CENTRAL LIMIT THEOREMS FOR DIOPHANTINE APPROXIMANTS

Transcription

1 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS ABSTRACT. In this paper we study the counting functions that represent the number of solutions of systems of linear inequalities arising in the theory of Diophantine approximation. We develop a method that allows to justify random-like behaviour of these functions and prove that they satisfy the Central Limit Theorem. Our approach is based on analysis of higher-order correlations on the space of lattices and a novel technique for estimating cumulants of Siegel transforms. COTETS 1. Introduction and main results 1 2. Estimates on higher-order correlations 5 3. CLT for functions with compact support on-divergence estimates for Siegel transforms CLT for smooth Siegel transforms CLT for counting functions and the proof of Theorem References Motivation 1. ITRODCTIO AD MAI RESLTS Many objects arising in Diophantine Geometry exhibit random-like behaviour. For instance, the classical Khinchin theorem in Diophantine approximation can be interpreted as the Borel-Cantelli Property for quasi-independent events, and Schmidt s quantitative generalisation of the Khinchin theorem is analogous to the Law of Large umbers. One might wonder whether much deeper probabilistic phenomena take place in Diophantine approximation. In this paper, we develop a general framework which allows to capture a quasi-independence property governing behaviour of arithmetic counting functions. We expect that this new methods will have a wide range of applications in Diophantine Geometry, and here we apply it to study the distribution the functions counting Diophantine approximants. A basic problem in Diophantine approximation asks for finding good rational approximants of vectors u = u 1,..., u m R m. More precisely, given positive numbers w 1,..., w m, which we shall assume sum to one, and positive constants ϑ 1,..., ϑ m, we consider the system of inequalities uj p j ϑ j, for j = 1,..., m, 1.1 q q 1+w j with p, q Z m. It is well-known that for Lebesgue-almost all u R m, the system 1.1 has infinitely many solutions p, q Z m, so it is natural to try to count solutions in bounded regions, which leads us to the counting function T u := {p, q Z m : 1 q < T and W. Schmidt [16] proved that for Lebesgue-almost all u [0, 1] m, 1.1 holds}. T u = C m log T + O u,ε log T 1/2+ε, for all ε > 0, 1.2 1

2 2 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS where C m := 2 m ϑ 1 ϑ m. One may view this as an analogue of the Law of Large umbers, the heuristic for this analogy runs along the following lines. First, note that where T u log T s=0 s u, s u := {p, q Z m : e s q < e s+1 and 1.1 holds}. If one could prove that the functions s 1 and s 2 were quasi-independent random variables on [0, 1] m, at least when s 1, s 2, and s 1 s 2 are sufficiently large, then 1.2 would follow by some version of the Law of Large umbers. Moreover, the same heuristic further suggests that, in addition to the Law of Large umbers, a Central Limit Theorem and perhaps other probabilistic limit laws also hold for T. In this paper, we put the above heuristic on firm ground. We do so by representing s as a function on the space of unimodular lattices. Then it turns our that the quasi-independence of the family s, which we are trying to capture, can be translated into the dynamical language of higher-order mixing for a subgroup of linear transformations acting on the space of lattices Main results We are not the first to explore Central Limit Theorems for Diophantine approximants. The onedimensional case m = 1 has been thoroughly investigated by Leveque [12, 13], Philipp [14], and Fuchs [6], leading to the following result proved by Fuchs [6]: there exists an explicit σ > 0 such that the counting function T u := {p, q Z : 1 q < T, u p/q < ϑ q 2 } satisfies as T, where { u [0, 1] : } T u 2c log T log T log log T 1/2 < ξ orm σ ξ 1.3 orm σ ξ := 2πσ 1/2 ξ e s2 /2σ ds denotes the normal distribution with the variance σ. Central Limit Theorems in higher dimensions when w 1 = = w m = 1/m have recently been studied Dolgopyat, Fayad and Vinogradov [3]. In this paper, using very different techniques, we establish the following CLT for general exponents w 1,..., w m. Theorem 1.1. Let m 2. Then for every ξ R, { u [0, 1] m T u C m log T : log T 1/2 as T, where } < ξ orm σm ξ 1.4 and ζ denotes Riemann s ζ-function. σ m := 2C m 2ζmζm ,

3 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 3 Our proof of Theorem 1.1, as well as the proof in [3], proceeds by interpreting T as a function on a certain subset of the space of all unimodular lattices in R m+1, and then studies how the sequence a s, where a is a fixed linear transformation of R m+1, distributes inside this space. However, the arguments in the two papers follow very different routes. The proof in [3] contains a novel refinement of the martingale method this approach was initiated in this setting by Le Borgne [11]. Here, one crucially uses the fact that when w 1 = = w m = 1/m, then the set is an unstable manifold for the action of a on the space of lattices. For general weights, has strictly smaller dimension than the unstable leaves, and it seems challenging to apply martingale approximation techniques. Instead, our method involves a quantitative analysis of higher-order correlations for functions on the space of lattices. We establish an asymptotic formula for correlations of arbitrary orders and use this formula to compute limits of all the moments of T directly. One of the key innovations of our approach is an efficient way of estimating sums of cumulants alternating sums of moments developed in our recent work [2]. We also investigate the more general problem of Diophantine approximation for systems of linear forms. The space M m,n R of m linear forms in n real variables is parametrized by real m n matrices. Given u M m,n R, we consider the family L i u of linear forms defined by L i u x 1,..., x n = n u ij x j, i = 1,..., m. Let be a norm on R n. Fix ϑ 1,..., ϑ m > 0 and w 1,..., w m > 0 which satisfy j=1 w w m = n, and consider the system of Diophantine inequalities p i + L i u q 1,..., q n < ϑ i q w i, i = 1,..., m, 1.5 with p, q = p 1,..., p m, q 1,..., q n Z m Z n \{0}. The number of solutions of this system with the norm of the denominator q bounded by T is given by T u := {p, q Z m Z n : 0 < q < T and 1.5 holds}. 1.6 Our main result in this paper is the following generalization of Theorem 1.1. Theorem 1.2. If m 2, then for every ξ R, { T u C m,n log T u M m,n [0, 1] : log T 1/2 } < ξ orm σm,n ξ 1.7 as T, where C m,n := C m ω n with ω n := S n 1 z n dz and σ m,n := 2C m,n 2ζm + n 1ζm + n 1 1. The special case w 1 =... = w m = n/m was proved earlier in [3].

4 4 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 1.3. An outline of the proof of Theorem 1.2 We begin by observing that T can be interpreted as a function on the space of lattices in R m+n. Given u M m,n [0, 1], we define the unimodular lattice Λ u in R m+n by n n Λ u := p 1 + u 1j q j,..., p m + u mj q j, q : p, q Z m Z n, 1.8 and we see that j=1 where Ω T denotes the domain j=1 T u = Λ u Ω T + O1, for T > 0, Ω T := { x, y R m+n : 1 y < T, x i < ϑ i y w i, i = 1,..., m }. 1.9 The space of unimodular lattices in R m+n is naturally a homogeneous space of the group SL m+n R equipped with the invariant probability measure µ. The set := {Λ u : u M m,n [0, 1]} is a mn-dimensional torus embedded in, and we equip with the Haar probability measure µ, interpreted as a Borel measure on. We further observe see Section 6 for more details that each domain Ω T can be tessellated using a fixed diagonal matrix a in SL m+n R, so that for a suitable function ˆχ : R, we have Λ Ω T 1 s=0 ˆχa s Λ for Λ. Hence we are left with analyzing the distribution of values for the sums s=0 ˆχas y with y. This will allow us to apply techniques developed in our previous work [2], as well as in [1] joint with M. Einsiedler. Intuitively, our arguments will be guided by the hope that the observables ˆχ a s are quasi-independent with respect to µ. Due to the discontinuity and unboundedness of the function ˆχ on, it gets quite technical to formulate this quasi-independence directly. Instead, we shall argue in steps. We begin in Section 2 by establishing quasi-independence for observables of the form φ a s, where φ is a smooth and compactly supported function on. This amounts to an asymptotic formula Corollary 2.4 for the higher-order correlations φ 1 a s 1 y φ r a s r y dµ y with φ 1,..., φ r C c It will be crucial for our arguments later that the error term in this formula is explicit in terms of the exponents s 1,..., s r and in certain norms of the functions φ 1,..., φ r. In Section 3, we use these estimates to prove the Central Limit Theorems for sums of the form F y := 1 s=0 φa s y µ φ with φ C c and y. To do this, we use an adaption of the classical Cumulant Method see Proposition 3.4, which provides bounds on cumulants alternating sums of moments given estimates on expressions as in 1.10, at least in certain ranges of the parameters s 1,..., s r. Here we shall exploit the decomposition 3.7 into seperated / clustered tuples. We stress that the cumulant Cum r F of order r can be expressed as a sum of O r terms, normalized by r/2, so that in order to prove that it vanishes asymptotically, we require more than just square-root cancellation; however, the

5 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 5 error term in the asymptotic formula for 1.10 is rather weak. onetheless, by using intricate combinatorial cancellations of cumulants, we can establish the required bounds. In order to extend the method in Section 3 to the kind of unbounded functions which arise in our subsequent approximation arguments we have to investigate possible escapes of mass for the sequence of tori a s inside the space. In Section 4, we prove several results in this direction see e.g. Proposition 4.5, as well as L p -bounds see Propositions 4.6 and 4.7. We stress that the general non-divergence estimates for unipotent flows developed by Kleinbock-Margulis [8] are not sufficient for our purposes, and in particular, the exact value of the exponent in Proposition 4.5 will be crucial for our argument. The proof of the L 2 -norm bound in Proposition 4.7 is especially interesting in this regard since it uncovers that the escape of mass is related to delicate arithmetic questions; our arguments require careful estimates on the number of solutions of certain Diophantine equations. To make the technical passages in the final steps of the proof of Theorem 1.2 a bit more readable, we shall devote Section 5 to Central Limit Theorems for sums of the form 1 s=0 ˆfa s y for y, where f is a smooth and compactly supported function on R m+n, and ˆf denotes the Siegel transform of f see Section 4.3 for definitions. We stress that even though f is assumed to be bounded, ˆf is unbounded on. To prove the Central Limit Theorems in this setting, we approximate ˆf by compactly supported functions on and then use the estimates from Section 3. However, the bounds in these estimates crucially depend on the order of approximation, so this step requires a delicate analysis of the error terms. The non-divergence results established in Section 4 play important role here. Finally, to prove the Central Limit Theorem for the function ˆχ which is the Siegel transform of an indicator function on a nice bounded domain in R m+n, and thus establish Theorem 1.2, we need to approximate χ with smooth functions, and show that the arguments in Section 5 can be adapted to certain sequences of Siegel transforms of smooth and compactly supported functions. This will be done in Section ESTIMATES O HIGHER-ORDER CORRELATIOS Let denote the space of unimodular lattices in R m+n. Setting G := SL m+n R and Γ := SL m+n Z, we may consider the space as a homogeneous space under the linear action of the group G, so that G/Γ. Let µ denote the G-invariant probability measure on. We fix m, n 1 and denote by the subgroup { } Im u := : u M 0 I m,n R < G, 2.1 n and set := Z m+n. Geometrically, can be visualized as a mn-dimensional torus embedded in the spaces of lattices. We denote by µ the probability measure on induced by the Lebesgue probability measure on M m,n [0, 1], and we note that corresponds to the collection of unimodular lattices Λ u, for u M m,n [0, 1, introduced earlier in 1.8.

6 6 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS Let us further fix positive numbers w 1,..., w m+n satisfying m m+n w i = w i, and denote by a t the one-parameter semi-group i=m+1 a t := diag e w 1t,..., e w mt, e w m+1t,..., e w m+nt, t > The aim of this section is to analyze the asymptotic behavior of a t as t, and investigate decoupling of correlations of the form φ 1 a t1 y... φ r a tr y dµ y for φ 1,..., φ r C c, 2.3 for large t 1,..., t r > 0. It will be essential for our subsequent argument that the error terms in this decoupling are explicit in terms of the parameters t 1,..., t r > 0 and suitable norms of the functions φ 1,..., φ r, which we now introduce. Every LieG defines a first order differential operator D on C c by D φx := d dt φexptx t=0. If we fix an ordered basis { 1,..., r } of LieG, then every monomial Z = l 1 1 l r r defines a differential operator by D Z := D l 1 1 D l r r, 2.4 of degree degz = l l r. For k 1 and φ C c, we define the norm φ L 2 k := 1/2 D Z φx 2 dµ x. 2.5 degz k The starting point of our discussion is a well-known quantitative estimate on correlations of smooth functions on : Theorem 2.1. There exist γ > 0 and k 1 such that for all φ 1, φ 2 C c and g G, φ 1 gxφ 2 x dµ x = φ 1 dµ φ 2 dµ + O g γ φ 1 L 2 k φ 1 L 2 k. This theorem has a very long history that we will not attempt to survey here, but only mention that a result of this form can be found, for instance, in [7, 9]. Let us from now on we fix k 1 so that Theorem 2.1 holds. Our goal is to decouple the higher-order correlations in 2.3, but in order to state our results we first need to introduce a family of finer norms on C c than L 2 k. Let us denote by C 0 the uniform norm on C c. If we fix a right-invariant Riemannian metric on G, then it induces a metric d on G/Γ, which allows us to define the norms { } φx1 φx 2 φ Lip := sup : x 1, x 2, x 1 x 2, dx 1, x 2 and for φ C c. We shall prove: { } k φ := max φ C 0, φ Lip, φ L 2 k, 2.6

7 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 7 Theorem 2.2. There exists δ > 0 such that for every compact Ω, f C c with suppf Ω, φ 1,..., φ r C c, x 0, and t 1,..., t r > 0, we have r r fu φ i a ti ux 0 du = fu du φ i dµ r + O x0,ω,r e δdt 1,...,t r f C k k φ i, where Dt 1,..., t r := min{t i, t i t j : 1 i j r}. Remark 2.3. The case r = 1 was proved by Kleinbock and Margulis in [10], and our arguments are inspired by theirs. We stress that the constant δ in Theorem 2.2 is independent of r. We also record the following corollary of Theorem 2.2. Corollary 2.4. There exists δ > 0 such that for every φ 0 C, φ 1,..., φ r C c, x 0, and t 1,..., t r > 0, we have r r φ 0 y φ i a ti y dµ y = φ 0 dµ φ i dµ r + O r e δ Dt 1,...,t r φ 0 C k k φ i. Proof of Corollary 2.4 assuming Theorem 2.2. Let x 0 denote the identity coset in = G/Γ, which corresponds to the standard lattice Z m+n, and recall that = x 0 / Γ. Let φ 0 C denote the lift of the function φ 0 to, and χ the characteristic function of the subset { } Im u 0 := : u M 0 I m,n [0, 1]. n Given ε > 0, let χ ε C c be a smooth approximation of χ with uniformly bounded support which satisfy χ χ ε 1, χ χ ε L 1 ε, χ ε C k ε k. We observe that if f ε := φ 0 χ ε and f 0 := φ 0 χ, then and which implies that r φ 0 y φ i a ti y dµ y = f 0 f ε L 1 ε φ 0 C 0 f ε C k φ 0 C k χ ε C k ε k φ 0 C k, = r f 0 u φ i a ti ux 0 du r f ε u φ i a ti ux 0 du + O ε r φ i C 0, i=0

8 8 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS and φ 0 dµ = f 0 u du = Therefore, Theorem 2.2 implies that r r φ 0 y φ i a ti y dµ y = f ε u du r + O r ε i=0 r = φ 0 dµ f ε u du + O ε φ 0 C 0. φ i dµ φ i C 0 + e δdt 1,...,t r f ε C k φ i dµ + O r ε + ε k e δdt 1,...,t r φ 0 C k The corollary with δ = δ/k + 1 follows by choosing ε = e δdt 1,...,t r /k Preliminary results r k φ i r k φ i. We recall that d is a distance on = G/Γ induced from a right-invariant Riemannian metric on G. We denote by B G ρ the ball of radius ρ centered at the identity in G. For a point x, we let ιx denote the injectivity radius at x, that is to say, the supremum over ρ > 0 such that the map B G ρ B G ρx : g gx is injective. Given ε > 0, let K ε = { Λ : v ε, for all v Λ \ {0} }. 2.7 By Mahler s Compactness Criterion, K ε is a compact subset of. Furthermore, using Reduction Theory, one can show: Proposition 2.5 [10], Prop ιx ε m+n for any x K ε. An important role in our argument will be played by the one-parameter semi-group b t := diag e t/m,..., e t/m, e t/n,..., e t/n, t > 0, 2.8 which coincides with the semi-group a t as defined in 2.2 with the special choice of exponents w 1 =... = w m = 1 m and w m+1 =... = w m+n = 1 n. The submanifold is an unstable manifold for the flow b t which makes the analysis of the asymptotic behavior of b t significantly easier than that of a t for general parameters. sing Theorem 2.1, Kleinbock and Margulis proved in [7] a quantitative equidistribution result for the family b t as t, we shall use a version of this result from their later work [10]. Theorem 2.6 [10]; Th There exist ρ 0 > 0 and c, γ > 0 such that for every ρ 0, ρ 0, f C c satisfying suppf B G ρ, x with ιx > 2ρ, φ C c, and t 0, fuφb t ux du = fu du φ dµ + O ρ f L 1 φ Lip + ρ c e γt f C k φ L 2 k.

9 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 9 Remark 2.7. Although the dependence on φ is not stated in [10, Theorem 2.3], the estimate is explicit in the proof. We will prove Theorem 2.2 through successive uses of Theorem 2.6. In order to make things more transparent, it will be convenient to embed the flow a t as defined in 2.2 in a multiparameter flow as follows. For s = s 1,..., s m+n R m+n, we set as := diag e s 1,..., e s m, e s m+1,..., e s m+n. 2.9 We denote by S + the cone in R m+n consisting of those s = s 1,..., s m+n which satisfy m m+n s 1,..., s m+n > 0 and s i = s i. For s = s 1,..., s m+n S +, we set s := mins 1,..., s m+n, and thus, with s t := w 1 t,..., w m+n t, we see that a t = as t. i=m+1 In addition to Theorem 2.6, we shall also need the following quantitative non-divergence estimate for unipotent flows established by Kleinbock and Margulis in [10]. Theorem 2.8 [10]; Cor There exists θ = θm, n > 0 such that for every compact L and a Euclidean ball B centered at the identity, there exists T 0 > 0 such that for every ε 0, 1, x L, and s S + satisfying s T 0, one has 2.2. Proof of Theorem 2.2 {u B : asux / K ε } ε θ B. Let us fix r 1 and a r-tuple t 1,..., t r. pon re-labeling, we may assume that t 1... t r, so that D := Dt 1,..., t r = min{t 1, t 2 t 1,..., t r t r 1 } The next lemma provides an additional parameter s S +, which depends on the r-tuple t 1,..., t r. This parameter will be used throughout the proof of Theorem 2.2, and we stress that the accompanying constants c 1, c 2 and c 3 are independent of r and the r-tuple t 1,..., t r. Lemma 2.9. There exist c 1, c 2, c 3 > 0 such that given any t r > t r 1 > 0, there exists s S + satisfying: i s c 1 t r t r 1, ii s s tr 1 c 2 t r t r 1, iii s tr s = z m,..., z m, z n,..., z n for some z c 3 mint r 1, t r t r 1. Proof. We start the proof by defining s by the formula in iii, where the parameter z will be chosen later, that is to say, we set s = w 1 t r z m,..., w mt r z m, w m+1t r z n,..., w m+nt r z. n Then i holds provided that A 1 t r z m c 1t r t r 1 and A 2 t r z n c 1t r t r 1, where A 1 := minw i : 1 i m and A 2 := minw i : m + 1 i m + n, so if we set c 1 = mina 1, A 2, then i holds when z c 1 minm, nt r

10 10 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS To arrange ii, we observe that s s tr 1 = w 1 t r t r 1 z m,..., w mt r t r 1 z m, and thus ii holds provided that w m+1 t r t r 1 z n,..., w m+nt r t r 1 z n A 1 t r t r 1 z m c 2t r t r 1 and A 2 t r t r 1 z n c 2t r t r 1. If we let c 2 = mina 1, A 2 /2, then ii holds when z c 2 minm, nt r t r So far we have arranged so that i and ii hold provided that z satisfies 2.11 and Let c 3 = minc 1, c 2 minm, n, and note that if we pick z = c 3 mint r 1, t r t r 1, then i, ii and iii are all satisfied. Let us now continue with the proof of Theorem 2.2. With the parameter s provided by Lemma 2.9 above, we have s c 1 D, 2.13 s s ti c 2 D for all i = 1,..., r 1, 2.14 as tr s = b z for some z c 3 D, 2.15 where b z is defined as in 2.8 and D as in 2.10 Let us throughout the rest of the proof fix a compact set Ω. Our aim now is to estimate integrals of the form r I r := fu φ i a ti ux 0 du, where f ranges over C c with suppf Ω. Our proof will proceed by induction over r, the case r = 0 being trivial. Before we can start the induction, we need some notation. Let ρ 0 and k be as in Theorem 2.6, and pick for 0 < ρ < ρ 0, a non-negative ω ρ C c such that suppω ρ B G ρ, ω ρ C k ρ σ, ω ρ v dv = 1, 2.16 for some fixed σ = σm, n, k > 0. The integral I r can now be rewritten as follows: r I r =I r ω ρ v dv = fuω ρ v φ i a ti ux 0 dudv = fa svasuω ρ v φ i a ti a svasux 0 dudv. If we set then f s,u v := fa svasuω ρ v and x s,u := asux 0, I r = f s,u v r φ i as ti svx s,u dv du.,

11 We observe that if f s,u v 0, then so that CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 11 v suppω ρ B G ρ and a svasu suppf, u a sv 1 assuppf a ssuppω ρ 1 asω. Since s S +, the linear map v a svas, for v = R mn, is non-expanding, and thus we can conclude that there exists a fixed Euclidean ball B in depending only on Ω and ρ 0, and centered at the identity, such that if f s,u v 0, then u B. This implies that the integral I r can be written as r I r = f s,u v φ i as ti svx s,u dv du, 2.17 and Furthermore, B f s,u C k f C k ω ρ C k ρ σ f C k f s,u L 1 du = = We decompose the integral I r in 2.17 as where I rε is the integral over and I r ε is the integral over B\B ε. fa svasuω ρ v dvdu 2.19 ω ρ v dv = f L 1. fu du I r = I rε + I r ε, B ε := {u B : x s,u / K ε }, To estimate I rε, we first recall that s c 1 D by 2.13, so if D T 0 /c 1, where T 0 is as in Theorem 2.8 applied to L = K ε and B, then the same theorem implies that there exists θ > 0 such that B ε ε θ B 2.20 for every ε 0, 1, and thus r I rε Ω ε θ B f C 0 ω ρ v dv φ i C 0 Ω ε θ f C 0 r φ i C Let us now turn to the problem of estimating I r ε. Since the Riemannian distance d on G, restricted to, and the Euclidean distance on are equivalent on a small open identity neighborhood, we see that d a tvat, e e t dv, e. for v B G ρ 0 and any t S +. Hence, using 2.14, we obtain that for all i = 1,..., r 1, and thus, for all v B G ρ 0, φ i as ti svx s,u = φ i as ti sx s,u + O e c 2D φ i Lip, r 1 r 1 φ i as ti svx s,u = φ i as ti sx s,u + O r r 1 e c 2D φ i Lip, 2.22

12 12 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS where φ Lip := max φ C 0, φ i Lip. This leads to the estimate r 1 I r ε = φ i as ti sx s,u f s,u vφ r as tr svx s,u dv du B\B ε r 1 + O r e c 2D φ i Lip f s,u L 1 du φ r C 0. B\B ε Hence, using 2.19, we obtain that Since by 2.15, r 1 I r ε = φ i as ti sx s,u f s,u vφ r as tr svx s,u dv du B\B ε r + O r e c2d f L 1 φ i Lip. f s,u vφ r as tr svx s,u dv = f s,u vφ r b z vx s,u dv, we apply Theorem 2.6 to estimate this integral. We recall that suppf s,u B G ρ. For u B\B ε, we have x s,u K ε, so that ιx s,u ε m+n by Proposition 2.5. In particular, we may take ε ρ 1/m+n to arrange that ιx s,u > 2ρ. Hence, by applying Theorem 2.6, we deduce that there exist c, γ > 0 such that f s,u vφ r b z vx s,u dv = f s,u v dv φ r dµ + O ρ f s,u L 1 φ r Lip + ρ c e γz f s,u C k φ r L 2 k, for all u B\B ε. sing and z c 3 D, we deduce that r 1 I r ε = φ i as ti sx s,u f s,u v dv du B\B ε r 1 + O r,b φ i C 0 + O r e c 2D f L 1 Since f L 1 Ω f C k, I r ε = r 1 B\B ε φ r dµ ρ f L 1 φ r Lip + ρ c+σ e γc 3 D f C k φ r L 2 k r φ i Lip. φ i as ti sx s,u f s,u v dv du φ r dµ r + O r,ω ρ + ρ c+σ e γc 3 D + e c 2D f C k k φ i. 2.23

13 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 13 Applying 2.22 one more time in the backward direction, we get r 1 φ i as ti sx s,u f s,u v dv du B\B ε r 1 = f s,u v φ i as ti svx s,u dv du It follows from 2.20 that = B\B ε + O r B\B ε B e c 2D + O r f s,u L 1 du B φ i Lip. r 1 r 1 f s,u v φ i as ti svx s,u dv r 1 f s,u v φ i as ti svx s,u dv ε θ B f s,u L 1 du r 1 du φ i C 0, where we recognize the first term as I r 1. sing 2.17 and 2.19, we now conclude that r 1 φ i as ti sx s,u f s,u v dv du B\B ε ε =I r 1 + O θ r + e c 2D r 1 f L 1 φ i Lip. Hence, combining this estimate with 2.23, we deduce that I r ε =I r 1 φ r dµ and thus, in view of 2.21, + O r,ω ε θ + ρ + ρ c+σ e γc 3 D + e c 2D f C k I r =I r 1 ε + I r 1 ε = I r 1 φ r dµ + O r,ω ε θ + ρ + ρ c+σ e γc 3 D + e c 2D f C k du r k φ i, r k φ i. This estimate holds whenever ρ < ρ 0 and ε ρ 1/m+n. Taking ρ = e c4d for sufficiently small c 4 > 0 and ε ρ 1/m+n, we conclude that there exists δ > 0 such that for all sufficiently large D, r I r =I r 1 φ r dµ + O r,ω e δd f C k k φ i The exponent δ depends on the constants c 2 and c 3 given by Lemma 2.9 and the parameters θ, c, σ, γ. In particular, δ is independent of r. By possibly enlarging the implicit constants we can

14 14 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS ensure that the estimate 2.24 also holds for all r-tuples t 1,..., t r, and not just the ones with sufficiently large Dt 1,..., t r. By iterating the estimate 2.24, using that I 0 is a constant, the proof of Theorem 2.2 is finished. 3. CLT FOR FCTIOS WITH COMPACT SPPORT Let a = diaga 1,..., a m+n be a diagonal linear map of R m+n with a 1,..., a m > 1, 0 < a m+1,..., a m+n < 1, and a 1 a m+n = 1. The map a defines a continuous self-map of the space, which preserves µ. We recall that the torus = Z m+n is equipped with the probability measure µ. In this section, we shall prove a Central Limit Theorem for the averages F := 1 1 φ a s µ φ a s, 3.1 s=0 restricted to, for a fixed function φ C c. We stress that this result is not needed in the proof of Theorem 1.2, but we nevertheless include it here because we feel that its proof might be instructive before entering the proof of the similar, but far more technical, Theorem 6.1. Theorem 3.1. For every ξ R, as, where µ {y : F y < ξ} orm σφ ξ σ 2 φ := s= φ a s φ dµ µ φ 2. Remark 3.2. It follows from Theorem 2.1 that the variance σ φ is finite. Our main tool in the proof of Theorem 3.1 will be the estimates on higher-order correlations established in Section 2. To make notations less heavy, we shall use a simplified version of Corollary 2.4 stated in terms of C k -norms note that k C k: Corollary 3.3. There exists δ > 0 such that for every φ 0,..., φ r C c and t 1,..., t r > 0, we have r r φ 0 y φ i a ti y dµ y = φ 0 dµ φ i dµ r + O r e δdt 1,...,t r φ i C k The method of cumulants Let, µ be a probability space. Given bounded measurable functions φ 1,..., φ r on, we define their joint cumulant as Cum r µ φ 1,..., φ r = 1 P 1 P 1! φ i dµ, P I P i I where the sum is taken over all partitions P of the set {1,..., r}. When it is clear from the context, we skip the subscript µ. For a bounded measurable function φ on, we also set Cum r φ = Cum r φ,..., φ. We shall use the following classical CLT-criterion see, for instance, [5]. i=0

15 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 15 Proposition 3.4. Let F T be a sequence of real-valued bounded measurable functions such that F T dµ = 0 and σ 2 := lim F 2 T dµ < 3.2 T and Then for every ξ R, lim T Cumr F T = 0, for all r µ{f T < ξ} orm σ ξ as T. Since all the moments of a random variable can be expressed in terms of its cumulants, this criterion is equivalent to the more widely known Method of Moments. However, the cumulants have curious, and very convenient, cancellation properties that will play an important role in our proof of Theorem 3.1. For a partition Q of {1,..., r}, we define the conditional joint cumulant with respect to Q as Cum r µ φ 1,..., φ r Q = 1 P 1 P 1! φ i dµ, P I P J Q i I J In what follows, we shall make frequent use of the following proposition. Proposition 3.5 [2], Prop For any partition Q with Q 2, for all φ 1,..., φ r L, µ Estimating cumulants Cum r µ φ 1,..., φ r Q = 0, 3.4 Fix φ C c. It will be convenient to write ψ s y := φa s y µ φ a s, so that F := 1 1 and F dµ = 0. s=0 In this section, we shall estimate cumulants of the form Cum r µ F = 1 r/2 ψ s 1 s 1,...,s r =0 Cum r µ ψ s1,..., ψ sr 3.5 for r 3. Since we shall later need to apply these estimate in cases when the function φ is allowed to vary with, it will be important to keep track of the dependence on φ in our estimates. We shall decompose 3.5 into sub-sums where the parameters s 1,..., s r are either separated or clustered, and it will also be important to control their sizes. For this purpose, it will be convenient to consider the set {0,..., 1} r as a subset of R r+1 + via the embedding s 1,..., s r 0, s 1,..., s r. Following the ideas developed in the paper [2], we define for non-empty subsets I and J of {0,..., r} and s = s 0,..., s r R+ r+1, ρ I s := max { s i s j : i, j I } and ρ I,J u := min { s i s j : i I, j J }, and if Q is a partition of {0,..., r}, we set ρ Q s := max { ρ I s : I Q } and ρ Q s := min { ρ I,J s : I J, I, J Q }.

16 16 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS For 0 α < β, we define and Q α, β := { s R r+1 + : ρ Q s α, and ρ Q s > β } α := { s R r+1 + : ρs i, s j α for all i, j }. The following decomposition of R r+1 + was established in our paper [2, Prop. 6.2]: given we have 0 = α 0 < β 1 < α 1 = 3β 1 < β 2 < < β r < α r = 3β r < β r+1, 3.6 r R r+1 + = β r+1 j=0 Q 2 Q α j, β j+1, 3.7 where the union is taken over the partitions Q of {0,..., r} with Q 2. pon taking restrictions, we also have r {0,..., 1} r = Ωβ r+1 ; Ω Q α j, β j+1 ;, 3.8 for all 2, where j=0 Q 2 Ωβ r+1 ; := {0,..., 1} r β r+1, Ω Q α j, β j+1 ; := {0,..., 1} r Q α j, β j+1. In order to estimate the cumulant 3.5, we shall separately estimate the sums over Ωβ r+1 ; and Ω Q α j, β j+1 ;, the exact choices of the sequences α j and β j will be fixed at the very end of our argument Case 0: Summing over Ωβ r+1 ;. In this case, s i β r+1 for all i, and thus Hence, s 1,...,s r Ωβ r+1 ; Ωβ r+1 ; β r r. Cum r µ ψ s1,..., ψ sr β r r φ r C 0, 3.9 where the implied constants may depend on r, but we shall henceforth omit this subscript to simplify notations Case 1: Summing over Ω Q α j, β j+1 ; with Q = {{0}, {1,..., r}}. In this case, we have so that it follows that Hence, 1 r/2 s 1,...,s r Ω Q α j,β j+1 ; s i1 s i2 α j for all i 1, i 2, Ω Q α j, β j+1 ; α r 1 j. Cum r µ ψ s1,..., ψ sr 1 r/2 α r 1 j φ r C

17 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS Case 2: Summing over Ω Q α j, β j+1 ; with Q 2 and Q {{0}, {1,..., r}}. In this case, the partition Q defines a non-trivial partition Q = {I 0,..., I l } of {1,..., r} such that for all s 1,..., s r Ω Q α j, β j+1 ;, we have and In particular, s i1 s i2 α j if i 1 Q i 2 and s i1 s i2 > β j+1 if i 1 Q i 2, 3.11 s i α j for all i I 0, and s i > β j+1 for all i / I 0. Let I be an arbitrary subset of {1,..., r}; we shall show that l ψ si dµ i I Ds i1,..., s il β j+1, 3.12 h=0 i I I h ψ si dµ, 3.13 where we henceforth shall use the convention that the product is equal to one when I I h =. Let us estimate the right hand side of We begin by setting Φ 0 := ψ si. i I I 0 It is easy to see that there exists ξ = ξm, n, k > 0 such that Φ 0 C k φ a si µ φ a si C k e I I0 ξ αj φ I I C k i I I 0 To prove 3.13, we expand ψ si = φ a s i µ φ a s i for i I\I 0 and get ψ si dµ = 1 I\J I0 i I J I\I 0 Φ 0 φ a s i dµ φ a s i dµ. i J i I\J I 0 We recall that when i / I 0, we have s i β j+1, and thus it follows from Corollary 3.3 with r = 1 that φ a s i dµ = µ φ + O e δβ j+1 φ C k, with i / I To estimate the other integrals in 3.15, we also apply Corollary 3.3. Let us first fix a subset J I\I 0 and for each 1 h l, we pick i h I h, and set Φ h := φ a si sih. i J I h Then l Φ 0 φ a s i dµ = Φ 0 Φ h a s i h dµ. i J h=1 We note that for i I h, we have s i s ih α j, and thus there exists ξ = ξm, n, k > 0 such that Φ h C k φ a si sih C k e J Ih ξ αj φ J Ih C k i J I h

18 18 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS sing 3.12, Corollary 3.3 implies that Φ 0 l h=1 Φ h a s i h l dµ = Φ 0 dµ Φ h dµ h=1 l + O e δβ j+1 Φ h C k. sing 3.14 and 3.16 and invariance of the measure µ, we deduce that l l Φ 0 Φ h a s i h dµ = Φ 0 dµ φ a si dµ h=1 h=1 i J I h + O e δβ j+1 rξα j φ I I 0 J. C k Hence, we conclude that l Φ 0 φ a s i dµ = Φ 0 dµ φ a si dµ 3.17 i J h=1 i J I h + O e δβ j+1 rξα j φ I I 0 J. C k We shall choose the parameters α j and β j+1 so that h=0 δβ j+1 rξα j > Substituting 3.15 and 3.17 in 3.15, we deduce that = ψ si dµ i I J I\I 0 1 I\J I0 l Φ 0 dµ h=1 + O e δβ j+1 rξα j φ I. C k i J I h φ a si dµ µ φ I\J I ext, we estimate the right hand side of Let us fix 1 h l and for a subset J I I h, we define Φ J := i J φ a s i s ih. As in 3.16, for some ξ > 0, Φ J C k i J φ a s i s ih C k e J ξ α j φ J C k.

19 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 19 Applying Corollary 3.3 to the function Φ J and using that s ih > β j+1, we get φ a s i dµ = Φ J a s i h dµ 3.20 i J = = Φ J dµ + O e δβ j+1 Φ J C k φ a s i dµ + O e δβ j+1 e rξ α j φ J, C k i J where we have used a-invariance of µ. Combining 3.15 and 3.20, we deduce that dµ = φ a s i dµ µ φ I I h\j i I I h ψ si J I I h 1 I Ih\J i J C k + O e δβ j+1 e rξ α j φ I I h = φ a si µ φ dµ + O i I I h e δβ j+1 rξα j φ I I h, C k 3.21 which implies l h=0 i I I h ψ si dµ l = Φ 0 dµ φ a si µ φ dµ h=1 i I I h + O e δβ j+1 rξα j φ r C. k Furthermore, multiplying out the products over I I h, we get l dµ 3.22 h=0 i I I h ψ si = Φ 0 dµ + O e δβ j+1 rξα j φ I. C k J I\I 0 1 I\I0 J l h=1 i I h J Comparing 3.19 and 3.22, we finally conclude that l ψ si dµ = i I h=0 + O φ a s i dµ µ φ I\I 0 J i I I h ψ si dµ e δβ j+1 rξα j φ I C k

20 20 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS when s 1,..., s r Ω Q α j, β j+1 ;. This establishes 3.13 with an explicit error term. This estimate implies that for the partition Q = {I 0,..., I l }, Cum r µ ψ s1,..., ψ sr = Cum r µ ψ s1,..., ψ sr Q + O e δβ j+1 rξα j φ r C k By Proposition 3.5, Cum r µ ψ s1,..., ψ sr Q = 0, so it follows that for all s 1,..., s r Ω Q α j, β j+1 ;, Cum r µ ψ s1,..., ψ sr e δβ j+1 rξα j φ r C k Final estimates on the cumulants. Let us now return to 3.5. pon decomposing this sum into the regions discussed above, and applying the estimates 3.9, 3.10 and 3.23 to respective region, we get the bound Cum r µ F β r r r/2 + max j α r 1 j 1 r/2 φ r C r/2 max j e δβ j+1 rξα j φ r C k. This estimate holds provided that 3.6 and 3.18 hold, namely when α j = 3β j < β j+1 and δβ j+1 rξα j > 0 for j = 1,..., r Given any γ > 0, we define the parameters β j inductively by the formula β 1 = γ and β j+1 = max γ + 3β j, γ + 3δ 1 rξβ j It easily follows by induction that β r+1 r γ, and we deduce from 3.24 that Cum r µ F γ + 1 r r/2 + γ r 1 1 r/2 φ r C + r/2 e δγ φ r 0 C. k Taking γ = r/δ log, we conclude that when r 3, 3.3. Estimating the variance In this section, we wish to compute the limit Cum r µ F 0 as F 2 L 2 = s 1 =0 s 2 =0 Setting s 1 = s + t and s 2 = t, we rewrite the above sums as where ψ s1 ψ s2 dµ. 1 F 2 L 2 =Θ Θ s, 3.28 Θ s := 1 1 s t=1 s=1 ψ s+t ψ t dµ. Since ψ t = φ a t µ φ a t, ψ s+t ψ t dµ = φ a s+t φ a t dµ µ φ a s+t µ φ a t. 3.29

21 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 21 It follows from Corollary 3.3 that for fixed s and as t, φ a s+t φ a t dµ φ a s φ dµ, and µ φ a t µ φ. We conclude that as t, and for fixed s, ψ s+t ψ t dµ Θ s := Θ s Θ s as. φ a s φ dµ µ φ 2, If one carelessly interchange limits above, one expects that as, F 2 L 2 Θ Θ s = Θ s s=1 s= To prove this limit rigorously, we need to say a bit more to ensure, say, dominated convergence. It follows from Corollary 3.3 that φ a s+t φ a t dµ = µ φ 2 + O φ a s+t dµ = µ φ + O e δs+t φ C k φ a t dµ = µ φ + O e δt φ C k. e δ mins,t φ 2 C k, and thus, in combination with 3.29, ψ s+t ψ t dµ e δ mins,t φ 2 C k This integral can also be estimated in a different way. If we set φ t = φ a t, then we deduce from Corollary 3.3 that φ a s+t φ a t dµ = φ t a s φ t dµ = µ φ t µ φ t + O e δs φ t 2 C, k and µ φ a s+t = µ φ t a s = µ φ t + O e δs φ t C k. It is easy to see that for some ξ = ξm, n, k > 0, φ t C k e ξt φ C k and µ φ t φ C k, and thus ψ s+t ψ t dµ e δs e ξt φ 2 C k Let us now combine 3.31 and 3.32: When t δ/2ξ s, we use 3.32, and when t δ/2ξ s, we use If we set δ = minδ/2, δ 2 /2ξ > 0, then ψ s+t ψ t dµ e δ s φ 2 C k,

22 22 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS for all s 0, whence Θ s 1 1 s t=1 ψ s+t ψ t dµ e δ s φ 2 C k uniformly in. Hence, the Dominated Convergence Theorem applied to 3.28 yields Proof of Theorem 3.1 In this subsection we shall check that the conditions of Proposition 3.4 hold for the sequence F defined in 3.1. First, by construction, It is easy to check that F T dµ = 0, and by 3.30, Furthermore, by 3.27, F 2 dµ for every r 3, which finishes the proof. s= Θ s as. Cum r µ F 0 as, 4.1. Siegel transforms 4. O-DIVERGECE ESTIMATES FOR SIEGEL TRASFORMS We recall that the space of unimodular lattices in R m+n can be identified with the quotient space G/Γ, where G = SL m+n R and Γ = SL m+n Z, which is endowed with the G-invariant probability measures µ. We denote by m G a bi-g-invariant Radon measure on G. Given a bounded measurable function f : R m+n R with compact support, we define its Siegel transform ˆf : R by ˆfΛ := fz for Λ. z Λ\{0} We stress that ˆf is unbounded on, its growth is controlled by an explicit function α which we now introduce. Given a lattice Λ in R m+n, we say that a subspace V of R m+n is Λ-rational if the intersection V Λ is a lattice in V. If V is Λ-rational, we denote by d Λ V the volume of V/V Λ, and define αλ := sup { d Λ V 1 : V is a Λ-rational subspace of R m+n}. It follows from the Mahler Compactness Criterion that α is a proper function on. Proposition 4.1 [17], Lem. 2. If f : R m+n R is a bounded function with compact support, then ˆfΛ suppf f C 0 αλ for all Λ. sing Reduction Theory, it is not hard to derive the following integrability of α: Proposition 4.2 [4], Lem α L p for 1 p < m + n. In particular, µ {α L} p L p for all p < m + n. In what follows, dz denotes the volume element on R m+n which assigns volume one to the unit cube. In our arguments below, we will make heavy use of the following two integral formulas:

23 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 23 Proposition 4.3 Siegel Mean Value Theorem; [18]. If f : R m+n R is a bounded Riemann integrable function with compact support, then ˆf dµ = fz dz. R m+n Proposition 4.4 Rogers Formula; [15], Theorem 5. If F : R m+n R m+n R is a non-negative measurable function, then Fgz 1, gz 2 dµ gγ =ζm + n 2 Fz 1, z 2 dz 1 dz 2 R z 1,z 2 Z m+n m+n R m+n + ζm + n 1 R m+n Fz, z dz + ζm + n 1 R m+n Fz, z dz, where the sum is taken over primitive integral vectors, and ζ denotes Riemann s ζ-function on-divergence estimates We retain the notation from Section 2. Given we denote by a the self-map on induced by 0 < w 1,..., w m < n and w w m = n, a = diage w 1,..., e w m, e 1,..., e Our goal in this subsection is to analyze the escape of mass for the submanifolds a s and bound the Siegel transforms ˆfa s y for y. The following proposition will play a very important role in our arguments. Proposition 4.5. There exists κ > 0 such that for every L 1 and s κ log L, µ {y : αa s y L} p L p for all p < m + n. Proof. Let χ L be the characteristic function of the subset {α < L} of. By Mahler s Compactness Criterion, χ L has a compact support. We further pick a non-negative ρ C c G with G ρ dm G = 1. Let η L x := ρ χ L x = ρgχ L g 1 x dm G g, x. Since µ is G-invariant, η L dµ = G χ L dµ = µ {α < L}. It follows from invariance of m G that if D Z is a differential operator as defined as in 2.4, then D Z η L = D Z ρ χ L. Hence, η L C c, and η L C k ρ C k. ote that there exists c > 1 such that for every g suppρ and all x, we have αg 1 x c 1 αx, and thus {α g 1 < L} {α < cl} and η L χ cl. This implies the lower bound µ {y : αa s y < cl} = χ cl a s y dµ y η L a s y dµ y.

24 24 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS By Corollary 3.3, there exists δ > 0 and k 1 such that η L a s y dµ y = η L dµ + O e δs η L C k and by Proposition 4.2, Combining these bounds, we get and thus = µ {α < L} + O e δs, µ {α L} p L p for all p < m + n. µ {y : αa s y < cl} µ {α < L} + O e δs = 1 + O p L p + e δs, µ {y : αa s y cl} p L p + e δs. By taking s κ log L where κ = p δ, the proof is finished. Proposition 4.6. Let f be a bounded measurable function on R m+n with compact support contained in the open set {x m+1,..., x m+n 0}. Then, with a as in 4.1, ˆf a s dµ <. sup s 0 Proof. We note that there exist 0 < υ 1 < υ 2 and ϑ > 0 such that the support of f is contained in the set { x, y R m+n : υ 1 y υ 2, x i ϑ y w i, i = 1,..., n }, 4.2 and without loss of generality we may assume that f is the characteristic function of this set. We recall that can be identified with the collection of lattices { Λu : u = u ij : i = 1,... m, j = 1,..., n [0, 1 m n}. We set u i = u i1,..., u in. Then by the definition of the Siegel transform, ˆfa s Λ u = f e w1s p 1 + u 1, q,..., e wms p m + u m, q, e s q. p,q Z m+n \{0} Denoting by χ i q the characteristic function of the interval [ ϑ q w i, ϑ q w i], we rewrite this sum as m ˆfa s Λ u = χ i q p i + u i, q 4.3 Hence, ˆf a s dµ = = υ 1 e s q υ 2 e s p Z m υ 1 e s q υ 2 e s υ 1 e s q υ 2 e s m χ i q p i + u i, q. pi Z m pi Z [0,1] n χ i We observe that for each i and p i Z, the volume of the set { u [0, 1] n : p + u, q ϑ q w i } q p i + u i, q du i.

25 is estimated from above by CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 25 2ϑ q w i max j q j q 1 w i, and we note that the set is empty whenever p > j q j + ϑ q w i. In particular, it is non-empty for at most O q choices of p Z. Hence, we deduce that m ˆf a s dµ q w i = q n 1, υ 1 e s q υ 2 e s υ 1 e s q υ 2 e s uniformly in s. This completes the proof. Proposition 4.7. Let f be a bounded measurable function on R m+n with compact support contained in the open set {x m+1,..., x m+n 0}. Then sup1 + s ν m ˆf a s L 2 <, s 0 where ν 1 = 1 and ν m = 0 when m 2. Proof. As in the proof of Proposition 4.6, it is sufficient to consider the case when f is the characteristic function of the set 4.2. Then ˆfa s y is given by 4.3, and we get ˆf a s 2 L 2 = ˆfa s y ˆfa s y dµ y = υ 1 e s q, l υ 2 e s For fixed q, l Z n, we wish to estimate I i q, l := p,r Z m p i,r i Z χ i q [0,1] n χ i q [0,1] n p i + u i, q χ i ri + u l i, l du i. p + u, q χi r + u, l du. First, we consider the case when q and l are linearly independent. Then there exist indices j, k = 1,..., n such that q j l k q k l j 0. Let us consider the function ψ on R 2 defined by ψx 1, x 2 = χ i q x 1χ i x l 2 as well as the periodized function ψ on R 2 /Z 2 defined by ψx = z Z2 ψz + x. If we set ω := q ζ u ζ and ρ := l ζ u ζ, then we denote by S the affine map ζ j,k l ζ j,k S : x 1, x 2 q j x 1 + q k x 2 + ω, l j x 1 + l k x 2 + ρ, which induces an affine endomorphism of the torus R 2 /Z 2. We note that p + u, q χi r + u, l duj du k = ψsx dµx, p,r Z χ i q [0,1] 2 l where µ denotes the Lebesgue probability measure on the torus R 2 /Z 2. Since the endomorphism S preserves µ, we see that ψsx dµx = ψx dµx = ψx dx = 4ϑ 2 q w i l w i. R 2 /Z 2 R 2 /Z 2 R 2 R 2 /Z 2

26 26 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS Therefore, we conclude that in this case, I i q, l q w i l w i. 4.4 pon re- Let us now we consider the second case when q and l are linearly dependent. arranging indices if needed, we may assume that q 1 = max q 1,..., q n, l 1,..., l n. 4.5 In particular, q 1 0, and thus l 1 0, since q and l are linearly dependent, so we can define the new variables n n v 1 = q j /q 1 u j = l j /l 1 u j and v 2 = u 2,..., v n = u n, and thus where j=1 J i q, l := p,r Z j=1 n n I i q, l J i q, l χ i q p + q 1v 1 χ i r + l l 1 v 1 dv 1. We note that the last integral is non-zero only when p q 1 and r l 1. We set q 1 = q d and l 1 = l d where d = gcdq 1, l 1. Then q 1 r l 1 p = jd for some j Z. We observe that when j is fixed, then the integers p and r satisfy the equation q r l p = j. Since gcdq, l = 1, all the solutions of this equation are of the form p = p 0 + kq, r = r 0 + kl for k Z. In particular, it follows that the number of such solutions satisfying p q 1 and r l 1 is at most Od. We write J i q, l = J 1 i q, l + J 2 i q, l, where the first sum is taken over those p, r with q 1 r l 1 p 0, and the second sum is taken over those p, r with q 1 r l 1 p = 0. pon applying a linear change of variables, we obtain J 1 i q, l = n+p/q1 χ i p,r: q 1 r l 1 p 0 n+p/q 1 χ i l j Z\{0} d Let us consider the function ρ i x := q q 1v 1 χ i q 1 r l 1 p/q 1 + l 1 v 1 dv 1 q q 1v 1 χ i jd/q 1 + l 1 v 1 dv 1. χ i q q 1v 1 χ i xd/q 1 + l 1 v 1 dv 1. We note that the integrand equals the indicator function of the intersection of the intervals [ ϑ q1 1 q w i, ϑ q 1 1 q w i ] l l and [ xd/q1 l 1 ϑ l 1 1 l w i, xd/q 1 l 1 + ϑ l 1 1 l w i ], and thus it follows that ρ i is non-increasing when x 0, and non-decreasing when x 0. This implies that ρ i j ρ i x dx. j Z\{0}

27 Since we conclude that ρ i x dx = CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 27 χ i q q 1v 1 dv 1 J 1 i q, l d j Z\{0} χ i xd/q l 1 dx d 1 q w i l w i, ρ i j q w i l w i. ext, we proceed with estimation of J 2 i q, l. Let c 0 := min{ q : q Z n \{0}}. Denoting by q 1, l 1 the number of solutions p, r of the equation we also obtain J 2 i q, l = q 1 r l 1 p = 0 n+p/q1 χ i p,r: q 1 r l 1 p=0 n+p/q 1 χ i l q 1, l 1 with p c n 0 ϑ + n q 1, q q 1v 1 χ i l 1 v 1 dv 1 q q 1v 1 χ i l 1 v 1 dv 1 q 1, l 1 q 1 1 q w i q 1, l 1 max q, l 1+w i, where we used that q 1 is chosen according to 4.5. Combining the obtained estimates for J 1 i q, l and J 2 i q, l, we conclude that when q and l are linearly dependent, where q 1 is chosen according to 4.5. ow we proceed to estimate J i q, l q w i l w i + q 1, l 1 max q, l 1+w i, 4.6 ˆf a s 2 L 2 υ 1 e s q, l υ 2 e s l m I i q, l. 4.7 sing 4.4, the sum in 4.7 over linearly independent q and l can be estimated as m q w i l w i q n l n 1. υ 1 e s q, l υ 2 e s υ 1 e s q, l υ 2 e s For a subset I of {1,..., m}, we set wi := i I w i. Then using 4.6, we deduce that the sum in 4.7 over linearly dependent q and l is bounded by q wi l wi q 1, l 1 Ic max q, l I c +wi c υ 1 e s q, l υ 2 e s I {1,...,m} I {1,...,m} e s n+ Ic +wi υ 1 e s q, l υ 2 e sq 1, l 1 Ic. 4.8 The star indicates that the sum is taken over linearly dependent q and l. When I c =, then wi = n. Since the number of q, l satisfying υ 1 e s q, l υ 2 e s is estimated as Oe 2ns, it is clear that the corresponding term in the above sum is uniformly bounded.

28 28 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS ow we suppose that I c. Since q and l are linearly dependent, the vector l is uniquely determined given l 1 and q, and we obtain that for some υ 1, υ 2 > 0, υ 1 e s q, l υ 2 e sq 1, l 1 Ic e s n 1 q 1, l 1 Ic. We shall use the following lemma: Lemma 4.8. For every k 1, where ν 1 = 1 and ν k = 0 when k 2. 1 q T 1 l q υ 1 es q 1 υ 2 es 1 l 1 q 1 q, l k T k+1 log T ν k Proof. We observe that the sum of q, l k over 1 l q is equal to the number of solutions p 1,..., p k, r 1,..., r k, l of the system of equations qr 1 lp 1 = 0,..., qr k lp k = satisfying p 1,..., p k c n 0 ϑ + nq and 1 l q. We order these solutions according to d := gcdq, l. Let q = q d and l = l d. Then q and l are coprime, and the system 4.9 is equivalent to q r 1 l p 1 = 0,..., q r k l p k = Because of coprimality, each p i have to be divisible by q, so that the number of such p i s is at most Oq/q = Od. We note that given d the number of possible choices for l is at most q/d, and p 1,..., p k, l uniquely determine r 1,..., r k. Hence, the number of solutions of 4.9 is estimated by d qq/dd k = qσ k 1 q, where σ k 1 q = d q dk 1. Writing q = q d, we conclude that 1 q T 1 l q This proves the lemma. q, l k 1 q T T k+1 qσ k 1 q T 1 q T 1 q T T/q q=1 d k 1 q k T k+1 log T ν k. A simple modification of this argument also gives that q, l k T k+1 log T ν k. 1 q T 1 l q Hence, it follows that υ 1 e s q, l υ 2 e sq 1, l 1 Ic e s n+ Ic 1 + s νi, where νi = 1 when I c = 1 and νi = 0 otherwise. Thus, the sum 4.8 is estimated as 1 + e swi 1 + s νi. I {1,...,m}

29 CETRAL LIMIT THEOREMS FOR DIOPHATIE APPROIMATS 29 The terms in this sum are uniformly bounded unless I = and I c = 1, namely, when m > 1. When m = 1, we obtain the bound O1 + s. This proves the proposition Truncated Siegel transform The Siegel transform of a compactly supported function is typically unbounded on ; to deal with this complication, it is natural to approximate ˆf by compactly supported functions on, the so called truncated Siegel transforms, which we shall denote by ˆf L. They will be constructed using a smooth cut-off function η L, which will be defined in the following lemma. Lemma 4.9. For every c > 1, there exists a family η L in C c satisfying: 0 η L 1, η L = 1 on {α c 1 L}, η L = 0 on {α > c L}, η L C k 1. Proof. Let χ L denote the indicator function of the subset {α L}, and pick a non-negative φ C c G with G φ dm G = 1 and with support in a sufficiently small neighbourhood of identity in G to ensure that for all g suppφ and x, We now define η L as c 1 αx αg ±1 x c αx. η L x := φ χ L x = G φgχ L g 1 x dm G g. Since φ 0 and G φ dm G = 1, it is clear that 0 η L 1. If αx c 1 L, then for g suppφ, we have αg 1 x L, so that η L x = G φ dm G = 1. If αx > c L, then for g suppφ, we have αg 1 x > L, so that η L x = 0. To prove the last property, we observe that it follows from invariance of m G that for a differential operator D Z as in 2.4, we have D Z η L = D Z φ χ L. Therefore, suppd Z η L {α c L} and D Z η L C 0 D Z φ L 1 G, whence η L C k 1. For a bounded function f : R m+n R with compact support, we define the truncated Siegel transform of f as ˆf L := ˆf η L. We record some basic properties of this transform that will be used later in the proofs. Lemma For f C c R m+n, the truncated Siegel transform ˆf L is in C c, and it satisfies ˆf L L ˆf L p p suppf,p f C 0 for all p < m + n, 4.11 ˆf L C 0 suppf L f C 0, 4.12 ˆf L C k suppf L f C k, 4.13 ˆf ˆf L L 1 suppf,τ L τ f C 0 for all τ < m + n 1, 4.14 ˆf ˆf L L 2 suppf,τ L τ 1/2 f C 0 for all τ < m + n Moreover, the implied constants are uniform when suppf is contained in a fixed compact set.