Central Limit Theorem for Diophantine approximation on spheres
Zouhair Ouaggag

§ ABSTRACT 
We prove a Central Limit Theorem and an effective estimate for the counting function of Diophantine approximants on the sphere S^n using homogeneous dynamics on the space of orthogonal lattices.



Keywords: intrinsic Diophantine approximation; Central Limit Theorem; Siegel transform; effective equidistribution.



    Marco Mellia
     
================





§ INTRODUCTION AND MAIN RESULTS


It is well-known in metric Diophantine approximation that for any c>0 and Lebesgue-almost all α∈ℝ^m, there exist infinitely many solutions (p,q) ∈ℤ^m×ℕ to the inequality[|| ·|| will denote the Euclidean norm.]

    ‖α - p/q‖  <  c/q^1+1/m   .


A refinement of this problem is to count solutions up to a certain bound for the complexity q of the approximants, which leads to consider counting functions such as

    N_T,c(α) |{ (p,q) ∈ℤ^m×ℕ : 1 ≤ q < e^T  and  (<ref>)  holds }| .


An accurate estimate of the counting function N_T,c was given in the work of W. Schmidt <cit.>, who proved for more general approximating functions, that for Lebesgue-almost all α∈ [0,1]^m, 

    N_T,c(α)=𝖢_c,m T + O_α,ε(  T^1/2+ε) ,

for all ε>0, with a constant 𝖢_c,m>0 depending only on c and m.


In recent years, there has been significant interest in the problems of so-called intrinsic Diophantine approximation, where one considers approximation by rational points on algebraic varieties, addressing analogues of classical questions in the geometry of numbers and metric Diophantine approximation. Important progress has been achieved in this setting. In particular, Kleinbock and Merrill in <cit.> developed the theory of Diophantine approximation on spheres, which was subsequently generalized to quadratic surfaces with general signatures by Fishman, Kleinbock, Merrill and Simmons in <cit.>. These works established in particular analogues of the classical Dirichlet's and Khinchin's theorems. Then Alam and Ghosh in <cit.> proved an asymptotic formula for the number of rational approximants on spheres (<ref>). We also mention the works of Ghosh, Gorodnik and Nevo who developed the metric theory of Diophantine approximation on simple algebraic groups, providing estimates for uniform and almost sure Diophantine exponents in <cit.>, establishing analogues of Khinchin's and Jarnik's theorems in <cit.>, and deriving an asymptotic formula with an error term for the number of approximants for a range of uniform Diophantine exponents in <cit.>.

 
In this paper we are interested in the following intrinsic Diophantine approximation problem. Given T, c>0 and α∈ S^n, we consider the inequality (<ref>) (with the critical Dirichlet exponent for intrinsic Diophantine approximation on S^n)

    ‖α - p/q‖  <  c/q   ,

and the counting function given by (<ref>) for intrinsic rational approximants

    𝖭_T,c(α) |{ (p,q) ∈ℤ^n+1×ℕ : p/q∈S^n, 1 ≤ q < cosh T  and  (<ref>)  holds }| .




 §.§ Effective estimate for the counting function

Alam and Ghosh gave in <cit.> a first quantitative estimate of 𝖭_T,c, using Birkhoff pointwise ergodicity on the space of orthogonal lattices to show that there exists a computable constant C_c,n>0, depending only on c and n, such that for almost every α∈S^n, 

    lim_T →∞𝖭_T,c(α)/T = C_c,n .


With a different approach, using effective equidistribution of translated orbits and important non-divergence estimates on the space of orthogonal lattices due to Eskin, Margulis and Mozes in <cit.>, we gave in <cit.> an effective estimate for 𝖭_T,c, showing that there exists a constant γ <1 depending only on the dimension n, such that for almost every α∈S^n, 

    𝖭_T,c(α) = C_c,nT + O_α(T^γ) .


 In order to improve the estimate of the error term in (<ref>) to the order T^1/2+ε as in  (<ref>) for the Euclidean space, we were missing some analog of Roger's formula for the space of orthogonal lattices. A crucial result in this direction was given recently by Kelmer and Yu in <cit.>, using spectral theory of spherical Eisenstein series to give an estimate of the second moment of the Siegel transform (see Theorem <ref>).
 They also derived an effective estimate for 𝖭_T,ψ(q), for more general quadratic surfaces S and general approximation function ψ:ℕ→ (0,+∞), decreasing and satisfying ∑_q≥ 1q^-1ψ(q)^n=∞, showing that in dimension n≢1 (mod  8), for almost every α∈S, 
 
    𝖭_T,ψ(q)(α) = C_nJ_ψ(T) + O_α,ψ(J_ψ(T)^n+3/n+4log (J_ψ(T))+I_ψ(T)) ,

with J_ψ(T):=∑_1≤ q < Tq^-1ψ(q)^n, I_ψ(T)=∑_1≤ q < Tq^-3ψ(q)^n+2 and C_n>0.

 
 Developing our approach in <cit.> and using methods derived from <cit.> to analyse the second moment of the Siegel transform, we prove in this paper an effective estimate of 𝖭_T,c with an error term of the same order as in (<ref>), for all dimensions n≥ 3.
  
Let n≥ 3. For almost every α∈S^n, for all ε>0, we have 

    𝖭_T,c(α) = C_c,nT + O_α,ε(T^1/2+ε) .

	


    Some remarks related to Theorem <ref>:
    
        
  * The constant C_c,n>0 in (<ref>), (<ref>) and in Theorems <ref>, <ref> is equal to the volume of a domain F_1,c⊂ℝ^n+2 given explicitly in (<ref>).
        
  * Although the estimate of the error term T^1/2+ε is optimal in the case of the Euclidean space, we can not conclude about the optimality of this estimate for the sphere. Nevertheless, our analysis of the limit distribution of 𝖭_T,c (see Theorem <ref>) suggests that T^1/2 is the correct normalisation and the error term would be optimal if one could show that the variance σ^2 is positive.
        
  * Our method fails for dimension n=2 because of the escape of mass in the space of orthogonal lattices for this dimension (see Proposition <ref>). 
    




 §.§ Distribution of the counting function

Another interesting question is to study the limit distribution of 𝖭_T,c as a random variable on the sphere. If we consider counting functions of approximants with denominators q in ranges [cosh (t) , cosh (t+1) ], i.e. functions

    𝒩_t,c := 𝖭_t+1,c- 𝖭_t,c

and if the random variables 𝒩_t_1,c and 𝒩_t_2,c "decorrelate" for large t_1, t_2 and |t_2-t_1|, then Theorem <ref> would follow by a Law of large numbers for the random variable 𝖭_T,c=∑_t=0^T-1𝒩_t,c. This heuristic was developed by Bjöklund and Gorodnik in <cit.> to show that the counting function N_T,c for the Euclidean space follows a Central limit theorem, using higher-order mixing in the space of unimodular lattices as the dynamical translation of quasi-independent random variables. Using a similar approach as in <cit.> and the recent results by Kelmer and Yu in <cit.> about the second moment of the Siegel transform, we show that the counting function 𝖭_T,c on the sphere also follows a Central limit theorem.
	
 

Let n≥ 3. Then for every ξ∈ℝ, 

    μ_S^n( {α∈ S^n:𝖭_T,c(α)-C_c,n· T/T^1/2<ξ}) →Norm_σ(ξ) ,  as T→∞,

where Norm_σ denotes the normal distribution with variance σ^2≥ 0.
	



The variance σ^2 in Theorem <ref> is given explicitly in (<ref>).




 §.§ Outline of the paper

Using the classical Dani correspondence, we first interpret the counting function 𝖭_T,c in terms of ergodic averages of a function over a subset of the space of unimodular lattices in ℝ^n+2, developing the approach in <cit.>, <cit.> and <cit.>. To do so, we embed the sphere S^n in the positive light cone 𝒞{ x ∈ℝ^n+1×ℝ_+ :Q(x)=0} of a quadratic form Q of inertia (n+1,1), and identify good approximants p/q∈ S^n for α∈ S^n with integer points (p,q) in Λ_0:=ℤ^n+2∩𝒞 whose images under certain rotations k_α∈ K=SO(n+1) lie in a specific domain E_T,c⊂𝒞 (we recall more details about this correspondence in Section <ref>). The number of solutions 𝖭_T,c is then related to the number of lattice points in the domain E_T,c, which can be approximated by a more convenient domain F_T,c and tessellated by the action of a hyperbolic subgroup {a_t, t∈ℝ}⊂ SO(Q). 

For a bounded and compactly supported function f on 𝒞, we denote by f its light-cone Siegel transform, defined for any lattice Λ⊂𝒞 by

    f(Λ):= ∑_x∈Λ∖{0}f(x) .
 
The counting function 𝖭_T,c can then be related to the light-cone Siegel transform of the characteristic function χ of an elementary domain F_1,c⊂𝒞, using averages of the form

    𝖭_T,c(α) ≈∑_t=0^T-1χ∘ a_t (k_αΛ_0),

i.e. ergodic averages of the light-cone Siegel transform χ along K-orbits pushed by {a_t}. The analysis of these averages can then be carried out using dynamics on the space of orthogonal lattices.


Because the approximation (<ref>) is not precise (see "sandwiching" in (<ref>)), we need a version of Borel-Cantelli argument for a family of functions. This argument was overlooked in our previous work <cit.> and is now considered in Proposition <ref>.  

In order to show that 𝖭_T,c follows a Central Limit Theorem, we will use the method of cumulants (Section <ref>), which is equivalent to the more widely known method of moments. The normal distribution is characterized by vanishing cumulants of orders r≥ 3, which can be expressed in the dynamical language as higher-order of an averaging function of the form
    
    𝖥_N ≈1/N^1/2∑_t=0^N-1χ∘ a_t .

The "quasi-independence" of sampling observables χ∘ a_t along K-orbits pushed by {a_t} corresponds to multiple equidistribution of these orbits in the space of orthogonal lattices, which we establish in Section <ref> (Proposition <ref>). However, using effective equidistribution requires to consider smooth and compactly supported test functions, whereas the Siegel transform has typically none of these regularities. We address this issue in two steps.


We first use that the Siegel transform f can be approximated by a truncated Siegel transform f^(L) in a way to control the approximation on translated K-orbits, i.e. control | f∘ a_t - f^(L)∘ a_t| with respect to the probability measure on these orbits (Proposition <ref>). In a second step (Proposition <ref>), we use that the characteristic function χ of the elementary domain F_1,c can be approximated by a family of smooth and compactly supported functions f_ε, again in a way to keep control of the approximated Siegel transform f_ε over translated K-orbits. In this process we also need non-divergence results for the Siegel transform with respect to the probability measure on these orbits (Propositions <ref> and <ref>). We also need to show that all these approximations still give the same limit distribution for the averaging function 𝖥_N (Section <ref>). 


In Section <ref> we use exponential multiple equidistribution established previously to show that the cumulants of 𝖥_N of orders r≥ 3 vanish as N→∞. To do so, we follow the argument developed in  <cit.> and <cit.>, analysing the joint cumulants through a decomposition into sub-sums of correlations corresponding to “separated" or “clustered" tuples t_1,…,t_r and controlling their size in terms of the parameters related to the smooth and truncated approximation of the counting function.


In Section <ref> we estimate the limit variance of 𝖥_N as N→∞ using resent results of Kelmer and Yu in <cit.> on the second moment of the Siegel transform. Convergence of the second variance  and vanishing of all cumulants of orders r≥ 3 complete the characterisation of the normal distribution for 𝖥_N. In Section <ref> we relate the distribution of 𝖥_N to the distribution of our counting function 𝖭_T,c and conclude the proof of the CLT-Theorem.


Our analysis of double correlations  from Section <ref> allows us to derive an "almost-everywhere"-bound for the ergodic averages ∑_tχ∘ a_t (Proposition <ref>) and to improve the effective estimate for the counting function 𝖭_T,c (proof of Theorem <ref> in Section <ref>).




  
Acknowledgement I am grateful to Dubi Kelmer for sharing his ideas and for the useful discussions. Many thanks also to René Pfitscher for our discussions and his useful observations. I am very grateful to Alex Gorodnik for his guidance all along this research work.




§ DIOPHANTINE APPROXIMATION ON SPHERES AND DYNAMICS ON THE SPACE OF LATTICES



We recall the correspondence presented in <cit.> and <cit.> between Diophantine approximation on the sphere S^n and the dynamics of orthogonal lattices in ℝ^n+2.


We consider the quadratic form Q: ℝ^n+2→ℝ defined by

    Q(x) := ∑_i=1^n+1 x_i^2 - x_n+2^2,    for  x= (x_1, … , x_n+2) ,

and the embedding of S^n in the positive light cone

    𝒞 := {  x ∈ℝ^n+2 : Q(x)=0,  x_n+2 > 0 } ,

via α↦ (α,1), which yields a one-to-one correspondence between primitive integer points on the positive light cone, (p,q) ∈𝒞∩ℤ^n+2_prim, and rational points on the sphere, p/q∈ S^n.


We denote by G=SO(Q)^∘≅ SO(n+1,1)^∘ the connected component of the group  of orientation-preserving linear transformations which preserve Q. We denote by Λ_0 := 𝒞∩ℤ^n+2 the set of integer points on the positive light cone. By a lattice Λ in 𝒞 we mean a set of the form gΛ_0 for some g∈ G. 
If we denote by Γ the stabilizer of Λ_0 in G, then Γ is a lattice in G containing the subgroup SO(Q)_ℤ^∘ of integer points in G, as a finite index subgroup.
The space of lattices in 𝒞 can be identified with the homogeneous space 𝒳 := G/Γ, endowed with the G-invariant probability measure μ_𝒳.


Let K denote the subgroup of G that preserves the last coordinate in ℝ^n+2, i.e.

    K= [ SO(n+1)        ;               1 ]≅SO(n+1) ,

equipped with the Haar probability measure μ_K.


The sphere S^n can be realized as a quotient of K, endowed with a unique left K-invariant probability measure, giving a natural correspondence between full-measure sets in K and those in S^n. 


For k ∈ K we define α_k ∈ S^n by k(α_k,1) = (0,…,0,1,1) ∈𝒞. For (p,q) ∈Λ_0, we write k(p,q)= (x_1,x_2, … , x_n+2) ∈𝒞, with x_n+2=q, and observe the following correspondence (<cit.>, Lemma 2.2.):

    ‖α_k - p/q‖ < c/q     ⇔  ‖ q( α_k,1) -(p,q) ‖ < c, 
       ⇔  ‖ qk(α_k,1) - k(p,q)‖ < c, 
       ⇔  ‖(x_1,x_2, … , x_n, x_n+1-x_n+2,0)‖ < c, 
       ⇔   2x_n+2(x_n+2-x_n+1) < c^2   (since k(p,q) ∈𝒞).

Hence, if we denote 

    E_T,c{ x ∈𝒞 : 2x_n+2(x_n+2-x_n+1) <c^2, 1≤ x_n+2 < cosh T },

then we have:

    𝖭_T,c(α_k) =| E_T,c∩ kΛ_0 |.


We denote 𝒴 K Λ_0, equipped with the Haar probability measure μ_𝒴.


We also consider elements 

    a_t =  [     I_n                ;          cosh t -sinh t;         -sinh t  cosh t ]  ∈ G

and the corresponding one-parameter subgroup

    A = {  a_t : t ∈ℝ}

endowed with the natural measure dt.


We will denote by χ_E the characteristic function of a given set E and use the notation a ≍ b (resp. a ≪ b) when there exist positive constants C_1 and C_2 such that C_1b ≤ a ≤ C_2 b (resp. a ≤ C_2 b).


In order to use the dynamics of translates of 𝒴 for the Diophantine approximation problem (<ref>), we first approximate E_T,c by a domain offering a convenient tessellation under the action of the subgroup A. We start with a similar approach as in <cit.> and improve the approximation by F_T,c in order to satisfy the accuracy obtained later for the counting function.




  
Approximation of E_T,c
We consider the following domain on a light-cone

    F_T,c := { x ∈𝒞 : x_n+2^2- x_n+1^2 < c^2 ,  c≤ x_n+2+ x_n+1 < ce^T},

and a sequence of domains (F_T,c,𝓁)_𝓁≥ 1 defined by 

    F_T,c,𝓁 := { x ∈𝒞 : x_n+2^2- x_n+1^2 < c_𝓁^2 ,  c≤ x_n+2+ x_n+1 < ce^T},

with c_𝓁 c·( 𝓁/𝓁+1)^1/2.


Up to the domains 
    C_0 { x ∈𝒞: x_n+2≤c^2+c/2+1}  and   C_𝓁{ x ∈𝒞: |x_n+1|/x_n+2≤𝓁/𝓁+1}∪ C_0 ,
 we can approximate E_T,c by F_T,c in the following sense. 
 There exist positive constants T_0 and r_0 depending only on c such that, for all T≥ T_0 and all integers 𝓁≥ 1, we have

    F_T-r_0,c,𝓁∖ C_𝓁  ⊆  
    E_T,c∖ C_0   ⊆   F_T+r_0,c .
 



For x∈ E_T,c∖ C_0,

    2x_n+2(x_n+2-x_n+1)<c^2   and   x_n+2≥ 1
       imply   x_n+2-x_n+1< c^2 ,

hence 

    x_n+2+x_n+1 >  2x_n+2-c^2  > c .

Also 

    x_n+1≤ x_n+2  implies   x_n+2+x_n+1≤ 2x_n+2<2cosh T < c e^T+r_0,

for some r_0>0 depending only on c.
 

We also have

    x_n+2^2-x_n+1^2= (x_n+2+x_n+1)(x_n+2-x_n+1) ≤ 2x_n+2(x_n+2-x_n+1) < c^2.

The inequalities (<ref>), (<ref>) and (<ref>) prove the second inclusion in (<ref>).
 

For x∈ F_T-r_0,c,𝓁∖ C_𝓁, we have 

    x_n+2^2-x_n+1^2<c_l^2<c^2   and   x_n+2+x_n+1≥ c
       imply   x_n+2-x_n+1< c ,

hence 

    2x_n+2 < x_n+2+x_n+1+c  <  ce^T-r_0+c ≤ 2cosh T

for all T>T_0, for some T_0>0 depending on c, and for r_0 large enough.
 

We also have

    x_n+1 > x_n+2-c > c^2+c/2+1 -c > 0 
    ,

which implies

    2x_n+2(x_n+2-x_n+1)    = (x_n+2^2-x_n+1^2)2x_n+2/x_n+2+x_n+1
       < c_l^22x_n+2/x_n+2+x_n+1
       = c^2(𝓁/𝓁+1)2x_n+2/x_n+2+x_n+1
       < c^2x_n+2+|x_n+1|/x_n+2+x_n+1
       = c^2 .

The inequalities (<ref>) and (<ref>) prove the first inclusion in (<ref>).



It follows from (<ref>) and (<ref>) that, for T>0 large enough,

    |(F_T-r_0,c,𝓁∖ C_𝓁)∩ kΛ_0|  ≤ 𝖭_T,c(α_k) + O(1)  ≤ |F_T+r_0,c∩ kΛ_0|.


We will need later an estimate of the counting error related to the “sandwiching" (<ref>). We estimate in the following lemma this error in terms of the integer parameter 𝓁≥ 1 to be specified later.


We have

    vol(F_1,c) =vol( F_1,c,𝓁)+ O(𝓁^-1),
    and        |F_T-r_0,c,𝓁∩ C_𝓁∩ kℤ^n+2| = O(𝓁^1/2),  uniformly in  k∈ K.



 Since 

    F_1,c∖ F_1,c,𝓁= { x ∈𝒞 : c_𝓁≤ (x_1^2+ … + x_n^2)^1/2 < c ,  c< x_n+2+ x_n+1 < ce } ,

we have 
    vol(F_1,c) = vol( F_1,c,𝓁)+ O( c-c_𝓁) ,

and 

    c^2-c_𝓁^2=c^2(1-𝓁/𝓁+1)=c^21/𝓁+1 ,
       c-c_𝓁=c^2/c+c_𝓁1/𝓁+1=O(𝓁^-1) ,

hence the first estimate.


For the second estimate, we observe that for x ∈ C_𝓁∩ F_T-r_0,c,𝓁 we have

    x_n+2^2  <  c_𝓁^2+x_n+1^2  ≤  c_𝓁^2+( 𝓁/𝓁+1)^2x_n+2^2  ,
    hence       x_n+2 < 𝓁+1/(2𝓁+1)^1/2c_𝓁 ,

thus for 𝓁 large enough, F_T-r_0,c,𝓁∩ C_𝓁 is contained in {x∈𝒞 :  x_n+2≪_c𝓁^1/2 and  x_1^2+ … + x_n^2 < c_𝓁^2}. Moreover, since k∈ K preserves the coordinate x_n+2 and is Lipschitz continuous in the other coordinates, by compactness there exist C>0 such that for every k∈ K,  k(F_T-r_0,c,𝓁∩ C_𝓁) is contained in {x∈𝒞 :  x_n+2≪_c𝓁^1/2 and  x_1^2+ … + x_n^2 < C}, which yields the uniform bound O(𝓁^1/2) for the number of its integer points.







  
Tessellation of F_T,c We observe further that the domain F_T,c can be tessellated using translates of the set F_1,c under the action of {a_t}. We have, for all N ≥ 1,

    F_N,c= _j=0^N-1 a_-j(F_1,c).


We denote by χ_1,c the characteristic function of F_1,c, and χ_1,c its Siegel transform defined by

    χ_1,c(Λ) ∑_z ∈Λ∖{0}χ_1,c (z),  for all Λ∈𝒳.


The tessellation (<ref>) implies, for all T>0 and all Λ∈𝒳,

    ∑_t=0^⌊ T ⌋-1χ_1,c(a_tΛ)  ≤  |F_T,c∩Λ|  ≤ ∑_t=0^⌊ T ⌋χ_1,c(a_tΛ).


It follows from (<ref>), (<ref>) and (<ref>), for large enough T>0 and k ∈ K as in (<ref>),

    ∑_t=0^⌊ T-r_0 ⌋-1χ_1,c,𝓁(a_t kΛ_0) +O( 𝓁^1/2)  ≤ 𝖭_T,c(α_k) +O(1)  ≤ ∑_t=0^⌊ T+r_0 ⌋χ_1,c(a_t kΛ_0).


Thus, estimating 𝖭_T,c(α) amounts to analyzing ergodic sums of the form ∑_t=0^Nχ_1,c∘ a_t on 𝒴= KΛ_0. We will use for this purpose effective higher order equidistribution results for unimodular lattices, specialized to 𝒴, which we discuss on the following section.
	


§ ESTIMATES ON HIGHER ORDER CORRELATIONS



	
In this section we prove an effective equidistribution of K-orbits by relating it to effective equidistribution of unstable horospherical orbits established in a more general setting in <cit.>. We recall the notations

    G    = SO(Q)^∘≅SO(n+1,1)^∘,
    
    K    = [ SO(n+1)        ;               1 ],
    
    a_t    =  [     I_n                ;          cosh t -sinh t;         -sinh t  cosh t ] ∈ G,   and   A = {  a_t : t ∈ℝ}.


We also consider the corresponding horospherical subgroups

    U = {  g∈ G   :   a_-tga_t→ e  as t→∞},
       U^- = {  g∈ G   :   a_tga_-t→ e  as t→∞},
       H = {  g∈ G   :   a_tg=ga_t},

and the Haar measures dμ_K, dt and dμ_U on K, A and U respectively.

It will be important in our argument later that the error term in the effective equidistribution is explicit in terms of the C^l-norm, for some l≥ 1, of the test functions on 𝒳. We introduce below the required notations.


Every Y ∈ Lie(G) defines a first order differential operator D_Y on C_c^∞(𝒳) by

    D_Y(ϕ)(x) d/dtϕ(exp (tY)x)|_t=0.


If { Y_1,…, Y_r} is a basis of Lie(G), then every monomial Z=Y_1^l_1… Y_f^l_r defines a differential operator by 

    D_Z D_Y_1^l_1… D_Y_r^l_r,

of degree deg(Z)=l_1+…+l_r. For integers l≥ 0 and ϕ∈ C_c^∞(𝒳), we write

    ||ϕ||_l  ||ϕ||_C^l= ∑_deg(Z)≤ l ||D_Z(ϕ)||_∞


A crucial ingredient for our analysis is the following effective equidistribution result for higher order correlations on translated U-orbits (Theorem <ref>) and the analogous result we derive for translated K-orbits (Proposition <ref>)



For every r≥ 1 there exist γ_r>0 and l_r≥ 1 such that, for every f ∈ C_c^∞(U) and φ_1,…φ_r ∈ C_c^∞(𝒳) and every compact subset L ⊂𝒳, there exists C>0 such that for every Λ∈ L and t_1,… t_r > 0, we have

    |  ∫_U f(u)(∏_i=1^rφ_i (a_t_i uΛ) ) dμ_U(u)- (∫_Ufdμ_U) (∏_i=1^r∫_𝒳φ_i dμ_𝒳) | ≤ C e^-γ_r D(t_1,…,t_r)||f||_l_r∏_i=1^r||φ_i||_l_r ,
 
where D(t_1,…,t_r)min{t_i,|t_i-t_j|:1≤ i≠ j≤ r}.




For every r≥ 1 there exist δ_r>0 and l_r≥ 1 such that, for every f ∈ C^∞(K) and φ_1,…φ_r ∈ C_c^∞(𝒳) and every compact subset L ⊂𝒳, there exists C>0 such that for every Λ∈ L and t_1,… t_r > 0, we have

    |  I_Λ,f,φ_1,…,φ_r(t_1,…,t_r) - (∫_Kfdμ_K) (∏_i=1^r∫_𝒳φ_i dμ_𝒳) | ≤ C e^-δ_r D(t_1,…,t_r)||f||_l_r∏_i=1^r||φ_i||_l_r ,
 
where I_Λ,f,φ_1,…,φ_r(t_1,…,t_r) ∫_K f(k)(∏_i=1^rφ_i (a_t_i kΛ)) dμ_K(k).



We consider the centralizer of A in K,  

    Mcent_K(A)=K∩ H = [ SO(n)      ;         I_2 ]≅SO(n),

and the submanifold S⊂ K defined via the exponential map by

    Lie(S)= {[  0_n    s          ; -s^T    0;              0 ] : s∈ℝ^n}.

We have Lie(K)=Lie(M)⊕Lie(S) and the map M× S→ K is a diffeomorphism in a neighborhood of the identity, giving a unique decomposition k= m(k)s(k) and also a decomposition of the measure dμ_K, in the sens that ∫_K f dμ_K=∫_M× S  f dμ_S dμ_M for any f bounded and compactly supported in this neighborhood, where we denote by dμ_M the Haar measure on M and by dμ_S a smooth measure defined on a neighborhood of the identity in S.

Further, we consider the decomposition  of G as the product U^-HU in a neighborhood of the identity, giving a unique decomposition s=u^-(s)h(s)u(s). We verify that the coordinate map S→ U, s↦ u(s) is a diffeomorphism in a neighborhood of the identity. We first observe that

    dim(S)=dim(K)-dim(M)=(n+1)n/2-(n-1)n/2=n=dim(U).

Moreover, for the product map p:U^-× H × U → G,  (u^-,h,u)↦ u^-hu, the derivative at the identity is given by D(p)_e(x,y,z)=x+y+z, for all (x,y,z)∈Lie(U^-) ×Lie(H) ×Lie(U). Hence, for all w ∈Lie(G), the U-component of D(p)_e^-1(w) is zero if and only if w ∈Lie(U^-)+Lie(H). Since Lie(S) ∩(Lie(U^-)+Lie(H))= 0, the derivative of s↦ u(s) is injective. Since dim(S)=dim(U), this is a local diffeomorphism.

We denote B_𝓇^K the ball of radius 𝓇>0 centered at the identity in K and localize the problem to a neighborhood of the identity by considering the partition of unity 1= ∑_j=1^Nϕ_j(kk_j^-1) for all k ∈supp(f) and some k_j ∈ supp(f), with non-negative functions ϕ_j ∈ C^∞(K) such that supp(ϕ_j) ⊆ B_𝓇^K, || ϕ_j ||_l ≪𝓇^-ν and N ≪𝓇^-λ, for some ν, λ >0, and for 𝓇>0 small enough to be fixed later.

We write for simplicity k=m_ks_k=m_ku_s_k^-h_s_ku_s_k, the unique decompositions of k and s in a neighborhood of the identity in K and S. We also write f_j(k) f(kk_j) and Λ_j  k_jΛ. We compute

    I_Λ,f,φ_1,…,φ_r(t_1,…,t_r) = ∑_j=1^N ∫_K ϕ_j(k) f(kk_j)(∏_i=1^rφ_i(a_t_ikk_jΛ))dμ_K(k)
       =∑_j=1^N ∫_K ϕ_j(k) f_j(k)(∏_i=1^rφ_i(m_ka_t_iu^-_s_ka_-t_ih_s_ka_t_iu_s_kΛ_j))dμ_K(k).

By Lipschitz continuity of the coordinate maps m_k, u^-_s_k and h_s_k on B_𝓇^K with 𝓇 small enough, there exists a constant C_1>0 such that for all k ∈ B_𝓇^K, we have 

    a_tu^-_s_ka_-t∈ B_C_1𝓇e^-2t^K    and    m_k,h_s_k∈ B_C_1𝓇^K.

By Lipschitz continuity of φ_1,…,φ_r, it follows

    | I_Λ,f,φ_1,…,φ_r(t_1,…,t_r) - ∑_j=1^N ∫_K ϕ_j(k) f_j(k)(∏_i=1^rφ_i(a_t_iu_s_kΛ_j))dμ_K(k)  | 
       = |∑_j=1^N ∫_K ϕ_j(k) f_j(k)(∏_i=1^rφ_i(a_t_iu_s_kΛ_j)-∏_i=1^rφ_i(m_ka_t_iu^-_s_ka_-t_ih_s_ka_t_iu_s_kΛ_j)dμ_K(k)  | 
       ≪_l  𝓇‖ f ‖_l‖∏_i=1^rφ_i‖_l ∫_K |∑_j=1^Nϕ_j(k)|dμ_K(k)
       =  𝓇‖ f ‖_l‖∏_i=1^rφ_i‖_l     (by K-invariance and since ϕ_j is a partition of unity)
       ≪_r  𝓇‖ f ‖_l∏_i=1^r‖φ_i‖_l.


We use now the decomposition of μ_K and apply the change of variable u ↦ s(u)=s_u, with a density ρ defined in a neighborhood of the identity in U by

    ∫_S Φ(s)dμ_S(s)=∫_UΦ(s(u))ρ(u)dμ_U(u)   for all Φ∈ C_c(S)  with supp(Φ)⊂ B_𝓇^S.

We have

    ∑_j=1^N ∫_K ϕ_j(k) f_j(k)(∏_i=1^rφ_i(a_t_iu_s_kΛ_j))dμ_K(k) 
       = ∑_j=1^N ∫_M× Sϕ_j(ms) f_j(ms)(∏_i=1^rφ_i(a_t_iu_sΛ_j))dμ_S(s)dμ_M(m) 
       =  ∫_M(∑_j=1^N ∫_U ϕ_j(ms_u) f_j(ms_u)(∏_i=1^rφ_i(a_t_iuΛ_j))ρ(u) dμ_U(u))dμ_M(m).

Using Theorem <ref> with the function f_m,j(u) ϕ_j(ms_u)ρ(u)f_j(mu) and observing that ||f_m,j ||_l ≪ ||ϕ_j||_l||ρ||_l ||f_j||_l and that ||ρ||_l≪ 1, it follows that the integral (<ref>) is equal to

    ∫_M∑_j=1^N (∫_U ϕ_j(ms_u) f_j(ms_u)ρ(u)dμ_U(u)(∏_i=1^r ∫_𝒳φ_i dμ_𝒳) .
                                           + . O( e^-γ D(t_1,…,t_r)||ϕ_j||_l||f||_l∏_i=1^r||φ_i||_l))dμ_M(m)
       = ∫_M(∑_j=1^N ∫_S ϕ_j(ms)f_j(ms)dμ_S(s))dμ_M(m) (∏_i=1^r ∫_𝒳φ_idμ_𝒳)  
                                           +  O( Ne^-γ D(t_1,…,t_r)||ϕ_j||_l||f||_l∏_i=1^r||φ_i||_l).

Using again the decomposition of μ_K, K-invariance and the partition of unity, we have

    ∫_M(∑_j=1^N ∫_S ϕ_j(ms)f_j(ms)dμ_S(s))dμ_M(m) = ∫_K(∑_j=1^N  ϕ_j(kk_j^-1))f(k)dμ_K(k) = ∫_K f dμ_K

which simplifies the estimate (<ref>) to

    ( ∫_K fdμ_K) (∏_i=1^r ∫_𝒳φ_idμ_𝒳)  + O( 𝓇^-λ e^-γ D(t_1,…,t_r)𝓇^-ν||f||_l∏_i=1^r||φ_i||_l).

Altogether we obtain

    I_Λ,f,φ_1,…,φ_r(t_1,…,t_r) =(∫_K fdμ_K) (∏_i=1^r ∫_𝒳φ_idμ_𝒳)   + O( ( 𝓇^-λ-ν e^-γ D(t_1,…,t_r)+𝓇)||f||_l∏_i=1^r||φ_i||_l).
 
We take 𝓇=e^-δ D(t_1,…,t_r) with δ = γ/1+λ+ν, which yields the claim. 


We will use the following simplified version of Proposition <ref>.



For every r≥ 1, there exist δ_r>0 and l_r≥ 1 such that for every φ_0,…φ_r ∈ C_c^∞(𝒳) and t_1,… t_r > 0, we have

    ∫_𝒴φ_0(y)(∏_i=1^rφ_i (a_t_i y)) dμ_𝒴(y) = ∫_𝒴φ_0 dμ_𝒴(∏_i=1^r∫_𝒳φ_i dμ_𝒳)  + O(e^-δ_r D(t_1,…,t_r)∏_i=0^r||φ_i||_l_r) .
 


We recall in the following section some properties of the Siegel transform that we use later to analyse the ergodic averages ∑_t=0^N χ∘ a_t.



§ SIEGEL TRANSFORM AND APPROXIMATION OF THE COUNTING FUNCTION





 §.§ Siegel transform



Given a bounded measurable function f:ℝ^n+2→ℝ with compact support, its (standard) Siegel transform on the space ℒ of unimodular lattices in ℝ^n+2 is defined by

    f^st.(Λ) ∑_z ∈Λ∖{0} f(z),   for Λ∈ℒ.

Its restriction to 𝒳 is called the light-cone Siegel transform, defined for a bounded and compactly supported function f on 𝒞 by

    f(Λ) ∑_z ∈Λ∖{0} f(z),   for Λ∈𝒳.

The Siegel transform of a bounded function is typically unbounded, but its growth rate is controlled by an explicit function α defined as follows.


Given a lattice Λ∈ℒ, we say that a subspace V of ℝ^n+2 is Λ-rational if the intersection V∩Λ is a lattice in V. If V is Λ-rational, we denote d_Λ(V) the covolume of V∩Λ in V. We define then

    α(Λ) sup{ d_Λ(V)^-1: V  is a Λ-rational subspace of ℝ^n+2}.
 

It follows from Mahler's Compactness Criterion that α  is a proper map ℒ→ [1, +∞). We recall below some important properties.



If f:ℝ^n+2→ℝ is a bounded function with compact support, then

    |f^st.(Λ)| ≪_supp(f) ||f||_∞α(Λ),    for all Λ∈ℒ.



We restrict this function to the space 𝒳 of lattices on the positive light cone and denote it also by α. An important property of α is its L^p-integrability in ℒ (see <cit.>) and also in 𝒳 with an explicit non-escape of mass.



The function α is in L^p(𝒳) for 1≤ p < n. In particular,

    μ_𝒳 ({α≥ L }) ≪_p L^-p,
       for all  p<n.



We recall the analogous for the space 𝒳 of the Siegel Mean Value Theorem in the space of unimodular lattices ℒ (see <cit.>).



If f:𝒞→ℝ is a bounded Riemann integrable function with compact support, then

    ∫_𝒳f(Λ) dμ_𝒳(Λ) = ∫_𝒞 f(z) dz

for some G-invariant measure dz on 𝒞.




 §.§ Non-divergence estimates


We recall here important estimates for the Siegel transform f on translated K-orbits by analyzing the escape of mass on submanifolds a_t𝒴⊂𝒳.

Following the same argument as in <cit.> and using effective equidistribution of translated K-orbits and L^p-integrability of the function α, we verified in <cit.> an analogous non-escape of mass for a_t𝒴.



There exists κ >0 such that for every L≥ 1 and t≥κlog L, 

    μ_𝒴 ({ y ∈𝒴 : α(a_t y) ≥ L  }) ≪_p L^-p,    for all  p<n.




A crucial ingredient in our argument later is the integrability of the Siegel transform f on a_t𝒴 uniformly in t. This is an important result of Eskin, Margulis and Mozes in <cit.> establishing the following estimate for the function α.



If n ≥ 2 and 0<p<2, then for any lattice Λ in ℝ^n+2,

    sup_t> 0∫_Kα ( a_t k Λ)^p dμ_K(k) < ∞.





 §.§ Truncated Siegel transform

	The Siegel transform of a smooth compactly supported function is typically not bounded. To be able to apply equidistribution results, we truncate the Siegel transform using a smooth cut-off function η_L built on the function α. We use the same construction as in [Lemma 4.9]bjoerklund2018central which yields the following lemma.


For every ξ∈ (0,1), there exists a family (η_L) in C_c^∞(𝒳) satisfying:

    0 ≤η_L ≤ 1 ,   η_L = 1  on {α≤ξ^-1L } ,   η_L = 0  on {α > ξ L } ,    || η_L||_C^l≪ 1.



For a bounded function f:𝒞→ℝ with compact support, we define the truncated Siegel transform of f by

    f^(L)f·η_L.


We recall in the following proposition some properties of the truncated Siegel transform f^(L) which we use later in our arguments.  



For a bounded measurable function f:𝒞→ℝ with compact support, the truncated Siegel transform f^(L) satisfies the following bounds:

    ||f^(L) ||_L^p_𝒳≤ ||f ||_L^p_𝒳≪_supp(f),p ||f ||_∞ ,   for all   p<n,
       sup_t≥ 0||f^(L)∘ a_t ||_L^p_𝒴≤sup_t≥ 0||f∘ a_t ||_L^p_𝒴 < ∞ ,   for all  1≤ p<2,
        ||f^(L) ||_∞≪_supp(f) L||f ||_∞,
        ||f - f^(L) ||_L^1_𝒳≪_supp(f),τ L^-(τ-1)||f ||_∞  , for all τ < n,
       ‖f-f^(L)‖_L^2_𝒳≪_supp(f),τ  L^-τ-2/2||f ||_∞  , for all τ < n,
       ||f∘ a_t - f^(L)∘ a_t ||_L^p_𝒴≪_supp(f),τ L^-τ(2-p)/2p||f ||_∞ ,   for all  1≤ p<2,  τ<n  and  t≥κlog L.

Moreover, if f ∈ C_c^∞(𝒞) then f^(L)∈ C_c^∞(𝒳) and satisfies

    ||f^(L) ||_C^l≪_supp(f) L||f ||_C^l  , for all  l≥ 1.




All but estimate (<ref>) were proven in <cit.>.

To show (<ref>), we apply Hölder's Inequality with 1 ≤ p < n and q=(1/2-1/p)^-1 and deduce

    ‖f-f^(L)‖_L^2_𝒳≪_supp(f)  ||α ||_L_𝒳^p μ_𝒳({α≥ξ^-1L })^1/q ||f ||_∞.

Then Proposition <ref> implies

    ‖f-f^(L)‖_L^2_𝒳≪_supp(f),p  L^-p-2/2||f ||_∞.





 §.§ Smooth approximation

For simplicity we write χχ_F_1,c and χ_𝓁χ_F_1,c,𝓁 for the characteristic functions of the sets F_1,c and F_1,c, 𝓁 respectively. We approximate χ and χ_𝓁 by a family of non-negative functions f_ε , f_𝓁,ε∈ C_c^∞(𝒞) with support in an ε-neighborhood of F_1,c and F_1,c, 𝓁 respectively, such that

    χ≤ f_ε≤ 1,    ||f_ε - χ||_L^1_𝒞≪ε ,    ||f_ε - χ||_L^2_𝒞≪ε^1/2,    ||f_ε||_C^l≪ε^-l,
    and       χ_𝓁≤ f_𝓁,ε≤ 1,    ||f_𝓁,ε - χ_𝓁||_L^1_𝒞≪ε ,    ||f_𝓁,ε - χ_𝓁||_L^2_𝒞≪ε^1/2,    ||f_𝓁,ε||_C^l≪ε^-l.


We reformulate in the following proposition a previous result in <cit.>, in order to take into account the parameter 𝓁≥ 1, and show that the smooth approximation of χ, χ_𝓁 also yields a good approximation of their Siegel transforms χ, χ_𝓁 on translated K-orbits, uniformly in the parameter 𝓁≥ 1.


There exists θ>0 such that for every 𝓁≥ 1 and every ε>0,

    ∫_𝒴 |f_𝓁,ε∘ a_t - χ_𝓁∘ a_t| dμ_𝒴≪_c,nε +e^-θ t.




Let 𝓁≥ 1. We first recall the definition of the set F_1,c,𝓁,

    F_1,c,𝓁 = { x ∈𝒞 : x_n+2^2- x_n+1^2 < c_𝓁^2 ,  c≤ x_n+2+ x_n+1 < ce },
    with      c_𝓁=c·(𝓁/𝓁+1)^2,

and observe that there exists c_𝓁,ε>c_𝓁 such that c_𝓁,ε=c_𝓁+ O(ε) and f_𝓁,ε≤χ_𝓁,ε, where χ_𝓁,ε denotes the characteristic function of the set

    { x ∈𝒞 :   c-ε≤ x_n+2+x_n+1≤ ce+ε ,    x_n+2^2-x_n+1^2 < c_𝓁,ε^2    }.

The difference χ_𝓁,ε-χ_𝓁 is bounded by the sum χ^(1)_𝓁,ε+χ^(2)_𝓁,ε+χ^(3)_𝓁,ε of the characteristic functions of the sets

    { x ∈𝒞 :  c-ε≤ x_n+2+x_n+1≤ c ,    x_n+2^2-x_n+1^2 < c_𝓁,ε^2    },
       { x ∈𝒞 :  ce ≤ x_n+2+x_n+1≤ ce+ε,    x_n+2^2-x_n+1^2 < c_𝓁,ε^2    },
       { x ∈𝒞 :  c ≤ x_n+2+x_n+1≤ ce ,    c_𝓁^2<x_n+2^2-x_n+1^2 < c_𝓁,ε^2    }.
 
Since 0≤χ_𝓁≤ f_𝓁,ε≤χ_𝓁,ε, it follows in particular 

    f_𝓁,ε (a_tΛ) - χ_𝓁(a_tΛ) ≤χ_𝓁,ε^(1)(a_tΛ)+χ_𝓁,ε^(2)(a_tΛ)+χ_𝓁,ε^(3)(a_tΛ).

We first consider χ^(1)_𝓁,ε. For x in the corresponding set, we also have

    0≤ x_n+2-x_n+1 < c_𝓁,ε^2/(c-ε)   and   x_1^2+ … +x_n^2 < c_𝓁,ε^2.

We write I_0,𝓁,ε [0,c_𝓁,ε], I_1,𝓁,ε [-c_𝓁,ε^2/(c-ε),0], I_2,ε [c-ε,c], k=(k_1, … , k_n+2)^T ∈ K, and compute

    ∫_𝒴 |χ_𝓁,ε^(1)∘ a_t|  dμ_𝒴 = ∫_Kχ_𝓁,ε^(1)(a_tkΛ_0)  dμ_K(k) = ∫_K∑_z ∈Λ_0χ_𝓁,ε^(1)(a_tkz)  dμ_K(k) 
       =   ∑_z ∈Λ_0 ∫_Kχ^(1)_𝓁,ε(⟨ k_1,z ⟩
    …
    ⟨ k_n,z ⟩
    ⟨ k_n+1,z ⟩cosh t - z_n+2sinh t
    ⟨ k_n+1,z ⟩(-sinh t) + z_n+2cosh t 
    ) dμ_K(k) 
       =∑_z ∈Λ_0∫_K 
    χ_I_0,𝓁,ε( ||⟨ k_1,z⟩ ,…, ⟨ k_n,z ⟩ || ) 
    χ_I_1,𝓁,ε(e^t(⟨ k_n+1,z ⟩ -z_n+2) )
    χ_I_2,ε(e^-t( ⟨ k_n+1,z ⟩ +z_n+2) )
    dμ_K(k).

We observe that the intersection (e^-tI_1,𝓁,ε+z_n+2) ∩ (e^tI_2,ε-z_n+2) is non-empty only if (c-ε)e^t≤ 2z_n+2≤ ce^t+c_𝓁,ε^2/c-εe^-t, i.e. z_n+2=ce^t/2 +O_c(ε e^t+e^-t) where the implicit constant is independent from 𝓁≥ 1. Moreover, writing each z ∈Λ_0 as z = z_n+2k_zv_0 with some k_z ∈ K and v_0=(0,…,0,1,1)∈𝒞, and using invariance under k_z, we have

    ∫_𝒴 |χ_𝓁,ε^(1)∘ a_t|  dμ_𝒴
       ≤∑_z∈Λ_0
     z_n+2=ce^t/2 +O(ε e^t+e^-t)∫_K 
     χ_I_0,𝓁,ε( z_n+2||⟨ k_1,v_0⟩ ,…, ⟨ k_n,v_0 ⟩ || )
    χ_e^-tI_1,𝓁,ε(z_n+2(⟨ k_n+1,v_0 ⟩ -1) )·
                                                       ·χ_e^tI_2,ε(z_n+2(⟨ k_n+1,v_0 ⟩ +1) )
    dμ_K(k) 
       ≤∑_z∈Λ_0
     z_n+2=ce^t/2 +O(ε e^t+e^-t)∫_K 
    χ_e^-t2/c-εI_0,𝓁,ε(||⟨ k_1,v_0⟩ ,…, ⟨ k_n,v_0 ⟩ || )
    χ_e^-2t2/c-ε I_1,𝓁,ε(⟨ k_n+1,v_0⟩ - 1 ) ·
                                                       ·χ_2/c-εI_2,ε(⟨ k_n+1,v_0 ⟩ +1 )
    dμ_K(k) 
       ≤∑_z∈Λ_0
     z_n+2=ce^t/2 +O(ε e^t+e^-t)μ_K ( { k ∈ K   :   |k_i,n+1| ≪_c e^-t,   i=1,…,n  , 
    
    |k_n+1,n+1-1| ≪_c min(e^-2t,ε).
    })
       ≤∑_z∈Λ_0
     z_n+2=ce^t/2 +O(ε e^t+e^-t)μ_K ( { k ∈ K   : ||kv_0-v_0 ||≪_c e^-t})
       ≪_n ∑_z∈Λ_0
     z_n+2=ce^t/2 +O(ε e^t+e^-t)μ_S^n( { v ∈S^n   :  ||v-v_0 ||≪ e^-t})
       ≪_c,n∑_z∈Λ_0
     z_n+2=ce^t/2 +O(ε e^t+e^-t) e^-nt.

We use further that there exist positive constants C and θ such that, for all n≥ 2, we have

    |{z ∈𝒞∩ℤ^n+2 : 0≤ z_n+2<T }| = CT^n+O(T^n-θ),

hence 

    |{z ∈𝒞∩ℤ^n+2 : (c-ε)e^t ≤ 2z_n+2<ce^t+c_𝓁,ε^2/c-εe^-t}| ≤ε e^nt+O_c(e^(n-θ)t).

It follows

    ∫_𝒴 |χ_𝓁,ε^(1)∘ a_t|  dμ_𝒴≪_c,nε + e^-θ t.



We proceed similarly for χ^(3)_𝓁,ε. For x in the corresponding set, we also have

    c_𝓁^2/ce≤ x_n+2-x_n+1 < c_𝓁,ε^2   and   c_𝓁^2<x_1^2+ … +x_n^2 < c_𝓁,ε^2.

We write I'_0,𝓁,ε [c_𝓁,c_𝓁,ε], I'_1,𝓁,ε [-c_𝓁,ε^2,-c_𝓁^2/ce], I'_2 [c,ce] and compute similarly

    ∫_𝒴 |χ_𝓁,ε^(3)∘ a_t|  dμ_𝒴  = ∫_K∑_z ∈Λ_0χ_𝓁,ε^(3)(a_tkz)  dμ_K(k) 
       =∑_z ∈Λ_0∫_K 
    χ_I'_0,𝓁,ε( ||⟨ k_1,z⟩ ,…, ⟨ k_n,z ⟩ || ) 
    χ_I'_1,𝓁,ε(e^t(⟨ k_n+1,z ⟩ -z_n+2) )
    χ_I'_2(e^-t( ⟨ k_n+1,z ⟩ +z_n+2) )
    dμ_K(k).

We observe again that the intersection (e^-tI'_1,𝓁,ε+z_n+2) ∩ (e^tI'_2-z_n+2) is non-empty only if C_1e^t ≤ z_n+2≤ C_2e^t for some positive constants C_1 and C_2 depending only on c>0. Moreover, writing each z ∈Λ_0 as z = z_n+2k_zv_0 with some k_z ∈ K and v_0=(0,…,0,1,1)∈𝒞, and using invariance under k_z, we have

    ∫_𝒴 |χ_𝓁,ε^(3)∘ a_t|  dμ_𝒴
       ≤∑_z∈Λ_0
     z_n+2≍_c e^t∫_K 
     χ_I'_0,𝓁,ε( z_n+2||⟨ k_1,v_0⟩ ,…, ⟨ k_n,v_0 ⟩ || )
    χ_e^-tI'_1,𝓁,ε(z_n+2(⟨ k_n+1,v_0 ⟩ -1) )·
                                                       ·χ_e^tI'_2(z_n+2(⟨ k_n+1,v_0 ⟩ +1) )
    dμ_K(k) 
       ≤∑_z∈Λ_0
     z_n+2≍_c e^t∫_K 
    χ_e^-t1/C_1I'_0,𝓁,ε(||⟨ k_1,v_0⟩ ,…, ⟨ k_n,v_0 ⟩ || )
    χ_e^-2t1/C_1I'_1,𝓁,ε(⟨ k_n+1,v_0⟩ - 1 ) ·
                                                       ·χ_1/C_1I'_2(⟨ k_n+1,v_0 ⟩ +1 )
    dμ_K(k) 
    
    
    
       ≤∑_z∈Λ_0
     z_n+2≍_c e^tμ_K ( { k ∈ K   : ||kv_0-v_0 ||≪_c ε
    e^-t})
       ≪_n ∑_z∈Λ_0
     z_n+2≍_c e^tμ_S^n( { v ∈S^n   :  ||v-v_0 ||≪_c ε e^-t})
       ≪_c,n∑_z∈Λ_0
     z_n+2≍_c e^tε^n e^-nt.

We use again the estimate

    |{z ∈𝒞∩ℤ^n+2 : z_n+2≍ e^t}|= O(e^nt),

hence 

    ∫_𝒴 |χ_𝓁,ε^(3)∘ a_t|  dμ_𝒴 ≪_c,n ε.

The bound for ||χ_𝓁,ε^(2)∘ a_t ||_L^1_𝒴 is obtained similarly as for χ^(1)_ε.

Altogether we obtain,

    ||f̂_ε∘ a_t -χ̂∘ a_t ||_L^1_𝒴≪_c,nε+ e^-θ t.





 §.§ Averaging function

As explained in Section <ref>, analysing the counting function 𝖭_T,c reduces to analysing ergodic averages of the form ∑_t χ∘ a_t. We define for this purpose the following averaging function. 

    𝖥_N 1/√(N)∑_t=0^N-1( χ∘ a_t -μ_𝒴(χ∘ a_t) ) .


To study the distribution of 𝖥_N we shall use in the following arguments the basic observation that if we approximate 𝖥_N by a sequence 𝖥_N in such a way that ||𝖥_N-𝖥_N ||_L^1_𝒴 0 and the limit distribution of 𝖥_N is continuous, then 𝖥_N and 𝖥_N have the same convergence in distribution.



  
Truncated averages

We first observe that 𝖥_N has the same convergence in distribution as the truncated averages

    𝖥_N,M1/√(N-M)∑_t=M^N-1( χ∘ a_t -μ_𝒴(χ∘ a_t) ),

for some M=M(N)→∞ to be specified later.

Indeed, we have

    ‖𝖥_N -𝖥_N,M‖_L^1_𝒴   ≤1/√(N)∑_t=0^M-1‖χ∘ a_t -μ_𝒴(χ∘ a_t) ‖_L^1_𝒴
               + ( 1/√(N-M)-1/√(N)) ∑_t=M^N-1‖χ∘ a_t -μ_𝒴(χ∘ a_t) ‖_L^1_𝒴
       ≪M/√(N)sup_t≥ 0‖χ∘ a_t  ‖_L^1_𝒴,

hence, by (<ref>) and provided that 

    M=o(N^1/2)

we have 

    ‖𝖥_N -𝖥_N,M‖_L^1_𝒴→ 0,  as N→∞.




  
Averages for the Siegel transform of a smooth approximation
Further, we observe that the averages 𝖥_N,M has the same convergence in distribution if the characteristic function χ is replaced by the smooth approximation f_ε introduced earlier. Indeed, if we consider the averages

    𝖥_N,M^(ε)1/√(N-M)∑_t=M^N-1( f_ε∘ a_t -μ_𝒴(f_ε∘ a_t) ),

with the parameter ε = ε(N), ε(N)  0 to be specified later, then Proposition <ref> implies

    ‖𝖥_N,M- 𝖥_N,M^(ε)‖_L^1_𝒴≤2/√(N-M)∑_t=M^N-1‖f_ε∘ a_t -χ∘ a_t ‖_L^1_𝒴≪ (N-M)^1/2(ε+e^-θ M) .

We will choose ε and M such that

    (N-M)^1/2ε→ 0   and   (N-M)^1/2e^-θ M→ 0 ,

which yields

    ‖𝖥_N,M- 𝖥_N,M^(ε)‖_L^1_𝒴→ 0   as   N→∞ .




  
Averages for the truncated Siegel transform
Finally, we also have the same convergence in distribution for the averages of the truncated Siegel transform

    𝖥_N,M^(ε, L)1/√(N-M)∑_t=M^N-1( f_ε^(L)∘ a_t -μ_𝒴(f_ε^(L)∘ a_t) ),

defined for parameters ε(N)  0 and  L(N) ∞ to be specified later. 

We assume that 

    M≫log L

such that Proposition <ref> applies when t≥ M. Since the family of functions f_ε is uniformly bounded by a compactly supported function, the estimate (<ref>) gives

    ‖𝖥_N,M^(ε) -𝖥_N,M^(ε, L)‖_L^1_𝒴   ≤1/√(N-M)∑_t=M^N-1‖(f_ε∘ a_t -f_ε^(L)∘ a_t)-μ_𝒴(f_ε∘ a_t -f_ε^(L)∘ a_t) ‖_L^1_𝒴
       ≤2/√(N-M)∑_t=M^N-1‖f_ε∘ a_t -f_ε^(L)∘ a_t ‖_L^1_𝒴
       ≪_τ (N-M)^1/2L^-τ/2,    for all  τ<n.

We will choose L(N) ∞ such that 

    N-M   =o(L^p)   for some p<n,

to obtain 

    ‖𝖥_N,M^(ε) -𝖥_N,M^(ε, L)‖_L^1_𝒴→ 0,  as N→∞.


Hence if we prove the CLT for the sequence (𝖥_N,M^(ε, L)), then the CLT for (𝖥_N) would follow.




§ CUMULANTS OF THE COUNTING FUNCTION




 §.§ The method of cumulants


We recall in this section the general approach of the method of cumulants (presented in <cit.> and <cit.>) to establish the convergence to a normal distribution using a characterisation by the cumulants.

Given a probability space (X,μ) and bounded measurable functions φ_1,⋯,φ_r on X, we define their joint cumulant as

    Cum_μ^(r)(φ_1,⋯,φ_r)= ∑_𝒫(-1)^|𝒫|-1(|𝒫|-1)!∏_I∈𝒫∫_X(∏_i∈ Iφ_i )dμ ,

where the sum is over all partitions 𝒫 of the set {1,⋯,r}. For a bounded measurable function φ on X we write 

    Cum_μ^(r)(φ)=Cum_μ^(r)(φ,⋯,φ) .

We will use the following classical CLT-criterion (see <cit.>).



Let (f_N)_N≥ 1 be a sequence of real-valued bounded measurable functions such that

    ∫_X f_N dμ=0   ,     σ^2lim_N→∞∫_X f_N^2 dμ < ∞

and

    lim_N→∞Cum_μ^(r)(f_N)=0 ,   for all  r≥ 3 .

Then for every ξ∈ℝ,

    μ( { f_N<ξ}) →Norm_σ(ξ)   as  N→∞ .

 
The method of cumulants is equivalent to the more widely known “method of moments", but the cumulants offer the following convenient cancellation property.


For a partition 𝒬 of {1,⋯,r }, we define the conditional joint cumulant with respect to 𝒬 by

    Cum_μ^(r)(φ_1,⋯,φ_r|𝒬)= ∑_𝒫(-1)^|𝒫|-1(|𝒫|-1)!∏_I∈𝒫∏_J∈𝒬∫_X(∏_i∈ I∩
    Jφ_i )dμ .


 <cit.>
 For any partition 𝒬 with |𝒬|≥ 2, 

    Cum_μ^(r)(φ_1,⋯,φ_r|𝒬)=0 ,

for all φ_1,⋯,φ_r ∈ L^∞(X,μ).




 §.§ Estimating the cumulants




It will be convenient to write

    ψ^(ε,L)_t(y):=f_ε^(L)(a_ty)-μ_𝒴(f_ε^(L)∘ a_t),

so that the averaging function is 

    𝖥_N,M^(ε, L)=1/√(N-M)∑_t=M^N-1ψ^(ε,L)_t   with  ∫_𝒴𝖥_N,M^(ε, L) dμ_𝒴=0.

Our aim in this section is to estimate the following joint cumulants for r≥ 3,

    Cum_μ_𝒴^(r)(𝖥_N,M^(ε, L))=1/(N-M)^r/2∑_t_1,…,t_r=M^N-1Cum_μ_𝒴^(r)(ψ^(ε,L)_t_1,…, ψ^(ε,L)_t_r).

We reproduce below the argument as developed in  <cit.> and <cit.>, taking into account the dependence on the parameters L and ε coming from the truncated Siegel transform and the smooth approximation respectively. The main idea in estimating these joint cumulants is to decompose (<ref>) into sub-sums corresponding to “separated" or “clustered" tuples t_1,…,t_r and to control their sizes.



  §.§.§ Separated and clustered times t_1,…,t_r

It will be convenient to consider {0,…,N-1}^r as a subset of ℝ_+^r+1 with the embedding (t_1,…,t_r)→ (0,t_1,…,t_r).

Following the approach developed in <cit.>, we define for 
non-empty subsets I and J of {0,…, r} and t = (t_0,…,t_r) ∈_+^r+1, 

    ρ^I(t) := max{ |t_i-t_j|   :   i,j ∈ I }  and  ρ_I,J(t):= min{ |t_i-t_j|   :   i ∈ I,   j ∈ J },

and if  is a partition of {0,…,r}, we set

    ρ^(t) := max{ρ^I(t)   :   I ∈}  and  ρ_(t) := min{ρ_I,J(t)   :   I ≠ J,   I, J ∈}.

For 0 ≤α < β, we define

    Δ_(α,β) 
    := 
    {t∈_+^r+1  :  ρ^(t) ≤α, 
    ρ_(t) > β}

and

    Δ(α):= 
    {t∈_+^r+1  :  ρ(t_i,t_j) ≤α for all i,j}.

The following decomposition of _+^r+1 was established in <cit.>: given 

    0=α_0<β_1<α_1=(3+r)β_1<β_2<⋯ <β_r<α_r=(3+r)β_r<β_r+1,

we have

    _+^r+1 = Δ(β_r+1) ∪( ⋃_j=0^r⋃_|| ≥ 2Δ_(α_j,β_j+1) ),

where the union is taken over the partitions  of {0,…,r} with ||≥ 2. Upon taking restrictions, we also have 

    {M,…,N-1}^r =Ω(β_r+1;M,N) ∪( ⋃_j=0^r⋃_|| ≥ 2Ω_(α_j,β_j+1;M,N) ),

for all N> M ≥ 0, where 

    Ω(β_r+1;M,N)    :={M,…,N-1}^r ∩Δ(β_r+1),
    Ω_(α_j,β_j+1;M,N)    :={M,…,N-1}^r ∩Δ_(α_j,β_j+1).

In order to estimate the cumulant (<ref>), we shall separately estimate the sums over Ω(β_r+1;M,N) and Ω_(α_j,β_j+1;M,N), the exact choices of the sequences (α_j) and (β_j) will be fixed later.

We may first choose 

    M>β_r+1

so that Ω(β_r+1;M,N)=∅ and does not contribute to the sum.




  §.§.§ Case 1: Summing over (t_1,…,t_r)∈Ω_(α_j,β_j+1;M,N) with ={{0},{1,…,r}}.

We shall first show that, in this case, we have

    Cum_μ_^(r)(ψ^(ε,L)_t_1,…,ψ^(ε,L)_t_r)≈Cum_μ_^(r)(ϕ^(ε,L)∘ a_t_1,…,ϕ^(ε,L)∘ a_t_r)

where ϕ^(ε,L):= f_ε^(L)-μ_(f_ε^(L)).
This reduces to estimating the integrals

    ∫_(∏_i∈ Iψ^(ε,L)_t_i) dμ_
    
    =   ∑_J⊂ I (-1)^|I\ J|(∫_(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_)
    ∏_i∈ I\ J(∫_ (f_ε^(L)∘ a_t_i) dμ_).

If (t_1,…,t_r)∈Ω_(α_j,β_j+1;M,N), and thus

    |t_i_1-t_i_2|≤α_j   and   t_i_1≥β_j+1    for all 1≤ i_1,i_2≤ r,

it follows from Corollary <ref> with r=1 that there exists δ>0 such that

    ∫_ (f_ε^(L)∘ a_t_i) dμ_
    =μ_(f_ε^(L))+O(e^-δβ_j+1 f_ε^(L)_C^l).

For a fixed J ⊂ I, we define

    Φ^(ε,L):=∏_i∈ Jf_ε^(L)∘ a_t_i-t_1,

and note that for some ξ=ξ(n,l)>0, we have

    Φ^(ε,L)_C^l≪∏_i∈ Jf_ε^(L)∘ a_t_i-t_1_C^l≪ e^|J|ξ α_j f_ε^(L)_C^l^|J|.

If we again apply Corollary <ref> to the function Φ^(ε,L), we obtain 

    ∫_(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_   =∫_ (Φ^(ε,L)∘ a_t_1) dμ_
       =∫_Φ^(ε,L)  dμ_+O(e^-δβ_j+1 Φ^(ε,L)_C^l) 
       =∫_(∏_i∈ Jf_ε^(L)∘ a_t_i)  dμ_+O(e^-δβ_j+1 e^rξ α_j f_ε^(L)_C^l^|J|),

where we used that μ_ is invariant under the transformation a_t.
Let us now choose the exponents α_j and β_j+1 so that
δβ_j+1-rξα_j>0. 
Combining (<ref>), (<ref>) and (<ref>), we deduce that 

    ∫_(∏_i∈ Iψ^(ε,L)_t_i) dμ_
    =   ∑_J⊂ I (-1)^|I\ J|(∫_(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_)
    μ_(f_ε^(L))^|I\ J|
       +O(e^-δβ_j+1 e^rξ α_j f_ε^(L)_C^l^|I|) 
    
    =   ∫_∏_i∈ I(f_ε^(L)∘ a_t_i-μ_(f_ε^(L))) dμ_
    +O(e^-(δβ_j+1-rξα_j ) f_ε^(L)_C^l^|I|),

and thus, for any partition ,

    ∏_I∈∫_(∏_i∈ Iψ^(ε,L)_t_i) dμ_
    =
    ∏_I∈∫_(∏_i∈ Iϕ^(ε,L)∘ a_t_i) dμ_
    +O(e^-(δβ_j+1-rξα_j ) f_ε^(L)_C^l^r),

and consequently,

    Cum_μ_^(r)(ψ^(ε,L)_t_1,…,ψ^(ε,L)_t_r)=    Cum_μ_^(r)(ϕ^(ε,L)∘ a_t_1,…,ϕ^(ε,L)∘ a_t_r)
       +O(e^-(δβ_j+1-rξ α_j) f_ε^(L)_C^l^r)

whenever (t_1,…,t_r)∈Ω_(α_j,β_j+1;M,N) with ={{0},{1,…,r}},
from which (<ref>) follows. 


We now claim that 

    |Cum_μ_^(r)(ϕ^(ε,L)∘ a_t_1,…,ϕ^(ε,L)∘ a_t_r)| ≪f_ε^(L)_C^0^(r-(n-1))^+f_ε^(L)_L^n-1()^min(r,n-1),

where we use the notation x^+=max(x,0). The implied constant in (<ref>) and below depend only on supp(f_ε), so that it is uniform in ε. By the definition of the cumulant, to prove  (<ref>), it suffices to show 
that for every z≥ 1 and indices i_1,…,i_z,

    ∫_|(ϕ^(ε, L)∘ a_t_i_1)⋯(ϕ^(ε,L)∘ a_t_i_z)|  dμ_≪f_ε^(L)_C^0^(z-(n-1))^+f_ε^(L)_L^n-1()^min(z,n-1).

Using 
the generalized Hölder inequality, we deduce that when z≤ n-1,

    ∫_|(ϕ^(ε,L)∘ a_t_i_1)⋯(ϕ^(ε,L)∘ a_t_i_z)|  dμ_   ≤ϕ^(ε,L)∘ a_t_i_1_L^n-1()⋯ϕ^(ε,L)∘ a_t_i_z_L^n-1()
       ≪f_ε^(L)_L^n-1()^z.

Also when z>n-1, 

    ∫_|(ϕ^(ε,L)∘ a_t_i_1)⋯(ϕ^(ε,L)∘ a_t_i_z)|  dμ_
    ≤   ϕ^(ε,L)_C^0^z-(n-1)∫_|(ϕ^(ε,L)∘ a_t_i_1)⋯(ϕ^(ε,L)∘ a_t_i_n-1)|  dμ_
    ≪   f_ε^(L)_C^0^z-(n-1)f_ε^(L)_L^n-1()^n-1.

This implies (<ref>) and (<ref>). 


Finally we recall that if (t_1,…,t_r)∈Ω_(α_j,β_j+1;M,N) with ={{0},{1,…,r}}, then we have |t_i_1-t_i_2|≤α_j
for all i_1 i_2, and thus

    |Ω_(α_j,β_j+1;M,N)|≪ (N-M)α_j^r-1.


Combining  (<ref>), (<ref>) and (<ref>) in (<ref>), and using Proposition <ref> with (<ref>), we conclude that

    1/(N-M)^r/2∑_t∈Ω_(α_j,β_j+1;M,N)Cum_μ_𝒴^(r)(ψ^(ε,L)_t_1,…, ψ^(ε,L)_t_r)   
       ≪  (N-M)^r/2  e^-(δβ_j+1- rα_jξ) f_ε^(L)_C^l^r +(N-M)^1-r/2α_j^r-1f_ε^(L)_C^0^(r-(n-1))^+f_ε^(L)_L^n-1()^min(r,n-1)
    
    
       ≪  (N-M)^r/2  e^-(δβ_j+1- rα_jξ)  L^r ε^-rl 
    + (N-M)^1-r/2α_j^r-1 
    L^(r-(n-1))^+.




  §.§.§ Case 2: Summing over (t_1,…,t_r)∈Ω_(α_j,β_j+1;M,N) with ||≥ 2 and 
	{{0},{1,…,r}}.

	
In this case, the partition  defines a non-trivial partition '={I_0,…,I_ℓ} of {1,…,r} such that 
for all (t_1,…, t_r)∈Ω_Q(α_j,β_j+1;M,N), we have 

    |t_i_1-t_i_2|≤α_j   if i_1∼_' i_2
    |t_i_1-t_i_2|> β_j+1  if i_1≁_' i_2,

and

    t_i≤α_j  for all i∈ I_0, t_i>β_j+1  for all i∉ I_0.

In particular,

    D(t_i_1,…, t_i_ℓ)≥β_j+1,

Let I be an arbitrary subset of {1,…,r}; we shall show that 

    ∫_( ∏_i∈ Iψ_t_i^(ε,L))  dμ_≈∏_h=0^ℓ(∫_(∏_i∈ I∩ I_hψ_t_i^(ε,L)) dμ_),

where we henceforth shall use the convention that the product is equal to one when I∩ I_h=∅. 


Let us estimate  the right hand side of (<ref>). We begin by setting

    Φ_0^(ε,L):=∏_i∈ I∩ I_0ψ_t_i^(ε,L).

It is easy to see that there exists ξ=ξ(n,l)>0 such that

    Φ_0^(ε,L)_C^l≪∏_i∈ I∩ I_0f_ε^(L)∘ a_t_i-μ_(f_ε^(L)∘ a_t_i)_C^l≪ e^|I∩ I_0|ξ α_j f_ε^(L)_C^l^|I∩ I_0|.

To prove (<ref>), 
we expand ψ_t_i^(ε,L)=f_ε^(L)∘ a_t_i-μ_(f_ε^(L)∘ a_t_i)
for i∈ I\ I_0 and get

    ∫_(∏_i∈ Iψ^(ε,L)_t_i) dμ_   =   ∑_J⊂ I\ I_0 (-1)^|I\ (J∪ I_0)|·
       ·   (∫_Φ_0^(ε,L)(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_) 
    ∏_i∈ I\ (J∪ I_0)(∫_ (f_ε^(L)∘ a_t_i) dμ_).

We recall that when i∉ I_0, we have t_i≥β_j+1, and thus it follows from Corollary <ref> with r=1 that

    ∫_ (f_ε^(L)∘ a_t_i) dμ_
    =μ_(f_ε^(L))+O(e^-δβ_j+1 f_ε^(L)_C^l),    with i∉ I_0.

To estimate the other integrals in (<ref>), we also apply Corollary <ref>.
Let us  first fix a subset J ⊂ I ∖ I_0 and for each 1 ≤ h ≤ l, we pick i_h∈ I_h, 
and set

    Φ_h^(ε,L):=∏_i∈ J∩ I_hf_ε^(L)∘ a_t_i-t_i_h.

Then 

    ∫_Φ_0^(ε,L)(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_=
    ∫_Φ_0^(ε,L)( ∏_h=1^ℓΦ_h^(ε,L)∘ a_t_i_h) dμ_.

We note that for i∈ I_h, we have |t_i-t_i_h|≤α_j, and 
thus there exists ξ=ξ(n,l)>0 such that
 

    Φ_h^(ε,L)_C^l≪∏_i∈ J∩ I_hf_ε^(L)∘ a_t_i-t_i_h_C^l≪ e^|J∩ I_h|ξ α_j f_ε^(L)_C^l^|J∩ I_h|.

Using (<ref>), Corollary <ref> implies that

    ∫_Φ_0^(ε,L)( ∏_h=1^ℓΦ_h^(ε,L)∘ a_t_i_h) dμ_
    =    (∫_Φ_0^(ε,L)  dμ_)∏_h=1^ℓ(∫_Φ_h^(ε,L)  dμ_)
       +O( e^-δβ_j+1 ∏_h=0^ℓΦ_h^(ε,L)_C^l).

Using (<ref>) and (<ref>) and invariance of  the measure μ_, we deduce that

    ∫_Φ_0^(ε,L)( ∏_h=1^ℓΦ_h^(ε,L)∘ a_t_i_h) dμ_
    =    (∫_Φ_0^(ε,L)  dμ_)∏_h=1^ℓ(∫_𝒳(∏_i∈ J∩ I_hf_ε^(L)∘ a_t_i)  dμ_)
       +O( e^-(δβ_j+1-rξα_j) f_ε^(L)_C^l^|(I∩ I_0)∪ J|).

Hence, we conclude that

    ∫_Φ_0^(ε,L)(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_
    =    (∫_𝒴Φ_0^(ε,L)  dμ_)∏_h=1^ℓ(∫_𝒳(∏_i∈ J∩ I_hf_ε^(L)∘ a_t_i)  dμ_) 
       +O( e^-(δβ_j+1-rξα_j) f_ε^(L)_C^l^|(I∩ I_0)∪ J|).

We shall choose the parameters α_j and β_j+1 so that 

    δβ_j+1-rξα_j>0.

Substituting (<ref>) and (<ref>) in (<ref>), we deduce that 

    ∫_(∏_i∈ Iψ_t_i^(ε,L)) dμ_
     
    =   ∑_J⊂ I\ I_0 (-1)^|I\ (J∪ I_0)|(∫_𝒴Φ_0^(ε,L)  dμ_)∏_h=1^ℓ(∫_𝒳(∏_i∈ J∩ I_hf_ε^(L)∘ a_t_i)  dμ_) μ_(f_ε^(L))^|I\ (J∪ I_0)|
        +O ( e^-(δβ_j+1-rξα_j) f_ε^(L)_C^l^|I|).


Next, we estimate the right hand side of (<ref>).
Let us fix 1 ≤ h ≤ l and for a subset J ⊂ I ∩ I_h, we define 

    Φ_J^(ε,L):=∏_i∈ Jf_ε^(L)∘ a_t_i-t_i_h.

As in (<ref>), for some ξ>0,

    Φ_J^(ε,L)_C^k≪∏_i∈ Jf_ε^(L)∘ a_t_i-t_i_h_C^l≪ e^|J|ξ α_j f_ε^(L)_C^l^|J|.

Applying Corollary <ref> to the function Φ_J^(ε,L) and using that t_i_h>β_j+1, we get 

    ∫_(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_   =∫_ (Φ_J^(ε,L)∘ a_t_i_h) dμ_
       =∫_Φ_J^(ε,L)  dμ_+O(e^-δβ_j+1 Φ_J^(ε,L)_C^l) 
       =∫_(∏_i∈ Jf_ε^(L)∘ a_t_i)  dμ_+O(e^-δβ_j+1 e^rξ α_j f_ε^(L)_C^l^|J|),

where we have used a-invariance of μ_.

Combining (<ref>) and (<ref>), we deduce that 

    ∫_(∏_i∈ I∩ I_hψ_t_i^(ε,L)) dμ_
    =   ∑_J⊂ I∩ I_h (-1)^|(I∩ I_h)\ J|(∫_(∏_i∈ Jf_ε^(L)∘ a_t_i) dμ_)
    μ_(f_ε^(L))^|(I∩ I_h)\ J|
       +O(e^-δβ_j+1 e^rξ α_j f_ε^(L)_C^l^|I∩ I_h|) 
    
    =   ∫_∏_i∈ I∩ I_h(f_ε^(L)∘ a_t_i-μ_(f_ε^(L))) dμ_
    +O(e^-(δβ_j+1-rξα_j ) f_ε^(L)_C^l^|I∩ I_h|),

which implies

    ∏_h=0^ℓ(∫_(∏_i∈ I∩ I_hψ_t_i^(ε,L)) dμ_)
    
    =   (∫_𝒴Φ_0^(ε,L)  dμ_) ∏_h=1^ℓ(∫_∏_i∈ I∩ I_h(f_ε^(L)∘ a_t_i-μ_(f_ε^(L))) dμ_)
       +O(e^-(δβ_j+1 - rξα_j) f_ε^(L)_C^l^r).

Furthermore, multiplying out the products over I∩ I_h, we get

    ∏_h=0^ℓ(∫_(∏_i∈ I∩ I_hψ_t_i^(ε,L)) dμ_)
    
    =   (∫_𝒴Φ_0^(ε,L)  dμ_) 
    ∑_J⊂ I\ I_0 (-1)^|I\ (I_0∪ J)|∏_h=1^ℓ(∫_∏_i∈ I_h∩ Jf_ε^(L)∘ a_t_i dμ_)
    μ_(f_ε^(L))^|I\ (I_0∪ J)|
       +O(e^-(δβ_j+1 - rξα_j) f_ε^(L)_C^l^|I|).

Comparing (<ref>) and (<ref>), we finally conclude that

    ∫_(∏_i∈ Iψ_t_i^(ε,L)) dμ_ =   ∏_h=0^ℓ(∫_(∏_i∈ I∩ I_hψ_t_i^(ε,L)) dμ_)
       +O(e^-(δβ_j+1 - rξα_j) f_ε^(L)_C^l^|I|)

when (t_1,…,t_r)∈Ω_(α_j,β_j+1;M,N).
This establishes (<ref>) with an explicit error term.
This estimate implies that for the partition '={I_0,…, I_ℓ},

    Cum_μ_^(r)(ψ_t_1^(ε,L),…,ψ_t_r^(ε,L))=
    Cum_μ_^(r)(ψ^(ε,L)_t_1,…,ψ^(ε,L)_t_r|')+
    O(e^-(δβ_j+1 - rξα_j) f_ε^(L)_C^l^r)

By Proposition <ref>,

    Cum_μ_^(r)(ψ^(ε,L)_t_1,…,ψ^(ε,L)_t_r|')=0,

so it follows that for all (t_1,…, t_r)∈Ω_Q(α_j,β_j+1;M,N),

    |Cum_μ_^(r)(ψ^(ε,L)_t_1,…,ψ_t_r)|
    ≪ e^-(δβ_j+1 - rξα_j) f_ε^(L)_C^l^r.

It follows

    1/(N-M)^r/2∑_(t_1,…, t_r)∈Ω_(α_j,β_j+1;M,N)   |Cum_μ_^(r)(ψ^(ε, L)_t_1,…,ψ^(ε, L)_t_r)|
       ≪ (N-M)^r/2e^-(δβ_j+1 - rξα_j) f_ε^(L)_C^l^r
       ≪ (N-M)^r/2e^-(δβ_j+1 - rξα_j) L^r ε^-rl,

where we used Lemma <ref> and (<ref>).



  §.§.§ Final estimates on the cumulants

Finally, we combine the established bounds to get the following estimate

    |Cum_μ_^(r)(𝖥_N,M^(ε, L))|≪      (N-M)^1-r/2(max_j α_j^r-1) L^(r-(n-1))^+
          +
      (N-M)^r/2(max_j   e^-(δβ_j+1 - rξα_j)) L^r ε^-rl.

This estimate holds provided that (<ref>) and (<ref>) hold, namely when

    α_j=(3+r)β_j<β_j+1  and  δβ_j+1-rξα_j>0     for j=1,…,r.

Given any γ>0, we define the parameters β_j inductively by the formula

    β_1=γ  and  β_j+1=max(γ+(3+r)β_j, γ+δ^-1r(3+r)ξβ_j).

It easily follows by induction that β_r+1≪_r γ, and choosing 

    M≫_r γ

we deduce from (<ref>) that

    |Cum_μ_^(r)(𝖥_N,M^(ε, L))|
    ≪ (N-M)^r/2  e^-δγ L^rε^-rl
    +(N-M)^1-r/2γ^r-1 L^(r-(n-1))^+.

We observe that since n≥ 2, 

    (r-(n-1))^+/n<r/2-1
      for all r≥ 3,

Hence, we can choose q>1/n such that

    q(r-(n-1))^+<r/2-1
      for all r≥ 3.

Then we select

    L=(N-M)^q,

so that, in particular, the condition (<ref>) is satisfied.
 

We recall that δ=δ(r) and l=l(r) and write  (<ref>) as 

    |Cum_μ_^(r)(𝖥_N,M^(ε, L))|
    ≪ (N-M)^r/2+rq e^-δγ ε^-rl
    +(N-M)^q(r-(n-1))^+-(r/2-1)γ^r-1 .


Choosing γ of the form

    γ=c_r·log (N-M)

with sufficiently large c_r>0, and assuming

    (N-M)^r/2L^rε^-rl=o(e^δγ)

we conclude that

    Cum_μ_^(r)(𝖥_N,M^(ε, L))→ 0   as N→∞

for all r≥ 3. 


The choice of the parameters L, ε, M, K and γ satisfying all the conditions mentioned earlier is discussed at the beginning of section <ref>. 




§ ESTIMATING THE VARIANCE


In this section we shall show the convergence of the variance of the averaging function 𝖥_N,M^(ε, L), given by

    𝖥_N,M^(ε, L)_L^2()^2 =   1/N-M∑_t_1=M^N-1∑_t_2=M^N-1∫_ψ_t_1^(ε, L)ψ_t_2^(ε, L)  dμ_,

with

    ψ_t^(ε, L) :=  f_ε^(L)∘ a_t -μ_𝒴(f_ε^(L)∘ a_t).

We first observe that this expression is symmetric with respect to t_1 and t_2,
writing t_1=s+t and t_2=s with 0≤ t≤ N-M-1 and M≤ s ≤ N-t-1,
we obtain that

    𝖥_N,M^(ε, L)_L^2()^2 
    =   Θ_N,M^(ε,L)(0)
    +2∑_t=1^N-M-1Θ_N,M^(ε,L)(t),

where

    Θ_N,M^(ε,L)(t):=1/N-M∑_s=M^N-1-t∫_ψ_s+t^(ε, L)ψ_s^(ε, L)  dμ_,

with

    ∫_ψ_s+t^(ε, L)ψ_s^(ε, L)  dμ_
    =
    ∫_ (f_ε^(L)∘ a_s+t)(f_ε^(L)∘ a_s)  dμ_ -
    μ_(f_ε^(L)∘ a_s+t)μ_(f_ε^(L)∘ a_s).

We shall first show that with a suitable choice of parameters ε and L we have:

    𝖥_N,M^(ε, L)_L^2()^2 
    
    =Θ_∞^(ε, L) (0) + 2∑_t=1^N-1Θ_∞^(ε, L) (t)+o(1),

where

    Θ_∞^(ε, L)(t):=∫_ (f_ε^(L)∘ a_t)f_ε^(L)  dμ_ -μ_(f_ε^(L))^2.

To estimate Θ_N,M(t), we introduce an additional parameter K=K(N)→∞ (to be specified later) with K≤ M and consider separately the cases when t< K and when t≥  K. 


First, we consider the case when t≥ K. 
By Corollary <ref>, we have 

    ∫_ (f_ε^(L)∘ a_s+t)(f_ε^(L)∘ a_s)  dμ_   =μ_(f_ε^(L))^2
     +O(e^-δmin(s,t)f_ε^(L)_C^l^2 ).

and also

    ∫_  (f_ε^(L)∘ a_s)  dμ_   =μ_ (f_ε^(L)) +O( e^-δ sf_ε^(L)_C^l).

Hence, combining (<ref>) and (<ref>), we deduce that

    ∫_ψ_s+t^(ε, L)ψ_s^(ε, L)  dμ_
    	=O (e^-δmin(s,t)f_ε^(L)_C^l^2).

Since

    ∑_t=K^N-M-1(∑_s=M^N-1-te^-δmin(s,t))
       ≤∑_t=K^N-1∑_s=M^N-1 (e^-δ t+e^-δ s)≪ N e^-δ K,

we conclude that

    ∑_t=K^N-M-1Θ_N,M(t) ≪  e^-δ K f_ε^(L)_C^l^2
    ≪  e^-δ K L^2 ε^-2l,

where we used Lemma <ref> and (<ref>).
The implied constants here and below in the proof depend only on (f_ε), which is uniformly bounded, hence the dependence is only on the constant c from the diophantine approximation (<ref>). 


Let us now consider the case t<K. We observe that Corollary <ref> (for r = 1)
applied to the function ϕ_t:= (f_ε^(L)∘ a_t)f_ε^(L) yields, 

    ∫_ (f_ε^(L)∘ a_s+t)(f_ε^(L)∘ a_s)  dμ_   =∫_ (ϕ_t∘ a_s)   dμ_
       =∫_ϕ_t  dμ_+O(e^-δ s ϕ_t_C^l).

Furthermore, for some ξ=ξ(n,l)>0, we have

    ϕ_t_C^l≪f_ε^(L)∘ a_t_C^l f_ε^(L)_C^l≪ e^ξ t f_ε^(L)_C^l^2.

Therefore, we deduce that

    ∫_ (f_ε^(L)∘ a_s+t)(f_ε^(L)∘ a_s)  dμ_
    =∫_ (f_ε^(L)∘ a_t)f_ε^(L) dμ_ +O (e^-δ s e^ξ t f_ε^(L)_C^l^2 ).

Combining this estimate with (<ref>), we obtain that

    ∫_ψ_s+t^(ε,L)ψ_s^(ε,L)  dμ_
    =Θ_∞^(ε, L)(t)
    + O (e^-δ s e^ξ t f_ε^(L)_C^l^2  ).

Using further the estimates from Lemma <ref> and (<ref>), it follows, for the case t<K,

    Θ_N,M^(ε,L)(t)=    N-M-t/N-MΘ_∞^(ε,L)(t)+O ((N-M)^-1 e^-δ M e^ξ t f_ε^(L)_C^l^2  )
    
    =    Θ_∞^(ε,L)(t)
    +O ((N-M)^-1tf_ε^(L)_L^2()^2+ (N-M)^-1 e^-δ M e^ξ t f_ε^(L)_C^l^2   )
    
    =    Θ_∞^(ε,L)(t)
    +O ((N-M)^-1t+ (N-M)^-1 e^-δ M e^ξ t ε^-2lL^2  ).

It follows

    Θ_N,M^(ε,L)(0)+2∑_t=1^K-1Θ_N,M^(ε,L)(t) =    Θ_∞^(ε,L)(0)+2∑_t=1^K-1Θ_∞^(ε,L) (t) 
       +O  ((N-M)^-1K^2+ (N-M)^-1 e^-δ M  e^ξ Kε^-2l L^2).

Combining (<ref>) and (<ref>), it follows from (<ref>) that 

    𝖥_N,M^(ε,L)_L^2()^2 
    
       =  Θ_∞^(ε,L)(0)+2∑_t=1^K-1Θ_∞^(ε,L) (t) 
        + O ((N-M)^-1K^2  + ((N-M)^-1 e^-δ M  e^ξ K+e^-δ K) L^2 ε^-2l).

We will choose later in (<ref>) to (<ref>) the parameters K(N), M(N), ε(N) and L(N) so that

    e^-δ K L^2ε^-2l→ 0, 
    
    (N-M)^-1 e^-δ M  e^ξ K L^2ε^-2l→ 0, 
    
    (N-M)^-1K^2 → 0,

as N→∞, which gives 

    𝖥_N,M^(ε,L)_L^2()^2 =Θ_∞^(ε,L) (0) + 2∑_t=1^K-1Θ_∞^(ε,L) (t)+o(1).

We shall show next that with a suitable choice of parameters we have

    𝖥_N,M^(ε,L)_L^2()^2 =Θ_∞^(ε) (0) + 2∑_t=1^K-1Θ_∞^(ε) (t)+o(1),

where 

    Θ_∞^(ε)(t):=∫_ (f_ε∘ a_t)f_ε  dμ_ -μ_(f_ε)^2.

Using again the estimates from Lemma <ref> 

    ||f_ε - f_ε^(L) ||_L^1_𝒳≪_supp(f_ε),τ L^-(τ-1)  and   ‖f_ε-f_ε^(L)‖_L^2_𝒳≪_supp(f_ε),τ  L^-τ-2/2

we have (since the supports of the functions f_ε are uniformly bounded) 

    μ_(f_ε^(L))   =μ_(f_ε)+O_τ( L^-τ-1),
    ∫_ (f_ε^(L)∘ a_t)f_ε^(L)  dμ_   = ∫_ (f_ε∘ a_t)f_ε  dμ_ + O_τ( L^-τ-2/2),

which yields

    Θ_∞^(ε,L)(t)= Θ_∞^(ε)(t) + + O_τ( L^-τ-2/2),

and (<ref>) then gives

    𝖥_N,M^(ε,L)_L^2()^2 
    
       =  Θ_∞^(ε)(0)+2∑_t=1^K-1Θ_∞^(ε) (t) 
        + O ((N-M)^-1K^2  + ((N-M)^-1 e^-δ M  e^ξ K+e^-δ K) L^2 ε^-2l +KL^-τ-2/2).


We will choose the parameters K(N) and L(N) such that

    KL^-τ-2/2→ 0  as  N→∞,   for some  τ<n,

which gives (<ref>). 

 

In order to analyse further the correlations ∫_ (f_ε∘ a_t)f_ε  dμ_ and show the convergence of the series ∑_t=1^K-1Θ_∞^(ε) (t), we shall use results from a recent work by Kelmer and Yu in <cit.>,  where incomplete Eisenstein series are used to analyse the second moment of the light-cone Siegel transform. We recall briefly in the following section some preliminaries to this approach.


Moment formulas of incomplete Eisenstein series


Before recalling the approach and results of Kelmer and Yu, we reproduce below some preliminaries from <cit.> about Eisenstein's series and adapt the notations to our coordinate system from Section <ref>.

We will denote in this section the elements in the subgroup A by 
    a_y:= [       I_n                    ;            y+y^-1/2 -y-y^-1/2;           -y-y^-1/2  y+y^-1/2 ]  , for  y>0,


the ℝ-split torus with a_y acting on e_0=(0,…,1,1)∈ℝ^n+2 as e_0a_y=y^-1e_0 and 

    K:={k=[ k̃   ;     1 ]: k̃∈_n+1(ℝ)}

a maximal compact subgroup. Let L be the stabilizer of e_0 in G and let P be the parabolic subgroup fixing the line spanned by e_0. More precisely, P=UAM and L=UM with

    M={m=([ m̃      ;     1   ;        1 ]): m̃∈_n(ℝ)}
 
the centralizer of A in K.
 

Any g∈ G can be written as g=u_xa_yk with k∈ K and in these coordinates the Haar measure of G is given (up to scaling) by

    dμ_G(g)=y^-(n+1)  d x d y dμ_K(k),

where dx is the usual Lebesgue measure on ℝ^n and μ_K is the probability Haar measure of K. 

The subgroup L is unimodular with its Haar measure given by 

    dμ_L(u_xm)=dxdμ_M(m),

where μ_M is the probability Haar measure of M≅_n(ℝ). 
Since L is the stabilizer of e_0 and G acts transitively on 𝒞, we can identify 𝒞 with the homogeneous space L \ G, which gives a natural right G-invariant measure on 𝒞. Explicitly, further identifying L\ G with A× M\ K gives natural polar coordinates on 𝒞: Every x∈𝒞 can be written uniquely as x=e_0a_yk for some y>0 and k∈ M\ K. In these coordinates the measure 

    d μ_𝒞(e_0a_yk):=y^-(n+1) d ydμ_M\ K(k)

is such an invariant measure. Here μ_M\ K is the unique right K-invariant probability measure on the homogeneous space M\ K which is homeomorphic to the unit sphere S^n. The measure μ_𝒞 is unique up to scaling, related to the G-invariant measure dz introduced in Proposition <ref> by dz=ω_Q dμ_𝒞.


 We have further the Langlands decomposition P=UAM (with the unipotent subgroup U given by the Iwasawa decomposition G=UAK) and L=UM. The cusps of Γ are the Γ-conjugacy classes of rational parabolic subgroups of G. Let m be the number of these cusps and P_1,…,P_m a set of representatives of these classes, each of which having a Langlands decomposition P_i=U_iA_iM_i, i=1,…,m. We denote by Γ_P_i:=Γ∩ P_i and by Γ_U_i:=Γ∩ U_i, where Γ_U_i is by definition a finite index subgroup of Γ_P_i (see <cit.>).

 For each P_i we fix the scaling matrix τ_i=k_ia_y_i, where k_i∈ K is such that P_i=k_iPk_i^-1 and where y_i>0 is the unique number such that μ_L(τ_i^-1Γ_P_iτ_i\ L)=1. 

We define the (spherical) Eisenstein series corresponding to the i-the cusp for Re(s)>n and g ∈ G by the convergent series
 
    E_i(s,g) := ∑_γ∈Γ_P_i\Γ y(τ_i^-1γ g)^s,

 where y(g) is given by the Iwasawa decomposition g=u_xa_y(g)k∈ UA K.

 For each 1≤  j≤ m the constant term of E_i(s,g) with respect to the j-th cusp is defined by

    c_ij(s,g):=1/vol(τ_j^-1Γ_U_jτ_j\ U)∫_τ_j^-1Γ_U_jτ_j\ UE_i(s, τ_j u_x g) dx,


which is known to be of the form

    c_ij(s,g)=δ_ijy(g)^s+φ_ij(s)y(g)^n-s

for some holomorphic function φ_ij defined for Re(s)>n.

The series E_i(s, g) (and hence also φ_ij) has a meromorphic continuation to the whole s-plane, which on the half plane Re(s)≥n/2 is holomorphic except for a simple pole at s=n (called the trivial pole) and possibly finitely many simple poles on the interval (n/2, n) (called exceptional poles). We denote by C_Γ⊆ (n/2, n) the finite set of exceptional poles of all Eisenstein series of Γ. 

The residue of E_i(s,g) at s=n is a constant which is the same for Eisenstein series at all cusps, given by the reciprocal of the measure of the homogeneous space Γ\ G, that is, for each 1≤ i≤ m and g∈ G,

    ω_Γ:=Res_s=nE_i(s,g)=μ_G(Γ\ G)^-1,


 For any bounded and compactly supported function f:𝒞→ℂ, the incomplete Eisenstein series attached to f at P_i is defined for any g ∈ G by
 
    E_i(g,f) := ∑_γ∈Γ_P_i\Γ f(e_0τ_i^-1γ g).

 Since E_i(·,f) is left Γ-invariant, it can be viewed as a function on the homogeneous space 𝒳. The light-cone Siegel transform of f can then be expressed in terms of incomplete Eisenstein series as follows.
 


    There exist constants λ_1,…,λ_m >0 such that for any bounded and compactly supported function f:→ℂ, 
    
    f = ∑_i=1^m E_i(·,f_λ_i),

  where f_λ(x):=f(λ^-1x) for any λ>0.


By the classical spherical harmonic analysis, the function space L^2(S^n) decomposes into irreducible _n+1(ℝ)-representations as following:

    L^2(S^n)=⊕_d≥ 0L^2(S^n,d),

where L^2(S^n, d) is the space of degree d harmonic polynomials in n+1 variables restricted to S^n. This in turn induces the following decomposition of L^2(M\ K) into irreducible K-representations

    L^2(M\ K)=⊕_d≥ 0 L^2(M\ K, d),

where L^2(M\ K, d) is the pre-image of L^2(S^n,d) under the isomorphism between L^2(M\ K) and L^2(S^n).
For each d≥ 0, we fix an orthonormal basis 
{ψ_d,l: 0≤ l≤_ℂL^2(M\ K, d)-1}
for L^2(M\ K, d). 
For any f: 𝒞→ℂ bounded and compactly supported, let 

    f_d,l(y):=∫_M\ Kf(e_0a_yk)ψ_d,l(k)dμ_M\ K(k).

so that f has a spherical expansion 

    f(e_0a_yk)=∑_d, l ≥ 0f_d,l(y)ψ_d,l(k)

in L^2 and also pointwise if f is smooth.

For any function f on ℝ^+, we denote by 
    f(s):=∫_0^∞f(y)y^-(s+1) d y,   , for s ∈ℂ
 its Mellin transform,  whenever this defining integral is absolutely convergent.

Using the spherical expansion we define the following bilinear form for any f,f': 𝒞→ℂ bounded and compactly supported and any s∈  (n2, n),

    M_f,f'(s):=∑_d,l≥ 0P_d(s)f_d,l(s)f'_d,l(s),

with P_0(s):=1 and P_d(s):=∏_i=0^d-1n-s+i/s+i if d≥ 1.


The following lemmas give estimates related to the operator M_f,f' which will be useful later for the analysis of Θ_∞(t). We write for simplicity M_f:=M_f,f.


    Let f be a bounded function on the light-cone with bounded support. For every s∈(n/2,n), we have
    
    | M_f(s)| ≪_s,supp(f)‖ f ‖_2^2  .



By definition of the Mellin transform and using the spherical expansion  f(e_0a_yk)=∑_d,lf_d,l(y)ψ_d,l(k), we have for any s∈ (n/2,n) 

    M_f(s) = ∑_d,lP_d(s)|f_d,l(s)|^2
       = ∑_d,lP_d(s) |∫_0^+∞f_d,l(y)y^-(s+1) dy|^2
       = ∑_d,lP_d(s) |∫_0^+∞(∫_M\ Kf(e_0a_yk)ψ_d,l(k)  dk)y^-(s+1) dy|^2.

Using the decomposition = ℝ_+ × M\ K given by x=e_0a_yk with the spherical coordinates y∈ℝ_+ and k∈ M\ K, we can write f(e_0a_yk)=ϕ_y(k)ρ(y), where (ϕ_y)_y>0 is a family of bounded function on M\ K and ρ is the characteristic function of an interval away from y=0 (since by the parametrization of 𝒞 we have e_0a_y=y^-1e_0). 

We also introduce the projection operator pr_d:L^2(M\ K)→ L^2(M\ K,d) on the space of degree d harmonic polynomials in n+1 variables restricted to M\ K and write f_d:=pr_d(f) for f∈ L^2(M\ K). Using that (ψ_d,l)_l≥ 0 is an orthonormal basis of L^2(M\ K,d) for every d≥ 0, it follows

    M_f(s) = ∑_d,l≥ 0P_d(s) |∫_0^+∞(∫_M\ Kϕ_y(k)ρ(y)ψ_d,l(k)  dk)y^-(s+1) dy|^2 
       =∑_d≥ 0P_d(s)∫_0^+∞∫_0^+∞⟨ϕ_y_1_d, ϕ_y_2_d⟩__M\ K ρ(y_1)ρ(y_2)y_1^-(s+1)y_2^-(s+1)  dy_1dy_2.

Using that |P_d(s) |≪ 1, Cauchy-Schwarz inequality, the decomposition in spherical harmonics given by L^2(M\ K)= ⊕_d≥ 0 L^2(M\ K,d), that ρ^2=ρ and that the support of ρ is an interval away from y=0,  we obtain

    |M_f(s)|    ≪∫_0^+∞∫_0^+∞∑_d≥ 0‖ϕ_y_1_d‖_2‖ϕ_y_2_d‖_2 ρ(y_1)ρ(y_2)y_1^-(s+1)y_2^-(s+1)  dy_1dy_2
       ≤∫_0^+∞∫_0^+∞‖ϕ_y_1‖_2 ‖ϕ_y_2‖_2 ρ(y_1)ρ(y_2)y_1^-(s+1)y_2^-(s+1)  dy_1dy_2
    
           = (∫_0^+∞‖ϕ_y‖_2  ρ(y)y^-(s+1)  dy)^2
       ≤(∫_0^+∞‖ϕ_y‖_2^2  ρ(y)^2y^-(n+1)  dy)(∫_0^+∞ρ(y)^2y^n-2s-1  dy)
       ≪_s,supp(ρ)(∫_0^+∞∫_M\ K|ϕ_y(k)ρ(y)|^2y^-(n+1) dk dy)
       = ‖ f ‖_2^2.



For any s∈ (n/2,n), we have P_d(s)≍_s (d+1)^n-2s.    

It will be useful for our argument later to have an estimate of P_d(s) also for s∈ℂ with real part in (n/2,n).

For any s=r+it ∈ℂ, with r∈(n/2,n), we have |P_d(s)|≪_δ |t|^1/2(d+1)^n-2r+δ for any δ>0.   



    We have
    
    |P_d(s) |^2
               =∏_k=0^d-1|n-s+k|^2/|s+k|^2
       =∏_k=0^d-1(n-r+k)^2/(r+k)^2·1+t^2/(n-r+k)^2/1+t^2/(r+k)^2
       ≪ (d+1)^2(n-2r)·∏_k=0^d-11+t^2/(n-r+k)^2/1+t^2/(r+k)^2     (by Lemma <ref>)

    Further we have
    
    log( ∏_k=0^d-11+t^2/(n-r+k)^2/1+t^2/(r+k)^2)    = ∑_k=1^d-1( log(1+t^2/(n-r+k)^2)- log(1+t^2/(r+k)^2))
                   +log(1+t^2/(n-r)^2/1+t^2/r^2)
       ≪∑_k=1^d-1t^2/1+t^2/(r+k)^2( 1/(n-r+k)^2-1/(r+k)^2)+O(1)
       ≪∑_k=1^d-1t^2k^2/k^2+t^2·1/k^3+O(1)
       ≪∑_k≤α(t)1/k+∑_k= α(t)+1^d-1t^2/(k^2+t^2)k +O(1)
       ≪logα(t)+t^2/α(t)^2+t^2log (d+1)+O(1).

    Choosing α(t)=⌈β |t|⌉ with β >0 large enough, we obtain
    
    ∏_k=0^d-11+t^2/(n-r+k)^2/1+t^2/(r+k)^2≪β |t| · (d+1)^1/β^2+1,

    hence |P_d(s)|≪_δ |t|^1/2·(d+1)^n-2r+δ for any δ>0.


We have further (see details in <cit.>)

    M_f_λ_i,f'_λ_j(s)   =λ_i^sλ_j^sM_f,f'(s),  for any  s∈ (n/2,n),
    μ_ (f_λ)    = λ^n μ_ (f),  for any  λ>0,

and denote by

    ω_Q := Res_s=nE_Q(s,g)   = ω_Γ∑_i=1^mλ_i^n,  with ω_Γ=μ_G(G/Γ)^-1
    
        c_Q:= ω_Γ||Res_s=s_nE_Q(s,g)||_L^2_𝓍^2   = ω_Γ∑_i,j=1^mλ_i^s_nλ_j^s_nRes_s=s_nφ_ij(s),

where E_Q(s,g) is the light-cone Eisenstein series of the quadratic form Q defined by

    E_Q(s,g):= ∑_i=1^mλ_i E_i(s,g),

which has at most one exceptional pole at s_n=⌊n+2/2⌋ (see <cit.>).
 

The correlations of incomplete Eisenstein series can then be estimated as follows.



    For any 1≤ i,j≤ m, there exist a bounded linear operator _ij: L^2()→ L^2() with operator norm || _ij||_op≤ 1 such that for any  f,f' ∈ C_c^∞(𝒞),
    
    ⟨ E_i(·,f),E_j(·,f') ⟩= ω_Γ^2 μ_(f)μ_(f') +ω_Γ⟨δ_ijf+_ij(f),f'⟩+ω_Γ∑_s_l ∈ C_ΓM_f,f'(s_l)Res_s=s_lφ_i,j(s) ,

    where the two inner products are with respect to μ_ and μ_ respectively.
    

From Theorem <ref> Kelmer and Yu derived the following mean value theorem and effective estimate of the second moment of the light-cone Siegel transform.


    
    Let f:→ℂ be a measurable, bounded and compactly supported function. Then we have
    
    ∫_f d μ_ = ω_Q μ_(f).

    Further assume f is smooth, then
    
    ∫_|f|^2 d μ_ = |ω_Q μ_(f)|^2+c_Q M_f,f(s_n) +O( μ_( |f|^2)),

    where s_n:= ⌊n+2/2⌋, the term M_f,f(s) is a quadratic form on f given by (<ref>) and c_Q given by (<ref>).


We generalize the second moment formula in Theorem <ref> to measurable, bounded and compactly supported functions in the following Proposition, using a similar argument as in <cit.>.


    
    Let f be measurable, bounded and compactly supported functions on 𝒞. Then we have
    
    ∫_|f|^2 d μ_ = |ω_Q μ_(f)|^2+c_Q M_f,f(s_n) +O( μ_( |f|^2)).




    There exist a sequence (f_i)_i∈ℕ in C_c^∞(𝒞) converging to f in L^1(𝒞) and in L^2(𝒞). By the mean value identity (<ref>) it follows that (f_i)_i∈ℕ converges to f in L^1(𝒳), hence also pointwise almost everywhere for some subsequence. To show that this convergence is also in L^2, we use Theorem <ref> and write for any i,j ∈ℕ
    
    f_i - f_j _L_𝒳^2   ≪ f_i - f_j _L^1_𝒞^2 +  f_i - f_j _L^2_𝒞^2 + M_f_i-f_j(s_n)
       ≪ f_i - f_j _L^1_𝒞^2 +  f_i - f_j _L^2_𝒞^2    (by Lemma <ref>).

    Since (f_i)_i∈ℕ is a Cauchy sequence in L^1(𝒞)∩ L^2(𝒞), it follows that (f_i)_i∈ℕ is a Cauchy sequence in L^2(𝒳) and therefore converges to f in L^2(𝒳). Hence the second moment formula for f follows from the second moment formula for (f_i)_i∈ℕ.  


We shall show next that

    𝖥_N,M^(ε,L)_L^2()^2 =Θ_∞ (0) + 2∑_t=1^K-1Θ_∞ (t)+o(1),

where 

    Θ_∞(t):=∫_ (χ∘ a_t)χ  dμ_ -μ_(χ)^2.

Using Proposition <ref>, Lemma <ref> and the estimates in (<ref>), we have for any t≥ 0

    |Θ_∞(t) - Θ_∞^(ε)(t) |    ≤∫_|(f_ε∘ a_t )f_ε- (χ∘ a_t )χ| dμ_ + | μ_( f_ε)^2- μ_( χ)^2|
       ≪f_ε-χ_L_^2(f_ε_L_^2 +χ__L_^2)+ f_ε-χ_L_^1
       ≪f_ε-χ_L_^2+  f_ε-χ_L_^1
       ≪μ_( | f_ε - χ|)^2+ |M_f_ε-χ(s_n)|+O( μ_(|f_ε-χ|^2)) + ε
       ≪ε,

and (<ref>) then gives

    𝖥_N,M^(ε,L)_L^2()^2
       =  Θ_∞(0)+2∑_t=1^K-1Θ_∞(t) 
        + O (N^-1(M+K)K  + (N^-1 e^-δ M  e^ξ K+e^-δ K) L^2 ε^-2l +KL^-τ-2/2 + ε K ),

which implies (<ref>), provided 

    ε K → 0,    as N→∞.


It remains to show that the series ∑_t=1^K-1Θ_∞ (t) converges as K→∞.
 

We write

    Θ_∞ (t) = ∫_χ_t·χ dμ_ - μ_( χ_t)μ_( χ).

with χ_t:=χ∘ a_t.
 


Using Lemma <ref>, we can express the correlations in (<ref>) in terms of incomplete  Eisenstein series attached to χ_t and χ:

    ∫_χ_t·χ dμ_ = ∑_i,j=1^m⟨ E_i(·,χ_t,λ_i),E_j(·,χ_λ_j) ⟩,

where the inner product is with respect to μ_.

It follows from (<ref>) that 

    ∫_χ_t·χ dμ_ = ∑_i,j=1^m⟨ E_i(·,χ_t,λ_i),E_j(·,χ_λ_j) ⟩_μ_
       = ∑_i,j=1^m( ω_Γ^2 μ_(χ_t,λ_i)μ_(χ_λ_j) +ω_Γ⟨δ_ijχ_t,λ_i+_ij(χ_t,λ_i),χ_λ_j⟩_μ_+ω_Γ∑_s_l ∈ C_ΓM_χ_t,λ_i,χ_λ_j(s_l)Res_s=s_lφ_ij(s))
       = ω_Q^2 μ_(χ_t)μ_(χ) +ω_Q ⟨χ_t, χ⟩_μ_ +∑_i,j=1^m⟨_ij(χ_t,λ_i),χ_λ_j⟩_μ_+c_Q M_χ_t,χ(s_n).

Since  χ_t and χ have disjoint supports for all t≥ 1, it follows from the mean value identity in Theorem <ref> and  from (<ref>) that (<ref>) reduces for all t≥ 1 to 

    Θ_∞(t) = ∑_i,j=1^m⟨_ij(χ_t,λ_i),χ_λ_j⟩_μ_+c_Q M_χ_t,χ(s_n).


We shall estimate next the terms M_χ_t,χ(s_n) and ⟨_ij(χ_t,λ_i),χ_λ_j⟩_μ_.


  
 For every s∈ (n/2,n), there exists σ>0 (depending on n and s) such that for every  t≥ 1 we have
 
    |M_χ_t,χ(s)| ≪ e^-σ t.


 
By definition of the Mellin transform, we have for any s∈ (n/2,n) 

    M_χ_t,χ(s) = ∑_d,lP_d(s)χ__d,l(s)(χ_t)_d,l(s)
       = ∑_d,lP_d(s) (∫_0^+∞χ__d,l(y)y^-(s+1) dy)(∫_0^+∞(χ_t)_d,l(y)y^-(s+1) dy)
       = ∑_d,lP_d(s) (∫_0^+∞(∫_M\ Kχ(e_0a_yk)ψ_d,l(k)  dk)y^-(s+1) dy)·
                                   ·(∫_0^+∞(∫_M\ Kχ_t(e_0a_yk)ψ_d,l(k)  dk)y^-(s+1) dy).

Using the decomposition = ℝ_+ × M\ K given by x=e_0a_yk with the spherical coordinates y∈ℝ_+ and k∈ M\ K, we write 

    χ(e_0a_yk)=ϕ_y(k)ρ(y)  and  χ_t(e_0a_yk)=ϕ_t,y(k)ρ_t(y),

where ρ and ρ_t are the characteristic functions of intervals given by the projections on the last coordinate of the domains F_1,c and a_e^-t(F_1,c) respectively (i.e. intervals of the form [α,β] and [α e^-t,β e^-t] for some positive constants α and β depending on the diophantine constant c from (<ref>)), while ϕ_y and ϕ_y,t are characteristic functions on M\ K resulting from the decomposition = ℝ_+ × M\ K.
We also introduce the projection operator  on the space of degree d harmonic polynomials in n+1 variables restricted to M\ K and write 

    pr_d:L^2(M\ K)→ L^2(M\ K,d)   and   f_d:=pr_d(f), for any  f∈ L^2(M\ K).

Using that (ψ_d,l)_l≥ 0 is an orthonormal basis of L^2(M\ K,d) for every d≥ 0, it follows

    M_χ_t,χ(s) = ∑_d,l≥ 0P_d(s) (∫_0^+∞(∫_M\ Kϕ_y(k)ρ(y)ψ_d,l(k)  dk)y^-(s+1) dy)·
                                   ·(∫_0^+∞(∫_M\ Kϕ_t,y(k)ρ_t(y)ψ_d,l(k)  dk)y^-(s+1) dy) 
       =∫_0^+∞∫_0^+∞∑_d≥ 0P_d(s)⟨(ϕ_y_1)_d, (ϕ_t,y_2)_d⟩__M\ K ρ(y_1)ρ_t(y_2)y_1^-(s+1)y_2^-(s+1)  dy_1dy_2.

We separate the summation above into two parts with a parameter D≥ 1 to be fixed later, and estimate the two parts using a similar approach as in <cit.>, in particular the estimate of P_d(s) from Lemma <ref> and the following inequality from spherical harmonic analysis  

    || ϕ_d||_2 ≪ (d+1)^n-1/2||ϕ||_1,   for any ϕ∈ L^2(M\ K).

For the first part of the summation, we use orthogonality of the projections pr_d and Cauchy-Schwarz inequality to obtain

    |∑_d≤ D P_d(s)⟨(ϕ_y_1)_d, (ϕ_t,y_2)_d⟩__M\ K|
          ≪∑_d≤ D  (d+1)^n-2s||ϕ_y_1||_2||ϕ_t,y_2||_2 
       ≪∑_d≤ D  (d+1)^n-2s+n-1/2||ϕ_y_1||_2||ϕ_t,y_2||_1 
       ≪ D^n-2s+n-1/2+1||ϕ_y_1||_2||ϕ_t,y_2||_1.

For the second part of the summation, we use Cauchy-Schwarz inequality for the sum and the convergence given by L^2(M\ K)= ⊕_d≥ 0 L^2(M\ K,d), which gives

    |∑_d> D P_d(s)⟨(ϕ_y_1)_d, (ϕ_t,y_2)_d⟩__M\ K|    ≪∑_d> D  (d+1)^n-2s||(ϕ_y_1)_d||_2||(ϕ_t,y_2)_d||_2
       ≤ D^n-2s∑_d≥ 0||ϕ_y_1_d||_2||ϕ_t,y_2_d||_2
       ≤ D^n-2s||ϕ_y_1||_2||ϕ_t,y_2||_2.

Using that ||ϕ||_2=||ϕ||_1^1/2 for any characteristic function ϕ, we optimize both estimates (<ref>) and (<ref>) by taking D=max (1,||ϕ_t,y_2||_1^-1/n+1 ) and obtain

    |∑_d≥ 0 P_d(s)⟨(ϕ_y_1)_d, (ϕ_t,y_2)_d⟩__M\ K|    ≪ ||ϕ_y_1||_2||ϕ_t,y_2||_1^1/2+2s-n/n+1.

We note that 1/2+2s-n/n+1>s/n for any s∈ (n/2,n) and n≥ 2, and write σ:= 1/2+2s-n/n+1-s/n>0. 


We observe further that, since e_0a_y=y^-1e_0 and since F_1,c⊆{x ∈𝒞: x_1^2+… + x_n+2^2 <c^2 }, we have for any y>0

    ϕ_y _1    = ∫_M \ Kχ (e_0a_yk)dk
       =∫_M \ Kχ (y^-1e_0k)dk
       ≤∫_M \ Kχ_{x_1^2+… +x_n^2 <y^2c^2}(e_0k)dk
       ≪_c y^n.

From (<ref>) and (<ref>), using that supp(ρ) is bounded away from 0 and supp(ρ_t) =[α e^-t,β e^-t], it follows 

    | M_χ_t,χ(s)|    ≪(∫_0^+∞y_1^n/2y_1^-(s+1)ρ(y_1) dy_1)(∫_0^+∞y_2^s+nσy_2^-(s+1)ρ_t(y_2)  dy_2)
       ≪_supp(ρ)∫_0^+∞y_2^-1+nσ ρ_t(y_2)dy_2 
       ≪∫_α e^-t^β e^-ty_2^-1+nσ dy_2
       ≪ e^-nσ t.


We give next an estimate of the term ∑_i,j=1^m⟨_ij(χ_λ_i),χ_t,λ_j⟩_μ_. In order to simplify the notations, we will omit without loss of generality the scaling coefficients λ_1,…, λ_m and consider only a single cusp.



    There exists γ >0 such that for every t≥ 1 we have
    
    |⟨(χ),χ_t⟩_μ_| ≪ e^-γ t.



    
    For a smooth and compactly supported function f∈ C_c^∞(), the operator _ can be expressed explicitly in terms  of the Mellin transform of the spherical harmonic coefficients of f as follows (see <cit.>):
    
    _(f)(e_0a_yk) = ∑_d,l(1/2π i∫_( n/2) P_d(s)φ(s)f_d,l(s)y^n-sds)ψ_d,l(k)

    where the contour integration is along the line of complex numbers with real part n/2.

    We shall approximate χ by a smooth and compactly supported function f_ε_t in the sense of (<ref>), with a parameter ε_t>0 to be fixed later. We note that ε_t is independent from the parameter ε=ε(N) introduced in (<ref>). Since  is a bounded linear operator on L^2() with operator norm ‖‖_op≤ 1, we have:
    
    | ⟨_(χ),χ_t⟩|    = | ⟨_(χ -f_ε_t)+_(f_ε_t),χ_t⟩|
       ≤χ -f_ε_t_2χ_t _2+| ⟨_(f_ε_t),χ_t⟩| 
       ≪ε_t^1/2 + | ⟨_(f_ε_t),χ_t⟩|   .

    Using the decomposition dμ_=y^-(n+1)dydk, the spherical expansion f(e_0a_yk)=∑_d,lf_d,l(y)ψ_d,l(k) and the decomposition L^2(M\ K)= ⊕_d≥ 0 L^2(M\ K,d), it follows:
    
    ⟨_(f_ε_t),χ_t⟩_μ_
       = ∫_M\ K∫_0^+∞(∑_d,l1/2π i( ∫_(n/2)P_d(s)φ(s)(f_ε_t)_d,l(s)y^n-sds)ψ_d,l(k))·
                                           ·(∑_d',l'(χ_t)_d',l'(y)ψ_d',l'(k))y^-(n+1)dydk
       = ∫_M\ K1/2π i∫_(n/2)( ∑_d,lP_d(s)φ(s)(f_ε_t)_d,l(s)ψ_d,l(k))·
                                           ·( ∑_d',l'( ∫_0^+∞(χ_t)_d',l'(y)y^-(s+1)dy)ψ_d',l'(k))dsdk
       =∑_d,l1/2π i(∫_(n/2)P_d(s)φ(s)(f_ε_t)_d,l(s)(χ_t)_d,l(s) ds ).

    
    We use again the same decomposition as in (<ref>) 
    χ_t(e_0a_yk)=ϕ_t,y(k)ρ_t(y)
  and introduce the function
     
    F_ε_t(y,k):= f_ε_t(e_0a_yk).
 Moreover, using the fact that there is at most one exceptional pole at s_n=⌊n+2/2⌋ in (n/2,n), we can move the contour of integration to the line ( n/2+𝓇) for some 𝓇>0 small enough. By expanding the integrand similarly to (<ref>) we have 

    ⟨_(f_ε_t),χ_t⟩_μ_
       =1/2π i∫_(n/2+𝓇)φ(s)∑_d≥0P_d(s)(∫_ℝ_+∫_ℝ_+⟨ F_ε_t(y_1,· ), (ϕ_t,y_2)_d⟩_μ_M\ K ρ_t(y_2)y_1^-(s+1)y_2^-(s̅+1)  dy_1dy_2 ) ds
       =1/2π i∫_(n/2+𝓇)φ(s)∑_d≥0P_d(s)
        
        (∫_ℝ_+∫_ℝ_+⟨∂^l F_ε_t/∂^l y_1, (ϕ_t,y_2)_d⟩_μ_M\ K ρ_t(y_2)y_1^-(s+1)+l/∏_j=0^l-1(s+j)y_2^-(s̅+1)  dy_1dy_2 ) ds,

where we applied an integration by parts for the l-th partial derivative with respect to y_1, with l≥ 1 (even) large enough to be fixed later.

We use the same computation as in the proof of Lemma  <ref> with the estimate |P_d(s)|≪ (d+1)^n-2𝓇+δ, for s=r+it ∈ℂ, from Lemma <ref>, and choose a fixed 𝓇>0 small enough such that n/2+𝓇<s_n. It follows 

    |⟨_(f_ε_t),χ⟩_μ_| 
       ≪∫_(n/2+𝓇)|φ(s)s^-l| (∫_ℝ_+∫_ℝ_+∑_d≥0|P_d(s)||⟨(∂^l F_ε_t/∂^l y_1)_d, (ϕ_t,y_2)_d⟩_μ_M\ K|  ·. 
                                               · ρ_t(y_2)y_1^-(n/2+𝓇+1)+ly_2^-(n/2+𝓇+1) dy_1dy_2 )ds 
       ≪_δ∫_(n/2+𝓇)|φ(s)s^-l+1/2|ds (∫_ℝ_+∫_ℝ_+∑_d≥0(d+1)^n-2𝓇+δ|⟨(∂^l F_ε_t/∂^l y_1)_d, (ϕ_t,y_2)_d⟩_μ_M\ K| ·. 
                                               · ρ_t(y_2)y_1^-(n/2+𝓇+1)+ly_2^-(n/2+𝓇+1) dy_1dy_2 ).

Using further the same estimate as in (<ref>) and the fact that I:=pr_ℝ_+(supp(F_ε_t)) is uniformly bounded away from y=0, we have

    ∫_ℝ_+∫_ℝ_+∑_d≥0(d+1)^n-2𝓇+δ|⟨(∂^l F_ε_t/∂^l y_1)_d, (ϕ_t,y_2)_d⟩_μ_M\ K| ρ_t(y_2)y_1^-(n/2+𝓇+1)+ly_2^-(n/2+𝓇+1) dy_1dy_2 
       ≪(∫_ℝ_+‖∂^l F_ε_t/∂^l y_1‖_L^2_M\ K y_1^-(n/2+𝓇+1)+ldy_1)(∫_ℝ_+‖ϕ_t,y_2‖_L^1_M\ K^1/2+2𝓇-δ/n+1 ρ_t(y_2)y_2^-(n/2+𝓇+1) dy_2)
       ≪‖ f_ε_t‖_C^l(∫_I y_1^-(n/2+𝓇+1)+ldy_1)(∫_ℝ_+y_2^n(1/2+2𝓇-δ/n+1) ρ_t(y_2)y_2^-(n/2+𝓇+1) dy_2)
       ≪_𝓇‖ f_ε_t‖_C^l(∫_ℝ_+ ρ_t(y_2)y_2^-1-δn/n+1+𝓇n-1/n+1 dy_2)
       ≪ε_t^-l  ∫_α e^-t^β e^-t  y_2^-1-δn/n+1+𝓇n-1/n+1 dy_2
       ≪ε_t^-le^-t(𝓇n-1/n+1-δn/n+1) .

We write σ:=𝓇n-1/n+1-δn/n+1 and choose 0<δ<n-1/n+1𝓇 such that σ >0. We use further the following estimates for the scattering matrix φ(s) near the critical line (n/2) in terms of a function W(t)≥ 1, with s=r+it, introduced in <cit.>:

    |φ(s)|^2=1+O( (r-n/2)W(t)) , for n/2<r<r_0<n,
    and       ∫_0^T W(t) dt≪ T^n+1 ,   (see Propositions 7.11 and 7.13 in <cit.>).

For fixed r=n/2+𝓇 and l≥ 1 large enough, using Cauchy-Schwarz inequality then integration by parts, we have

    (∫_(n/2+𝓇)|φ(s)s^-l+1/2|ds )^2  ≤(∫_(n/2+𝓇)|φ(s)s^-l/2|^2ds )(∫_(n/2+𝓇)|s^-l+1|ds )
       = (∫_ℝ(1+O(𝓇W(t)))|t^2+r^2|^-l/2dt)(∫_ℝ|t^2+r^2|^-l+1/2 dt ) 
       ≪_𝓇,l∫_ℝW(t)|t^2+r^2|^-l/2dt 
       ≪∫_ℝ |t|^n+2|t^2+r^2|^-l+2/2dt = O(1)     with some fixed l> n+1 .

All together we obtain

    | ⟨_(χ),χ_t⟩| ≪ε_t^1/2 + ε_t^-le^-σ t,

and choose ε_t = e^-2γ t with γ :=σ/1+2l.

Putting together (<ref>), (<ref>) and the estimates showed in Lemmas <ref> and <ref>, we obtain


    ‖𝖥_N^(ε,L)‖_L^2_𝒴^2    = ∑_t=-K+1^K-1Θ_∞(t) + o(1)
       = ∑_t=-K+1^K-1(∑_i,j=1^m⟨_ij(χ_t,λ_i),χ_λ_j⟩_μ_+c_Q M_χ_t,χ(s_n)) + o(1)

and 

    ‖𝖥_N^(ε,L)‖_L^2_𝒴^2N→∞⟶ σ^2  := ∑_i,j=1^m⟨_ij(χ_∞,λ_i),χ_λ_j⟩_μ_+c_Q M_χ_∞,χ(s_n),

where we denote by χ_∞ the characteristic function of the domain

    { x ∈𝒞 : x_n+2^2- x_n+1^2 < c^2}  =  ⋃_t=-∞^∞a_-t(F_1,c).





§ PROOF OF THE CLT FOR THE COUNTING FUNCTION
	

Using the characterisation by the cumulants (Proposition <ref>), we first show that the sequences (𝖥_N,M^(ε,L))_N≥ 1, hence also the sequence (𝖥_N)_N≥ 1, converge in distribution to the normal law Norm_σ.



Let m≥ 2. For every ξ∈ℝ,

    μ_𝒴({y ∈𝒴:𝖥_N(y)<ξ}) →Norm_σ(ξ)

as N→∞, for some variance σ^2<∞.





By Proposition <ref> and considering that 𝖥_N and 𝖥_N,M^(ε,L) have the same limit distribution, it is enough to show that

    Cum_μ_𝒴^(r)(𝖥_N,M^(ε,L))→ 0   as   N→∞

when r≥ 3, and

    ‖𝖥_N,M^(ε,L)‖_L^2_𝒴^2→σ^2   as  N→∞ .

We showed in sections <ref> and <ref> that these two conditions hold provided that the parameters 

    ε=ε(N), L=L(N), M=M(N), γ=γ(N), K=K(N)≤ M(N)

satisfy the conditions we recall here

    M   =o(N^1/2) ,<ref>
    
    (N-M)^1/2ε   → 0 ,<ref>
    
    (N-M)^1/2e^-θ M   → 0 ,<ref>
    
    M   ≫log L ,<ref>
    
    (N-M)   =o(L^p)  , for some p<n ,<ref>
    
    M   ≫_r γ ,<ref>
    
    (N-M)^r/2L^rε^-rl   =o(e^δγ) ,<ref>
    
    e^-δ K L^2ε^-2l   → 0 , <ref>
    
    (N-M)^-1 e^-δ M  e^ξ K L^2ε^-2l   → 0 ,<ref>
    
    (N-M)^-1K^2    → 0<ref>,
    
    KL^-τ-2/2   → 0  , for some  τ<n <ref>
    ε  K    → 0<ref>.

One verifies easily that the following choice of parameters, with n≥ 3,

    M   =(log N) (loglog N), 
    ε   =(N-M)^-q_1,   for some q_1>1/2,
    
    L   =(N-M)^q_2  for some q_2>0 large enough to satisfy (<ref>),
    
    K   =c_1log (N-M)  for some   c_1>0  large enough to satisfy (<ref>),
    γ   =c_rlog (N-M)  for some   c_r>0  large enough to satisfy (<ref>)

verify the required conditions.
 
Hence, Theorem <ref> follows from Proposition <ref>.


Next we relate the function 𝖥_N to the counting function 𝖭_T,c and show that (𝖥_N)_N≥ 1 has the same limit distribution as (𝖣_T)_T>0 defined in the following. 

For k∈ K and α_k ∈ S^n defined by k(α_k,1)=(0,…,0,1,1)∈ S^n, we consier

    𝖣_T(k)  :=𝖭_T,c(α_k)-C_c,n· T/T^1/2

where C_c,n:=∫_𝒞χ dμ_𝒞=vol(F_1,c).


For all N≥ 1, we have

    ∑_t=0^N-1∫_𝒴χ∘ a_t dμ_𝒴 = C_c,nN+O(1) .




    By the mean value identity in (<ref>), we have
    
    C_c,n=∫_𝒞χ dμ_𝒞= ∫_𝒳χ̂d μ_𝒳.

    It follows

    | ∑_t=0^N-1∫_𝒴χ∘ a_t dμ_𝒴 - C_c,nN|    = | ∑_t=0^N-1∫_𝒴χ∘ a_t dμ_𝒴 - ∑_t=0^N-1∫_𝒳χdμ_𝒳|
       ≤∑_t=0^N-1∫_𝒴|  χ∘ a_t- μ_𝒳(χ) |dμ_𝒴.

Introducing a parameter L_t>0 such that L_t ∞ and using the estimates for the truncated Siegel transform from Proposition <ref>, we have for any 2≤τ <n and t≥κlog L_t,

    ‖(χ∘ a_t - μ_𝒳(χ)) -(χ^(L_t)∘ a_t - μ_𝒳(χ^(L_t) )) ‖_L^1(𝒴)   ≤‖χ∘ a_t  -χ^(L_t)∘ a_t‖_L^1(𝒴) + μ_𝒳( | χ- χ^(L_t)|)
       ≪ L_t^-τ/2+ L_t^-(τ-1)
       ≪ L_t^-τ/2.

Introducing further a parameter ε_t>0 such that ε_t  0 and using the estimates for the smooth approximation of χ from Proposition <ref> and from (<ref>) we have

    ‖(χ^(L_t)∘ a_t - μ_𝒳(χ^(L_t))) -(f_ε_t^(L_t)∘ a_t - μ_𝒳(f_ε_t^(L_t) )) ‖_L^1(𝒴)
                   ≤‖χ^(L_t)∘ a_t - f_ε_t^(L_t)∘ a_t‖_L^1(𝒴) + μ_𝒳( | χ^(L_t)- f_ε_t^(L_t)|)
                   ≪ε_t +e^-θ t.

Using further the effective equidistribution estimate from Proposition <ref>, we have

    ‖f_ε_t^(L_t)∘ a_t - μ_𝒳(f_ε_t^(L_t) ) ‖_L^1(𝒴)   ≪ e^-δ t‖f_ε_t^(L_t)‖_C^l
       ≪ e^-δ tε_t^-lL_t.

We choose L_t=t^a and ε_t=t^-b for some a>2/τ and b>1/l, then fix an integer N_0=N_0(κ,a)≥ 1 such that t≥κlog L_t for all t≥ N_0. 

Altogether we obtain

    | ∑_t=0^N-1∫_𝒴χ∘ a_t dμ_𝒴 - C_c,nN|    ≪| ∑_t=0^N_0-1∫_𝒴χ∘ a_t dμ_𝒴 - C_c,nN|
            + ∑_t=N_0^N-1( L_t^-τ/2+ε_t+ e^-θ t + e^-δ tε_t^-lL_t)
       = O(1) .




We will also need the following estimate related to the approximation in (<ref>). We recall our equivalent notations χ=χ_1,c=χ__F_1,c and use in the following Lemma the notation χ_1,c for more clarity.



We have

    ∫_K | 𝖭_N,c(α_k)-∑_t=0^N-1χ_1,c∘ a_t(kΛ_0)  | dμ_K(k) = O(N^1/3) .



From (<ref>) we have

    ∫_K | 𝖭_N,c(α_k)-∑_t=0^N-1χ_1,c∘ a_t(kΛ_0)  | dμ_K(k)
       ≤∫_K | ∑_t=0^⌊ N+r_0 ⌋χ_1,c∘ a_t(kΛ_0)-∑_t=0^⌊ N-r_0 ⌋-1χ_1,c_𝓁∘ a_t(kΛ_0)  |dμ_K(k)+O(𝓁^1/2)
       ≤∑_t=0^⌊ N-r_0 ⌋-1∫_𝒴χ__F_1,c∖ F_1,c_𝓁∘ a_tdμ_𝒴+ ∑_t=⌊ N-r_0 ⌋^⌊ N+r_0 ⌋∫_𝒴χ_1,c∘ a_tdμ_𝒴 +O(𝓁^1/2).

By Lemma <ref> we have

    ∑_t=⌊ N-r_0 ⌋^⌊ N+r_0 ⌋∫_𝒴χ_1,c∘ a_t=O(1).

Further, we estimate the volume of the set

    F_1,c∖ F_1,c_𝓁= { x ∈𝒞 : c_𝓁≤ (x_1^2+ … + x_n^2)^1/2 < c ,  c< x_n+2+ x_n+1 < ce } ,

using that |c-c_𝓁|=O(l^-1), which gives 

    ∫_𝒞χ__F_1,c∖ F_1,c_𝓁dμ_𝒞 =O(𝓁^-1).

By Lemma <ref> we have 

    ∑_t=0^⌊ N-r_0 ⌋-1∫_𝒴χ__F_1,c∖ F_1,c_𝓁∘ a_tdμ_𝒴= O( N𝓁^-1+1),

which yields the claim with 𝓁=⌊ N^2/3⌋.


It follows from Lemmas <ref> and <ref> that

    ∫_K| 𝖣_N(k) -𝖥_N(kΛ_0) |dμ_K(k) 
       = 1/N^1/2∫_K| 𝖭_N,c(α_k)-∑_t=0^N-1χ∘ a_t(kΛ_0)  +∑_t=0^N-1∫_𝒴χ∘ a_t -C_c,nN |dμ_K(k) 
       =o(1) ,

hence (𝖣_N) and (𝖥_N) have the same limit distribution, i.e. for all ξ∈ℝ we have

    | {k ∈ K:𝖣_N(k)<ξ}|→Norm_σ (ξ) ,    as  N→∞.
 
If we take N_T=⌊ T ⌋, then N_T ≤ T< N_T+1, hence

    𝖣_T(k)= 𝖭_T,c(α_k)-C_c,nT/T^1/2≤𝖭_N_T+1,c(α_k)-C_c,nN_T/T^1/2=a_T𝖣_N_T+1+b_T ,   with  a_T→ 1  and  b_T→ 0.

It follows

    | {k ∈ K:𝖣_T(k)<ξ}| ≥ | {k ∈ K:𝖣_N_T+1(k)<(ξ-b_T)/a_T }|.

Therefore, for any ε>0 and sufficiently large T,

    | {k ∈ K:𝖣_T(k)<ξ}| ≥ | {k ∈ K:𝖣_N_T+1(k)<ξ-ε}|,

thus

    lim inf_T→∞| {k ∈ K:𝖣_T(k)<ξ}| ≥Norm_σ(ξ-ε) ,

for all ε>0, which implies

    lim inf_T→∞| {k ∈ K:𝖣_T(k)<ξ}| ≥Norm_σ(ξ) .

One shows similarly 

    lim sup_T→∞| {k ∈ K:𝖣_T(k)<ξ}| ≤Norm_σ(ξ) ,

which finishes the proof of Theorem <ref>.



§ PROOF OF THE EFFECTIVE ESTIMATE FOR THE COUNTING FUNCTION
	

To obtain an effective estimate for the counting function 𝖭_T,c, the central argument in our approach is to derive an almost-everywhere-bound for averages ∑_t=0^T-1(χ∘ a_t-μ_𝒳(χ)) from an L^2-bound on these averages. We generalized this argument in <cit.> to L^p-bounds, p>1, following the approach in <cit.> based on an original idea of Schmidt in <cit.>. We generalize in the following proposition our result from <cit.> in order to take into account the approximation of χ_1,c by the sequence (χ_1,c,𝓁)_𝓁≥ 1 coming from the sandwiching (<ref>).



Let (Y,ν) be a probability space, and let (f_𝓁)_𝓁≥ 1 be a sequence of measurable functions f_𝓁: Y ×ℕ→ℝ. Suppose there exist p> 1 and C>0 such that, for all 𝓁≥1 and any integers 0≤ a<b, we have

    ∫_Y|  ∑_t=a^b-1f_𝓁(y,t) |^pdν(y) ≤ C(b-a) .

Then, there exist C_p>0, depending only on p>1, such that for any subsequence (f_𝓁_N)_N≥ 1 there exists a full-measure set Y_0⊆ Y such that for all y∈ Y_0, all ε >0, there exists N_y≥1 such that for all N≥ N_y, we have

    ∑_t=0^N-1 f_𝓁_N(y,t) ≤ C_p
    N^1/plog ^1+1/p+ ε/p N.



In the argument as formulated in <cit.>, the estimate in (<ref>) is satisfied for all pairs (a,b) of the form (2^i j,2^i(j+1)) coming from the dyadic decomposition of N-1. In our previous work (see proof of Proposition 4.2 in <cit.>), we observed that for our argument it is actually enough to consider only a reduced selection of such pairs, denoted below by L(N), which still builds a partition of [1,N)∩ℕ and allows moreover to satisfy the conditions t≥κlog L and t≥ -1/θlogε required by Proposition <ref> and Proposition <ref>, for t ∈{a,b-1} and for parameters L and ε to be defined as functions of (a,b).

 For non-negative integers a<b we write [a..b) [a,b)∩ℕ. For an integer s≥ 2 we consider the following set of dyadic subsets,

    L_s{[2^i..2^i+1) :  0≤ i≤ s-2 }∪{[2^ij..2^i(j+1)) :  2^ij≥ 2^s-1, 2^i(j+1)≤ 2^s }∪{ [0..1) },

where the sets of the first type [2^i..2^i+1), 0≤ i≤ s-2, together with [0..1), are a decomposition of the set [0..2^s-1).

We observe that for any integer N≥ 2 with 2^s-1< N ≤ 2^s, the set [0..N) is the disjoint union of at most 2s-1 subsets in L_s (namely [0..1), the s-1 subsets of the first type and at most s-1 sets of the second type which can be constructed from the binary expansion of N-1). We denote by L(N) this set of subsets, i.e. [0..N)=_I∈ L(N)I.


In the following lemmas, the notations and assumptions are the same as in Proposition <ref> .
 For every l≥ 1, we have

    ∑_I∈ L_s∫_Y|∑_t∈ I f_l(y,t) |^p dν(y) ≤ Cs2^s.



 Since L_s is a subset of the set of all dyadic sets  [2^ij..2^i(j+1)) where i,j are non-negative integers and 2^i(j+1)≤ 2^s, we have for any l≥ 1

    ∑_I∈ L_s∫_Y|∑_t∈ I f_l(y,t) |^p dν(y)     ≤∑_i=0^s-1∑_j=0^2^s-i-1∫_Y|∑_t∈ I f_l(y,t) |^p dν(y) 
       ≤∑_i=0^s-1∑_j=0^2^s-i-1 C2^i
       ≤ Cs2^s.



For every ε >0, there exists a sequence of measurable subsets { Y_s}_s ≥ 1 of Y such that:


  * ν(Y_s) ≤ C s^-(1+pε).

  * For every integer N≥ 2 with 2^s-1≤ N-1<2^s and for every y ∉ Y_s one has

    | ∑_t=0^N-1f_𝓁_N(y,t) | ≪_p 
    2^s/ps^1+1/p+ε.





For every s≥1, consider the function f_s:Y×ℕ→ℝ defined by

    f_s(y,t) := max_2^s-1<N≤2^s∑_I∈ L_s| ∑_t∈ If_𝓁_N(y,t)|^p,

and the measurable set 

    Y_s=
    { y ∈ Y : f_s(y,t)  >  2^s s^2+pε}.

The first assertion follows from Lemma <ref> and Markov's Inequality.

Further, for any N≥ 2 such that 2^s-1≤ N-1<2^s and any y∉ Y_s, using the partition [0..N)=_I∈ L(N)I with L(N) of cardinality at most 2s-1, we have

    |∑_t=0^N-1 f_𝓁_N(y,t)|^p    = |∑_I∈ L(N)∑_t∈ I f_𝓁_N(y,t)|^p
       ≤ (2s-1)^p-1∑_I∈ L(N)|∑_t∈ I f_𝓁_N(y,t)|^p    (by Hölder's Inequality)
       ≤ (2s-1)^p-1f_s(y,t) 
       ≪_p s^1+p+pε2^s ,    (since y∉ Y_s)

which yields the claim by raising to the power 1/p.


Let ε>0 and choose a sequence of measurable subsets {Y_s}_s≥ 1 as in (<ref>). Observe that

    ∑_s=1^∞ν(Y_s) ≤∑_s=1^∞ C s^-(1+pε) < ∞.

The Borel-Cantelli lemma implies that there exists a full-measure subset Y(ε)⊂ Y such that for every y ∈ Y(ε) there exists s_y ∈ℕ such that for all s > s_y we have y ∉ Y_s.

Let N≥ 2 and s= 1 + ⌊log (N-1) ⌋, so that 2^s-1≤ N-1 < 2^s. Then, for N-1 ≥ 2^s_y we have s> s_y and y ∉ Y_s, thus

    |∑_t=0^N-1f_𝓁_N(y,t) |    ≪_p 2^s/ps^1+1/p +ε
       ≤
    (2N)^1/plog^1+1/p+ ε(2N).

This implies the claim for y ∈∩_m∈ℕY(1/m).


We now apply Proposition <ref> to the counting function ∑_tχ∘ a_t, where we write for simplicity χ=χ__F_1,c for the characteristic function of the set F_1,c defined in (<ref>). We denote by vol(F_1,c) the average of the Siegel transform from Proposition <ref> for the function χ given by

    vol(F_1,c) ∫_𝒞χ(z)dz = ∫_𝒳χ(Λ)dμ_𝒳(Λ).




Let n≥ 3. For all ε>0, for almost every k ∈ K we have

    ∑_t=0^N-1χ(a_t k Λ_0) = N vol(F_1,c) + O_k,ε (N^1/2+ε).



Using Proposition <ref>, it is enough to show that for every 1<p<2, for all pairs of integers (a,b) from the dyadic decomposition of N specified in Remark <ref>, we have

    || ∑_t=a^b-1( χ-μ_𝒳(χ))∘ a_t ||_L^p(𝒴)^p≪ (b-a) .

Let 1<p<2. Using the estimates for the truncated Siegel transform from Proposition <ref>, we have, for 2p/3p-2≤τ < n and t≥κlog L,

    ||  ( χ∘ a_t-μ_𝒳(χ))- ( χ^(L)∘ a_t-μ_𝒳(χ^(L)))||_L^p_𝒴   ≤||   χ∘ a_t- χ^(L)∘ a_t||_L^p_𝒴+ ∫_𝒳|χ-χ^(L)| 
       ≪ L^-τ(2-p)/2p + L^-(τ-1)
       ≪ L^-τ(2-p)/2p.

Further, using Proposition <ref> and the estimates from Proposition <ref> and (<ref>), we have for t≥ -1/θlogε,

    ‖( χ^(L)∘ a_t-μ_𝒳(χ^(L))) - ( f_ε^(L)∘ a_t-μ_𝒳(f_ε^(L)))‖_L^p_𝒴≤||   χ^(L)∘ a_t- f_ε^(L)∘ a_t||_L^p_𝒴+ ∫_𝒳|χ^(L)-f_ε^(L)| 
                   ≤||(χ -f_ε)^(L)∘ a_t ||_∞^p-1/p·||(χ -f_ε)^(L)∘ a_t ||_L^1_𝒴^1/p+ ∫_𝒞|χ-f_ε| 
                   ≪ L^p-1/p ε^1/p  + ε
                   ≪ L^p-1/p ε^1/p.

Further, using effective equidistribution for smooth and compactly supported functions (Proposition <ref> for r=1), we have 

    ‖∑_t=a^b-1( f_ε^(L)-μ_𝒳(f_ε^(L)))∘ a_t ‖_L^p_𝒴≤‖∑_t=a^b-1( f_ε^(L)-μ_𝒳(f_ε^(L)))∘ a_t ‖_L^2_𝒴
           ≤‖∑_t=a^b-1( f_ε^(L)∘ a_t-μ_𝒴(f_ε^(L)∘ a_t)) ‖_L^2_𝒴 + |∑_t=a^b-1( μ_𝒴(f_ε^(L)∘ a_t)-μ_𝒳(f_ε^(L))) |
           ≪ (b-a)^1/2‖𝖥_b,a^(ε,L)‖_L^2_𝒴 + ∑_t=a^b-1 e^-δ t‖f_ε^(L)‖_l,

where 𝖥_b,a^(ε, L) = 1/√(b-a)∑_t=a^b-1( f_ε^(L)∘ a_t -μ_𝒴(f_ε^(L)∘ a_t) ) as defined in (<ref>) and estimated in (<ref>) by 

    ‖𝖥_b,a^(ε, L)‖_L^2_𝒴   = ∑_-K^KΘ_∞(t)
       + O( (b-a)^-1K^2+((b-a)^-1e^-δ ae^ξ K+e^-δ K)L^2ε^-2l+KL^-τ-2/2+Kε).

Putting (<ref>), (<ref>), (<ref>) and (<ref>) together, and considering moreover that ∑_-K^KΘ_∞(t) is bounded uniformly in K (by the convergence showed in Section <ref>), we obtain

    || ∑_t=a^b-1( χ-∫_𝒳χ)∘ a_t ||_L^p(𝒴)
       ≪ (b-a)(L^-τ(2-p)/2p+L^p-1/pε^1/p) + e^-δ aLε^-l
       +(b-a)^1/2( 1+(b-a)^-1K^2+((b-a)^-1e^-δ ae^ξ K+e^-δ K)L^2ε^-2l+KL^-τ-2/2+Kε).

In order to bound the first term in (<ref>) by (b-a)^1/p, we choose 

    L=(b-a)^q_2 ,  with  q_2= 2(p-1)/τ (2-p),
    and       ε= L^-τ(2-p)+2(p-1)/2= (b-a)^-q_1,  with  q_1=q_2τ(2-p)+2(p-1)/2= (p-1)(1+q_2).

  We choose further
  
    K=c_p log (b-a),  with some    c_p>0,

and, in order to satisfy the condition K≤ a, we verify that for all but finitely many pairs (a,b) from the dyadic decomposition of N, i.e. for pairs of the forms (2^i,2^i+1) and (2^ij,2^i(j+1)) with i≥ i_0(c_p)≥ 1, we have

    K=c_p log(2^i) ≤ a.

With this choice of L, ε and K, we verify next that the terms in (<ref>) are bounded by (b-a)^1/p, for all but finitely many pairs (a,b) from the dyadic decomposition. We have indeed

    (b-a)^-1/2e^-δ ae^ξ KL^2ε^-2l≤ e^-δ 2^i_0(b-a)^-1/2+ξ c_p+2q_2+2lq_1≤  (b-a)^1/p,

and for some c_p>0 large enough, we also have

    (b-a)^1/2e^-δ KL^2ε^-2l≤ (b-a)^1/2-δ c_p+2q_2+2lq_1≤  (b-a)^1/p.

One verifies easily that the other terms in (<ref>) are also bounded by (b-a)^1/p.

Finally, we verify that the conditions t≥κlog  L and t≥ -1/θlogε are also verified, since we have, for all but finitely many pairs (a,b),

    κlog L = κ q_2 log (b-a) = κ q_2 log (2^i) ≤ a ≤ t
    and     -1/θlogε = q_1/θlog (b-a) = q_1/θlog (2^i) ≤ a ≤ t.

We obtain, for n>τ>2,

    || ∑_t=a^b-1( χ-∫_𝒳χ)∘ a_t ||_L^p_𝒴≪_p (b-a)^1/p,

which ends the prove.

Since the bound in Proposition <ref> is uniform for 𝓁≥ 1, since all the implicit constants in the estimates used in Proposition <ref> depend only on the support of χ and since the supports of χ_l are uniformly bounded for 𝓁≥ 1, the same argument as in Proposition <ref> with Proposition <ref> applied to χ_𝓁-μ_𝒳(χ_𝓁) yields the same asymptotic given in the following proposition.


Let n≥ 3. For any subsequence (𝓁_N)_N≥ 1, for almost every k ∈ K and all ε>0, we have

    ∑_t=0^N-1χ_𝓁_N(a_t k Λ_0) = N vol(F_1,c,𝓁_N) + O_k,ε (N^1/2+ε).





Combining Propositions <ref> and <ref> with the estimate (<ref>) we have, for almost every k∈ K, for all T> T_k for some T_k≥ 2,

    T vol(F_1,c,𝓁)+O(T ^1/2+ε)+O(𝓁^1/2)≤𝖭_T,c(α_k) +O(1) ≤ Tvol(F_1,c)+O(T ^1/2+ε).

From (<ref>) we also have

    Tvol(F_1,c,𝓁)+O(T ^1/2+ε) + O(𝓁^1/2)
       = T ( vol(F_1,c) + O(𝓁^-1)) +O(T ^1/2+ε)+ O(𝓁^1/2)
       = T  vol(F_1,c) + O(Tl^-1+T^1/2+ε+l^1/2).

By choosing the subsequence 𝓁_N=⌊ N^2/3⌋, with N=⌊ T ⌋, we obtain

    𝖭_T,c(α_k)
    =Tvol(F_1,c)+O(T ^1/2+ε).

Since full-measure sets in K correspond to full-measure sets in S^n, we conclude that this last estimate holds for almost every α∈ S^n.






