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

Abstract

Let $\psi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ be a non-increasing function. A pair $(A,{\textbf{b}}),$ where A is a real $m\times n$ matrix and ${\textbf{b}}\in \mathbb {R}^{m},$ is said to be $\psi$-Dirichlet improvable, if the system

$$\begin{aligned} \Vert A{\textbf{q}} +{\textbf{b}}-{\textbf{p}}\Vert ^m<\psi (T), \quad \Vert {\textbf{q}}\Vert ^n<T \end{aligned}$$

is solvable in ${\textbf{p}}\in \mathbb {Z}^{m},$ ${\textbf{q}}\in \mathbb {Z}^{n}$ for all sufficiently large T where $\Vert \cdot \Vert$ denotes the supremum norm. For $\psi$-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 $\psi$-Dirichlet non-improvable sets.

1 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 $m \times n$ matrices.

Theorem 1.1

(Dirichlet’s Theorem) Given any $A~ \in ~X^{mn}$ and $T>1,$ there exist ${\textbf{p}}\in \mathbb {Z}^{m}$ and ${\textbf{q}}\in \mathbb {Z}^n \setminus \{0\}$ such that

$$\begin{aligned} \Vert A{\textbf{q}}-{\textbf{p}}\Vert ^m \le \frac{1}{T} \text { and } \Vert {\textbf{q}}\Vert ^n<T. \end{aligned}$$

(1.1)

Here $\Vert \cdot \Vert$ denotes the supremum norm in $\mathbb {R}^{i},$ $i\in \mathbb N.$ 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 $A \in X^{mn}$ there exist infinitely many integer vectors ${\textbf{q}}\in \mathbb {Z}^n$ such that

$$\begin{aligned} \Vert A{\textbf{q}}-{\textbf{p}}\Vert ^m \le \frac{1}{\Vert {\textbf{q}}\Vert ^{n}} \text { for some } {\textbf{p}}\in \mathbb {Z}^{m}. \end{aligned}$$

(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 $A\in X^{mn},$ which serves as the beginning of the metric theory of Diophantine approximation, a field concerned with understanding sets of $A\in X^{mn}$ satisfying similar conclusions but with the right hand sides replaced by faster decaying functions of T and $\Vert {\textbf{q}}\Vert ^{n}$ respectively. Those sets are well studied in the asymptotic setup (1.2) long ago.

Indeed, for a function $\psi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ a matrix $A\in X^{mn}$ is said to be $\psi$-approximable if the inequality¹

$$\begin{aligned} \Vert A{\textbf{q}}-{\textbf{p}}\Vert ^m < \psi ({\Vert {\textbf{q}}\Vert ^{n}}) \text { for some } {\textbf{p}}\in \mathbb {Z}^{m} \end{aligned}$$

(1.3)

is satisfied for infinitely many integer vectors ${\textbf{q}}\in \mathbb {Z}^n.$ As the set of $\psi$-approximable matrices is translation invariant under integer vectors, we can restrict attention to mn-dimensional unit cube $[0,1]^{mn}.$ Then the set of $\psi$-approximable matrices in $[0,1]^{mn}$ will be denoted by $W_{m,n}(\psi ).$

The following result gives the size of the set $W_{m,n}(\psi )$ in terms of Lebesgue measure.

Theorem 1.3

(Khintchine–Groshev Theorem, [11]) Given a non-increasing $\psi$, the set $W_{m, n}(\psi )$ has zero (respectively full) Lebesgue measure if and only if the series $\sum _k\psi (k)$ converges (respectively, diverges).

Let us now briefly describe what is known in the setting of (1.1). For a non-increasing function $\psi :[T_{0},\infty )\rightarrow \mathbb {R}_{+}$ with $T_{0}>1$ fixed, consider the set $D_{m,n}(\psi )$ of $\psi$-Dirichlet improvable matrices consisting of $A\in X^{mn}$ such that the system

$$\begin{aligned} \Vert A{\textbf{q}}-{\textbf{p}}\Vert ^m \le \psi {(T)} \text { and } \Vert {\textbf{q}}\Vert ^n<T \end{aligned}$$

has a nontrivial integer solution for all large enough T. Elements of the complementary set, $D_{m,n}(\psi )^{c},$ will be referred as $\psi$-Dirichlet non-improvable matrices.

With the notation $\psi _{a}(x):=x^{-a},$ (1.1) implies that $D_{1, 1}(\psi _{1})=\mathbb {R},$ and that for any m, n every matrix is $\psi _{1}$-Dirichlet improvable. It was observed in [8] that for $\min (m,n)=1$ and in [17] for the general case, that the Lebesgue measure of $D_{m, n}(c\psi _{1})$ of the set $c\psi _{1}$-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 $A{\textbf{q}}$ by those of a system of affine forms ${\textbf{q}}\mapsto A{\textbf{q}}+{\textbf{b}}$ where $A\in X^{mn}$ and ${\textbf{b}}\in \mathbb {R}^{m}.$ Following [15], for a non-increasing function $\psi :[T_{0},\infty )\rightarrow \mathbb {R}_{+}$ a pair $(A,{\textbf{b}})\in X^{mn}\times \mathbb {R}^{m}$ is called $\psi$-Dirichlet improvable if for all T large enough, one can find nonzero integer vectors ${\textbf{q}}\in \mathbb {Z}^{n}$ and ${\textbf{p}}\in \mathbb {Z}^{m}$ such that

$$\begin{aligned} \Vert A{\textbf{q}}+{\textbf{b}}-{\textbf{p}}\Vert ^m<\psi (T)\quad \text {and}\quad \Vert {\textbf{q}}\Vert ^n<T. \end{aligned}$$

(1.4)

Let $\widehat{D}_{m, n}(\psi )$ denote the set of $\psi$-Dirichlet improvable pairs in the unit cube $[0,1]^{mn+m}$. If the inhomogeneous vector ${\textbf{b}}\in \mathbb {R}^m$ is fixed then let $\widehat{D}^{{\textbf{b}}}_{m, n}(\psi )$ be the set of all $A\in X^{mn}$ such that (1.4) holds i.e. for a fixed ${\textbf{b}}\in \mathbb {R}^m$ we have $\widehat{D}^{\textbf{b}}_{m, n}(\psi )=\{ A\in X^{mn}:(A,{\textbf{b}})\in \widehat{D}_{m, n}(\psi )\}.$

The Lebesgue measure criterion for the set $\widehat{D}_{m, n}(\psi )$ i.e. doubly metric case has been proved by Kleinbock–Wadleigh [16] by reducing the problem to the shrinking target problem on the space of grids in $\mathbb {R}^{m+n}$. The proof of their theorem is based on a correspondence between Diophantine approximation and homogenous dynamics.

Theorem 1.4

(Kleinbock–Wadleigh, [16]) Given a non-increasing $\psi$, the set $\widehat{D}_{m, n}(\psi )$ has zero (respectively full) Lebesgue measure if and only if the series $\sum _j\frac{1}{j^2\psi (j)}$ diverges (respectively converges).

Recently (2022), Kim–Kim [12] established the Hausdorff measure analogue of Theorem 1.4.

Theorem 1.5

(Kim–Kim, [12]) Let $\psi$ be non-increasing with $\lim _{T\rightarrow \infty }\psi (T)=0$ and $0 \le s\le mn+m.$ Then

$$\begin{aligned} \mathcal {H}^{s}( \widehat{D}_{m, n}(\psi )^{c} )= {\left\{ \begin{array}{ll} 0\ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn+m-s} \,<\,\infty ; \\ \mathcal {H}^{s}([0,1]^{mn+m}) \ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn+m-s} \,=\,\infty . \end{array}\right. } \end{aligned}$$

In the same article Kim–Kim also provided the Hausdorff measure criterion for the singly metric case.

Theorem 1.6

(Kim–Kim, [12]) Let $\psi$ be non-increasing with $\lim _{T\rightarrow \infty }\psi (T)=0.$ Then for any $0\le s\le mn$

$$\begin{aligned} \mathcal {H}^{s}( \widehat{D}^{{\textbf{b}}}_{m, n}(\psi ) ^{c} )= {\left\{ \begin{array}{ll} 0\ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn-s} \,<\,\infty ; \\ \mathcal {H}^{s}([0,1]^{mn}) &{} \text { if } \quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn-s} \,=\,\infty , \end{array}\right. } \end{aligned}$$

for every ${\textbf{b}}\in \mathbb {R}^{m}\setminus \mathbb {Z}^{m}.$

Naturally one can ask about the generalization of Theorems 1.5 and 1.6 in terms of f-dimensional Hausdorff measure. Recall that a natural generalization of the s-dimensional Hausdorff measure $\mathcal {H}^{s}$ is the f-dimensional Hausdorff measure $\mathcal {H}^{f}$ where f is a dimension function, that is an increasing, continuous function $f:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ such that $f(r)\rightarrow 0$ as $r\rightarrow 0.$

In this article we extend the results of Kim–Kim [12] by establishing the zero-full law for the sets $\widehat{D}_{m,n}(\psi )$ and $\widehat{D}^{{\textbf{b}}}_{m,n}(\psi )$ in terms of generalised f-dimensional Hausdorff measure. We obtain the following main results.

Theorem 1.7

Let $\psi$ be non-increasing and f be a dimension function with

$$\begin{aligned} f(xy)\asymp x^{s}f(y) \quad \forall \quad y^{\alpha }\le x\le y^{\frac{1}{\alpha }} \end{aligned}$$

(1.5)

where $mn+m-n<s<mn+m$ and $\alpha >1$ is some absolute constant independent of x and y and suppose that

$$\begin{aligned} f'(x)= a(x){\frac{f(x)}{x}} \end{aligned}$$

(1.6)

such that $a(x)\rightarrow s$ as $x\rightarrow 0$. Further, let

$$\begin{aligned} (q^{-\frac{m}{n}})^\alpha \le \psi (q)\le (q^{-\frac{m}{n}})^{\frac{1}{\alpha }}. \end{aligned}$$

(1.7)

Then

$$\begin{aligned} \mathcal {H}^{f}( \widehat{D}_{m, n}(\psi ) ^{c} )= {\left\{ \begin{array}{ll} 0\ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn+m} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) \,<\,\infty ; \\ \mathcal {H}^{f}([0,1]^{mn+m}) \ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn+m} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) \,=\,\infty . \end{array}\right. } \end{aligned}$$

For the singly metric case we have the following result.

Theorem 1.8

Let $\psi$ be non-increasing and f be a dimension function such that $r^{-mn} f(r) \rightarrow \infty$ as $r\rightarrow 0.$ Suppose that (1.5)–(1.7) holds and $mn-n<s<mn.$ Then

$$\begin{aligned} \mathcal {H}^{f}(\widehat{D}^{{\textbf{b}}}_{m, n}(\psi )^{c})= {\left\{ \begin{array}{ll} 0\ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) \,<\,\infty ; \\ \mathcal {H}^{f}([0,1]^{mn}) \ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) \,=\,\infty \end{array}\right. } \end{aligned}$$

for every ${\textbf{b}}\in \mathbb {R}^{m}\setminus \mathbb {Z}^{m}.$

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 $t\in \mathbb {R}$. Indeed, (1.5) follows since $f(xy) = (xy)^s \log ^t(xy) \asymp x^s y^s \log ^t(y) = x^s f(y)$, and (1.6) follows since

$$\begin{aligned} \frac{x f'(x)}{f(x)} = x \frac{d}{dx}[s\log (x) + t \log \log (x)] = x\left( \frac{s}{x} + \frac{t}{x\log (x)}\right) \rightarrow s \text { as }{x\rightarrow 0}. \end{aligned}$$

2 Preliminaries and auxiliary results

2.1 Hausdorff measure and dimension

Let $f:~{\mathbb {R}}_+\rightarrow ~{\mathbb {R}}_+$ be a dimension function i.e. an increasing continuous function such that $f(r)\rightarrow 0$ as $r\rightarrow ~0$ and let ${\mathcal {V}}$ be an arbitrary subset of $\mathbb {R}^n.$ For $\rho >0,$ a $\rho$-cover for a set ${\mathcal {V}}$ is defined as a countable collection $\{U_i\}_{i\ge 1}$ of sets in $\mathbb {R}^n$ with diameters $0<\textrm{diam} (U_i)\le \rho$ such that ${\mathcal {V}}\subseteq \bigcup _{i=1}^{\infty } U_i$. Then for each $\rho >0$ define

$$\begin{aligned} {\mathcal {H}}_{\rho }^f({\mathcal {V}})=\inf \left\{ \sum _{i=1}^{\infty } f\big (\textrm{diam}(U_i)\big ): \{U_{i}\} \text { is a }\rho -\text {cover of } {\mathcal {V}} \right\} . \end{aligned}$$

Note that ${\mathcal {H}}_\rho ^f({\mathcal {V}})$ is non-decreasing as $\rho$ decreases and therefore approaches a limit as $\rho \rightarrow 0$. Accordingly, the f-dimensional Hausdorff measure of ${\mathcal {V}}$ is defined as

$$\begin{aligned} {\mathcal {H}}^f({\mathcal {V}}):=\lim _{\rho \rightarrow 0}{\mathcal {H}}_\rho ^f({\mathcal {V}}). \end{aligned}$$

This limit could be zero or infinity, or take a finite positive value.

If $f(r)=r^s$ where $s > 0$, then ${\mathcal {H}}^{f}$ is the s-dimensional Hausdorff measure and is represented by ${\mathcal {H}}^s.$ It can be easily verified that Hausdorff measure is monotonic, that is, if E is contained in F then ${\mathcal {H}}^{s}(E)\le {\mathcal {H}}^{s}(F)$, countably sub-additive, and satisfies ${\mathcal {H}}^s(\emptyset )=0$.

The following property

$$\begin{aligned} {\mathcal {H}}^{s}({\mathcal {V}})<\infty \implies {\mathcal {H}}^{s'}({\mathcal {V}})=0 \quad {\text { if }} s'>s, \end{aligned}$$

implies that there is a unique real point s at which the Hasudorff s-measure drops from infinity to zero (unless ${\mathcal {V}}$ is finite so that ${\mathcal {H}}^{s}({\mathcal {V}})$ is never infinite). The value taken by s at this discontinuity is referred to as the Hausdorff dimension of a set ${\mathcal {V}}$ and is defined as

$$\begin{aligned} \dim _{\textrm{H}} {\mathcal {V}}:=\inf \{s > 0:\; {\mathcal {H}}^s({\mathcal {V}})=0\}. \end{aligned}$$

For establishing the convergent part of Theorems 1.7 and 1.8 we will apply the following Hausdorff measure version of the famous Borel–Cantelli lemma [6, Lemma 3.10]:

Lemma 2.1

Let $\{B_{i}\}_{i\ge 1}$ be a sequence of measurable sets in $\mathbb {R}^{n}$ and suppose that for some dimension function f,  $\sum _{i}f(\textrm{diam}(B_{i}))<\infty .$ Then $\mathcal {H}^{f}( \limsup _{i\rightarrow \infty } B_{i} )=0.$

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 $\mu$ be a probability measure supported on a subset ${\mathcal {V}}$ of $\mathbb {R}^{k}.$ Suppose there are positive constants $c>0$ and $\varepsilon >0$ such that

$$\begin{aligned} \mu (U)\le c f(\textrm{diam}(U)) \end{aligned}$$

for all sets U with $\textrm{diam}(U)\le \varepsilon .$ Then $\mathcal {H}^{f}({\mathcal {V}})\ge \mu ({\mathcal {V}})/c.$

Theorem 2.3

([1, Theorem 2]) Let $\psi :\mathbb N\rightarrow \mathbb {R}_{+}$ be any approximating function and let $mn>1.$ Let f and $g:r\rightarrow g(r):=r^{-m(n-1)}f(r)$ be dimension functions such that $r \mapsto r^{-mn}f(r)$ is monotonic. Then

$$\begin{aligned} \mathcal {H}^{f}( {W}_{m, n}(\psi ) )= {\left\{ \begin{array}{ll} 0\ &{} \text { if }\quad \sum \limits _{q=1}^{\infty } q^{m+n-1}g\left( \frac{\widehat{\psi }(q)}{q}\right) \,<\,\infty ; \\ \mathcal {H}^{f}([0,1]^{mn}) &{} \text { if } \quad \sum \limits _{q=1}^{\infty } q^{m+n-1} g\left( \frac{\widehat{\psi }(q)}{q}\right) \,=\,\infty , \end{array}\right. } \end{aligned}$$

where $\widehat{\psi }(q)=\psi (q^{n})^{\frac{1}{m}}.$

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 [5, §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 [5] 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 $\Vert \cdot \Vert$. Let $\mathcal {R}=\{ R_{\kappa }\subseteq [0,1]^{mn}:\kappa \in J\}$ be a family of subsets, referred to as resonant sets $R_{\kappa }$ of $[0,1]^{mn}$ indexed by an infinite, countable set J. Let $\beta :J\rightarrow \mathbb {R}_{+}:\kappa \mapsto \beta _{\kappa }$ be a positive function on J i.e. the function $\beta$ attaches the weight $\beta _{\kappa }$ to the set $R_{\kappa }.$ Next assume that the number of terms $\kappa$ in J with $\beta _{\kappa }$ bounded above is always finite. Following the ideas from [5, §12.1] and [12] let us assume that the family ${\mathcal {R}}$ of resonant sets $R_{\kappa }$ consists of $(m-1)n$-dimensional, rational hyperplanes and define the following notations. For a set $S\subseteq [0,1]^{mn},$ let

$$\begin{aligned} \Delta (S,r):=\{V\in [0,1]^{mn}:\text {dist}(V,S)<r\}, \end{aligned}$$

where $\text {dist}(V,S):=\inf \{ \Vert V-Y \Vert :Y\in S \}.$ Fix a decreasing function $\Psi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ let

$$\begin{aligned} \Lambda (\Psi )=\{ V\in [0,1]^{mn}:V \in \Delta (R_{\kappa },\Psi (\beta _{\kappa })) \text { for i.m. } \kappa \in J \} \end{aligned}$$

(2.1)

The set $\Lambda (\Psi )$ is a lim sup set; it consists of elements of $[0,1]^{mn}$ which lie in infinitely many of the thickenings $\Delta (R_{\kappa },\Psi (\beta _{\kappa })).$ It is natural to call $\Psi$ 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 $\Lambda (\Psi )$. Let us rewrite the set $\Lambda (\Psi )$ in a way which brings its lim sup nature to the forefront.

For $N\in \mathbb N,$ let

$$\begin{aligned} \Delta (\Psi ,N):=\bigcup _{\kappa \in J:2^{N-1}<\beta _{\kappa }\le 2^{N}}\Delta (R_{\kappa },\Psi (\beta _{\kappa })). \end{aligned}$$

Thus $\Lambda (\Psi )$ is the set consisting elements of $[0,1]^{mn}$ which lie in infinitely many $\Delta (\Psi ,N),$ that is,

$$\begin{aligned} \Lambda (\Psi ):=\limsup _{N\rightarrow \infty }\Delta (\Psi ,N) \end{aligned}$$

(2.2)

Next let $\rho :\mathbb {R}_{+} \rightarrow \mathbb {R}_{+}$ be a function with $\rho (t)\rightarrow 0$ as $t\rightarrow \infty$ and let

$$\begin{aligned} \Delta (\rho ,N):=\bigcup _{\kappa \in J:2^{N-1}<\beta _{\kappa }\le 2^{N}}\Delta (R_{\kappa },\rho (\beta _{\kappa })). \end{aligned}$$

(2.3)

Definition 2.4

Let B be an arbitrary ball in $[0,1]^{mn}.$ Suppose there exist a function $\rho$ and an absolute constant $\kappa >0$ such that

$$\begin{aligned} |B\cap \Delta (\rho ,N)|\ge \kappa |B| \text { for } N\ge N_{0}(B), \end{aligned}$$

(2.4)

where $|\cdot |$ denotes the Lebesgue measure on $[0,1]^{mn}.$ Then the pair $({\mathcal {R}},\beta )$ is said to be a ‘local ubiquitous system’ relative to $\rho$ and the function $\rho$ will be referred to as the ‘ubiquitous function’.

A function h is said to be 2-regular if there exists a strictly positive constant $\lambda <1$ such that for N sufficiently large

$$\begin{aligned} h(2^{N+1})\le \lambda h(2^{N}). \end{aligned}$$

The next theorem is a simplified version of Theorem 1 and Theorem 2 from [5]. To state the result we define notions similar to those in [5]. Note that with notions in [5], we have $\Omega :=[0,1]^{mn}$, the Lebesgue measure on $[0,1]^{mn}$ is of type (M2) with $\delta =mn$ and $\gamma =(m-1)n$ and the local ubiquitous system $({\mathcal {R}},\beta )$ satisfies the intersection conditions with $\gamma =(m-1)n$ (see [5, section 12.1]). Given that the Lebesgue measure is comparable with $\mathcal {H}^{\delta }-$ a simple consequence of (M2), we have the following combined version of Theorem 1 and Theorem 2 from [5].

Theorem 2.5

Suppose that $({\mathcal {R}},\beta )$ is a local ubiquitous system relative to $\rho$ and that $\Psi$ is an approximating function. Let f be a dimension function such that $r^{-nm}f(r)$ is monotonic, $r^{-nm}f(r)\rightarrow \infty$ as $r\rightarrow 0$ and $r^{-n(m-1)}f(r)$ is increasing. Furthermore, suppose that $\rho$ is 2-regular and

$$\begin{aligned} \sum ^{\infty }_{n=1}\frac{(\Psi (2^{N}))^{-n(m-1)}f(\Psi (2^{N}))}{\rho (2^{N})^{n}}=\infty . \end{aligned}$$

(2.5)

Then

$$\begin{aligned} \mathcal {H}^{f}( \Lambda (\Psi ) )= \mathcal {H}^{f}( [0,1]^{mn}). \end{aligned}$$

(2.6)

Proof

With $\delta =mn,$ and $\gamma =(m-1)n$ the function g in [5, Theorem 2] becomes $g(r):=f(\Psi (r))\Psi (r)^{-\gamma }\rho (r)^{\gamma - \delta }=f(\Psi (r))\Psi (r)^{-(m-1)n}\rho (r)^{-n}.$ Also $\rho$ is 2-regular, thus from [5, Theorem 2] it follows that

$$\begin{aligned} \mathcal {H}^{f}( \Lambda (\Psi ) )=\infty \quad \text { if } \quad \sum ^{\infty }_{n=1}g(2^{N})=\infty , \end{aligned}$$

which is same as the divergent sum condition in (2.5).

Note that as the dimension function $r^{-nm}f(r) \rightarrow \infty$ as $r\rightarrow 0$ then $H^f (\Omega )=\infty$ and Theorem 2.5 leads to the same conclusion as Theorem 2 in [5]. $\square$

2.3 Dirichlet improvability and homogenous dynamics

In one dimensional settings, continued fraction expansions have been useful in characterising $\psi$-Dirichlet improvable numbers [15]. However this machinery is not applicable in higher dimensions. For general dimensions, building on ideas from [7] (also see [13]), a dynamical approach was proposed in [15], reformulating the homogenous approximation problem as a shrinking target problem and a similar approach was used in [16] to solve an analogous inhomogeneous problem. Following the ideas from [12, 16] 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.

Fix $d=m+n.$ Let

$$\begin{aligned} G_{d}=SL_{d}(\mathbb {R}) \text { and } \widehat{G}_{d}=ASL_{d}(\mathbb {R})=G_{d} \rtimes \mathbb {R}^{d} \end{aligned}$$

and put

$$\begin{aligned} \Gamma _{d}=SL_{d}(\mathbb {Z}) \text { and } \widehat{\Gamma }_{d}=ASL_{d}(\mathbb {Z})=\Gamma _{d} \rtimes \mathbb {Z}^{d}. \end{aligned}$$

Denote by $\widehat{Y}_{d}$ the space of affine shifts of unimodular lattices in $\mathbb {R}^{d}$ (i.e. space of unimodular grids). Clearly, $\widehat{Y}_{d}$ is canonically identified with $\widehat{G}_{d}/ \widehat{\Gamma }_{d}$ via

$$\begin{aligned} <g,{\textbf{w}}>\widehat{\Gamma }_{d}\in \widehat{G}_{d}/ \widehat{\Gamma }_{d} \longleftrightarrow g\mathbb {Z}^{d}+{\textbf{w}}\in \widehat{Y}_{d} \end{aligned}$$

where $<g,{\textbf{w}}>$ is an element of $\widehat{G}_{d}$ such that $g\in G_{d}$ and ${\textbf{w}}\in \mathbb {R}^{d}.$ Similarly, $Y_{d}:=G_{d}/{\Gamma }_{d}$ is identified with the space of unimodular lattices in $\mathbb {R}^{d}$ (i.e. the space of unimodular grids containing zero vector). Note that ${\Gamma }_{d}$ (respectively, $\widehat{\Gamma }_{d}$) is a lattice in $G_{d}$ (respectively, $\widehat{G}_{d}$). Denote by $m_{Y_{d}}$ the Haar probability measure on $Y_{d}.$ For any $t\in \mathbb {R},$ the flow of interest $a_{t}$ is given by the diagonal matrix

$$\begin{aligned} a_{t}:=\text {diag}(e^{t/m},\cdots ,e^{t/m},e^{-t/n},\cdots , e^{-t/n}). \end{aligned}$$

Let

$$\begin{aligned}{} & {} u_{A}:=\left( \begin{array}{cc} I_{m} &{} A \\ 0 &{} I_{n} \end{array} \right) \in G_{d},\\{} & {} u_{A,{\textbf{b}}}:=\left<\left( \begin{array}{cc} I_{m} &{} A \\ 0 &{} I_{n} \end{array} \right) , \left( \begin{array}{c} {\textbf{b}}\\ 0 \end{array} \right) \right>\in {\widehat{G}}_{d} \end{aligned}$$

for $A\in X^{mn}$ and $(A,{\textbf{b}})\in X^{mn}\times \mathbb {R}^{m}.$ Let us also denote by

$$\begin{aligned} \Lambda _{A}:=u_{A}\mathbb {Z}^{d}\in Y_{d} \text { and }\Lambda _{A,{\textbf{b}}}:=u_{A,{\textbf{b}}}\mathbb {Z}^{d}\in \widehat{Y}_{d}, \end{aligned}$$

where $u_{A,{\textbf{b}}}\mathbb {Z}^{d}=\left\{ \left( \begin{array}{cc} A{\textbf{q}}+{\textbf{b}}-{\textbf{p}}\\ {\textbf{q}}\end{array} \right) :{\textbf{p}}\in \mathbb {Z}^{m},{\textbf{q}}\in \mathbb {Z}^{n}\right\} .$

Following [16], define $\Delta :\widehat{Y}_{d}\rightarrow [-\infty ,+\infty )$ by

$$\begin{aligned} \Delta (\Lambda ):=\log \inf _{{\textbf{v}}\in \Lambda }\Vert {\textbf{v}}\Vert . \end{aligned}$$

Lemma 2.6

([14]) Let $\psi :[T_{0},\infty )\rightarrow \mathbb {R}_{+}$ be a continuous, non-increasing function where $T_{0}\in \mathbb {R}_{+}$ and m, n be positive integers. Then there exists a continuous function

$$\begin{aligned} z=z_{\psi }:[t_{0},\infty )\rightarrow \mathbb {R}, \end{aligned}$$

where $t_{0}:=\frac{m}{m+n}\log T_{0}-\frac{n}{m+n}\log \psi (T_{0}),$ such that

• (i) the function $t\mapsto t+nz(t)$ is strictly increasing and unbounded;

• (ii) the function $t\mapsto t-mz(t)$ is non-decreasing;

• (iii) $\psi (e^{t+nz(t)})=e^{-t+mz(t)}$ for all $t\ge t_{0}.$

Note that, properties (i) and (ii) of Lemma 2.6 imply that any $z=z_{\psi }$ does not oscillate too wildly. Namely, $z(s)-\frac{1}{m}\le z(u)\le z(s)+\frac{1}{n} \text { whenever } s\le u\le s+1.$

The following lemma, which rephrases $\psi$-Dirichlet improvable properties of $(A,{\textbf{b}})\in X^{mn}\times \mathbb {R}^{m}$ as the statement about the orbit of $\Lambda _{A,{\textbf{b}}}$ in the dynamical space $(\widehat{Y}_{d},a_{t})$, is the general version of the correspondence between the improvability of the inhomogeneous Dirichlet theorem and dynamics on $\widehat{Y}_{d}.$

Lemma 2.7

([16]) Let $z=z_{\psi }$ be the function associated to $\psi$ by Lemma 2.6. Then $(A,{\textbf{b}})\in \widehat{D}_{m,n}(\psi )$ if and only if $\Delta (a_{t}\Lambda _{A,{\textbf{b}}})<z_{\psi }(t)$ for all sufficiently large t.

This equivalence is usually called the Dani Correspondence. In view of this interpretation a pair fails to be $\psi$-Dirichlet improvable if and only if the associated grid visits the target $\Delta ^{-1}([z_{\psi }(t),\infty ))$ at unbounded times t under the flow $a_{t}.$ Note that from the above lemma in the definitions $\widehat{D}_{m,n}(\psi )^{c}=\limsup \limits _{t\rightarrow \infty } \left\{ (A,{\textbf{b}}):\Delta (a_{t}\Lambda _{A,{\textbf{b}}})\ge z_{\psi }(t)\right\}$ and $\widehat{D}^{{\textbf{b}}}_{m,n}(\psi )^{c}=\limsup \limits _ {t\rightarrow \infty }\left\{ A:\Delta (a_{t}\Lambda _{A,{\textbf{b}}})\ge z_{\psi }(t)\right\} ,$ the limsup is taken for real values $t\in \mathbb {R}$. However to prove the convergent part, we need to use Hausdorff–Cantelli lemma (Lemma 2.1), therefore we will consider limsup sets taken for $t\in \mathbb N.$ Thus we will use the following definitions: there exists a non-zero positive constant $C_{0}$ such that

$$\begin{aligned}&\widehat{D}_{m,n}(\psi )^{c}\subseteq \limsup \limits _{t\rightarrow \infty ,t\in \mathbb N}\left\{ (A,{\textbf{b}}):\Delta (a_{t}\Lambda _{A,{\textbf{b}}})\ge z_{\psi }(t)-C_{0}\right\} , \end{aligned}$$

(2.7)

$$\begin{aligned}&\widehat{D}^{{\textbf{b}}}_{m, n}(\psi )^{c}\subseteq \limsup \limits _ {t\rightarrow \infty ,t\in \mathbb N}\left\{ A:\Delta (a_{t}\Lambda _{A,{\textbf{b}}})\ge z_{\psi }(t)-C_{0}\right\} . \end{aligned}$$

(2.8)

The validity of these definitions can be observed by the fact that $z_{\psi }$ does not oscillate wildly by [16, Remark 3.3] and $\Delta$ is uniformly continuous on the set $\Delta ^{-1}([z,\infty ))$ for any $z\in \mathbb {R},$ ([16, Lemma 2.1]).

3 Proof of Theorems 1.7 and 1.8: the convergent case

Lemma 3.1

Let $\psi :[T_{0},\infty )\rightarrow {\mathbb {R}_{+}}$ be a non-increasing function, and let $z=z_{\psi }$ be the function associated to $\psi$ by Lemma 2.6. Let f be a dimension function satisfying (1.5) and (1.6) where $nm-n<s\le nm.$ Also suppose that (1.7) holds. Then we have

$$\begin{aligned}{} & {} \sum \limits _{q=\lceil {T_{0}}\rceil }^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right)<\infty \\{} & {} \quad \iff \sum \limits _{t=\lceil {t_{0}}\rceil }^{\infty }e^{-(m+n)z(t)}e^{(m+n)t}f(e^{-\frac{(m+n)t}{mn}})<\infty . \end{aligned}$$

Proof

The proof of this lemma uses ideas introduced in [14, Lemma 8.3] and [16]. Using the monotonicity of $\psi$ and [16, Remark 3.3], let us replace the sums with integrals

$$\begin{aligned} \int _{T_{0}}^{\infty } \frac{1}{\psi (x)x^{2}} \left( \frac{ x^{\frac{1}{n}}}{\psi (x)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (x)^\frac{1}{m}}{ x^{\frac{1}{n}}}\right) dx \text { and } \int _{t_{0}}^{\infty }e^{-(m+n)z(t)}e^{(m+n)t}f(e^{-\frac{(m+n)t}{mn}})dt. \end{aligned}$$

Define

$$\begin{aligned} P:=-\log \circ \psi \circ \exp :[T_{0},\infty )\rightarrow \mathbb {R}\text { and } \lambda (t):=t+nz(t). \end{aligned}$$

Since $\psi (e^{\lambda })=e^{-P(\lambda )}$, letting $\log x=\lambda$ we have

$$\begin{aligned} \int _{T_{0}}^{\infty } \frac{1}{\psi (x)x^{2}} \left( \frac{ x^{\frac{1}{n}}}{\psi (x)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (x)^\frac{1}{m}}{ x^{\frac{1}{n}}}\right) dx&= \int _{\log T_{0}}^{\infty } \frac{1}{\psi (e^{\lambda })e^{2\lambda }} \left( \frac{ e^{\lambda m}}{\psi (e^{\lambda })^{n}}\right) f\left( \frac{\psi (e^{\lambda })^\frac{1}{m}}{ e^{\frac{\lambda }{n}}}\right) e^{\lambda }d\lambda \nonumber \\&= \int _{\log T_{0}}^{\infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} ) d\lambda . \end{aligned}$$

(3.1)

Using $P(\lambda (t))=t-mz(t),$ we have

$$\begin{aligned}&\int _{t_{0}}^{\infty }e^{-(m+n)z(t)}e^{(m+n)t}f(e^{-\frac{(m+n)t}{mn}})dt\nonumber \\&\quad = \int _{t_{0}}^{\infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} ) d\left[ \frac{m}{m+n}\lambda +\frac{n}{m+n}P(\lambda )\right] \nonumber \\&\quad =\frac{m}{m+n} \int _{\log T_{0}}^{\infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} ) d\lambda \nonumber \\&\qquad + \frac{n}{m+n}\int _{\log T_{0}}^{\infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} ) d(P(\lambda )) . \end{aligned}$$

(3.2)

The term in the last line can be expressed by

$$\begin{aligned}&\frac{n}{m+n}\int _{\log {T}_{0}}^{\infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m}} e^{\frac{-\lambda }{n}} ) d(P(\lambda ))\nonumber \\&\quad \asymp \frac{n}{m+n} \int _{\log {T}_{0}}^{\infty } e^{(m-1)\lambda }f(e^{\frac{-\lambda }{n}})e^{(1+n)P(\lambda )}e^{-s\frac{P(\lambda )}{m} } d(P(\lambda )), \nonumber \\&\quad =\frac{n}{m+n}\Big (1+\frac{mn-s}{m}\Big )^{-1} \int _{\log {T}_{0}}^{\infty } e^{(m-1)\lambda }f(e^{\frac{-\lambda }{n}}) d(e^{((1+n) -\frac{s}{m})P(\lambda )}), \end{aligned}$$

(3.3)

the second last equation follows from (1.5) and (1.7). Since by using (1.7) and the fact that $\psi (e^{\lambda })=e^{-P(\lambda )}$ we obtain the condition

$$\begin{aligned} (e^{\frac{-\lambda }{n}})^{\alpha }\le e^{\frac{-P(\lambda )}{m} } \le (e^{\frac{-\lambda }{n}})^{\frac{1}{\alpha }}, \end{aligned}$$

therefore by using (1.5) we can write

$$\begin{aligned} f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} )\asymp e^{-s\frac{P(\lambda )}{m} } f( e^{\frac{-\lambda }{n}}). \end{aligned}$$

Next we will use integration by parts to evaluate the integral in (3.3).

$$\begin{aligned}&\int _{\log T_{0}}^{\infty } e^{(m-1)\lambda }f( e^{\frac{-\lambda }{n}} ) d(e^{( (1+n) -\frac{s}{m} )P(\lambda )})\nonumber \\&\quad =- \int _{\log T_{0}}^{\infty } [(m-1) e^{(m-1)\lambda }f(e^{-\frac{\lambda }{n}}) - \frac{1}{n} e^{(m-1)\lambda } e^{\frac{-\lambda }{n}} f'(e^{\frac{-\lambda }{n}}) ] e^{((1+n)-\frac{s}{m})P(\lambda )}d\lambda \nonumber \\&\qquad +e^{(m-1)\lambda }f(e^{-\frac{\lambda }{n}}) e^{((1+n)-\frac{s}{m})P(\lambda )}\Big |^{\infty }_{\log T_{0}}\nonumber \\&\quad = \int _{\log T_{0}}^{\infty }\left( (1-m) e^{(m-1)\lambda } f(e^{-\frac{\lambda }{n}})e^{((1+n)-\frac{s}{m})P(\lambda )} + \frac{1}{n} e^{(m-1)\lambda }f'(e^{-\frac{\lambda }{n}})e^{-\frac{\lambda }{n} } e^{((1+n)-\frac{s}{m})P(\lambda )}\right) d\lambda \nonumber \\&\qquad +\lim _{\lambda \rightarrow \infty } e^{(m-1)\lambda } f(e^{-\frac{\lambda }{n}})e^{((1+n)-\frac{s}{m})P(\lambda )}-T^{m-1}_{0}f(T^{-\frac{1}{n}}_{0})\psi (T_{0})^{-((1+n)-\frac{s}{m})},\nonumber \\&\quad \text { by } (1.6), \text { we have } f'(e^{-\frac{\lambda }{n}})= a(e^{-\frac{\lambda }{n}})\frac{f(e^{-\frac{\lambda }{n}})}{e^{-\frac{\lambda }{n}}},\nonumber \\&\quad = \int _{\log T_{0}}^{\infty } \left( 1-m + \frac{1}{n} a(e^{-\frac{\lambda }{n}}) \right) e^{(m-1)\lambda } f(e^{-\frac{\lambda }{n}})e^{((1+n)-\frac{s}{m})P(\lambda )} d\lambda \nonumber \\&\qquad +\lim _{\lambda \rightarrow \infty } e^{(m-1)\lambda } f(e^{-\frac{\lambda }{n}})e^{((1+n)-\frac{s}{m})P(\lambda )}-T^{m-1}_{0}f(T^{-\frac{1}{n}}_{0})\psi (T_{0})^{-((1+n)-\frac{s}{m})}\nonumber \\&\qquad \asymp \int _{\log T_{0}}^{\infty } (1-m + \frac{1}{n} a(e^{-\frac{\lambda }{n}}))e^{(m-1)\lambda } e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} ) d\lambda \end{aligned}$$

(3.4)

$$\begin{aligned}&\qquad +\lim _{\lambda \rightarrow \infty } e^{(m-1)\lambda } e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} e^{\frac{-\lambda }{n}} )} -T^{m-1}_{0}f(T^{-\frac{1}{n}}_{0})\psi (T_{0})^{-((1+n)-\frac{s}{m})}. \end{aligned}$$

(3.5)

Note that as $\lambda \rightarrow \infty ,$ $e^{-\frac{\lambda }{n}}\rightarrow 0$ thus by assumption $a(e^{-\frac{\lambda }{n}})\rightarrow s$ and therefore

$$\begin{aligned} \left( 1-\frac{mn-a(e^{-\frac{\lambda }{n}})}{n}\right) \rightarrow \left( 1-\frac{mn-s}{n}\right) , \end{aligned}$$

which is finite and positive for $T_{0}$ large enough (since $s > nm - n$). Observe that

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}}) =0 \end{aligned}$$

if the integral

$$\begin{aligned} \int _{\log T_{0}}^{\infty } e^{(m-1)\lambda }e^{(1+n)P(\lambda )}f(e^{\frac{-P(\lambda )}{m} } e^{\frac{-\lambda }{n}} ) d\lambda \end{aligned}$$

converges. Thus the convergence of

$$\begin{aligned} \int _{T_{0}}^{\infty } \frac{1}{\psi (x)x^{2}} \left( \frac{ x^{\frac{1}{n}}}{\psi (x)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (x)^\frac{1}{m}}{ x^{\frac{1}{n}}}\right) dx \text { or } \int _{t_{0}}^{\infty }e^{-(m+n)z(t)}e^{(m+n)t}f(e^{-\frac{(m+n)t}{mn}})dt \end{aligned}$$

implies the convergence of other since all summands are positive except the finite value $-T^{m-1}_{0}f(T^{-\frac{1}{n}}_{0})\psi (T_{0})^{-((1+n)-\frac{s}{m})}.$ $\square$

In order to apply the Hausdorff–Cantelli lemma (Lemma 2.1) we need a sequence of coverings for the sets $\widehat{D}_{m,n}(\psi )^{c}$ and $\widehat{D}^{{\textbf{b}}}_{m,n}(\psi )^{c}.$ Recall that we are considering the supremum norm $\Vert \cdot \Vert$ on $[0,1]^{mn}$ and let $\lambda _{j}(\Lambda )$ denote the j-th successive minimum of a lattice $\Lambda \subseteq \mathbb {R}^{d}$ i.e. the infimum of $\lambda$ such that the ball $B^{\mathbb {R}^{d}}_{\lambda }(0)$ contains j independent vectors of $\Lambda .$ Then:

Proposition 3.2

(Kim–Kim, [12, Proposition 3.6]) Let $C_{0}$ be the same constant as in (2.7) and (2.8). For $t\in \mathbb N,$ let $Z_{t}:=\{ A\in [0,1]^{mn}:\log (d\lambda _{d}(a_{t}\Lambda _{A}))\ge z_{\psi }(t)-C_{0} \}.$ Then $Z_{t}$ can be covered with $Ke^{(m+n)(t-z_{\psi }(t))}$ balls in $X^{mn}=M_{m,n}(\mathbb {R})$ of radius $\frac{1}{2}e^{-(\frac{1}{m}+\frac{1}{n})t}$ for a constant $K>0$ not depending on t.

We are now in a position to prove the following statement.

Proposition 3.3

Let $mn-n< s\le mn.$ If $\sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) <\infty ,$ then $\mathcal {H}^{f}( \limsup \limits _{t\rightarrow \infty } Z_{t} )=0$ and $\mathcal {H}^{f+m}( \limsup \limits _{t\rightarrow \infty } Z_{t} \times [0,1]^{m})=0.$ (Note that ${\mathcal {H}}^{f+m}$ represents the Hausdorff measure of a set when we take $(f+m)(r)=r^{m}f(r)).$

Proof

By Lemma 3.1, the assumption $\sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) <\infty$ is equivalent to

$$\begin{aligned}&\sum \limits _{t=1}^{\infty }e^{-(m+n)(z(t)-t)}f(e^{-(\frac{1}{m}+\frac{1}{n})t})<\infty . \end{aligned}$$

(3.6)

For each $t\in \mathbb N,$ let $D_{t,{1}}, D_{t,{2}},\cdots ,D_{t,{p_{t}}}$ be the balls of radius $\frac{1}{2}e^{-(\frac{1}{m}+\frac{1}{n})t}$ covering $Z_{t}$ as in Proposition 3.2. Note that $p_{t},$ the number of the balls, is not greater than $Ke^{(m+n)(t-z_{\psi }(t))}$ by Proposition 3.2. By applying Lemma 2.1 to the sequence of balls $\{D_{t,{j}}\}_{t\in \mathbb N,1\le j \le p_{t}},$ we have $\mathcal {H}^{f}( \limsup \limits _{t\rightarrow \infty } Z_{t} )\le \mathcal {H}^{f}( \limsup \limits _{t\rightarrow \infty } D_{t_{j}} )=0.$

We prove the second statement by a similar argument. Proposition 3.2 implies that $Z_{t}\times [0,1]^{m}$ can be covered with $Ke^{\frac{m+n}{n}t}e^{(m+n)(t-z_{\psi }(t))}$ balls of radius $\frac{1}{2}e^{-(\frac{1}{m}+\frac{1}{n})t}.$ Applying Lemma 2.1 again, we have $\mathcal {H}^{f +m}( \limsup \limits _{t\rightarrow \infty } Z_{t} \times [0,1]^{m})=0.$ $\square$

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 $\log (d\lambda _{d}(a_{t}\Lambda _{A}))\ge \Delta (a_{t}\Lambda _{A,{\textbf{b}}})$ for every ${\textbf{b}}\in \mathbb {R}^{m}.$ Let $v_{1},\dots ,v_{d}$ be linearly independent vectors satisfying $\Vert v_{i}\Vert \le \Lambda _{d}(a_{t}\Lambda _{A})$ for $1\le i \le d.$ The shortest vector of $a_{t}\Lambda _{A,{\textbf{b}}}$ can be written as a form of $\sum _{1}^{d}\alpha _{i}v_{i}$ for some $-1\le \alpha _{i}\le 1$, so the length of the shortest vector is less than $\sum _{1}^{d}\Vert v_{i}\Vert .$ Thus, $\Delta (a_{t}\Lambda _{A,{\textbf{b}}})\le \log \sum \limits _{i}^{d}\Vert v_{i}\Vert \le \log (d\lambda _{d}(a_{t}\Lambda _{A})).$ This implies $\widehat{D}^{{\textbf{b}}}_{m, n}(\psi )^{c}\subseteq \limsup \limits _{t\rightarrow \infty } \{ A\in [0,1]^{mn}:\Delta (a_{t}\Lambda _{A,{\textbf{b}}})\ge z_{\psi }(t)-C_{0}\}\subseteq \limsup \limits _{t\rightarrow \infty } Z_{t}$ by Lemma 2.7 and Proposition 3.3, thus we obtain $\mathcal {H}^{f}( \hat{D}^{{\textbf{b}}}_{m, n}(\psi )^{c} )\le \mathcal {H}^{f}( \limsup \limits _{t\rightarrow \infty } Z_{t} )=0$.

Similarly for the doubly metric case, together with the second statement of Proposition 3.3, $\widehat{D}_{m, n}(\psi )^{c} \subseteq \limsup \limits _{t\rightarrow \infty } \{ (A,{\textbf{b}})\in [0,1]^{mn+m}: \Delta (a_{t}\Lambda _{A,{\textbf{b}}})\ge z_{\psi }(t)-C_{0}\}\subseteq \limsup \limits _{t\rightarrow \infty }Z_{t}\times [0,1]^{m}$ provides the proof of the convergent part of Theorem 1.7. $\square$

4 Proof of Theorems 1.7 and 1.8: the divergent case

Recall that $d=m+n$ and assume that $\psi :[T_{0},\infty )\rightarrow \mathbb {R}_{+}$ is a decreasing function satisfying $\lim _{T\rightarrow \infty }\psi (T)=0.$ Denote by $\Vert \cdot \Vert _{\mathbb {Z}}$ and $|.|_{\mathbb {Z}}$ the distance to the nearest integer vector and number, respectively. Define the function $\widetilde{\psi }:[S_{0},\infty )\rightarrow \mathbb {R}_{+}$ by

$$\begin{aligned} \widetilde{\psi }(S)=(\psi ^{-1}(S^{-m})))^{\frac{-1}{n}} \end{aligned}$$

where $S_{0}=\psi (T_{0})^{\frac{-1}{m}}.$ The next lemma associates $\psi$-Dirichlet non-improvability with $\widetilde{\psi }$-approximability via a transference lemma as follows.

Lemma 4.1

[12, Lemma 4.2] Given $(A,{\textbf{b}})\in X^{mn}\times \mathbb {R}^{m},$ if the system

$$\begin{aligned} \Vert A^{t}{\textbf{x}}\Vert _{\mathbb {Z}}<d^{-1}|{\textbf{b}}\cdot {\textbf{x}}|_{\mathbb {Z}}\widetilde{\psi }(S) \text { and } \Vert {\textbf{x}}\Vert <d^{-1}|{\textbf{b}}\cdot {\textbf{x}}|_{\mathbb {Z}}S \end{aligned}$$

has a nontrivial solution ${\textbf{x}}\in \mathbb {Z}^{m}$ for an unbounded set of $S\ge S_{0},$ then $(A,{\textbf{b}})\in \widehat{D}_{m,n}(\psi )^{c}.$

Following [12] we adopt some notations. Let $W_{S,\varepsilon }$ be the set of $A\in [0,1]^{mn}$ such that there exists ${\textbf{x}}_{A,S}\in \mathbb {Z}^{m}\setminus \{{\textbf{0}}\}$ satisfying

$$\begin{aligned} \Vert A^{t}{\textbf{x}}_{A,S}\Vert _{\mathbb {Z}}<d^{-1}\varepsilon \widetilde{\psi }(S) \text { and } \Vert {\textbf{x}}_{A,S}\Vert <d^{-1}\varepsilon S \end{aligned}$$

and let

$$\begin{aligned} \widehat{W}_{S,\varepsilon }:=\{ (A,{\textbf{b}})\in [0,1]^{mn+m}: A\in W_{S,\varepsilon }\text { and } |{\textbf{b}}\cdot {\textbf{x}}_{A,S}|_{\mathbb {Z}}>\varepsilon \}. \end{aligned}$$

For fixed ${\textbf{b}}\in \mathbb {R}^{m},$ consider the set $W_{{\textbf{b}},S,\varepsilon }$ of matrices $A\in [0,1]^{mn}$ such that there exists ${\textbf{x}}\in \mathbb {Z}^{m}\setminus \{{\textbf{0}}\}$ satisfying

• $|{\textbf{b}}\cdot {\textbf{x}}|_{\mathbb {Z}}>\varepsilon$

• $\Vert A^{t}{\textbf{x}}\Vert _{\mathbb {Z}}<d^{-1}\varepsilon \widetilde{\psi }(S)$ and $\Vert {\textbf{x}}\Vert <d^{-1}\varepsilon S.$

Let $W_{{\textbf{b}},\varepsilon }:=\limsup \limits _{S\rightarrow \infty }W_{{\textbf{b}},S,\varepsilon }.$ Note that $A\in W_{S,\varepsilon }$ if and only if

$$\begin{aligned} \Vert A^{t}{\textbf{x}}_{A,S}\Vert _{\mathbb {Z}}<\Psi _{\varepsilon }(U) \text { and } \Vert {\textbf{x}}_{A,S}\Vert <U \text { for some } {\textbf{x}}_{A,S}, \end{aligned}$$

where

$$\begin{aligned} \Psi _{\varepsilon }(U):=d^{-1}\varepsilon \widetilde{\psi }(d\varepsilon ^{-1}U), \;\;\; U = d^{-1} \varepsilon S. \end{aligned}$$

(4.1)

By Lemma 4.1$\limsup \limits _{S\rightarrow \infty }\widehat{W}_{S,\varepsilon }\subseteq \widehat{D}_{m,n}(\psi )^{c}$ and $W_{{\textbf{b}},\varepsilon }\subseteq \widehat{D}^{{\textbf{b}}}_{m,n}(\psi )^{c}.$

Further $\limsup \limits _{S\rightarrow \infty }{W}_{S,\varepsilon }=\{A\in [0,1]^{mn}: A^{t}\in W_{n,m}(\Psi _{\varepsilon })\}$ is the set of matrices whose transposes are $\Psi _{\varepsilon }$-approximable. From here onwards we use a slightly different definition of $\Psi _{\varepsilon }$-approximability; recall from footnote 1 where the inequality $\Vert A^{t}{\textbf{x}}\Vert _{\mathbb {Z}}<\Psi _{\varepsilon }(\Vert {\textbf{x}}\Vert )$ is used instead of (1.3). Then, $W_{{\textbf{b}},\varepsilon }$ can be considered as the set of matrices whose transposes are $\Psi _{\varepsilon }$-approximable with solutions restricted on the set $\{{\textbf{x}}\in \mathbb {Z}^{m}:|{\textbf{b}}\cdot {\textbf{x}}|_{\mathbb {Z}}>\varepsilon \}.$

4.1 Mass distributions on $\Psi _{\varepsilon }$-approximable matrices

In this subsection we prove the divergent part of Theorem 1.7 using mass distributions on $\Psi _{\varepsilon }$-approximable matrices following [1].

Lemma 4.2

For each $mn-n< s\le mn$ and $0<\varepsilon < 1/2,$ let $U_{0}=d^{-1}\varepsilon S_{0}$ and f be a dimension function satisfying (1.5) and (1.6). Suppose that (1.7) holds. Then

$$\begin{aligned}{} & {} \sum \limits _{q=\lceil {T_{0}}\rceil }^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right)<\infty \\{} & {} \quad \iff \sum \limits _{h=\lceil {U_{0}}\rceil }^{\infty }h^{m+n-1}\Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )^{-n(m-1)}f \Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big ) <\infty . \end{aligned}$$

Proof

Similar to Lemma 3.1, we may replace the sums with integrals

$$\begin{aligned} \int ^{\infty }_{T_{0}}\frac{1}{\psi (x)x^{2}} \left( \frac{ x^{\frac{1}{n}}}{\psi (x)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (x)^\frac{1}{m}}{ x^{\frac{1}{n}}}\right) dx \text { and } \int \limits _{U_0}^{\infty }h^{m+n-1}\Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )^{-n(m-1)}f \Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )dh, \end{aligned}$$

respectively.

Note that since $\Psi _{\varepsilon }(h)=d^{-1}\varepsilon {\widetilde{\psi }}(d\varepsilon ^{-1}h),$ if we consider the term $\int \limits _{U_0}^{\infty }h^{m+n-1}\Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )^{-n(m-1)} f \Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )dh,$ then

$$\begin{aligned}&\int \limits _{U_0}^{\infty }h^{m+n-1}\Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )^{-n(m-1)}f \Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big ) dh<\infty \\&\qquad \iff \int \limits _{S_0}^{\infty }y^{m+n-1}\Big (\frac{{\widetilde{\psi }}(q)}{y} \Big )^{-n(m-1)}f \Big (\frac{{\widetilde{\psi }}(q)}{y} \Big ) dy <\infty . \end{aligned}$$

Also, since ${\widetilde{\psi }}(y)=\psi ^{-1}(y^{-m})^{-\frac{1}{n}},$ we have

$$\begin{aligned}&\int ^{\infty }_{S_{0}}y^{m+n-1} \left( \frac{{\widetilde{\psi }}(y)}{y} \right) ^{-n(m-1)}f \left( \frac{{\widetilde{\psi }}(y)}{y} \right) dy\\&\quad =\int ^{\infty }_{S_{0}} y^{mn+m-1}(\psi ^{-1}(y^{-m}))^{m-1} f\left( \frac{(\psi ^{-1}(y^{-m}))^{-\frac{1}{n}}}{y}\right) dy \\&\quad =\frac{1}{m}\int _{S^{m}_{0}}^{\infty }t^{n} (\psi ^{-1}(t^{-1}))^{m-1}f\left( \frac{(\psi ^{-1}(t^{-1}))^{-\frac{1}{n}}}{t^{\frac{1}{m}}}\right) dt\\&\quad = \frac{1}{m}\int _{\psi ^{-1}(S^{-m}_{0})}^{\infty }x^{m-1} (\psi (x)^{-1})^{n}f\left( \frac{x^{-\frac{1}{n}}}{(\psi (x)^{-1})^{\frac{1}{m}}}\right) d\psi (x)^{-1}\\&\quad \asymp \frac{1}{m}\left( n-\frac{s}{m} +1\right) ^{-1}\int ^{\infty }_{T_{0}}x^{m-1}f(x^{-\frac{1}{n}}) d(\psi (x)^{-1})^{n-\frac{s}{m}+1}, \end{aligned}$$

where in the second last line we used the change of variables $x= \psi ^{-1}(t^{-1}),$ $t=\psi (x)^{-1}$ and in the last line we used (1.5) and (1.7). Since it follows from (1.7) that $(x^{-\frac{1}{n}})^{\alpha }\le (\psi (x)^{-1})^{-\frac{1}{m}} \le (x^{-\frac{1}{n}})^\frac{1}{\alpha }.$ Therefore by using (1.5) we can write

$$\begin{aligned} f((\psi (x)^{-1})^{-\frac{1}{m}}x^{-\frac{1}{n}}) \asymp (\psi (x)^{-1})^{-\frac{s}{m}}f(x^{-\frac{1}{n}}). \end{aligned}$$

Using integration by parts

$$\begin{aligned}&\int ^{\infty }_{T_{0}}x^{m-1}f(x^{-\frac{1}{n}}) d(\psi (x)^{-1})^{n-\frac{s}{m}+1}\\&\quad =(\lim _{x\rightarrow \infty }x^{m-1}\psi (x)^{-n-1+\frac{s}{m}}f(x^{-\frac{1}{n}})-T^{m-1}_{0}\psi (T_{0})^{-n-1+\frac{s}{m}}f(T_{0}^\frac{-1}{n}) )\\&\qquad + \int _{T_{0}}^{\infty }\left[ -(m-1)x^{m-2}f(x^{-\frac{1}{n}})+\frac{1}{n} x^{m-1}x^{-\frac{1}{n}-1}f'(x^{-\frac{1}{n}})\right] \psi (x)^{-n-1+\frac{s}{m}}dx\\&\quad =\lim _{x\rightarrow \infty }x^{m-1}\psi (x)^{-n-1+\frac{s}{m}}f(x^{-\frac{1}{n}})-T^{m-1}_{0}\psi (T_{0})^{-n-1+\frac{s}{m}}f(T_{0}^\frac{-1}{n}) \\&\qquad +\int _{T_{0}}^{\infty }\left[ -(m-1)+\frac{1}{n}a(x^{-\frac{1}{n}})\right] x^{m-2}f(x^{-\frac{1}{n}})\psi (x)^{-n-1+\frac{s}{m}}dx, \text { by }(1.6) \\&\qquad \asymp \lim _{x\rightarrow \infty }x^{m-1}\psi (x)^{-n-1} f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) -T^{m-1}_{0}\psi (T_{0})^{-n-1+\frac{s}{m}}f\left( \frac{\psi (T_{0})^{\frac{1}{m}}}{T_{0}^{\frac{1}{n}}}\right) \\&\qquad + \int _{T_{0}}^{\infty }\left[ \frac{1}{n}a(x^{-\frac{1}{n}}) - (m-1)\right] x^{m-2}\psi (x)^{-n-1}f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) dx. \end{aligned}$$

Note that

$$\begin{aligned}&\int _{T_{0}}^{\infty }x^{m-2}\psi (x)^{-n-1}f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) dx =\int _{T_{0}}^{\infty }x^{m-1}\psi (x)^{-n-1}f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) d\log x. \end{aligned}$$

(4.2)

Thus the convergence of $\int ^{\infty }_{T_{0}}x^{m-2}\psi (x)^{-1-n}f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) dx$ gives that

$$\begin{aligned} \lim _{x\rightarrow \infty }x^{m-1}\psi (x)^{-n-1} f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) <\infty . \end{aligned}$$

Also observe that as $x\rightarrow \infty ,$ $a(x^\frac{-1}{n})\rightarrow s.$ Therefore

$$\begin{aligned} \frac{1}{n}a(x^\frac{-1}{n})-(m-1)\rightarrow \frac{s-n(m-1)}{n} \end{aligned}$$

which is finite and positive (since $s>mn-n$). Therefore the convergence of $\int ^{\infty }_{T_{0}}x^{m-2}\psi (x)^{-1-n}f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) dx$ gives the convergence of

$$\begin{aligned} \frac{1}{n} \int _{T_{0}}^{\infty }a(x^{-\frac{1}{n}})x^{m-2}\psi (x)^{-n-1}f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) dx -(m-1)\int _{T_{0}}^{\infty }x^{m-2}\psi (x)^{-n-1} f\left( \frac{\psi (x)^{\frac{1}{m}}}{x^{\frac{1}{n}}}\right) dx. \end{aligned}$$

Hence the convergence of

$$\begin{aligned} \int ^{\infty }_{T_{0}}\frac{1}{\psi (x)x^{2}} \left( \frac{ x^{\frac{1}{n}}}{\psi (x)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (x)^\frac{1}{m}}{ x^{\frac{1}{n}}}\right) dx \text { or } \int ^{\infty }_{S_{0}} y^{m+n-1} \left( \frac{{\widetilde{\psi }}(y)}{y} \right) ^{-n(m-1)}f \left( \frac{{\widetilde{\psi }}(y)}{y} \right) dy, \end{aligned}$$

implies the convergence of other one since for $T_{0}$ large enough all summands in (4.2) are positive except the finite value

$$\begin{aligned} -T^{m-1}_{0}\psi (T_{0})^{-n-1+\frac{s}{m}}f\left( \frac{\psi (T_{0})^{\frac{1}{m}}}{T_{0}^{\frac{1}{n}}}\right) . \end{aligned}$$

$\square$

Lemma 4.3

([1, Section 5]) Assume that $\sum \limits _{q=1}^{\infty } \frac{1}{\psi (q)q^{2}} \left( \frac{ q^{\frac{1}{n}}}{\psi (q)^\frac{1}{m}}\right) ^{mn} f\left( \frac{\psi (q)^\frac{1}{m}}{ q^{\frac{1}{n}}}\right) =\infty .$ Fix $0<\varepsilon <\frac{1}{2}.$ Then, for any $\eta >1$ there exists a probability measure $\mu$ on $\limsup W_{S,\varepsilon }$ satisfying the condition that for an arbitrary ball D of sufficiently small radius r(D) we have

$$\begin{aligned} \mu (D)\ll \frac{f(r(D))}{\eta }, \end{aligned}$$

where the implied constant does not depend on D or $\eta .$

Proof

Note that $\limsup _{S\rightarrow \infty }W_{S,\varepsilon }=\{ A\in [0,1]^{mn}: A^{t} \in W_{n,m} (\Psi _{\varepsilon }) \}.$ By Lemma 4.2

$$\begin{aligned} \sum \limits _{h=1}^{\infty }h^{m+n-1}\Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big )^{-n(m-1)}f \Big (\frac{\Psi _{\varepsilon }(h)}{h} \Big ) =\infty , \end{aligned}$$

which is the divergent assumption of Theorem 2.3 for $W_{n,m} (\Psi _{\varepsilon }).$ 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 $\mu$ on $\limsup _{S\rightarrow \infty }W_{S,\varepsilon }$ satisfying the above condition. $\square$

Let us prove the divergent part of Theorem 1.7.

Proof

Assume that $mn+m-n<s<mn+m$ and fix $0<\varepsilon <\frac{1}{2}.$ For any fixed $\eta >1,$ let $\mu$ be a probability measure on $\limsup _{S\rightarrow \infty }W_{S,\varepsilon }$ as in Lemma 4.3 with f(r(D)) replaced by $r(D)^{-m}f(r(D)).$

Here we remark that since f(r) satisfies (1.5) and (1.6) it is not hard to check that the new function $f^{*}(r):=\frac{f(r)}{r^{m}}$ satisfies conditions (1.5) and (1.6) with s replaced by $s-m.$ Indeed, (1.5) (with s replaced by $s-m$) follows since $f^{*}(xy)=\frac{f(xy)}{(xy)^m}\asymp \frac{x^{s}f(y)}{x^{m}y^{m}} ={x^{s-m}}f^{*}(y),$ and (1.6) (with s replaced by $s-m$) follows since

$$\begin{aligned} \frac{rf^{*'}(r)}{f^{*}(r)}=\frac{r}{f^{*}(r)}[r^{-m}f'(r)-mr^{-m-1}f(r)]=\left[ r\frac{f'(r)}{f(r)}-m\right] \rightarrow ( s-m)\text { as } r\rightarrow 0. \end{aligned}$$

Now consider the product measure $\nu =\mu \times m_{\mathbb {R}^{m}},$ where $m_{\mathbb {R}^{m}}$ is the canonical Lebesgue measure on $\mathbb {R}^{m}$ and let $\pi _{1}$ and $\pi _{2}$ be the natural projections from $\mathbb {R}^{mn+m}$ to $\mathbb {R}^{mn}$ and $\mathbb {R}^{m},$ respectively.

For any fixed integer $N\ge 1,$ let $V_{S,\varepsilon }=W_{S,\varepsilon }{\setminus } \bigcup _{k=N}^{S-1}W_{k,\varepsilon }$ and $\widehat{V}_{S,\varepsilon }=\{(A,{\textbf{b}})\in \widehat{W}_{S,\varepsilon }: A\in V_{S,\varepsilon }\}.$ Then $\nu (\bigcup _{S\ge N}\widehat{W}_{S,\varepsilon })=\nu (\bigcup _{S\ge N}\widehat{V}_{S,\varepsilon })\ge 1-2\varepsilon$, see [12, p.21].

Since $N\ge 1$ is arbitrary, we have $\nu (\limsup _{S\rightarrow \infty }\widehat{W}_{S,\varepsilon })\ge 1-2\epsilon .$ For an arbitrary ball $B\subseteq \mathbb {R}^{mn+m}$ of sufficiently small radius r(B),  we have

$$\begin{aligned} \nu (B)\le \mu (\pi _{1}(B))\times m_{\mathbb {R}^{m}}(\pi _{2}(B))\ll \frac{f(r(B))}{\eta }, \end{aligned}$$

where the implied constant does not depend on B or $\eta .$ By using the Mass Distribution Principle i.e. Lemma 2.2 and the Transference Lemma i.e. Lemma 4.1, we have

$$\begin{aligned} \mathcal {H}^{f}( \widehat{D}_{m, n}(\psi )^{c} )\ge \mathcal {H}^{f}( \limsup _{S\rightarrow \infty }\widehat{W}_{S,\varepsilon }) \gg (1-2\varepsilon )\eta , \end{aligned}$$

and by letting $\eta \rightarrow \infty$ we obtain the desired result. $\square$

4.2 Local ubiquity for $W_{{\textbf{b}},\varepsilon }$

We will use the idea of local ubiquity for $W_{{\textbf{b}},\varepsilon }$ to prove the divergent part of Theorem 1.8. Following [12] we define

$$\begin{aligned} \varepsilon ({\textbf{b}})=\min _{1\le j\le m, \ |b_{j}|_{\mathbb {Z}}>0}\frac{|b_{j}|_{\mathbb {Z}}}{4}, \end{aligned}$$

(4.3)

for ${\textbf{b}}=(b_{1},\cdots ,b_{m})\in \mathbb {R}^{m}\setminus \mathbb {Z}^{m}.$ Note that $\varepsilon ({\textbf{b}})>0$ is due to the fact that ${\textbf{b}}\in \mathbb {R}^{m}\setminus \mathbb {Z}^{m}.$

The following lemma is used when we count the number of integral vectors ${\textbf{z}}\in \mathbb {Z}^{m}$ such that

$$\begin{aligned} \left| {\textbf{b}}\cdot {\textbf{z}}\right| _{\mathbb {Z}}\le \varepsilon ({\textbf{b}}). \end{aligned}$$

(4.4)

Lemma 4.4

([12, Lemma 4.4]) For ${\textbf{b}}=(b_{1},\cdots ,b_{m})\in \mathbb {R}^{m}\setminus \mathbb {Z}^{m},$ let $\varepsilon ({\textbf{b}})$ be as in (4.3) and $1\le i\le m$ be an index such that $\varepsilon ({\textbf{b}})=\frac{|b_{i}|_{\mathbb {Z}}}{4}.$ Then, for any ${\textbf{x}}\in \mathbb {Z}^{m},$ at most one of ${\textbf{x}}$ and ${\textbf{x}}+{\textbf{e}}_{i}$ satisfies (4.4) where ${\textbf{e}}_{i}$ denotes the vector with a 1 in the ith coordinate and 0’s elsewhere.

For a fixed ${\textbf{b}}\in \mathbb {R}^{m}\setminus \mathbb {Z}^{m},$ let $\varepsilon _{0}:=\varepsilon ({\textbf{b}}),$ $\Psi _{0}:=\Psi _{\varepsilon _{0}}$ and $\Psi (h)=\frac{\Psi _{0}(h)}{h}.$ With notions in the Subsection 2.2, which are defined for the ubiquitous system construction, let

$$\begin{aligned}{} & {} J:=\{ ({\textbf{x}},{\textbf{y}})\in \mathbb {Z}^{m}\times \mathbb {Z}^{n}: \Vert {\textbf{y}}\Vert \le m\Vert {\textbf{x}}\Vert \text { and } |{\textbf{b}}\cdot {\textbf{x}}|_{\mathbb {Z}}>\varepsilon _{0}\} \text { and } \end{aligned}$$

(4.5)

$$\begin{aligned}{} & {} \text { for } \kappa := ({\textbf{x}},{\textbf{y}})\in J \text { denote } \beta _{\kappa }:=\Vert {\textbf{x}}\Vert \text { and }R_{\kappa }:=\{A\in [0,1]^{mn}:A^{t}{\textbf{x}}={\textbf{y}}\}. \end{aligned}$$

(4.6)

Note that $W_{{\textbf{b}},\varepsilon _{0}}\subset \Lambda (\Psi )$ and the family ${\mathcal {R}}$ of resonant sets $R_{\kappa }$ consists of $(m-1)n$-dimensional, rational affine subspaces.

By Lemma 4.2, now we assume that the divergence part of Theorem 1.8 is satisfied. Then we can find a strictly increasing sequence of positive integers $\{h_{i}\}_{i\in \mathbb N}$ such that

$$\begin{aligned} \sum \limits _{h_{i-1}<h\le h_{i}}^{\infty }h^{m+n-1}\Big (\frac{\Psi _{0}(h)}{h} \Big )^{-n(m-1)}f \Big (\frac{\Psi _{0}(h)}{h} \Big ) >1 \end{aligned}$$

(4.7)

and $h_{i}>2h_{i-1}.$ Put $\omega (h):=i^{\frac{1}{n}}$ if $h_{i-1}<h\le h_{i}.$ Then

$$\begin{aligned} \sum \limits _{h=1}^{\infty }h^{m+n-1}\Big (\frac{\Psi _{0}(h)}{h} \Big )^{-n(m-1)}f \Big (\frac{\Psi _{0}(h)}{h} \Big ) \omega (h)^{-n} =\infty . \end{aligned}$$

For a constant $c>0,$ define the ubiquitous function $\rho _{c}:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}$ by

$$\begin{aligned} \rho _{c}(h)= {\left\{ \begin{array}{ll} ch^{-\frac{1+n}{n}} \ &{} \text { if }\quad m=1; \\ ch^{-\frac{m+n}{n}}\omega (h) \ &{} \text { if } \quad m\ge 2. \end{array}\right. } \end{aligned}$$

(4.8)

Clearly the ubiquitous function is 2-regular.

Theorem 4.5

([12, Theorem 4.5]) The pair $({\mathcal {R}},\beta )$ is a locally ubiquitous system relative to $\rho =\rho _{c}$ for some constant $c>0.$

The divergent part of Theorem  1.8.

Assume that $(m-1)n<s\le mn$ and $r^{-nm}f(r)\rightarrow \infty$ as $r\rightarrow 0.$ It follows from Theorem 2.5 and Theorem 4.5 that

$$\begin{aligned} \mathcal {H}^{f}( \widehat{D}^{{\textbf{b}}}_{m, n}(\psi )^{c} )\ge \mathcal {H}^{f}( W_{{\textbf{b}},\varepsilon _{0}})=\mathcal {H}^{f}( [0,1]^{mn}). \end{aligned}$$

Similar as in [12] here we have used the fact that the divergence and convergence of the sums

$$\begin{aligned} \sum _{N=1}^{\infty }2^{\kappa N}{\mathcal {F}}(2^{N}) \text { and } \sum _{h=1}^{\infty }h^{\kappa -1}{\mathcal {F}}(h) \end{aligned}$$

coincide for any monotonic function ${\mathcal {F}}:\mathbb {Z}_{+}\rightarrow \mathbb {Z}_{+}$ and $\kappa \in \mathbb {R}.$ This completes the proof of the divergent part of Theorem 1.8.