Inhomogeneous dual Diophantine approximation on affine subspaces

We prove the convergence and divergence cases of an inhomogeneous Khintchine-Groshev type theorem for dual approximation restricted to affine subspaces in $\mathbb{R} ^n$. The divergence results are proved in the more general context of Hausdorff measures.


Introduction
Throughout the paper, ψ : R + → R + is a non-increasing function and W n (ψ) is the set of x ∈ R n for which there exist infinitely many a ∈ Z n \ {0} such that |a 0 + x · a| < ψ( a n ) (1.1) for some a 0 ∈ Z. Here and throughout, denotes the supremum norm of a vector and the dot stands for the standard inner product of vectors. For obvious reasons, the set W n (ψ) is often referred to as the (dual) set of "ψ-approximable" vectors in R n . The fundamental Khintchine-Groshev theorem [20,21] in the metric theory of Diophantine approximation, provides an elegant characterization of the n-dimensional Lebesgue measure of W n (ψ) in terms of the convergence/divergence properties of a 'volume sum' associated with the approximating function ψ. We reinforce the fact that ψ will always be assumed to be non-increasing. Theorem 1.1 (Khintchine-Groshev). Let ψ be an approximating function. Then (1.2) We will use | | to denote the absolute value of a real number as well as the Lebesgue measure of a measurable subset X of R n ; the context will make the use clear.
VB and SV are supported by EPSRC Programme grant EP/J018260/1. Ghosh is supported by UGC. 1 (2) A point x ∈ R n is called very well approximable (abbr. VWA) if there exists ε > 0 such that x ∈ W n (ψ ε ) where Thus, the essence of the definition of VWA points is that for these points the 'Dirichlet exponent' can be improved beyond the trivial. Note that in view of Theorem 1.1, we have that |W n (ψ ε )| = 0 for any ε > 0. In other words, almost every point x ∈ R n is not VWA.
(3) The more general Hausdorff measure version of the Khintchine-Groshev theorem has been established in [12]. For a general background to the classical theory of metric Diophantine approximation, we refer the reader to the survey type articles [4,7].
We now consider the setting of inhomogeneous Diophantine approximation. Let θ : R n → R be a function and, given ψ, we let W θ n (ψ) be the set of x ∈ R n for which there exist infinitely many a ∈ Z n \ {0} such that |a 0 + x · a + θ(x)| < ψ( a n ) (1.3) for some a 0 ∈ Z. The set W θ n (ψ) is often referred to as the (dual) set of "(ψ, θ)inhomogeneously approximable" vectors in R n . The following inhomogeneous version of Theorem 1.1 is established in [1]. We denote by C n the set of n-times continuously differentiable functions. Theorem 1.2. Let ψ be an approximating function and θ : R n → R be a function such that θ ∈ C 2 . Then (1. 4) We remark that the choice of θ = constant is the setting of traditional inhomogeneous Diophantine approximation and in that case the above result was well known, see for example [11]. For the more general Hausdorff measure version of Theorem 1.2 within the traditional setting see [6, 7, §12.1].
In this paper, we consider the theory of Diophantine approximation on manifolds, specifically inhomogeneous approximation on affine subspaces. The subject of metric Diophantine approximation on manifolds studies the conditions under which a smooth submanifold of R n inherits Diophantine properties of R n which are generic for Lebesgue measure. Examples include the resolution of the famous Baker-Sprindžuk conjecture [25] due to Kleinbock and Margulis [24] using homogeneous dynamics on the space of unimodular lattices. Their result implies that almost every point on a nondegenerate submanifold M of R n is not very well approximable; that is |W n (ψ ε ) ∩ M | M = 0 ∀ ε > 0 . (1.5) Here and elsewhere | . | M denotes the induced Lebesgue measure on M. It is worth mentioning that any manifold M of R n satisfying (1.5) is called extremal and that Kleinbock and Margulis proved the stronger "multiplicative" extremal statement for nondegenerate manifolds. Essentially, nondegenerate manifolds are smooth manifolds of R n which are sufficiently curved so as to deviate from any hyperplane, see [2,24] for a formal definition.
The convergence case of the Khintchine-Groshev theorem was shown to hold for nondegenerate submanifolds of R n in [10] and independently in [2]. Indeed, in [10] Bernik, Kleinbock and Margulis established the stronger 'multiplicative' version. The complementary divergence case was subsequently proved in [5] and, as a result, we have the following complete analogue of the Khintchine-Groshev theorem for nondegenerate manifolds. Theorem 1.3. Let M be a nondegenerate submanifold of R n and let ψ be an approximating function. Then (1. 6) We note that the convergence case of above theorem implies the extremal statement (1.5) for nondegenerate manifolds. In this paper, we are concerned with affine subspaces which are the main examples of manifolds which are not nondegenerate. Theorem 1.3 above is therefore not applicable to them. Nevertheless, the analogue of the Baker-Sprindžuk conjecture for affine subspaces was studied by Kleinbock in [22] (see also [23]), and the Khintchine-Groshev theorem in the series of papers [3,14,16,17]. We refer the reader to the recent survey [18] for further details on this subject. The key goal of this paper is to develop the analogous inhomogeneous theory for affine subspaces.
We now briefly describe the current state of the inhomogeneous theory of Diophantine approximation on manifolds. In [8,9], the authors discovered a transference principle which allowed them to establish the inhomogeneous version of the Baker-Sprindžuk conjecture for nondegenerate manifolds from the original homogeneous statement. Indeed, the inhomogeneous "multiplicative" extremal statement established in [8] implies that for any nondegenerate submanifold M of R n and θ = constant, It is worth mentioned that in [19], it has been shown that the homogeneous to inhomogeneous transference principle of [8] is flexible enough to be used for arbitrary Diophantine exponents, not just the critical or "extremal" one. As demonstrated in [19], this naturally extends the scope of potential applications of the original transference principle. Beyond extremal statements such as (1.7), the complete inhomogeneous version of the Khintchine-Groshev theorem, both convergence and divergence cases, for nondegenerate manifolds is established in [1]. In other words, with mild conditions imposed on the 'inhomogeneous' function θ, the statement of Theorem 1.3 is shown to be valid with W n (ψ) replaced by W θ n (ψ). In fact, in the divergence case, for any θ ∈ C 2 the more general Hausdorff measure version is established. As is to be expected, the convergence case of the Khintchine-Groshev theorem established in [1] implies the inhomogeneous extremal statement (1.7) for nondegenerate manifolds. To the best of our knowledge, unlike in the homogeneous setting, an inhomogeneous theory of Diophantine approximation on affine subspaces is yet to be developed. As already alluded to above, the purpose of this work is to address this imbalance by establishing an inhomogeneous version of the Khintchine-Groshev theorem for affine subspaces of R n . As a consequence, we obtain the inhomogeneous extremal statement (1.7) for affine subspaces. Indeed, our results go some way towards developing a coherent inhomogeneous theory for degenerate manifolds as outlined in [1, §1.4].
In the study of Diophantine approximation on affine subspaces, one needs to assume some condition on the slope of the affine subspace in order to ensure that the affine subspace inherits generic Diophantine properties from its ambient Euclidean space. We will now introduce certain Diophantine exponents of matrices which play a key role in this regard. Indeed, we need these exponents in order to even state our main convergence theorem. Given a column θ ∈ R d+1 and a matrix A ∈ Mat d+1×n−d (R), the inhomogeneous Diophantine exponent ω(A; θ) of (A; θ) is defined to be the supremum of v > 0 for which there are infinitely many a ′ ∈ Z n−d \ {0} such that for some a ′′ ∈ Z d+1 . In the case θ = 0, ω(A) := ω(A; 0) is the usual (homogeneous) Diophantine approximation exponent of the matrix A. It is well known that (n − d)/(d + 1) ≤ ω(A) ≤ ∞ for all A ∈ Mat d+1×n−d (R) and that ω(A) = (n − d)/(d + 1) for Lebesgue almost every A.
We now introduce the higher Diophantine exponents of A as defined by Kleinbock in [23]. For A ∈ Mat d+1×n−d (R), we set where Id d+1 denotes the (d + 1) × (d + 1) identity matrix. Let e 0 , . . . , e n denote the standard basis of R n+1 and set where i ≤ j, be the linear subspace of R n+1 spanned by vectors e i , . . . , e j . Clearly, W 0→n = R n+1 . Now let w ∈ j (W 0→n ) represent a discrete subgroup Γ of Z n+1 , that is w is the wedge product of vectors from any given basis of Γ. Define the map c : for 0 ≤ i ≤ n, and let π • denote the projection (W 0→n ) → (W d+1→n ). For each j = 1, . . . , n − d, define (1.14) It is shown in Lemma 5.3 of [23] that ω 1 (A) = ω(A). We shall see in the next section that the Diophantine exponents ω j (A) play a key role in the convergence case of the Khintchine-Groshev theorem for affine subspaces.
1.2. Our Main Theorems. As before H will denote an open subset of a ddimensional affine subspace of R n parametrised as in (1.9). Then, given ψ and θ, the object of study is the set W θ n (ψ) ∩ H ; i.e. the set of "(ψ, θ)-inhomogeneously approximable" vectors on H . 1 Our first result establishes the convergence case of the inhomogeneous Khintchine-Groshev theorem for affine subspaces. Let ψ be an approximating function and θ : R n → R be a function such that θ| H is analytic. Further in the case θ| H is a linear function so that for some column θ = (θ 0 , . . . , θ n ) t , assume that ω(A; θ) < n . (1.17) Recall that | | H denotes the induced Lebesgue measure on H . Clearly the above convergence theorem implies the inhomogeneous extremal statement (1.7) for any affine subspace satisfying (1.15) and any analyticθ that additionally satisfies (1.17) in the case it is linear.

Remarks:
(1) In the case θ| H is linear, condition (1.17) on the exponent of (A; θ) is optimal. Indeed, suppose that holds for infinitely many a ′ ∈ Z n−d \ {0} and some a ′′ ∈ Z d+1 , but holds only for finitely many a ′ ∈ Z n−d \ {0} and a ′′ ∈ Z d+1 . Clearly, in this case ω(A; θ) = n. The existence of such pairs (A; θ) can be proved by using the inhomogeneous version of Jarnik's theorem for systems of linear forms [4,Theorem 19]. Assuming a ′ and a ′′ satisfy (1.20), write a ′ as (a d+1 , . . . , a n ) t and a ′′ as (a 0 , . . . , a d ) t . Then, on identifying a with (a 1 , . . . , a n ) t one readily verifies thatθ and that a ≪ a ′ . Therefore Then, clearly for every x the inequality holds for infinitely many a ∈ Z n−d \ {0} and some a 0 ∈ Z. In this case ). An obvious modification of the above argument shows that, given an approximating function ψ satisfying (1.19), in the case θ| H is linear, (1.18) necessarily implies the existence of c > 0 such that How close this is to being a sufficient condition remains an interesting question that is open even in the homogeneous case.
For the divergence counterpart to Theorem 1.4, we shall prove the following more general statement in terms Hausdorff measures. Throughout, H s (X) will denote the s-dimensional Hausdorff measure of a subset X of R n and dim X the Hausdorff dimension, where s > 0 is a real number.
Let H an open subset of an affine subspace of R n of dimension d and let s > d − 1. Let ψ be an approximating function and θ : R n → R be a function such that θ| H ∈ C 2 . Suppose that (1.15) holds and that Given an approximating function ψ, the lower order at infinity τ ψ of 1/ψ is defined by and indicates the growth of 1/ψ 'near' infinity. Now observe that the divergent sum condition (1.23) is satisfied whenever Therefore, it follows from the definition of Hausdorff dimension that We therefore obtain the following dimension statement concerning the set W θ n (ψ) ∩ H . Let ψ be an approximating function and θ : R n → R be a function such that θ| H ∈ C 2 . Suppose that (1.15) holds and that 1 ≤ τ ψ < ∞. Then (1.26)

Remarks:
(1) To the best of our knowledge, the above findings constitute the first known results in the context of inhomogeneous Diophantine approximation on affine subspaces. In fact, Theorem 1.5 is new even for Lebesgue measure (i.e. when s = d) and in many cases for the homogeneous setting (i.e. when θ ≡ 0). The only previously known cases in the homogeneous setting were: (a) the case of lines passing through the origin which was treated in [3], and (b) the case of affine hyperplanes (d = n − 1) which was treated in [15].
(2) In the case d = n − 1 condition (1.15) represents a single inequality imposed on the main Diophantine exponent ω(A) of A. (4) In [1], the smoothness condition imposed on the inhomogeneous function θ is weaker than what we have assumed to establish the convergence statement of Theorem 1.4. In short we have imposed the stronger analyticity condition to deal with a technical problem involving (C, α)-good functions (see below for the definition). It is plausible that this condition can be relaxed and brought at par with that imposed in [1].
(5) The homogeneous results in [10] and the inhomogeneous convergence results in [1] are proved in the context of more general multivariable approximating functions. This setting includes the case of "multiplicative" Diophantine approximation. Both our main theorems should hold for nondegenerate submanifolds of affine subspaces and our convergence theorem should, in addition, be true in the multivariable setting. We plan to return to this extension in a separate work.

The Gradient Division
In this section, we prepare the groundwork to prove Theorem 1.4, the "convergence case". Let U be the same as in (1.8) and as before defineθ : U −→ R by settingθ Clearlyθ is an analytic function since θ| H is analytic. For a ∈ Z n \ {0}, we define L(a) := {x ∈ U : |θ(x) + a 0 + (x,xA)a| < ψ( a n ) for some a 0 ∈ Z} .
Observe that lim sup L(a) is the projection of W θ n (ψ) ∩ H onto R d . Here and elsewhere, unless stated otherwise, a ∈ Z n \ {0} and any unspecified limsup is taken over such a. Since the projection from H to R d is bi-Lipschitz, Theorem 1.4 will follow on showing that | lim sup L(a)| = 0. In fact, it is sufficient to show that for each x ∈ U, we can choose an open ball B centred at x with 11B ⊆ U such that To prove the above measure zero statement, it is natural to consider separately the case when we have a 'large derivative' and the case when we do not. More precisely, we split the set L(a, B) into two subsets depending on the size of the quantity ∇(θ(x) + (x,xA)) = ∇(θ(x)) + [Id d A ′ ]a -here A ′ is as introduced at the start of §1.1 and as usual ∇ denotes the gradient operator. With this in mind, for any sufficiently small open ball B with 11B ⊆ U, we define and r is the radius of B. Here for a multi-index β = (i 1 , . . . , i d ) of non-negative integers |β| := i 1 +· · ·+i d and ∂ β denotes the corresponding differentiation operator, that is . We will prove that for Clearly, on combining the measure zero statements (2.4) and (2.5) we obtain the desired measure zero statement (2.1).
3. Estimating the measure of lim sup L large (a, B) In this section, we will establish (2.4) as a simple consequence of the following statement.
where 2B is the ball with the same centre as B and radius 2r. Let and Then for every δ ′ > 0, the set of all x ∈ B such that |p + F (x)| < δ ′ for some To prove (2.4) from Proposition 3.1, we start with any sufficiently small open ball small B in U. We fix a ∈ Z n \ {0} and take F (x) = ((x,xA),θ(x)) · (a, 1) for x ∈ 2B and δ ′ = ψ( a n ). Clearly M = L, where L is given by (2.3). Hence by Proposition 3.1, we get that and thus Since the latter sum is convergent, on applying the Borel-Cantelli lemma we obtain (2.4), as desired. In the above as well as elsewhere ≪ means an inequality with an unspecified multiplicative constant.
In order to establish (2.5) we will use the 'inhomogeneous transference principle' introduced in [8, Section 5], which simplified version is recalled in the next section. H t (α, φ(t)) .
We now discuss the two main properties which enables one to transfer zero ̺-measure statements for the 'homogeneous' limsup sets Λ H (φ) to the 'inhomogeneous' limsup sets Λ I (φ).
(1) Intersection property: The triplet (H, I, Φ) is said to satisfy the intersection property if for any φ ∈ Φ, there exists φ * ∈ Φ such that for all but finitely many t ∈ T and for all distinct α, α ′ ∈ A, we have . (2) Contracting property: We say that ̺ is contracting with respect to (I, φ) if for any φ ∈ Φ, there exists φ + ∈ Φ, a sequence of positive numbers {k t } t∈T with t∈T k t < ∞ and for all but finitely many t ∈ T and all α ∈ A, a collection C t,α of balls B centred in B satisfying the three conditions given below: and The main transference for our purpose, which follows easily from [8,Theorem 5], can be stated as follows.
Recall, that in view of §3, the proof of Theorem 1.4 has been reduced to showing the truth of (2.5). The above transference principle plays a key role in carrying out this task. With this in mind, the proof of (2.5) splits naturally into three main steps. Let B ⊆ U be an open ball and recall that ̺ is the d-dimensional Lebesgue measure restricted to B.
Step 1: We show that if γ is appropriately chosen, then regardless of the choice B. This will be the subject of §6 - §8.
Step 2: We show that the triplet (H, I, Φ) as defined above satisfies the intersection property. This will be the subject of §9.
Step 3: We show that ̺ is contracting with respect to (I, φ δ ). This will be the subject of §10.
The upshot of successfully carry out these steps, is that on applying Theorem 4.1, we are able to conclude that This in turn implies (2.5), since To carry out Step 1 we shall employ dynamical tools. For that, we need to recall a few elementary properties of the so called 'good functions' introduced by Kleinbock and Margulis [24].  . . . , f n ), a map from an open subset U ⊆ R d to R n . We will say that f is good at x 0 ∈ U if there exists a neighbourhood V ⊆ U and C, α > 0 such that any linear combination of 1, f 1 , . . . , f n is (C, α) − good on V . The map f is said to be good if it is good at every point of U. Note that C, α need not be uniform.
We will make use of the following properties of (C, α)-good functions, e.g. see [22].
(G1) If f is (C, α)-good on an open set V , so is λf for all λ ∈ R.
(G2) If f i , i ∈ I are (C, α)-good on V , so is sup i∈I |f i |.
One can note that from (G2), it follows that the supremum norm of a vector valued function f is (C, α)-good whenever each of its components is (C, α)-good. Furthermore, in view of (G3), we can replace the norm by an equivalent one, only affecting C but not α.
The following result provides us with an important class of good functions.
In particular, constant and linear polynomials are The main dynamical tool that we will be exploiting to show (5.3) is commonly known as the "quantitative nondivergence" estimate in the space of unimodular lattices. This constitutes our next section.

A quantitative nondivergence estimate
Let W be a finite dimensional real vector space. For a discrete subgroup Γ of W , we set Γ R to be the minimal linear subspace of W containing Γ. A subgroup Γ of Λ is said to be primitive in Λ if Γ = Γ R ∩ Λ. We denote the set of all nonzero primitive subgroups of Γ by L(Γ). Let j := dim(Γ R ) be the rank of Γ. We say that w ∈ j (W ) represents Γ if In fact, one can easily see that such a representative of Γ is always unique up to a sign.
A function ν : (W ) −→ R + is called submultiplicative if (i) ν is continuous with respect to natural topology on (W ), (ii) ∀ t ∈ R and w ∈ (W ), ν(tw) = |t|ν(w), In view of property (ii) above, without any confusion, we can define ν(Γ) := ν(w) where w represents Γ. Armed with the notion of submultiplicative, we are in the position to state the "quantitative nondivergence" estimate that we will require in establishing (5.3). α > 0, 0 < ρ < 1 and let ν be a submultiplicative function on (W ). Assume that for any Γ ∈ L(Λ), Then for every ε ′′ > 0 we have that Fix a ball B ⊂ U such that 11B ⊂ U. For t ∈ Z + and δ ∈ [0, γ), we define the set Then, by definition Λ H (φ δ ) ∩ B ⊆ lim sup t→∞ A t and so (5.3) follows on showing that With this in mind, pick β ∈ 0, 1 2(n+1) and set and ε := 2 βt ε ′ = 2 2n √ ndL 1 n+1 2 βt 2 t/2(n+1) . (8.4) Furthermore, for x ∈ R d , let and for t ∈ Z + , let where ε, δ ′ , T, K are defined above. Note that these parameters depend on t and some fixed constants. Also, denote by Λ the subgroup of Z 1+d+n consisting of vectors of the form: Then, it readily follows from the above definitions that In view of the Borel-Cantelli lemma, this will follow on showing that With the intention of using Theorem 7.1 to prove (8.9), we take W = R 1+d+n with basis e 0 , e * 1 , . . . , e * d , e 1 , . . . , e n , Λ as given by (8.7) and H(x) = g t u x . The submultiplicative function ν on W is chosen as described in [10, §7]. Namely, let W * be the d-dimensional subspace of W spanned by e * 1 , . . . , e * d so that Λ given by (8.7) is equal to the intersection of Z 1+d+n and W ⊥ * . Here we identify W ⊥ * with R n+1 canonically. Also let W be the ideal of (W ) generated by 2 (W * ) and let π * be the orthogonal projection with kernel W. Then w e is defined to be the Euclidean norm of π * (w). In other words, if w is written as a sum of exterior products of the base vectors e i and e * i , to compute ν(w) we ignore the components containing exterior products of the type e * i ∧ e * j , 1 ≤ i = j ≤ d, and simply take the Euclidean norm of the sum of the remaining components. By definition, it is immediate that ν| W agrees with the Euclidean norm.
For appropriately determined quantities C, α, ρ we now validate, one by one, the conditions (KM1)-(KM3) associated with Theorem 7.1. Condition (KM3) can be verified for any ρ ≤ 1 in exactly the same manner as in [10, §7]. For the verification of the remaining conditions, we begin with the explicit computation of the quantity H(x)w for any w ∈ k (W ⊥ * ) and k = 1, . . . , n + 1. On writing x = (x 1 , . . . , x d ) and (x,xA) = (f 1 (x), . . . , f n (x)), we see that Note that each f i (x) is a polynomial in x 1 , . . . , x d with degree at most 1 so that each partial derivative 8.1. Checking (KM1). Since Λ = Z 1+d+n ∩ W ⊥ * , any representative w ∈ k (W ) of any subgroup of Λ of rank k, 1 ≤ k ≤ n + 1, can be written as I a I e I , where each a I ∈ Z and e I = e i i ∧ · · · ∧ e i k with i 1 , . . . , i k ∈ {0, 1, . . . , n}, i 1 < · · · < i k .
Since each component of π * (H(x)w) is a polynomial in x 1 , . . . , x d with degree at most 1 and in view of Proposition 6.1, each of them is it follows from property (G4) of good functions that ν(π * (H(x)w)) is (C, α)-good onB with This verifies condition (KM1).
Assume now that 1 ≤ k ≤ n. To bound the norm of π * (H(x)w) from below, we will proceed along the lines of [17, §5.3] using a technique from [23]. As observed in [17, §5.3], for any x ∈ B π * (H(x)w) ≥ g tũx w Thus, the name of the game is to bound sup x∈B g tũx w from below. It follows from (4.6) in [23], that where c is the function given by (1.13), π is the projection from (W ⊥ * ) to (W 1→n ) and W 1→n stands for the span of the vectors e 1 , . . . , e n . Now, recall that where R A is given by (1.11). Therefore, we can replace sup x∈B (x,xA)c(w) in the above norm calculation by sup x∈B xR A c(w) . As the functions 1, x 1 , . . . , x d are linearly independent over R on B, the map v → sup x∈B xv defines a norm on ( (W 1→n )) d+1 which must be equivalent to the supremum norm on ( (W 1→n )) d+1 . Thus, there is a constant K 2 > 0 depending on d, n and B, such that sup and consequently, via (8.14), that . (8.15) To continue, we consider two separate cases depending on the size of the rank k. We first note that from Lemma 5.1 in [23], we get that for any n − d < k ≤ n for all but finitely many w ∈ k (Λ) we have that R A c(w) ≥ 1. Also note that R A c(w) does not vanish, as otherwise if R A c(w 0 ) were zero, then, by the linearity of the map c(w), we would get R A c(λw) = 0 for all integers λ, contrary to what we have already seen. Consequently, there is a constant K 3 > 0 depending only on A, such that R A c(w) ≥ K 3 . (8.16) Therefore, by (8.15), we get that It follows from (8.2)-(8.4), that On picking β such that 1 2(n + 1) − 1 n < β < 1 2(n + 1) , (8.18) we obtain via (8.17), that for all subgroups Γ of Λ with rank n − d < k ≤ n We now obtain an analogous lower bound result for subgroups Γ of Λ with rank 1 ≤ k ≤ n − d. In this case, a consequence of (1.15) is that there exist constants 0 < θ ′ , K 4 < 1, depending only on A, such for any w ∈ k (Λ) where κ is the solution to the equation In other words, κ := (K 2 K 4 ) k n−θ ′ +1 T δ ′ k n−θ ′ +1 and so it follows from (8.2)-(8.4) that On redefining β if necessary; namely so that both (8.18) and , (8.23) hold, it follows that Note that (8.23) has a solution β since 0 < θ ′ < 1. This together with (8.21) implies that for all subgroups Γ of Λ with rank 1 ≤ k ≤ n − d, we have that On combining (8.11), (8.13), (8.19) and (8.24), we have verified condition (KM2) with We are now in the position to apply Theorem 7.1 to establish the desired convergent sum statement (8.9).

By (10.3) and the fact that
On combining (10.6), (10.9) and the fact that B is bounded, we find that there exists a scaling factor τ ≥ 1 such that the ball B(x) := τ B ′ (x) satisfies and 5B(x) ⊂ 11 B . (10.11) For t ≥ t 0 and α ∈ A, we now let Then by construction and the l.h.s. of (10.10), any such collection of balls automatically satisfies conditions (4.2) and (4.3) with φ = φ δ and φ + = φ + δ . Regarding condition (4.4), we proceed as follows.
11. The divergence theory: proof of Theorem 1.5 The proof of Theorem 1.5 makes use of the following statement, which is a special and simplified version of Theorem 3 appearing in [1].
Theorem 11.1. Let M := {f(x) : x ∈ U} ⊆ R n be a manifold of dimension d parameterized by a smooth map f : U → R n defined on a ball U ⊆ R d . Suppose there exists an absolute constant C 0 ≥ 1 such that for any ball B with 2B ⊆ U and any κ ∈ (0, 1), we have that x ∈ B : ∃ (a, a 0 ) ∈ Z n \ {0} × Z s.t.
|a 0 + f(x) · a| < κ Q n a ≤ Q ≤ C 0 κ |B| (11.1) for all sufficiently large Q. Let ψ be an approximation function and θ : R n → R be a function such that θ| M ∈ C 2 . Let s > d − 1 and suppose that a∈Z n \{0} a ψ( a n ) a Note that, by the monotonicity of ψ, (11.2) is equivalent to (1.23). Hence, armed with Theorem 11.1, the proof of Theorem 1.5 reduces to establishing (11.1) with U ⊂ R d being an open subset, M = H and f given by (1.9). With this in mind, for any ball B such that 11B ⊆ U, any κ ∈ (0, 1) and Q > 1, let (11.5) We note that the set appearing in (11.1) is contained in the union of the "large derivative" set L 1 (B, κ, Q) and the "small derivative" set L 2 (B, κ, Q). Thus l.h.s. of (11.1) ≤ |L 1 (B, κ, Q)| + |L 2 (B, κ, Q)| . (11.6) As in the proof of Theorem 1.4, estimating the measure of the large derivative set is relatively easy and makes use of Proposition 3.1. To begin with, observe that Now fix a ∈ Z n \ {0} and with reference to Proposition 3.1, let F (x) = (x,xA) · a for x ∈ 2B and δ ′ = κ Q −n . By definition, M ≥ 1 4r 2 where M is given (3.2) and so it follows from Proposition 3.1 that |L 1 (a, B, κ, Q)| ≤ K d κ Q n |B| . In turn, this implies that |L 1 (B, κ, Q)| ≤ K d κ Q n (2Q + 1) n |B| ≤ 3 n K d κ |B| . (11.7) We now turn our attention to estimating the measure of the small derivative set L 2 (B, κ, Q). Let t be the unique integer satisfying 2 t ≤ Q < 2 t+1 and δ satisfy 0 < δ < γ, where γ is as defined earlier. For t sufficiently large we obviously have that L 2 (B, κ, Q) ⊂ A t , (11.8) where A t is defined at the beginning of §8. Then, as a result of (8.8) and (8.27), we have that |L 2 (B, κ, Q)| ≤ κ |B| (11.9) provided that t is sufficiently large.
This thereby completes the proof of Theorem 1.5.