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
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.
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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
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
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 inequalityFootnote 1
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
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
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
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 \(0\le s\le mn\)
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
where \(mn+m-n<s<mn+m\) and \(\alpha >1\) is some absolute constant independent of x and y and suppose that
such that \(a(x)\rightarrow s\) as \(x\rightarrow 0\). Further, let
Then
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
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
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
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
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
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
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
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
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
where \(\text {dist}(V,S):=\inf \{ \Vert V-Y \Vert :Y\in S \}.\) Fix a decreasing function \(\Psi :\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) let
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
Thus \(\Lambda (\Psi )\) is the set consisting elements of \([0,1]^{mn}\) which lie in infinitely many \(\Delta (\Psi ,N),\) that is,
Next let \(\rho :\mathbb {R}_{+} \rightarrow \mathbb {R}_{+}\) be a function with \(\rho (t)\rightarrow 0\) as \(t\rightarrow \infty \) and let
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
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
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
Then
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
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
and put
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
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
Let
for \(A\in X^{mn}\) and \((A,{\textbf{b}})\in X^{mn}\times \mathbb {R}^{m}.\) Let us also denote by
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
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
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
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
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
Define
Since \(\psi (e^{\lambda })=e^{-P(\lambda )}\), letting \(\log x=\lambda \) we have
Using \(P(\lambda (t))=t-mz(t),\) we have
The term in the last line can be expressed by
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
therefore by using (1.5) we can write
Next we will use integration by parts to evaluate the integral in (3.3).
Note that as \(\lambda \rightarrow \infty ,\) \(e^{-\frac{\lambda }{n}}\rightarrow 0\) thus by assumption \(a(e^{-\frac{\lambda }{n}})\rightarrow s\) and therefore
which is finite and positive for \(T_{0}\) large enough (since \(s > nm - n\)). Observe that
if the integral
converges. Thus the convergence of
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
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
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
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
and let
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
where
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
Proof
Similar to Lemma 3.1, we may replace the sums with integrals
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
Also, since \({\widetilde{\psi }}(y)=\psi ^{-1}(y^{-m})^{-\frac{1}{n}},\) we have
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
Using integration by parts
Note that
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
Also observe that as \(x\rightarrow \infty ,\) \(a(x^\frac{-1}{n})\rightarrow s.\) Therefore
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
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
\(\square \)
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
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
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
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
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
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
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
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
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
and \(h_{i}>2h_{i-1}.\) Put \(\omega (h):=i^{\frac{1}{n}}\) if \(h_{i-1}<h\le h_{i}.\) Then
For a constant \(c>0,\) define the ubiquitous function \(\rho _{c}:\mathbb {R}_{+}\rightarrow \mathbb {R}_{+}\) by
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
Similar as in [12] here we have used the fact that the divergence and convergence of the sums
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.
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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.
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Bakhtawar, A., Simmons, D. Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions. Res. number theory 9, 54 (2023). https://doi.org/10.1007/s40993-023-00436-y
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DOI: https://doi.org/10.1007/s40993-023-00436-y