Equidistribution of primitive vectors, and the shortest solutions to their GCD equations
EEquidistribution of primitive vectors, and the shortest solutionsto their GCD equations
Tal Horesh ∗ Yakov Karasik † Abstract
We prove effective joint equidistribution of several natural parameters associated to primitivevectors in Z n , as the norm of these vectors tends to infinity. These parameters include the direction,the orthogonal lattice, and the length of the shortest solution to the associated gcd equation. Weshow that the first two parameters equidistribute w.r.t. the Haar measure on the correspondingspaces, which are the unit sphere and the space of unimodular rank n − R n respectively.The main novelty is the equidistribution of the shortest solutions to the gcd equations: we showthat, when normalized by the covering radius of the orthogonal lattice, the lengths of these solutionsequidistribute in the interval [0 ,
1] w.r.t. a measure that is Lebesgue only when n = 2, and non-Lebesgue otherwise. These equidistribution results are deduced from effectively counting latticepoints in domains which are defined w.r.t. a generalization of the Iwasawa decomposition in simplealgebraic Lie groups, where we apply a method due to A. Gorodnik and A. Nevo. Contents
I From Z n to SL n ( Z ) SL n ( R ) g ∈ SL n ( R ), and their interpretation . . . . . . . . . . . 8 m ( Z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Relation between fundamental domains and quotient spaces . . . . . . . . . . . . . 13 n ( Z ) . . . . . . . . . 134.2 A fundamental domain for Γ that captures the shortest solutions . . . . . . . . . . 14 II Counting lattice points 23 ∗ IST Austria, [email protected] . † Department of Mathematics and Computer Science, Justus-Liebig-Universität Gießen, Germany, [email protected] . a r X i v : . [ m a t h . N T ] F e b Counting lattice points in well rounded families of sets inside Lie groups 23 A f for the Iwasawa roundomorphism . . . . . . . . . . . . . . . . . . . 32
11 The base sets 3312 The family Y αr ( F n − ) is BLC 3413 Concluding the proofs of the theorems 40References 41 An integral vector v = ( a , . . . , a n ) is called primitive if gcd ( a , . . . , a n ) = 1. Equidistribution problemsconcerning primitive vectors first arose under the umbrella of Linnik type problems [Lin68, EH99,Duk03, Duk07, EMV13], a unifying name for questions that concern the distribution of the projectionsof integral vectors to the unit sphere. These projections can also be thought of as directions ofprimitive vectors, which we denote by ˆ v := v/ k v k . Another equidistribution problem of primitivevectors concerns their orthogonal lattices Λ v := Z n ∩ v ⊥ , where v is a primitive vector, and v ⊥ isits orthogonal hyperplane. Note that one can achieve a one-to-one correspondence between primitivevectors and their orthogonal lattices by either identifying v with − v , or by choosing an orientation onthe lattices Λ v ; we opt for the latter. With this one-to-one correspondence in mind, we associate toeach primitive vector the shape of the lattice Λ v , which is the equivalence class of rank n − R n that can be obtained from Λ v by an orientation preserving linear transformation, i.e. by a rotationand multiplication by a positive scalar. The equidistribution of shapes of Λ v , denoted shape (Λ v ), inthe finite volume space X n − := SO n − ( R ) \ SL n − ( R ) / SL n − ( Z )has been considered in [Mar10, Sch98]; the joint equidistribution of shape (Λ v ), along with the direc-tions of v , denoted ˆ v , has been studied in [AES16b, AES16a, EMSS16, ERW17].Another equidistribution question for primitive vectors has been suggested by Risager and Rudnickin [RR09], and it concerns the normalized shortest solutions to gcd equations: given a primitive v = ( a , . . . , a n ), the gcd equation of v is the Diophantine equation a x + · · · + a n x n = 1 , (1.1)whose set of solutions is the grid w +Λ v , with w being any solution to (1.1). Let w v denote the shortestsolution to the equation (1.1) w.r.t. the L norm. The length k w v k is unbounded as k v k → ∞ , soin order to formulate an equidistribution question for k w v k , it should be normalized to a boundedquantity. Risager and Rudnick (see also [Tru13, HN16]) have considered the case of n = 2, and showedthat the quotients k w v k / k v k uniformly distribute in the interval [0 , ] as k v k → ∞ . This raises thequestion of what would be the analogous phenomenon in higher dimensions. It turns out that onecan not expect equidistribution of k w v k / k v k when n ≥
3, since these quotients tend to zero on afull-density subset of the set of all n –primitive vectors, denoted Z n prim .2igure 1: The density of ν Theorem A.
There exists a subset A of Z n prim with lim R →∞ A ∩ B R ) Z n prim ∩ B R ) = 1 , where B R = { v ∈ R n : k v k ≤ R } , such that for every sequence { v m } ⊂ A , the quotients k w v m k / k v m k tend to zero as m → ∞ . Indeed, the above theorem (as well as Corollary 1.2 below) suggests that in dimension greater than2, the “correct” normalization of the shortest solution is not by the norm of v . Hence, approachingRisager and Rudnick’s problem in higher dimensions consists in fact of three questions:(i) What is the correct normalization of the shortest solutions in dimension n ≥ k w v k is by the covering radius ρ v of the lattice Λ v (the covering radius of a latticeis the radius of a bounding sphere for its Dirichlet domain), and construct a measure ν n with respectto which the quotients k w v k /ρ v equidistribute in the interval [0 , ν n on [0 ,
1] is non-uniform (see Figure 1 for the density function of ν ), except for the case of n = 2: there, the measure ν is Lebesgue and the covering radius is ρ v = k v k /
2, hence we recover theresult of Risager and Rudnick.In fact we do more, and show that the equidistribution of k w v k /ρ v occurs jointly with the uni-form distribution of ˆ v in S n − . We also obtain the previously known joint equidistribution of shapesshape (Λ v ) and directions ˆ v from the equidistribution of another parameter of Λ v , that encodes infor-mation of both shape (Λ v ) and ˆ v . Consider the space L n − ,n := SL n ( R ) / (cid:18)(cid:20) SL n − ( Z ) R n − × n (cid:21) × (cid:26)(cid:20) α − n − I n − n × × n α (cid:21) : α > (cid:27)(cid:19) , which is the space of homothety classes of ( n − R n . We identify this space with thespace of unimodular (i.e. covolume one) ( n − R n , L n − ,n ’ SO n ( R ) (cid:20) P n −
00 1 (cid:21) / (cid:20) SL n − ( Z ) 00 1 (cid:21) , (1.2)3here P n − < SL n − ( R ) is the group of upper triangular matrices with positive diagonal entries. Theidentification is by associating to each equivalence class [Λ] the unique representative of covolume one,which we also denote by [Λ]. The space L n − ,n is canonically projected to X n − and to S n − , bymodding out from the left by SO n ( R ) or by SO n − ( R ) (cid:2) P n −
00 1 (cid:3) respectively, and the projections of[Λ v ] to X n − and S n − are exactly shape (Λ v ) and ˆ v .From the equidistribution of [Λ v ] in L n − ,n , we will also conclude the joint equidistribution of thedirections ˆ v together with the projections of Λ v to the following space: U n − := SL n − ( R ) / SL n − ( Z ) , which is the space of unimodular lattices of rank n −
1. We denote these projections by (cid:74) Λ v (cid:75) (thisprojection is in fact not canonical, and depends on a choice of coordinates that will be made in Section2.2).The equidistribution in the spaces X n − , U n − , L n − ,n and S n − is a uniform distribution, namelyw.r.t. a finite uniform invariant measure, which is unique up to a choice of normalization. We denotethese measures by µ X n − , µ U n − , µ L n − ,n and µ S n − , and expand about them below, after the statementof our main result. The measure µ S n − , for example, is the Lebesgue measure on the sphere.The equidistribution of the quotients k w v k /ρ v inside [0 ,
1] is, as we have already mentioned, notuniform. The proportion of primitive vectors v for which the quotients k w v k /ρ v fall within the interval[0 , α ] with 0 ≤ α ≤ α : X n − → R + which is defined by associating to every z ∈ X n − the following quantity. Recall that z ∈ X n − is a unimodular lattice in R n − up to rotation.Recall also that the Dirichlet domain of a lattice is symmetric around the origin, and so the Lebesguevolume of Dir( z ) ∩ B , where Dir( z ) is the Dirichlet domain of any lattice in the class z and B is a ballcentered at the origin, is independent of the choice of a representative from z . LetL α ( z ) = Leb(Dir( z ) ∩ B αρ ( z ) ) , where Leb is the Lebesgue measure, ρ ( z ) is the covering radius of (any representative from) z , and B αρ ( z ) is an origin centered ball in R n − with radius αρ ( z ).Finally, we derive our equidistribution results by counting primitive vectors v (resp. primitive( n − v ) whose projections to the aforementioned spaces lie in subsets that have controlledboundary : this is a rather soft condition on the boundary of subsets of orbifolds that is definedexplicitly in Section 3, and is met, e.g., when the boundary of the set is contained in a finite unionof C submanifolds of strictly lower dimension than the one of the orbifold. We refer to a set withcontrolled boundary as a boundary controllable set , or a BCS. Our main result is the following. Theorem B.
Assume that Φ ⊆ S n − , E ⊆ X n − e E ⊆ U n − and Ψ ⊂ L n − ,n are BCS’s.1. The number of v ∈ Z n prim with k v k ≤ e T , ˆ v ∈ Φ , shape (Λ v ) ∈ E and k w v k /ρ v ∈ [0 , α ] is µ S n − (Φ) · R E L α ( z ) dµ X n − ( z ) n Q ni =2 ζ ( i ) · Q n − i =1 Leb( S i ) ι ( n − · e nT + error termwhere ι ( m ) = [SO m ( R ) : Z (SO m ( R ))] = ( if m is even if m is odd . (1.3)
2. The number of v ∈ Z n prim with k v k ≤ e T , ˆ v ∈ Φ , (cid:74) Λ v (cid:75) ∈ e E and k w v k /ρ v ∈ [0 , α ] is µ S n − (Φ) · Re E L α ( π U→X (˜ z )) dµ U n − (˜ z ) n Q ni =2 ζ ( i ) · e nT + error term , where π U→X is the projection from U n − to X n − . . The number of v ∈ Z n prim with k v k ≤ e T , [Λ v ] ∈ Ψ and k w v k /ρ v ∈ [0 , α ] is R Ψ L α ( π L→X ( y )) dµ L n − ,n ( y ) n Q ni =2 ζ ( i ) · e nT + error term , where π L→X is the projection from L n − ,n to X n − .The error term is O (cid:15) ( e nT (1 − τ n + (cid:15) ) ) with τ n = d ( n − / e / n for every (cid:15) > when E (resp. Ψ , e E ) isbounded, and O (cid:15) ( e nT (1 − η n τ n + (cid:15) ) ) with η n = n / (2 n − n − n + 4) when it is not. The lattice Λ v has covolume k v k and it is primitive, where a lattice Λ in Z n is said to be primitive ifit is of the form V ∩ Z n , with V being a linear subspace of R n of dimension rank (Λ). Then, Theorem Bcan also be read as a counting result for primitive ( n − ≤ α ≤
1, let ν n ([0 , α ]) = Z z ∈X n − L α ( z ) dµ X n − ( z ) . The following is now straightforward from part (1) of Theorem B:
Corollary 1.1.
For primitive vectors v ∈ Z n with n ≥ , the normalized shortest solutions k w v k /ρ v and the directions ˆ v jointly equidistribute as k v k → ∞ : the quotients k w v k /ρ v inside [0 , w.r.t. ν n ,and the directions ˆ v inside the unit sphere w.r.t. the Lebesgue measure. As we have already mentioned, for the case of n = 2 the above corollary recovers the result ofRisager and Rudnick for uniform distribution of k w v k / k v k in the interval [0 , / (cid:0) n (cid:1) embeddings of R into R n that are of the form( x, y ) (0 , . . . , , x, , . . . , , y, , . . . (cid:0) n (cid:1) sequences of primitive vectors v ∈ Z n for which the quotients k w v k / k v k uniformlydistribute in the interval [0 , /
2] as k v k → ∞ . Combining this with Theorem A, we conclude: Corollary 1.2.
For primitive vectors v ∈ Z n with n ≥ , there is no Borel measure on R w.r.t. whichthe quotients k w v k / k v k equidistribute as k v k → ∞ . The measures µ X n − , µ U n − , µ L n − ,n and µ S n − . The measure µ S n − is the Lebesgue measure onthe sphere. The measures µ X n − and µ U n − are the unique Radon invariant measures arriving from aHaar measure on SL n − ( R ) that are normalized as follows: the µ U n − volume of U n − is n − Y i =2 ζ ( i ) , and the µ X n − -volume of X n − is ι ( n − n − Y i =2 ζ ( i ) / ( n − Y i =1 Leb( S i )) , where ι : N → { , } was defined in (1.3). The justification for the volume of U n − is the computationin [Gar14] along our choice of Haar measure on SL n ( R ) that is explained in Subsection 2.1. Thischoice determines the volumes of X n − , as shown in Lemma 3.9. On L n − ,n , however there is noinvariant measure induced from SL n ( R ), and instead we view this space as the quotient in (1.2), wherea submanifold of SL n ( R ) quotiented by a discrete group. This submanifold supports a transitive actionof the product group SO n ( R ) × (cid:2) P n −
00 1 (cid:3) , and µ L n − ,n is the unique Radon measure that is invariantunder this action and satisfies that the µ L n − ,n -volume of L n − ,n is the product of volumes of S n − and U n − . 5 omparison with previous work. Let us comment on related work that preceded the theoremabove. As already mentioned, equidistribution of the k w v k / k v k was known for n = 2; it was firstproved in [RR09], and effective versions were later established in [Tru13] and [HN16], where the errorterm coincides with the one of Theorem B for n = 2. The equidistribution (in a non-effective manner)of shapes of primitive lattices of any rank was established in [Sch98]; the case of rank n − U n − and L n − ,n (as apposed to just X n − ), and most importantly, the equidistribution related tothe gcd problem. Another significant addition is the fact that we allow the projections to the relevantspaces ( E , e E , Ψ) to be unbounded; to this end, it is critical that the counting includes an error term,since it could be compromised to allow unboundedness. Our method can be used to consider the case ofgeneral co-dimension as well, which we will do in a forthcoming paper. Effective counting of primitivelattices was done in [Sch68],[Sch15], but the subsets E in the shape space were not general enoughto deduce equidistribution. Joint equidistribution of shapes and directions has been studied, e.g. in[AES16b, AES16a, EMSS16], in the case where the primitive vectors v are restricted to a large sphere k v k = e T , as apposed to a large ball k v k ≤ e T , the latter being the case considered in Theorem B.The sphere case is of course much more delicate, and this is the reason why almost all existing resultsdo not include an error term. The key to proving Theorem B is counting lattice points in the groupSL n ( R ) w.r.t. the Iwasawa coordinates; in the context of counting points of discrete subgroups insidesimple Lie groups w.r.t. a decomposition of the group, we mention [Goo83, GN12, GOS10, MMO14]. Outline of the paper.
The proof of Theorems A and B consist of two main ideas, and the paperis divided accordingly:1.
A reduction to a problem of counting lattice points in the group SL n ( R ) (Part I), whichis done by finding “isomorphic” copies of the spaces X n − , U n − , L n − ,n , S n − inside SL n ( R )(Section 3) and establishing a correspondence between primitive vectors v (resp. primitive latticesΛ v ) and integral matrices in SL n ( R ) (Section 4), such that the projections of the primitivelattices to the spaces X n − etc. will correspond to the projections of the integral matrices in theirisomorphic copies. This converts Theorem B into a counting lattice points problem in SL n ( R )(Section 5). A key role in this translation is played by a refinement of the Iwasawa coordinates ofSL n ( R ), introduced in Section 2. In section 6 we simplify the counting problem by reducing tocounting in a family of compact subsets of SL n ( R ), by providing a rather direct estimate for thenumber of lattice up to a given covolume that lie far up the cusp in the space of ( n − n ( R ) that is required in order to complete the proof Theorem B, and then use itto prove Theorems A and B.2. Solving the counting problems (Part II ). This part is devoted to proving the aforementionedProposition 7.1. The main ingredient is a method due to A. Gorodnik and A. Nevo [GN12], whichconcerns counting lattice points in increasing families {B T } T > inside non-compact algebraicsimple Lie groups. In Section 8 we describe this method, and sketch a plan for completing theproof of Proposition 7.1 according to it. In Sections 9, 10, 11, 12 we follow that plan, and theproofs are concluded in Section 13. Notations for inequalities.
We will use the following conventions for inequalities. If a = ( a , . . . , a n )and b = ( b , . . . , b n ) are two n -tuples of real numbers, we denote a ≤ b if a i ≤ b i for every i = 1 , . . . , n .If f and g are two non-negative functions then we denote f (cid:28) g if there exists a positive constant C and some t such that for t < t one has f ( t ) ≤ Cg ( t ). We denote f (cid:16) g if g (cid:28) f (cid:28) g . Acknowledgement.
This work was done when both authors were at IHES (Institut des Hautes ÉtudesScientifiques, France), and we are grateful for the opportunity to work there, and for the outstanding In [ERW17] an error term is established for dimensions n = 4 , ν , and to Ami Paz for his help with preparing the figures. We would also like to thankBarak Weiss and Amos Nevo for helpful discussions in early stages of the project, and to MichealBersudsky for referring us to Schmidt’s work on effective counting of primitive lattices. Part I
From Z n to SL n ( Z ) SL n ( R ) Set G := SL n ( R ) and let K be SO n ( R ), A the diagonal subgroup in G , and N the subgroup of upperunipotent matrices. Then, G = KAN is the Iwasawa decomposition of G . Consider yet anothersubgroup of G , G := SL n − ( R ) 0...0 · · · , which is clearly an isomorphic copy of SL n − ( R ) inside G . Write G = K A N for the Iwasawadecomposition of G , i.e. K := K ∩ G = SO n − ( R ) 000 0 1 ,A := A ∩ G = diag ( α , . . . , α n − ,
1) with α · · · α n − = 1 ,N := N ∩ G = upper unipotentof order n − . The crux of the RI decomposition is that it completes the Iwasawa decomposition of G to the Iwa-sawa decomposition of G . For this we define K , A , N that complete K , A , N to K , A and N respectively. Define N := (cid:20) I n − R n − (cid:21) , A := a − n − I n −
000 0 a and note that N = N N , A = A A , and that A is a one-parameter subgroup of A which commuteswith G . Fix a transversal K of the diffeomorphism K/K → S n − with the following property: Condition 2.1.
If Φ ⊆ S n − and Φ ⊆ K are BCS, then so does Φ K Φ ⊆ K , where K Φ is theinverse image of Φ in K .The existence of such a transversal K is proved in Lemma 3.4. Let P := A N and Q := KP ;note that Q is not a group, but that it is a smooth manifold that is diffeomorphic to the group K × P .The RI decomposition is given by G = K G A N = K K A A N N , and we also have G = QA N . 7 arameterizations of the RI components. Clearly the groups
A, A , A and N, N , N are pa-rameterized by the Euclidean spaces of the corresponding dimensions. For t ∈ R , s = ( s , . . . , s n − ) ∈ R n − and x ∈ R n − , we let a t := diag( e tn − I n − , e − t ), a s := diag( e − s , e s − s , . . . , e sn − ,
1) and n x = (cid:2) I n − x (cid:3) . Similarly, since K parameterizes the unit sphere S n − , we let k u denote the elementin K corresponding to a unit vector u ∈ S n − . In addition to the above, we will show in Section 3.3that certain subsets of Q , G and P parameterize the spaces L n − ,n , U n − and X n − . When an RIcomponent S (or a subset of it) parameterizes a space X , and B ⊂ X is a subset, we let S B denote theimage of B under the parameterization. For example, if D ⊂ R n − , then N denotes its image in N ,namely the set of n x where x ∈ D . Measures on the RI components.
For every S ⊂ G appearing as a component in the Iwasawa orRefined Iwasawa decompositions of G , we let µ S denote a measure on S as follows: µ K , µ N are Haarmeasures, and so do µ K , µ N , µ P , µ G , µ G and µ N . The measures µ N , µ N and µ N are Lebesgue;as N = N (cid:110) N and all three groups are unimodular, µ N = µ N × µ N . Since K parameterizes S n − ,we can endow it with a measure µ K that is the pullback of the Lebesgue measure on the sphere. Weassume that the Haar measures µ K and µ K are normalized such that µ K = µ K × µ K . Then, bychoosing the measure of K to be Q n − i =1 Leb( S i ), we have that the measure of K is Q n − i =1 Leb( S i ). Themeasures µ A , µ A , µ A are Radon measures such that µ A = e nt dt, µ A = n − Y i =1 e − s i ds i as we compute in Example 9.4, and µ A = µ A × µ A by Remark 9.3. Note that these measures arenon-Haar. Since Q is diffeomorphic to the group K × P , we endow it with the Haar measure on thisgroup: µ Q = µ K × µ P . Since µ K = µ K × µ K , we also have that also µ Q = µ K × µ G . All in all,the Haar measure on G , which can be written in Iwasawa coordinates as µ G = µ K × µ A × µ N , (e.g.[Kna02, Prop. 8.43]), can be also decomposed according to the Refined Iwasawa coordinates: µ G = µ K × µ G × µ A × µ N = µ Q × µ A × µ N = µ K × µ K × µ A × µ N × µ A × µ N . (2.1)Where it should be clear from the context, we will occasionally denote µ instead of µ G g ∈ SL n ( R ) , and their interpretation The following proposition reveals the role of the RI decomposition of SL n ( R ) in studying the param-eters k v k , ˆ v , [Λ v ], (cid:74) Λ v (cid:75) and shape (Λ v ) of a vector v . Let us observe that the projection [Λ v ] to U n − is now well defined, following the choice of a transversal K , which determines a unique way to rotateany hyperplane in R n to span { e , . . . , e n − } ∼ = R n − .It will be convenient to set the following notations: for any invertible matrix g , let Λ g denote thelattice spanned by the columns of g , and Λ jg denote the lattice spanned by the first j columns of g .Also, for 0 = v ∈ R n define: G v = { g = [ v | · · · | v n ] ∈ SL n ( R ) : v ∧ · · · ∧ v n − = v } . (2.2) Proposition 2.2.
Let g = [ v | · · · | v n − | v n ] ∈ SL n ( R ) and write g = kan = qa t n x with q = k u g and g = k a s n . Let w = v n , and p = a s n = [ z
00 1 ] . If g ∈ G v , then the RI components of g are asfollows: ( i ) u = ˆ v ( ii ) e t = k v k ( iii ) e − si + itn − = covol(Λ ig )( iv ) Λ q ∈ (cid:2) Λ n − g (cid:3) ( v ) Λ g ∈ (cid:113) Λ n − g (cid:121) ( vi ) Λ z ∈ shape (cid:0) Λ n − g (cid:1) nd (vii) x = ( x , . . . , x n − ) is such that w v ⊥ = P n − i =1 x i v i . The proof of this proposition requires two short lemmas regarding the elements of G v . Lemma 2.3.
For g = [ v | · · · | v n − | v n ] ∈ SL n ( R ) , the following are equivalent:1. g ∈ G v .2. The columns { v , . . . v n − } form a basis of co-volume k v k to v ⊥ such that { v , . . . , v n − , v } is apositively oriented basis w.r.t. the standard basis of R n .3. h v n , v i = 1 , and h v i , v i = 0 for i = 1 , . . . , n − .Proof. (1) ⇐⇒ (2) by definition. The direction (1) = ⇒ (3) follows from1 = det( g ) = h ( v ∧ · · · ∧ v n − ) , v n i = h v, v n i . Conversely, (3) implies v = α · v ∧ · · · ∧ v n − for some α = 0, and that h α − v, v n i = α − . But since(as above) 1 = h α − v, v n i , this forces α = 1. Lemma 2.4. If g ∈ G v , the last column of g is w and w v ⊥ is the orthogonal projection of w on thehyperplane v ⊥ , then w = w v ⊥ + k v k − v. Proof.
Write w = w v ⊥ + αv . By part (3) of Lemma 2.3, 1 = h w, v i = h w v ⊥ + αv, v i = h αv, v i , hence α = k v k . proof of Proposition 2.2. Write k = [ φ | · · · | φ n − | φ n ]. Since the columns of k are the orthonormal basisobtained by the Gram-Schmidt algorithm on the columns of g , we have that span { φ , . . . , φ n − } =span { v , . . . , v n − } = v ⊥ . By orthonormality and part (2) of Lemma 2.3, φ n = ˆ v = v/ k v k . Since k and k have the same last column, then ˆ v is also the last column of k , i.e. k = k ˆ v , which proves (i).It is clear that if a = diag( a , . . . , a n ), then Q i a j = k v ∧ · · · ∧ v i − ∧ v i k = covol(Λ ig ). Since g ∈ G v , and a has determinant 1, we get that the last diagonal entry of a (hence also of a ) is 1 / k v k .This proves (ii) and (iii). In particular a = diag( k v k / ( n − , . . . , k v k / ( n − , k v k − ).Write g ( n ) − ( a ) − = k g = q ; right multiplication by an element of N does not change the first n − g , and right multiplication by ( a ) − multiplies these columns by k v k − / ( n − . Thisproves (iv), and (v), (vi) immediately follow.Write w as the sum of its projections to the orthogonal spaces R v and v ⊥ : w = w v + w v ⊥ . Observethat g ( n ) − = kp a . The last column of kp a is φ n / k v k , where from the calculation on k we knowthat φ n = ˆ v ; by Lemma 2.4, we get that the last column of kp a is w v . The last column of g ( n ) − is w − P n − i =1 x i v i , so we conclude that w − P n − i =1 x i v i = w v = w − w v ⊥ , which implies (vii). In this section we find “isomorphic” copies of the spaces X n − , U n − , L n − ,n , S n − inside SL n ( R ).The property we are after in these isomorphic copies, is that the images of sets satisfying a boundarycondition, will also satisfy it. This boundary condition is the following: Definition 3.1.
A subset B of an orbifold M will be called boundary controllable set , or a BCS, if forevery x ∈ M there is an open neighborhood U x of x such that U x ∩ ∂B is contained in a finite union ofembedded C submanifolds of M , whose dimension is strictly smaller than dim M . In particular, B is a BCS if its (topological) boundary consists of finitely many subsets of embedded C submanifolds.9he goal of this section is to prove the following: Proposition 3.2.
There exist full sets of representatives in SL n ( R ) : • K ⊂ K parameterizing S n − ∼ = K \ K • (cid:93) F n − ⊂ G parameterizing U n − = G /G ( Z )• F n − ⊂ P parameterizing X n − ∼ = K \ G /G ( Z )• K (cid:93) F n − ⊂ Q parameterizing L n − ,n ∼ = Q/G ( Z ) that are BCS’s and with the properties that (i) a BCS is parameterized by a BCS and vice versa; for K , a product of BCS’s in K and K is a BCS in K . (ii) The pullbacks of the invariant measureson the parameterized spaces to their set of representatives coincide with the measures that the sets ofrepresentatives inherent from their ambient manifolds: for all cases but K it is the restriction of themeasure µ S on the ambient manifold, and for K it is the measure µ K defined in Subsection 2.1. A full proof of Proposition 3.2 can be found in [HK20, Prop. 8.1] Here, we will only prove it fullyfor K and S n − (this case is easier since K /K is compact), and for the remaining spaces we willsettle for constructing fundamental domains that are BCS, with the property that the Haar measurerestricted to them coincides with the unique (up to a scalar) invariant measure on the quotient (whichis the space parameterized by the fundamental domain in question). We start by constructing sets ofrepresentatives for the sphere (Subsection 3.1), then for the spaces of lattices (Subsection 3.2), and weconclude with a partial proof of Proposition 3.2 in Subsection 3.3. In order to construct a set of representatives K for S n − , we observe the following. Fact 3.3.
Since ∂ ( A ∪ B ) , ∂ ( A ∩ B ) ⊆ ∂A ∪ ∂B , the union, intersection and subtraction of BCSs arein themselves BCS’s. Also, a finite product of BCSs is a BCS in the product of the ambient manifolds,and a diffeomorphic image of a BCS is a BCS. Now the existence of a transversal K for S n − is a consequence of the lemma below. Lemma 3.4.
Let K be a Lie group. Assume that K < K a closed subgroup such that the quotientspace K/K is compact. There exists subset K ⊆ K which is a BCS such that:1. π | K : K → K/K is a bijection;2. if Φ ⊆ K/K and Φ ⊆ K are BCS, then the product π | − K (Φ) | {z } ⊂ K · Φ |{z} ⊂ K in K is also a BCS.Proof of Lemma 3.4. Since π : K → K/K is a principal K fiber bundle, there exists an open covering { U α } of K/K with K -equivariant diffeomorphisms τ α : π − ( U α ) → U α × K , where τ α ( x ) = ( π ( x ) , ∗ ). We can assume that there is a BCS covering { W α } of K/K such that W α ⊆ U α (e.g., by reducing to open balls contained in U α ); by compactness, we may also assume thatthis covering is finite. Finally, by replacing every W α with W α \ ∪ α − i =1 W i , we may assume that the sets W α are disjoint, maintaining the BCS property (Remark 3.3). Set K = t α τ − α ( W α × id K )10igure 2: F : a fundamental domain for SL ( Z ) in P (the hyperbolic upper half plane).(note that the interior is a manifold). Since the union is disjoint, π | K : K → K/K is a bijection.Moreover, since W α is a BCS, then so does W α × id K , and then so does τ − α ( W α × id K ); by Remark3.3, K is a BCS.Finally, by definition of K one has that k ∈ U α ∩ K maps under τ α to ( π ( k ) , id K ). If Φ ⊆ K/K and Φ ⊆ K then π | − K (Φ ∩ W α ) · Φ = τ − α ((Φ ∩ W α ) × Φ) , where by Remark 3.3 the right hand side is a BCS. Then π | − K (Φ) · Φ is a BCS, as a finite union ofsuch. SL m ( Z ) We recall a construction for fundamental domains for the SL m ( Z ) action on SL m ( R ) and on SO m ( R ) \ SL m ( R )),and list some of their properties. Definition 3.5.
Let { v , . . . , v m } be a basis for R m , and let { φ , . . . , φ m } be the orthonormal basisobtained from it by the Gram-Schmidt orthogonalization algorithm. We say that { v , . . . , v m } is reduced if1. the projection of v j to V ⊥ j − has minimal non-zero length a j (here V = { } ), where V j − =span R { v , . . . , v j − } ;2. the projection of v j to V j − is P j − i =1 n i,j a i · φ i with | n ij | ≤ for all i = 1 , . . . , j − m × m matrix with a reduced basis in its columns is also called reduced.Observe that if a real m × m matrix g is reduced, then it lies in SL m ( R ) and satisfies g = kan where k = [ φ · · · φ m ], a = diag ( a , . . . , a m ) and n = (cid:2) n i,j (cid:3) , with φ j , a j and n i,j as in the definitionabove. In particular, whether g is reduced or not, depends only on an . By the work of Siegel [BM00],the set of reduced matrices contains a fundamental domain for the action of SL m ( Z ). A specific choiceof such a domain was made by Schmidt [Sch98] (see also [Gre93]), and it is defined as follows; we willuse the notation Sym + (Λ) for the group of orientation preserving isometries of Λ (sometimes referredto a the “point group” of Λ). 11 efinition 3.6. We let f F m ⊂ SL m ( R ) = SO m ( R ) P m , where P m is the subgroup consisting of uppertriangular matrices, denote a choice of a fundamental domain lying inside the set of g = kan ∈ SL m ( R )such that: (i) g is reduced; (ii) n ,j ≥ j > ι ( m ) (see Notation (1.3)); (iii) k lies inside afundamental domain of Sym + (Λ an ) < SO m ( R ), where Λ an is the lattice spanned by the columns of an . The projection of f F m to P m is denoted F m (Figure 2).Note that conditions (i) and (ii) are on an , whereas condition (iii) is on k . Thus, the projection of f F m to P m ∼ = SO m ( R ) \ SL m ( R ) is a fundamental domain for the action of SL m ( Z ) on SO m ( R ) \ SL m ( R )that lies inside the set of triangular m × m matrices satisfying conditions (i) and (ii) in Definition 3.6,and the relation between f F m and F m is given by: Proposition 3.7.
The relation between the fundamental domains f F m and F m is given by f F m = [ z ∈ F m K z · z, where K z is a fundamental domain for the finite group Sym + (Λ z ) . Note that f F m is not a product of F m ⊂ P m with a subset of SO m ( R ), since different lattices Λ z have different point groups Sym + (Λ z ). However, there is only a finite number (that depends on m ) ofpossible fibers, since there are finitely many possible symmetry groups for lattices in R m . Moreover,for generic z ’s the point groups are identical: Proposition 3.8 ([Sch98]) . For z ∈ int ( F m ) , Sym + (Λ z ) = Z (SO m ( R )) , the center of SO m ( R ) . Thus suggests that for a full-measure set of z ∈ F m , a uniform fiber in K m can be chosen; hence f F m can be approximated by F m times that generic fiber. Lemma 3.9.
Let G = SL n ( R ) and P < G the subgroup of upper triangular matrices. Assume e E ⊆ L n is the lift of E ⊆ X n . If E is a BCS then e E is, and µ L n ( e E ) = µ X n ( E ) · Q n − i =1 Leb( S i ) /ι ( n ) . Assume Ψ ⊆ L n − ,n projects to e E ⊆ U n − and Φ ⊆ S n − , in the sense that Q Ψ = K Φ G e E (e.g. if Ψ is theinverse image of e E ). If e E and Φ are BCS’s, then so is Q Ψ , and µ L n,n − (Ψ) = µ L m ( e E ) µ S n − (Φ) .Proof. By Proposition 3.7, G e E = S z ∈ P E K z · z . Since there are only finitely many possible fibers, thenby Proposition 3.8 G e E = ( K gen · ( P E ∩ int ( F n ))) ∪ ( q ( n ) [ i =1 K z i · (cid:8) z ∈ P E ∩ ∂F m : Sym + (Λ z ) = Sym + (Λ z i ) (cid:9) )where K gen is the generic fiber and (cid:8) z ∈ P E ∩ ∂F n : Sym + (Λ z ) = Sym + (Λ z i ) (cid:9) is contained in ∂F n ,and is therefore a BCS of measure zero in P . Since the fibers in SO n ( R ) are BCS’s in SO n ( R )due to Lemma 3.4, and since P E ∩ int ( F n ) is a BCS by Proposition 3.2 and Fact 3.3, and sinceSO n ( R ) × P is diffeomorphic to SL n ( R ) with µ SL n ( R ) = µ SO n ( R ) × µ P , we have that G e E is a BCSin SL n ( R ) and has the same measure as K gen · ( P E ∩ int ( F n )), which is µ SO n ( R ) ( K gen ) · µ P ( P E ) = µ SO n ( R ) (SO n ( R )) µ P ( P E ) /ι ( n ) = µ P ( P E ) · Q n − i =1 Leb( S i ) /ι ( n ) (recall choice of the volume of SO n ( R )in Subsection 2.1). According to Proposition 3.2, which says that BCS’s and the measures in the“good” sets of representatives and in the spaces that they represent correspond, we get that we getthat e E is a BCS and that µ U n ( e E ) = µ X n ( E ) · Q n − i =1 Leb( S i ) /ι ( n ).The proof for Ψ is a direct consequence of [HK20, Propositions 6.15 and 6.16]For future reference, we list some properties of f F m , F m that will be useful in the proof of our maintheorem; in fact, the following applies to every reduced matrix, and in particular to the elements of f F m , F m . The notations for a j and V j are as in Definition 3.5.12 emma 3.10. Suppose g = kan is reduced and that its columns span a lattice Λ . Then1. n is a unipotent upper triangular matrix with non-diagonal entries in [ − / , / ; in particular, (cid:13)(cid:13) n ± (cid:13)(cid:13) , k n ± t k (cid:28) .2. a = diag ( a , . . . , a m ) satisfies that a (cid:28) · · · (cid:28) a m . Specifically, √ a j ≤ a j +1 .3. If λ ∈ Λ satisfies λ / ∈ V j − , then k λ k ≥ dist( λ, V j − ) ≥ dist( v j , V j − ) = a j .4. If x ∈ V j , then k ax k (cid:28) a j k x k . In order to deduce that the invariant measures on the fundamental domains K , (cid:93) F n − , F n − etc. arethe Haar measures on the spaces that they represent, we require the following result: Theorem 3.11 ([Jüs18, Thm 2.2]) . Let G be a unimodular Radon lcsc group with a Haar measure µ G , and let ν be a G -invariant Radon measure on an lcsc space Y . Assume that the G action on Y isstrongly proper. Then there exists a unique Radon measure ν on G \ Y such that for all f ∈ L ( Y, ν ) , Z Y f ( y ) dν ( y ) = Z G \ Y (cid:18)Z G f ( gy ) dµ G ( g ) (cid:19) dν ( Gy ) . Proof of Proposition 3.2.
By construction, K , (cid:93) F n − and F n − are sets of representatives for S n − , U n − and X n − respectively, and K ⊂ K is a BCS according to Lemma 3.4. F n − ⊂ P is a BCSsince its boundary is contained in a finite union of lower-dimeansional manifolds in P (see [Sch98, pp.48-49], and (cid:93) F n − ⊂ G is a BCS by Lemma 3.9. Finally, K (cid:93) F n − ⊂ Q , it is a set of representatives for L n − ,n since SL n ( R ) /N G ( Z ) A ’ K G A N /G ( Z ) A N ’ K G /G ( Z )and (cid:93) F n − is a set of representatives for G /G ( Z ) ∼ = U n − . It is a BCS by Lemma 3.9. For part(i) of the proposition, a BCS in S n − , U n − , X n − or L n − ,n is mapped to a BCS in K , (cid:93) F n − , F n − and K × (cid:93) F n − respectively: for K it holds because of Lemma 3.4, and for the remaining sets this isproved in [HK20, Prop. 8.1]. The correspondence of measures is a consequence of Theorem 3.11 above(but one can find more details in [HK20, Prop. 6.10]). We begin in Subsection 4.1 by establishing a 1 to 1 correspondence between primitive vectors in Z n and integral matrices in fundamental domains for the discrete subgroup defined asΓ := ( N (cid:111) G ) ( Z ) = (cid:20) SL n − ( Z ) Z n (cid:21) . Then, in Subsection 4.2, we define an explicit such fundamental domain in which the integral repre-sentative of a primitive vector v , has the shortest solution w v in its last column. SL n ( Z ) Recall G v was defined in Formula 2.2. We first prove:13 roposition 4.1. If Ω ⊂ SL n ( R ) is a fundamental domain for the right action of Γ , then there existsa bijection that depends on Ω (cid:0) Z n ∩ v ⊥ (cid:1) ↔ v ↔ γ v (Ω) := the unique element in Ω ∩ G v ( Z ) , between (cid:26) primitive oriented ( n − -lattices in Z n (cid:27) ↔ n primitive vectorsin Z n o ↔ n integral matricesin Ω o . Proof.
The correspondence (cid:0) Z n ∩ v ⊥ (cid:1) ↔ v is explained in the Introduction, and it suffices to show thecorrespondence v ↔ γ v (Ω) . We first claim that G v ∩ SL n ( Z ) = {∅} ⇐⇒ v ∈ Z n primitive. (4.1)The direction = ⇒ is a consequence of (1) ⇒ (3) in Lemma 2.3. Conversely, if v is primitive, thenthere exists w ∈ Z n such that h v, w i = 1. Let { v , . . . , v n − } be an integral basis for v ⊥ such that { v , . . . , v n − , v } is a positively oriented basis for R n . Then, by (3) ⇒ (1) in Lemma 2.3, the resultingmatrix [ v | · · · | v n − | w ] is in G v . Since its columns are integral, it is also in SL n ( Z ).Observe that G v is an orbit of the group N (cid:111) ( G ( Z )), acting by right multiplication on G =SL n ( R ), and that Γ is the subgroup of integral elements in this group. According to (4.1), v ∈ Z n isprimitive if and only if there exists an integral γ in G v . This is equivalent to all the points in the orbit γ · Γ being integral. Since Ω is a fundamental domain for Γ, the coset γ · Γ intersects Ω in a singlepoint { γ v } = Ω ∩ ( γ · Γ). We claim that γ · Γ = G v ( Z ); indeed, G v ( Z ) = G v ∩ SL n ( Z ) = ( γ · N G ) ∩ SL n ( Z ) = γ · (( N G ) ∩ SL n ( Z )) = γ · Γ . Γ that captures the shortest solutions Having shown that the primitive vectors in R n correspond to integral matrices in a fundamental domainof Γ, we proceed to construct a specific such domain, with the property that every representative γ v has in its last column the shortest solution w v to the gcd equation of v . We begin with a more general(even if not as general as possible) construction for a fundamental domain of Γ; but first, a notation. Notation . For g ∈ SL n ( R ), we let z g denote the upper triangular ( n − × ( n −
1) matrix suchthat the P component of g is (cid:2) z g
11 0 (cid:3) . Proposition 4.3.
Let e F ⊂ SL n − ( R ) be a fundamental domain of SL n − ( Z ) , and D = {D ( z ) } z ∈ F n − be a family of fundamental domains for Z n − in R n − . Then Ω = Ω D := [ g ∈ e F K · g · A · N ( z g ) is a fundamental domain for the action of Γ on SL n ( R ) by multiplication from the right. The proof is rather standard, and we skip it.
Remark . Clearly, if all the domains D ( z ) are the same domain D , then Ω is the product set K G e F A N .For g in SL n ( R ), consider the linear map L g that sends the first n − g to the (ordered)standard basis for R n − . Note that the ( n − n − g , spanned by the first n − g ,is mapped under L g onto Z n − = span Z { e . . . , e n − } . As a result, a fundamental domain for Λ n − g in v ⊥ is mapped under L g onto a fundamental domain of Z n − in R n − . We consider the image of theDirichlet domain for Λ n − g , which is Y ( z g ) := L g (Dir(Λ n − g )). Note that indeed the right-hand sidedepends only on the P component of g : since g = ka p n , then the RHS is L n L p L ka (Dir(Λ n − g )).Now L ka acts as a rotation and multiplication by scalar such that Λ n − g maps to Λ n − p = Λ z g and14he Dirichlet domains map to one another. Since L n is identity map, then L g (Dir(Λ n − g )) equals L p (Dir(Λ n − p )). Then Y F n − := { Y ( z ) } z ∈ F n − (4.2)is a family of fundamental domains for Z n − in R n − , and so by Proposition 4.3 and by the notationfor K z appearing in Proposition 3.7, the following is a fundamental domain for Γ:Ω short := Ω Y = [ g ∈ (cid:94) F n − K · g · A N Y ( z g ) = [ z ∈ F n − K K z · [ z
00 1 ] |{z} ∈ P · A N Y ( z ) . (4.3)Recall from Proposition 2.2 that g = ( | · · · | w ) ∈ G v ∩ Ω( Y ) = ⇒ w v ⊥ ∈ L − g ( Y ( z g )) ⇐⇒ w v ⊥ ∈ Dir(Λ n − g ) , namely w v ⊥ (such that w = v/ k v k + w v ⊥ , see Lemma 2.4) is the shortest representative of the coset w v ⊥ + Λ n − g in the hyperplane v ⊥ . This means that w is the shortest representative of the coset w + Λ n − g (which lies in the affine hyperplane { u : h u, v i = 1 } ). As a result, for every primitive vector v , the representative γ v = G v ( Z ) ∩ Ω short , has last column w which is w v := the shortest integral w which satisfies h w, v i = 1 . The relation between the norm of w v , which is what we are interested in for Theorems A and B, andbetween the norm of w v ⊥ v , which is what is captured by Ω short , is given by the following lemma. Lemma 4.5. If { v n } is a divergent sequence of primitive vectors, then lim n →∞ (cid:12)(cid:12)(cid:12) k w v m k − (cid:13)(cid:13)(cid:13) w v ⊥ m v m (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12) = O ( k v m k − ) . Proof.
By Lemma 2.4, | k w v m k − (cid:13)(cid:13)(cid:13) w v ⊥ m v m (cid:13)(cid:13)(cid:13) | ≤ (cid:13)(cid:13)(cid:13) w v m − w v ⊥ m v m (cid:13)(cid:13)(cid:13) = 1 k v m k . The goal of this section is to reduce the proof of Theorem B into a problem of counting integralmatrices in subsets of SL n ( R ), and specifically of Ω short . We begin by defining these subsets. First,consider the covering radius of the lattice spanned by the first n − ρ (Λ n − g ) = radius of bounding circle for Dir(Λ n − g ) . Clearly, the norm k w v ⊥ k lies in the interval [0 , ρ (Λ g )], i.e. k w v ⊥ k ρ (Λ g ) ∈ [0 , . We consider sub-families Y α ⊆ Y for which this quotient is restricted to a sub-interval [0 , α ], with0 ≤ α ≤
1. Let B v ⊥ r denote an origin-centered n − v ⊥ whose radius is r . For α ∈ [0 , Y α ( z g ) = L g ( B v ⊥ αρ (Λ g ) ∩ Dir(Λ g )) Y αF n − = { Y α ( z ) } z ∈ F n − . (5.1)We now turn to define the subsets of Ω short such that the integral matrices inside them represent viathe bijection γ v ↔ v the primitive vectors that are counted in Theorem B.15igure 3: The domain F S . Notation . For
T >
0, Φ ⊆ S n − , e E ⊆ U n − , E ⊆ X n − , Ψ ⊆ L n − ,n and α ∈ [0 , short ) T (Φ , E , α ) = Ω short ∩ ( g = k k p a n : k ∈ K Φ , a ∈ A [0 ,T ] ,p ∈ P , n ∈ N Y α ( z g ) ) = [ p ∈ P K Φ · K z p · p · A T N Y α ( z p ) , (Ω short ) T (Φ , e E , α ) = Ω short ∩ ( g = k g a n : k ∈ K Φ , a ∈ A [0 ,T ] ,g ∈ G e E , n ∈ N Y α ( z g ) ) = [ g ∈ G e E K Φ · g · A T N Y α ( z g ) , and (Ω short ) T (Ψ , α ) = Ω short ∩ ( g = qa n : q ∈ Q Ψ , a ∈ A [0 ,T ] ,n ∈ N Y α ( z g ) ) = [ q ∈ Q Ψ q · A T N Y α ( z q ) . The following notation is for sets in G whose A component is restricted to a compact box. Notation . For every S = ( S , . . . , S n − ) > B ⊂ G , let B S denote the subset B ∩ { g : π A i ( g ) ≤ S i ∀ i } , where π A i is the projection to the A i component (see Figure 3 for F S ).Recall from the Introduction that Λ v = Λ n − γ v and ρ v = ρ (Λ n − v ). The following is now immediatefrom Proposition 4.1, Proposition 2.2, and the construction of Ω short , and concludes the translation ofTheorem B into a problem of counting lattice points in SL n ( R ): Corollary 5.3.
Consider the correspondence v ↔ γ v where γ v = ( G v ( Z )) ∩ Ω short . For T > , Φ ⊆ S n − e E ⊆ U n − , E⊆ X n − , Ψ ⊆ L n − ,n and α ∈ [0 , :1. The SL n ( Z ) matrices in (Ω short ) T (Φ , E , α ) correspond under γ v ↔ v to the elements of n v ∈ Z n primitive : k v k ≤ e T , ˆ v ∈ Φ , shape (Λ v ) ∈ E , k w v ⊥ k /ρ v ∈ [0 , α ] o .
2. The SL n ( Z ) matrices in (Ω short ) T (Φ , e E , α ) correspond under γ v ↔ v to the elements of n v ∈ Z n primitive : k v k ≤ e T , ˆ v ∈ Φ , (cid:74) Λ v (cid:75) ∈ e E , k w v ⊥ k /ρ v ∈ [0 , α ] o . . The SL n ( Z ) matrices in (Ω short ) T (Ψ , α ) correspond under γ v ↔ v to the elements of n v ∈ Z n primitive : k v k ≤ e T , [Λ v ] ∈ Ψ , k w v ⊥ k /ρ v ∈ [0 , α ] o . For the cases where E (resp. e E , Ψ ) is the set parametertized by F Sn − (resp. (cid:93) F n − S , K (cid:93) F n − S ) with S = ( S , . . . , S n − ) , then the condition shape (Λ v ) ∈ E (resp. (cid:74) Λ v (cid:75) ∈ e E , [Λ v ] ∈ Ψ ) is equivalent to k ( v ∧···∧ v i ) kk v k i/ ( n − ≥ e − S i / for every ≤ i ≤ n − . In the previous section we reduced the proof of Theorem B to counting SL n ( Z ) points inside the subsets(Ω short ) T as T → ∞ . These sets have the disadvantage of not being compact, despite their finitevolume; this is apparent from the fact that they contain A , which is unbounded. Since our countingmethod (described in Subsection 8.1) does not allow non-compact sets, the aim of this section is toreduce counting in (Ω short ) T to counting in a compact subset of it. Here we will allow a fundamentaldomain of Γ as general as in Corollary 4.3, and not restrict just to Ω short . Notation . Let Ω be a fundamental domain for Γ in SL n ( R ) as in Corollary 4.3. For every T, S > ST := Ω ∩ n g = ka t a s n : s j ≤ S j , t ∈ [0 , T ] o . Note that µ (Ω T − Ω ST ) is in O ( e nT − S min ), where S min = min j S j .The goal of this section is to prove the following: Proposition 6.2.
Let Ω a fundamental domain for Γ in G , and σ = ( σ , . . . , σ n − ) where < σ i < ∀ i . Denote Ω [ σT ; ∞ ] T := Ω T − Ω σTT and σ min = min ( σ , . . . , σ n − ) . Then for every (cid:15) > [ σT ; ∞ ] T ∩ SL n ( Z )) = O (cid:15) ( e T ( n − σ min + (cid:15) ) ) . Two auxiliary claims are required for the proof.
Lemma 6.3.
Assume that Λ is a full lattice in R n with covolume v , and set ≤ d ≤ n . Let therebe d − intervals [ α i , β i ] with ≤ α i < β i . The number of rank d subgroups of Λ whose covolumeis ≤ X and who satisfy that covol(Λ i ) ∈ [ X α i , X β i ] for some reduced basis { v , . . . , v d } such that Λ i := span Z { v , . . . , v i } , is O n ( v − d X e ( α,β )) , where e ( α, β ) = n − d + 1 + 2 d − X i =1 β i + d − X i =1 ( n − i )( β i − α i ) . Proof.
Let Λ < Λ be a rank d subgroup, and write Λ = span Z { v , . . . , v n − } where { v , . . . , v n − } is a reduced basis for Λ. We use the notations introduced in Definition 3.5: { φ , . . . , φ n − } is theGram-Schmidt basis obtained from { v , . . . , v n − } , V i is span { v , . . . , v i } = span { φ , . . . , φ i } , and a i is the projection of v i on the line orthogonal to V i − inside the space V i , where span {∅} is set to bethe trivial subspace { } . In other words, a i is the distance of v i from the subspace V i − (Figure 4a).If Λ is such that covol(Λ i ) ∈ (cid:2) X α i , X β i (cid:3) , then a i ≤ R i = X β i − α i − . Denote the number of possibilitiesfor choosing v i given that Λ i − is known by v i | Λ i − . We first claim that for every 1 ≤ i ≤ d v i | Λ i − = O (cid:0) ( R i ) n − i +1 · v − · covol(Λ i − ) (cid:1) . (6.1)17 a) v , v and a in the planespan R { v , v } = span R (Λ) (b) A Dirichlet domain for ˜Λ ,[ − / , / v , multiplied by a ballof radius R . Figure 4: Lemma 6.3.Indeed, for i = 1, the number v i | Λ i − is simply the number of possibilities for choosing a Λ vector v inside a ball of radius a = k v k in R n , and therefore v | Λ = ∩ B R ) = O ( v − · R n ) . For i >
1, the orthogonal projection of v i to the subspace V i − must lie inside a Dirchlet domain ofthe lattice ˜Λ i − := span Z { a φ , . . . , a i − φ i − } . Thus, v i has to be chosen from the set of Λ pointswhich are of distance ≤ a i ≤ R i from the Dirichlet domain for ˜Λ i − in span R (Λ i − ). These are the Λ points that lie in a domain which is the product of the Dirichlet domain for ˜Λ i − (in V i − ) with a ballof radius R i in the n − ( i −
1) dimensional subspace V ⊥ i − (Figure 4b). Denote this ball by B n − ( i − R i ,and then v i | Λ i − ≤ ∩ { B n − ( i − R i × Dirichlet domain for ˜Λ i − } )= O ( v − · vol( B n − ( i − R i ) · covol(Λ i − ))= O ( v − · ( R i ) n − i +1 · covol(Λ i − )) . This establishes Equation (6.1). Now, the number of possibilities for Λ is given by: d Y i =1 ( v i | Λ i − ) = O ( d Y i =1 ( v − · ( R i ) n − i +1 · covol(Λ i − )))= O ( v − d d Y i =1 ( X ( β i − α i − )( n − i +1) · X β i − ))where α = 0 and β d = 1 (as covol(Λ ) = k v k ≥ X , and covol(Λ d ) = covol(Λ) ≤ X ). Since d X i =1 (( n − i + 1)( β i − α i − ) + β i − ) = n − d + 1 + d − X i =1 ( n − i )( β i − α i ) + 2 d − X i =1 β i = e ( α, β )18hen the number of lattices Λ is bounded by O ( v − d X e ( α,β ) ). Corollary 6.4.
Assume that Λ is a full lattice in R n with covolume v , let ≤ d ≤ n , and < ω < · · · < ω d − . For every (cid:15) > , the number of rank d subgroups of Λ with covolume ≤ X which satisfy covol(Λ i ) ∈ [1 , X ω i ] is O (cid:15) ( v − d · X n − d +1+2 ω + (cid:15) ) , where ω := P d − i =1 ω i .Proof. Divide every interval [0 , ω i ] into N i = N i ( ω i ) sub-intervals0 = β i < β i < . . . < β iN i = ω i such that | β ij − β ij − | ≤ (cid:15) for every j = 1 , . . . , N i . By refining these partitions, we may assume withoutloss of generality that N = . . . = N d − := N . Fix j ∈ { , . . . , N } ; according to Lemma 6.3, thenumber of rank d subgroups Λ of Λ with covol(Λ) ≤ X and covol(Λ i ) ∈ [ X β ij − , X β ij ] for every i = 1 . . . d − X to the power of n − d + 1 + 2 d − X i =1 β ij + d − X i =1 ( n − i ) ( β ij − β ij − ) ≤ n − d + 1 + 2 d − X i =1 ω i + d − X i =1 ( n − i ) · (cid:15) = n − d + 1 + 2 ω + (cid:15) · ( d −
1) ( n − d/ , where we have used | β ij − β ij − | ≤ (cid:15) and β ij ≤ ω i .Let Λ < Λ be as in the statement. Since covol(Λ i ) lies in [ X , X ω i ] for every i = 1 , . . . , d − i there exist j i , . . . j id − such that covol(Λ i ) ∈ [ X β ij − , X β ij ]. It follows that O (cid:15) ( v − d X { j ,...j n − }⊂{ ,...,N } X n − d +1+2 ω + (cid:15) · O n (1) ) = O (cid:15) ( v − d X n − d +1+2 ω + (cid:15) ) . Proof of Proposition 6.2.
Let γ v = ka s a t n ∈ γ ∈ Ω T ∩ SL n ( Z ). By definition of Ω [ σT ; ∞ ] T we havethat γ ∈ Ω [ σT ; ∞ ] T if and only if t ∈ [0 , T ] and s i ≥ σ i T for some i . According to Proposition 2.2,(by which covol(Λ iv ) = e itn − − si ), and since Λ iv is integral, we have that 1 ≤ covol(Λ iv ) ≤ e ( in − − σi ) T .Thus, the number of SL n ( Z )-elements γ in Ω [ σT ; ∞ ] T is bounded by the number of ( n − v of Z n of co-volume ≤ e T := X , for which there exists i ∈ { , . . . , n − } such thatcovol(Λ iv ) ∈ [1 , X in − − σi ], where for j = i covol(Λ jv ) ∈ [1 , X jn − ]. In other words, [ σT ; ∞ ] T ∩ SL n ( Z )) = [ u =( u ,...,u n − ) ∈{ , } n − −{ } { Λ v : ∀ i, covol(Λ iv ) ∈ [1 , X in − − σiui ] } )which by Corollary 6.4 with e T = X , d = n − in − − σ i u i = ω i equals to X u ∈{ , } n − −{ } O (cid:15) ( X n − (cid:15) − P n − i =1 σ i u i ) = O (cid:15) ( X n − σ min + (cid:15) )where σ min = min { σ i } . In Section 5 (see Corollary 5.3), the proof of Theorem B was reduced to counting integral matrices inthree families of subsets of Ω short ⊂ SL n ( R ). But, as we shall see, it is in fact sufficient to count integralmatrices in only one of these families, the one corresponding to part 3 of Theorem B: (Ω short ) T (Ψ , α ).The content of this section is a proof of Theorem B, assuming the following (yet to be proved) countingstatement in this family: 19 roposition 7.1. For α ∈ (0 , , assume that Ψ ⊆ L n − ,n is BCS. Set λ n = n / (cid:0) (cid:0) n − (cid:1)(cid:1) , and let τ n be as in Theorem B.1. For (cid:15) ∈ (0 , τ n ) , S = ( S , . . . , S n − ) , S = P n − i =1 S i and every T ≥ S nλ n τ n + O (1) , short ) ST (Ψ , α ) ∩ SL n ( Z )) = µ ((Ω short ) ST (Ψ , α )) µ (SL n ( R ) / SL n ( Z )) + O (cid:15), Ψ ( e S /λ n e nT (1 − τ n + (cid:15) ) ) .
2. For (cid:15) > , δ ∈ [0 , τ n − (cid:15) ) , T ≥ O (1) and S ( T ) such that P S i ( T ) ≤ nδλ n T + O Ψ (1) , short ) S ( T ) T (Ψ , α ) ∩ SL n ( Z )) = µ ((Ω short ) S ( T ) T (Ψ , α )) µ (SL n ( R ) / SL n ( Z )) + O (cid:15), Ψ ( e nT (1 − τ n + δ + (cid:15) ) ) . Notice that the difference between parts 1 and 2 of the proposition above is that in the first part S is fixed, while in the second part, at the cost of compromising the error term, we allow the sum of S i -s to grow proportionally to T . Remark . If Ψ ⊂ L n − ,n is also bounded, then for suitable S one has that (Ω short ) ST (Ψ , α ) =(Ω short ) T (Ψ , α ); thus in this case part 1 of Proposition 7.1 can be written as short ) T ∩ SL n ( Z )) = µ ((Ω short ) T ) /µ (SL n ( R ) / SL n ( Z )) + O (cid:15) ( e nT (1 − τ n + (cid:15) ) ) , where the implied constant depends on Ψ.The proof of Proposition 7.1 is in Section 13. Let us now prove Theorem B based on this proposition: Proof of Theorem B.
According to Corollary 5.3 and to Lemma 4.5, the quantities we seek to estimatein parts (1), (2) and (3) of the theorem is in one to one correspondence with the integral matrices inthe following subsets of SL n ( R ): (1) (Ω short ) T (Φ , E , α ), (2) (Ω short ) T (Φ , e E , α ), or (3) (Ω short ) T (Ψ , α ).Observe that, indeed the main terms in the theorem are the volumes of these sets, divided by themeasure of SL n ( R ) / SL n ( Z ). Let us demonstrate the computation for the case of the family (1), forwhich we recall the notation for the fibers K z i and the generic fiber K gen appearing in the proof ofLemma 3.9: µ ((Ω short ) T (Φ , E , α )) = µ [ p ∈ P K Φ · K z p · p · A T N Y α ( z p ) == µ K ( K Φ ) µ K (cid:0) K gen (cid:1) µ A ( A T ) Z P ∩ int( F n − ) µ N (cid:16) N Y α ( z p ) (cid:17) dµ P ( p )+ X i µ K (cid:0) K Φ K z i (cid:1) µ A ( A T ) Z P ∩ ∂F n − ∩ (cid:8) p : Sym + ( Λ p ) = Sym + ( Λ zi ) (cid:9) µ N (cid:16) N Y α ( z p ) (cid:17) dµ P ( p ) , where we have used: the definition 5.1 for (Ω short ) T (Φ , E , α ), Formula 2.1 for the decomposition of µ to RI components and Proposition 3.8 which tells us that all the interior points in F n − have thegeneric fiber. Now, the second summand is of measure zero, since the boundary of F n − is such, so weare left only with the first summand.Since, by “Measures on the RI components” in Section 2, µ N is the Lebesgue measure on R n − , µ K is the Lebesgue measure on S n − , the volume of SO n − ( R ) is Q n − i =1 Leb( S i ) (implying that themeasure of K gen is Q n − i =1 Leb( S i ) /ι ( n − µ A ( A T ) = e nT /n , and since by Proposition 3.2we can pass from integration on F n − to integration on X n − , we have that the above equals to e nT · µ S n − (Φ) · Q n − i =1 Leb( S i ) n · ι ( n − Z E L α ( z ) dµ X n − ( z ) ,
20s wanted.We claim that it is sufficient to prove part (3) of the theorem, since parts (1) and (2) are specialcases. Indeed, family (1) is a special case of family (2), when taking e E ⊆ U n − to be the inverse imageof E ⊆ U n − . This is because Lemma 3.9 gives that the lift e E is a BCS when E is, so the assumptionof part 1 of the theorem implies the assumption in part 2 for the lifts; moreover, this Lemma givesthat µ U n − ( e E ) = µ X n − ( E ) Q n − i =1 Leb( S i ) /ι ( n − e E and Φcoincides with the one provided in part 1 of this theorem for E and Φ. Similarly, family (2) is a specialcase of family (3), when taking Ψ such that Q Ψ = K Φ G e E . By Lemma 3.9, Ψ is a BCS when e E and Φare, and µ L n,n − (Ψ) = µ U n − ( e E ) µ S n − (Φ). We therefore prove only part 3 of Theorem B. (i) Let us first consider the case where Ψ is not bounded , and therefore (Ω short ) T (Ψ , α ) is notbounded also. Fix (cid:15) ∈ (0 , τ n ), δ ∈ (0 , τ n − (cid:15) ), and σ n := ( δnλ n n − − (cid:15) ) ·
1, where λ n = n n − ; note thatthe sum of the coordinates of σ n is δnλ n − ( n − (cid:15) , which is smaller than δnλ n + O Ψ (1) /T for T largeenough. Using Proposition 6.2, we reduce to counting in compact sets (Ω short ) σ n TT (Ψ , α ), and pay withan error term of O (cid:15) ( e nT (1 − δλnn − + (cid:15)n + (cid:15) ) ), which we can write as O (cid:15) ( e nT (1 − δλnn − + (cid:15) ) ) since (cid:15) is arbitrary. (ii) Counting integral matrices in the sets (Ω short ) σ n TT (Ψ , α ) will complete the proof, and it is performedusing (the second part of) Proposition 7.1, based on which it equals µ ( (Ω short ) σ n TT (Ψ , α )) + O (cid:15) ( e nT (1 − τ n + δ + (cid:15) ) ) . Since µ ((Ω short ) T (Ψ , α )) = µ ((Ω short ) σ n TT (Ψ , α )) + O ( e nT (1 − δλnn − ) ) (see remark about the measure inNotation 6.1), and the error term is swallowed in the one obtained in step (i), we obtain short ) T (Ψ , α ) ∩ SL n ( Z )) = µ ((Ω short ) T (Ψ , α )) + O (cid:15) ( e nT (1 − τ n + δ + (cid:15) ) ) + O (cid:15) ( e nT (1 − δλnn − + (cid:15) ) ) . (iii) We now choose δ that will balance the two error terms above: 1 − τ n + δ = 1 − δλ n n − if andonly if δ = τ n / (1 + λ n n − ) = τ n · (cid:16) − n n − n − n +4 (cid:17) . Then the final error term for non boundedΨ is e nT (cid:0) − τ n · n n − n − n +4 (cid:1) . (iv) Moving forward to bounded
Ψ, we repeat a similar strategy asin the unbounded case, performing only step (ii). Fix (cid:15) ∈ (0 , τ n ) and apply Proposition 7.1 (case ofRemark 7.2) to obtain that the number of integral matrices in (Ω short ) T (Ψ , α ) is µ ((Ω short ) T (Ψ , α )) + O (cid:15) (cid:0) e nT (1 − τ + (cid:15) ) (cid:1) . This completes the proof for the bounded case in the theorem. Proof of Theorem A.
Let B = Ω short ∩ n g = ka s a t n : s n − ∈ [0 , t/ o . We define A = { v ∈ Z n prim : γ v ∈ B} , and claim that it is a set of full density in Z n prim . In fact, we show that Z n prim − A is a set of densityzero. For this, we note that the set Ω short − B is contained in the set e B = lim T →∞ e B T where e B T = T a t =1 ((Ω short ) t − (Ω short ) t − ) ∩ (cid:26) g = ka a s n : 0 ≤ s , . . . , s n − , ( t − ≤ s n − (cid:27) . Note that e B T can also be written as the disjoint union e B T = T a t =1 ((Ω short ) t − (Ω short ) (0 ,..., , ( t − t ) − ((Ω short ) t − − (Ω short ) (0 ,..., , ( t − t − ) . (7.1)21s a result, the volume of e B T can be bounded as follows: µ (cid:16) e B T (cid:17) ≤ T X t =1 µ ((Ω short ) t − (Ω short ) (0 ,..., , ( t − t ) (cid:28) T X t =1 Z ∞ ( t − e − s n − ds n − ! Z tt − e nτ dτ = 1 n T X t =1 e − t + ( e nt − e n ( t − ) ≤ T e ( n − ) T . The presentation in (7.1) can also be used to estimate the number of SL n ( Z ) elements in e B T , bycounting SL n ( Z ) elements in each of the summands separately. For this, Let δ and σ n = ( σ , . . . , σ n − )be as in the proof of the unbounded case in Theorem B. Going along the lines of this proof, we canreduce counting in each (non-compact) summand to counting in the truncated set(Ω short ) σ n tt − (Ω short ) ( σ t,...,σ n − t, min { σ n − t, ( t − } ) t − (cid:16) (Ω short ) σ n tt − − (Ω short ) ( σ t,...,σ n − t, min { σ n − t, ( t − } ) t − (cid:17) , since according to Proposition 6.2, the difference in the amount of lattice points inside each summandand its truncation lies in O (cid:15) ( e nt (1 − δλnn − + (cid:15) ) ), and the difference between their measures is also swallowedin this error estimate (see remark about the measure in Notation 6.1). Using Proposition 7.1(ii) toestimate the amount of lattice points in each truncated summand, we obtain that the number of latticepoints in each (full) summand is its measure divided by µ (SL n ( R ) / SL n ( Z )), up to an error term oforder O (cid:15) ( e nt (1 − δλnn − + (cid:15) ) ). As a result, e B T ∩ SL n ( Z )) = µ ( e B T ) µ (SL n ( R ) / SL n ( Z )) + O (cid:15) ( T e nT (1 − δλnn − + (cid:15) ) ) . According to Corollary 5.3,lim T →∞ Z n prim − A ) ∩ B e T ) Z n prim ∩ B e T ) = lim T →∞ short ) T − B ) ∩ SL n ( Z )) short ) T ∩ SL n ( Z ))which by definition of e B is at most ≤ lim T →∞ e B T ∩ SL n ( Z )) short ) T ∩ SL n ( Z )) . The denominator in the above limit is, according to Proposition 4.1 and Theorem B , asymptotic to µ ((Ω short ) T ) µ (SL n ( R ) / SL n ( Z )) (cid:16) e nT . We then have thatlim T →∞ Z n prim − A ) ∩ B e T ) Z n prim ∩ B e T ) ≤ lim T →∞ µ ( e B T ) µ ((Ω short ) T ) = lim T →∞ T e ( n − ) T e nT = 0 , which establishes that A is a set of full density in Z n prim .22or a primitive vector v with large enough norm, we have by Lemma 4.5 and definition of w v that0 < k w v kk v k (cid:28) (cid:13)(cid:13) w ⊥ v (cid:13)(cid:13) k v k ≤ ρ v k v k = ρ v covol(Λ v ) . Minkowski’s 2nd Theorem gives us that ρ v = ρ (Λ v ) (cid:16) m n − (Λ v ), where m i denotes the i th successiveminima. From [GM02, Theorem 7.9] we have that covol(Λ v ) (cid:16) m (Λ v ) · · · m n − (Λ v ). Thus the abovecan be further estimated as: (cid:28) m n − (Λ v ) m (Λ v ) · · · m n − (Λ v ) = 1 m (Λ v ) · · · m n − (Λ v ) (cid:28) covol(Λ n − v ) − , where by Proposition 2.2(iii), (cid:28) e sn − − n − n − t ≤ e − n − n − t . The above decays to 0 if (and only if) n >
2, and we are done – since if { v m } ⊂ A diverges, then γ v m = k m a m a t m n m with t m → ∞ as m → ∞ , implying that k w v m k / k v m k → Part II
Counting lattice points
This second part is the technical part of the paper, where we prove Proposition 7.1, in order to concludethe proof of Theorem B. This proposition concerns counting lattice points in SL n ( R ); our main tool forthis purpose is a method introduced in [GN12] for counting lattice points in increasing families of setsinside semisimple Lie groups. The advantages of this method is that it produces an error term, andthat it allows counting in quite general families, requiring only that these families are well rounded ,which is a regularity condition. The cost of this generality is that the property of well roundednessis often hard to verify. In [HK20] we develop a machinery to somewhat simplify this process, mainlyby allowing us to replace the underlying simple group G = KAN with the much-easier-to-work-inCartesian product K × A × N ; we will refer to some technical results from there in the course of PartII. We begin in Subsection 8.1 by describing the counting lattice points method that we will use, andproceed in Subsection 8.2 with laying out a plan of proof for Proposition 7.1. From now on, we use Γto denote a general lattice in a Lie group, hence abandoning the notation in Section 4.
In this subsection we briefly describe the counting method developed in [GN12]. This approach,aimed at counting lattice points in increasing families of sets inside non-compact algebraic simple Liegroups, consists of two ingredients: a regularity condition on the sets involved, and a spectral estimateconcerning the unitary G representation π G/ Γ : G → L ( G/ Γ) (the orthogonal complement of the G invariant L functions). Before stating the counting theorem 8.4 from [GN12], we describe the twoingredients, starting with the regularity condition.23 a) The set B T (b) The set B T is perturbed by O (cid:15) (c) B ( − (cid:15) ) T and B (+ (cid:15) ) T Figure 5: Well roundedness.
Definition 8.1.
Let G be a Lie group with a Borel measure µ , and let {O (cid:15) } (cid:15)> be a family of identityneighborhoods in G . Assume {B T } T > ⊂ G is a family of measurable domains and denote B (+ (cid:15) ) T := O (cid:15) B T O (cid:15) = [ u,v ∈O (cid:15) u B T v, B ( − (cid:15) ) T := \ u,v ∈O (cid:15) u B T v (see Figure 5). The family {B T } is Lipschitz well rounded (LWR) with (positive) parameters ( C , T ) iffor every 0 < (cid:15) < / C and T > T : µ (cid:16) B (+ (cid:15) ) T (cid:17) ≤ (1 + C (cid:15) ) µ (cid:16) B ( − (cid:15) ) T (cid:17) . (8.1)The parameter C is called the Lipschitz constant of the family {B T } .The definition above allows any family {O (cid:15) } (cid:15)> of identity neighborhoods; in this paper we shallrestrict to the following: Assumption 8.2.
We will assume that O G(cid:15) = exp ( B (cid:15) ) , where B (cid:15) is an origin-centered (cid:15) -ball insidethe Lie algebra of G , and exp is the Lie exponent.Remark . We allow the case of a constant family {B T } = B : we say that B is a Lipschitz wellrounded set (as apposed to a Lipschitz well rounded family ) with parameters ( C , (cid:15) ) if µ ( B (+ (cid:15) ) ) ≤ (1 + C (cid:15) ) µ ( B ( − (cid:15) ) ) for every 0 < (cid:15) < (cid:15) . It is proved in [HK20, Prop. 3.5] that if a set B is BCS andbounded, then it is LWR.We now turn to describe the second ingredient, which is the spectral estimation. In certain Liegroups, among which algebraic simple Lie groups G , there exists p ∈ N for which the matrix coefficients h π G/ Γ u, v i are in L p + (cid:15) ( G ) for every (cid:15) >
0, with u, v lying in a dense subspace of L ( G/ Γ) (see [GN09,Thm 5.6]). Let p (Γ) be the smallest among these p ’s, and denote m (Γ) = ( p = 2,2 d p (Γ) / e otherwise.The parameter m (Γ) appears in the error term exponent of the counting theorem below, which is thecornerstone of the counting results in this paper. Theorem 8.4 ([GN12, Theorems 1.9, 4.5, and Remark 1.10]) . Let G be an algebraic simple Lie groupwith Haar measure µ , and let Γ < G be a lattice. Assume that {B T } ⊂ G is a family of finite-measure omains which satisfy µ ( B T ) → ∞ as T → ∞ . If the family {B T } is Lipschitz well rounded withparameters ( C B , T ) , then ∃ T > such that for every δ > and T > T : B T ∩ Γ) − µ ( B T ) /µ ( G/ Γ) (cid:28) G, Γ ,δ C dim G G B · µ ( B T ) − τ (Γ)+ δ , where µ ( G/ Γ) is the measure of a fundamental domain of Γ in G and − τ (Γ) = 1 − (2 m (Γ) (1 + dim G )) − ∈ (0 , . The parameter T is such that T ≥ T and for every T ≥ T µ ( B T ) τ (Γ) (cid:29) G, Γ C dim G G B . (8.2)Bounds on the parameter p (Γ) (i.e. on m (Γ)) clearly imply bounds on the parameter τ (Γ) ap-pearing in the error term exponent. We refer to [Li95], [LZ96] and [Sca90] for upper bounds on p (Γ)in simple Lie groups. Specifically for the group SL n ( R ), the current known bound for n > n ( R ) is 2 ≤ p (Γ) ≤ n − n ( Z ), p (SL n ( Z )) = 2 n − m (SL n ( Z )) = 2 d ( n − / e and therefore τ (SL n ( Z )) is exactly τ n fromTheorem B. Proposition 7.1 is concerned with counting in the sets:(Ω short ) ST (Ψ , α ) = [ q ∈ ( Q Ψ ) S q · A T N Y α ( z q ) . According to Theorem 8.4, in order to prove Proposition 7.1, it is sufficient to claim that the familiesabove are LWR with parameters that do not depend on S . This will be done by following the twosteps below. In each step, we mention technical results from [HK20], and conclude with a summary ofhow and where the goal of the step will be proved in this paper, and which role will it assume in theproof of Propostion 7.1. Step 1: Reduction from LWR in SL n ( R ) = KA A N N to LWR in K × A × A × N × N . It is much easier to verify well roundedness in the (resp. compact, abelian, unipotent) subgroups
K, A, N of SL n ( R ), and their subgroups, than in the simple SL n ( R ). Let r denote the map fromSL n ( R ) to the product, that sends g = ka a n n to ( k, a , a , n , n ) . Then r (cid:16) (Ω short ) ST (Ψ , α ) (cid:17) = [ ( ( k,a ,n ) ,a ) ∈ r (( Q Ψ ) S ) × A T ( k, a , n , a ) × N Y α ( a n ) . We will apply the following result from [HK20], that will enable us to reduce to verifying the wellroundedness of r (cid:16) (Ω short ) ST (Ψ , α ) (cid:17) ; but first, a definition. Definition 8.5 ([HK20, Def. 4.1]) . Let G and Y be two Lie groups with Borel measures µ G and µ Y .A Borel measurable map r : G → Y will be called an f -roundomorphism if it is:1. Measure preserving: r ∗ ( µ G ) = µ Y . When a component is omitted, it means that it is the identity.
Locally Lipschitz: r ( O G(cid:15) g O G(cid:15) ) ⊆ O Yf(cid:15) r ( g ) O Yf(cid:15) for some continuous f = f ( g ) : G → R > andfor every 0 < (cid:15) < f .In [HK20, Prop. 4.2] we prove that if a family B T ⊆ Y is LWR and r : G → Y is a roudomorphismsuch that r − ( B T ) is bounded uniformly in T , than r − ( B T ) is LWR. Here we only need the casewhere Y is a direct product of groups: Proposition 8.6 ([HK20, Corollary 4.3]) . Let r : G → Y = Y × · · · × Y q be an f -roundomorphismand let B T = B T × · · · × B qT ⊆ Y . Set µ Y = µ Y × · · · × µ Y q , O Y(cid:15) = O Y (cid:15) × · · · × O Y q (cid:15) and assume that1. For j = 1 , . . . , q : B jT ⊆ Y j is LWR w.r.t. the parameters ( T j , C j ) ;2. f is bounded uniformly by a real number F on the sets r − ( B T ) .Then r − ( B T ) is LWR, w.r.t. the parameters T = max { T , . . . , T q } , C (cid:16) q F · max { C , . . . , C q , } . In particular, a direct product of LWR families is LWR in the direct product of the corresponding group.
For the proof of Proposition 7.1:
In Section 10 we will prove that the map r is a roundomorphismand establish a bound on f , reducing well roundedness of (Ω short ) ST (Ψ , α ) to well roundedness of r ((Ω short ) ST (Ψ , α )). Step 2: Verifying LWR property in a product of groups.
The sets r (cid:16) (Ω short ) ST (Ψ , α ) (cid:17) ) in Step 1 are of the general form B T = [ z ∈E T z × D z ⊆ P × R m , ( ? )where P is a Lie group. We require the following Lipschitzity condition on the family {D z } : Definition 8.7 ([HK20, Definition 5.1 and Proposition 5.6]) . Let P be a Lie group and O (cid:15) a familyof coordinate balls. Let E be a subset of P , and consider the family D E = {D z } z ∈E , where D z ⊆ R m ( m is uniform for all z ). We say that the family D E is bounded Lipschitz continuous (or BLC ) w.r.t O (cid:15) if there exists C > < (cid:15) < C − the following hold:1. For a norm ball B (cid:15) ⊂ R m of radius (cid:15) , D z + B (cid:15) ⊆ (1 + C(cid:15) ) D z .2. If z ⊆ O (cid:15) z O (cid:15) for z, z ∈ E , then D z ⊆ (1 + C(cid:15) ) D z .3. The Lebesgue volume of D z is bounded uniformly from below by a positive constant V min .4. D z ⊆ B R for some uniform R > z ∈ E .The following result relates the BLC property of the family {D z } , to the LWR property of the setsin ( ? ). Proposition 8.8 ([HK20, Proposition 5.5]) . Let {E T } T > be an increasing family inside a Lie group P , and E := ∪ T > E T . Let D E = {D z } z ∈E where D z ⊂ R m , and consider the family B T = [ z ∈E T z × D z ⊆ P × R m . If {E T } T > is LWR with parameters ( T , C E ) , and D E is BLC w.r.t. the family {O P(cid:15) } (cid:15)> and withparameters ( C D , V min , R ) , then B T is LWR w.r.t the family O P(cid:15) × B R m (cid:15)/ ⊂ P × R m and with parameters ( T , C B ) where C B ≺ C D + ( V max /V min ) C E + 1 and V max = µ R m ( B R ) . or the proof of Proposition 7.1: Following Proposition 8.8, in order to prove that the sets r ((Ω short ) ST (Ψ , α )) from Step 1 are LWR, one should show that:• The family { Y α ( a , n ) } ( a ,n ) ∈ r ( F Sn − ) is BLC (Definition 8.7), which is done in Section 12.• The family E T = r ( Q S Ψ ) × A T over which the union is taken is LWR (Definition 8.1). For this,by Remark 8.6, it is sufficient to show that each of the factors is LWR. The two factors will behandled as follows: – In Section 9 we show that { A T } is LWR; – in Section 11 we show that r ( Q S Ψ )) is LWR.The proof of Proposition 7.1 is completed in Section 13. A In this section and the one that follows, we extend our discussion from G = SL n ( R ) to G being a realsemi-simple Lie group with finite center and Iwasawa decomposition G = KAN . Here we focus on thesubgroup A , and consider subgroups of it that are the image of subspaces in a , the Lie algebra of A ,under the exponent map. To introduce them, we first set some notations. Notation . For vectors H , . . . , H q ∈ a , we write H := ( H , . . . , H q ) ∈ a q . If s = ( s , . . . , s q ) ∈ R q we let s · H := P qi =1 s i H i . We say that H is linearly independent if H , . . . , H q are.We let { φ , . . . φ p } ⊂ a ∗ denote the positive roots, counted with multiplicities, and we use thestandard notation for their sum: 2 ρ = p X i =1 φ i ∈ a ∗ . Definition 9.2.
Given linearly independent H = ( H , . . . , H q ), we define the subgroup A ( H ) < A tobe A ( H ) := { exp ( s · H ) : s ∈ R q } , and endow it with the (non-Haar!) measure µ A ( H ) := e ρ ( H ) s · · · e ρ ( H q ) s q ds · · · ds q . When q = 1, we omit the underlines: H = H and s = s . Remark . Every closed connected subgroup of A is of the form A ( H ). Furthermore, A ( H ) ∩ A ( H ) = { A } if and only if H is linearly independent of H . In that case, A ( H × H ) = A ( H ) × A ( H ) as bothgroups and measure spaces. In particular, if H is a basis for a , then A ( H ) = A and µ A ( H ) = µ A . Example 9.4.
In the case of G = SL n ( R ), N = " ··· R ... ... and A = " e α ... e αn , where P α i = 0.The roots φ i,j ∈ a ∗ are defined via φ i,j ( P nk =1 α k e k,k ) = α j − α i , where the positive roots (w.r.t. which N is defined) are the ones with j < i . For H = P nk =1 α k e k,k ∈ a ,2 ρ ( H ) = 2 ρ n X k =1 α k e k,k ! = n X k =1 ( n + 1 − k ) α k . A and A as defined in Section 2, the bases for the Lie algebras are H = (1 / ( n − , . . . , / ( n − , − H i = ( − e i,i + e i +1 ,i +1 ) / i = 1 , . . . , n −
2. For A , according to the formula above for 2 ρ , wehave that 2 ρ ( H ) = n and therefore µ A = µ A ( H ) = e nt dt, and for A , 2 ρ ( H i ) = − i and therefore µ A = n − Y i =1 e − s i ds i . Definition 9.5.
We consider the following subsets of A :1. For S = ( S , . . . , S q ), A S ( H ) = { exp( s · H ) : s ∈ q Q i =1 [0 , S i ] } ⊆ A ( H ) .
2. When all S i are equal to T , we simply write A T ( H ) ⊆ A ( H ).The goal of this subsection is to prove the following: Proposition 9.6.
The family { A T ( H ) } T > is LWR with parameters which depend only on H , andthe fixed set A S ( H ) is well rounded with parameters which depend only on H , when S , . . . , S q arelarger from some δ > . E.g. δ = 4 / ρ ( H i ) if ρ ( H i ) = 0 , and δ = 1 otherwise.Remark . Notice that the sets A S ( H ) are clearly BCS and bounded, and are therefore (Remark 8.3)LWR; hence the content of the proposition for these sets is that their LWR parameters are uniform(i.e., do not depend on S ). Proof.
We only prove the proposition for the family { A T ( H ) } T >δ since the proof for the set A S ( H )is identical. Moreover, it is sufficient to consider the case of q = 1, and then the general case followsfrom Proposition 8.6. Notice that ln(( A T ( H ) (+ (cid:15) ) ) = [ − (cid:15) , T + (cid:15) ] , ln(( A T ( H )) ( − (cid:15) ) ) = [ (cid:15) , T − (cid:15) ] . We shall prove LWR of { A T ( H ) } T > computationally, by splitting to different cases according to thesign of ρ ( H ). Assume first that 2 ρ ( H ) = 0, and then µ A ( H ) (( A T ( H )) (+ (cid:15) ) ) = Z t = T + (cid:15)t = − (cid:15) e ρ ( H ) t dt = ( e ρ ( H )( T + (cid:15) ) − e − ρ ( H ) (cid:15) ) / ρ ( H ) , and µ A ( H ) (( A T ( H )) ( − (cid:15) ) ) = Z t = T − (cid:15)t = (cid:15) e ρ ( H ) t dt = ( e ρ ( H )( T − (cid:15) ) − e ρ ( H ) (cid:15) ) / ρ ( H ) . It follows that, µ A ( H ) (( A T ( H )) (+ (cid:15) ) ) − µ A H (( A T ( H )) ( − (cid:15) ) ) µ A ( H ) (( A T ( H )) ( − (cid:15) ) ) = (cid:0) e ρ ( H )( T + (cid:15) ) − e − ρ ( H ) (cid:15) (cid:1) − (cid:0) e ρ ( H )( T − (cid:15) ) − e ρ ( H ) (cid:15) (cid:1) e ρ ( H )( T − (cid:15) ) − e ρ ( H ) (cid:15) . • If 2 ρ ( H ) > e ρ ( H ) T + 1 e ρ ( H ) T | {z } ≤ · e ρ ( H ) (cid:15) − e − ρ ( H ) (cid:15) e − ρ ( H ) (cid:15) − e − ρ ( H ) T · e ρ ( H ) (cid:15) . (cid:15) ≤ · ρ ( H ) and T ≥ ρ ( H ) it holds that e ρ ( H ) (cid:15) − e − ρ ( H ) (cid:15) ≤ · ρ ( H ) (cid:15) and e − ρ ( H ) (cid:15) − e − ρ ( H ) T · e ρ ( H ) (cid:15) ≥ /
2; then, µ A ( H ) (( A T ( H )) (+ (cid:15) ) ) − µ A ( H ) (cid:0) (( A T ( H )) ( − (cid:15) ) ) (cid:1) µ A ( H ) (( A T ( H )) ( − (cid:15) ) ) ≤ · · ρ ( H ) (cid:15) / · ρ ( H ) . • If 2 ρ ( H ) <
0, we have= ( e − ρ ( H ) (cid:15) − e ρ ( H )( T + (cid:15) ) ) − ( e ρ ( H ) (cid:15) − e ρ ( H )( T − (cid:15) ) ) e ρ ( H ) (cid:15) − e ρ ( H )( T − (cid:15) ) = ( e ρ ( − H ) (cid:15) − e − ρ ( − H ) (cid:15) ) + ( e − ρ ( − H )( T − (cid:15) ) − e − ρ ( − H )( T + (cid:15) ) ) e ρ ( H ) (cid:15) − e ρ ( H )( T − (cid:15) ) = (1 + e − ρ ( − H ) T ) | {z } ≤ · e ρ ( − H ) (cid:15) − e − ρ ( − H ) (cid:15) e − ρ ( − H ) (cid:15) − e − ρ ( − H ) T · e ρ ( − H ) (cid:15) . So, the same computation as in the previous case shows that the last expression is ≤ · · | ρ ( H ) | (cid:15) / =12 · | ρ ( H ) | (cid:15) when (cid:15) ≤ · ρ ( H ) and T ≥ ρ ( H ) .Finally, when 2 ρ ( H ) = 0, µ A ( H ) (( A T ( H )) (+ (cid:15) ) ) µ A ( H ) (( A T ( H )) ( − (cid:15) ) ) = T + 2 (cid:15)T − (cid:15) = 1 + 4 T − (cid:15) (cid:15) ≤ (cid:15), when T − (cid:15) >
1, which holds when for (cid:15) < / T >
10 The Iwasawa roundomorphism
In Subsection 8.2 we defined maps called roundomorphisms, for which the pre-image of a well roundedfamily is in itself well rounded. We also introduced a map r on SL n ( R ), and the aim of this section isto prove that r is a roundomorphism, allowing us to reduce the well roundedness of families in SL n ( R )to well roundedness of their projections to K , A , A N and N . We begin by showing that (themore crude) map G → K × A × N projecting to the Iwasawa coordinates of a semisimple group is aroundomorphism. Recall that we let G denote a semisimple Lie group with finite center and Iwasawa decomposition G = KAN . The subgroups K , A and N are equipped with measures µ K , µ A and µ N respectively,such that for a given Haar measure µ G of G , µ G = µ K × µ A × µ N . Note that while µ K and µ N areHaar measures of their corresponding group, µ A is not (see Definition 9.2 for µ A ).Let a be the Lie algebra of A , n the Lie algebra of N , and recall that { φ , . . . φ p } ⊂ a ∗ are thepositive (restricted) roots w.r.t. n . Here the φ i ’s are not necessarily different, but with multiplicities.For a = exp ( H ) ∈ A define m ( H ) := max i {− φ i ( H ) , } , err ( a ) := C e m( H ) , (10.1)where C norm ≥ k·k on n in the followingmanner: (1 /C norm ) k Z k ∞ ≤ k Z k ≤ C norm k Z k ∞ for every Z ∈ n . Remark . Notice that err ( · ) is sub-multiplicative:err ( a a ) ≤ err ( a ) err ( a ) . Proposition 10.2 (Effective Iwasawa decomposition) . Let G be a semisimple Lie group with finitecenter. The diffeomorphism defining the Iwasawa decomposition r : G → K × A × N , r ( g ) = ( k, a, n ) is a f -roundomorphism w.r.t. O G(cid:15) , O K × A × N(cid:15) and f ( g ) (cid:28) C ( n ) · err ( a ) , where C ( n ) = k Ad n k op .The proof requires the following auxiliary lemma. Lemma 10.3.
Let N − := Θ ( N ) , where Θ is a global Cartan involution compatible with the givenIwasawa decomposition. Then A acts on both N, N − by conjugation such that the following holds: a − O N(cid:15) a ⊆ O N err ( a ) (cid:15) ,a O N − (cid:15) a − ⊆ O N − err ( a ) (cid:15) . Proof.
First we introduce some notations. Let Z , . . . , Z p be the corresponding linearly independenteigenvectors in g of φ , . . . φ p respectively. Denote n x = n [ x ,...,x p ] := exp( p P i =1 x i Z i ) . Then N = { n x : x ∈ R p } ; N − = { n − x = Θ( n x ) : x ∈ R p } . For every H ∈ a and Z ∈ n the action of a − = exp ( − H ) on exp ( Z ) is given byConj exp( − H ) (exp ( Z )) = exp(Ad e − H ( Z )) = exp( e ad − H ( Z )) . In particular, if Z = P pi =1 x i Z i then (since ad − H ( Z i ) = [ − H, Z i ] = φ i ( − H ) · Z i and therefore e ad − H ( Z i ) = e φ i ( − H ) · Z i ):Conj exp( − H ) (exp( p P i =1 x i Z i )) = exp(Ad e − H ( p P i =1 x i Z i )) = exp( p P i =1 x i Ad e − H ( Z i ))= exp( p P i =1 x i · e ad − H ( Z i )) = exp( p P i =1 x i · e φ i ( − H ) · Z i ) . As a result, a − · n x · a = exp ( − H ) · n x · exp ( H ) = n [ x e φ − H ) ,...,x p e φp ( − H ) ] = n h x, ( e − φi ( H ) ) pi =1 i . If a − · n x · a = n y , then for n x ∈ O N(cid:15) and k x k < (cid:15) it holds for y that k y k = kh x, ( e − φ i ( H ) ) pi =1 ik ≤ C norm kh x, ( e − φ i ( H ) ) pi =1 ik ∞ ≤ C norm k x k ∞ k ( e − φ i ( H ) ) pi =1 k ∞ ≤ (cid:15) · err ( a ) . Thus, a − O N(cid:15) a ⊆ O N err ( a ) (cid:15) . The second part follows from the first by applying Θ (the global Cartan involution) to the above.As a final preparation to the proof of Proposition 10.2, we list some properties of the families ofidentity neighborhoods O G(cid:15) = exp G ( B (cid:15) ) appearing in the statement of the proposition. We let G be ageneral Lie group. Then O G(cid:15) has the following properties :30. (
Conjugation by g dilates by k Ad g k ) If the Lie algebra of G is g then for every g ∈ G , g − O G(cid:15) g ⊆ O G(cid:15) ·k Ad g k op = exp { Z ∈ g : k Z k ≤ (cid:15) · k Ad g k op } , where k·k is any euclidean norm on g and k·k op is the norm on the space of linear g -operators.2. ( Connectivity ) O G(cid:15) is a connected subset of G .3. ( Additivity ) for small enough (cid:15) and δ , there exists c > O G(cid:15) O Gδ ⊆ O Gc ( (cid:15) + δ ) .4. ( Decomposition of G allows decomposition of O G(cid:15) ) If G is semi-simple (as it is in Proposition10.2), hence has Iwasawa decomposition, the family O G(cid:15) is equivalent to the family O K(cid:15) O A(cid:15) O N(cid:15) =exp K × A × N ( B (cid:15) ) of identity neighborhoods in G in the sense that there exist (cid:15) , c, C > < (cid:15) < (cid:15) it holds that O Gc(cid:15) ⊆ O
K(cid:15) O A(cid:15) O N(cid:15) ⊆ O
GC(cid:15) . Using Bruhat coordinateson identity neighborhood in G , the family O G(cid:15) is also equivalent to the family O M(cid:15) O N − (cid:15) O A(cid:15) O N(cid:15) ,where M = ( Z K ( A )) ; we may assume that the parameter (cid:15) is the same. proof of Proposition 10.2. Clearly, we only need to show that r (cid:0) O G(cid:15) g O G(cid:15) (cid:1) ⊆ O K × A × Nf(cid:15) r ( g ) O K × A × Nf(cid:15) , where f is as in the statement. This will be accomplished in three steps. Step 1: Left perturbations.
According to Properties 4 and 1, there exist (cid:15) , c , c > (cid:15) < (cid:15) O G(cid:15) kan = k (cid:0) k − O G(cid:15) k (cid:1) an ⊆ k O Gc (cid:15) an ⊆ k O Kc (cid:15) O Ac (cid:15) O Nc (cid:15) an = k O Kc (cid:15) · O Ac (cid:15) a · a − O Nc (cid:15) an. By Lemma 10.3, a − O N(cid:15) a ⊆ O N err ( a ) (cid:15) , hence r (cid:0) O G(cid:15) g (cid:1) ⊆ O K × A × Nc err ( a ) (cid:15) r ( g ) O K × A × Nc (cid:15) . Step 2: Right perturbations.
By Properties 4 (for the Bruhat coordinates) and 1, kan O G(cid:15) = ka (cid:0) n O G(cid:15) n − (cid:1) n ⊆ ka O GC ( n ) (cid:15) n ⊆ ka · O Mc C ( n ) (cid:15) O N − c C ( n ) (cid:15) O Ac C ( n ) (cid:15) O Nc C ( n ) (cid:15) · n = k O Mc C ( n ) (cid:15) · a · O N − c C ( n ) (cid:15) O Ac C ( n ) (cid:15) O Nc C ( n ) (cid:15) n = k O Mc C ( n ) (cid:15) (cid:16) a O N − c C ( n ) (cid:15) a − (cid:17) a O Ac C ( n ) (cid:15) O Nc C ( n ) (cid:15) n. By Lemma 10.3, a O N − c C ( n ) (cid:15) a − ⊆ O N − c C ( n ) err ( a ) (cid:15) ⊆ O Gc C ( n ) err ( a ) (cid:15) . Moreover, for (cid:15) ≤ (cid:15) / ( c C ( n ) err ( a ))we have O Mc C ( n ) (cid:15) O Gc C ( n ) err ( a ) (cid:15) ⊆ O Kc C ( n ) (cid:15) O Kc C ( n ) err ( a ) (cid:15) O Ac C ( n ) err ( a ) (cid:15) O Nc C ( n ) err ( a ) (cid:15) ⊆ O Kc C ( n ) err ( a ) (cid:15) O Ac C ( n ) err ( a ) (cid:15) O Nc C ( n ) err ( a ) (cid:15) . As a result, kan O G(cid:15) ⊆ k O Kc C ( n ) err ( a ) (cid:15) O Ac C ( n ) err ( a ) (cid:15) O Nc C ( n ) err ( a ) (cid:15) a O Ac C ( n ) (cid:15) O Nc C ( n ) (cid:15) n. Let a (cid:15) ∈ O Ac C ( n ) (cid:15) . Write a = aa (cid:15) . By sub-multiplicativity of err ( · ) (Remark 10.1) we get, O Nc C ( n ) err ( a ) (cid:15) a = a a − O Nc C ( n ) err ( a ) (cid:15) a ⊆ a O Nc C ( n ) err ( a ) err ( a ) (cid:15) ⊆ a O Nc C ( n ) err ( a ) (cid:15) . Combining all of the above, we conclude kan O G(cid:15) ⊆ k O Kc C ( n ) err ( a ) (cid:15) O Ac C ( n ) err ( a ) (cid:15) a O Ac C ( n ) (cid:15) O Nc C ( n ) err ( a ) (cid:15) O Nc C ( n ) (cid:15) n. In other words, r (cid:0) g O G(cid:15) (cid:1) ⊆ O K × A × Nc C ( n )(err ( a ) +1) (cid:15) r ( g ) O K × A × Nc C ( n )(err ( a )+1) (cid:15) . tep 3: Combining left and right perturbations. Finally, using the additivity property 3 on O K × A × N(cid:15) we conclude that r (cid:0) O G(cid:15) g O G(cid:15) (cid:1) ⊆ O K × A × Nf ( g ) (cid:15) r ( g ) O K × A × Nf ( g ) (cid:15) for (cid:15) ≤ /f ( g ) and f ( g ) (cid:28) C ( n ) · err ( a ) . After having established that the map G → K × A × N projecting to the KAN coordinates is aroundomorphism, we deduce it for the KA A N and RI decompositions as well (Corollary 10.5). Lemma 10.4.
Let N be a connected nilpotent Lie group with Haar measure µ N . Suppose that N = N (cid:110) N , where N and N are two closed subgroups of N equipped with Haar measures µ N and µ N .Then each element in N can be decomposed in a unique way as n = n n , and the map r ( n ) = ( n , n ) ∈ N × N is a f -roundomorphism for some continuous f : N → R ≥ . If N is abelian, then f ≡ .Proof. The first condition in the definition of a roundomorphism is a consequence of the nilpotencyassumption (see [Kna02, Corollary 8.31, Theorem 8.32]). The second condition, local Lipschitzity,follows from the fact that r in the lemma is a diffeomorphism (see [HK20, Prop. 4.6]). Corollary 10.5 (Effective RI decomposition) . Let G be a semisimple Lie group with finite center andIwasawa decomposition G = KAN . Assume that N and N are closed subgroups of N equipped withHaar measures µ N , µ N such that N = N (cid:110) N and µ N = µ N × µ N . Similarly, let A and A beclosed subgroups of A such that A = A × A and µ A = µ A × µ A . The projection map r : G → K × A × A × N × N , r ( g ) = ( k, a , a , n , n ) is an f -roundomorphisms w.r.t. f ( g ) (cid:28) c ( n , n ) · err ( a a ) where c ( n , n ) is a continuous functions on N × N .Proof. This follows from Proposition 10.2 combined with Lemma 10.4 and the fact that a compositionof roundomorphisms is a roundomorphism ([HK20, Lemma 4.5]). f for the Iwasawa roundomorphism Assume the setting of Corollary 10.5, where we have shown that “the Refined Iwasawa decompositionmap”, r , is a roundomorphism, and expressed the error function f in terms of a , a . We now proceedto compute f under some assumptions on A , A , which are satisfied for the A , A introduced inSection 2 and are relevant for the counting problem in Proposition 7.1. The discussion is concluded inLemma 10.8, where we deduce the correct f for our counting problem, and it will be used in the proofof the proposition.Denote L := dim ( A ). Let H , . . . , H l , H , . . . , H L − l be a basis for a , and denote A = A ( H ) , A = A ( H ) , where H = ( H , . . . , H l ) and H = ( H , . . . , H L − l ). We compute f under the following assumption.: Assumption 10.6.
For every i = 1 , . . . , l assume H i ∈ C − { } , where C is the positive Weil chamberw.r.t. N . For every i = 1 , . . . , L − l , assume ρ ( H i ) < . The latter can be achieved, for example, byrequiring that H i ∈ −C − { } for every i . otation . For H = ( H , . . . , H q ) and m( H j ) as defined in Formula (10.1), denotem H = max j { m( H j ) } = max i,j {− φ i ( H j ) , } . The content of the following Lemma is that under assumption 10.6, the error function of theIwasawa roundomorphism is only affected by the A component of A . Lemma 10.8.
Under assumption 10.6, a t a s = exp( t · H + s · H ) satisfies that err ( a t a s ) ≤ C e m H s ,where s := s · (1 , . . . ,
1) = P s i . In particular, for G = SL n ( R ) and A , A as defined in Section 2, err ( a t a s ) ≤ C e s .Proof. If the elements H j are in C − { } , then m( t · H + s · H ) = m( s · H ) ≤ m H s . As for G = SL n ( R ) and A , A as defined in Section 2, the basis elements in a that correspond to A , A are H = (1 / ( n − , . . . , / ( n − , −
1) and H j = ( − e j,j + e j +1 ,j +1 ) / j = 1 , . . . , n − H j = max { , − ( − ) , − , − − } = 1for every j = 1 , . . . , n −
2, hence m H = 1.
11 The base sets
We return our focus to G = SL n ( R ). The aim of this section is to prove that r (( Q Ψ ) S ) is LWR, andtherefore (see second step in the plan on Section 8.2) the base set in r ((Ω short ) ST (Ψ , α )) = [ ( ( k,a ,n ) ,a ) ∈ r (Ψ S ) × A T ( k, a , n , a ) × N Y α ( a n ) is LWR independently of S . From now on, H and H j for j = 1 , . . . , n − Lemma 11.1.
For any Ψ ⊆ KF n − ⊂ Q that is a BCS, the set r (Ψ) is LWR in K × A × N . As aresult, r (Ψ S ) ) is LWR with parameters that do not depend on S .Remark . The set KF n − (resp. F n − ) itself is not LWR in Q (resp. P ), only its image under r is. Since Lemma 11.1 is about counting in a group that is a direct product, it is proved by working ineach of the components separately. Among the two components A and N , the problematic one is ofcourse A ; the role of the following two lemmas is to handle this component. Lemma 11.3.
The projection to the A ( H i ) component of F n − is bounded from below for every i = 1 , . . . , n − .Proof. We need to show that for every H ∈ a such that exp ( H ) · n ∈ F n − , it holds that thecoefficients of H in its presentation of a linear combination of { H j } are bounded from below. Thesecoefficients are given by linear functionals: H = P n − j =1 ψ j ( H ) H j (actually, { ψ j } n − j =1 ⊂ ( a ) ∗ is the dualbasis to { H j } n − j =1 ⊂ a ). Denote φ i := φ i +1 ,i where { φ i,j } are the roots for SL n ( R ) defined in Example9.4. Clearly { φ i } form a basis to ( a ) ∗ , and by Lemma 3.10 they satisfy that φ i ( H ) ≥ ln( √ /
2) forevery i = 1 , . . . , n − H as above. It is therefore sufficient to show that in the presentation ofevery ψ j as a linear combination of { φ i } , the coefficients are non-negative. Write ψ i = 2 P n − j =1 x i,j φ j and evaluate at each of H , . . . , H n − we obtain the following system of linear equations − − − − ... ...... − − x i, ... x i,n − = e i . A computation shows that the solution ( x i,j ) n − j =1 is indeed non-negative.33o see how the following lemma concerns the A component, notice that the group ( A H i , dµ A H i )is measure preserving isomorphic to ( R > , · , dx/x ) for every i = 1 , . . . , n − Lemma 11.4.
The map ψ : ( R > , · , dx/x ) → ( R , + , (0 , ∞ ) ( x ) · dx ) given by ψ ( x ) = 1 /x is a f -roundomorphism with f ( x ) = 2 /x .Proof. A standard computation shows that ϕ pushes dx/x to (0 , ∞ ) ( x ) · dx . Moreover, for (cid:15) < / ψ ( O R > (cid:15) x O R > (cid:15) ) ⊆ ψ ( x · [1 − (cid:15), (cid:15) ]) ⊆ x − · [1 − (cid:15), (cid:15) ]= ψ ( x ) + 2 f ( x ) [ − (cid:15), (cid:15) ] = O R f(cid:15) ψ ( x ) O R f(cid:15) . Proof of Lemma 11.1.
We start by showing that r ( KF n − ) ⊂ K × A × N is LWR. Consider the map ϕ : K × A × N → K × ( R , + , (0 , ∞ ) ( x ) · dx ) ( n − × N , induced by the map given in the previous Lemma. It is an f -roundomorphism with f ( k, x , . . . , x n − , n ) = n − x ··· x n − . Since, by Lemma 11.3, the projection to A H i of r ( F n − ) (hence of r ( KF n − )) is boundedfrom below for every i , we conclude that ϕ ( r ( KF n − )) is a bounded set.By Proposition 3.2, ∂ϕ ( r ( F n − )) ⊆ ϕ ( r∂ ( F n − )) ∪ K × ∂ ( R × ( n − > ) × N is contained in a finiteunion of lower dimensional embedded submanifolds, and therefore so is the boundary of KF n − ;so, according to Remark 8.3, ϕ ( r ( KF n − )) is LWR. Finally, since f | r ( KF n − ) is bounded, then byProposition 8.6 we conclude that r ( KF n − ) is LWR.As the boundary of Ψ is also contained in a finite union of lower dimensional embedded submani-folds, then r (Ψ) is LWR by the same considerations.We now turn to prove that the set r ( KF Sn − ) is LWR; this set is the intersection of r ( KF n − ) withthe set K × A S × π N ( F n − ), where π N ( F n − ) is the projection of F n − to N . According to [HK20,Lemma 3.4], LWR property is maintained under intersections, and so it is sufficient to show that K × A S × π N ( F n − ) is LWR. This is indeed the case since A S is LWR with a parameter independentof S (by Proposition 9.6), π N ( F n − ) and K are LWR since they are bounded BCS (see Lemma 3.10),and LWR is maintained under taking products by Remark 8.6. Thus r (cid:0) Ψ S (cid:1) = r (Ψ) ∩ r ( KF Sn − ) isagain LWR, as the intersection of two such sets.
12 The family Y αr ( F n − ) is BLC The goal of this Section is to show that the family Y αr ( F n − ) is BLC for all 0 < α ≤
1, according to theplan of proof for Proposition 7.1, described in Subsection 8.2.The domain F n − is a subset of P , which is a diffeomorphic and group isomorphic copy of P n − ,the group of ( n − × ( n −
1) upper triangular matrices with positive diagonal entries and determinant1. To simplify the notation, we consider the situation in general dimension with F m ⊂ P m , and write P m = A m N m where A m is the diagonal subgroup of SL m ( R ) and N m is the subgroup of uppertriangular unipotent matrices. In particular, we abandon the notations of P , A , N and keep inmind that for our purpose, one takes m = n −
1. The roundomorphism r introduced in Corollary 10.5now becomes r : P m → A m × N m z = an ( a, n ) . Let us recall some further notations that were introduced previously, perhaps with n − m .For z = [ z | · · · | z m ] ∈ F m we let Λ z denote the lattice spanned by the columns of z , and consider thelinear map L z : R m → R m given by z j e j for every j = 1 , . . . , m . Note that L z maps Λ z to Z n .34 emark . L − z ( x ) = zx for every x ∈ R m (i.e., the linear map L − z is given by the matrix z ). Hence, L z ( zx ) = x , namely the image under L z of a vector is its coordinates w.r.t. the basis { z , . . . z m } ,which is also clear from the definition of L z .We begin by considering the case of α = 1. Proposition 12.2.
The family Y r ( F m ) = { Y ( an ) } ( a,n ) ∈ r ( F m ) = { Y ( z ) } r ( z ) ∈ r ( F m ) is BLC w.r.t. O A m × N m (cid:15) . In the proof, Lemma 3.10, will play a key role. In particular, we note that the last part of thislemma implies shrinking property of conjugation of upper triangular matrices by elements of F m , andwe formulate this in the following corollary. Corollary 12.3.
Let [ z | · · · | z m ] = z = a z n z ∈ F m . Then for any upper triangular matrix p ,1. k a z pa − z k (cid:28) k p k ;2. k zpz − k , k z t pz − t k (cid:28) k p k .Proof. part 1 follows from the fact that if i ≤ j then a i (cid:28) a j and therefore (cid:12)(cid:12) a i p i,j a − j (cid:12)(cid:12) = | a i | | p i,j | | a j | − ≺ | a j | | p i,j | | a j | − = | p i,j | . Since p ij = 0 for i > j , then k a z pa − z k ≺ k a z pa − z k ≺ k p k (cid:28) k p k . For the second part notice that: k zpz − k = k a z n z pn − z a − z k ≤ k a z n z a − z k | {z } ≺ k a z pa − z k | {z } ≺k p k k a z n − z a − z k | {z } ≺ ≺ k p k , and k z t pz − t k = k n t z a z pa − z n − t z k ≤ k n t z k | {z } ≺ k a z pa − z k | {z } ≺k p k k n − t z k | {z } ≺ (cid:28) k p k . The following fact indicates the relation between the norms of z = a z n z and its columns, to theentries of a z and the covering radius of Λ z . Fact 12.4.
Let [ z | · · · | z m ] = z = a z n z in F m .1. For j = 1 , . . . , m , k z j k (cid:16) a j .2. ρ (Λ z ) (cid:16) a m (cid:16) k z k .Notation. Set Let E j := span R { e , . . . , e j } , where { e , . . . , e m } is the standard basis to R m . Proof.
According to Corollary 12.3 and Lemma 3.10, a i = dist( z i , E i − ) ≤ k z i k = k a z n z e i k = k a z n z a − z a z e i k ≤ k a z n z a − z k | {z } ≺ k a z e i k | {z } a i (cid:28) a i , which proves the first part. As for the second part, we have on the one hand that (by Lemma 3.10,parts (1) and (2)) k z k = k a z n z k ≺ k a z k (cid:16) a m and on the other hand that a m (cid:16) k a z k = k a z n z n − z k ≺ k a z n z k = k z k . The fact that a m (cid:16) ρ z is proved in [GM02, Theorem 7.9].35 emma 12.5. Let ( a , n ) ∈ O A m × N m (cid:15) ( a, n ) O A m × N m (cid:15) . If z = an, z = a n and z ∈ F m , then k z z − k , k z − z k ≤ C (cid:15) for some C > .Proof. Clearly ( a , n ) ∈ O A m × N m (cid:15) ( a, n ) O A m × N m (cid:15) is equivalent to z ∈ O A m (cid:15) a O A m (cid:15) O N m (cid:15) n O N m (cid:15) . Usingthe fact that O P m (cid:15) is equivalent to O A m (cid:15) O N m (cid:15) and Corollary 12.3 we obtain, O A m (cid:15) a O A m (cid:15) O N m (cid:15) n O N m (cid:15) = an (cid:16) n − O A m (cid:15) O N m (cid:15) n (cid:17) O N m (cid:15) ⊆ an · n − O P m c (cid:15) n · O N m (cid:15) ⊆ an O P m c (cid:15) O N m (cid:15) ⊆ z O P m c (cid:15) . Again using Corollary 12.3, one also obtains z O P m c (cid:15) = (cid:0) z O P m c (cid:15) z − (cid:1) z ⊆ O P m c (cid:15) z. Finally, fix C > O P m c (cid:15) ⊆ { p ∈ P m : k p k ≤ C (cid:15) } . The following lemma is the technical core of the proof of Proposition 12.2.
Lemma 12.6.
Suppose z, z ∈ F m and that r ( z ) ∈ O (cid:15) r ( z ) O (cid:15) . Let v ∈ Z m and write λ = zv, λ = z v .Then the following hold:1. k z t λ k (cid:28) k λ k ;2. k λ k ≤ (1 + C (cid:15) ) k λ k for the constant C > from Lemma 12.5;3. k z t λ − z t λ k (cid:28) (cid:15) k λ k .Proof. For the first part, recall that L − z ( x ) = zx and then k L − t z ( λ ) k = k z t λ k = k n t z a z λ k ≤ k n t z k | {z } ≺ k a z λ k ≺ k a z λ k . Next, let j ∈ { , . . . , m } such that λ ∈ E j \ E j − . By parts (4) and (3) respectively of Lemma 3.10: k aλ k ≺ a j k λ k ≤ k λ k . All in all, k L − t z ( λ ) k (cid:28) k λ k .For the second part, use Lemma 12.5: k λ k = k z v k = k z z − zv k ≤ k z z − k k zv k ≤ (1 + C ε ) k λ k . For the third part, it is clear that k z t λ − z t λ k ≤ k z t ( λ − λ ) k + k ( z t − z t ) λ k and we shall bound each of these two summands. The first one is bounded by k z t ( λ − λ ) k = k z t ( z − z ) v k = k z t ( I − z z − ) | {z } p ∈ P m zv k = k ( z t pz − t ) z t zv |{z} λ k ≤ k z t pz − t kk z t λ k where by Corollary 12.3, Lemma 12.5, and the first part of the current Lemma, ≺ k p k k z t λ k (cid:28) (cid:15) k λ k . The second summand is bounded by See fourth property of O G(cid:15) in Section 10 ( z t − z t ) λ k = k ( z t z t − I ) z t λ k = k (( z z ) t − I ) z t λ k ≤ k ( z z ) t − I k · k z t λ k . By Lemma 12.5 and the first part of the current Lemma, the above is ≺ C (cid:15) · (cid:15) k λ k , and by the secondpart of the current lemma the latter is ≤ C (cid:15) · (cid:15) · ((1 + C (cid:15) ) k λ k ) ≺ (cid:15) k λ k . Towards proving Proposition 12.2, stating that the family Y F m is BLC, we prove that this familysatisfies the fourth property of BLC. Lemma 12.7.
The family Y is bounded uniformly from above. Namely, there exists R > thatdepends only on m such that Y ( z ) = L z ( Dir ( z )) is contained in B R for every z ∈ F m . We introduce a notation, to be used in the proofs of Lemma 12.7 and Proposition 12.2. For λ ∈ Λ z ,write H | λ | for the strip H | λ | := n x : |h x, λ i| ≤ k λ k / o . It is easy to check that it consists of all the vectors in R m which are closer to the origin than to ± λ .As a result, Dir(Λ z ) = \ = λ ∈ Λ z H | λ | . (12.1) Proof.
According to (12.1) and definition of H | λ | , an element x ∈ Dir (Λ z ) satisfies that |h λ, x i| ≤k λ k / = λ ∈ Λ z . In particular, this holds for λ ∈ { z , . . . , z m } ⊂ Λ z (the columns of z ).Recall that by Remark 12.1, x = zL z ( x ). The inequality |h z j , x i| ≤ k z j k / |h z j / k z j k , zL z ( x ) i| ≤ /
2, i.e. |h z t z j / k z j k , L z ( x ) i| ≤ / |k z j k − z t j z | {z } row · L z ( x ) | {z } column | ≤ / . Considering all m inequalities, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − k z k − z t1 − ... − k z m k − z t m − · z · L z ( x ) | {z } column (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 / , . . . , / t (where one should understand ≤ and |·| as referring to the components), namely (cid:12)(cid:12)(cid:12) diag( k z j k ) mj =1 · z t z · L z ( x ) (cid:12)(cid:12)(cid:12) ≤ (1 / , . . . , / t . Let g := diag( k z j k ) mj =1 · z t z ; based on the last inequality, in order to show that k L z ( x ) k is boundedby some constant R = R ( m ), it is sufficient to prove that k g − k (cid:28) m . Indeed, k g − k = k z − z − t diag( k z j k ) mj =1 k Fact 12.4part (1) (cid:28) k z − z − t diag( a j ) mj =1 k = k z − z − t a z k = k n − z a − z n − t z a z k = k n − z (cid:0) a z n − z a − z (cid:1) t k ≤≤ k n − z k | {z } (cid:28) · k a z n − z a − z k | {z } (cid:28)k n − z k(cid:28) (cid:28) k a z n − z a − z k (cid:28) k n − z k is also due to Corollary 12.3.37e are now ready to prove Proposition 12.2. proof of Proposition 12.2. We begin by verifying property
BLC (I) . According to (12.1), it is sufficientto prove that this property holds for each strip H | λ | separately, namely that L z (cid:0) H | λ | (cid:1) + B (cid:15) ⊆ (1 + C(cid:15) ) L z (cid:0) H | λ | (cid:1) . Since (Remark 12.1) L z (cid:0) H | λ | (cid:1) = n y : (cid:12)(cid:12)(cid:10) L − t z ( λ ) , y (cid:11)(cid:12)(cid:12) ≤ k λ k / o = n y : (cid:12)(cid:12)(cid:10) z t λ, y (cid:11)(cid:12)(cid:12) ≤ k λ k / o , and L z (cid:0) H | λ | (cid:1) + B (cid:15) ⊆ n x : (cid:12)(cid:12)(cid:10) x, L − t z ( λ ) (cid:11)(cid:12)(cid:12) ≤ k λ k / (cid:13)(cid:13) L − t z ( λ ) (cid:13)(cid:13) · (cid:15) o , the desired inclusion is equivalent to k λ k / (cid:15) (cid:13)(cid:13) L − t z ( λ ) (cid:13)(cid:13) ≤ (1 + C(cid:15) ) k λ k / . This indeed holds, since by part 1 of Lemma 12.6, k L − t z ( v ) k = k z t v k (cid:28) k v k .We turn to prove property BLC (II) . As with property
BLC (I) , it is sufficient to verify it foreach strip H | λ | separately. Assume that r P ( z ) ∈ O (cid:15) r P ( z ) O (cid:15) . Let y ∈ Dir( z ) ⊂ R m , namely (cid:12)(cid:12)(cid:10) z t λ , y (cid:11)(cid:12)(cid:12) ≤ k λ k / = λ ∈ Λ z . We need to prove that y ∈ (1 + C(cid:15) ) L z (cid:0) H | λ | (cid:1) , for all 0 = λ ∈ Λ z , namely that (cid:12)(cid:12)(cid:10) z t λ, y (cid:11)(cid:12)(cid:12) ≤ (1 + C(cid:15) ) k λ k / . Now, (cid:12)(cid:12)(cid:10) z t λ, y (cid:11)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:10) z t λ , y (cid:11)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:10) z t λ − z t λ , y (cid:11)(cid:12)(cid:12) ≤ k λ k / k y k · k z t λ − z t λ k . According to Lemma 12.7, ≤ k λ k / R · k z t λ − z t λ k = ( k λ k / k λ k + 2 R (cid:13)(cid:13) z t λ − z t λ (cid:13)(cid:13) / k λ k ) · k λ k / k λ k / k λ k | {z } ≤ C (cid:15) + 2 R (cid:13)(cid:13) z t λ − z t λ (cid:13)(cid:13) / k λ k | {z } ≺ (cid:15) ) k λ k / ≤ (1 + C(cid:15) ) · k λ k / . The
BLC (III) is trivial since Y ( z ) = L z (Dir( z )) are fundamental domains for Z m in R m , hence theirvolume is exactly 1. Property BLC (IV) for the family Y r ( F m ) is the content of Lemma 12.7.The following is the main result of this section. Proposition 12.8.
For every < α ≤ the family Y αr ( F m ) defined in Formula (5.1) is BLC w.r.t. O (cid:15) as in Proposition 12.2.Proof. Set ρ z := ρ (Λ z ), and similarly for z . To prove the first property, it is sufficient to show thatfor some C > B αρ z + L − z ( B (cid:15) ) ⊆ (1 + C(cid:15) ) B αρ z . By Fact 12.4, there is a constant
C > L − z ( B (cid:15) ) = z ( B (cid:15) ) ⊆ B k z k (cid:15) ⊆ B C ( αρ z ) (cid:15) .
38s a result, B αρ z + L − z ( B (cid:15) ) ⊆ B αρ z + B Cαρ z (cid:15) ⊆ B αρ z (1+ C(cid:15) ) = (1 + C(cid:15) ) B ρ z . As for the second property, since it is maintained under intersections, it is sufficient to prove that L z ( B αρ z ) ⊆ (1 + C(cid:15) ) L z ( B αρ z ) . Or in other words, L − z L z ( B αρ z ) ⊆ (1 + C(cid:15) ) B αρ z . To this end, we first claim that there exists C > ρ z ≤ (1 + C (cid:15) ) (1 + C (cid:15) ) ρ z ; (12.2)indeed, by property BLC (II) for Y r ( F m ) (Proposition 12.2), we have that L z (Dir(Λ z )) ⊆ (1 + C (cid:15) ) · L z (Dir(Λ z ))and therefore Dir(Λ z ) ⊆ (1 + C (cid:15) ) · L − z L z (Dir(Λ z ))(Lem. 12.1) ⊆ (1 + C (cid:15) ) · z z − · Dir(Λ z ) ⊆ (1 + C (cid:15) ) · k z z − k Dir(Λ z )(Lem. 12.5) ⊆ (1 + C (cid:15) ) · (1 + C ) Dir(Λ z ) . Now, L − z L z ( B αρ z ) ⊆ k zz k · B Rmk. 12.5 ⊆ αρ z (1 + C (cid:15) ) · B αρ z eq. (12.2) ⊆ (1 + C (cid:15) ) (1 + C (cid:15) ) · B αρ z which establishes that L − z L z ( B αρ z ) ⊆ (1 + C(cid:15) ) B αρ z and completes the proof of the second property.Property BLC (IV) is a direct consequence of Lemma 12.7, and so we turn to prove the thirdproperty. First, we claim that for z = a z n z ∈ F m , the vectors ± a j := ( a z / e j = ( a j / e j lie in Dir(Λ z ). Indeed, suppose otherwise that there exists λ ∈ Λ z such that k a j + λ k < k a j k . Then λ cannot lie inside V j − = span { z , . . . z j − } , because if it did then it would have been orthogonal toa j , which implies k a j k + k λ k = k a j + λ k assumption < k a j k , a contradiction. Hence λ / ∈ V j − , implying λ = λ j − + λ ⊥ j − with 0 = λ ⊥ j − ∈ V ⊥ j − . Now, a j = dist( z j , V j − ) ( λ/ ∈ Vj − ) ≤ dist( λ, V j − ) = k λ ⊥ j − k ≤ k λ k ≤≤ k λ + a j k + k a j k assumption < k a j k = a j . This is clearly a contradiction, establishing that the vectors ± a j indeed lie inside Dir(Λ z ).Let c > k c a j k = c a j ≤ αρ z for every j = 1 , . . . , m ; such c exists and is independentof z because a (cid:28) · · · a m (cid:28) ρ z (according to Fact 12.4 and part (2) of Lemma 3.10). We may assumethat c ≤ z ) is convex and contains the origin and the points a j ), that thepoints c a j are also contained in Dir(Λ z ). They are obviously contained in B αρ z , hence by convexity[ − c, c ] a × · · · × [ − c, c ] a m = c m · m Y j =1 h − a j , a j i ⊆ Dir(Λ z ) ∩ B αρ z . The above shape has volume c m · Q mj =1 a i = c m · det ( z ); its image under L z = z − has therefore volume c m . It follows that the volume of L z (cid:0) Dir(Λ z ) ∩ B αρ z (cid:1) is bounded from below by c m , which does notdepend on z . 39 We will prove Proposition 7.1 in a slightly greater generality, when the lattice Γ < SL n ( R ) is general,and when the sets Ω ST are fibered over a family D r ( F n − ) = {D ( a n ) } ( a ,n ) ∈ r ( F n − ) that is notnecessarily Y α . Indeed, we consider:Ω ST (Ψ) = [ q ∈ Q S Ψ q · A T N ( z q ) . Proposition 7.1 is a consequence of Proposition 12.2, combined with the following:
Theorem 13.1.
Let Ω ST (Ψ) be as above, where Ψ ⊆ L n − ,n is a BCS, and D r ( F n − ) is a BLC familyof subsets of R n . Set λ n = n n − . Let Γ < SL n ( R ) be a lattice and τ = τ (Γ) .1. For < (cid:15) < τ , S = ( S , . . . , S n − ) , S = P n − i =1 S i and every T ≥ S nλ n τ + O D (1) , (cid:16) Ω ST (Ψ) ∩ Γ (cid:17) = µ ( Ω ST (Ψ)) µ ( G/ Γ) · e nT n + O Ψ ,(cid:15) ( e S /λ n e nT (1 − τ + (cid:15) ) ) .
2. For < (cid:15) < τ , δ ∈ (0 , τ − (cid:15) ) , S ( T ) = ( S ( T ) , . . . , S n − ( T )) such that S ( T ) = P S i ( T ) Part 1 . Consider the image of Ω ST (Ψ) under r , which is of the form r (cid:16) Ω ST (Ψ) (cid:17) = [ ( k, ( a ,n ) ,a ) ∈ r (cid:0) Q S Ψ (cid:1) × A T ( k, a , n , a ) × N ( a ,n ) (see Subsection 8.2). We claim that it is a well rounded family with increasing parameter T in thegroup K × A × N × A × N . First, since the family D r ( F n − ) is assumed to be BLC, and theprojection of r (cid:16) Q S Ψ (cid:17) to A × N is contained in r ( F n − ), then the restriction of D to this projectionis also BLC. Since D is independent of the k, a components, we may extend the set over which it isparameterized to include these components ([HK20, Cor. 5.3]), hence the family D r (cid:0) Q S Ψ (cid:1) × A T is BLC.As for the base set, Ψ is a BCS by assumption, and so Q Ψ is also a BCS, by Proposition 3.2. Thus, r (cid:16) Q S Ψ (cid:17) ⊂ K × A × N is LWR according to Lemma 11.1, with parameters that do not depend on S . Since A T is LWR (Proposition 9.6), then Remark 8.6 implies that r (cid:16) Q S Ψ (cid:17) × A T is LWR inside K × A × N × A . By Proposition 8.8, this implies that the family r (cid:16) Ω ST (Ψ) (cid:17) is LWR with Lipschitzconstant that is (cid:16) r is an f -roundomorphism with f ( ka a s n n ) (cid:28) e s , it follows from Proposition 8.6 that Ω ST (Ψ) ⊂ SL n ( R ) is LWR with C ≺ Ψ e S and T that is in-dependent of S and of the family D . The first part of the theorem now follows from Theorem 8.4;it is only left to observe, for the error term, that µ (Ω ST (Ψ)) (cid:16) Ψ e nT by Assumption 10.6, and toverify the lower bound on T . The latter is obtained by substituting the bound on the parameter C into the condition 8.2 in Theorem 8.4. Indeed, using the notation of Theorem 8.4, this condi-tion is equivalent to τ (Γ) ln µ ( B T )) ≥ dim G G ln C B + O (1). Substituting C B = C ≺ Ψ e S and µ ( B T ) = µ (Ω ST (Ψ)) ≤ µ (Ω T (Ψ)) (cid:16) Ψ e nT , the condition translates into τ (Γ) · nT ≥ dim G G · S + O Ψ (1) = S /λ n + O Ψ (1)40.e. to T ≥ S / ( nτ (Γ) λ n ) + O Ψ (1) . Part 2 . Let S = S ( T ) > 0. In order for the main term in part (1) to be of lower order than themain term, we require the existence of a parameter γ ∈ (0 , 1) for which S /λ n + (1 − τ (Γ) + (cid:15) ) · nT < γ · nT. This is equivalent to S < λ n · ( γ + τ (Γ) − (cid:15) − nT. Hence, if we denote by δ the number γ + τ (Γ) − (cid:15) − 1, we must require that δ > γ = δ + (1 + (cid:15) − τ (Γ)) lies in (0 , < (cid:15) < τ (Γ), then clearly 0 < (cid:15) − τ (Γ) < 1, so the conditionon δ becomes δ ∈ (0 , τ (Γ) − (cid:15) ) . The condition on T in part (1) is equivalent to S ≤ nλ n τ (Γ) · T + O Ψ (1), i.e. S ≤ min { nλ n δ · T + O Ψ (1) , nλ n τ (Γ) · T } = nλ n δ · T + O Ψ (1)for T large enough and δ ∈ (0 , τ (Γ) − (cid:15) ). References [AES16a] M. Aka, M. 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