Poissonian pair correlation for directions in multi-dimensional affine lattices, and escape of mass estimates for embedded horospheres

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Str\"ombergsson and the second author, and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname{SL}(d,\mathbb{R})$-horospheres in the space of affine lattices.


Introduction
It is often difficult to rigorously determine the pseudorandom properties of a given sequence of real numbers modulo one, including even the simplest second-order correlation functions.In the present paper we consider the problem in higher dimension and construct an explicit sequence of points υ 1 , υ 2 , υ 3 , . . . on the unit sphere S d´1 whose two-point statistics converge to that of a Poisson point process.This sequence is given by the unit vectors υ j " }y j } ´1y j representing the directions of vectors y j in a fixed affine lattice in R d of unit covolume.Here the y j are listed in increasing length }y j }, where } ¨} denotes the Euclidean norm.If there are two or more vectors of the same length we take them in arbitrary order (our results will not depend on the choice made).If there are several lattice points with the same direction, they will appear repeatedly in the sequence.Our approach extends the results of [EMV15], which in turn builds on [MS10], from d " 2 to higher dimensions.We are furthermore able to relax the Diophantine hypotheses imposed in [EMV15].
A sequence pυ j q 8 j"1 on S d´1 is called uniformly distributed, if for any set D Ď S d´1 with vol S d´1 pBDq " 0 we have that lim where V S d´1 " vol S d´1 pS d´1 q.The pair correlation function of the partial sequence pυ j q N j"1 is defined as where c d " V ´1 d´1 S d´1 and d S d´1 is the standard geodesic distance for the unit sphere S d´1 .The scaling by c d N 1 d´1 ensures we are measuring correlations in units where the mean density of points is one (note that the scaled sphere c d N 1 d´1 S d´1 has volume N ).The function R 2 N psq is known as Ripley's Kfunction in the statistical literature.
We say the pair correlation of the sequence pυ j q 8 j"1 is Poissonian, if for any s ą 0 (1.1) lim , which is the volume of a ball in R d´1 of radius s.This limit holds for example almost surely for a sequence of independent and uniformly distributed random points on S d´1 .It coincides with the pair correlation function of a Poisson point process in R d´1 of intensity one, hence the term "Poissoninan".Every affine lattice of unit covolume can be explicitly written as L ξ " pZ d `ξqM 0 , where ξ P R d and M 0 P G " SLpd, Rq.For integer shift ξ P Z d we obtain the underlying lattice L " Z d M 0 .It follows from classical asymptotics for the number of affine lattice points in expanding sectors with a fixed opening angle that the sequence of directions is uniformly distributed on S d´1 , for any shift ξ P R d .
It is an interesting observation that a Poissonian pair correlation implies uniform distribution on general compact manifolds [M20].This fact was first proved in the case of S 1 by Aistleitner, Lachmann, and Pausinger [ALP18], and independently by Larcher and Stockinger [LS20].For the convergence of the pair correlation function, we will however, unlike the case of uniform distribution, require Diophantine conditions on the lattice shift ξ.For κ ě d, we say that ξ P R d is Diophantine of type κ if there exists C κ ą 0 such that
Theorem 1.1.Let d ě 2 and ξ P R d zQ d ; furthermore if d " 2 assume that ξ is p0, 0, 2q-vaguely Diophantine.Then the pair correlation function of the sequence pυ j q 8 j"1 of directions is Poissonian.We note that the hypothesis on ξ (in the case d " 2) is satisfied for all Brjuno vectors, and thus in particular for Diophantine vectors of any type.In Appendix B we prove that there is a set of second Baire category of ξ P R 2 zQ 2 for which the pair correlation function diverges.This shows that the Diophantine condition is indeed required.
The pair correlation function also exists for ξ P Q d if d ě 3 and is closely related to the two-point statistics of multi-dimensional Farey sequences [BZ05,M13] and visible lattice points [BCZ00,MS10].The deeper reason why we see a Poisson pair correlation for ξ R Q d is that the limit distribution is expressed through the Haar measure on the semi-direct product group SLpd, Rq ˙Rd , where the averages over double-lattice sums reduces to Siegel's mean value formula; cf.Proposition 7.1 and [EMV15, Proposition 14].In the case of ξ P Q d , we need to apply Rogers' formulas which exhibit non-trivial correlations in the lattice sums, which explains the non-Poissonian correlations in this case.This is also the reason why three-point and higher-order correlation functions for directions in affine lattices with ξ R Q d are non-Poissonian.Fine-scale statistics of directions have also been studied in the context of quasicrystals [BGHJ14,MS15,H22].
It is worth highlighting that in the analogous problem of directions in hyperbolic lattices, the pair correlation statistics are not Poissonian; see [MV18] and references therein.
Theorem 1.1 provides an example of a deterministic sequence in higher dimension whose pair correlation density is Poissonian.Other local statistics, however, deviate from the Poisson distribution in other statistical tests, as shown in [MS10].By deterministic we mean here that convergence is proved not just almost surely or in probability, but for a fixed, explicit sequence.An interesting non-Poisson random point process with Poissonian pair correlation is discussed in [BS84].
Theorem 1.1 generalises results of [EMV15] for two-dimensional affine lattices and under a stronger Diophantine condition on ξ, as well as earlier work by Boca and Zaharescu [BZ06] which was limited to almost every ξ P R 2 .Other examples of deterministic sequences with Poissonian pair correlation in one dimension include ?n mod 1 (excluding n that are perfect squares) [EM04,EMV15b] and the recent paper by Lutsko, Sourmelidis and Technau [LST21] on αn θ mod 1 which holds for every α ą 0 and θ ď 1{3.Sequences such as αn 2 mod 1 [RS98] or α2 n mod 1 [RZ99] have Poissonian pair correlation for almost every α, but with no explicit instances of α currently known.For more references on recent developments on metric pair correlation problems, we refer the reader to [AEM21,LS20b] and references therein.
Finally we mention work of Bourgain, Rudnick and Sarnak [BRS16,BRS17], who considered the fine-scale statistics of lattice points (without a shift) on large spheres, rather than radially projected points as in our setting.Remarkably, in dimension two, Kurlberg and Lester have recently been able to prove that all correlation functions converge to Poisson along density-one subsequences of eligible radii [KL22].
The next section will recall the convergence in distribution for the directions in affine lattices from [MS10], and then state an extension to convergence of mixed moments (Theorem 2.2), which is the main result of this paper.An application of the Siegel mean value formula gives explicit expressions for all second-order statistics, and in particular shows that the pair correlation function is Poissonian (Corollaries 2.3 and 2.4).These results thus immediately imply Theorem 1.1.Section 3 introduces the space of affine lattices.In Section 5 we prove escape-of-mass estimates for spherical averages that allow us to pass from convergence in distribution to convergence of moments.Sections 6 and 7 supply the proofs of our Main Lemma, which immediately implies Theorem 2.2, and Corollaries 2.3 and 2.4, respectively.
Acknowledgements.We thank João Lopes Dias, Andreas Strömbergsson and the anonymous referee for helpful comments.

Limit distribution and higher moments
We consider the set P T of affine lattice points y P L ξ inside the ball of radius B d T or, more generally, P c,T the lattice points in the spherical shell The well known asymptotics for the number of lattice points in a large ball yields for T Ñ 8, For σ ą 0 and υ P S d´1 , we define D c,T pσ, υq Ď S d´1 to be the open disc with center υ and volume vol S d´1 pD c,T pσ, υqq " σd 1 ´cd T ´d.
Then the radius of D c,T pσ, υq is -T ´d d´1 , and for T Ñ 8, Thus σ measures the disc's volume in terms of the average density of points on the sphere; this scale is compatible with the one introduced above for the pair correlation function.
We define the counting function N c,T pσ, υq :" # y P P c,T : }y} ´1y P D c,T pσ, υq ( for the number of affine lattice points whose direction is contained in D c,T pσ, υq.Note that on average over υ, uniform distribution implies (cf.[MS10, Section 2.3]) that for any Borel probability measure λ on S d´1 with continuous density, we have (2.1) lim N c,T pσ, υqλpdυq " σ.
This says that the expected number of affine lattice points, with direction contained in D c,T pσ, υq for random υ, is σ.
We recall the following result from [MS10], which provides the full limit distribution of N c,T pσ, υq with random υ distributed according to a general Borel probability measure λ.
The limit distribution satisfies the following properties, cf.Section 4: (a) E c,ξ pr, σq is independent of λ and L.
A key ingredient of the proof of Theorem 2.1 is Ratner's measure classification theorem, which allows one to prove equidistribution of horospheres embedded in the space of affine lattices.An effective version of this statement was established only recently [K21].
We denote the positive real part of z P C by Re `pzq :" max tRepzq, 0u.
The following is the principal theorem of this paper.
Theorem 2.2.Let σ 1 , . . ., σ m ą 0, and λ a Borel probability measure on S d´1 with continuous density.Choose ξ P R d and z " pz 1 , . . ., z m q P C m , such that one of the following hypotheses holds: (A1) Re `pz 1 q `¨¨¨`Re `pz m q ă d.
We will prove this in Section 6.
The following corollaries of Theorem 2.2 state that in particular the second moment and pair correlation converge and are Poisonnian.
Corollary 2.4 implies Theorem 1.1 by approximating the characteristic function from above/below by C 0 functions.The additional dependence of f pυ 1 , υ 2 , sq on υ 1 , υ 2 P S d´1 can be used to generalise Theorem 1.1 to pair counting where υ j 1 and υ j 2 are restricted to different subsets of We then have the following.
and let Γ 1 " Γ ˙Zd denote the corresponding arithmetic subgroup.The right action of g " pM, bq P G 1 on R d is defined by xg :" xM `b.We embed G in G 1 via the homomorphism M Þ Ñ pM, 0q.In the following we will identify G with the corresponding subgroup in G 1 and use the shorthand M for pM, 0q.Given σ ą 0 and 0 ď c ă 1, define the cone ) .
For g P G 1 and any bounded set By construction, we can view N p¨, Cq as a function on the space of affine lattices, Γ 1 zG 1 .For y " py 2 , . . ., y d q P R d´1 and t ě 0, let .
By an elementary geometric argument, given σ ą 0 and ǫ ą 0, there exists T 0 ą 0 such that for all υ P S d´1 z t´e 1 u, ξ P R d , M 0 P G and T " e The argument is the same as in the two-dimensional case discussed in [EMV15]; see in particular Fig. 3 (the yellow and red domains should now be viewed as higher-dimensional cones with symmetry axis along e 1 ).For u " pu 12 , . . ., u 1d , u 23 , . . ., u pd´1qd q P R dpd´1q 2 and v " pv 1 , v 2 , . . ., v d q P T :" The Iwasawa decomposition of M P G is given by where u P R dpd´1q 2 , v P T and k P SOpdq.Consider the Siegel set (3.8) S :" This set has the property that it contains a fundamental domain of G and can be covered with a finite number of fundamental domains.Throughout this paper, we fix a fundamental domain of G contained in S, and denote it by F. For x P ΓzG, there exists a unique M P F such that x " ΓM .Define ι : ΓzG Ñ F so that ιpΓM q " M .We extend the above to define a fundamental domain F 1 and Siegel set S 1 of the Γ 1 action on G 1 by F 1 " tp1, bqpM, 0q : b P r´1 2 , 1 2 q d , M P Fu, S 1 " tp1, bqpM, 0q : b P r´1 2 , 1 2 s d , M P Su.As before, we define the map ι : Γ 1 zG 1 Ñ F 1 by ιpΓ 1 gq " g.
Given M P G, we define vpM q as the v coordinate of the Iwasawa decomposition (3.9) ιpΓM q " npuqapvqk.
Proof.Let D r be the smallest closed ball of radius r centered at 0 which contains C.
From the fact that (3.16) #pr´c d rv ´1 i , c d rv ´1 i s X pZ `bi qq P t0, 1u for any 1 ď i ď s, (3.13) follows.
The case of mixed moment will be dealt with by the inequality (3.17) ˇˇN pg, C 1 q z 1 ¨¨¨N pg, C m q zm ˇˇď ˇˇN pg, C 1 Y ¨¨¨Y C m q z 1 `¨¨¨`zm ˇˇ.

Properties of the limiting distribution
In this section we prove the properties (a)-(e) of the limiting distribution in Theorem 2.1.We denote by m X 1 , m X 1 , and m Xq the Haar probability measures on the homogeneous spaces respectively.Here Γ q denotes the congruence subgroup Γ q :" tγ P Γ q : γ " idpmod qqu for q ě 2. According to [MS10, Theorems 6.3, 6.5 and subsequent remarks, and Lemma 9.5], the limiting distribution E c,ξ p¨, σq in Theorem 2.1 is given as follows: (4.1) where C c pσq is as in (3.1).Property (a) follows from the observation that the distribution described in (4.1) is independent of λ and L.
For ξ P R d zQ d the distribution is also independent of ξ, so property (d) follows.
Property (b) follows from (2.1).Property (c) and (e) follows from calculations of [M00].We write g " p1, bqpM, 0q P G 1 with M " npuqapvqk P S as in (3.7) and (3.8).For s P t1, . . ., d ´1u, put and for s " 0, d, We have N pg, C c pσqq " ÿ where w i pmq " pm i `bi q `i´1 ÿ j"1 u ji pm j `bj q.Without loss of generality we may assume b 1 , . . ., b d P r´1 2 , 1 2 s.For M P S s with sufficiently large v 1 ¨¨¨v s , N pg, C c pσqq " ÿ We first consider the case of E c,ξ for ξ P Z d .For r 0 Ñ 8 and m G denoting the Haar measure of G (with arbitrary normalisation), In the last line we are using the continuity of φ s,max with respect to k P SOpdq.According to the calculation of [M00, Proof of Theorem 3.11] with n " 2, the sum in the last line isr ´d 0 .This proves property (c) for ξ P Z d .The case of other ξ P Q d is analogous.
In the case of ξ P R d zQ d we get In this case we use the calculation of [M00, Proof of Theorem 4.3] which implies that the sum in the last line isr ´d´1 0 .This proves property (e).

Escape of mass
Denote by χ I the characteristic function of a subset I Ď R. For R ě 1 and η, r ą 0, define the Γ 1 -invariant function F R,η,r : G 1 Ñ R by (5.1) F R,η,r pgq :" χ rR,8q ˆsrpgq ź i"1 In view of Lemma 3.1, (3.16) and (3.17), we note that for (5.2) η " Repz 1 q `¨¨¨`Repz m q, r " rpC 1 Y ¨¨¨Y C m q, and all g P G such that ś srpgq i"1 v i pgq ě R with R sufficiently large, we have that (5.3) ˇˇN pg, C 1 q z 1 ¨¨¨N pg, C m q zm ˇˇď pC d r d q η F R,η,r pgq.
The following proposition establishes under which conditions there is no escape of mass in the equidistribution of horospheres with respect to the function F R,η,r and thus also for N pg, C 1 q z 1 ¨¨¨N pg, C m q zm .Proposition 5.1.Let ξ P R d , M 0 P G, η, r ą 0, and ψ P C 0 pR d´1 q.Assume that one of the following hypotheses hold: F R,η,r `Γ1 p1, ξqM 0 r npyqΦ t ˘ψpyqdy ˇˇˇ" 0.
To prepare for the proof of this statement, put K :" supp ψ.Without loss of generality we may assume K Ă r´1, 1s d´1 .Indeed, there exists s 0 ě 0 such that e ´s0 K Ă r´1, 1s d´1 , so we may replace M 0 , y, and Φ t in (5.4) by M 0 Φ ´s0 , e s 0 y, and Φ t`s 0 , respectively, and reduce it to the case K Ă r´1, 1s d´1 .
Next we define two maps γ " γ t : R d´1 Ñ Γ and h " h t : R d´1 Ñ F as follows.For y P R d´1 , t P R, there exist unique γpyq " γ t pyq P Γ and hpyq " h t pyq P F such that M 0 r npyqΦ t " γpyqhpyq.
For 1 ď s ď d ´1 and l " pl 1 , . . ., l s q P Z s ě0 , we let (5.5) Ξ s l :" ) with δ d " d4 d .Then for g " p1, ξγpyqqhpyq with hpyq P Ξ s l we have where l " l 1 `¨¨¨`l s .It follows that the integral in (5.4) is bounded by (5.6) This will be sufficient for proving case (B1).For (B2) we need a refinement that also considers the size of ξγpyq; see (5.23) below.
Let us write β i pyq :" e i t hpyq ´1 for 1 ď i ď d and y P R d´1 , and consider the Iwasawa decomposition of hpyq, hpyq " npupyqqapvpyqqkpyq.
Lemma 5.2.If hpyq P Ξ s l for l P Z s ě0 , then |β i pyq| ă 2 ´li for all 1 ď i ď s.Proof.For the sake of simplicity we write v i " v i phpyqq for 1 ď i ď d and u ij " u ij phpyqq for 1 ď i ă j ď d.We also define r upyq " pr u ij q 1ďiăjďd by npr upyqq " npupyqq ´1.Note that each r u ij can be expressed in terms of at most 2 d monomials of u 12 , . . ., u pd´1qd with coefficients ˘1, hence |r u ij | ď 2 d for any 1 ď i ă j ď d.
If hpyq P Ξ s l , then we have for any 1 ď j ă i ď d, for all 1 ď i ď s.
Denote by π 1 : R d Ñ R and the orthogonal projection to the first coordinate and π 1 : R d Ñ R d´1 the orthogonal projection to the remaining pd ´1q coordinates.Let Λ " Z d t M ´1 0 and, for k P Z, let (5.7) R k :" Then for sufficiently large k where the implied constants are independent of k but depend on the fixed M 0 P G. Throughout this section, let K 0 P Z be the largest integer such that !x P Λ : for all k ď K 0 .Note that K 0 only depends on the choice of Λ.We define the norm } ¨} for the wedge product by For k " pk 1 , . . ., k s q P Z s ěK 0 and p " pp 1 , . . ., p s q P Z s ě0 , we denote by Λ k ppq the set of px 1 , . . ., x s q P Λ k 1 ˆ¨¨¨ˆΛ ks such that (5.9) for j " 1, . . ., s.Then any px 1 , . . ., x s q P Λ k 1 ˆ¨¨¨ˆΛ ks such that x 1 , . . ., x s are R-linearly independent is contained in Ť pPZ s ě0 Λ k ppq.
Lemma 5.3.For any k P Z s ěK 0 and p P Z s ě0 , where ω s pk, pq " Moreover, if there exists 1 ď j ď s such that p j ě jk j `K, then #Λ k ppq " 0, where K is a sufficiently large constant depending only on the choice of lattice Λ.
Proof.Given x 1 , . . ., x j´1 , let V be the subspace spanned by x 1 , . . ., x j´1 and denote by Υ the region of Note that Υ X R k has width -2 k j along the directions in V , and width -2 k j ´pj along the directions perpendicular to V .If p j ď k j , then the number of possible x j P Λ k j satisfying (5.9) is therefore at most !p2 k j q j´1 p2 k j ´pj q d´pj´1q " 2 dk j ´pd`1´jqp j .
In case p j ą k j , let j 1 be the maximal number of R-linearly independent vectors in Υ X Λ k j .We may assume j ď j 1 ď d since there is no x j satisfying (5.9) in Υ otherwise.Then we can take a j 1 -dimensional parallelepiped Q generated by s x 1 , . . ., s x j 1 P Υ X Λ k j such that there is no element of Υ X Λ k j inside Q.Since s x 1 , . . ., s x j 1 are R-linearly independent and contained in Λ " Z d t M ´1 0 , the j 1 -dimensional volume of Q t M 0 is ě 1.Hence, the j 1dimensional volume of Q is " 1 independently of p j and x 1 , . . ., x j´1 .Also, the interior of the sets x j `Q with x j P Λ k j are pairwise disjoint.Note that Q is contained in j 1 pΥ X R k j q since the generators are in Υ X R k j .Thus for any x j P Υ X Λ k j , the set x j `Q is contained in pj 1 `1qpΥ X R k j q and the j 1 -dimensional volume of this region is !pj 1 `1q j 1 p2 k j q j´1 p2 k j ´pj q j 1 ´pj´1q ! 2 jk j ´pj .
Because of this and the uniform lower bound on the volume of Q, it follows that the number of possible x j P Λ k j satisfying (5.9) is at most ! 2 jk j ´pj .In particular, there is no such x j P Λ k j if p j ě jk j `K.
We have shown that for fixed x 1 , . . ., x j´1 , the number of possible x j P Λ k j satisfying (5.9) is ! 2 ω j pk j ,p j q ď 2 ωspk j ,p j q .Hence the desired estimate follows.
Lemma 5.4.There exists T 0 ě 0 such that the following holds for any t ą T 0 .For l P Z s ě0 and px 1 , . . ., x s q P Λ s , the set Ω l px 1 , . . ., x s q is the empty set if there exists 1 ď i ď s such that If K 0 ďk i ď t t d log 2 ´li u for all 1 ď i ď s and px 1 , . . ., x s q P Λ k ppq, then (5.10) vol R d´1 pΩ l px 1 , . . ., x s qq ! e Proof.For y P Ω l px 1 , . . ., x s q, by definition of β i pyq we have for i " 1, . . ., s.By a straightforward computation with x i " pπ 1 x i , π 1 x i q P R ˆRd´1 , it implies that (5.11) (5.12) for i " 1, . . ., s.
If there exists i such that x i P Λ k i with k i ą t d log 2 ´li , then it contradicts (5.12).If there exists i such that which also contradicts (5.11) and (5.12) since they imply This proves the first claim of the lemma.Suppose now that k i ď t d log 2 ´li for all i " 1, . . ., s and px 1 , . . ., x s q P Λ k ppq.To prove the estimate (5.10), we may assume Ω l px 1 , . . ., x s q ‰ H and pick any y P Ω l px 1 , . . ., x s q.Let p 1 1 , . . ., p 1 s P Z ě0 be the integers such that 2 for j " 1, . . ., s.
On the other hand, we have for any 1 ď j ď s.It follows that 2 ´pj " 2 ´ki e ´j´1 d t ě 2 ´ki e ´d´2 d t for all 1 ď i ď j, so for C as above, holds for sufficiently large t.Combining with (5.18), we have p 1 j ď p j `Op1q for all 1 ď j ď s.
For each i, the set of y P K satisfying (5.11) is a 2 ´li e ´d´1 d t }π 1 x i } ´1thickened hyperplane in R d´1 which is perpendicular to π 1 x i .Therefore, vol R d´1 pΩ l px 1 , . . ., x s qq is the volume of the intersection of such s-number of hyperplanes and the compact set K. The intersection has width ! 2 ´li e ´d´1 d t }π 1 x i } ´1 along the direction of π 1 x i for 1 ď i ď s.It follows that the volume of the intersection is bounded above by Proof of Proposition 5.1 under (B1).By Lemma 5.2, (5.20) vol R d´1 ´!y P K : hpyq P Ξ s l )¯ď vol R d´1 ´!y P K : |β i pyq| ă 2 ´li pi " 1, . . ., sq )¯.
The remaining task is thus to estimate the measure of the set of y P K such that hpyq P Ξ s l and γpyq P Γ s,r l .Recall that Z d " Λ t M 0 .From now on we fix r ą 0 as in (5.2) and no longer record the implicit dependence of constants on this parameter.For k P Z s ěK 0 , l P Z s ě0 , and p P Z s ě0 , we denote by Λ l k ppq the set of elements in px 1 , . . ., x s q P Λ k ppq satisfying (5.24) for all 1 ď i ď s.As we counted the number of lattice points of Λ k ppq in Lemma 5.3, here we count the number of lattice points of Λ l k ppq as follows.Lemma 5.5.
For any k P Z s ěK 0 , l P Z s ě0 , and p P Z s ě0 , 2 ω s,ξ pk j ,p j ,l j q .
Moreover, if there exists 1 ď j ď s such that p j ě jk j `K or k j ă log 2 ζpξ,2 l´1 q 4}M 0 } , then #Λ k ppq " 0. Here, K is a sufficiently large constant depending on the choice of lattice Λ.
Proof.For px 1 , . . ., x s q P Λ l k ppq Ď Λ k ppq, recall that (5.25) for j " 1, . . ., s.Given x 1 , . . ., x j´1 , in the proof of Lemma 5.3 we already showed that the possible number of x j P Λ k j is ! 2 ωspk j ,p j q " 2 sk j ´pj if p j ą k j .Hence, it is enough to show that this bound can be improved under the assumption p j ď k j .
We first consider the case p j ď k j ´log 2 ζpξ,2 l´1 q 4}M 0 } .In this case, if x j P Λ k j satisfies (5.25), then x j must be ! 2 k j ´pj -close to the subspace V spanned by x 1 , . . ., x j´1 .Hence, the region of x j satisfying (5.25) has width 2 k j ´pj along the directions perpendicular to V , and width 2 k j along the directions of V .This region can be covered with at most ! 2 dk j ´pd`1´jqp j ζpξ, δ ´1 d,r 2 l j ´1q ´d ! 2 dk j ´pd`1´jqp j ζpξ, 2 l j ´1q ´d cubes with sidelength }M 0 } ´1ζ pξ, δ ´1 d,r 2 l j ´1q.We claim that there is at most one point of Λ k j satisfying (5.24) in each cube with sidelength }M 0 } ´1ζ pξ, δ ´1 d,r 2 l j ´1q.To see this, suppose that there are two distinct points x, x 1 P Λ k j with distance ă }M 0 } ´1ζ pξ, δ ´1 d,r 2 l j ´1q satisfying (5.24).Then we have |ξ ¨px t M 0 ´x1 t M 0 q| Z ď δ d,r 2 ´lj `1 and However, by definition (1.2) there is no m P Z d z t0u with |m| ă ζpξ, δ ´1 d,r 2 l j ´1q and |ξ ¨m| Z ď δ d,r 2 ´lj `1.Hence the claim is proved.It follows from the claim that the number of possible x j P Λ k j satisfying (5.24) and (5.25) is at most ! 2 dk j ´pd`1´jqp j ´ζpξ,2 l j ´1q 4}M 0 } ¯´d " 2 ω s,ξ pk j ,p j ,l j q .
We have shown that for fixed x 1 , . . ., x j´1 , the number of possible x j P Λ k j satisfying (5.24) and (5.25) is ! 2 ω s,ξ pk j ,p j ,l j q .Hence we obtain the desired estimate.
As we have shown in Lemma 5.3, #Λ k ppq " 0 if there exists 1 ď j ď s such that p j ě jk j `K.On the other hand, if there exists 1 ď j ď s such that k j ă log 2 ζpξ,2 l´1 q 4}M 0 } , then we have |x j t M 0 | ď }M 0 }2 k j `2 ă ζpξ, 2 l´1 q since x j P Λ k j .It follows that |ξ ¨px j t M 0 q| ą 2 ´l`1 by definition of ζpξ, T q.In other words, there is no x j P Λ k j satisfying (5.24), hence #Λ k ppq " 0.
We now discuss the case that d " 2 and ξ is p0, η ´2, 2q-vaguely Diophantine.If d " 2, then s " 1, and in view of the definition (5.9) the set Λ k ppq is the empty set unless p " 0. Hence, the double sum in the last line of (5.26) is written and bounded above by ! e t 2 2 ´2l ζpξ, 2 l´1 q ´2.Plugging this in (5.26), we get vol R ´!y P K : Note that here we gained additional decay of ζpξ, 2 l´1 q ´1 in comparison to (5.29 Let us denote by S d´1 `and S d´1 ´the upper hemisphere and the lower hemisphere, respectively, i.e.
By the construction of the map in (3.4), k is smooth and its differential is non-singular and bounded on S d´1 `. pN c,T pσ ˚, υq`1q Repz 1 q`¨¨¨`Repzmq λpdυq where σ ˚" max 1ďjďm σ j .We now split the integral over the upper and lower hemispheres.The integral over the upper hemisphere S d´1 `vanishes in the limit in view of Proposition 5.7 and the upper bounds (3.5) and (5.3).The analogous statement for S d´1 ´follows by symmetry, since the quantity N c,T pσ ˚, υq for a given ξ has the same value as N c,T pσ ˚, ´υq for ´ξ, with everything else (including M 0 ) being fixed.This completes the proof of Theorem 2.2. 7. Proof of Corollary 2.3 and Corollary 2.4 7.1.Proof of Corollary 2.3.We will need the following variant of Siegel's formula.
Here (and below) we view g " ιpΓ 1 gq P G 1 as the representative of the coset Γ 1 g in the fundamental domain F 1 .
For the proof of Corollary 2.3, we first prove for σ " pσ 1 , σ 2 q.To deduce (7.1), note that Recall that for σ ą 0, the area of C c pσq is precisely σ.Applying Proposition 7.1, the off-diagonal part of the right-hand side is equal to ż R d ˆRd χ Ccpσ 1 q pxqχ Ccpσ 2 q pyq dx dy " σ 1 σ 2 .
For the diagonal part, let C " C c pσ 1 qXC c pσ 2 q and note that C " C c pmin tσ 1 , σ 2 uq.
Then the diagonal part is evaluated as follows: ż This completes the proof of (7.1).
For the proof of Corollary 2.3, observe that for z 1 " z 2 " 1 one of the hypotheses of Theorem 2.2 holds under the assumption of Corollary 2.3.Combining (7.1) with the property (b) above, Corollary 2.3 then follows from Theorem 2.2.7.2.Proof of Corollary 2.4.In this section we show that Corollary 2.3 implies Corollary 2.4.Throughout this section we assume that the statement of Corollary 2.3 holds.Lemma 7.2.Let h P CpS d´1 q and σ 1 , σ 2 ą 0. Then hpαqdα.
On the other hand, the diagonal part j 1 " j 2 of the left-hand side of (7.2) is Since h is continuous and dpα j , αq !σ 1 ,σ 2 N ´1 d´1 for any 1 ď j ď N , for any ǫ ą 0 there exists N 0 such that for all N ě N 0 we have |hpα j q ´hpαq| ă ǫ for any α P S d´1 and 1 ď j ď N .It follows that the integral (7.4) is approximated by hpαqdα, since the α j 's are uniformly distributed over S d´1 .Therefore, the second summand of (7.3) is the off-diagonal contribution appearing in (7.2) as desired.

Appendix A. Brjuno type condition
Following [LDG19] (cf.also [BF19]) we say that ξ P R d is a s-Brjuno vector if The classical Brjuno condition corresponds to s " 1.
In this section, we prove that for s ą ρ`1 ν , every s-Brjuno vector is pρ, 0, νq-vaguely Diophantine.Given ξ P R d let us define φ : N Ñ R ą0 by φpN q :" max Then the definition of ζpξ, T q can be written in terms of φpN q as follows: ζpξ, T q " min !N P N : e ´φpN q ď T ´1) " min tN P N : φpN q ě log T u .
Suppose that ξ is s-Brjuno type for some s ą ρ`1 ν .Then we have ř n 2 ´n s φp2 n q ă 8, hence φptq ď t 1 s log 2 for sufficiently large t.It follows that ζpξ, 2 l´1 q " min tN P N : φpN q ě pl ´1q log 2u ě pl ´1q s |ξ ¨m| Z ą C κ |m| ´κ for any m P Z d z t0u, where | ¨| denotes the supremum norm of R d , and | ¨|Z denotes the supremum distance from 0 P T d .It is known that Lebesguealmost all ξ P R d are of type κ for any κ ą d.We will in fact only require a milder Diophantine condition.Define the function ζ : R d ˆRą0 Ñ N by ( ) η ă d `1 and ξ is p0, η ´2, 2q-vaguely Diophantine if d " 2 and pd ´1, η ´d, 1q-vaguely Diophantine if d ě 3. Then
Let g " p1, bqpM, 0q with M P G and b P R d .For each 1 ď i ď d, we have v i pgq " }e i vpM q} " }e i M } and v i `gph, 0q ˘" }e i vpM hq} " }e i M h}, hence there exists C " CpCq ą 1 such that C ´1v i `gph, 0q ˘ď v i pgq ď Cv i `gph, 0q ˘for any h P C. We also have b i pgq " b i `gph, 0q ˘for all i since b i is invariant under SLpd, Rq-action.Let g 1 " gph, 0q.It follows that Lemma 5.6.For any compact set C Ă G, there exists C " CpCq ą 1 such that F R,η,r `gph, 0q ˘ď C pd´1qη F C ´pd´1q R,η,Cr pgq for any h P C and g P G 1 .Proof.