Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions

Let ψ:R+→R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$$\end{document} be a non-increasing function. A pair (A,b),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A,{\textbf{b}}),$$\end{document} where A is a real m×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\times n$$\end{document} matrix and b∈Rm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{b}}\in \mathbb {R}^{m},$$\end{document} is said to be ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Dirichlet improvable, if the system ‖Aq+b-p‖m<ψ(T),‖q‖n<T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert A{\textbf{q}} +{\textbf{b}}-{\textbf{p}}\Vert ^m<\psi (T), \quad \Vert {\textbf{q}}\Vert ^n<T \end{aligned}$$\end{document}is solvable in p∈Zm,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{p}}\in \mathbb {Z}^{m},$$\end{document}q∈Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{q}}\in \mathbb {Z}^{n}$$\end{document} for all sufficiently large T where ‖·‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \cdot \Vert $$\end{document} denotes the supremum norm. For ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Dirichlet non-improvable sets, Kleinbock–Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim–Kim (2022) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff f-measure version of Kim–Kim (2022) results for ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}-Dirichlet non-improvable sets.


Introduction
To begin with, we recall the higher dimensional general form of Dirichlet's Theorem (1842).Let m, n be positive integers and let X mn denotes the space of real m × n matrices.
Theorem 1.1 (Dirichlet's Theorem).Given any A ∈ X mn and T > 1, there exist p ∈ Z m and q ∈ Z n \ {0} such that Aq − p m ≤ 1 T and q n < T. ( Here • denotes the supremum norm in R i , i ∈ N. Theorem 1.1 guarantees a nontrivial integer solution for all T. The standard application of (1.1) is the following corollary, guaranteeing that such a system is solvable for an unbounded set of T.
Corollary 1.2.For any A ∈ X mn there exist infinitely many integer vectors q ∈ Z n such that Aq − p m ≤ 1 q n for some p ∈ Z m . (1. 2) The two statements above give rise to two possible ways to pose Diophantine approximation problems sometimes referred to as uniform vs asymptotic approximation results: that is, looking for solvability of inequalities for all large enough T vs. for some arbitrarily large T. The rate of approximation given in above two statements works for all real matrices A ∈ X mn , which serves as the beginning of the metric theory of Diophantine approximation, a field concerned with understanding sets of A ∈ X mn satisfying similar conclusions but with the right hand sides replaced by faster decaying functions of T and q n respectively.Those sets are well studied in the asymptotic setup (1.2) long ago.Indeed, for a function ψ : R + → R + a matrix A ∈ X mn is said to be ψ-approximable if the inequality 1Aq − p m < ψ( q n ) for some p ∈ Z m (1.3) is satisfied for infinitely many integer vectors q ∈ Z n .As the set of ψ-approximable matrices is translation invariant under integer vectors, we can restrict attention to mn-dimensional unit cube [0, 1] mn .Then the set of ψ-approximable matrices in [0, 1] mn will be denoted by W m,n (ψ).
The following result gives the size of the set W m,n (ψ) in terms of Lebesgue measure.
Let us now briefly describe what is known in the setting of (1.1).For a non-increasing function ψ : [T 0 , ∞) → R + with T 0 > 1 fixed, consider the set D m,n (ψ) of ψ-Dirichlet improvable matrices consisting of A ∈ X mn such that the system Aq − p m ≤ ψ(T ) and q n < T has a nontrivial integer solution for all large enough T .Elements of the complementary set, D m,n (ψ) c , will be referred as ψ-Dirichlet non-improvable matrices.
With the notation ψ a (x) := x −a , (1.1) implies that D 1,1 (ψ 1 ) = R, and that for any m, n every matrix is ψ 1 -Dirichlet improvable.It was observed in [8] that for min(m, n) = 1 and in [15] for the general case, that the Lebesgue measure of D m,n (cψ 1 ) of the set cψ 1 -Dirichlet improvable matrices is zero for any c < 1, .
The theory of inhomogeneous Diophantine approximation starts by replacing the values of a system of linear forms Aq by those of a system of affine forms q → Aq + b where A ∈ X mn and b ∈ R m .Following [13], for a non-increasing function ψ : T large enough, one can find nonzero integer vectors q ∈ Z n and p ∈ Z m such that Aq + b − p m < ψ(T ) and q n < T.
(   by reducing the problem to the shrinking target problem on the space of grids in R m+n .The proof of their theorem is based on a correspondence between Diophantine approximation and homogenous dynamics. Theorem 1.4 (Kleinbock-Wadleigh, [14]).Given a non-increasing ψ, the set D m,n (ψ) has zero (respectively full) Lebesgue measure if and only if the series j 1 j 2 ψ(j) diverges (respectively converges).
Theorem 1.5 (Kim-Kim, [12]).Let ψ be non-increasing with lim T →∞ ψ(T ) = 0 and 0 ≤ s ≤ mn + m.Then In the same article Kim-Kim also provided the Hausdorff measure criterion for the singly metric case.
Theorem 1.6 (Kim-Kim, [12]).Let ψ be non-increasing with lim T →∞ ψ(T ) = 0. Then for any 0 ≤ s ≤ mn Naturally one can ask about the generalization of Theorem 1.5 and Theorem 1.6 in terms of f -dimensional Hausdorff measure.Recall that a natural generalization of the s-dimensional Hausdorff measure H s is the fdimensional Hausdorff measure H f where f is a dimension function, that is an increasing, continuous function f : R + → R + such that f (r) → 0 as r → 0.
In this article we extend the results of Kim-Kim [12] by establishing the zero-full law for the sets D m,n (ψ) and D b m,n (ψ) in terms of generalised f -dimensional Hausdorff measure.We obtain the following main results.Theorem 1.7.Let ψ be non-increasing and f be a dimension function with where mn + m − n < s < mn + m and α > 1 is some absolute constant independent of x and y and suppose that such that a(x) → s as x → 0. Further, let For the singly metric case we have the following result.
Theorem 1.8.Let ψ be non-increasing and f be a dimension function such that r −mn f (r) → ∞ as r → 0. Suppose that (1.5) -(1.7) holds and mn − n < s < mn.Then We remark that the conditions (1.5) and (1.6) are satisfied in a wide variety of cases, for example f (x) = x s log t (x) for some s > 0 and t ∈ R. Indeed, (1.5) follows since f (xy) = (xy) s log t (xy) ≍ x s y s log t (y) = x s f (y), and (1.6) follows since Acknowledgements.The research of first-named author was supported by the Australian Mathematical Society Lift-off Fellowship and the Australian Research Council Grant DP180100201.The second-named author was supported by a Royal Society University Research Fellowship, URF\R1\180649.The authors would like to thank the anonymous reviewer for the careful reading of the paper which has led to multiple improvements.

Preliminaries and auxiliary results
2.1.Hausdorff measure and dimension.Let f : R + → R + be a dimension function i.e. an increasing continuous function such that f (r) → 0 as r → 0 and let V be an arbitrary subset of R n .For ρ > 0, a ρ-cover for a set V is defined as a countable collection Note that H f ρ (V) is non-decreasing as ρ decreases and therefore approaches a limit as ρ → 0. Accordingly, the f -dimensional Hausdorff measure of V is defined as This limit could be zero or infinity, or take a finite positive value.
If f (r) = r s where s > 0, then H f is the s-dimensional Hausdorff measure and is represented by H s .It can be easily verified that Hausdorff measure is monotonic, that is, if E is contained in F then H s (E) ≤ H s (F ), countably sub-additive, and satisfies H s (∅) = 0.
The following property implies that there is a unique real point s at which the Hasudorff s-measure drops from infinity to zero (unless V is finite so that H s (V) is never infinite).The value taken by s at this discontinuity is referred to as the Hausdorff dimension of a set V and is defined as For establishing the convergent part of Theorem 1.7 and Theorem 1.8 we will apply the following Hausdorff measure version of the famous Borel-Cantelli lemma [4, Lemma 3.10]: Lemma 2.1.Let {B i } i≥1 be a sequence of measurable sets in R n and suppose that for some dimension function We will use the following principle known as Mass Distribution Principle [10, §4.1] for the divergent part of Theorem 1.7.
Lemma 2.2.Let µ be a probability measure supported on a subset V of R k .Suppose there are positive constants c > 0 and ε > 0 such that Theorem 2]).Let ψ : N → R + be any approximating function and let mn > 1.Let f and g : r → g(r) := r −m(n−1) f (r) be dimension functions such that r → r −mn f (r) is monotonic.Then 2.2.Ubiquitous systems.To prove the divergent parts of Theorem 1.8 we will use the ubiquity technique developed by Beresnevich, Dickinson, and Velani, see [3, §12.1].The idea and concept of ubiquity was originally formulated by Dodson, Rynne, and Vickers in [9] and coincided in part with the concept of 'regular systems' of Baker and Schmidt [2].Both have proven to be extremely useful in obtaining lower bounds for the Hausdorff dimension of limsup sets.The ubiquity framework in [3] provides a general and abstract approach for establishing the Lebesgue and Hausdorff measure of a large class of limsup sets.
Consider the mn-dimensional unit cube [0, 1] mn with the supremum norm ] mn : κ ∈ J} be a family of subsets, referred to as resonant sets R κ of [0, 1] mn indexed by an infinite, countable set J. Let β : J → R + : κ → β κ be a positive function on J i.e. the function β attaches the weight β κ to the set R κ .Next assume that the number of terms κ in J with β κ bounded above is always finite.Following the ideas from [3, §12.1] and [12] let us assume that the family R of resonant sets R κ consists of (m − 1)n-dimensional, rational hyperplanes and define the following notations.For a set S ⊆ [0, 1] mn , let The set Λ(Ψ) is a lim sup set; it consists of elements of [0, 1] mn which lie in infinitely many of the thickenings ∆(R κ , Ψ(β κ )).It is natural to call Ψ the approximating function as it governs the 'rate' at which the elements of [0, 1] mn must be approximated by resonant sets in order to lie in Λ(Ψ).Let us rewrite the set Λ(Ψ) in a way which brings its lim sup nature to the forefront.
Thus Λ(Ψ) is the set consisting elements of [0, 1] mn which lie in infinitely many ∆(Ψ, N ), that is, Next let ρ : R + → R + be a function with ρ(t) → 0 as t → ∞ and let Definition 2.4.Let B be an arbitrary ball in [0, 1] mn .Suppose there exist a function ρ and an absolute constant where | • | denotes the Lebesgue measure on [0, 1] mn .Then the pair (R, β) is said to be a 'local ubiquitous system' relative to ρ and the function ρ will be referred to as the 'ubiquitous function'.
A function h is said to be 2-regular if there exists a strictly positive constant λ < 1 such that for N sufficiently large The next theorem is a simplified version of Theorem 1 and Theorem 2 from [3].To state the result we define notions similar to those in [3].Note that with notions in [3], we have Ω := [0, 1] mn , the Lebesgue measure on [0, 1] mn is of type (M2) with δ = mn and γ = (m − 1)n and the local ubiquitous system (R, β) satisfies the intersection conditions with γ = (m−1)n (see [3, section 12.1]).Given that the Lebesgue measure is comparable with H δ − a simple consequence of (M2), we have the following combined version of Theorem 1 and Theorem 2 from [3].
Theorem 2.5.Suppose that (R, β) is a local ubiquitous system relative to ρ and that Ψ is an approximating function.Let f be a dimension function such that r −nm f (r) is monotonic, r −nm f (r) → ∞ as r → 0 and r −n(m−1) f (r) is increasing.Furthermore, suppose that ρ is 2-regular and Proof.With δ = mn, and γ = (m−1)n the function g in [3, Theorem 2] becomes g(r) which is same as the divergent sum condition in (2.5).
Note that as the dimension function r −nm f (r) → ∞ as r → 0 then H f (Ω) = ∞ and Theorem 2.5 leads to the same conclusion as Theorem 2 in [3].
2.3.Dirichlet improvability and homogenous dynamics.In one dimensional settings, continued fraction expansions have been useful in characterising ψ-Dirichlet improvable numbers [13].However this machinery is not applicable in higher dimensions.For general dimensions, building on ideas from [7] (also see [16]), a dynamical approach was proposed in [13], reformulating the homogenous approximation problem as a shrinking target problem and a similar approach was used in [14] to solve an analogous inhomogeneous problem.Following the ideas from [12,14] we will use the standard argument usually known as the 'Dani correspondence' which serves as a connection between Diophantine approximation and homogenous dynamics.In order to describe how Dirichlet-improvability is related to dynamics we will start by recalling the dynamics on space of grids.To describe this dynamical interpretation, let us fix some notation.
Denote by Y d the space of affine shifts of unimodular lattices in R d (i.e.space of unimodular grids).Clearly, For any t ∈ R, the flow of interest a t is given by the diagonal matrix Let us also denote by Following [14], define ∆ : Lemma 2.6 ( [17]).Let ψ : [T 0 , ∞) → R + be a continuous, non-increasing function where T 0 ∈ R + and m, n be positive integers.Then there exists a continuous function where t 0 := m m+n log T 0 − n m+n log ψ(T 0 ), such that (i) the function t → t + nz(t) is strictly increasing and unbounded; (ii) the function t → t − mz(t) is non-decreasing; (iii) ψ(e t+nz(t) ) = e −t+mz(t) for all t ≥ t 0 .
Note that, properties (i) and (ii) of Lemma 2.6 imply that any z = z ψ does not oscillate too wildly.Namely, z(s) − 1 m ≤ z(u) ≤ z(s) + 1 n whenever s ≤ u ≤ s + 1.The following lemma, which rephrases ψ-Dirichlet improvable properties of (A, b) ∈ X mn × R m as the statement about the orbit of Λ A,b in the dynamical space ( Y d , a t ), is the general version of the correspondence between the improvability of the inhomogeneous Dirichlet theorem and dynamics on Y d .Lemma 2.7 ( [14]).Let z = z ψ be the function associated to ψ by Lemma 2.6.Then (A, b) ∈ D m,n (ψ) if and only if ∆(a t Λ A,b ) < z ψ (t) for all sufficiently large t.
This equivalence is usually called the Dani Correspondence.In view of this interpretation a pair fails to be ψ-Dirichlet improvable if and only if the associated grid visits the target ∆ −1 ([z ψ (t), ∞)) at unbounded times t under the flow a t .Note that from the above lemma in the definitions D m,n (ψ } , the limsup is taken for real values t ∈ R. However to prove the convergent part, we need to use Hausdorff-Cantelli lemma (Lemma 2.1), therefore we will consider limsup sets taken for t ∈ N. Thus we will use the following definitions: there exists a non-zero positive constant C 0 such that The validity of these definitions can be observed by the fact that z ψ does not oscillate wildly by [14, Remark 3.3] and ∆ is uniformly continuous on the set ∆ −1 ([z, ∞)) for any z ∈ R, ([14, Lemma 2.1]).
Proof.The proof of this lemma uses ideas introduced in [17, Lemma 8.3] and [14].Using the monotonicity of ψ and [14, Remark 3.3], let us replace the sums with integrals dx and ∞ t0 e −(m+n)z(t) e (m+n)t f (e − (m+n)t mn )dt. Define Since ψ(e λ ) = e −P (λ) , letting log x = λ we have  Next we will use integration by parts to evaluate the integral in (3.3).
We are now in a position to prove the following statement.
(Note that H f +m represents the Hausdorff measure of a set when we take Proof.By Lemma 3.1, the assumption For each t ∈ N, let D t,1 , D t,2 , • • • , D t,pt be the balls of radius 1 2 e −( 1 m + 1 n )t covering Z t as in Proposition 3.2.Note that p t , the number of the balls, is not greater than Ke (m+n)(t−z ψ (t)) by Proposition 3.2.By applying Lemma 2.1 to the sequence of balls {D t,j } t∈N,1≤j≤pt , we have We prove the second statement by a similar argument.Proposition 3.2 implies that Z t ×[0, 1] m can be covered with Ke The convergence parts of Theorems 1.7 and 1.8 follow from this proposition.We will adapt a similar method as in [12].
Proof.We first prove the singly metric case i.e., the convergent part of Theorem 1.8.We claim that log(dλ d (a t Λ A )) ≥ ∆(a t Λ A,b ) for every b ∈ R m .Let v 1 , . . ., v d be linearly independent vectors satisfying v i ≤ Λ d (a t Λ A ) for 1 ≤ i ≤ d.The shortest vector of a t Λ A,b can be written as a form of d 1 α i v i for some −1 ≤ α i ≤ 1, so the length of the shortest vector is less than Similarly for the doubly metric case, together with the second statement of Proposition 3.
Following [12] we adopt some notations.Let W S,ε be the set of A ∈ [0, 1] mn such that there exists x A,S ∈ Z m \ {0} satisfying For fixed b ∈ R m , consider the set W b,S,ε of matrices A ∈ [0, 1] mn such that there exists x ∈ Z m \ {0} satisfying Further lim sup } is the set of matrices whose transposes are Ψ εapproximable.From here onwards we use a slightly different definition of Ψ ε -approximability; recall from footnote 1 where the inequality A t x Z < Ψ ε ( x ) is used instead of (1.3).Then, W b,ε can be considered as the set of matrices whose transposes are Ψ ε -approximable with solutions restricted on the set {x ∈ Z m : |b•x| Z > ε}.
Note that since Ψ ε (h) = d −1 ε ψ(dε −1 h), if we consider the term where in the second last line we used the change of variables x = ψ −1 (t −1 ), t = ψ(x) −1 and in the last line we used (1.5) and (1.7).Since it follows from (1.7) that (x . Therefore by using (1.5) we can write Using integration by parts which is finite and positive (since s > mn − n).Therefore the convergence of Hence the convergence of implies the convergence of other one since for T 0 large enough all summands in (4.2) are positive except the finite value Then, for any η > 1 there exists a probability measure µ on lim sup W S,ε satisfying the condition that for an arbitrary ball D of sufficiently small radius r(D) we have where the implied constant does not depend on D or η.
Proof.Note that lim sup S→∞ W which is the divergent assumption of Theorem 2.3 for W n,m (Ψ ε ).From the proof of Jarnik's Theorem in [1] and the construction of probability measure in [1, Section 5] we can obtain a probability measure µ on lim sup S→∞ W S,ε satisfying the above condition.
Let us prove the divergent part of Theorem 1.7.
Here we remark that since f (r) satisfies (1.5) and (1.6)  ) where e i denotes the vector with a 1 in the ith coordinate and 0's elsewhere.

. 4 )
Let D m,n (ψ) denote the set of ψ-Dirichlet improvable pairs in the unit cube [0, 1] mn+m .If the inhomogeneous vector b ∈ R m is fixed then let D b m,n (ψ) be the set of all A ∈ X mn such that (1.4) holds i.e. for a fixed b ∈ R m we have D b m,n (ψ) = {A ∈ X mn : (A, b) ∈ D m,n (ψ)}.The Lebesgue measure criterion for the set D m,n (ψ) i.e. doubly metric case has been proved by identified with the space of unimodular lattices in R d (i.e. the space of unimodular grids containing zero vector).Note that Γ d (respectively, Γ d ) is a lattice in G d (respectively, G d ).Denote by m Y d the Haar probability measure on Y d . 2λ

4 .
Proof of Theorem 1.7 and 1.8: the divergent case Recall that d = m+n and assume that ψ : [T 0 , ∞) → R + is a decreasing function satisfying lim T →∞ ψ(T ) = 0. Denote by • Z and |.| Z the distance to the nearest integer vector and number, respectively.Define the function ψ and x A,S < d −1 εS and let W S,ε := {(A, b) ∈ [0, 1] mn+m : A ∈ W S,ε and |b • x A,S | Z > ε}.