A Mass Transference Principle for systems of linear forms and its applications

In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications of this result to problems in Diophantine approximation. These include a general transference of Lebesgue measure Khintchine-Groshev type theorems to Hausdorff measure statements. The statements we obtain are applicable in both the homogeneous and inhomogeneous settings as well as allowing transference under any additional constraints on approximating integer points. In particular, we establish Hausdorff measure counterparts of some Khintchine-Groshev type theorems with primitivity constraints recently proved by Dani, Laurent and Nogueira [8].


Introduction
The main goal of this paper is to settle a problem posed in [BBDV09] regarding the mass transference principle, a technique in geometric measure theory that was originally discovered in [BV06a] having primarily been motivated by applications in metric number theory. To some extent the present work is also driven by such applications.
To begin with, recall that the sets of interest in metric number theory often arise as the upper limit of a sequence of 'elementary' sets, such as balls, and satisfy elegant zero-one laws. Recall that if (E i ) i∈N is a sequence of sets then the upper limit or lim sup of this sequence is defined as These zero-one laws usually involve simple criteria, typically the convergence or divergence of a certain sum, for determining whether the measure of the lim sup set is zero or one. To give an The conditions (i) and (ii) in Theorem BV1 arise as a consequence of the particular proof strategy employed in [BV06b]. However, it was conjectured [BBDV09,Conjecture E] that Theorem BV1 should be true without conditions (i) and (ii). By adopting a different proof strategy (one similar to that used to prove the mass transference principle in [BV06a] rather than 'slicing') we are able to remove conditions (i) and (ii) and, consequently, prove the following.
Theorem 1. Let R and Υ be as given above. Let f and g : r → g(r) := r −l f (r) be dimension functions such that r −k f (r) is monotonic and let Ω be a ball in R k . Suppose that, for any ball B in Ω, H k (B ∩ Λ(g(Υ) 1/m )) = H k (B). (2) Then, for any ball B in Ω, H f (B ∩ Λ(Υ)) = H f (B).
At first glance, conditions (i) and (ii) in Theorem BV1 do not seem particularly restrictive. Indeed, there are a number of interesting consequences of this theorem, see [BBDV09,BV06b]. However, in the following section we present applications of Theorem 1 which may well be out of reach when using Theorem BV1. In § 3 and § 4 we establish necessary preliminaries and some auxiliary lemmas before presenting the full proof of Theorem 1 in § 5.

Some applications of Theorem 1
In this section we highlight merely a few applications of Theorem 1 which we hope give an idea of the breadth of its consequences. In § 2.1 we show that, using Theorem 1, with relative ease we are able to remove the last remaining monotonicity condition from a Hausdorff measure analogue of the classical Khintchine-Groshev theorem. We also show how the same outcome may be achieved, albeit with a somewhat longer proof, by using Theorem BV1 instead of Theorem 1. In § 2.2 we obtain a Hausdorff measure analogue of the inhomogeneous version of the Khintchine-Groshev theorem.
In § 2.3 we present Hausdorff measure analogues of some recent results of Dani, Laurent and Nogueira [DLN15]. They have established Khintchine-Groshev type statements in which the approximating points (p, q) are subject to certain primitivity conditions. We obtain the corresponding Hausdorff measure results. On the way to realising some of the results outlined above, in § 2.2 and § 2.3 we develop several more general statements which reformulate Theorem 1 in terms of transferring Lebesgue measure statements to Hausdorff measure statements for very general sets of Ψ-approximable points (see Theorems 4, 5 and 6). The recurring theme throughout this section is that, given more-or-less any Khintchine-Groshev type statement, Theorem 1 can be used to establish the corresponding Hausdorff measure result.

The Khintchine-Groshev theorem for Hausdorff measures
Let n 1 and m 1 be integers. Denote by I nm the unit cube [0, 1] nm in R nm . Throughout this section we consider R nm equipped with the norm · : R nm → R defined as follows: where x = (x 1 , . . . , x m ) with each x representing a column vector in R n for 1 m, and | · | 2 is the usual Euclidean norm on R n . The role of the norm (3) will become apparent soon, namely through the proof of Theorem 2 below.
Given a function ψ : N → R + , let A n,m (ψ) denote the set of x ∈ I nm such that |qx + p| < ψ(|q|) for infinitely many (p, q) ∈ Z m × Z n \{0}. Here, | · | denotes the supremum norm, x = (x i ) is regarded as an n × m matrix and q and p are regarded as a row vectors. Thus, qx represents a point in R m given by the system q 1 x 1 + · · · + q n x n (1 m) of m real linear forms in n variables. We will say that the points in A n,m (ψ) are ψ-approximable. That A n,m (ψ) satisfies an elegant zero-one law in terms of nm-dimensional Lebesgue measure when the function ψ is monotonic is the content of the classical Khintchine-Groshev theorem. We opt to state here a modern version of this result which is best possible (see [BV10]). In what follows |X| will denote the k-dimensional Lebesgue measure of X ⊂ R k .
Theorem BV2. Let ψ : N → R + be an approximating function and let nm > 1. Then The earliest versions of this theorem were due to Khintchine and Groshev and included various extra constraints including monotonicity of ψ. A famous counterexample constructed by Duffin and Schaeffer [DS41] shows that, while Theorem BV2 also holds when m = n = 1 and ψ is monotonic, the monotonicity condition cannot be removed when m = n = 1 and so it is natural 1017 D. Allen and V. Beresnevich to exclude this situation by letting nm > 1. In the latter case, the monotonicity condition has been removed completely, leaving Theorem BV2. That monotonicity may be removed in the case n = 1 is due to a result of Gallagher and in the case where n > 2 it is a consequence of a result due to Schmidt. For further details we refer the reader to [BBDV09] and references therein. The final unnecessary monotonicity condition to be removed was the n = 2 case. Formally stated as Conjecture A in [BBDV09], this case was resolved in [BV10].
Regarding the Hausdorff measure theory we shall show the following.
Theorem 2. Let ψ : N → R + be any approximating function and let nm > 1. Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that r −nm f (r) is monotonic. Then, Theorem 2 is not entirely new and was in fact previously obtained in [BBDV09] via Theorem BV1 subject to ψ being monotonic in the case that n = 2. The deduction there was relying on a theorem of Sprindžuk rather than Theorem BV2 (which is what we shall use). In fact, with several additional assumptions imposed on ψ and f , the result was first obtained by Dickinson and Velani [DV97]. Indeed, the proof of the convergence case of Theorem 2 makes use of standard covering arguments that, with little adjustment, can be drawn from [DV97].
In what follows we shall give two proofs for the divergence case of Theorem 2, one using Theorem BV1 and the other using Theorem 1. The reason for this is to show the advantage of using Theorem 1 on the one hand, and to explicitly exhibit obstacles in using Theorem BV1 in other settings on the other hand. In the proofs we will use the following notation. For (p, q) ∈ Z m × Z n \{0} let R p,q := {x ∈ R nm : qx + p = 0}.
Note that, throughout the proofs of Theorem 2, (p, q) will play the role of the index j appearing in Theorem BV1 and Theorem 1. Also note that for δ 0 we have We note that if ψ(r) 1 for infinitely many r ∈ N, then A n,m (ψ) = I nm and the divergence case of Theorem 2 is trivial. Hence, without loss of generality we may assume that ψ(r) 1 for all r ∈ N. First we show how Theorem BV1 and Theorem BV2 imply the divergence case of Theorem 2. (4) To use Theorem BV1 we have to restrict the approximating integer points q in order to meet conditions (i) and (ii) of Theorem BV1. We will use the same idea as in [BBDV09]; namely, we will 1018 impose the requirement that |q| = |q K | for a fixed K ∈ {1, . . . , n}. Sprindžuk's theorem that is used in [BBDV09] allows for the introduction of this requirement almost instantly. Unfortunately, this is not the case when one is using Theorem BV2 and hence we will need a new argument. For each 1 i n define the auxiliary functions Ψ i : Z n \{0} → R + by setting In what follows, similarly to A n,m (ψ), we consider sets A n,m (Ψ) of points x ∈ I nm such that for infinitely many pairs (p, q) ∈ Z m × Z n \{0}, where Ψ : Z n \{0} → R + is a multivariable function. Since, by definition, Ψ i (q) ψ(|q|) for each 1 i n and each q ∈ Z n \{0}, it follows that A n,m (Ψ i ) ⊂ A n,m (ψ) for each 1 i n.
By (6), to complete the proof of (4), it is sufficient to show that Without loss of generality we will assume that K = 1. Define Note that, since g is increasing and ψ(r) 1, the constant M is finite. Let Υ p,q := Ψ 1 (q)/|q| for each (p, q) ∈ S. The purpose for introducing this auxiliary set S will become apparent later. Now, for each (p, q) ∈ S, ∆(R p,q , Υ p,q ) ∩ I nm = x ∈ I nm : and, in taking this limit, (p, q) ∈ S can be arranged in any order. Therefore, (7) will follow on showing that Showing (9) will rely on Theorem BV1. First of all observe that conditions (i) and ( Indeed, regarding condition (i), we have that R p,q ∩ V consists of the single element and so is non-empty. Regarding condition (ii), for (p, q) ∈ S we have that x 1, + p q 1 < 1 and x i = 0 for i = 1 since |q 1 | = |q| and |q| 2 √ n|q|. Hence diam(V ∩ ∆(R p,q , 1)) 2 and we are done. Now let θ : N → R + be given by and, for each 1 i n, let Θ i : Z n \{0} → R + be given by Similarly to (6), we have that A n,m (Θ i ) ⊂ A n,m (θ) for each 1 i n. Furthermore, Indeed, the '⊃' inclusion follows from the above. To show the converse, note that for any x ∈ A n,m (θ) the inequality |qx + p| < θ(|q|) is satisfied for infinitely many (p, q) ∈ Z m × Z n \{0}. Clearly, for each q ∈ Z n \{0} we have that θ(|q|) = Θ i (q) for some 1 i n. Therefore, there is a fixed i ∈ {1, . . . , n} such that |qx + p| < θ(|q|) = Θ i (q) is satisfied for infinitely many (p, q) ∈ Z m × Z n \{0}. This means that x ∈ A n,m (Θ i ) for some i, thus verifying (10). Next, observe that, by (5), the sum diverges. Therefore, by Theorem BV2, we have that |A n,m (θ)| = 1. Hence, by (10), there exists some 1 K n such that |A n,m (Θ K )| > 0. By the zero-one law of [BV08, Theorem 1], we know that |A n,m (Θ K )| ∈ {0, 1}. Hence, |A n,m (Θ K )| = 1.
Without loss of generality we will suppose that K = 1, the same as in (7).

1020
A mass transference principle for systems of linear forms Now, using the fact that |q| |q| 2 , for (p, q) ∈ S we have that ∆(R p,q , g(Υ p,q ) 1/m ) ∩ I nm = x ∈ I nm : Furthermore, observe that if {x ∈ I nm : |qx+p| < Θ 1 (q)} = ∅, then |p| M |q| and so (p, q) ∈ S. Therefore, In particular, |Λ(g(Υ) 1/m ) ∩ I nm | = 1 and so for any ball B ⊂ I nm we have that H nm (Λ(g(Υ) 1/m ) ∩ B) = H nm (B). Hence, we may apply Theorem BV1 with k = nm, l = m(n − 1) and m to conclude that, for any ball B ⊂ I nm , we have H f (B ∩ Λ(Υ)) = H f (B). In particular, H f (Λ(Υ) ∩ I nm ) = H f (I nm ) and the proof is thus complete. 2 We now show how Theorem 1 and Theorem BV2 imply the divergence case of Theorem 2.
For such pairs (p, q) we have that where the lim sup is taken over Allen and V. Beresnevich where this penultimate inclusion follows since |q| |q| 2 . Observe that if {x ∈ I nm : |qx + p| < θ(|q|)} = ∅, then |p| M |q|. It follows that Now, by Theorem BV2 and the divergence condition (5), we know that |A n,m (θ)| = 1 since Hence, |Λ(g(Υ) 1/m ) ∩ I nm | = 1 and so we may apply Theorem 1 with k = nm, l = m(n − 1) and m to conclude that, for any ball and so the proof is complete. 2 Remark 1. Note that the proof of (11) is not only shorter and simpler than that of (4) but it also does not rely on the zero-one law [BV08,Theorem 1]. This seemingly minor point becomes a substantial obstacle in trying to use the same line of argument as for (4) in other settings, for example, in inhomogeneous problems. The point is that, as of now, we do not have an inhomogeneous zero-one law similar to [BV08, Theorem 1], see [Ram17] for partial results and further comments. The approach based on using Theorem 1, on the other hand, works with ease in the inhomogeneous and other settings.

Inhomogeneous systems of linear forms
In this section we will be concerned with the inhomogeneous version of the Khintchine-Groshev theorem presented in the previous subsection. Given an approximating function Ψ : Z n \{0} → R + and a fixed y ∈ I m , we denote by A y n,m (Ψ) the set of x ∈ I nm for which holds for infinitely many (p, q) ∈ Z m × Z n \{0}. In the case that Ψ(q) = ψ(|q|) for some function ψ : N → R + we write A y n,m (ψ) for A y n,m (Ψ). Regarding inhomogeneous Diophantine approximation, we have the following statement which can be deduced as a corollary of [Spr79,ch. 1,Theorem 15]. In the case that ψ is monotonic this statement also follows as a consequence of the ubiquity technique, see [BDV06,§ 12.1].
Inhomogeneous Khintchine-Groshev theorem. Let m, n 1 be integers and let y ∈ I m . If ψ : N → R + is an approximating function which is assumed to be monotonic if n = 1 or n = 2, then The following is the Hausdorff measure version of the above statement.

1022
A mass transference principle for systems of linear forms Theorem 3. Let m, n 1 be integers, let y ∈ I m , and let ψ : N → R + be an approximating function. Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that r −nm f (r) is monotonic. In the case that n = 1 or n = 2 suppose also that rg(ψ(r)/r) 1/m is monotonic. Then, Remark 2. Although the condition that rg(ψ(r)/r) 1/m being monotonic when n = 1 or n = 2 is the one that we naturally arrive at upon combining Theorem 1 with the inhomogeneous Khintchine-Groshev theorem, it is worth noting here that this condition may be relaxed. In the case when n = 2, by appealing to the more general theorem of Sprindžuk [Spr79, ch. 1, Theorem 15] (from which the inhomogeneous Khintchine-Groshev theorem stated above can be deduced for n 2), it is possible to replace monotonicity of rg(ψ(r)/r) 1/m in the statement of Theorem 3 with the more aesthetically pleasing assumption that ψ is monotonically decreasing.
When n = 1 it should be possible to make the same assumption replacement by using ideas from ubiquity (see [BDV06,§ 12.1] and references within).
The proof of the convergence case of Theorem 3 once again makes use of standard covering arguments. The divergence case is a consequence of the inhomogeneous Khintchine-Groshev theorem and Theorem 1. The proof of the divergence case is almost identical to that of (11) and we therefore leave the details out. Furthermore, exploiting this same argument a little further, we can use Theorem 1 to prove the following two more general statements from which both Theorems 2 and 3 follow as corollaries. In some sense Theorems 4 and 5 below are reformulations of Theorem 1 in terms of sets of Ψ-approximable (and ψ-approximable) points.
Theorem 4. Let Ψ : Z n \{0} → R + be an approximating function and let y ∈ I m . Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that r −nm f (r) is monotonic. Let The following statement is a special case of Theorem 4 with Ψ(q) := ψ(|q|).
Theorem 5. Let ψ : N → R + be an approximating function, let y ∈ I m and let f and g :

D. Allen and V. Beresnevich
The proof of Theorem 4 is similar to that of (11). We shall explicitly deduce it from the even more general result of § 2.3, where the approximating function will be allowed to depend on p as well as q. Theorem 3 now trivially follows on combining the inhomogeneous Khintchine-Groshev theorem with Theorem 5. Furthermore, any progress in removing the monotonicity constraint on ψ from the inhomogeneous Khintchine-Groshev theorem can be instantly transferred into a Hausdorff measure statement upon applying Theorem 5. Indeed, we suspect that a full inhomogeneous analogue of Theorem BV2 must be true. Recall that it is open only in the case when n = 1 or n = 2.

Approximation by primitive points and more
The key goal of this section is to present Hausdorff measure analogues of some recent results obtained by Dani, Laurent and Nogueira in [DLN15]. The setup they consider assumes certain coprimality conditions on the (m + n)-tuple (q 1 , . . . , q n , p 1 , . . . , p m ) of approximating integers. To achieve our goal we will first prove a very general statement which further extends Theorems 4 and 5 and is of independent interest. In particular, we will allow for the approximating function to depend on (p, q) and will also introduce a 'distortion' parameter Φ that allows certain flexibility within our framework. This allows us, for example, to incorporate the so-called 'absolute value theory' [Dic93,HK13,HL13].
Within this section Ψ : Z m × Z n \{0} → R + will be a function of (p, q), y ∈ I m will be a fixed point and Φ ∈ I mm will be a fixed m × m square matrix. Further, define M y,Φ n,m (Ψ) to be the set of x ∈ I nm such that holds for (p, q) ∈ Z m × Z n \{0} with arbitrarily large |q|. Based upon Theorem 1, we now state and prove the following generalisation of Theorems 4 and 5. Theorem and let y ∈ I m and Φ ∈ I mm be fixed. Let f and g : By the monotonicity of g and condition (12), we have that M is finite. Let and let S Φ be any fixed subset of S such that for each (p , q) ∈ S there exists (p, q) ∈ S Φ such that pΦ = p Φ and Θ(p , q) 2Θ(p, q).
1024 A mass transference principle for systems of linear forms The existence of S Φ is easily seen. For each (p, q) ∈ S Φ , let Also note that for each q ∈ Z n \{0} there are only finitely many p ∈ Z m such that (p, q) ∈ S Φ . Therefore where, when defining Λ(Υ), the lim sup is taken over (p, q) ∈ S Φ . Hence, by (14), it would suffice for us to show that H f (Λ(Υ) ∩ I nm ) = H f (I nm ). Consider Λ(g(Υ) 1/m ), where the lim sup is again taken over (p, q) ∈ S Φ . Take any (p , q) ∈ S and let (p, q) ∈ S Φ satisfy (13). Then, since |q| |q| 2 , we have that Also observe that if Recall that |M y,Φ n,m (Θ)| = 1. Furthermore, in view of [BV08, Lemma 4], we have that Together with (15) this implies that |Λ(g(Υ) 1/m )∩I nm | = 1. Further, note that, by (12), Υ p,q → 0 as |q| → ∞. Therefore, Theorem 1 is applicable with k = nm, l = m(n−1) and m and we conclude that for any ball B ⊂ I nm we have that H f (B ∩ Λ(Υ)) = H f (B). In particular, this means that H f (I nm ∩ Λ(Υ)) = H f (I nm ), as required. 2 1025 D. Allen and V. Beresnevich Proof of Theorem 4. Let Ψ be as in Theorem 4. First observe that if Ψ(q) 1 for infinitely many q ∈ Z n , then A y n,m (Ψ) = I nm and there is nothing to prove. Otherwise we obviously have that Ψ(q)/|q| → 0 as |q| → ∞. In this case, extending Ψ and Θ to be functions of (p, q) so that Ψ(p, q) := Ψ(q) and Θ(p, q) := Θ(q), we immediately recover Theorem 4 from Theorem 6. 2 Theorem 6 can be applied in various situations beyond what has already been discussed above. For example, divergence results of [DH13] can be obtained by using Theorem 6 with where I u is the identity matrix. In what follows we shall give applications of Theorem 6 in which the dependence of Ψ on both p and q becomes particularly useful. Namely, we shall extend the results of Dani, Laurent and Nogueira [DLN15] to Hausdorff measures. First we establish some notation. For any d 2 let P (Z d ) be the set of points v = (v 1 , . . . , v d ) ∈ Z d such that gcd(v 1 , . . . , v d ) = 1. For any subset σ = {i 1 , . . . , i ν } of {1, . . . , d} with ν 2, let P (σ) be the set of points v ∈ Z d such that gcd(v i 1 , . . . , v iν ) = 1. Next, given a partition π of {1, . . . , d} into disjoint subsets π of at least two elements, let P (π) be the set of points v ∈ Z d such that v ∈ P (π ) for all components π of π.
Theorem 7. Let ψ : N → R + be an approximating function such that ψ(q)/q → 0 as q → ∞. Let π be any partition of {1, . . . , m + n} and let Φ ∈ I mm and y ∈ I m be fixed. Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that r −nm f (r) is monotonic and let θ : N → R + be defined by θ(q) = qg(ψ(q)/q) 1/m . Then Now, let us turn our attention to the results of Dani, Laurent and Nogueira from [DLN15]. For the moment, we will return to the homogeneous setting. Given a partition π of {1, . . . , m+n} and an approximating function ψ : N → R + we will denote by A π n,m (ψ) the set of x ∈ I nm such that |qx + p| < ψ(|q|) holds for (p, q) ∈ Z m × Z n \{0} with arbitrarily large |q| and (q 1 , . . . , q n , p 1 , . . . , p m ) ∈ P (π). We note that in this case the inequality holds for (p, q) ∈ Z m × Z n \{0} with arbitrarily large |q| if and only if the inequality holds for infinitely many (p, q) ∈ Z m × Z n \{0}. The notation A n,m (ψ) will be used as defined in § 2.1. The following statement is a consequence of [DLN15, Theorem 1.2].

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A mass transference principle for systems of linear forms Theorem DLN1. Let n, m ∈ N and let π be a partition of {1, . . . , m + n} such that every component of π has at least m + 1 elements. Let ψ : N → R + be a function such that the mapping x → x n−1 ψ(x) m is non-increasing. Then, The following Hausdorff measure analogue of Theorem DLN1 follows from Theorem 7.
Theorem 8. Let n, m ∈ N and let π be a partition of {1, . . . , m + n} such that every component of π has at least m + 1 elements. Let ψ : N → R + be an approximating function. Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that the function r −nm f (r) is monotonic and q n+m−1 g(ψ(q)/q) is non-increasing. Then, Proof. First note that in light of the fact that q n+m−1 g(ψ(q)/q) is non-increasing we may assume without loss of generality that ψ(q)/q → 0 as q → ∞. To see this, suppose that ψ(q)/q 0. Therefore, there must exist some ε > 0 such that ψ(q)/q ε infinitely often. In turn, since g is a dimension function, and hence non-decreasing, this means that q n+m−1 g(ψ(q)/q) q n+m−1 g(ε) infinitely often. However, since this expression is non-increasing, we must have that g(ε) = 0. In particular, this means that g(r) = 0 and, hence, also f (r) = 0 for all r ε. Thus H f (X) = 0 for any X ⊂ I nm and so the result is trivially true.
In view of the conditions imposed on π, we must have that nm > 1. Furthermore, since A π n,m (ψ) ⊂ A n,m (ψ), it follows from Theorem 2 that H f (A π n,m (ψ)) = 0 when ∞ q=1 q n+m−1 g(ψ(q)/q) < ∞. Alternatively, one can use a standard covering argument to obtain a direct proof of the convergence part of Theorem 8.
Hence the proof is complete. 2 1027 D. Allen and V. Beresnevich If ψ(q) := q −τ for some τ > 0 let us write A π n,m (τ ) := A π n,m (ψ). The following result regarding the Hausdorff dimension of A π n,m (τ ) is a corollary of Theorem 8.
Corollary 1. Let n, m ∈ N and let π be a partition of {1, . . . , m+n} such that every component of π has at least m + 1 elements. Then Proof. For τ n/m the result follows on applying Theorem 8 with f δ (r) := r s 0 +δ where s 0 = m(n − 1) + m + n τ + 1 .
Indeed, with δ sufficiently small, all the conditions of Theorem 8 are met and furthermore, as is easily seen, we have from Theorem 8 that This means that H s 0 +δ (A π n,m (τ )) = 0 for δ > 0 and H s 0 +δ (A π n,m (τ )) = H s 0 +δ (I nm ) for δ 0. Therefore, if s 0 nm then dim H (A π n,m (τ )) = s 0 since, in this case, H s 0 +δ (I nm ) = ∞ whenever δ < 0. Finally, note that s 0 nm if and only if τ n/m.
In the case where τ < n/m observe that A π n,m (τ ) ⊃ A π n,m (n/m) so dim H (A π n,m (τ )) dim H A π n,m n m = nm.
Combining this with the trivial upper bound gives dim H (A π n,m (τ )) = nm when τ < n/m, as required. 2 Next we consider two results of Dani, Laurent and Nogueira regarding inhomogeneous approximation. As before, for a fixed y ∈ I m we let A y n,m (ψ) denote the set of points x ∈ I nm for which holds for infinitely many (p, q) ∈ Z m ×Z n \{0}. Given a partition π of {1, . . . , m+n}, let A π,y n,m (ψ) be the set of points x ∈ I nm for which (17) holds for infinitely many (p, q) ∈ Z m × Z n \{0} with (q 1 , . . . , q n , p 1 , . . . , p m ) ∈ P (π).
Rephrasing it in a way which is more suitable for our current purposes, a consequence of [DLN15, Theorem 1.1] reads as follows.
Theorem DLN2. Let n, m ∈ N and let π be a partition of {1, . . . , m + n} such that every component of π has at least m + 1 elements. Let ψ : N → R + be a function such that the mapping x → x n−1 ψ(x) m is non-increasing. Then, (i) if ∞ q=1 q n−1 ψ(q) m = ∞ then for almost every y ∈ I m we have |A π,y n,m (ψ)| = 1; (ii) if ∞ q=1 q n−1 ψ(q) m < ∞ then for any y ∈ I m we have |A y n,m (ψ)| = 0.
The corresponding Hausdorff measure statement we obtain in this case is as follows.
Theorem 9. Let n, m ∈ N and let π be a partition of {1, . . . , m + n} such that every component of π has at least m + 1 elements. Let ψ : N → R + be an approximating function. Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that the function r −nm f (r) is monotonic and q n+m−1 g(ψ(q)/q) is non-increasing. Then, Proof. This is similar to the proof of Theorem 8 with the only difference being the introduction of y. 2 Finally, let us reintroduce the parameter Φ ∈ I mm . In this case, considering the sets M π,y,Φ n,m (ψ) (as defined on p. 13), it follows from [DLN15, Theorem 1.3] that we have: Theorem DLN3. Let n, m ∈ N and let π be a partition of {1, . . . , m + n} such that every component of π has at least m + 1 elements. Let ψ : N → R + be a function such that the mapping x → x n−1 ψ(x) m is non-increasing. Then, for any y ∈ I m , Combining this with Theorem 7 we obtain the following Hausdorff measure statement.
Proof. Once again the proof is similar to that of Theorem 8. The Hausdorff f -measure with respect to the dimension function f will be denoted throughout by H f and is defined as follows. Suppose F is a subset of R k . For ρ > 0, a countable collection {B i } of balls in R k with radii r(B i ) ρ for each i such that F ⊂ i B i is called a ρ-cover for F . Clearly such a cover exists for every ρ > 0. For a dimension function f define A simple consequence of the definition of H f is the following useful fact (see, for example, [Fal03]).
Lemma 1. If f and g are two dimension functions such that the ratio f (r)/g(r) → 0 as r → 0, then H f (F ) = 0 whenever H g (F ) < ∞.
In the case that f (r) = r s (s 0), the measure H f is the usual s-dimensional Hausdorff measure H s and the Hausdorff dimension, dim H F , of a set F is defined by dim H F := inf{s 0 : H s (F ) = 0}.
For subsets of R k , H k is comparable to the k-dimensional Lebesgue measure. Actually, H k is a constant multiple of the k-dimensional Lebesgue measure (but we shall not need this stronger statement).
Furthermore, for any ball B in R k we have that V k (B) is comparable to |B|. Thus there are constants 0 < c 1 < 1 < c 2 < ∞ such that for any ball B in R k we have A general and classical method for obtaining a lower bound for the Hausdorff f -measure of an arbitrary set F is the following mass distribution principle. This will play a central role in our proof of Theorem 1 in § 5.
Lemma 2 (Mass distribution principle). Let µ be a probability measure supported on a subset F of R k . Suppose there are positive constants c and r o such that for any ball B with radius r r o . If E is a subset of F with µ(E) = λ > 0 then H f (E) λ/c.
The above lemma is stated as it appears in [BV06a] since this version is most useful for our current purposes. For further information in general regarding Hausdorff measures and dimension we refer the reader to [Fal03,Mat95].

The 5r-covering lemma
Let B := B(x, r) be a ball in R k . For any λ > 0, we denote by λB the ball B scaled by a factor λ; i.e. λB := B(x, λr).
We conclude this section by stating a basic, but extremely useful, covering lemma which we will use throughout [Mat95].
Lemma 3 (The 5r-covering lemma). Every family F of balls of uniformly bounded diameter in R k contains a disjoint subfamily G such that

The K G,B covering lemma
Our strategy for proving Theorem 1 is similar to that used for proving the mass transference principle for balls in [BV06a]. There are however various technical differences that account for the different shape of approximating sets. First of all we will require a covering lemma analogous to the K G,B -lemma established in [BV06a,§ 4]. This appears as Lemma 4 below. The balls obtained from Lemma 4 correspond to planes in the lim sup set Λ(g(Υ) 1/m ). Furthermore, for the proof of Theorem 1 it is necessary for us to obtain from each of these 'larger' balls a collection of balls which correspond to the 'shrunk' lim sup set Λ(Υ). The desired properties of this collection and the existence of such a collection are the contents of Lemma 5 of this section.
To save on notation, throughout letΥ j := g(Υ j ) 1/m . For an arbitrary ball B ∈ R k and for Analogously to [BV06a, Lemma 5] we will require the following covering lemma.
Lemma 4. Let R, Υ, g and Ω be as in Theorem 1 and assume that (2) is satisfied. Then for any ball B in Ω and any G ∈ N, there exists a finite collection satisfying the following properties: Remark 3. Essentially, K G,B is a collection of balls drawn from the families Φ j (B). We write (A; j) for a generic ball from K G,B to 'remember' the index j of the family Φ j (B) that the ball A comes from. However, when we are referring only to the ball A (as opposed to the pair (A; j)) we will just write A. Keeping track of the associated j will be absolutely necessary in order to be able to choose the 'right' collection of balls within A that at the same time lie in an Υ j -neighbourhood of the relevant R j . Indeed, for j = j we could have A = A for some A ∈ Φ j (B) and A ∈ Φ j (B).
Proof of Lemma 4. For each j ∈ N, consider the collection of balls By (2), for any G 1 we have that Observe that D. Allen and V. Beresnevich and that the difference of the two sets lies within 3Υ j of the boundary of B. Then, since Υ j → 0, and consequentlyΥ j → 0, as j → ∞, we have that In particular, there exists a sufficiently large G ∈ N such that for any G G we have However, for any G < G we also have L.
Thus, for any G ∈ N we must have In fact, using the same argument as above it is possible to show that for any for any 0 < ε < 1 and hence that we must have (19) is sufficient for our purposes here. By Lemma 3, there exists a disjoint subcollection G ⊂ {(L; j) : j G, L ∈ Φ 3 j (B)} such that Now, let G consist of all the balls from G but shrunk by a factor of 3; so the balls in G will still be disjoint when scaled by a factor of 3. Formally, G := {( 1 3 L; j) : (L; j) ∈ G}. Then, we have that From (19) and (20)  Next note that, since the balls in G are disjoint and contained in B andΥ j → 0 as j → ∞, we have that Therefore, there exists a sufficiently large N 0 ∈ N such that Thus, taking K G,B to be the subcollection of (A; j) ∈ G with G j < N 0 ensures that K G,B is a finite collection of balls while still satisfying the required properties (i)-(iii). 2 Lemma 5. Let R, Υ, g, Ω and B be as in Lemma 4 and assume that (2) is satisfied. Furthermore, assume that r −k f (r) → ∞ as r → 0. Let K G,B be as in Lemma 4. Then, provided that G is sufficiently large, for any (A; j) ∈ K G,B there exists a collection C(A; j) of balls satisfying the following properties: ; and (v) there exist some constants d 1 , d 2 > 0, independent of G and j, such that Proof. First of all note that, by the assumption that r −k f (r) → ∞ as r → 0, we have that In particular we can assume that G is sufficiently large so that 6Υ j <Υ j for any j G.
Let x 1 , . . . , x t ∈ R j ∩ 1 2 A be any collection of points such that and t is maximal possible. The existence of such a collection follows immediately from the fact that R j ∩ 1 2 A is bounded and, by (23), the collection is discrete. Let Thus, property (i) is trivially satisfied for this collection C(A; j). Recall that, by construction, A ∈ Φ j (B), which means that the radius of 1 2 A is 1 2Υ j . If L ∈ C(A; j), say L := B(x i , Υ j ), and A is centred at x 0 , then for any y ∈ 3L we have that y−x i < 3Υ j while x i −x 0 1 2Υ j . Then, using (22) and the triangle inequality, we get that y − x 0 y − x i + x i − x 0 3Υ j + 1 2Υ j <Υ j . Hence 3L ⊂ A whence property (ii) follows. Further, property (iii) follows immediately from condition (23).
By the maximality of the collection x 1 , . . . , x t , for any x ∈ R j ∩ 1 2 A there exists an x i from this collection such that x − x i 6Υ j . Hence, 7L. Thus On the other hand, by property (ii), we have that which together with the previous inequality establishes property (iv). Finally, property (v) is an immediate consequence of property (iv) upon noting that

Proof of Theorem 1
As with the proof of the mass transference principle given in [BV06a] and the proof of Theorem BV1 given in [BV06b], we begin by noting that we may assume that r −k f (r) → ∞ as r → 0. To see this we first observe that, by Lemma 1, if r −k f (r) → 0 as r → 0 we have that H f (B) = 0 for any ball B in R k . Furthermore, since B ∩ Λ(Υ) ⊂ B, the result follows trivially. Now suppose that r −k f (r) → λ as r → 0 for some 0 < λ < ∞. In this case, H f is comparable to H k and so it would be sufficient to show that H k (B ∩ Λ(Υ)) = H k (B). Since r −k f (r) → λ as r → 0 we have that the ratio f (r)/r k is bounded between positive constants for sufficiently small r. In turn, this implies that, in this case, the ratio of the values g(Υ j ) 1/m and Υ j is uniformly bounded between positive constants. It then follows from [BV08,Lemma 4] that This together with (2) then implies the required result in this case.
Thus, for the rest of the proof we may assume without loss of generality that r −k f (r) → ∞ as r → 0. With this assumption it is a consequence of Lemma 1 that H f (B 0 ) = ∞ for any ball B 0 in Ω, which we fix from now on. Therefore, our goal for the rest of the proof is to show that To this end, for each η > 1, we will construct a Cantor subset K η of B 0 ∩ Λ(Υ) and a probability measure µ supported on K η satisfying the condition that for any arbitrary ball D of sufficiently small radius r(D) we have where the implied constant does not depend on D or η. By the mass distribution principle (Lemma 2) and the fact that K η ⊂ B 0 ∩ Λ(Υ), we would then have that H f (B 0 ∩ Λ(Υ)) H f (K η ) η and the proof is finished by taking η to be arbitrarily large.

The desired properties of K η
We will construct the Cantor set K η := ∞ n=1 K(n) so that each level K(n) is a finite union of disjoint closed balls and the levels are nested, that is K(n) ⊃ K(n + 1) for n 1. We will denote the collection of balls constituting level n by K(n). As with the Cantor set in [BV06a], the construction of K η is inductive and each level K(n) will consist of local levels and sub-levels. So, suppose that the (n − 1)th level K(n − 1) has been constructed. Then, for every B ∈ K(n − 1) we construct the (n, B)-local level, K(n, B), which will consist of balls contained in B. The collection of balls K(n) will take the form K(n) := B∈K(n−1) K(n, B).
Looking even more closely at the construction, each (n, B)-local level will consist of local sub-levels and will be of the form Here, K(n, B, i) denotes the ith local sub-level and l B is the number of local sub-levels. For n 2 each local sub-level will be defined as the union where B will lie in a suitably chosen collection of balls G(n, B, i) within B, K G ,B will arise from Lemma 4 and C(A; j) will arise from Lemma 5. It will be apparent from the construction that the parameter G becomes arbitrarily large as we construct levels. The set of all pairs (A; j) that contribute to (26)  If additionally we start with K(1) := B 0 then, in view of the definition of the sets C(A; j), the inclusion K η ⊂ B 0 ∩ Λ(Υ) is straightforward. Hence the only real part of the proof will be to show the validity of (24) for some suitable measure supported on K η . This will require several additional properties which are now stated.
The properties of levels and sub-levels of K η with c 1 and c 2 as defined in (18). (P4) For any n 2, B ∈ K(n − 1), any i ∈ {1, . . . , l B − 1} and any L ∈ K(n, B, i) and M ∈ K(n, B, i + 1) we have f (r(M )) 1 2 f (r(L)) and g(r(M )) 1 2 g(r(L)). (P5) The number of local sub-levels is defined by and satisfies l B 2 for B ∈ K(n) with n 2.
Properties (P1) and (P2) are imposed to make sure that the balls in the Cantor construction are sufficiently well separated. On the other hand, Properties (P3) and (P5) make sure that there are 'enough' balls in each level of the construction of the Cantor set. Property (P4) essentially ensures that all balls involved in the construction of a level of the Cantor set are sufficiently small compared with balls involved in the construction of the previous level. All of the Properties (P1)-(P5) will play a crucial role in the measure estimates we obtain in § 5.4 and § 5.5.

The existence of K η
In this section we show that it is possible to construct a Cantor set with the properties outlined in § 5.1. In what follows we will use the following notation: Level 1. The first level is defined by taking the arbitrary ball B 0 . Thus, K(1) := B 0 and Property (P0) is trivially satisfied. We proceed by induction. Assume that the first (n − 1) levels K(1), K(2), . . . , K(n − 1) have been constructed. We now construct the nth level K(n).
Level n. To construct the nth level we will define local levels K(n, B) for each B ∈ K(n − 1). Therefore, from now on we fix some ball B ∈ K(n − 1) and a sufficiently small constant ε := ε(B) > 0 which will be determined later. Recall that each local level K(n, B) will consist of local sub-levels K(n, B, i) where 1 i l B and l B is given by Property (P5). Let G ∈ N be sufficiently large so that Lemmas 4 and 5 are applicable. Furthermore, suppose that G is large enough so that 3Υ j < g(Υ j ) 1/m whenever j G, and where c 3 is the constant appearing in Property (P3) above. Note that the existence of G satisfying (27)-(29) follows from the assumptions that r −k f (r) → ∞ as r → 0 and Υ j → 0 as j → ∞.
Sub-level 1. With B and G as above, let K G,B denote the collection of balls arising from Lemma 4. Define the first sub-level of K(n, B) to be By the properties of C(A; j) (Lemma 5), it follows that (P1) is satisfied within this sub-level. From the properties of K G,B (Lemma 4) and Lemma 5 it follows that (P2) and (P3) are satisfied for i = 1.
Higher sub-levels. To construct higher sub-levels we argue by induction. For l < l B , assume that the sub-levels K(n, B, 1), . . . , K(n, B, l) satisfying Properties (P1)-(P4) with l B replaced by l have already been defined. We now construct the next sub-level K(n, B, l + 1). As every sub-level of the construction has to be well separated from the previous ones, we first verify that there is enough 'space' left over in B once we have removed the sub-levels K(n, B, 1), . . . , K(n, B, l) from B. More precisely, let 4L.
We will show that H k (A (l) ) 1 2 H k ( 1 2 B).
Now, to each ball B ∈ G(n, B, l + 1) we apply Lemma 4 to obtain a collection of balls K G ,B and define K(n, B, l + 1) := B ∈G(n,B,l+1) (A;j)∈K G ,B C(A; j).
Regarding (P1), we first observe that it is satisfied for balls in (A;j)∈K G ,B L∈C(A;j) L by the properties of C(A; j) and the fact that the balls in K G ,B are disjoint. Next, since any balls in K G ,B are contained in B and the balls B ∈ G(n, B, l + 1) are disjoint, it follows that (P1) is satisfied for balls L in K(n, B, l + 1). Finally, combining this with (32), we see that (P1) is satisfied for balls L in K l+1 (n, B). That (P2) is satisfied for this sub-level is a consequence of Lemma 4(i) and (ii) and the fact that the balls B ∈ G(n, B, l + 1) are disjoint.
To establish (P3) for i = l + 1 note that Finally, (P4) is trivially satisfied as a consequence of the imposed condition (34) and (P5), that l L 2 for any ball L in K(n, B, l + 1), follows from (29).
Hence, Properties (P1)-(P5) are satisfied up to the local sub-level K(n, B, l + 1) thus establishing the existence of the local level K(n, B) = K l B (n, B) for each B ∈ K(n − 1). In turn, this establishes the existence of the nth level K(n) (and also K(n)).

5.3
The measure µ on K η In this section, we define a probability measure µ supported on K η . We will eventually show that the measure satisfies (24). For any ball L ∈ K(n), we attach a weight µ(L) defined recursively as follows.