k} \sum _{u = 2}^{\infty } \sum _{D \in \mathfrak {D}_{r, u}^k} \frac {N(D, u)^d}{u^{dr}} \cdot I(D, u) = O\left (2^{-d}\right )$ ,

2. (ii) $\displaystyle \sum _{r = 1}^{k} \sum _{D \in \mathfrak {D}_{r, 1}^{k, 1}} I(D, u) = O\left (2^{-d}\right )$ , where $\mathfrak {D}_{r, 1}^{k, 1} \subseteq \mathfrak {D}_{r, 1}^k$ contains matrices D such that $\max |d_{ij}| \geq 2$ ,

3. (iii) $\displaystyle \sum _{r = 1}^{k} \sum _{D \in \mathfrak {D}_{r, 1}^{k, 2}} I(D, u) = O\left ({\left (3/4\right )}^{d/2}\right )$ , where $\mathfrak {D}_{r, 1}^{k, 2} \subseteq \mathfrak {D}_{r, 1}^k$ contains matrices D such that $\max |d_{ij}| = 1$ and at least one row of D has at least two nonzero entries.

Proof. The proof of this lemma can be found in [23, Proposition 2, Lemma 1 and Lemma 2]. The main ingredients in Södergren’s proof are [14, §9] and the contents of [16, §4].

We remark that we only need the fact that $N(D,u)^d/u^{dr}\le 1/u^d$ to prove Property (i). Hence, one can use Lemma 3.4 for applications with the space $Y_{{\mathbf {p}}/q}$ as well as the space Y in Section 3 and Section 5.

Rogers’ estimate shows the following:

Lemma 3.5.

$$\begin{align*}R_{d, k}^{\text{affine}} = O{\left({\left(3/4\right)}^{d/2}\right)}. \end{align*}$$

Proof. It follows from (3.2) that $R_{d, k}^{\text {affine}}$ is less than the sum of LHS’s in Lemma 3.4.

The lemmas above combine to give us the following theorem:

Theorem 3.6.

(3.5) $$ \begin{align} \mathbb{E}{\left(\prod_{i = 1}^{k} N_i\right)} \to \sum_{(\alpha, \beta)} M_{\alpha, \beta}^{\text{affine}} \prod_{i = 1}^{r} V_{\alpha_i} \end{align} $$

as $d \to \infty $ .

3.1.1 Proof of Theorem 1.1

This proof closely follows the proof of Theorem 1 in [23, §4]. Let us discuss the Poisson process $\{N^{\lambda }(t): t \geq 0\}$ . By definition, $N^{\lambda }(t)$ denotes the number of points falling in the interval $[0, t]$ and $N^{\lambda }(t)$ is Poisson distributed with expectation $\lambda t$ . By $0 \leq T_1 \leq T_2 \leq T_3 \leq \cdots $ , let us denote the points of the Poisson process.

Lemma 3.7. Let $k \geq 1$ and let $\mathscr {P}(k)$ denote the set of partitions of $\{1, \ldots , k\}$ . For $1 \leq i \leq k$ , let $f_i : \mathbb {R}_{\geq 0} \to \mathbb {R}$ be functions satisfying $\prod _{i \in B} f_i \in L^1(\mathbb {R}_{\geq 0})$ for every nonempty subset $B \subseteq \{1, \ldots , k\}$ . Then

(3.6) $$ \begin{align} \mathbb{E}{\left(\prod_{i = 1}^{k} {\left(\sum_{\ell = 1}^{\infty} f_i(T_{\ell})\right)}\right)} = \sum_{P \in \mathscr{P}(k)} \lambda^{\#P} {\left(\int_{0}^{\infty} \prod_{i \in B} f_i(x) {\hspace{0.5mm} {\mathrm{d}}}{x}\right)}. \end{align} $$

Proof. The proof of this lemma is similar to [23, Proposition 3].

We apply Lemma 3.7 with functions , where is the characteristic function of the interval $[0, V_i]$ . Thus, we get

(3.7)

where $i_B = \min _{i \in B} i$ .

The following lemma helps us compare the RHS’s of (3.5) and (3.7) for $\lambda = 1$ .

Lemma 3.8 ([23], Lemma 3)

There is bijection $g : \mathfrak {M}^k \to \mathscr {P}(k)$ with the property that if $\widetilde {D} \in \mathfrak {M}^k$ is an $k \times m$ matrix and $g(\widetilde {D}) = P = \{B_1, \ldots , B_{\#P}\}$ , then $\#P = m$ and $\{\alpha _1 < \cdots < \alpha _m\} = \{i_{B_1} < \cdots < i_{B_m}\}$ .

Proof. Other than switching the rows and columns of the matrices D, the proof of this lemma is the same as [23, Lemma 3].

Theorem 3.6, (3.7) and Lemma 3.8 imply the following result:

Theorem 3.9.

$$\begin{align*}\mathbb{E}{\left(\prod_{i = 1}^{k} N_i\right)} \to \mathbb{E}{\left(\prod_{i = 1}^{k} N^1(V_i)\right)} \end{align*}$$

as $d \to \infty $ .

Corollary 3.10. Let $\mathbf {V} = (V_1, \ldots , V_k)$ and consider the random vectors

$$\begin{align*}\mathbf{N}(\Lambda, \mathbf{V}) = {\left(N_1(\Lambda), \ldots, N_k(\Lambda)\right)} \end{align*}$$

and

$$\begin{align*}\mathbf{N}(\mathbf{V}) = {\left(N^1(V_1), \ldots, N^1(V_k)\right)}. \end{align*}$$

Then $\mathbf {N}(\Lambda , \mathbf {V})$ converges in distribution to $\mathbf {N}(\mathbf {V})$ as $d \to \infty $ .

Proof. This proof follows a similar line of argument as [23, Corollary 1]. We omit it for the sake of brevity.

Corollary 3.10 implies that all finite dimensional distributions coming from the process $\{N_t(\Lambda ) : t \geq 0\}$ converge to the corresponding finite dimensional distributions of the Poisson process $\{N^1(t) : t \geq 0\}$ as $d \to \infty $ . By [3, Theorem 12.6 and Theorem 16.7], we see that the process $\{N_t(\Lambda ) : t \geq 0\}$ converges weakly to the process $\{N^1(t) : t \geq 0\}$ as $d \to \infty $ .

Corollary 3.10, with $k = 1$ , is a generalisation of [16, Theorem 3] to the affine case.

3.2 Congruence Case

In this section, we prove Theorem 1.2. We recall the notation. For $d \geq 2$ , let $\mathcal {S} = \{S_t: t> 0\}$ , an increasing family of subsets of $\mathbb {R}^{d}$ and $\mathbf {p}/q \in \mathbb {Q}^{d}$ . For $\Lambda \in Y_{\mathbf {p}/q}$ , set

$$\begin{align*}N_t(\Lambda) = \#(S_t \cap \Lambda). \end{align*}$$

For $\Lambda \in Y_{\mathbf {p}/q}$ , let us order the lengths of nonzero vectors in $\Lambda $ as $0 < \ell _1 \leq \ell _2 \leq \ell _3 \leq \cdots $ , and let $\mathscr {V}_i$ denote the volume of the closed ball of radius $\ell _i$ centered at origin. Taking $\mathcal {S} = \{B_t: t> 0\}$ to be the increasing family of closed balls with $\operatorname {\mathrm {vol}}(B_t) = t$ around the origin, we see that

$$\begin{align*}N_t(\Lambda) = \#\{i : \mathscr{V}_i \leq t\}. \end{align*}$$

Thus, Theorem 1.2 is equivalent to the following:

Theorem 3.11. For any fixed n, the n-dimensional random variable $(\mathscr {V}_1, \ldots , \mathscr {V}_n)$ converges in distribution to the distribution of the first n points of a Poisson process on the non-negative real line with intensity

$$\begin{align*}\begin{cases} 1 & \text{if} \ q \geq 3, \\ \tfrac{1}{2} & \text{if} \ q = 2. \\ \end{cases} \end{align*}$$

As in the affine case, we approach Theorem 1.2 via a joint moment formula for $N_t( \cdot )$ . Let $k \geq 1$ and $0 < V_1 \leq V_2 \leq \cdots \leq V_k$ . Define $N_i$ ’s, $\rho _i$ ’s and F similar to the affine case. We apply Theorem 2.13 to the function F. We first consider the sub-collection of the RHS of (2.12) denoted by $M_{d, k}^{\text {cong}}$ , defined as

(3.8) $$ \begin{align} M_{d, k}^{\text{cong}} := \sum_{\widetilde{D} \in \mathfrak{M}^k} \int_{(\mathbb{R}^{d})^m} \prod_{i = 1}^{k} \rho_i {\left(\widetilde{D} \begin{pmatrix} \mathbf{y}_1 \\ \vdots \\ \mathbf{y}_m \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_m}, \end{align} $$

where

$$ \begin{align*} \mathfrak{R}_1^k &= \bigcup_{1 \leq m \leq k-1} {\left({\left(\bigcup_{u \geq 2} \mathfrak{C}_{m, u}^k\right)} \cup {\left\{\widetilde{D} = {\left(\widetilde{D}_{ij}\right)} \in \mathfrak{C}_{m, 1}^k : \exists |\widetilde{D}_{ij}| \geq 2\right\}}\right)}, \\ \mathfrak{R}_{2}^k &= {\left\{\widetilde{D} \in {\left(\bigcup_{1 \leq m \leq k-1} \mathfrak{C}_{m, 1}^k\right)} \smallsetminus \mathfrak{R}_1^k: \substack{\displaystyle\exists~\text{row such that at least} \\ \\ \displaystyle\text{two entries are nonzero}}\right\}}, \\ \mathfrak{M}^k &= {\left(\bigcup_{1 \leq m \leq k-1} \mathfrak{C}_{m, 1}^k \smallsetminus {\left(\mathfrak{R}_1^k \cup \mathfrak{R}_{2}^k\right)}\right)} \cup \left\{\mathrm{Id}_k\right\}. \end{align*} $$

The rest of the terms in (2.12) will be denoted as $R_{d, k}^{\text {cong}}$ ; that is,

$$\begin{align*}\mathbb{E}{\left(\prod_{i = 1}^{k}N_i\right)} = M_{d, k}^{\text{cong}} + R_{d, k}^{\text{cong}}. \end{align*}$$

Define $\mathfrak {M}_{\alpha , \beta }^{\text {cong}}$ , for $(\alpha , \beta )$ a division of $\{1, \ldots , k\}$ , similar to the affine case and let $M_{\alpha , \beta }^{\text {cong}}$ denote the cardinality of $\mathfrak {M}_{\alpha , \beta }^{\text {cong}}$ . We can rewrite (3.8) as

(3.9) $$ \begin{align} M_{d, k}^{\text{cong}} = \sum_{(\alpha, \beta)} \sum_{\widetilde{D} \in \mathfrak{M}_{\alpha, \beta}^{\text{cong}}} \int_{(\mathbb{R}^{d})^m} \prod_{i = 1}^{k} \rho_i {\left(\widetilde{D} \begin{pmatrix} \mathbf{y}_1 \\ \vdots \\ \mathbf{y}_m \end{pmatrix}\right)} {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{\mathbf{y}_m}, \end{align} $$

where the outer sum is over all possible divisions $(\alpha , \beta )$ of $\{1, \ldots , k\}$ .

Remark 3.12. For $q \geq 3$ , it follows from the definition of $\widetilde {D}$ and similar arguments as in Remark 3.2 that for $\widetilde {D} \in \mathfrak {M}^k$ , the nonzero entries of $\widetilde {D}$ can only be 1. But for $q = 2$ , the nonzero entries can be $\pm 1$ . In particular, this is the reason why we need the condition that $S_d$ is symmetric for the case when $q=2$ (see the second last equality in (3.11) below).

Lemma 3.13. For $q \geq 3$ and for $q = 2$ with $S_t$ being symmetric around the origin, we have

(3.10) $$ \begin{align} M_{d, k}^{\text{cong}} = \sum_{(\alpha, \beta)} M_{\alpha, \beta}^{\text{cong}} \prod_{i = 1}^{m} V_{\alpha_i}. \end{align} $$

Proof. For $q \geq 3$ , the proof of this lemma is identical to that of Lemma 3.3. Hence, we only focus on the case when $q = 2$ .

Consider any matrix $\widetilde {D} = \left (\widetilde {D}_{ij}\right ) \in \mathfrak {M}_{\alpha , \beta }^{\text {{cong}}}$ and let $\lambda _{\ell }$ be such that $\widetilde {D}_{\beta _{\ell }, \lambda _{\ell }} = 1$ , $1 \leq \ell \leq k - m$ . Then, as $S_t$ ’s are symmetric and $V_i$ ’s are increasing, the following calculation finishes the proof:

(3.11) $$ \begin{align}\begin{aligned} &\int_{(\mathbb{R}^{d})^{m}} \prod_{i = 1}^{k} \rho_i\left(\widetilde{D}\begin{pmatrix} {\mathbf{y}}_1 \\ \vdots \\ {\mathbf{y}}_{m} \end{pmatrix}\right) {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} = \int_{(\mathbb{R}^{d})^{m}} \prod_{i = 1}^k \rho_i{\left(\sum_{j = 1}^{m} \widetilde{D_{ij}} {\mathbf{y}}_j\right)} {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} \\[5pt] =& \int_{(\mathbb{R}^{d})^{m}} \prod_{i = 1}^{m} \rho_{\alpha_i}({\mathbf{y}}_i) \prod_{\ell = 1}^{k - m} \rho_{\beta_{\ell}}(\pm\mathbf{y}_{\lambda_{\ell}}) {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_1} \cdots {\hspace{0.5mm} {\mathrm{d}}}{{\mathbf{y}}_{m}} = \int_{(\mathbb{R}^{d})^m} \prod_{i = 1}^{m} \rho_{\alpha_i}(\mathbf{y}_i) = \prod_{i = 1}^{m} V_{\alpha_i}.\\[-30pt] \end{aligned}\end{align} $$

From Lemma 3.13 and Lemma 3.4, we find the following:

Theorem 3.14.

(3.12) $$ \begin{align} \mathbb{E}{\left(\prod_{i = 1}^{k} N_i\right)} \to \sum_{(\alpha, \beta)} M_{\alpha, \beta}^{\text{cong}} \prod_{i = 1}^{m} V_{\alpha_i}. \end{align} $$

3.2.1 Proof of Theorem 1.2

For $q \geq 3$ , the proof of Theorem 1.2 follows the proof of Theorem 1.1. We need a small modification in Lemma 3.8 for the case $q = 2$ because in this case, the entries of matrices in $\mathfrak {M}^k$ can be negative. From now on, we only focus on the case $q = 2$ .

Let $\mathfrak {M}_{\alpha , \beta , +}^{\text {cong}}$ denote the subset of $\mathfrak {M}_{\alpha , \beta }^{\text {cong}}$ of matrices with positive entries, and similarly, let $\mathfrak {M}_+^k$ denote the subset of $\mathfrak {M}^k$ of matrices with positive entries. With $M_{\alpha , \beta , +}^{\text {cong}} := \#(\mathfrak {M}_{\alpha , \beta , +}^{\text {cong}})$ , note that

$$\begin{align*}M_{\alpha, \beta}^{\text{cong}} = \#{\left(\mathfrak{M}_{\alpha, \beta}^{\text{{cong}}}\right)} = 2^{k - \#\alpha} M_{\alpha, \beta, +}^{\text{{cong}}}. \end{align*}$$

Thus, from (3.12), we find

(3.13) $$ \begin{align} \mathbb{E}{\left(\prod_{i = 1}^{k} \widetilde{N}_i\right)} \to \sum_{(\alpha, \beta)} 2^{-\#\alpha} M_{\alpha, \beta, +}^{\text{{cong}}} \prod_{i = 1}^{m} V_{\alpha_i}, \end{align} $$

where $\widetilde {N}_i = \frac {1}{2}N_i$ for $1 \leq i \leq k$ , i.e., $\widetilde {N}_t = \frac {1}{2}N_t$ .

With the following lemma, we can compare the RHS’s of Theorem 3.13 and (3.7) for $\lambda = 1/2$ .

Lemma 3.15. There is bijection $g : \mathfrak {M}_+^k \to \mathscr {P}(k)$ with the property that if $\widetilde {D} \in \mathfrak {M}_+^k$ is an $k \times m$ matrix and $g(\widetilde {D}) = P = \{B_1, \ldots , B_{\#P}\}$ , then $\#P = m$ and $\{\alpha _1 < \cdots < \alpha _m\} = \{i_{B_1} < \cdots < i_{B_m}\}$ .

Proof. The same argument with Lemma 3.8 holds.

(3.7), (3.13) and Lemma 3.15 combine to show the following:

Theorem 3.16. For $q = 2$ ,

$$\begin{align*}\mathbb{E}{\left(\prod_{i = 1}^{k} \widetilde{N}_i\right)} \to \mathbb{E}{\left(\prod_{i = 1}^{k} N^{1/2}(V_i)\right)}. \end{align*}$$

Corollary 3.17. Let $q = 2$ , $\mathbf {V} = (V_1, \ldots , V_k)$ and consider the random vectors

$$\begin{align*}\mathbf{\widetilde{N}}(\Lambda, \mathbf{V}) = {\left(\widetilde{N}_1(\Lambda), \ldots, \widetilde{N}_k(\Lambda)\right)} \end{align*}$$

and

$$\begin{align*}\mathbf{N}(\mathbf{V}) = {\left(N^{1/2}(V_1), \ldots, N^{1/2}(V_k)\right)}. \end{align*}$$

Then $\mathbf {\widetilde {N}}(\Lambda , \mathbf {V})$ converges in distribution to $\mathbf {N}(\mathbf {V})$ as $d \to \infty $ .

Proof. This proof follows a similar line of argument as [23, Corollary 1]. We omit it for the sake of brevity.

Corollary 3.17 implies that all finite dimensional distributions coming from the process $\{\widetilde {N}_t(\Lambda ) : t \geq 0\}$ converge to the corresponding finite dimensional distributions of the Poisson process $\{N^{1/2}(t) : t \geq 0\}$ as $d \to \infty $ . By [3, Theorem 12.6 and Theorem 16.7], we see that the process $\{N_t(\Lambda ) : t \geq 0\}$ converges weakly to the process $\{N^{1/2}(t) : t \geq 0\}$ as $d \to \infty $ .

Corollary 3.17, with $k = 1$ , is a generalisation of [16, Theorem 3] to the congruence case.

4 New Moment Formulae

In this section, we want to simplify Theorem 2.12 and Theorem 2.13 for the special case as considered by Strömbergsson and Södergren in [22]. Theorems 4.1 and 4.2 below will be used in Section 5.

For bounded and compactly supported functions $f_i: {\mathbb {R}}^d \rightarrow {\mathbb {R}}_{\ge 0}$ ( $1\le i \le k$ ), define

(4.1) $$ \begin{align} F_i(\mathbf{v})=f_i(\mathbf{v}) -\int_{{\mathbb{R}}^d} f_i {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}. \end{align} $$

We want to compute the integrals of ${\mathcal S}_{k}(\prod _{i=1}^k F_i)=\prod _{i=1}^k \widehat {F_i}$ over Y and $Y_{{\mathbf {p}}/q}$ .

We first observe that by applying Theorem 2.12,

(4.2) $$ \begin{align} &\int_{Y} {\prod_{i=1}^k \widehat F_i}(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_Y (\Lambda) =\int_{Y} \prod_{i=1}^k \left(\widehat{f_i}(\Lambda) - \int_{{\mathbb{R}}^d} f_i {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v} \right)\nonumber\\ &=\sum_{A\subseteq \{1, \ldots, k\}} (-1)^a \left(\prod_{i"\in A}\int_{{\mathbb{R}}^d} f_{i"} {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}\right)\int_{Y} {\prod_{i\in A^c} \widehat{f_i}}(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_Y(\Lambda) \nonumber\\&=\sum_{A\subseteq \{1, \ldots, k\}} (-1)^a \prod_{i"\in A}\int_{{\mathbb{R}}^d} f_{i"} {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}\;\times\\ &\left(\sum_{m=1}^{k-a} \sum_{u\in {\mathbb{N}}} \sum_{\widetilde {D} \in {\mathfrak{A}}^{k-a}_{m,u}} \frac {N( \widetilde{D},u)^d} {u^{dm}}\int_{({\mathbb{R}}^d)^m} \left(\prod_{i \in A^c}f_i\right)\left(\frac {\widetilde{D}}{u}\left(\begin{array}{c} \mathbf w_1\\ \vdots\\ \mathbf w_m \end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_m \right), \nonumber\end{align} $$

where $a=\# A$ and $A^c=\{1,\ldots ,k\}-A$ .

Note that for a given $A\subseteq \{1, \ldots , k\}$ and $\widetilde {D}\in {\mathfrak {A}}^{k-a}_{m,u}$ , one can find a unique matrix $D"=D"(A, \widetilde {D})\in {\mathfrak {D}}^k_{m+a,u}$ (in fact, ${\mathfrak {A}}^k_{m+a,u}$ ) for which

(4.3) $$ \begin{align} \begin{aligned} &\left(\prod_{i" \in A} \int_{{\mathbb{R}}^d} f_{i"}\: {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}\right)\cdot \int_{({\mathbb{R}}^d)^m} \left(\prod_{i \in A^c} f_i\right) \left(\frac {\widetilde{D}} u\left(\begin{array}{c} \mathbf w_1\\ \vdots\\ \mathbf w_m \end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_m\\ &\hspace{1in}=\int_{({\mathbb{R}}^d)^{m+a}} \left(\prod_{i=1}^k f_i\right) \left(\frac {D"} u \left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{m+a}\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{m+a}. \end{aligned}\end{align} $$

Moreover, from the definitions of $N(D", u)$ and $N(\widetilde {D}, u)$ in Notation 2.1 (3), one can directly obtain the following equality:

(4.4) $$ \begin{align} \frac {N(D",u)^d} {u^{dn}}=\frac {N(\widetilde{D},u)^d}{u^{dm}}. \end{align} $$

We claim the following:

Theorem 4.1. For $1\le i\le k$ , let $F_i$ be the function defined as in (4.1) for a bounded and compactly supported function $f_i:{\mathbb {R}}^d\rightarrow {\mathbb {R}}_{\ge 0}$ ( $1\le i\le k$ ). It follows that

(4.5) $$ \begin{align} \begin{aligned} \int_{Y} {\prod_{i=1}^k \widehat{F_i}}(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_Y (\Lambda) =\sum_{n=1}^{k-1} \sum_{u\in {\mathbb{N}}} \sum_{D"\in {\mathfrak{S}}^k_{n,u}} \frac {N(D",u)^d} {u^{dn}}\int_{({\mathbb{R}}^d)^n} \prod_{i=1}^k f_i \left(\frac {D"} {u} \left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_n\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_n, \end{aligned}\end{align} $$

where ${\mathfrak {S}}^k_{n,u}\subseteq {\mathfrak {A}}^k_{n,u}$ is the set of $D"$ which is one of the following:

1. (a) Each column of $[D"]$ has at least two nonzero elements.

2. (b) There are $0\le a\le n-2$ and $D\in {\mathfrak {D}}^{k-a-1}_{n-a-1,u}-{\mathfrak {A}}^{k-a-1}_{n-a-1,u}$ for which

   $$\begin{align*}D"=\left(\begin{array}{c|c} u\mathrm{Id}_a & \\ \hline & \begin{array}{c|c} u & 0 \cdots 0\\ \hline \begin{array}{c} 0 \\ \vdots \\ 0 \end{array} & D \end{array} \end{array}\right), \end{align*}$$
   where each column of D has at least two nonzero elements.

Similarly, from Theorem 2.13, we have that $\int _{Y_{{\mathbf {p}}/q}} \prod _{i=1}^k \widehat {F_i} (\Lambda ) d\mu _q$ is the sum of integrals given as in (4.2) with replacing ${\mathfrak {A}}^{k-a}_{m,u}$ by ${\mathfrak {C}}^{k-a}_{m,u}$ . For $D"=D"(A,\widetilde D)\in {\mathfrak {D}}^k_{m+a,u}$ defined using $A\subseteq \{1, \ldots , k\}$ and $\widetilde D\in {\mathfrak {C}}^{k-a}_{m,u}$ as in (4.3), we will see that $D"\in {\mathfrak {C}}^k_{m+a,u}$ . It is easily seen that the equality (4.4) holds in the congruence case.

Theorem 4.2. For $1\le i\le k$ , let $F_i$ be the function defined as in (4.1) for a bounded and compactly supported function $f_i:{\mathbb {R}}^d\rightarrow {\mathbb {R}}_{\ge 0}$ ( $1\le i\le k$ ). It follows that

$$\begin{align*}\begin{aligned} &\int_{Y_{{\mathbf{p}}/q}} {\prod_{i=1}^k \widehat F_i}(\Lambda) {\hspace{0.5mm} {\mathrm{d}}}\mu_q (\Lambda)\\ &\hspace{0.4in}=\sum_{n=1}^{k-1} \sum_{u\in {\mathbb{N}}} \sum_{D"\in \mathfrak T^k_{n,u}} \frac {N(D",u)^d} {u^{dn}} \int_{({\mathbb{R}}^d)^n} \prod_{i=1}^k f_i \left(\frac {D"} {u} \left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_n\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_n, \end{aligned}\end{align*}$$

where $\mathfrak T^k_{n,u}$ is a subset of ${\mathfrak {C}}^k_{n,u}$ collecting $D"$ which is one of the following:

1. (a) Each column of $D"$ has at least two nonzero elements.

2. (b) There are $0\le a\le n-2$ and $\widetilde D\in {\mathfrak {C}}^{k-a}_{n-a,u}$ so that

   $$\begin{align*}D"=\left(\begin{array}{cc} u\mathrm{Id}_a & \\ & \widetilde{D} \end{array} \right), \end{align*}$$
   where $[\widetilde {D}]^1={{}^{\mathrm {t}}{(u, 0, \ldots , 0)}}$ and any other columns of $\widetilde {D}$ have at least two nonzero elements. Moreover, the right-bottom minor of $[\widetilde {D}]$ with size $(k-a-1)\times (n-a-1)$ is not an element of ${\mathfrak {C}}^{k-a-1}_{n-a-1,u}$ (or any ${\mathfrak {C}}^{k-a-1}_{n-a-1,*}$ ).

Proof of Theorem 4.1 and Theorem 4.2

As described in (4.3), a possible matrix $D"$ among elements of ${\mathfrak {D}}^k_{n,u}$ is constructed by using $A\subseteq \{1, \ldots , k\}$ and $\widetilde {D}\in {\mathfrak {A}}^{k-a}_{n-a,u}$ . Conversely, we want to consider all possible pairs $(A, \widetilde {D})$ which give the same $D"$ .

Let such a $D"=(d^{\prime \prime }_{ij})$ be given. Denote

$$\begin{align*}B=\left\{1\le i"\le k : \begin{array}{l} 1\le \exists j_0\le n \text{ for which}\\ \hspace{0.3in}d^{\prime\prime}_{i" j}=0 \text{ for all } j \text{ except } d^{\prime\prime}_{i" j_0}=u \text{ and}\\[0.05in] \hspace{0.3in}d^{\prime\prime}_{i j_0}=0 \text{ for all } i \text{ except } d^{\prime\prime}_{i" j_0}=u \end{array}\right\}. \end{align*}$$

After changing the (last $(k-b_1)$ ) coordinates of ${\mathbb {R}}^k$ , we may assume that

(4.6) $$ \begin{align} \frac {D"} {u}=\left(\begin{array}{ccc} \mathrm{Id}_{b_1} & & \\ & \begin{array}{c} \dfrac {\widetilde {D_0}} {u} \end{array}& \\ & & \mathrm{Id}_{b_2} \end{array}\right), \end{align} $$

where $b_1$ and $b_2$ could be $0$ (then $D"/u$ will be one- or two-block diagonal matrix) and $\widetilde D_0\in {\mathfrak {A}}^{k-b_1-b_2}_{n-b_1-b_2,u}$ (or ${\mathfrak {C}}^{k-b_1-b_2}_{n-b_1-b_2,u}$ , respectively) is the minimal size among possible $(A,\widetilde {D})$ for which $D"(A,\widetilde {D})=D"$ ; that is,

$$ \begin{align*} \text{each column of } \widetilde D_0 \text{ except } [\widetilde{D_0}]^1 \text{ has at least two nonzero elements.} \end{align*} $$

Notice that any matrix constructed by choosing more than $k-b_1-b_2$ rows and more than $n-b_1-b_2$ columns from $D"/u$ and having $\widetilde {D_0}/u$ as its minor is element of ${\mathfrak {A}}^{*}_{*,u}$ (or ${\mathfrak {C}}^{*}_{*,u}$ , respectively). For example, $D"\in {\mathfrak {C}}^k_{n,u}$ since $D"$ is constructed by $(\overline {D},t,{{}^{\mathrm {t}}{(0,\ldots , 0, \boldsymbol {\ell }, 0, \ldots , 0)}})$ under the map in (2.10), where $\overline {D}$ is the right-bottom minor of $uD"$ with size $(k-1)\times (n-1)$ , and $(t, \boldsymbol {\ell })$ is a pair used for defining $\widetilde D$ .

Now let us check case by case. Denote by

$$\begin{align*}B_1=\{i\in B: i \le b_1+1\} \quad\text{and}\quad B_2=\{k-b_2+1, \ldots, k\}\end{align*}$$

so that $B=B_1\cup B_2$ . Note that $(b_1+1)$ could be not contained in B. Observe that possible A for constructing $D"$ is of the form $A_1 \cup A_2$ , where $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$ . The difference between $A_1$ and $A_2$ is that $A_1$ may have an extra condition according to the given $D"$ , but any subset of $B_2$ can be $A_2$ .

We first assume that $B_2\neq \emptyset $ . Since

$$\begin{align*}\begin{aligned} \sum_{\scriptsize \begin{array}{c} \text{"possible"}\\ A\subseteq B\end{array}} (-1)^{\# A} &=\sum_{\scriptsize \begin{array}{c} \text{"possible"}\\ A_1 \subseteq B_1\end{array}} (-1)^{\# A_1} \sum_{\forall A_2 \subseteq B_2} (-1)^{\# A_2}\\ &=\sum_{\scriptsize \begin{array}{c} \text{"possible"}\\ A_1 \subseteq B_1\end{array}} (-1)^{\# A_1} \cdot 0 = 0, \end{aligned}\end{align*}$$

with the observation in (4.3), the partial sum

(4.7) $$ \begin{align}\begin{aligned} &\sum_{\scriptsize \begin{array}{c} A, \widetilde{D}\text{ s.t.}\\ D"(A, \widetilde{D})=D"\end{array}}\hspace{-0.15in} (-1)^{\# A}\:\frac {N(D",u)^d} {u^{dr}} \int_{({\mathbb{R}}^d)^n} \prod_{i=1}^k f_i \left(\frac {D"} u \left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{m+a}\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1\cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{m+a} \end{aligned}\end{align} $$

associated with $D"$ in the right-hand side of (4.5) is zero.

Now let us assume that $b_2=0$ . If $B=\emptyset $ , that is,

$$ \begin{align*} \text{each column of } D" \text{ has at least two nonzero vectors}, \end{align*} $$

and only possible $(A, \widetilde {D})$ is $(\emptyset , D")$ . This is the case (a) in the theorem.

Suppose that $|B|=b_1\ge 1$ . Equivalently, suppose that

$$ \begin{align*} [\widetilde {D_0}] \text{ as well as other columns of } \widetilde D_0 \text{ has at least two nonzero elements.} \end{align*} $$

Then any subset A of B is possible for defining $D"$ ; hence, the partial sum (4.7) is zero.

The only left case is when $|B|=b_1+1\ge 2$ . Notice that

$$ \begin{align*} [\widetilde{D_0}]^1={{}^{\mathrm{t}}{(u,0,\ldots,0)}}, \text{and} \end{align*} $$

$$ \begin{align*} \text{the right-bottom minor of } \widetilde{D_0} \text{ with size } (k-b_1-1)\times (n-b_1-1) \end{align*} $$

$$ \begin{align*} \text{is not an element of } {\mathfrak{A}}^{k-b_1-1}_{n-b_1-1, u} ({\mathfrak{C}}^{k-b_1-1}_{n-b_1-1, u}, \text{ respectively}). \end{align*} $$

In this case, any subset of B except B itself can be possible A for defining $D"$ , and this is the case (b) in the theorem.

5 CLT and Brownian motion

As in Section 3, we will use the method of moments which is applicable with the normal distribution and Brownian motion, following [22]. Recall that the k-th moment of the normal distribution is $0$ when k is odd and $(k-1)!!$ when k is even.

For Brownian motion, it suffices to show that the induced measure $P^1_d$ and $P^{{\mathbf {p}}/q}_d$ from $Z^1_d(t)$ and $Z^{{\mathbf {p}}/q}_d(t)$ , respectively, on the space $C[0,1]$ of continuous real-valued functions on $[0,1]$ weakly converge to Wiener measure as d goes to infinity.

Let $\phi :{\mathbb {N}} \rightarrow {\mathbb {R}}_{>0}$ be a function for which $\lim _{d\rightarrow \infty } \phi (d)=\infty $ and $\phi (d)=O_{\varepsilon }(e^{\varepsilon d})$ for every $\varepsilon>0$ . Let $\iota \in {\mathbb {N}}$ and $c_1, \ldots , c_{\iota }>0$ be arbitrarily given. For each $d\in {\mathbb {N}}$ , consider $S_{i,d}\in {\mathbb {R}}^{d}$ to be a Borel measurable set satisfying $\operatorname {\mathrm {vol}}(S_{i,d})=c_i \phi (d)$ for $1\le i\le \iota $ and $S_{i,d}\cap S_{i',d}=\emptyset $ if $i\neq i'$ . If we consider the case that $\Lambda \in Y_{{{\mathbf {p}}}/q}$ with $q=2$ , we further assume that $S_{i,d}=-S_{i,d}$ for $1\le i\le \iota $ and $d\in {\mathbb {N}}$ .

Let

$$\begin{align*}\begin{aligned} Z^1_{i,d}&:= \frac {\#(\Lambda \cap S_{i,d})-c_i\phi(d)}{\sqrt{\phi(d)}},\hspace{1.25in}\Lambda\in Y\;\text{and}\\[0.1in] Z^{{\mathbf{p}}/q}_{i,d}&:=\left\{\begin{array}{cl} \dfrac {\#(\Lambda \cap S_{i,d})-c_i\phi(d)}{\sqrt{\phi(d)}}, &\text{if } q\neq 2;\\[0.2in] \dfrac {\#(\Lambda \cap S_{i,d})-c_i\phi(d)}{\sqrt{2\phi(d)}}, &\text{if } q=2, \end{array} \right.\quad \Lambda \in Y^{{\mathbf{p}}/q}. \end{aligned}\end{align*}$$

Proposition 5.1. Let $\diamondsuit =1$ or ${\mathbf {p}}/q$ . For any fixed ${\mathbf {k}}=(k_1, \ldots , k_{\iota })\in {\mathbb {N}}^{\iota }$ , it follows that

$$\begin{align*}\begin{aligned} &\lim_{d\rightarrow \infty} {\mathbb{E}}\left((Z^{\diamondsuit}_{1,d})^{k_1} \cdots (Z^{\diamondsuit}_{\iota,d})^{k_{\iota}}\right)\\ &\hspace{0.5in}=\left\{\begin{array}{cl} \prod_{i=1}^{\iota} c_i^{k_i/2} (k_i-1)!!, &\text{if } k_1, \ldots, k_{\iota} \text{ are all even},\\[0.05in] 0, &\text{ otherwise.}\end{array} \right. \end{aligned}\end{align*}$$

Proof. Let $k=k_1+\cdots +k_{\iota }$ and consider $d> \lfloor k^2/4\rfloor +3$ . For each $d\in {\mathbb {N}}$ and $1\le i\le \iota $ , let $f_{i,d}$ be the indicator function of $S_{i,d}$ and define

$$ \begin{align*} F_{i,d}(\Lambda)=\widehat{f_{i,d}}(\Lambda) -\int_{{\mathbb{R}}^d} f_{i,d} {\hspace{0.5mm} {\mathrm{d}}} \mathbf{v}=\widehat{f_{i,d}}(\Lambda)- c_i\phi(d), \;\Lambda \in Y. \end{align*} $$

We will use Theorem 4.1 and Theorem 4.2.

Recall that we can divide $\bigcup _{1\le n \le k} \bigcup _{u \in {\mathbb {N}}}{\mathfrak {D}}^k_{n,u}$ as the union of ${\mathfrak {R}}^k_{1}$ , ${\mathfrak {R}}^k_{2}$ and ${\mathfrak {M}}^k$ , where

$$\begin{align*}\begin{aligned} {\mathfrak{R}}^k_{1}&= \bigcup_{1\le n \le k-1}\left(\left(\bigcup_{u\ge 2} {\mathfrak{D}}^k_{n,u}\right) \cup \left\{D=(d_{ij}) \in {\mathfrak{D}}^k_{n,1} : \exists |d_{ij}| \ge 2 \right\}\right);\\ {\mathfrak{R}}^k_{2}&= \left\{D\in \Big(\bigcup_{1\le n \le k-1}{\mathfrak{D}}^k_{n,1}\Big) - {\mathfrak{R}}^k_{1} : \begin{array}{l} \exists \text{ column such that}\\ \text{at least two entries are nonzero}\end{array}\right\};\\ {{\mathfrak{M}}}^k &= \Big(\bigcup_{1\le n\le k-1}{\mathfrak{D}}^k_{n,1}\Big) - \left({\mathfrak{R}}^k_{1} \cup {\mathfrak{R}}^k_{2}\right). \end{aligned}\end{align*}$$

(i) The space Y

By Theorem 4.1 and Theorem 3.4, one can deduce that

(5.1) $$ \begin{align} \begin{aligned} &{\mathbb{E}} \left(\prod_{i=1}^{\iota} (Z^1_{i,d})^{k_i}\right) =\frac 1 {\phi(d)^{k/2}} \int_{Y} {\prod_{i=1}^{\iota} \widehat F_{i,d}^{k_i}} (\Lambda){\hspace{0.5mm} {\mathrm{d}}}\mu_Y(\Lambda)\\ &=\frac 1 {\phi(d)^{k/2}}\sum_{n=1}^{k-1} \sum_{\scriptsize \begin{array}{c} D" \in\\ {\mathfrak{S}}^k_{n,1}\cap {\mathfrak{M}}^k\end{array}} \int_{({\mathbb{R}}^d)^n} \left(\prod_{i=1}^{\iota} f_{i,d}^{k_i}\right) \left(D" \left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_n \end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_n \\ &\hspace{0.2in}+ O\left(\left(\frac {\sqrt{3}} 2\right)^d \phi(d)^{k/2-1}\right). \end{aligned}\end{align} $$

Notice that if $D" \in {\mathfrak {S}}^k_{n,1}\cap {\mathfrak {M}}^k$ , then $D"$ is of type (a) in Theorem 4.1. Hence, for each column of $D"$ , there are at least two nonzero entries, and for each row of $D"$ , there is exactly one nonzero entry. Moreover, as mentioned in Remark 3.2, entries of $D"$ are $\{0,1\}$ .

We first claim that $D"$ for which the inner integral above is nontrivial is the block diagonal matrix of the form

$$\begin{align*}\left(\begin{array}{cccc} D^{\prime\prime}_{k_1, n_1} & & & \\ & D^{\prime\prime}_{k_2, n_2} & & \\ & & \ddots & \\ & & & D^{\prime\prime}_{k_{\iota}, n_{\iota}} \end{array}\right), \end{align*}$$

where $n_1+\cdots +n_{\iota }=n$ and $n_i\ge 1$ for each $1\le i\le \iota $ . Moreover, each $D^{\prime \prime }_{k_i,n_i} \in {\mathfrak {S}}^{k_i}_{n_i,1}\cap {\mathfrak {M}}^{k_i}$ . Indeed, since the set $\{S_{i,n}\}_{1\le i\le \iota }$ is mutually disjoint, for each column, it is only possible that nontrivial entries are located between the $\left ((\sum _{\ell =1}^{i-1} k_i)+1\right )$ -th row and the $\left (\sum _{\ell =1}^{i} k_i\right )$ -th row for some $1\le i \le \iota $ . In other words, nontrivial entries are concentrated in rows which correspond to $f_i$ . The fact that $D"$ is a block diagonal matrix comes from that $D"\in {\mathfrak {D}}^k_{n,u}$ , especially, from the first property of Notation 2.1 (2).

It is not hard to show that each $D^{\prime \prime }_{k_i,n_i}$ is in ${\mathfrak {S}}^{k_i}_{n_i,1} \cap {\mathfrak {M}}^{k_i}$ from the fact that $D" \in {\mathfrak {S}}^k_{n,1}\cap {\mathfrak {M}}^k$ . Hence, the main term in (5.1) is

(5.2) $$ \begin{align} \prod_{i=1}^{\iota} \frac 1 {\phi(d)^{k_i/2}} \hspace{-0.05in}\sum_{n_i=1}^{\lfloor k_i/2\rfloor} \hspace{-0.15in}\sum_{\scriptsize \begin{array}{c} D^{\prime\prime}_{k_i,n_i} \hspace{-0.05in}\in\\ {\mathfrak{S}}^{k_i}_{n_i,1} \cap {\mathfrak{M}}^{k_i}\end{array}} \hspace{-0.2in} \int_{({\mathbb{R}}^d)^{n_i}} f_i^{k_i}\left(D^{\prime\prime}_{k_i,n_i}\left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{n_i}\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{n_i}. \end{align} $$

The next claim is that for each i, there is a one-to-one correspondence between ${\mathfrak {S}}^{k_i}_{n_i,1} \cap {\mathfrak {M}}^{k_i}$ and the set of partitions $\mathcal P=\{P_1, \ldots , P_{n_i}\}$ of $\{1, \ldots , k_i\}$ such that

$$\begin{align*}|\mathcal P|=n_i \text{ and } |P_{\ell}|\ge 2 \text{ for } 1\le \ell \le n_i. \end{align*}$$

Let $\mathcal P$ be such a partition. Reordering if necessary, we may assume that $\min P_1 < \ldots < \min P_{n_i}$ . The corresponding element in ${\mathfrak {S}}^{k_i}_{n_i,1} \cap {\mathfrak {M}}^{k_i}$ is

(5.3) $$ \begin{align} \left[D^{\prime\prime}_{k_i,n_i}\right]_{\ell j}=\left\{\begin{array}{cl} 1, &\text{if } \ell \in P_j;\\ 0, &\text{otherwise.} \end{array} \right. \end{align} $$

It is obvious that from the first property of Notation 2.1 (2) and the definition of ${\mathfrak {M}}^{k_i}$ , any element in ${\mathfrak {S}}^{k_i}_{n_i,1} \cap {\mathfrak {M}}^{k_i}$ is a matrix of the form (5.3) for some partition $\{P_1, \ldots , P_{n_i}\}$ of $\{1, \ldots , k_i\}$ .

Let $N(k_i,n_i)$ be the number of such partitions. If $n_i< k_i/2$ , since $\lim _{d\rightarrow \infty } \phi (d)=\infty $ ,

(5.4) $$ \begin{align} \begin{aligned} &\frac 1 {\phi(d)^{k_i/2}} \sum_{\scriptsize \begin{array}{c} D^{\prime\prime}_{k_i,n_i}\hspace{-0.05in}\in\\ {\mathcal S}^{k_i}_{n_i,1}\cap {\mathfrak{M}}^{k_i}\end{array}} \int_{({\mathbb{R}}^d)^{n_i}} F_i^{k_i} \left(D^{\prime\prime}_{k_i,n_i}\left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{n_i}\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{n_i}\\ &\hspace{1in}\le \frac {c_i^{n_i}N(k_i,n_i)} {\phi(d)^{k_i/2-n_i}} \longrightarrow 0\;\text{as }d \rightarrow \infty. \end{aligned} \end{align} $$

If $n_i=k_i/2$ , by the induction on $k_i/2$ , one can show that

(5.5) $$ \begin{align} \begin{aligned} &\frac 1 {\phi(d)^{k_i/2}} \sum_{\scriptsize \begin{array}{c} D^{\prime\prime}_{k_i,n_i}\hspace{-0.05in}\in \\ {\mathcal S}^{k_i}_{n_i,1}\cap {\mathfrak{M}}^{k_i}\end{array}} \int_{({\mathbb{R}}^d)^{n_i}} F_i^{k_i} \left(D^{\prime\prime}_{k_i,n_i}\left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{n_i}\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{n_i}\\ &\hspace{1in}=c_i^{k_i/2} N(k_i, k_i/2)= c_i^{k_i/2} (k_i-1)!!. \end{aligned}\end{align} $$

The result follows from (5.2), (5.4) and (5.5).

(ii) The space $Y_{{\mathbf {p}}/q}$

The proof is similar to that of (i), where we use Theorem 4.2, Lemma 3.4. One can check that $D"\in \mathfrak T^k_{n,1}\cap {\mathfrak {M}}^k$ is of type (a) in Theorem 4.2.

One difference from the affine case is when $q=2$ , $D"\in \mathfrak T^k_{n,1}\cap {\mathfrak {M}}^k$ , which permits to have $-1$ as its entries. More precisely, the rows corresponding to $I_{D"}^c$ can have $\pm 1$ as their nonzero entries.

It follows that

$$\begin{align*}\begin{aligned} &\lim_{d\rightarrow \infty}{\mathbb{E}}\left(\prod_{i=1}^{\iota} Z^{k_i}_{i,d}\right)=\lim_{d\rightarrow \infty} \prod_{i=1}^{\iota} \frac 1 {(2\phi(d))^{k_i/2}}\times\\ &\hspace{0.3in} \sum_{n_i=1}^{\lfloor k_i/2 \rfloor}\sum_{\scriptsize \begin{array}{c} D^{\prime\prime}_{k_i,n_i}\in\\ \mathfrak T^{k_i}_{n_i,1}\cap {\mathfrak{M}}^{k_i}\end{array}} \int_{({\mathbb{R}}^d)^{n_i}} f_i^{k_i}\left(D^{\prime\prime}_{k_i,n_i} \left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{n_i}\end{array}\right)\right){\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{n_i}. \end{aligned}\end{align*}$$

As in the affine case, the limit is nontrivial only if all $k_i$ ’s are even and is determined by summation over $\mathfrak T^{k_i}_{k_i/2,1}\cap {\mathfrak {M}}^{k_i}$ . Hence, if $q=2$ , since $\# I_{D^{\prime \prime }_{k_i,k_i/2}}^c=k_i/2$ , the number $\#\left (\mathfrak T^{k_i}_{k_i/2,1}\cap {\mathfrak {M}}^{k_i}\right )$ is $2^{k_i/2}N(k_i,k_i/2)$ . Therefore,

$$\begin{align*}\begin{aligned} &\prod_{i=1}^{\iota}\frac 1 {(2\phi(d))^{k_i/2}} \sum_{\scriptsize \begin{array}{c} D^{\prime\prime}_{k_i,n_i}\hspace{-0.05in}\in \\ \mathfrak T^{k_i}_{n_i,1}\cap {\mathfrak{M}}^{k_i}\end{array}} \int_{({\mathbb{R}}^d)^{n_i}} F_i^{k_i} \left(D^{\prime\prime}_{k_i,n_i}\left(\begin{array}{c} \mathbf w_1 \\ \vdots \\ \mathbf w_{n_i}\end{array}\right)\right) {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_1 \cdots {\hspace{0.5mm} {\mathrm{d}}}\mathbf w_{n_i}\\ &=\prod_{i=1}^{\iota} \frac 1 {(2\phi(d))^{k_i/2}} (c_i\phi(d))^{k_i/2} \cdot 2^{k_i/2}N(k_i, k_i/2) =\prod_{i=1}^{\iota} c_i^{k_i/2} (k_i-1)!!.\\[-25pt] \end{aligned} \end{align*}$$

Proofs of Theorem 1.3 and 1.4

As a corollary of Proposition 5.1 with $\iota =1$ , for $\diamondsuit =1$ and ${\mathbf {p}}/q$ , it follows that for any $k\in {\mathbb {N}}$ ,

$$\begin{align*}\lim_{d\rightarrow \infty} {\mathbb{E}}\left( (Z^{\diamondsuit}_d)^k\right) =\left\{\begin{array}{cl} (k-1)!!, &\text{if } k\in 2{\mathbb{N}};\\ 0, &\text{otherwise,} \end{array} \right. \end{align*}$$

which shows that $Z^{\diamondsuit }_d\rightarrow {\mathcal {N}}(0,1)$ as $d\rightarrow \infty $ in distribution by the method of moments.

Proofs of Theorem 1.5 and 1.6

For any $0<t_1<\ldots <t_{\iota }<1$ , set

$$\begin{align*}S_{i,d}=(t_i)^{1/d}S_d - (t_{i-1})^{1/d}S_d,\; 2\le i\le \iota \end{align*}$$

and $S_{1,d}=(t_1)^{1/d}S_d$ . Since $S_d$ is star-shaped, all $S_{i,d}$ ’s are mutually disjoint. By Proposition 5.1, for $\diamondsuit =1$ and ${\mathbf {p}}/q$ , the random vector

$$\begin{align*}\left(Z^{\diamondsuit}_d (t_1), Z^{\diamondsuit}_d(t_2)- Z^{\diamondsuit}_d(t_1), \ldots, Z^{\diamondsuit}_d(t_{\iota})-Z^{\diamondsuit}_d(t_{\iota-1})\right) \end{align*}$$

converges weakly as finite-dimensional distributions to

$$\begin{align*}\left({\mathcal{N}}(0,t_1), {\mathcal{N}}(0, t_2-t_1), \ldots, {\mathcal{N}}(0, t_{\iota}-t_{\iota-1})\right) \end{align*}$$

by the method of moments.

The rest of the proof is to show the tightness. As in the proof of Theorem 1.6 in [22], by [3, Theorem 13.3 and (13.14)], it suffices to show that for any $0\le r \le s \le t \le 1$ ,

$$ \begin{align*} {\mathbb{E}}\left((Z^{\diamondsuit}_d(s)-Z^{\diamondsuit}_d(r))^2 (Z^{\diamondsuit}_d(t)-Z^{\diamondsuit}_d(s))^2\right) \ll (\sqrt t - \sqrt r)^2. \end{align*} $$

We omit the proof since it is almost the same as in the proof of [22, Theorem 1.6] (see especially equations from (4.4) to (4.9)), where the arguments are applicable to a star-shaped set $S_d\subseteq {\mathbb {R}}^d$ centered at the origin without any modification. Here, we want to remark that we need the argument in [22] only for the congruence case. For the affine case, since $\bigcup _{u\in {\mathbb {N}}} {\mathfrak {S}}^4_{1,u} \cap ({\mathfrak {R}}_1 \cup {\mathfrak {R}}_2)=\emptyset $ , it is deduced directly from (4.5) in [22] that

$$\begin{align*}\begin{aligned} &{\mathbb{E}}\left((Z_d(s)-Z_d(r))^2(Z_d(t)-Z_d(s))^2\right)\\ &\ll (t-r)^2 + \max\left(\left(\frac 3 4\right)^{n/2}(t-r)^2, \left(\frac 3 4\right)^{n/2}(t-r)^3 \phi(d)\right)\\ &\ll (t-r)^2 < (\sqrt t - \sqrt r)^2.\\[-18pt] \end{aligned} \end{align*}$$