Unipotent flows and counting lattice points on homogeneous varieties

Alex Eskin^{∗}, Shahar Mozes^{†}, and Nimish Shah^{‡}

February 1, 1996

1 Introduction

In this paper, using ergodic properties of subgroup actions on homogeneous spaces of Lie groups, we study asymtotic behaviour of number of lattice points on certain affine varieties. Consider for instance the following.

Example 1 Let p(λ) be a monic polynomial of degree n ≥ 2 with integer coefficients and irreducible over ℚ. Let M_{n}(ℤ) denote the set of n×n integer matrices, and put

V_{p}(ℤ) = {A ∈ M_{n}(ℤ) : det(λI − A) = p(λ)}.

Hence V_{p}(ℤ) is the set of integral matrices with characteristic polynomial p(λ). Consider the norm on n×n real matrices given by ∥(x_{ij})∥ = \sqrt{∑_{ij} x_{ij}^{2}}, and let N(T, V_{p}) denote the number of elements of V_{p}(ℤ) with norm less than T.

Theorem 1.1 Suppose further that p(λ) splits over ℝ, and for a root α of p(λ) the ring of algebraic integers in ℚ(α) is ℤ[α]. Then, asymptotically as T → ∞,

N(T, V_{p}) ∼ \frac{2^{n−1}hRω_{n}}{\sqrt{D} · ∏_{k=2}^{n} Λ(k/2)}T^{n(n−1)/2}

∗Research partially supported by the NSF and the Alfred P. Sloan Foundation.

†Sponsored in part by the Edmund Landau Center for research in Mathematical Analysis supported by the Minerva Foundation (Germany).

‡Research at MSRI supported by NSF grant DMS 8505550.

where h is the class number of ℤ[α], R is the regulator of ℚ(α), D is the discriminant of p(λ), ω_{n} is the volume of the unit ball in ℝ^{n(n−1)/2}, and Λ(s) = π^{−s}Γ(s)ζ(2s).

Example 1 is a special case of the following counting problem which was first studied in [DRS] and [EM].

The counting problem: Let W be a real finite dimensional vector space with a ℚ structure and V a Zariski closed real subvariety of W defined over ℚ. Let G be a reductive real algebraic group defined over ℚ, which acts on W via a ℚ-representation ρ : G → GL(W). Suppose that G acts transitively on V. Let ∥ · ∥ denote a Euclidean norm on W. Let B_{T} denote the ball of radius T > 0 in W around the origin, and define

N(T, V) = |V ∩ B_{T} ∩ ℤ^{n}|,

the number of integral points on V with norm less than T. We are interested in the asymptotics of N(T, V) as T → ∞.

We use the rich theory of unipotent flows on homogeneous spaces developed in [Mar1], [DM1], [Rat1], [Rat2], [Rat3], [Rat4], [Sha1] and [DM3] to obtain results in this direction.

Let Γ be a subgroup of finite index in G(ℤ) such that W (ℤ)Γ ⊂ W (ℤ). By a theorem of Borel and Harish-Chandra [BH-C], V (ℤ) is a union of finitely many Γ-orbits. Therefore to compute the asymptotics of N(T, V) it is enough to consider each Γ-orbit, say O, separately and compute the asymptotics of

N(T, V, O) = |O ∩ B_{T} |.

Suppose that O = v_{0} · Γ for some v_{0} ∈ V (ℤ). Then the stabilizer H = {g ∈ G : v_{0}g = v_{0}} is a reductive real algebraic ℚ-subgroup, and V ≅ H\G. Define

R_{T} = {Hg ∈ H\G : v_{0}g ∈ B_{T} },

the pullback of the ball B_{T} to H\G.

Assume that G^{0} and H^{0} do not admit nontrivial ℚ-characters. Then by the theorem of Borel and Harish-Chandra, Γ\G admits a G-invariant (Borel) probability measure, say μ_{G}, and (Γ ∩ H)\H admits an H-invariant probability measure, say μ_{H}. Now the natural inclusion (Γ ∩ H)\H ↪ Γ\G is an H-equivariant proper map. Let π : G → Γ\G be the natural quotient map. Then the orbit π(H) is closed, (Γ ∩ H)\H ≅ π(H), and μ_{H} can be treated as a measure on Γ\G supported on π(H). Such finite invariant measures supported on closed orbits of subgroups are called homogeneous measures. Let λ_{H\G} denote the (unique) G-invariant measure on H\G induced by the normalization of the Haar measures on G and H.

The following result was proved in [DRS]; subsequently a simpler proof appeared in [EM].

Theorem 1.2 Suppose that V is affine symmetric and Γ is irreducible (equivalently, H is the set of fixed points of an involution of G, and G is ℚ-simple). Then asymptotically as T → ∞,

N(T, V, O) ∼ λ_{H\G}(R_{T}).

Translates of homogeneous measures. For any g ∈ G, let μ_{H}g denote the translated measure defined as

μ_{H}g(E) = μ_{H}(Eg^{−1}), ∀ Borel sets E ⊂ Γ\G.

Note that μ_{H}g is supported on π(H)g. A key ingredient in the proofs of Theorem 1.2 in [DRS] and [EM] is showing that if H is the set of fixed points of an involution of G then for any sequence {g_{i}} ⊂ G, such that {Hg_{i}} has no convergent subsequence in H\G, the translated measures μ_{H}g_{i} get ‘equidistributed’ on Γ\G as i → ∞; that is, the sequence {μ_{H}g_{i}} weakly converge to μ_{G}. The method of [DRS] uses spectral analysis on Γ\G, while the argument of [EM] uses the mixing property of the geodesic flow. However, both methods seem limited essentially to the affine symmetric case. It should be remarked that for the proof of Theorem 1.2 one needs only certain averages of translates of the form μ_{H}g to become equidistributed.

Motivated by this approach to the counting problem, we study the limit distributions of translates of homogeneous measures. We show that under certain conditions if for some sequence {g_{i}} we have lim μ_{H}g_{i} = ν then the measure ν is again homogeneous. We give exact algebraic conditions on the sequence {g_{i}} relating it to the limit measure ν. Using this analysis, we show that the counting estimates as in Theorem 1.2 hold for a large class of homogeneous varieties. The following particular cases of homogeneous varieties, which are not affine symmetric, are of interest. We first place Example 1 in this context.

Example 1 continued. Note that V_{p}(ℤ) is the set of integral points on the real subvariety V_{p} = {A ∈ M_{n}(ℝ) : det(λI − A) = p(λ)} contained in the vector space W = M_{n}(ℝ). Let G = {g ∈ GL_{n}(ℝ) : det g = ±1}. Then G acts on W via conjugations, and V_{p} is a closed orbit of G (see [New, Theorem III.7]). Put Γ = G(ℤ) = GL_{n}(ℤ). The companion matrix of p(λ) is

v_{0} = (\begin{matrix}0 & & 0 & −a_{n} \\ 1 & & 0 & −a_{n−1} \\ 0 & ⋯ & ⋮ & ⋮ \\ ⋮ & & 0 \\ 0 & & 1 & −a_{1} \end{matrix}) ∈ V_{p}(ℤ).  (1)

The centralizer H of v_{0} is a maximal ℚ-torus and H^{0} has no nontrivial ℚ-characters. We emphasize that H is not the set of fixed points of an involution, and the variety V_{p} = H\G is not affine symmetric. Nevertheless, we show that N(T, V_{p}, Γv_{0}) ∼ λ_{H\G}(R_{T}). By computing the volumes, we obtain the following estimate.

Theorem 1.3 Let N(T, V_{p}) be the number of points on V_{p}(ℤ) of norm less than T. Then asymptotically as T → ∞,

N(T, V_{p}) ∼ c_{p}T^{n(n−1)/2},

where c_{p} > 0 is an explicitly computable constant.

We obtain a ‘formula’ for calculating c_{p}; for the sake of simplicity we calculate it explicitly only under the additional assumptions on p(λ) of Theorem 1.1.

See [BR] for some deeper consequences of the above result.

Example 2. Let A be a nondegenerate indefinite integral quadratic form in n ≥ 3 variables and of signature (p, q), where p ≥ q, and B a definite integral quadratic form in m ≤ p variables. Let W = M_{m×n}(ℝ) be the space of m × n matrices. Consider the norm on W given by ∥(x_{ij})∥ = \sqrt{∑_{i,j} x_{ij}^{2}}. Define

V_{A,B} = {X ∈ M_{m×n}(ℝ) : XA^{t}X = B}.

Thus a point on V_{A,B}(ℤ) corresponds to a way of representing B by A over ℤ. We assume that V_{A,B}(ℤ) is not empty.

The group G = SO(A) acts on W via right multiplication, and the action is transitive on V_{A,B}. The stabilizer of a point ξ ∈ V_{A,B} is an orthogonal group H_{ξ} in n − m variables. Let Γ = G(ℤ). Then the number of Γ-orbits on V_{A,B}(ℤ) is finite. Let ξ_{1}, …, ξ_{h} be the representatives for the orbits.

Theorem 1.4 Let N(T, V_{A,B}) denote the number of points on V_{A,B}(ℤ) with norm less than T. Then asymptotically as T → ∞,

N(T, V_{A,B}) ∼ ∑_{i=1}^{h} \frac{vol(Γ ∩ H_{ξ_{i}}\H_{ξ_{i}})}{vol(Γ\G)}c_{A,B}T^{r(n−r−1)}

where r = min(m, q), and c_{A,B} > 0 is an explicitly computable constant (see equation (40)).

Remark 1.5 In some ranges of p, q, m, n this formula may be proved by the Hardy-Littlewood circle method, or by Θ-function techniques. Using our method one also obtains asymptotic formulas for the number of points in the individual orbits Γξ_{i}.

Remark 1.6 In the case m > q, the asymptotics of the number of integer points does not agree with the heuristic of the Hardy-Littlewood circle method, even if the number of variables mn is very large compared to the number of quadratic equations m(m+1)/2. The discrepancy occurs because the null locus {X : XA^{t}X = 0} does not contain a non-singular real point (cf. [Bir, Theorem 1]) and so the ‘singular integral’ vanishes.

Limiting distibutions of translates of homogeneous measures:

The following is the main result of this paper which allows us to investigate the counting problems. The result is also of general interest, especially from the view point of ergodic theory on homogeneous spaces of Lie groups.

Theorem 1.7 Let G be a connected real algebraic group defined over ℚ, Γ ⊂ G(ℚ) an arithmetic lattice in G with respect to the ℚ-structure on G, and π : G → Γ\G the natural quotient map. Let H ⊂ G be a connected real algebraic ℚ-subgroup admitting no nontrivial ℚ-characters. Let μ_{H} denote the H-invariant probability measure on the closed orbit π(H). For a sequence {g_{i}} ⊂ G, suppose that the translated measures μ_{H}g_{i} converges to a probability measure μ on Γ\G. Then there exists a connected real algebraic ℚ-subgroup L of G containing H such that the following holds:

(i) There exists c_{0} ∈ G such that μ is a c_{0}^{−1}Lc_{0}-invariant measure supported on π(L)c_{0}.

In particular, μ is a homogeneous measure.

(ii) There exist sequences {γ_{i}} ⊂ Γ and c_{i} → c_{0} in G such that γ_{i}^{−1}Hγ_{i} ⊂ L and Hg_{i} = Hγ_{i}c_{i} for all but finitely many i ∈ ℕ.

Our proof of this theorem is based on the following observation.

Proposition 1.8 Let the notation be as in Theorem 1.7. Then either there exists a sequence c_{i} → c in G such that μ_{i} = μ_{H}c_{i} for all i ∈ ℕ (in which case μ = μ_{H}c), or μ is invariant under the action of a nontrivial unipotent one-parameter subgroup of G.

Thanks to this proposition, we can apply the well-developed techniques of unipotent flows to study limit distributions of translates of homogeneous measures, and eventually to the problem of counting lattice points on homogeneous varieties. It is of interest to note that much of the motivation for the extensive study of the behavior of unipotent flows on homogeneous spaces came from number theory. The celebrated Oppenheim conjecture, which was proved by G.A. Margulis in [Mar1], concerning values of irrational indefinite quadratic forms at integer vectors, had led M.S. Raghunathan to formulate his conjecture concerning the closures of unipotent orbits. The Raghunathan conjecture as well as a measure theoretic version of it (conjectured by S.G. Dani and by G.A. Margulis) were proved in M. Ratner seminal work [Rat 1-4]. We refer the reader to the ICM addresses of Margulis [Mar2], Ratner [Rat5], and S.G. Dani [Dan2] and a survey article of Borel [Bor3] for a discussion of these and related questions.

In order to be able to apply Theorem 1.7 to the problem of counting, we need to know some conditions under which the sequence {μ_{H}g_{i}} of probability measures does not escape to infinity. Suppose further that G and H are reductive. Let Z(H) be the centralizer of H in G. By rationality π(Z(H)) is closed in Γ\G. Now if π(Z(H)) is noncompact, there exits a sequence {z_{i}} ⊂ Z(H) such that {π(z_{i})} is divergent; that is, it has no convergent subsequence. Then μ_{H}z_{i} escapes to the infinity; that is μ_{H}z_{i}(K) → 0 for any compact set K ⊂ Γ\G. The condition that π(Z(H)) is noncompact is equivalent to the condition that H is contained in a proper parabolic ℚ-subgroup of G. In the converse direction we have the following (see [EMS]).

Theorem 1.9 Let G be a connected real reductive algebraic group defined over ℚ, and H a connected real reductive ℚ-subgroup of G, both admitting no nontrivial ℚ-characters. Suppose that H is not contained in any proper parabolic ℚ-subgroup of G defined over ℚ. Let Γ ⊂ G(ℚ) be an arithmetic lattice in G and π : G → Γ\G the natural quotient map. Let μ_{H} denote the H-invariant probability measure on π(H). Then given an ϵ > 0 there exists a compact set K ⊂ Γ\G such that μ_{H}g(K) > 1 − ϵ, ∀g ∈ G.

The proof of this result uses generalizations of some results of Dani and Margulis [DM2]. Combining this theorem with Theorem 1.7, we deduce the following consequences.

Corollary 1.10 Suppose that H is reductive and a proper maximal connected real algebraic ℚ-subgroup of G. Then for any sequence {g_{i}} ⊂ G, if the sequence {Hg_{i}} is divergent (that is, it has no convergent subsequence) in H\G, then the sequence {μ_{H}g_{i}} gets equidistributed with respect to μ_{G} as i → ∞ (that is, μ_{H}g_{i} → μ_{G} weakly).

From this consequence we obtain the following estimate regarding the counting problem stated in the beginning of the introduction.

Theorem 1.11 Let G and H be as in the counting problem. Suppose that H^{0} is reductive and a proper maximal connected real algebraic ℚ-subgroup of G, where H^{0} denotes the connected component of identity in H. Then asymptotically as T → ∞

N(T, V, O) ∼ λ_{H\G}(R_{T}).

Remark 1.12 Suppose that H is the set of fixed point of an involution of G. Let L be a connected real reductive ℚ-subgroup of G containing H^{0}. Then there exists a normal ℚ-subgroup N of G such that L = H^{0}N. Now if G is ℚ-simple, then H^{0} is a maximal proper connected ℚ-subgroup of G (see [Bor3, Lemma 8.0]). Hence Theorem 1.2 follows from Theorem 1.11.

In the general case, we obtain the following analogue of Corollary 1.10. We note that the condition that H is not contained in any proper ℚparabolic subgroup of G, is also equivalent to saying that any real algebraic ℚ-subgroup L of G containing H is reductive.

Corollary 1.13 Let G be a connected real reductive algebraic group defined over ℚ, and H a connected real reductive ℚ-subgroup of G not contained in any proper parabolic ℚ-subgroup of G. Let Γ ⊂ G(ℚ) be an arithmetic lattice in G. Suppose that a sequence {g_{i}} ⊂ G is such that the sequence {μ_{H} · g_{i}} does not converge to the G-invariant probability measure. Then after passing to a subsequence, there exist a proper connected real reductive ℚ-subgroup L of G containing H and a compact set C ⊂ G such that

{g_{i}} ⊂ (Z(H) ∩ Γ)LC

For applying this result to the counting problem, we need to know that averages of translates of the measure μ_{H} along the sets R_{T} become equidistributed as T tends to infinity. I.e., we want the set of ‘singular sequences’, for which the limit measure is not G-invariant, to have negligible ‘measure’ in the sets R_{T} as T → ∞. This does not hold when the sets R_{T} are ‘focused’ along H\L (⊂ H\G):

Definition 1.14 Let G and H be as in the counting problem. For a sequence T_{n} → ∞, the sequence {R_{T_{n}}} of open sets in H\G is said to be focused, if there exist a proper connected reductive real algebraic ℚ-subgroup L of G containing H^{0} and a compact set C ⊂ G such that

lim sup_{n→∞} \frac{λ_{H\G}(q_{H}((Z(H^{0}) ∩ Γ)LC) ∩ R_{T_{n}})}{λ_{H\G}(R_{T_{n}})} > 0,

where q_{H} : G → H\G is the natural quotient map.

Note that since L is reductive and defined over ℚ, we have that π(L) is closed in Γ\G. In particular, (Z(H^{0}) ∩ Γ)L is closed in G. Also H^{0}zL = zL for any z ∈ Z(H^{0}). Now since C is compact, the set q_{H}((Z(H^{0}) ∩ Γ)LC) is closed in H\G.

Now if the focusing of {R_{T_{n}}} does not occur, then using Corollary 1.13 we can obtain the following analogue of Corollary 1.10.

Corollary 1.15 Let G and H be as in the counting problem. Suppose that H^{0} is not contained in any proper ℚ-parabolic subgroup of G^{0}, and for some sequence T_{n} → ∞, the sequence {R_{T_{n}}} is not focused. Then given ϵ > 0 there exists an open set A ⊂ H\G with the following properties:

lim inf_{n→∞} \frac{λ_{H\G}(A ∩ R_{T_{n}})}{λ_{H\G}(R_{T_{n}})} > 1 − ϵ  (2)

and given any sequence {g_{i}} ⊂ q_{H}^{−1}(A), if the sequence {q_{H}(g_{i})} is divergent in H\G then the sequence {μ_{H}g_{i}} converges to μ_{G}.

This corollary allows us to obtain the counting estimates like in Theorem 1.2 and Theorem 1.11 for a large class of homogeneous varieties.

Theorem 1.16 Let G and H be as in the counting problem. Suppose that H^{0} is not contained in any proper ℚ-parabolic subgroup of G^{0} (equivalently, (Z(H)∩Γ)\Z(H) is compact), and for some sequence T_{n} → ∞ with bounded gaps, the sequence {R_{T_{n}}} is not focused. Then asymptotically

N(T, V, O) ∼ λ_{H\G}(R_{T}).

From the proof of this theorem we also obtain the following version on counting points on closed Γ-orbits on homogeneous varieties in place of integral points.

Theorem 1.17 Let G be reductive real algebraic group defined over ℚ with no nontrivial ℚ-characters and Γ an arithmetic lattice with respect to the ℚ-structure on G. Let H be a reductive real algebraic ℚ-subgroup of G such that (Z(H^{0}) ∩ Γ)\Z(H^{0}) is compact. Suppose that G acts linearly on a Euclidean vector space W and there exists a point p ∈ W such that the orbit p · G is closed and H = {g ∈ G : pg = p}. Define B_{T} ⊂ W and R_{T} ⊂ H\G as in the counting problem. Suppose further that for some sequence T_{n} → ∞ with bounded gaps, the sequence {R_{T_{n}}} is not focused. Then asymptotically as T → ∞,

N(T, pΓ) = |pΓ ∩ B_{T} | ∼ λ_{H\G}(R_{T}).

For the examples 1 and 2 considered above, in section 6 we illustrate the use of Theorem 1.16 by verifying the non-focusing of {R_{T_{n}}}. We note that the non-focusing assumption in Theorem 1.16 is not vacuous. In section 7, we consider an example, where a sequence {R_{T_{n}}} is focused and the counting estimate is different from what is predicted by Theorem 1.17.

In the above setup one is required to verify the condition of nonfocusing in Theorem 1.16 separately for each application of the result. From our examples, it seems that the process of computing volumes of R_{T} itself shows how to verify the focusing condition.

The paper is organized as follows: In section 2 we prove Proposition 1.8, and describe the results of M. Ratner classifying measures invariant under unipotent flows. In section 3 we loosely follow some ideas and methods developed by S.G. Dani and G.A. Margulis to study behavior of the translated orbits π(H)g in Γ\G near images of certain algebraic subvarieties of G. The main result of this section is Proposition 3.13, which gives a condition in terms of an appropriate representation space which holds when the limit measure μ is not the G-invariant measure. In the course of completion of the proof of Theorem 1.7 in section 4, we develop a general method to derive ergodic theoretic information from the above mentioned condition. The counting Theorem 1.16 is proved in section 5.

We would like to thank M. Burger, S.G. Dani, G.A. Margulis, G. Prasad, M. Ratner, Z. Rudnick, and P. Sarnak for useful discussions. We also thank the referee for valuable suggestions leading to significant improvement in the readability of the paper. Parts of this work were carried at MSRI, University of Chicago, Princeton University, Institute for Advanced Study, and Stanford University. We would like to acknowledge their hospitality.

2 Invariance under unipotents and Ratner’s theorem

First we make some reductions regarding the proof of Theorem 1.7: Let c ∈ G be such that π(c) ∈ supp(μ). Since μ_{H}g_{i} → μ, there exist sequences {h_{i}} ⊂ H and {γ_{i}} ⊂ Γ such that γ_{i}^{−1}h_{i}^{−1}g_{i} → c as i → ∞. Therefore to prove the theorem, we may assume, without loss of generality, that g_{i} = γ_{i} for all i ∈ ℕ. Let L be a minimal connected real algebraic ℚ-subgroup of G containing γ_{i}^{−1}Hγ_{i} for infinitely many i ∈ ℕ. By passing to a subsequence we may assume that γ_{i}^{−1}Hγ_{i} ⊂ L for all i ∈ ℕ. Again without loss of generality we can replace H by γ_{1}^{−1}Hγ_{1} and γ_{i} by γ_{1}^{−1}γ_{i} for all i ∈ ℕ. Thus H ⊂ L. At this stage it is enough to prove the following result for G = L and the homomorphisms ρ_{i} : H → L given by ρ_{i}(h) = γ_{i}^{−1}hγ_{i} (∀h ∈ H).

Theorem 2.1 Let G and H be connected real algebraic groups defined over ℚ and with no nontrivial ℚ-characters. Let Γ ⊂ G(ℚ) and Λ ⊂ H(ℚ) be arithmetic lattices in G and H respectively. Let ρ_{i} : H → G (i ∈ ℕ) be a sequence of ℚ-homomorphisms with the following properties:

(1) No proper ℚ-subgroup of G contains ρ_{i}(H) for infinitely many i ∈ ℕ.

(2) For every h ∈ H(ℚ), there exists k ∈ ℕ such that {ρ_{i}(h) : i ∈ ℕ} ⊂ G(\frac{1}{k}ℤ);

(3) For any sequence h_{i} → e in H, all the eigenvalues for the action of Ad(ρ_{i}(h_{i})) on the Lie algebra of G tend to 1 as i → ∞;

(4) For any regular algebraic function f on G, the functions f ∘ ρ_{i} span a finite dimensional space of functions on H.

(5) For all i ∈ ℕ, ρ_{i}(Λ) ⊂ Γ.

Let π : G → Γ\G be the natural quotient map and let μ_{G} denote the Ginvariant probability measure on Γ\G. For each i ∈ ℕ, let μ_{i} denote the ρ_{i}(H)-invariant probability measure on π(ρ_{i}(H)). Then μ_{i} → μ_{G} weakly as i → ∞.

Remark 2.1’ . By property (5) the map π ∘ ρ_{i} : H → Γ\G factors through the canonical map Λ\H → Γ\G. Let μ_{H} denote the H-invariant probability measure on Λ\H. Then μ_{i} = (π_{i})_{∗}(μ_{H}), the pushforward of μ_{H} under π_{i} for all i ∈ ℕ.

It is straightforward to verify that the maps ρ_{i}(h) = γ_{i}^{−1}hγ_{i} satisfy the properties (1)-(5); use [Bor2, Corollary 1.9, pp. 54] to verify property (4).

Invariance under a unipotent flow

A basic observation in the proof of Theorem 2.1 is the following version of Proposition 1.8.

Proposition 2.2 Let the notation be as in Theorem 2.1. Assume that ρ_{i}(H) ≠ G for infinitely many i ∈ ℕ. Then μ is invariant under a nontrivial unipotent one-parameter subgroup of G.

Proof. Let 𝔤 and 𝔥 denote the Lie algebras of G and H respectively. Let Dρ_{i} : 𝔥 → 𝔤 denote the differential of ρ_{i} at the identity. First suppose that, after passing to a subsequence, there exists a sequence {X_{i}} ⊂ 𝔥 and Y ∈ 𝔤 \ {0} such that as i → ∞,

X_{i} → 0  and  Dρ_{i}(X_{i}) → Y.  (3)

Let t ∈ ℝ. By property (3) of the maps ρ_{i}, Ad(exp tY) is a unipotent transformation. Put h_{i} = exp(tX_{i}), ∀i ∈ ℕ. Then as i → ∞,

μ_{i} = μ_{i}ρ_{i}(h_{i}) = μ_{i} · exp(Dρ_{i}(tX_{i})) → μ · exp(tY).

Thus μ = μ exp(tY), and the conclusion of the proposition holds (cf. [Moz]).

If the condition in equation (3) does not hold, the set {Dρ_{i}(X) : i ∈ ℕ} is relatively compact in 𝔤 for all X ∈ 𝔥.

Since H is a connected real algebraic group defined over ℚ, by weak approximation (see [PR, Theorem 7.7]), H(ℚ) is dense in H. Hence there exists a finite set S ⊂ 𝔥 such that exp(S) ⊂ H(ℚ) and the subgroup generated by exp(S) is Zariski dense in H.

Take any h ∈ exp(S). Then the set D = {ρ_{i}(h) : i ∈ ℕ} is relatively compact, and by property (2) of the maps ρ_{i}, there exists k ∈ ℕ such that D ⊂ G(\frac{1}{k}ℤ). Since G(\frac{1}{k}ℤ) is discrete, D is finite. Hence by passing to a subsequence we get that ρ_{i}(h) = ρ_{j}(h) for all i, j ∈ ℕ, and h ∈ exp(S). Since the group generated by exp(S) is Zariski dense in H, this shows that ρ_{i} = ρ_{j} for all i, j. Now by property (1) of the maps ρ_{i}, we have ρ_{i}(H) = G for all i ∈ ℕ. This contradicts the hypothesis of the proposition, and the proof is complete.

This proposition allows us to use the nice algebraic behavior of unipotent flows in our investigation.

Ratner’s theorem on measure rigidity of unipotent flows

To study the measure μ as in Proposition 2.2, we reformulate Ratner’s description [Rat3] of finite ergodic invariant measures for the actions of unipotent subgroups on homogeneous spaces of Lie groups.

Let G be a Lie group, Γ a discrete subgroup of G, and π : G → Γ\G the natural quotient map. Let H be the collection of all closed subgroups F of G such that F ∩ Γ is a lattice in F and the subgroup generated by unipotent one-parameter subgroups of G contained in F acts ergodically on π(F) ≅ (F ∩ Γ)\F with respect to the F -invariant probability measure.

Note that for every F ∈ H, Ad(F ∩ Γ) is Zariski dense in Ad(F), where Ad denotes the Adjoint representation of G (see [MS, Prop. 2.1]).

Remark 2.3 Suppose that G is an algebraic ℚ-group and Γ ⊂ G(ℚ) is an arithmetic lattice in G. Then every F ∈ H is a ℚ-subgroup of G and the radical of F is unipotent (see [Sha1, Prop. 3.2]).

Proposition 2.4 The collection H is countable.

Proof. See [Rat3, Theorem 1.1] or [DM3, Proposition 2.1] for different proofs of this result.

Let U be a unipotent one-parameter subgroup of G and F ∈ H. Define

N(F, U) = {g ∈ G : U ⊂ g^{−1}F g}
S(F, U) = ⋃ {N(F′, U) : F′ ∈ H, F′ ⊂ F, dim F′ < dim F }.

Lemma 2.5 ([MS, Lemma 2.4]) Let g ∈ G and F ∈ H. Then g ∈ N(F, U)\ S(F, U) if and only if the group g^{−1}F g is the smallest closed subgroup of G which contains U and whose orbit through π(g) is closed in Γ\G. Moreover in this case the action of U on π(F)g is ergodic with respect to a finite g^{−1}F g-invariant measure.

As a consequence of this lemma,

π(N(F, U) \ S(F, U)) = π(N(F, U)) \ π(S(F, U)),  ∀F ∈ H. (4)

Remark 2.6 Let G and Γ be as in Remark 2.3, and g ∈ G(ℚ). Then g ∈ N(F, U)\S(F, U) if and only if the smallest ℚ-subgroup of G containing U is g^{−1}F g (see [Sha1, Proposition 3.2]).

Ratner’s theorem [Rat3] states that given any U-ergodic invariant probability measure on Γ\G, there exists F ∈ H and g ∈ G such that μ is g^{−1}F g-invariant and μ(π(F)g) = 1. Now decomposing any finite invariant measure into its ergodic component, and using Lemma 2.5, we obtain the following description for any U-invariant probability measure on Γ\G (see [MS, Theorem 2.2]).

Theorem 2.7 (Ratner) Let U be a unipotent one-parameter subgroup of G and μ be a finite U-invariant measure on Γ\G. For every F ∈ H, let μ_{F} denote the restriction of μ on π(N(F, U)\S(F, U)). Then μ_{F} is U-invariant and any U-ergodic component of μ_{F} is a g^{−1}F g-invariant measure on the closed orbit π(F)g for some g ∈ N(F, U) \ S(F, U).

In particular, for all Borel measurable subsets A of Γ\G,

μ(A) = ∑_{F ∈H^{∗}} μ_{F} (A),

where H^{∗} ⊂ H is a countable set consisting of one representative from each Γ-conjugacy class of elements in H.

3 Local behavior of translates of H-orbits

In this section we study the following situation: we have a relatively compact open set Ω ⊂ H with a probability measure ν such that the measure ρν on Ω extends to a Haar measure on H, where ρ is a continuous function which is bounded above and below by strictly positive constants on Ω. Given a sequence {g_{i}} ⊂ G, we have maps ϕ_{i} : Ω → Γ\G given by ϕ_{i}(h) = π(hg_{i}), ∀h ∈ Ω. For each i ∈ ℕ, let ν_{i} ∈ P(Γ\G) be such that ν_{i}(E) = ν(ϕ_{i}^{−1}(E)) for all Borel sets E ⊂ Γ\G. Suppose that ν_{i} → μ in P(Γ\G) and μ is invariant under some nontrivial unipotent one-parameter subgroup U of G.

We want to analyze the case when the limit measure μ is not the Ginvariant measure. By Ratner’s description of μ as in Theorem 2.7, there exists a proper subgroup F ∈ H, ϵ_{0} > 0, and a compact set C_{1} ⊂ N(F, U) \ S(F, U) such that μ(π(C_{1})) > ϵ_{0}. Thus for any neighborhood Φ of π(C_{1}), we have ν(ϕ_{i}^{−1}(Φ)) > ϵ_{0} for all large i ∈ ℕ; that is, the image of Ω ‘spends a fixed proportion of time in Φ’. To investigate the behavior of the maps ϕ_{i} near π(N(F, U) \ S(F, U)), we follow the methods of Dani and Margulis as in [DM3] (cf. [Sha1]). This involves constructing a linear representation of G associated with F as follows.

Linearisation of neighborhoods of singular subsets

Let F ∈ H. Let 𝔤 denote the Lie algebra of G and let 𝔣 denote its Lie subalgebra associated to F. For d = dim 𝔣, put V_{F} = ∧^{d}𝔤, the d-th exterior power, and consider the linear G-action on V_{F} via the representation ∧^{d} Ad, the d-th exterior power of the Adjoint representation of G on 𝔤. Fix p_{F} ∈ ∧^{d}𝔣 \ {0}, and let η_{F} : G → V_{F} be the map defined by η_{F} (g) = p_{F} · g = p_{F} · (∧^{d} Ad g) for all g ∈ G. Note that

η_{F}^{−1}(p_{F}) = {g ∈ N_{G}(F) : det(Ad g|_{𝔣}) = 1}.

Proposition 3.1 ([DM3, Theorem 3.4]) The orbit p_{F} · Γ is discrete in V_{F}.

Remark 3.2 In the arithmetic case described in Remark 2.3, the above proposition is immediate.

Now let U be a unipotent one-parameter subgroup of G. Let the notation be as in the previous section. First we recall a version of [DM3, Prop. 3.2].

Proposition 3.3 Let A_{F} be the linear span of η_{F} (N(F, U)) in V_{F}. Then

η_{F}^{−1}(A_{F}) = N(F, U).

Put Γ_{F} = N_{G}(F)∩Γ. Then for any γ ∈ Γ_{F}, we have π(F)γ = π(F), and hence γ preserves the volume of π(F). Therefore | det(Ad γ|_{𝔣})| = 1. Hence p_{F} · γ = ±p_{F}. Now define

V̄_{F} = \{ \begin{matrix}V_{F} /\{Id,-Id\} & if p_{F} · Γ_{F} = \{p_{F} , −p_{F} \} \\ V_{F} & if p_{F} · Γ_{F} = p_{F} \end{matrix}

The action of G factors through the quotient map of V_{F} onto V̄_{F}. Let p̄_{F} denote the image of p_{F} in V̄_{F}, and define η̄_{F} : G → V̄_{F} as η̄_{F} (g) = p̄_{F} · g for all g ∈ G. Then Γ_{F} = η̄_{F}^{−1}(p̄_{F}) ∩ Γ. Let A_{F} denote the image of A_{F} in V̄_{F}. Note that the inverse image of A_{F} in V_{F} is A_{F}.

For every x ∈ Γ\G, define the set of representatives of x in V̄_{F} to be

Rep(x) = η̄_{F} (π^{−1}(x)) ⊂ V̄_{F} .

Using Lemma 2.5 and Proposition 3.3, we get

Rep(π(g)) ∩ A_{F} = p̄_{F} · Γg ∩ A_{F} = p̄_{F} · g, ∀g ∈ N(F, U) \ S(F, U).

We extend this observation in the following result (cf. [Sha1, Prop. 6.5]).

Proposition 3.4 ([DM3, Corollary 3.5]) Let D be a compact subset of A_{F}. Then for any compact set K ⊂ X \ π(S(F, U)), there exists a neighborhood Φ of D in V̄_{F} such that any x ∈ K has at most one representative in Φ.

Using this proposition, we can uniquely represent in Φ the parts of the trajectories ϕ_{i}(Ω) lying in K. In order to understand the behavior of these trajectories in Φ, we need to study certain growth properties of the following class of functions.

Certain growth properties for a class of functions

Definition For any n ∈ ℕ and Λ > 0, let E(n, Λ) be the collection of functions ϕ : ℝ → ℂ of the form

ϕ(t) = ∑_{i=1}^{n} ∑_{l=0}^{n−1} a_{il}t^{l}e^{λ_{i}t}, ∀t ∈ ℝ,

where a_{il} ∈ ℂ and λ_{i} ∈ ℂ with |λ_{i}| ≤ Λ for all i.

Let E_{G}(n, Λ) be the collection of functions θ : ℝ → G such that the following holds: for any 1 ≤ d ≤ dim 𝔤 and any v ∈ ∧^{d}𝔤, if we define ϕ(t) = f(v · θ(t)) (∀t ∈ ℝ), where f is a linear functional on ∧^{d}𝔤, or if we define ϕ(t) = ∥v · θ(t)∥^{2} (∀t ∈ ℝ), where ∥ · ∥ denotes a Euclidean norm, then ϕ ∈ E(n, Λ).

Take m ∈ ℕ. Let E_{G}(m, n, Λ) be the set of functions Θ : ℝ^{m} → G such that for any x ∈ ℝ^{m} with ∥x∥ = 1 and any y ∈ ℝ^{m}, if we define θ(t) = Θ(y + tx), ∀t ∈ ℝ, then θ ∈ E_{G}(n, Λ).

Remark 3.5 These functions arise in our study in the following context: Let m ∈ ℕ, and {X_{1}, …, X_{m}} ⊂ G. Define Θ : ℝ^{m} → G by

Θ(\underline{t}) = exp(t_{1}X_{1}) ⋯ exp(t_{m}X_{m}) ∀\underline{t} = (t_{1}, …, t_{m}) ∈ ℝ^{m}.

Then there exist Λ > 0 and n ∈ ℕ such that for any g ∈ G, the map t ↦ Θ(t)g belongs to E_{G}(m, n, Λ). Here Λ depends on the choice of (X_{i})_{1≤i≤m} only up to an upper bound on the absolute values of eigenvalues of {Ad exp(X_{i}) : i = 1, …, m}; while n is independent of the choice.

The following growth property of functions in E(n, Λ) plays a important role in the sequel; see [EMS, Cor. 2.10] for a proof.

Proposition 3.6 For any n ∈ ℕ and Λ > 0, there exists a constant δ_{0} = δ_{0}(n, Λ) > 0 such that the following holds: given ϵ > 0, there exists M > 0 such that for any ϕ ∈ E(n, Λ) and any interval I of length at most δ_{0},

|{t ∈ I : |f(t)| < (1/M) sup_{t∈I} |f(t)|}| ≤ ϵ · |I|.

Such growth properties for polynomials of bounded degrees were used in many works on dynamics of unipotent flows. The next result is a basic tool for our analysis.

Case of one-dimensional trajectories

Proposition 3.7 Let n ∈ ℕ, Λ ≥ 0, a compact set C ⊂ A_{F} and an ϵ > 0 be given. Then there exists a (larger) compact set D ⊂ A_{F} with the following property: For any neighborhood Φ of D in V̄_{F} there exists a neighborhood Ψ of C in V̄_{F} (with Ψ ⊂ Φ) such that for any θ ∈ E_{G}(n, Λ), any v ∈ V̄_{F}, and any interval I of length at most δ_{0}(n, Λ), if \overline{v · θ(I)} ⊄ Φ, then

|{t ∈ I : v · θ(t) ∈ Ψ}| ≤ ϵ · |{t ∈ I : v · θ(t) ∈ Φ}|.  (5)

Proof. Let C be a finite collection of linear functionals on V_{F} such that

A_{F} = ∩_{f∈C}f^{−1}(0).

By Proposition 3.6, there exists M > 0 such that for any interval J of length at most δ_{0}(n, Λ) and ψ ∈ E(n, Λ), we have

|{t ∈ J : |ψ(t)| ≤ (1/M) sup_{t∈J} |ψ(t)|}| ≥ ϵ · |J|.

For R > 0, define B(R) = {v ∈ V̄_{F} : ∥v∥^{2} < R}. Let R > 0 be such that C ⊂ B(R). Put

D = A_{F} ∩ \overline{B(M · R)}.

For c > 0, let Z_{c}(C) be the image of {v ∈ V_{F} : |f(v)| < c, ∀f ∈ C} in V̄_{F}. Now given a neighborhood Φ of D, there exists c > 0 such that Z_{c}(C) ∩ \overline{B(M · R)} ⊂ Φ. Let

Ψ = Z_{c/M}(C) ∩ B(R).

Then Ψ is a neighborhood of C contained in Φ.

Fix any v ∈ V_{F}, let v̄ be its image in V̄_{F}. Let J be any connected component of \{t ∈ I : v̄ · θ(t) ∈ Φ\}. Suppose that \overline{v̄ · θ(I)} ⊄ Φ. Then there exists a_{1} ∈ \overline{J} such that v̄ · θ(a_{1}) ∉ Φ. Therefore either |f_{0}(v · θ(a_{1}))| ≥ c for some f_{0} ∈ C or ∥v · θ(a_{1})∥^{2} ≥ M · R. Hence by the choice of M > 0 and since θ ∈ E_{G}(n, Λ), we have that

|{t ∈ J : |f(v · θ(t))| < c/M and ∥v · θ(t)∥^{2} < R}| ≤ ϵ · |J|.

From this equation (5) follows.

The following result is one of the main components of our proof of Theorem 1.7. Similar results for unipotent trajectories were obtained in [DM1, Sha1, DM3, MS]

Proposition 3.8 Let n ∈ ℕ, Λ ≥ 0, ϵ > 0, a compact set K ⊂ Γ\G \ π(S(F, U)), and a compact set C ⊂ A_{F} be given. Then there exists a (larger) compact set D ⊂ A_{F} with the following property: for any neighborhood Φ of D in V̄_{F}, there exists a neighborhood Ψ of C in V̄_{F} (with Ψ ⊂ Φ) such that for any θ ∈ E_{G}(n, Λ) and any interval I ⊂ ℝ of length at most δ_{0}(n, Λ), one of the following conditions is satisfied:

1. There exists γ ∈ Γ such that p̄_{F} · γθ(I) ⊂ Φ.

2. |{t ∈ I : π(θ(t)) ∈ K and p̄_{F} · Γ · θ(t) ∩ Ψ ≠ ∅}| < ϵ · |I|.

Proof. Let a compact set D ⊂ A_{F} be as in Proposition 3.7. Let Φ be a given neighborhood of D in V̄_{F}. Replacing Φ by a smaller neighborhood of D, by Proposition 3.4 the set Rep(x) ∩ Φ contains at most one element for all x ∈ K. By the choice of D there exists a neighborhood Ψ of C contained in Φ such that equation (5) holds.

Now put

E = {t ∈ I : π(θ(t)) ∈ K and p̄_{F} · Γ · θ(t) ∩ Ψ ≠ ∅}.  (6)

Let t ∈ E. By the choice of Φ, there exists a unique v_{t} ∈ p̄_{F} ·Γ such that v_{t} · θ(t) ∈ Φ. Now suppose that condition 1 does not hold. Then for every t ∈ E, there exists a largest open interval I(t) ⊂ I containing t such that

v_{t} · θ(I(t)) ⊂ Φ and \overline{v_{t} · θ(I(t))} ⊄ Φ.  (7)

Put I = {I(t) : t ∈ E}. Then for any I_{1} ∈ I and s ∈ I_{1} ∩ E, we have I(s) = I_{1}. Therefore for any t_{1}, t_{2} ∈ E, if t_{1} < t_{2} then either I(t_{1}) = I(t_{2}) or I(t_{1}) ∩ I(t_{2}) ⊂ (t_{1}, t_{2}). Hence any t ∈ I is contained in at most two distinct elements of I. Thus

∑_{I_{1}∈I} |I_{1}| ≤ 2|I|.  (8)

Now by equations (5) and (7), for any t ∈ E,

|{s ∈ I(t) : v_{t} · θ(s) ∈ Ψ}| < ϵ · |I(t)|.  (9)

Therefore by equations (8) and (9), we get

|E| ≤ ϵ · ∑_{I_{1}∈I} |I_{1}| ≤ (2ϵ)|I|,

which is condition 2 for 2ϵ in place of ϵ.

Case of higher dimensional trajectories

To obtain a higher dimensional analogue of Proposition 3.8, we need the following results.

Let m ∈ ℕ. Let S be the unit sphere in ℝ^{m} centered at 0 and let σ be a rotation invariant measure on S.

Lemma 3.9 Given ϵ > 0 there exists ϵ_{1} > 0 such that for any measurable subset A ⊂ S with σ(A) < ϵ_{1} · σ(S), any ball B ⊂ ℝ^{m}, and any b_{0} ∈ B, we have

|{b_{0} + t · x ∈ B : x ∈ A, t ≥ 0}| < ϵ · |B|.

Lemma 3.10 Given ϵ > 0 there exists κ ∈ (0, 1) such that the following holds: For a ball B ⊂ ℝ^{m} and a measurable subset E ⊂ B, if |E| ≥ κ · |B| then for any b_{0} ∈ B,

σ({x ∈ S : (b_{0} + ℝ · x) ∩ E ≠ ∅}) ≥ (1 − ϵ)σ(S).

Proofs of Lemma 3.9 and Lemma 3.10. Observe that it is enough to prove the results for a unit ball B, for which the assertions are easily verified.

The following result is a generalization of Lemma 3.10.

Lemma 3.11 Given ϵ > 0 there exists η = η(m, ϵ) ∈ (0, 1) such that the following holds: For a ball B in ℝ^{m} and nonempty measurable subsets E, F ⊂ B, if |E ∪ F | ≥ (1 − η)|B| and |E| ≥ |F |, then there exists b_{0} ∈ F such that

σ({x ∈ S : (b_{0} + ℝ · x) ∩ E ≠ ∅}) ≥ (1 − ϵ)σ(S).  (10)

Proof. Let κ ∈ (0, 1) be as in Lemma 3.10 for the given ϵ > 0. Put η = 3^{−(m+1)}(1 − κ). Let E and F be nonempty measurable subsets of B such that |E ∪ F | ≥ (1 − η)|B| and |E| ≥ |F |.

If |E|/|B| ≥ κ, by Lemma 3.10, equation (10) follows for any t_{0} ∈ F. Therefore we can assume that |E|/|B| ≤ κ. Recall that if m_{E} denotes the Lebesgue measure of ℝ^{m} restricted to E then there exists a measurable subset E^{∗} ⊂ E such that |E \ E^{∗}| = 0 and the Radon-Nikodym derivative of m_{E} with respect to the Lebesgue measure is 1 at all t ∈ E^{∗} (see [Rud, Theorem 8.6]). Thus for a sufficeintly small ball B′ centered at a point of E^{∗}, |E ∩ B′| > κ|B′|. However, for sufficiently large balls B′ ⊆ B, |E ∩ B′| ≤ κ|B′|, because |E| ≤ κ|B|. Therefore there exists a covering C of E^{∗} consisting of balls B′ ⊂ B such that

|E ∩ B′|/|B′| = κ.  (11)

First suppose that F ∩B′ ≠ ∅ for some B′ ∈ C. Then for any t_{0} ∈ F ∩B′, the equation (10) follows from equation (11) and Lemma 3.10, applied to B′ in place of B.

Now suppose that F ∩ B′ = ∅, ∀B′ ∈ C. Put D = B \ (E ∪ F). Then due to equation (11) we get

|D ∩ B′|/|B′| = 1 − κ,  ∀B′ ∈ C.  (12)

By [Rud, Lemma 8.4], there exists a finite subcollection C′ ⊂ C consisting of disjoint balls such that

∑_{B′∈C′} |B′| > 3^{−m} · |E|.

Hence using equations (12), since |E| ≥ ((1 − η)/2)|B|, we get

|D| > (1 − κ)3^{−m} · |E| ≥ (1 − κ)3^{−(m+1)} · |B| = η · |B| ≥ |B| − |E ∪ F | = |D|,

which is a contradiction and the proof is complete.

In the next proposition we obtain the basic property regarding the dynamical behavior of E_{G}(m, n, Λ)-type trajectories (cf. [Sha2, Prop. 5.4] in the case of polynomials of several variables).

Proposition 3.12 Let m ∈ ℕ, n ∈ ℕ, Λ ≥ 0, ϵ > 0, a compact set K ⊂ Γ\G \ π(S(F, U)), and a compact set C ⊂ A_{F} be given. Then there exists a (larger) compact set D ⊂ A_{F}, and ∃η > 0 depending on ϵ and m, so that for any neighborhood Φ of D in V̄_{F}, there exists a neighborhood Ψ of C contained in Φ such that for any Θ ∈ E_{G}(m, n, Λ) and a ball B ⊂ ℝ^{m} of diameter at most δ_{0}(n, Λ) (see Proposition 3.6), one of the following possibilities holds:

1. Let B_{K} = {t ∈ B : π(Θ(t)) ∈ K}. Then |B_{K}| < (1 − η) · |B|.

2. There exists γ ∈ Γ such that p̄_{F} · γΘ(B) ⊂ Φ.

3. |{t ∈ B : Rep(π(Θ(t))) ∩ Ψ ≠ ∅}| < ϵ · |B|.

Proof. For the given compact sets C and K, let D be a compact subset of A_{F} as in Proposition 3.8 for ϵ/(4m) in place of ϵ. Let Φ be a given neighborhood of D. Replacing Φ by a smaller neighborhood of D, by Proposition 3.4 the set Rep(x) ∩ Φ contains at most one element for any x ∈ K. Let Ψ ⊂ Φ be a neighborhood of C in V̄_{F} such that the conclusion of Proposition 3.8 holds for ϵ/(4m) in place of ϵ.

Let ϵ_{1} > 0 be such that the conclusion of Lemma 3.9 holds for ϵ/2 > 0 in place of ϵ. Let η ∈ (0, ϵ/4) be such that the conclusion of Lemma 3.11 holds for ϵ_{1} in place of ϵ.

Take a Θ ∈ E_{G}(m, n, Λ) and a ball B ⊂ ℝ^{m} of diameter at most δ_{0}(n, Λ) (see Proposition 3.6). For every γ ∈ Γ, we define

A_{γ} = {t ∈ B_{K} : p̄_{F} · γΘ(t) ∈ Φ}.

Note that for γ_{1}, γ_{2} ∈ Γ if p̄_{F} · γ_{1} ≠ p̄_{F} · γ_{2} then A_{γ_{1}} ∩ A_{γ_{2}} = ∅.

There are three cases:

(1) There exists some t_{0} ∈ B_{K} \ ∪_{γ∈Γ}A_{γ}.

(2) A_{γ_{0}} = B_{K} for some γ_{0} ∈ Γ.

(3) There exist γ_{1}, γ_{2} ∈ Γ such that p̄_{F} γ_{1} ≠ p̄_{F} γ_{2}, A_{γ_{i}} ≠ ∅ and |A_{γ_{1}}| ≤ |A_{γ_{2}}|.

Define

F = \{ \begin{matrix}\{t_{0}\}, & in case (1) \\ \{t ∈ B : p̄_{F} · γ_{0}Θ(t) ∉ Φ\}, & in case (2) \\ A_{γ_{1}}, & in case (3). \end{matrix}

Put E = B_{K} \ F. Note that B_{K} ∩ F = ∅ in Case (2). Now suppose that Possibilities 1 and 2 of the proposition do not hold. Then

|E ∪ F | ≥ |B_{K}| ≥ (1 − η)|B|

and F ≠ ∅. Also |F | ≤ |E|. Therefore by Lemma 3.11, there exists t_{0} ∈ F such that if we define

Σ = {x ∈ S : (t_{0} + ℝ · x) ∩ E ≠ ∅}

then

|Σ| ≥ (1 − ϵ_{1})|S|.  (13)

For every x ∈ S, define B_{x} = {t ∈ ℝ : t · x + t_{0} ∈ B}. Then by equation (13) and Lemma 3.9,

| ⋃_{x∈S\Σ}(t_{0} + B_{x} · x)| ≤ (ϵ/2)|B|.  (14)

Take any x ∈ Σ. Apply Proposition 3.8 for the function θ(t) = Θ(t · x + t_{0}), ∀t ∈ ℝ and I = B_{x}. Then by the definition of Σ, in each of the three cases we conclude that Possibility 1 of Proposition 3.8 does not hold. Therefore according to Possibility 2 of Proposition 3.8, if we define

A_{x} = |{t ∈ B_{x} : π(θ(t)) ∈ K and Rep(π(θ(t))) ∩ Ψ ≠ ∅}|

Then |A_{x}|/|B_{x}| < ϵ/(4m). Put

A = ⋃_{x∈Σ}(t_{0} + A_{x} · x).

Then A is a measurable subset of B. By the polar decomposition of B with t_{0} as a pole, we get that

|A| ≤ (ϵ/4)|B|.  (15)

Now possibility 3 follows from equations (14), (15) and the assumption that possibility 1 of the proposition does not hold. This completes the proof.

In the proof of Theorem 2.1, we need the following consequence of Proposition 3.12.

Proposition 3.13 Let m, n ∈ ℕ and Λ > 0 be given. Let B be a ball of diameter at most δ_{0} = δ_{0}(n, Λ) (see Proposition 3.6) in ℝ^{m} around 0. Let Θ_{i} : ℝ^{m} → G (i ∈ ℕ) be mappings in E(m, n, Λ) and λ_{i} be the probability measure on π(Θ_{i}(B)) which is the pushforward under π ∘ Θ_{i} of the normalized Lebesgue measure on B. Suppose that λ_{i} → λ weakly in the space of probability measures on Γ\G. Suppose there exist a unipotent oneparameter subgroup U of G and F ∈ H such that λ(π(N(F, U))) > 0 and λ(π(S(F, U))) = 0. Then there exists a compact set D ⊂ A_{F} such that the following holds: For any sequence of neighborhoods {Φ_{i}} of D in V̄_{F}, there exists a sequence {γ_{i}} ⊂ Γ such that for all large i ∈ ℕ,

p̄_{F} · γ_{i}Θ_{i}(B) ⊂ Φ_{i}.  (16)

Proof. Choose a compact set C_{1} ⊂ N(F, U) \ S(F, U) and an ϵ > 0 such that λ(π(C_{1})) > ϵ. Let η > 0 be as in the statement of Proposition 3.12. There exists a compact set K ⊂ X \ π(S(F, U)) such that

λ(K) > 1 − η/2.

Put C = η̄_{F} (C_{1}) ⊂ A_{F}. Then there exists a compact set D of A_{F} containing C with the property that given a neighborhood Φ of D in V̄_{F} there exists a neighborhood Ψ of C in V̄_{F} contained in Φ such that the conclusion of Proposition 3.12 holds.

By the choice of K, there exists i_{0} ∈ ℕ such that possibility 1 of Proposition 3.12 does not hold for Θ_{i}, ∀i ≥ i_{0}. Now possibility 3 of Proposition 3.12 applied to Θ_{i}, for any i ≥ i_{0} says that

|{t ∈ B : π(Θ_{i}(t)) ∈ π(η̄_{F}^{−1}(Ψ))}| < ϵ.

Since π(η̄_{F}^{−1}(Ψ)) is a neighborhood of π(C_{1}), this means λ_{i}(π(C_{1})) ≤ ϵ, which is a contradiction for all large i ∈ ℕ.

Thus for any decreasing sequence {Φ_{i}} of neighborhoods of D in V̄_{F} such that ∩_{i∈ℕ}Φ_{i} = D, the possibility 2 of Proposition 3.12 holds for Θ_{i} and Φ_{i} for all large i ∈ ℕ. Hence there exists a sequence {γ_{i}} ⊂ Γ such that equation (16) holds.

Thus we can reduce the study of limits of translates of the measure μ_{Ω} to a simpler situation in terms of finite dimensional linear representations. We shall carry out further analysis of the condition given by equation (16) in the next section.

4 Proof of theorem 2.1

Let m = dim(H) and {X_{1}, …, X_{m}} be a basis of 𝔥. Define a map Θ : ℝ^{m} → H by

Θ(t_{1}, …, t_{m}) = exp(t_{1}X_{1}) ⋯ exp(t_{m}X_{m}), ∀(t_{1}, …, t_{m}) ∈ ℝ^{m}.

For any h ∈ H and i ∈ ℕ, define Θ_{i}^{h} : ℝ^{m} → G as Θ_{i}^{h}(t) = ρ_{i}(hΘ(t)), ∀t ∈ ℝ. By property (3) of the maps ρ_{i}, it follows that absolute values of all the eigenvalues of Ad exp(Dρ_{i}X_{k}) = Ad ρ_{i}(exp X_{k}) are bounded above by a positive constant independent of i ∈ ℕ and 1 ≤ k ≤ m. Therefore by Remark 3.5, there exist n ∈ ℕ and Λ > 0 such that Θ_{i}^{h} ∈ E_{G}(m, n, Λ), ∀i ∈ ℕ. Let B be a ball of diameter at most δ_{0}(n, Λ) in ℝ^{m} around 0 (see Proposition 3.6). Let λ_{i}^{h} denote the probability measure on π(Θ_{i}^{h}(B)) which is the push forward under π∘Θ_{i}^{h} of a multiple of the Lebesgue measure restricted to B.

It will be convenient to recall a result [EMS, Theorem 3.5], which is deduced from [DM3, Theorem 6.1].

Theorem. Given m ∈ ℕ, n ∈ ℕ, Λ > 0, a compact set K ⊂ Γ\G, R > 0, and ϵ > 0, there exists a larger compact set K′ ⊂ Γ\G such that for any ϕ ∈ E_{G}(m, n, Λ) and a ball B of diameter at most R in ℝ^{m}, one of the following holds:

1. π(ϕ(B)) ∩ K = ∅.

2. |{t ∈ B : π(ϕ(t)) ∈ K′}| ≥ (1 − ϵ)|B|.

Note that for each h ∈ H, there exists a finite sequence {h_{j}}_{j=0}^{r} ⊂ H such that h_{0} = e, h_{r} = h, and |{t ∈ B : h_{j−1}Θ(t) ∈ h_{j}Θ(B)}| > ϵ|B| for 1 ≤ j ≤ r. Now for each i ∈ ℕ, we apply the above Theorem for ϕ = Θ_{i}^{h_{j}} and K = K_{j} to obtain K_{j+1} = K′ such that λ_{i}^{h_{j}}(K_{j+1}) > 1 − ϵ, where K_{0} = {π(e)} and 0 ≤ j ≤ r. In particular, λ_{i}^{h}(K_{r+1}) > 1 − ϵ for all i ∈ ℕ.

Any compact subset of H is covered by finitely many sets of the form hΘ(B), where h ∈ H. Therefore by Remark 2.1’ there exists a compact set K′ ⊂ Γ\G such that μ_{i}(K′) > 1 − ϵ for all i ∈ ℕ. Now by passing to a subsequence, there exits a probability measure μ on Γ\G such that μ_{i} → μ weakly as i → ∞.

By Proposition 2.2, μ is invariant under a nontrivial unipotent oneparameter subgroup, say U, of G. By a version of Ratner’s theorem as in Theorem 2.7, there exists F ∈ H such that μ(π(N(F, U))) > 0 and μ(π(S(F, U))) = 0.

Since H(ℚ) is dense in H, a given compact subset of H can be covered by finitely many open sets of the form hΘ(B) with h ∈ H(ℚ). Therefore by passing to a subsequence, there exists h ∈ H(ℚ) such that {λ_{i}^{h}} converges weakly to a probability measure λ^{h} and λ^{h}(π(N(F, U))) > 0.

There exists a constant c > 0 such that for any Borel measurable set A ⊂ Γ\G, we have λ_{i}^{h}(A) < cμ_{i}(A) (∀i ∈ ℕ). Hence λ^{h} is absolutely continuous with respect to μ. Therefore λ^{h}(π(S(F, U)) = 0.

Now by Proposition 3.13, there exists a compact set D ⊂ A_{F}, a decreasing sequence of relatively compact neighborhoods Φ_{i} of D in V̄_{F} with ∩_{i=1}^{∞}Φ_{i} = D, and a sequence {γ_{i}} ⊂ Γ such that for all i ∈ ℕ,

p̄_{F} · γ_{i}Θ_{i}^{h}(B) ⊂ Φ_{i}.

Let Ω′ = hΘ(B) ∩ H(ℚ). Let S be the collection of all real valued functions on Ω′ of the form

ω ↦ f(p_{F} · γρ_{i}(ω)),

where γ ∈ G, i ∈ ℕ, and f is a real linear functional on V_{F} which factors through V_{F} → V̄_{F}. By property (4) of the maps ρ_{i}, S spans a finite dimensional space of functions on Ω′. Therefore there exists a finite set Σ ⊂ Ω′ such that for any ϕ ∈ S, if ϕ(Σ) = 0 then ϕ(Ω′) = 0.

Now for any s ∈ Σ and i ∈ ℕ,

p̄_{F} · γ_{i}ρ_{i}(s) ∈ p̄_{F} · γ_{i}Θ_{i}^{h}(B) ⊂ Φ_{1}.

By property (2) of the maps ρ_{i}, there exists k ∈ ℕ such that {ρ_{i}(s) : i ∈ ℕ, s ∈ Σ\} ⊂ G(\frac{1}{k}ℤ). Since p̄_{F} γ · G(\frac{1}{k}ℤ) is a discrete subset of V̄_{F} and Φ_{1} is bounded, by passing to a subsequence, we may assume that

p̄_{F} · γ_{i}ρ_{i}(s) = p̄_{F} γ_{1} · ρ_{1}(s), ∀i ∈ ℕ, s ∈ Σ.

Therefore by the choice of Σ, for all ω ∈ Ω′,

p̄_{F} γ_{i} · ρ_{i}(ω) = p̄_{F} γ_{1} · ρ_{1}(ω),  ∀i ∈ ℕ.  (17)

Since H(ℚ) is dense in H and hΘ(B) is open in H, equation (17) holds for all ω ∈ hΘ(B).

Now putting ω = e in equation (17), we get p̄_{F} γ_{i} = p̄_{F} γ_{1}. Thus

p̄_{F} γ_{1} · ρ_{i}(g) = p̄_{F} γ_{1} · ρ_{1}(g) ⊂ Φ_{i}, ∀i ∈ ℕ, g ∈ hΘ(B).  (18)

Since ∩_{i=1}^{∞}Φ_{i} = D, we have that

p̄_{F} γ_{1} · ρ_{i}(hΘ(B)) ⊂ A_{F} , ∀i ∈ ℕ.

Therefore by the Zariski density of hΘ(B) in H and Proposition 3.3,

γ_{1}ρ_{i}(H) ⊂ N(F, U), ∀i ∈ ℕ.

Replacing F by γ_{1}^{−1}F γ_{1}, we obtain that

ρ_{i}(H) ⊂ N(F, U),  ∀i ∈ ℕ.  (19)

Take any i ∈ ℕ. Let L ∈ H be of minimal dimension such that L ⊂ F and ρ_{i}(H) ⊂ N(L, U). By definition

S(L, U) = ⋃ {N(L_{1}, U) : L_{1} ∈ H, L_{1} ⊂ L, dim L_{1} < dim L}.

Since H is a connected real algebraic group, each ρ_{i}(H) is an irreducible real subvariety of G. Therefore if ρ_{i}(H) ⊂ S(L, U), then ρ_{i}(H) ⊂ N(L_{1}, U) for some L_{1} ∈ H with L_{1} ⊂ L and dim L_{1} < dim L. This contradicts the choice of L. Hence ρ_{i}(H) ⊄ S(L, U). Thus ρ_{i}(H(ℚ)) \ S(L, U) is Zariski dense in ρ_{i}(H).

Let w ∈ ρ_{i}(H(ℚ)) \ S(L, U). By Remark 2.6, the smallest ℚ-subgroup of G containing U is ω^{−1}Lω. Since U ⊂ L, by dimension considerations w^{−1}Lw ⊂ L. Thus w ∈ N(L). Now by Zariski density,

ρ_{i}(H) ⊂ N(L).  (20)

Since L is the smallest ℚ-subgroup of G containing U, it is determined independent of the choice of i. Therefore equation (20) holds for all i ∈ ℕ. Therefore by property (1) of the maps ρ_{i}, we have G = N(L). In particular, N(L, U) = G. Since L ⊂ F and N(F, U) \ S(F, U) ≠ ∅, we have L = F. Now since μ(π(S(F, U)) = 0, by Theorem 2.7 every U-ergodic component of μ is F -invariant. Hence μ is F -invariant.

Now we project everything onto F \G, and use induction on dim(G) as follows. Put Ḡ = F \G. Then Ḡ is a connected real reductive group defined over ℚ, and the quotient homomorphism ϕ : G → Ḡ is defined over ℚ. Put Γ̄ = ϕ(Γ). Then Γ̄ ⊂ Ḡ(ℚ) is an arithmetic lattice in Ḡ. Put ρ̄_{i} = ϕ ∘ ρ_{i} for all i ∈ ℕ. Note that the properties (1)-(5) continue to hold for the maps ρ̄_{i}. Let q : Γ\G → Γ̄\Ḡ and π̄ : Ḡ → Γ̄\Ḡ be the natural quotient maps. Then q ∘ π = π̄ ∘ ϕ.

Let q_{∗} : P(Γ\G) → P(Γ̄\Ḡ) be the pushforward map induced by q from the space of (Borel) probability measures on Γ\G to that on Γ̄\Ḡ. This map is continuous with respect to the topologies of weak-star convergence on both the spaces. Let μ̄_{i} = q_{∗}(μ_{i}) and μ̄ = q_{∗}(μ). Then μ̄_{i} is the ρ̄_{i}(H)-invariant probability measure supported on π̄(ρ̄_{i}(H)), and μ̄_{i} → μ̄.

Since U ⊂ F and dim U > 0, we have that dim Ḡ < dim G. Therefore to prove the theorem by induction on dim G, we can assume the validity of the theorem for Ḡ in place of G. Therefore μ̄ is the Ḡ-invariant probability measure on Γ̄\Ḡ. Since F = ker ϕ and μ is F -invariant, we have that μ is G-invariant. This completes the proof of the theorem.

Remark 4.1 From the above proof, it follows that G contains a normal subgroup F ∈ H such that G = HF.

5 Applications to the counting problem

In this section we deduce the consequences of Theorem 1.7 and Theorem 1.9 and prove the counting Theorem 1.16. First we need some lemmas.

Let G, H, and L be connected real algebraic groups such that H ⊂ L ⊂ G. Define

Z(H, L) = {g ∈ G : g^{−1}Hg ⊂ L}.

In view of Theorem 1.7, we want to understand the set Z(H, L) ∩ Γ, when G, H and L are defined over ℚ.

Lemma 5.1 Suppose that at least one of G, L and H is reductive. Then Z(H, L) is a union of finitely many closed double cosets of the form Z(H) · g · L, where g ∈ Z(H, L).

Proof. Observe that Z(H, L) is a real subvariety of G. Therefore it has finitely many closed connected components. Take any g ∈ Z(H, L). Put H_{1} = g^{−1}Hg. Then H_{1} ⊂ L and Z(H, L) = g · Z(H_{1}, L). It is enough to show that Z(H_{1})L contains a neighborhood of e in Z(H_{1}, L).

Let 𝔤, 𝔥, and 𝔩 denote the Lie algebras of G, H_{1}, and L, respectively. Let X ∈ 𝔤 be a vector tangent to Z(H_{1}, L) at e. Then

[X, 𝔥] ⊂ 𝔩.  (21)

If either G or L is reductive, let 𝔩^{⊥} be the ortho-complement of 𝔩 in 𝔤 with respect to the Killing form. Otherwise, if H_{1} is reductive, let 𝔩^{⊥} be an Ad(H_{1})-invariant complementary subspace of 𝔩 in 𝔤. In both the cases,

𝔤 = 𝔩 ⊕ 𝔩^{⊥}  and  Ad(H_{1})(𝔩^{⊥}) = 𝔩^{⊥}.

Therefore,

[𝔩, 𝔥] ⊂ 𝔩  and  [𝔩^{⊥}, 𝔥] ⊂ 𝔩^{⊥}.  (22)

Write X = X_{1} + X_{2}, where X_{1} ∈ 𝔩 and X_{2} ∈ 𝔩^{⊥}. Then by equations (21) and (22),

[X_{2}, 𝔥] = 0.

This shows that X ∈ 𝔩 + 𝔷, where 𝔷 is the Lie algebra of Z(H_{1}). Thus at the identity, the tangent space of Z(H_{1}, L) is contained in the tangent space of Z(H_{1}) · L. By definition Z(H_{1})L ⊂ Z(H_{1}, L). Therefore Z(H_{1})L contains a neighborhood of e in Z(H_{1}, L). This completes the proof.

Lemma 5.2 Let G, H, and L be connected reductive real algebraic groups defined over ℚ such that H ⊂ L ⊂ G. Let Γ ⊂ G(ℚ) be an arithmetic lattice in G. Then there exists a finite set D ⊂ Z(H, L) ∩ Γ such that

Z(H, L) ∩ Γ = (Z(H) ∩ Γ) · D · (L ∩ Γ).

Proof. By [Bor1, Prop. 7.7], there exists a vector space V with a ℚstructure, a point p ∈ V (ℚ), and a rational representation of G on V such that the orbit G·p is closed in V and L = {g ∈ G : g·p = p}. By Lemma 5.1, A = Z(H, L)p is a union of finitely many closed Z(H)-orbits. Since Z(H) is a reductive real algebraic group defined over ℚ, by [Bor1, Theorem 9.11], the set A ∩ Γ · p is a finite union of Z(H) ∩ Γ-orbits. Hence there exists a finite set D ⊂ Γ such that

Z(H, L) ∩ ΓL = (Z(H) ∩ Γ)DL.

Now the result follows.

Proof of Corollary 1.13. By Theorem 1.9, after passing to a subsequence, {μ_{H}g_{i}} converges weakly to a probability measure μ on Γ\G. By Theorem 1.7, after passing to a subsequence, there exist a ℚ-subgroup L′ of G containing H and sequences {γ′_{i}} ⊂ Γ ∩ Z(H, L′) and c′_{i} → c′_{0} such that Hg_{i} = Hγ′_{i}c′_{i}, ∀i ∈ ℕ. Since H is not contained in any proper ℚ-parabolic subgroup of G, we have that L′ is reductive. By Lemma 5.2, after passing to a subsequence, there exist γ ∈ Z(H, L) ∩ Γ and a sequence {γ_{i}} ⊂ Z(H) ∩ Γ such that γ′_{i} = γ_{i}γL′ for all i ∈ ℕ. Put L = γL′γ^{−1}, c_{0} = γc′_{0} and c_{i} = γc′_{i} for all i ∈ ℕ. Then H ⊂ L, c_{i} → c_{0}, and

Hγ_{i}^{−1}g_{i} = γ_{i}^{−1}Hg_{i} = γ_{i}^{−1}Hγ′_{i}c′_{i} ⊂ HγL′c′_{i} = Lc_{i},

∀i ∈ ℕ. Hence g_{i} ∈ γ_{i}Lc_{i}, ∀i ∈ ℕ.

Proof of Corollary 1.15. In view of Remark 4.1, let L be the collection of subgroups of G of the form HF, where F ∈ H and H ⊂ N(F). Since H is countable, we can write L = {L_{i}}_{i∈ℕ}. Let C = {C_{i}}_{i∈ℕ} be an increasing sequence of compact subsets of G such that G = ∪_{i∈ℕ}C_{i}. For every k ∈ ℕ, put

B_{k} = q_{H} ( ⋃_{i=1}^{k}(Z(H) ∩ Γ)L_{i}C_{i}) .

Since {R_{T_{n}}}_{n∈ℕ} is not focused, for every k ∈ ℕ there exists n_{k} ∈ ℕ such that

λ_{H\G}(B_{k} ∩ R_{T_{n}})/λ_{H\G}(R_{T_{n}}) < ϵ, ∀n ≥ n_{k}.

Therefore the set

A = ∪_{k=1}^{∞}(R_{n_{k+1}} \ B_{k})

satisfies equation (2).

Let a sequence {g_{i}} ⊂ q_{H}^{−1}(A) be such that {q_{H}(g_{i})} is divergent in H\G. Then for every n ∈ ℕ, there exists i_{n} ∈ ℕ such that q_{H}(g_{i}) ∉ R_{T_{n}} for every i ≥ i_{n}. Therefore for any compact set C in G and any L ∈ L′ we have that

{g_{i}}_{i∈ℕ} ⊄ (Z(H) ∩ Γ)LC.  (23)

Hence by Corollary 1.13, μ is G-invariant.

Correspondence between counting and translates of measures

We recall some observations from [DRS, Sect. 2]; see also [EM]. Let the notation be as in the counting problem stated in the introduction. For T > 0, define a function F_{T} on G by

F_{T} (g) = ∑_{γ∈(H∩Γ)\Γ} χ_{T} (v_{0} · γg),

where χ_{T} is the characteristic function of B_{T}. By construction F_{T} is left Γ-invariant, and hence it will be treated as a function on Γ\G. Note that

F_{T} (e) = ∑_{γ∈(H∩Γ)\Γ} χ_{T} (v_{0} · γ) = N(T, V, O).

Since we expect, as in Theorem 1.2, that

N(T, V, O) ∼ λ_{H\G}(R_{T}),

we define

F̂_{T} (g) = \frac{1}{λ_{H\G}(R_{T})}F_{T} (g).

Thus Theorem 1.16 is the assertion

F̂_{T} (e) → 1 as T → ∞.

The connection between Theorem 1.16 and Corollary 1.15 is the following proposition.

Proposition 5.3 Let the notation and conditions be as in Theorem 1.16. Then F̂_{T_{n}} → 1 in the weak-star topology on L^{∞}(Γ\G, μ_{G}); that is, ⟨F̂_{T_{n}}, ψ⟩ → ⟨1, ψ⟩ for any compactly supported continuous function ψ on Γ\G.

Proof. As in [DRS, Sect. 2],

⟨F̂_{T} , ψ⟩ = \frac{1}{λ_{H\G}(R_{T})} ∫_{R_{T}} \overline{ψ^{H}} dλ_{H\G},

where

ψ^{H}(Hg) = ∫_{Γ\ΓH} ψ(Γhg) dμ_{H}(Γh) = ∫_{Γ\G} ψ d(μ_{H}g)

is a function on H\G.

Let ϵ > 0 be given. Since the sequence {R_{T_{n}}} is not focused, we obtain a set A ⊂ H\G as in Corollary 1.15. Break up the integral over R_{T_{n}} into the integrals over R_{T_{n}} ∩ A and R_{T_{n}} \ A. By equation (2) and the boundedness of ψ, the second integral is O(ϵ). By Corollary 1.15, for any sequence {g_{i}} ⊂ q_{H}^{−1}(A), if {q_{H}(g_{i})} has no convergent subsequence in H\G, then μ_{H} ·g_{i} → μ_{G}. Hence

ψ^{H}(Hg_{i}) → ∫_{Γ\G} ψ dμ_{G} = ⟨ψ, 1⟩.

We use dominated convergence theorem to justify the interchange of limits. Now

lim_{n→∞}⟨F̂_{T_{n}}, ψ⟩ = lim_{n→∞} \frac{1}{λ_{H\G}(R_{T_{n}})} ∫_{R_{Tn}∩A} \overline{ψ^{H}} dλ_{H\G} + O(ϵ)
= lim_{n→∞} \frac{1}{λ_{H\G}(R_{T_{n}})} ∫_{R_{Tn}∩A} \overline{⟨ψ, 1⟩} dλ_{H\G} + O(ϵ)
= lim_{n→∞} \frac{λ_{H\G}(R_{T_{n}} ∩ A)}{λ_{H\G}(R_{T_{n}})}⟨1, ψ⟩ + O(ϵ)
= ⟨1, ψ⟩ + O(ϵ)

Since ϵ is arbitrary, the proof is complete.

Proposition 5.4 There are constants a(δ) and b(δ) tending to 1 as δ → 0 such that

b(δ) ≤ lim inf_{T →∞} \frac{λ_{H\G}(R_{(1−δ)T})}{λ_{H\G}(R_{T})} ≤ lim sup_{T →∞} \frac{λ_{H\G}(R_{(1+δ)T})}{λ_{H\G}(R_{T})} ≤ a(δ).

For a proof, see appendix A.

Proof of Theorem 1.16. Combining Proposition 5.3 and Proposition 5.4 exactly as in [DRS, Lemma 2.3], we obtain that F̂_{T_{i}} → 1 pointwise on Γ\G as i → ∞. As we observed before this completes the proof.

6 Examples

Given a “counting problem”, i.e., the problem of estimating the number of lattice points on a homogeneous variety V = pG as described in the introduction, in order to apply Theorem 1.16 we need to

1. Verify that the centralizer Z(H^{0}) of the component of the identity of the stabilizer is anisotropic over ℚ. That is π(Z(H^{0})) is compact.

2. Verify that for any sequence T_{i} → ∞, the sequence {R_{T_{i}}} in H\G of pull backs of balls of radii T_{i} around the origin is not focused (see Definition 1.14).

In this section we verify these conditions for the examples discussed in the introduction and complete the proofs of the counting estimates given there.

Example 1

Proof of Theorem 1.3. Let D be the centralizer of v_{0} in M_{n}(ℝ). Since the eigenvalues of v_{0} are distinct, D is an abelian algebra of dimension n. Also D has a natural ℚ-structure, and hence ℚ-dim(D(ℚ)) = n.

Let α be a root of p(λ). Then {1, α, …, α^{n−1}} is a ℚ-basis of the field ℚ(α). The multiplication by any x ∈ ℚ(α) can be expressed as a matrix C_{x} with respect to this basis. Since C_{α} = v_{0}, we have that C_{x} ∈ D(ℚ) for all x ∈ ℚ(α). By dimension considerations, the linear map C : ℚ(α) → D(ℚ) is an isomorphism of algebras over ℚ. Let Norm : ℚ(α) → ℚ be the norm map. Then Norm(x) = det(C_{x}) for all x ∈ Q(α).

Let H be the stabilizer of v_{0} in G. Then H = D ∩ G. Hence H(ℚ) = ker(| Norm |). Also C^{−1}(H(ℤ)) is a subgroup of finite index in the group of units in ℚ(α). Since the units in ℚ(α) form a lattice in the group of unit norm elements, we have that H(ℤ)\H is compact. Hence H^{0} is ℚanisotropic. Since the dimension of H is n−1, we have that H^{0} is a maximal connected real torus in G defined over ℚ. Since a maximal ℚ-torus of a proper ℚ-parabolic subgroup admits nontrivial ℚ-characters, H^{0} is not contained in a proper ℚ-parabolic subgroup of G.

Thus in order to apply Theorem 1.16 to complete to proof of the present theorem, it remains to compute the volume growth of the pullback sets R_{T}, and to verify that they are not focused.

Reduction to the diagonal case:

Suppose that p(λ) has r ≥ 0 pairs of complex conjugate roots, say λ_{1}, λ̄_{1}, …, λ_{r}, λ̄_{r}, and s ≥ 0 real roots, say μ_{1}, …, μ_{s}. Then 2r + s = n. Suppose that λ_{k} = x_{k} + iy_{k}, where x_{k}, y_{k} ∈ ℝ, 1 ≤ k ≤ r. There exists g_{0} ∈ G such that

g_{0}^{−1}v_{0}g_{0} = v_{1} = diag ((\begin{matrix}x_{1} & y_{1} \\ −y_{1} & x_{1} \end{matrix}) , …, (\begin{matrix}x_{r} & y_{r} \\ −y_{r} & x_{r} \end{matrix}) , μ_{1}, …, μ_{s}) . (24)

The stabilizer H_{1} of v_{1} in G consists of elements of the form

h = diag (t_{1} (\begin{matrix}cos θ_{1} & sin θ_{1} \\ − sin θ_{1} & cos θ_{1} \end{matrix}) , …, t_{r} (\begin{matrix}cos θ_{r} & sin θ_{r} \\ − sin θ_{r} & cos θ_{r} \end{matrix}) , t_{r+1}, …, t_{r+s}) ,

where ∏_{j=1}^{r+s} t_{i} = ±1, t_{j} ∈ ℝ^{×}, and θ_{k} ∈ ℝ. Note that the map ϕ : H_{1}\G → H\G, defined by ϕ(H_{1}g) = H(g_{0}g), ∀g ∈ G, is G-equivariant. For T > 0, define

R_{T}^{1} = {H_{1}g ∈ H_{1}\G : ∥g^{−1}v_{1}g∥ < T }.

Then R_{T} = ϕ(R_{T}^{1} ) and λ_{H\G}(R_{T}) = vol_{H_{1}\G}(R_{T}^{1} ). Note that conjugation by g_{0} preserves the Haar measures on H and H_{1}.

Haar integral on H_{1}\G:

Let N be the group of elements of the form n = (n_{ij})_{i,j=1,... ,r+s} ∈ SL_{n}(ℝ), where

n_{ij} ∈ \{ \begin{matrix}M_{2×2}(ℝ) & if i ≤ r, j ≤ r \\ M_{1×2}(ℝ) & if i > r, j ≤ r \\ M_{2×1}(ℝ) & if i ≤ r, j > r \\ M_{1×1}(ℝ) & if i > r, j > r, \end{matrix}  (25)

n_{ij} = 0 if i < j and n_{ii} is the identity matrix for each i.

Let

M = {diag(M_{1}, …, M_{r}, t_{1}, …, t_{s}) ∈ G : M_{i} ∈ GL_{2}(ℝ), t_{i} ∈ ℝ^{×}}.

Then MN is a standard parabolic subgroup of G. Hence G = MNK, where K = O(n).

Let A be the maximal ℝ-split torus in H_{1}. Then H_{1} = A(K ∩ H_{1}) and M ≅ A × SL_{2}(ℝ)^{r}. Let

B = {diag(a_{1}, a_{1}^{−1}, …, a_{r}, a_{r}^{−1}, 1, …, 1) ∈ M : a_{i} > 0}.  (26)

By Cartan decomposition of SL_{2}(ℝ), we have M = A(K ∩ M)B(K ∩ M). Since K ∩ M = K ∩ H_{1}, we get M = H_{1}B(K ∩ M). Let dm be a Haar integral on M such that for any f ∈ C_{c}(M),

∫_{M} f dm = ∫_{(h,b,k_{1})∈H_{1}×B×(K∩M)} f(hbk_{1}) (∏_{i=1}^{r} a_{i}^{2}) dh db dk_{1} (27)

where dh, db, and dk_{1} are Haar integrals on H_{1}, B and K ∩ M, respectively, and b as in equation (26).

Since M normalizes N, we have that G = H_{1}BNK. Now let λ_{H_{1}\G} be the G-invariant measure on H_{1}\G such that for any f ∈ C_{c}(H_{1}\G),

∫_{H_{1}\G} f dλ_{H_{1}\G} = ∫_{B×N×K} f(H_{1}bnk)) (∏_{i=1}^{r} a_{i}) da_{1} ⋯ da_{r} dn dk, (28)

where dn and dk are Haar integrals on N and K, respectively, b as in equation (26), and each da_{i} is the Lebesgue integral on ℝ.

Co-ordinate description of R_{T}^{1} :

Note that R_{T}^{1} corresponds to the set

{(b, n, k) ∈ B × N × K : ∥n^{−1}b^{−1}v_{1}bn∥ < T }.

If we express

b^{−1}v_{1}b = diag(z_{1}, …, z_{r+s})

then by equations (24),

z_{i} = \{ \begin{matrix}(\begin{matrix}a_{i}^{−1} & 0 \\ 0 & a_{i} \end{matrix}) (\begin{matrix}x_{i} & y_{i} \\ −y_{i} & x_{i} \end{matrix}) (\begin{matrix}a_{i} & 0 \\ 0 & a_{i}^{−1} \end{matrix}) & for 1 ≤ i ≤ r, \\ μ_{k} & for i = r + k, \end{matrix}

where b = (a_{1}, a_{1}^{−1}, …, a_{r}, a_{r}^{−1}, 1, …, 1).

As in equation (25), for n = (n_{ij}) ∈ N , if we express n^{−1} = (n′_{ij}) ∈ N then

n′_{ij} = −n_{ij} + f_{ij}, (i < j),

where f_{ij} depends only on {n_{kl} : k < l and l − k < j − i}.

Now if we express n^{−1}(b^{−1}v_{1}b)n = (w_{ij})_{i,j=1,... ,r+s}, then

w_{ij} = \{ \begin{matrix}0 & if i > j \\ z_{i} & if i = j \\ −n_{ij}z_{j} + z_{i}n_{ij} + A_{ij} & if i < j, \end{matrix}

where

A_{ij} depends only on {z_{k} : 1 ≤ k ≤ r + s} ∪ {n_{kl} : k < l and l − k < j − i}. (29)

Thus R_{T}^{1} corresponds to the set

{(b, n, k) ∈ B × N × K : ∑_{1≤i≤r+s} ∥z_{i}∥^{2} + ∑_{1≤i<j≤r+s} ∥w_{ij}∥^{2} < T^{2}}. (30)

Asymptotics of λ_{H_{1}\G}(R_{T}^{1} ):

For i < j, let L_{ij} denote the linear transformation which takes the matrix n_{ij} to the matrix ñ_{ij} = −n_{ij}z_{j} + z_{i}n_{ij}. Then the eigenvalues of L_{ij} are

\begin{matrix}λ_{i} − λ_{j}, λ_{i} − λ̄_{j}, λ̄_{i} − λ_{j}, λ̄_{i} − λ̄_{j} & if 1 ≤ i < j ≤ r \\ λ_{i} − μ_{j}, λ̄_{i} − μ_{j} & if i ≤ r < j \\ μ_{i} − μ_{j} & if r + 1 ≤ i < j. \end{matrix}

Hence the Jacobian of the transformation on N which sends (n_{ij})_{i<j} to (ñ_{ij})_{i<j} is

J = ∏_{1≤i<j≤r} |λ_{i} − λ_{j}|^{2}|λ_{i} − λ̄_{j}|^{2} · ∏_{i≤r<j} |λ_{i} − μ_{j}|^{2} · ∏_{r+1≤i<j} |μ_{i} − μ_{j}|.

Now w_{ij} = ñ_{ij} + A_{ij} for all i < j. By (29), the Jacobian of the transformation which takes (n_{ij})_{i<j} to (w_{ij})_{i<j} is the constant J. Note that ∥z_{i}∥ ∼ |y_{i}|a_{i}^{2} as a_{i} → ∞ for 1 ≤ i ≤ r and ∥z_{r+k}∥ = |μ_{k}| for 1 ≤ k ≤ s. Now using (30) and the change of variables (n_{ij})_{i<j} ↦ (w_{ij})_{i<j}, we get that asymptotically

λ_{H_{1}\G}(R_{T}^{1} ) ∼ \frac{ω_{n}}{J} · T^{dim B+dim N},

where ω_{n} is the volume of the unit ball in ℝ^{n(n−1)/2}.

Non-focusing of the sets R_{T}:

Now let L be the component of identity of the ℝ-points of a proper connected (reductive) ℚ-subgroup of G containing H^{0}. Put L_{1} = g_{0}^{−1}Lg_{0}. Then L_{1} ⊃ H_{1}. From the structure of H_{1}, one verifies that each irreducible subspace for the adjoint action of H_{1} on the Lie algebra of L_{1} is in fact invariant under the adjoint action of M. Put L′_{1} = M^{0}L_{1}. Then L′_{1} is a proper reductive subgroup of G. Since M^{0} contains the full diagonal subgroup of G^{0} = SL_{n}(ℝ), we have that L′_{1} is the Levi part of a proper parabolic subgroup of SL_{n}(ℝ). Hence L′_{1} is the Levi part of a standard parabolic subgroup associated to a permutation of the standard basis of ℝ^{n}. Therefore the simple components of L′_{1} are SL_{m}(ℝ), where m < n, and H_{1} intersects each one of them in a maximal torus. Therefore by the computation as above for each m in place of n, we conclude that for any compact set C ⊂ G, we have

λ_{H_{1}\G}(L′_{1}C ∩ R_{T}^{1} ) = O(T^{dim(B)+dim(N∩L_{1})}).

Since dim L′_{1} < dim(SL_{n}(ℝ)) and B ⊂ L′_{1}, we have that N ⊄ L_{1}, and hence

dim(N ∩ L_{1}) < dim(N).

Therefore

lim_{T →∞} λ_{H_{1}\G}(L′_{1}C ∩ R_{T}^{1} )/λ_{H_{1}\G}(R_{T}^{1} ) = 0.

Hence for any compact set C ⊂ G,

lim_{T →∞} λ_{H\G}(LC ∩ R_{T})/λ_{H\G}(R_{T}) = 0.

Thus the sets R_{T} are not focused along any ℚ-subgroup L of G containing H.

Applying the counting theorem:

Now we can apply Theorem 1.16 to conclude that

N(T, V_{p}, Γv_{0}) ∼ \frac{vol(H ∩ Γ\H)}{vol(Γ\G)} vol(R_{T}).

Hence

N(T, V_{p}) ∼ c_{p}T^{n(n−1)/2},

where c_{p} > 0 is an explicitly computable constant. This completes the proof of Theorem 1.3.

As we mentioned earlier in the introduction, for the sake of simplicity we assume that all roots of p(λ) are real and ℤ[α] is the ring of integers in ℚ(α), where p(α) = 0, and give a formula for c_{p}.

Proof of Theorem 1.1: Let λ_{G} and λ_{H} be any Haar measures on G and H, respectively. Let λ_{H\G} be the G-invariant measure on H\G such that for any f ∈ C_{c}(G), if we put f̄(Hg) = ∫_{H} f(hg)dλ_{H}(h) for all Hg ∈ H\G, then f̄ ∈ C_{c}(H\G) and

∫_{G} f dλ_{G} = ∫_{H\G} f̄dλ_{H\G}.

Similarly, the choice of λ_{G} and λ_{H} determines volume forms on Γ\G and H(ℤ)\H. Let c_{G} and c_{H} denote the volumes of Γ\G and H\G, respectively. In view of the normalizations of the Haar measures on G and H, as chosen in the counting Theorem 1.16, we obtain that asymptotically

N(T, V_{p}, v_{0}Γ) ∼ (c_{H}c_{G}^{−1})λ_{H\G}(R_{T}).  (31)

Choice of λ_{G} and λ_{H}:

Let D denote the full diagonal subgroup of GL_{n}(ℝ). Fix a Haar integral such that dx = ∏_{i=1}^{n} dx_{i}/x_{i} on D, where x = (x_{1}, …, x_{n}) and each dx_{i} is the Lebesgue integral on ℝ. Put H_{1} = D ∩ G. Fix a Haar integral da on H_{1} such that dx = t^{−1}dtda, where x = t^{1/n}a, t = | det x|, and a ∈ H_{1}. Let K = O(n) and N be the group of upper triangular unipotent matrices in GL_{n}(ℝ). Let dk be the Haar integral on K such that ∫_{K} 1dk = 1. Let dn = ∏_{i<j} dn_{ij} be the Haar integral on N, where n = (n_{ij}) and dn_{ij} is the Lebesgue integral on ℝ. In view of the Iwasawa decomposition G = KNA, the integral dg = dkdnda is a Haar integral on G. We choose λ_{G} such that dλ_{G}(g) = dg on G. Let g_{0} ∈ G such that H_{1} = g_{0}^{−1}Hg_{0}. We choose λ_{H} such that under the Ad g_{0} action the Haar integral dλ_{H} on H maps to the integral da on H_{1}.

Volume computations:

Using [Terras, Ex. 21, Eqn. 4.40, Theorem 4], we deduce that

c_{G} = 2^{−(n−1)} ∏_{k=2}^{n} Λ(k/2),  where Λ(s) = π^{−s}Γ(s)ζ(2s).  (32)

Since we assume that ℤ[α] is the ring of integers in ℚ(α), we have that C^{−1}(H(ℤ)) is the group of units in ℚ(α). By [Lang, Sect.V.1,pp.110], if R is the regulator of ℚ(α), then

c_{H} = 2^{n}R.  (33)

Since all roots of p(λ) are real, following the computations in the proof of Theorem 1.3, and obtain asymptotically as T → ∞,

λ_{H\G}(R_{T}) ∼ 2^{−n}ω_{n}D^{−1/2} · T^{n(n−2)/2},  (34)

where D = ∏_{i≠j} |μ_{i} − μ_{j}| is the discriminant of p(λ) and {μ_{1}, …, μ_{n}} are the roots of p(λ).

By [New, Theorem III.14], the number of distinct Γ-orbits in V_{p}(ℤ) is the class number, say h, of ℤ[α].

Thus by equations (31), (32), (33), and (34), asymptotically

N(T, V_{p}) ∼ \frac{2^{n−1}hRω_{n}}{\sqrt{D} ∏_{k=2}^{n} Λ(k/2)} · T^{n(n−1)/2}.

Example 2

Proof of Theorem 1.4. Fix ξ ∈ V_{A,B}(ℤ). Let W_{ξ} be the m-dimensional subspace of ℝ^{n} spanned by the rows of ξ. Since ξA^{t}ξ = B is definite, the restriction of A to W_{ξ} is definite. Choose an orthonormal basis for W_{ξ} with respect to A, and extend it to a basis of ℝ^{n} such that the matrix of A takes the form

A = (\begin{matrix}I & 0 \\ 0 & J_{0} \end{matrix}) ,

where J_{0} is a (n − m) × (n − m) diagonal matrix whose diagonal entries are ±1’s. In this basis ξ = (ξ_{0}, 0), where ξ_{0} ∈ M_{m×m}(ℝ). Let H be the stabilizer of ξ in G and M = Z(H^{0}). Then

H = (\begin{matrix}I & 0 \\ 0 & H_{0} \end{matrix})  and  M = (\begin{matrix}M_{0} & 0 \\ 0 & I \end{matrix}) ,  (35)

where H_{0} ∈ SO(J_{0}) and M_{0} ∈ SO(m). In particular π(Z(H^{0})) is compact.

In order to apply Theorem 1.16, it remains to show that the pullback sets {R_{T} } are not focused, and to calculate their volume growth.

Let θ denote the Cartan involution θ(g) = ^{t}g^{−1}. Let K = {g ∈ G : θ(g) = g}. Then K is a maximal compact subgroup of G and M ⊂ K. Let σ be the involution of G = SO(A) defined by

σ(g) = (\begin{matrix}I & 0 \\ 0 & −J_{0} \end{matrix}) θ(g) (\begin{matrix}I & 0 \\ 0 & −J_{0} \end{matrix}) .

Note that θσ = σθ. Also the fixed point set of σ is H̃ = HM. Thus H̃ is an affine symmetric subgroup of G. There exists an ℝ-split torus Ã such that σ(a) = a^{−1} = θ(a) (∀a ∈ Ã) and G = H̃ÃK (see [Sch, Ch. 7]). Thus H\G/K ≅ M × Ã. As we shall see, the non-focusing of the sets R_{T} will follow from the fact that they grow at the same rate in all directions of Ã.

Ã-root-space decomposition of Lie(G):

Let r = min(m, q). Write matrix of the quadratic form A as

A = (\begin{matrix}I & 0 & 0 \\ 0 & J_{1} & 0 \\ 0 & 0 & −I \end{matrix}) ,

where J_{1} is a diagonal matrix with ±1 in each diagonal entry and I is the identity matrix. Note that in the above matrix, and all that follow, the first and the third columns are r elements wide, and the second column is n − 2r elements wide. Then Lie(Ã) is

𝔞̃ = \{Λ̃_{t} = (\begin{matrix}0 & 0 & Λ_{t}E \\ 0 & 0 & 0 \\ EΛ_{t} & 0 & 0 \end{matrix}) : t = (t_{1}, …, t_{r}) ∈ ℝ^{r}, Λ_{t} = diag(t_{1}, ⋯, t_{r})\} ,

where E is a matrix with 1 in each anti-diagonal entries and 0 elsewhere. Put

J = (\begin{matrix}I & 0 & E \\ 0 & I & 0 \\ E & 0 & −I \end{matrix}) .

Then

J^{−1}AJ = (\begin{matrix}0 & 0 & E \\ 0 & J_{1} & 0 \\ E & 0 & 0 \end{matrix})  and  J^{−1}Λ̃_{t}J = (\begin{matrix}Λ_{t} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & −EΛ_{t}E \end{matrix}) .

Put 𝔤 = Lie(G) = 𝔰𝔬(A), then

J^{−1}𝔤J = (\begin{matrix}X & Y_{1} & Z_{1} \\ Y_{2} & W & −J_{1} ^{t}Y_{1}E \\ Z_{2} & −E^{t}Y_{2}J_{1} & −E^{t}XE \end{matrix}) ,

where X, and the Y_{i} are arbitrary, the Z_{i} ∈ 𝔰𝔬(E), and W ∈ 𝔰𝔬(J_{1}). Thus the positive roots Σ^{+}(𝔤, 𝔞̃) obtained by diagonalizing 𝔞̃ on 𝔤 are given by:

β(Λ̃_{t}) = \{ \begin{matrix}t_{i} − t_{j} & (dim 𝔤^{β} = 1) & (1 ≤ i < j ≤ r) \\ t_{i} + t_{j} & (dim 𝔤^{β} = 1) & (1 ≤ i < j ≤ r) \\ t_{i} & (dim 𝔤^{β} = n − 2r) & (1 ≤ i ≤ r). \end{matrix}  (36)

Therefore the half sum of the positive roots is

ρ(Λ̃_{t}) = (1/2) ∑_{i=1}^{r}(n − 2i)t_{i}.  (37)

Haar measure on H\G:

We will use a formula for a G-invariant integral d(H̃g) on the symmetric space H̃\G (see [Sch, Theorem 8.1.1]):

∫_{H̃\G} f(H̃g)d(H̃g) = ∫𝔞̃_{+} ∫_{K} f(H̃ exp(Λ̃_{t})k)δ(Λ̃_{t}) dt_{1} ⋯ dt_{r} dk,

where 𝔞̃_{+} is the positive (closed) Weyl chamber in 𝔞̃,

δ(Λ̃_{t}) = ∏_{β} (sinh β(Λ̃_{t}))^{p_{β}} (cosh β(Λ̃_{t}))^{q_{β}} ,

β runs over the positive roots Σ^{+}(𝔤, 𝔞), and dim 𝔤^{β} = p_{β} + q_{β}.

Since H̃ = HM, using the above formula a G-invariant integral d(Hg) on H\G is given by

∫_{H\G} f(Hg)d(Hg) = ∫_{M} ∫𝔞̃_{+} ∫_{K} f(Hm exp(Λ̃_{t})k)δ(Λ̃_{t}) dm dt_{1} ⋯ dt_{r} dk.

As t → +∞, both sinh t and cosh t are asymptotic to (1/2) exp t. Therefore in the interior of 𝔞̃_{+}, asymptotically

δ(Λ̃_{t}) ∼ exp(2ρ(Λ̃_{t})).  (38)

Co-Ordinate description of R_{T}:

Put a_{i} = exp(t_{i}), and express (a_{1}, …, a_{r}) = ηω, where ω ∈ S^{r−1}. We will use the notation D_{ηω} = exp(Λ̃_{t}).

Claim 1 The limit

λ(m, ω) = lim_{η→∞}(1/η)∥ξ · mD_{ηω}k∥  (39)

exists, it is independent of k ∈ K, and the convergence is uniform in the parameters m ∈ M and ω ∈ S_{+}^{r−1}, where S_{+}^{r−1} = S^{r−1} ∩ exp 𝔞̃_{+}. Moreover the function λ(m, ω) is continuous and bounded away from 0.

To prove the claim, write

∥ξ · mD_{ηω}k∥ = ∥(ξ_{0}m_{0}, 0) · JJ^{−1}D_{ηω}∥
= ∥(ξ_{0}m_{0}, ψ) · J^{−1}D_{ηω}∥
= η∥(ξ_{0}m_{0})_{1} · J^{−1}D_{ω}∥ + O(1),

where m_{0} ∈ M_{0}, ψ ∈ M_{m×(n−m)}(ℝ) is bounded, and (ξ_{0}m_{0})_{1} is the matrix obtained by replacing the last n − r columns of (ξ_{0}m_{0}, 0) by 0. Thus

lim_{η→∞}(1/η)∥ξ · mD_{ηω}k∥ = ∥(ξ_{0}m_{0})_{1} · J^{−1}D_{ω}∥

and the claim follows.

Claim 2 Asymptotically, vol(R_{T}) ∼ c_{A,B}T^{r(n−r−1)}, where

c_{A,B} = \frac{1}{r(n − r − 1)} ∫_{M} ∫_{S_{+}^{r−1}} \frac{∏_{1}^{r} ω_{j}^{n−2j−1}}{λ(m, ω)^{r(n−r−1)}} dm dω > 0.  (40)

To prove the claim we express

vol(R_{T}) = ∫_{M} ∫_{{t∈ℝ^{r}:∥ξ_{0}·m exp(Λ̃_{t})k∥<T }} δ(Λ̃_{t}) dm dt dk
= ∫_{M} ∫_{K} ∫_{S_{+}^{r−1}} ∫_{\{η>0:∥ξ_{0}·mD_{ηω}k∥<T \}} δ(D_{ηω})\frac{η^{−1}}{ω_{1} ⋯ ω_{r}} dm dk dω dη.

By Claim 1, and equations (38) and (37), asymptotically

vol(R_{T}) ∼ ∫_{M} ∫_{K} ∫_{S_{+}^{r−1}} ∫_{0}^{T/λ(m,ω)} η^{r(n−r−1)−1} ∏_{j=1}^{r} ω_{j}^{n−2j−1} dm dk dω dη
= c_{A,B} · T^{r(n−r−1)}.

This proves the claim.

Non-focusing of the sets R_{T}:

Claim 3 Let L be a connected reductive real algebraic subgroup of G containing H. Then θ(L) = L, (recall that θ(g) = ^{t}g^{−1}).

To prove the claim, let θ_{1} be a Cartan involution of G stabilizing L. Since H ⊂ L is reductive, there exists l ∈ L such that θ_{2} = i_{l}^{−1}θ_{1}i_{l} stabilizes H, where i_{l} is conjugation by l. Since θ restricted to H is a Cartan involution of H, there exists h ∈ H, such that θ′ = i_{h}^{−1}θ_{2}i_{h} agrees with θ on H. Note that θ′(L) = L. Since θ and θ′ are Cartan involutions of G, θ′ = i_{g}θi_{g}^{−1} for some g ∈ G. Let g_{1} = g^{−1}θ(g). Then θ′ = i_{g_{1}}θ. Since θ and θ′ agree on H, g_{1} ∈ Z(H) = M.

Let P = {p ∈ G : θ(p) = p^{−1}}. By Cartan decomposition, G = PK and K ∩ P = {e}. Write g = pk for p ∈ P and k ∈ K. Then g_{1} = p^{2} ∈ P.

But M ∩ P = {e}. Therefore g_{1} = e, and hence θ = θ′. Thus θ(L) = L and the claim holds.

Claim 4 The sequence {R_{T_{i}}} is not focused for any sequence T_{i} → ∞.

Now suppose that {R_{T_{i}}} is focused along a proper connected reductive real algebraic subgroup L of G containing H. Let 𝔩 = Lie(L), 𝔥 = Lie(H) and 𝔨 = Lie(K). With respect to the Killing form on 𝔤, let 𝔮 = 𝔥^{⊥}, 𝔭 = 𝔨^{⊥}, and 𝔮_{L} = 𝔩^{⊥}∩𝔭∩𝔮. Since θ(L) = L, by Theorem A.1, L = H exp(𝔭∩𝔮∩𝔩)K_{L} and G = L exp 𝔮_{L}K.

Now let C ⊂ G be a compact set. Then there exists a compact set 𝔠 ⊂ 𝔮_{L}, such that

LC ⊂ H exp(𝔮 ∩ 𝔭 ∩ 𝔩 + 𝔠)K  (41)

Put

Δ = {(m, ω) ∈ M × S_{+}^{r−1} : mD_{ω}m^{−1} ∈ L}.

Then

LK = ⋃{HmD_{ηω}K : (m, ω) ∈ Δ, η ≥ 0}.

For η > 0, put

U_{η} = {(m, ω) ∈ M × S^{r−1} : HmD_{ηω}K ∩ LC ≠ ∅}.

Then by equation (41), U_{η} ↓ Δ as η → ∞. Note that measure of Δ is zero with respect to the integral dmdk. Now by Claim 1 and the computations in Claim 2, we obtain that vol(R_{T} ∩ LC) = o(vol(R_{T})). This completes the proof of the claim.

Now applying Theorem 1.16, we obtain Theorem 1.4.

7 Effect of focusing on counting estimates

We present an example where the sets R_{T} (T ≫ 0) are focused along an intermediate subgroup H ⊂ L ⊂ G. Unlike in Theorem 1.17, here N(T, V, O) ≁ λ_{H\G}(R_{T}) as T → ∞. This example was presented without proof in [EM].

Let G = SL_{2}(ℂ) and H = {diag(a, a^{−1}) : a ∈ ℝ^{×}}. Let Γ = g^{−1}SL_{2}(ℤ[i])g, where g ∈ SL_{2}(ℝ) conjugates a hyperbolic element of SL_{2}(ℤ) into H. Then Γ ∩ H\H is compact. Here we consider the ℚ-structure on G such that G(ℚ) = g^{−1}SL_{2}(ℚ[i])g.

We will consider the following representation of G, with a closed orbit isomorphic to H\G. Fix N ≥ 4, and let (z_{1}, z_{2}) be co-ordinates on ℂ^{2}. Let W be the vector space of polynomial functions v(z_{1}, z_{2}, z̄_{1}, z̄_{2}) on ℂ^{2} which are homogeneous of degree N in z_{1} and z_{2}, and also in z̄_{1} and z̄_{2}. The monomials z_{1}^{m}z_{2}^{N−m}z̄_{1}^{n}z̄_{2}^{N−n}, where 0 ≤ m, n ≤ N, form a basis for W. Since G acts linearly on ℂ^{2}, it acts linearly on W by substitution; i.e. for g ∈ G and w ∈ W, we have [w · g](z_{1}, z_{2}) = w((z_{1}, z_{2})g^{−1}) (∀(z_{1}, z_{2}) ∈ ℂ^{2}).

Consider the polynomial

p(z_{1}, z_{2}) = [\frac{z_{1}z̄_{2} + z̄_{1}z_{2}}{2} ]^{2} [\frac{−z_{1}z̄_{2} + z̄_{1}z_{2}}{2i} ]^{N−2} .

Then Stab_{G}(p) = H. Let V = p · G. Put U = \{u(w) = (\begin{matrix}1 & w \\ 0 & 1 \end{matrix}) : w ∈ ℂ\} and K = SU(2). Then G = HUK. Since U is unipotent and K is compact, V is closed in the Hausdorff topology. Now since the G-action on W is linear, V is Zariski closed. Thus the variety V is naturally identified with H\G.

Next we consider the distribution of the subset p · Γ ⊂ V. Let ∥ · ∥ be a K-invariant norm on W. Then for T > 0 we have R_{T} = {Hg ∈ H\G : ∥p · g∥ < T }. Put L = SL_{2}(ℝ). Then L is a ℚ-subgroup of G containing H with no nontrivial ℚ-characters. We will show that for any sequence T_{n} → ∞, the sequence {R_{T_{n}}} is focused along L.

Co-ordinate description of R_{T}:

Since R_{T} is K-invariant, it can be treated as a subset of H\G/K. Since G = HUK, we can use U ≅ ℂ as the co-ordinate space for H\G/K. We identify U with ℝ^{2} by the mapping (x, y) ↦ u(x + iy). Thus we can identify R_{T} with a subset of ℝ^{2}.

Lemma 7.1 In the ℝ^{2}-co-ordinates R_{T} = {(x, y) : f(x, y) < T^{2}}, where

f(x, y) = c_{1}x^{2} + c_{2}y^{2M} + c_{3}x^{2}y^{2M} + O(x^{2}y^{2M−2}) + O(y^{2M−2}).

Here M = N − 2 ≥ 2, c_{i} > 0 (i = 1, 2, 3), and the big O-terms in the definition of f are sums of squares of monomials.

Proof. Since (z_{1}, z_{2}) · u(w)^{−1} = (z_{1}, z_{2} − wz_{1}), if w = x + iy, then

[p · u(w)](z_{1}, z_{2})

= [\frac{z_{1}(z̄_{2} − w̄z̄_{1}) + z̄_{1}(z_{2} − wz_{1})}{2}]^{2} [\frac{−z_{1}(z̄_{2} − w̄z̄_{1}) + z̄_{1}(z_{2} − wz_{1})}{2i}]^{N−2}
= [\frac{z_{1}z̄_{2} + z_{2}z̄_{1} − 2x|z_{1}|^{2}}{2} ]^{2} [\frac{z̄_{1}z_{2} − z_{1}z̄_{2} − 2iy|z_{1}|^{2}}{2i} ]^{N−2} .  (42)

Since the given norm on W is K-invariant, the distinct weight spaces are orthogonal. Therefore the norm can be express in the form

∥ ∑_{m,n} c_{mn}z_{1}^{m}z_{2}^{N−m}z̄_{1}^{n}z̄_{2}^{N−n}∥ = ∑_{m,n} λ_{mn}|c_{mn}|^{2},  (43)

where the λ_{mn}’s are positive constants.

Now the conclusion of the lemma follows from equations (42) and equation (43).

In view of the last lemma, by ignoring the lower order terms, we define

R′_{T} = {(x, y) : c_{1}x^{2} + c_{2}y^{2M} + c_{3}x^{2}y^{2M} < T^{2}}.

Focusing of {R_{T} } along the x-axis:

Lemma 7.2 For T ≫ 1, vol(R′_{T}) = O(T). Also for any ϵ > 0, there exists c_{ϵ} > 0 such that

lim sup_{T →∞} \frac{vol(R′_{T} ∩ \{|y| > c_{ϵ}\})}{vol(R′_{T})} ≤ ϵ.

Figure 1. The sets R′_{T} for N = 5 and T = 100, 200.

Proof. We have (x, y) ∈ R′_{T} ⟹ y^{2M}(c_{2} + c_{3}x^{2}) < T^{2} − c_{1}x^{2}. So

\frac{1}{T} vol(R′_{T}) = \frac{4}{T} ∫_{0}^{T/\sqrt{c_{1}}} [\frac{T^{2} − c_{1}x^{2}}{c_{2} + c_{3}x^{2}} ]^{1/(2M)} dx

= \frac{4}{\sqrt{c_{1}}} ∫_{0}^{1} [\frac{1 − u^{2}}{c_{2}/T^{2} + (c_{3}/c_{1})u^{2}} ]^{1/(2M)} du

→ \frac{4}{\sqrt{c_{1}}} (\frac{c_{1}}{c_{3}} )^{1/(2M)} ∫_{0}^{1} [\frac{1 − u^{2}}{u^{2}} ]^{1/(2M)} du

as T → ∞, and the limit is finite beause M ≥ 2.

In terms of the U × K-co-ordinates on H\G, the G-invariant integral is given by d(Hg) = du dk, where du is the Lebesgue integral on U ≅ ℝ^{2} and dk is a Haar integral on K.

Lemma 7.3 For T ≫ 1, vol_{H\G}(R_{T}) = O(T). Also for every ϵ > 0, there exists c_{ϵ} > 0 such that

lim sup_{T →∞} \frac{vol_{H\G}(R_{T} ∩ \{|y| > c_{ϵ}\})}{vol_{H\G}(R_{T})} ≤ ϵ.

Proof. For |y| ≫ 1, y^{2M−2} is dominated by y^{2M}, and x^{2}y^{2M−2} is dominated by x^{2}y^{2M}. Thus, in this region, R′_{T} is an excellent approximation to R_{T}. Hence the conclusion follows from Lemma 7.2.

Proposition 7.4 The sets R_{T} are focused along L = SL_{2}(ℝ). In fact, for any ϵ > 0 there exists a compact subset C of G such that

lim inf_{T →∞} \frac{vol_{H\G}(LC ∩ R_{T})}{vol_{H\G}(R_{T})} > 1 − ϵ.

Proof. For any c > 0 there exists a compact set C ⊂ G, such that R_{T} ∩ LC ⊃ R_{T} ∩ {|y| < c}. Now the conclusion of the proposition follows from Lemma 7.3.

Focusing function:

Since L is affine symmetric, G = LBK, where B = \{b(t) = (\begin{matrix}cosh t & i sinh t \\ −i sinh t & cosh t \end{matrix}) : t ∈ ℝ}. For c ≥ 0, express u(ic) = lb(t)k, where l ∈ L, t ≥ 0 and k ∈ K. By multiplying the equation on the left by the inverse of its complex conjugate, and then taking the Hilbert-Schmidt norm of both sides, we get 2+4c^{2} = e^{4t}+e^{−4t}, i.e c = sinh 2t. In particular, we have the decomposition, G = Lu(iℝ)K.

By [Sch, Sect. 8.4.2], a Haar integral on G in the L×B ×K-co-ordinates is cosh^{2}(2t) dl dt dk. Hence in the L × u(iℝ) × K-co-ordinates, the integral j(c) dl dc dk is G-invariant, where j(c) = \sqrt{1 + c^{2}}. Thus using the u(iℝ)×Kco-ordinates on L\G, the G-invariant integral on L\G is d(Lg) = j(c) dc dk.

For c ∈ ℝ, define X_{T} (c) = {x ∈ ℝ : (x, c) ∈ R_{T} }. By Lemma 7.1, |X_{T} (c)| ∼ O(c^{−M})T for T ≫ 1. Now define

λ(c) = lim_{T →∞} \frac{|X_{T} (c)|}{vol_{H\G}(R_{T})};  (44)

the limit exists because vol_{H\G}(R_{T}) = O(T), by Lemma 7.3. Also λ(c) is continuous and

λ(c) = λ(−c) = O(c^{−M})  as c → ∞.  (45)

For g = lu(ic)k, where l ∈ L, c ∈ ℝ, and k ∈ K, define λ̃(g) = λ(c)/j(c). Then λ̃ is a left L-invariant and right K-invariant continuous function on G.

We can also treat λ̃ as a function on L\G. Since M ≥ 2, by equation (45),

∫_{L\G} λ̃(Lg) d(Lg) = ∫_{−∞}^{∞} λ(c) dc = O (∫_{1}^{∞} c^{−M}dc) < ∞.  (46)

Define the ‘focusing map’ on Γ\G by

Λ(Γg) = ∑_{γ̇∈(L∩Γ)\Γ} λ̃(γg).

Then

⟨Λ, ψ⟩ = ∫_{Γ\G} d(Γg) ∑_{γ̇∈(L∩Γ)\Γ} λ̃(γg)\overline{ψ}(Γg)
= ∫_{L\G} d(Lg) ∫_{(L∩Γ)\L} λ̃(Lg)\overline{ψ}(Γlg) d((L ∩ Γ)l)
= ∫_{L\G} d(Lg) λ̃(Lg) (∫_{Γ\G} \overline{ψ} dμ_{L}g) ,  (47)

which is finite by equation (46).

Counting estimate:

Let ψ be a compactly supported continuous function on Γ\G. Then for any c ∈ ℝ, by Theorem 1.7,

lim_{x→∞} ∫_{Γ\G} \overline{ψ} d[μ_{H} · u(x + ic)] = ∫_{Γ\G} \overline{ψ} d[μ_{L} · u(ic)].  (48)

As in Proposition 5.3, define the ‘counting function’ on Γ\G by

F̂_{T} (Γg) = \frac{1}{vol_{H\G}(R_{T})} ∑_{γ̇∈(H∩Γ)\Γ} χ_{R_{T}} (p · γg).

Then

lim_{T →∞}⟨F̂_{T} , ψ⟩ = lim_{T →∞} \frac{1}{vol_{H\G}(R_{T})} ∫_{R_{T}} (∫_{Γ\G} \overline{ψ} dμ_{H}g) d(Hg)

= lim_{T →∞} \frac{1}{vol_{H\G}(R_{T})} ∫_{K} dk ∫_{−∞}^{∞} dc ∫_{X_{T} (c)} dx (∫_{Γ\G} \overline{ψ} d[μ_{H} · u(x + ic)k])

= lim_{T →∞} \frac{1}{vol_{H\G}(R_{T})} ∫_{K} dk ∫_{−∞}^{∞} |X_{T} (c)| (∫_{Γ\G} \overline{ψ} d[μ_{L} · u(ic)k]) dc (49)

= ∫_{K} dk ∫_{−∞}^{∞} λ(c) (∫_{Γ\G} \overline{ψ} d[μ_{L} · u(ic)k]) dc  (50)

= ∫_{L\G} λ̃(Lg) (∫_{Γ\G} \overline{ψ} dμ_{L}g) d(Lg)  (51)
= ⟨Λ, ψ⟩;  (52)

the equations (49), (50), (51) and (52) follow from equations (48), (44),

(46) and (47), respectively. Hence as in the proof of Theorem 1.16 given in section 5, we obtain the following.

Theorem 7.5 There exists a non constant continuous positive function Λ(g) on Γ\G such that asymptotically as T → ∞,

F_{T} (g) ∼ Λ(g) vol_{H\G}(R_{T}).

A Appendix: Asymptotics of volume growth of R_{T}

In this appendix, we prove Proposition 5.4. Our argument is more involved than that of [DRS, Appendix 1] because we do not assume that H is an affine symmetric subgroup, and we allow dilations of arbitrary symmetric convex sets B_{T}, not only K-invariant balls.

The following theorem is proved in [Mos], see also [Hel, Ch. IV, exercise A.2]

Theorem A.1 Let G be an algebraic reductive group, H a reductive subgroup. Let K be a maximal compact subgroup of G, associated to a Cartan involution stabilizing H. Let 𝔤, 𝔨 and 𝔥 denote the Lie algebras of G, H and K respectively. Let 𝔭 = 𝔨^{⊥} and 𝔮 = 𝔥^{⊥} with respect to the Killing form on 𝔤. Then

G = H exp(𝔭 ∩ 𝔮)K.

For g ∈ G, consider the corresponding decomposition g = hqk, where h ∈ H, q ∈ exp(𝔮∩𝔭), and k ∈ K. Then h, q, and k are uniquely determined by g up to the transformation (h, q, k) ↦ (hm, Ad m(q), m^{−1}k), where m ∈ H ∩ K.

Let M = K ∩ H and let π : K → M\K be the natural quotient map. There exists a submanifold K′ of K such that e ∈ K′, π is a local diffeomorphism on K′ and π(K′) has full K-invariant measure on M\K.

Let Φ : (𝔭 ∩ 𝔮) × K′ → H\G be the map given by Φ(Y, k) = H exp(Y)k. We will relate the invariant measure on H\G to the Euclidean measure on 𝔭 ∩ 𝔮, say dY, and the pullback of the K-invariant measure on M\K to K′, say dk. Let J Φ(Y, k) denote the Jacobian of Φ at (Y, k). Then the Jacobian is independent of k ∈ K′. We write J Φ(Y, k) = δ(Y).

Now a G-invariant integral d(Hg) on H\G can be given by

∫_{H\G} f(Hg) d(Hg) = ∫_{𝔭∩𝔮} ∫_{K′} f(H exp(Y)k)δ(Y) dY dk.  (53)

Using the polar co-ordinates on 𝔭 ∩ 𝔮, we write Y = rω, where r = ∥Y ∥ and ω = Y/∥Y ∥. We also write δ(Y) = δ(r, ω, k).

Lemma A.2 There exists n ∈ ℕ and Λ > 0 such that as a function of r, we have r^{l}δ(r, ω) ∈ E(n, Λ) for all ω ∈ 𝔭∩𝔮 with ∥ω∥ = 1, where l = dim(𝔭∩𝔮).

Proof. (cf. [Sch, Thm. 8.1.1]). Put q = exp(Y). For Z ∈ 𝔭 ∩ 𝔮 and X ∈ T (K′),

Φ(exp(Z + Y), exp Xk′) = H exp(Z + Y) exp(−Y) exp(Ad(q^{−1})X)qk′.

Therefore

dΦ_{(Y,k′)}(Z, X) = Pr_{𝔮}((d exp)_{Y} (Z) + Ad(q^{−1})(X)),

where Pr_{𝔮} : 𝔤 → 𝔮 is the projection parallel to 𝔥.

By [Var, Theorem 2.14.3],

(d exp)_{Y} (Z) = ∑_{n=0}^{∞} \frac{(−1)^{n}}{(n + 1)!}(ad Y)^{n}(Z).  (54)

Write Y = rω in polar co-ordinates, where r > 0 and ω ∈ S^{l−1}, the unit sphere in 𝔭 ∩ 𝔮. For ω ∈ S^{l−1}, let ν_{i}(ω) and W_{i}(ω) be the eigenvalues and eigenvectors, respectively, of ad ω ∈ End(𝔤). Since all vectors in 𝔭 ∩ 𝔮 are semisimple, we can choose the ν_{i} and W_{i} to depend continuously on ω. By equation (54),

(d exp)_{Y} (Z) + Ad_{q}(X) = ∑_{i=1}^{dim 𝔤} (\frac{1 − e^{−rν_{i}(ω)}}{rν_{i}(ω)} ⟨Z, W_{i}⟩ + e^{rν_{i}(ω)}⟨X, W_{i}⟩) W_{i}.

Thus as a function of r, each matrix entry of Pr_{𝔮}(r((d exp)_{Y} (Z) + Ad_{q}(X))) belongs to some E(n′, Λ′), where n′ and Λ′ can be chosen independent of ω. Therefore r^{l} times the Jacobian determinant δ(r, ω) belongs to E(n, Λ), for some n and Λ independent of ω.

Note that in the polar co-ordinates, the integral d(Hg) on H\G as in equation (53) can be expressed as

∫_{H\G} f(Hg) d(Hg) = ∫_{r>0} ∫_{S^{l−1}} ∫_{K′} f(H exp(rω)k)δ(r, ω)r^{l−1} dr dω (55)

Lemma A.3 There exists a constant C, independent of ω, such that for any r > 1,

δ(r, ω)r^{l−1} ≤ C ∫_{0}^{r} δ(s, ω)s^{l−1} ds  (56)

Proof. In view of Lemma A.2, it is enough to show the following: Let δ_{0} be as in Proposition 3.6. Then for all A ≤ r ≤ A + δ_{0} and f ∈ E(n, Λ), we have

f(r) ≤ C ∫_{A}^{A+δ_{0}} f(s) ds.  (57)

Since we can make the change of variable x → x + A, we may assume A = 0. We may also assume that f is the maximum at r on [0, δ_{0}]. Now equation (57) follows from Proposition 3.6.

Shape of R_{T} in (r, ω, k)-co-ordinates:

We are given a linear representation of G on a vector space ℝ^{N}, a vector v_{0} ∈ ℝ^{N} whose stabilizer is H, and a norm ∥ · ∥ on ℝ^{N} which may not be K-invariant.

Let the notation be as above. For any ω ∈ S^{n−1}, choose a basis consisting of eigenvectors {v_{i}(ω)}_{i=1}^{N} for exp(rω) such that

v_{i} · exp(rω) = e^{rλ_{i}(ω)}v_{i} .

The v_{i}(ω) can be made to depend continuously on ω. Then if g = h exp(rω)k,

v_{0} · g = ∑_{i=1}^{N} e^{rλ_{i}(ω)}v_{i}(ω) · k,

and hence

vol(R_{T}) = ∫_{K} ∫_{S_{T} (k)} δ(r, ω) r^{l−1} dr dω dk = ∫_{K} m(T, k) dk,  (58)

where

S_{T} (k) = {(r, ω) ∈ (𝔭 ∩ 𝔮) : ∥ ∑ e^{rλ_{i}(ω)}v_{i}(ω) · k∥ ≤ T },  (59)

and

m(T, k) = ∫_{S_{T} (k)} δ(r, ω)r^{l−1} dr dω.  (60)

Lemma A.4 For T ≫ 0 the following holds.

(i) S_{T} (k) is star-shaped. In particular, for each ω ∈ S^{n−1} and k ∈ K, there exists a unique s > 0, denoted by r(T, ω, k), such that

∥ ∑_{i=1}^{N} e^{sλ_{i}(ω)}v_{i}(ω) · k∥ = T.

(i) r(κT, ω, k,) ≤ r(T, ω, k) + β(κ), where β(κ) → 1 as κ → 1. The function β(κ) is independent of ω and k.

First assuming this lemma, we prove the following.

Proposition A.5 For some functions a(κ) → 1 and b(κ) → 1 as κ → 1, we have

a(κ) ≤ lim inf_{T →∞} \frac{m(κT, k)}{m(T, k)} ≤ lim sup_{T →∞} \frac{m(κT, k)}{m(T, k)} ≤ b(κ).  (61)

Proof. Because S_{T} (k) is star-shaped, we can do the integration as in equation (60) in polar co-ordinates:

m(T, ω, k) = ∫_{0}^{r(ω,k,T)} δ(r, ω)r^{l−1} dr,  and
m(T, k) = ∫_{ω} m(T, ω, k) dω.

Therefore

log \frac{m(κT, ω, k)}{m(T, ω, k)} = log ∫_{0}^{r(κT,ω,k)} δ(s, ω)s^{l−1} ds − log ∫_{0}^{r(T,ω,k)} δ(s, ω)s^{l−1} ds
≤ log ∫_{0}^{r(T,ω,k)+β(κ)} δ(s, ω)s^{l−1} ds  (62)
− log ∫_{0}^{r(T,ω,k)} δ(s, ω)s^{l−1} ds
= β(κ)\frac{δ(σ, ω)σ^{l−1}}{∫_{0}^{σ} δ(s, ω)s^{l−1} ds} ≤ Cβ(κ),  (63)

where equation (62) follows from Lemma A.4 and equation (63) follows from the mean value theorem, for some

σ ∈ [r(T, ω, k), r(T, ω, k) + β(κ)].

Thus

\frac{m(κT, ω, k)}{m(T, ω, k)} ≤ exp(Cβ(κ)) = b(κ)

with b(κ) → 1 as κ → 1. From this, equation (61) follows.

Proof of Proposition 5.4. The result is an immediate consequence of Proposition A.5 and equation (58).

Proof of Lemma A.4. We will prove both the statements simultaneously. First we need to define λ_{max}(ω) = max{λ_{i}(ω)}, ( λ_{i} are as in equation (59)). Since all norms are equivalent, there exist constants c_{1}, c_{2} such that

c_{1}e^{sλ_{max}(ω)} ≤ ∥ ∑ e^{sλ_{i}(ω)}v_{i}(ω) · k∥ = T ≤ c_{2}e^{sλ_{max}(ω)}.

This implies that if the region S_{T} (k) is not star-shaped, say a ray hits the boundary ∂S_{T} (k) at two points

p_{j} = ∑ e^{s_{j}λ_{i}(ω)}v_{i}(ω) · k (j = 1, 2),

|s_{1} − s_{2}| < (log c_{2} − log c_{1})/λ^{−} = c_{3},  (64)

where λ^{+} = min_{ω} λ_{max}(ω) > 0.

Claim. For β > 0 and T sufficiently large,

∥ ∑ e^{(r(ω,k,T)+β)λ_{i}(ω)}kv_{i}∥ ≥ T g(β)  (65)

with g(0) = 1, g′(0) > 0, and g(β) > 1 if 0 < β < c_{3} ( c_{3} is defined in equation (64)).

Assuming this for now, we immediately see that S_{T} (k) is star-shaped for sufficiently large T: since by equation (65)

∥ ∑ e^{(r(ω,k,T)+β)λ_{i}(ω)}v_{i}(ω) · k∥ > T,

if 0 < β < c_{3}, while by equation (64) this is the only range where coincidence may occur. Also equation (65) shows that, because S_{T} (k) is star-shaped,

r(ω, k, T) + β ≥ r(ω, k, g(β)T).

Since g is increasing near β = 0, for κ near 1 we can determine β by requiring g(β) = κ. Thus for sufficiently large T, and for κ near 1,

r(ω, k, T) + β(κ) ≥ r(ω, k, κT)

This proves Lemma A.4, assuming the claim.

Proof of the claim. Let ϵ > 0 be a small parameter to be chosen later. Put s = r(T, ω, k), X_{i} = e^{sλ_{i}(ω)}v_{i}(ω) · k, and δ_{ϵ}(ω) = λ_{max}(ω) − ϵ. Let Δ^{+}(ω) = {i : λ_{i}(ω) > δ_{ϵ}(ω)} and Δ^{−}(ω) = {i : λ_{i}(ω) < δ_{ϵ}(ω)}. For β > 0,

∥ ∑_{i} e^{(s+β)λ_{i}(ω)}v_{i}(ω) · k∥

= ∥e^{β(λ_{max}(ω)−ϵ)} ∑_{i} X_{i} + ∑_{i}(e^{βλ_{i}(ω)} − e^{β(λ_{max}(ω)−ϵ)})X_{i}∥
≥ e^{βδ_{ϵ}(ω)}∥ ∑_{i} X_{i}∥ − ∑_{i} |e^{βδ_{ϵ}(ω)} − e^{βλ_{i}(ω)}| ∥X_{i}∥
= e^{βδ_{ϵ}(ω)}T − ∑_{i∈Δ^{+}(ω)}(e^{βλ_{i}(ω)} − e^{βδ_{ϵ}(ω)})∥X_{i}∥ − ∑_{i∈Δ^{−}(ω)}(e^{βδ_{ϵ}(ω)} − e^{βλ_{i}(ω)})∥X_{i}∥

≥ e^{βδ_{ϵ}(ω)}T − c_{4}T ∑_{i∈Δ^{+}(ω)}(e^{βλ_{i}(ω)} − e^{βδ_{ϵ}(ω)}) − c_{5} ∑_{i∈Δ^{−}(ω)}(e^{βδ_{ϵ}(ω)} − e^{βλ_{i}(ω)})T^{\frac{δϵ(ω)}{λmax(ω)}}

≥ T [e^{β(λ^{−}−ϵ)}(1 − c_{4} ∑_{i∈Δ^{+}(ω)}(e^{ϵβ} − 1)) − c_{5} ∑_{i∈Δ^{−}(ω)}(e^{β(λ^{+}−ϵ)} − e^{βν}) T^{−ϵ/λ^{+}} ] ,

where λ^{−} = min_{ω} λ_{max}(ω) > 0, λ^{+} = max_{ω} λ_{max}(ω) > 0, and ν = min_{i,ω} λ_{i}(ω).

= T [e^{β(λ^{−}−ϵ)}(1 − c_{6}(e^{ϵβ} − 1)) − c_{7}(e^{β(λ^{+}−ϵ)} − e^{βν})T^{−ϵ/λ^{+}}]
= T (g_{1}(β) − g_{2}(β)T^{−ϵ/λ^{+}}).

We can choose ϵ small enough so that

g_{1}(β) = e^{β(λ^{−}−ϵ)}(1 − c_{6}(e^{ϵβ} − 1))

has positive derivative at β = 0, and g_{1}(β) > 1 for 0 < β < c_{3}. Then

∥ ∑ e^{(s+β)λ_{i}(ω)}v_{i}(ω) · k∥ ≥ T (g_{1}(β) − g_{2}(β)T^{−ϵ/λ^{+}}).

Since g_{2}(β) = c_{7}(e^{β(λ^{+}−ϵ)} − e^{βν}) vanishes at β = 0, we can adjust ϵ > 0 so that for sufficiently large T,

g_{1}(β) − g_{2}(β)T^{−ϵ/λ^{+}} ≥ g(β)

with g(0) = 1, g′(0) > 0, and g(β) > 1 for 0 < β < c_{3}. This proves the claim.

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Department of Mathematics, University of Chicago, Chicago IL 60637, USA; eskin@math.uchicago.edu.

Institute of Mathematics, Hebrew University, Jerusalem 91904, ISRAEL; mozes@math.huji.ac.il

School of Mathematics, Tata Institute of Fundamental Research, Bombay 400005, INDIA; nimish@tifrvax.tifr.res.in