On a Kurzweil type theorem via ubiquity

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.


Introduction
Kurzweil's theorem [Kur55] in inhomogeneous Diophantine approximation is concerned with well approximable target vectors.We start by introducing related definitions and notations.Given a decreasing function ψ : R `Ñ R `and an m ˆn matrix A P M m,n pRq, we say that b P R m is ψ-approximable for A if there exist infinitely many solutions q P Z n to the following inequality }Aq ´b} Z ă ψp}q}q.
Denote by W A pψq the set of such vectors in the unit cube r0, 1s m .Here and hereafter, }x} " max 1ďiďm |x i | and }x} Z " min nPZ m }x ´n} for x P R m .We say that A P M m,n pRq is badly approximable if lim inf }q}Ñ8 }q} n m }Aq} Z ą 0.
Kurzweil proved the following zero-one law for W A pψq.
Theorem 1.1.[Kur55] If A P M m,n pRq is badly approximable, then for any decreasing ψ : R `Ñ R `we have q n´1 ψpqq m ă 8, q n´1 ψpqq m " 8.
Here and hereafter, | ¨| stands for Lebesgue measure on R m .
We remark that Kurzweil showed that in fact the badly approximable condition is an equivalent condition for the zero-one law, not a sufficient condition.
2020 Mathematics Subject Classification: Primary 11J20 ; Secondary 11J83, 28A78 In this article, we will consider similar results when the badly approximable condition is relaxed.We say that A P M m,n pRq is singular if for any ǫ ą 0 for all large enough X ě 1 there exists q P Z n such that }Aq} Z ă ǫX ´n m and 0 ă }q} ă X.
Otherwise we call it non-singular (or regular following [Cas57]).One can check that A P M m,n pRq is singular if and only if for any ǫ ą 0 for all large enough ℓ P Z ě1 there exists q P Z n such that (1.1) }Aq} Z ă ǫ2 ´n m ℓ and 0 ă }q} ă 2 ℓ .
Hence A P M m,n pRq is non-singular if and only if there exists ǫ ą 0 such that the set Lpǫq :" tℓ P Z ě1 : there is no solution q P Z n to (1.1) with ℓu is unbounded.We call Lpǫq ǫ-return sequence for A.
(1) Note that A P M m,n pRq is badly approximable if and only if there exists ǫ ą 0 such that Lpǫq " Z ě1 .
(2) In a dynamical point of view as in [Dan85], the set Lpǫq corresponds to return times to a compact set related to ǫ of a certain diagonal flow in the space of lattices.
The following is the main theorem of this article.
Theorem 1.3.Let A P M m,n pRq be non-singular with ǫ-return sequence Lpǫq " tℓ i u iě1 .For any decreasing ψ : R `Ñ R `and 0 ď s ď m, the s-dimensional Hausdorff measure of W A pψq is given by For δ ą 0, let ψ δ pqq " δq ´n m .Denote Bad A pδq " r0, 1s m zW A pψ δ q and Bad A " Ť δą0 Bad A pδq. Theorem 1.3 with ψ " ψ δ and s " m directly implies the following corollary.
Corollary 1.4.If A P M m,n pRq is non-singular, then for any δ ą 0, the set Bad A pδq has Lebesgue measure zero, hence, Bad A has Lebesgue measure zero.
(1) There are some historical remarks about Corollary 1.4.The onedimensional result of the corollary was proved in [Kim07] using irrational rotations and the Ostrowski representation.The corollary in full generality was proved in [Sha13] using a certain mixing property in homogeneous dynamics.Simultaneous version (i.e.n " 1) of the corollary was proved in [Mos] using a certain well distributed property.Our method relies on a suitable construction of a ubiquitous system.
(2) A zero-one law for Lebesgue measure of W A pψq in one-dimensional case was investigated in [FK16].According to their results, Theorem 1.3 is not optimal.It seems very interesting to obtain zero-one laws for W A pψq in multidimensional case.(3) A weighted version of Kurzweil's theorem was investigated in [Har12].
There have been several recent results on the weighted ubiquity and weighted transference theorems (see [CGGMS20], [G20], and [WW21]).It seems plausible to utilize these results to obtain a weighted version of Theorem 1.3.
As stated in [BBDV09, Section 9], using Theorem 1.1 and Mass Transference Principle in [BV06], we are able to deduce Hausdorff measure version of Kurzweil's theorem.Theorem 1.3 which relies on a ubiquity method also implies the following corollary.
Corollary 1.6.If A P M m,n pRq is badly approximable, then for any decreasing ψ : R `Ñ R `and 0 ď s ď m, we have Moreover, the convergent part holds for every A P M m,n pRq.
Proof.The convergent part will be proved in Section 2. Since the divergence and convergence of the sums 2 ℓn ψp2 ℓ q s and 8 ÿ q"1 q n´1 ψpqq s coincide, the divergent part follows from Theorem 1.3 and Remark 1.2 (1).
We explore some counterparts of Kurzweil's theory.We denote by wpA, bq the supremum of the real numbers w for which, for arbitrarily large X, the inequalities }Aq ´b} Z ă X ´w and }q} ă X have an integral solution q P Z n .We also denote by p wpAq the supremum of the real numbers w for which, for all sufficiently large X, the inequalities }Aq} Z ă X ´w and }q} ă X have an non-zero integral solution q P Z n .If p wp t Aq ą m n , then by [BL05, Theorem], for almost all b P R m , wpA, bq " Thus for any δ ą 0, the Lebesgue measure of Bad A pδq is full.This is opposite to Corollary 1.4.Note that if p wp t Aq ą m n , then t A is singular, hence A is singular.So, they do not conflict with each other.
If we consider the case p wp t Aq " m n and t A is singular, then we cannot deduce from [BL05, Theorem] that Bad A pδq is of full Lebesgue measure for any δ ą 0. We will give a certain sufficient condition for Bad A pδq being of full Lebesgue measure for any δ ą 0.
If rk Z p t AZ m `Zn q ă m `n, then p wp t Aq " 8, hence we may assume that rk Z p t AZ m `Zn q " m `n.Then following [BL05, Section 3], there exists a sequence of best approximations then for any δ ą 0, the Lebesgue measure of Bad A pδq is full.
(1) The summability assumption implies that Y The structure of this paper is as follows: In Section 2, we prove the convergent part of Corollary 1.6.In Section 3, we introduce some preliminaries for the proof of Theorem 1.3 including ubiquitous systems, Transference Principle, and Weyl type uniformly distribution.We prove Theorem 1.3 and Theorem 1.7 in Section 4 and Section 5, respectively.

Convergent part: a warm up
In this section, we prove the convergent part of Corollary 1.6.We will use the following Hausdorff measure version of the Borel-Cantelli lemma [BD99, Lemma 3.10].
Hence, using Hausdorff-Cantelli lemma, we have that for any 0 ď s ď m, This proves the convergent part of Corollary 1.6.

Preliminaries for divergent part
3.1.Ubiquity systems.The proof of Theorem 1.3 is based on the ubiquity framework developed in [BDV06], which provides a very general and abstract approach for establishing the Hausdorff measure of a large class of limsup sets.In this subsection, we set up ubiquitous systems that suits our situation.
We consider T m with the supremum norm } ¨}.With notation in [BDV06] we set up the following: J " tq P Z n u, R q " Aq P T m , R " tR q : q P Ju, β q " }q}.
Let l " tl i u and u " tu i u be positive increasing sequences such that l i ă u i and lim iÑ8 l i " 8.

It follows that
Throughout, ρ : R `Ñ R `will denote a function satisfying lim rÑ8 ρprq " 0 and is referred to as the ubiquitous function.Let ∆ u l pρ, iq :" BpAq, ρpu i qq.
Definition 3.1 (Local ubiquity).Let B be an arbitrary ball in T m .Suppose that there exist a ubiquitous function ρ and an absolute constant κ ą 0 such that (3.1) |B X ∆ u l pρ, iq| ě κ|B| for i ě i 0 pBq.Then the pair pR, βq is said to be a locally ubiquitous system relative to pρ, l, uq.
Finally, a function h is said to be u-regular if there exists a positive constant λ ă 1 such that for i sufficiently large hpu i`1 q ď λhpu i q.
With notation in [BDV06], the Lebesgue measure on T m is of type (M2) with δ " m and the intersection conditions are also satisfied with γ " 0. These conditions are not stated here but these extra conditions exist and need to be established for the more abstract ubiquity.
Theorem 3.2.[BDV06] Suppose that pR, βq is a local ubiquitous system relative to pρ, l, uq and assume further that ρ is u-regular.Then for any 0 ď s ď m H s pW A pψqq " H s pT m q if 8 ÿ i"1 ψpu i q s ρpu i q m " 8.

Transference principle.
We need the following transference principle between homogeneous and inhomogeneous Diophantine approximation.See [Cas57, Chapter V, Theorem VI]).
Theorem 3.3 (Transference principle).Suppose that there is no solution q P Z n zt0u such that }Aq} Z ă C and }q} ă X.
Then for any b P R m , there exists q P Z n such that }Aq ´b} Z ď C 1 and }q} ď X 1 , where This principle implies the following corollary.
Corollary 3.4.Let A P M m,n pRq be non-singular and let Lpǫq " tℓ i u iě1 be the ǫ-return sequence for A. Then for any b P R m , there exists q P Z n such that }Aq ´b} Z ď 1 2 pǫ ´m `1qǫ2 ´n m ℓ i and }q} ď 1 2 pǫ ´m `1q2 ℓ i .
Proof.It follows directly from Theorem 3.3 with C " ǫ2 ´n m ℓ i and X " .
Since #tq P Z n : }q} ď N u -N n , this proves the claim.
Following the proof of classical Weyl's criterion (see e.g.[KN74, Theorem 2.1]), we can deduce that for any ball B Ă T m we have #tAq P B : }q} ď N u #tq P Z n : }q} ď N u Ñ |B| as N Ñ 8.
Remark 3.6.The above proposition is slightly different to the multidimensional Weyl's criterion.We do not take every partial sum but "radial" partial sum.

Proof of Theorem 1.3
Let A be non-singular and let Lpǫq " tℓ i u iě1 be the ǫ-return sequence.With the notations in Subsection 3.1, we take sequences l " lpǫq " tl i u and u " upǫq " tu i u as follows: with some positive constant c 1 " c 1 pǫq ă 1, which will be determined later.We first establish the following local ubiquity with the set-up in Subsection 3.1.
Theorem 4.1.The pair pR, βq is a locally ubiquitous system relative to ´ρprq " c 2 r ´n m , l, u ¯with the constant c 2 " ǫ `1 2 pǫ ´m `1q ˘1`n m .
Proof.Fix any ball B " Bpx, r 0 q in T m .By Corollary 3.4, we have that BpAq, ρpu i qq ˇˇˇˇǓ sing Proposition 3.5 with 2B " Bpx, 2r 0 q, there is an absolute constant C ą 0 independent of the choice of B such that for all large enough i ě 1 #tAq P 2B : }q} ď l i u ď Cl n i |B|.
Thus for all large enough i ě 1 so that ρpu i q ă r 0 , we have ˇˇˇˇˇB X ď qPZ n :}q}ďl i BpAq, ρpu i qq ˇˇˇˇˇď Cl n i |B|p2ρpu i qq m " p2c 2 q m Cc n 1 |B|.
By taking 0 ă c 1 ă 1 so that p2c 2 q m Cc n 1 ă 1 2 , which depends only on ǫ, it follows from (4.1) that for all large enough i ě 1 Proof of Theorem 1.3.It follows from ℓ i`1 ě ℓ i `1 that for any i ě 1 ρpu i`1 q " 1 2 pǫ ´m `1qǫ2 ´n m ℓ i`1 ď 2 ´n m 1 2 pǫ ´m `1qǫ2 ´n m ℓ i " 2 ´n m ρpu i q, hence ρ is u-regular.Since the divergence and convergence of the sums 8 ÿ i"1 ψpu i q s ρpu i q m and 8 ÿ i"1 ψp2 ℓ i q s ρp2 ℓ i q m coincide, Theorem 3.2 and Theorem 4.1 imply Theorem 1.3.

Proof of Theorem 1.7
In order to prove Theorem 1.7, we basically follow the proof of [BL05,Theorem].
As in the introduction, let py k q kě1 be a sequence of best approximations for t A. Let Y k " }y k }, M k " } t Ay k } Z , and Proof.Consider the following two sequences: This theorem is stronger than the previous observation because if p wp t Aq ą m n , then there is γ ą 0 such that Y m n `γ k`1 M k ă 1 for all sufficiently large k ě 1.Hence, m`n ă 8 since Y k increases at least geometrically (see [BL05, Lemma 1]).(3) It was proved in [BKLR21, KKL] that A is singular on average if and only if there exists δ ą 0 such that Bad A pδq has full Hausdorff dimension.Thus it seems very interesting to obtain an equivalent Diophantine property of A for Bad A pδq being of full Lebesgue measure.
3. Weyl type uniform distribution.In this subsection, we will show Weyl type uniform distribution result for the sequence tAqu qPZ n Ă T m .For A P M m,n pRq, Kronecker's theorem (see e.g.[Cas57, Chapter III, Theorem IV]) asserts that the sequence tAqu qPZ n is dense in T m if and only if the subgroupGp t Aq :" t AZ m `Zn Ă R n has maximal rank m`n over Z.If A is non-singular, then t A is non-singular, hence Gp t Aq has maximal rank m `n over Z.By Kronecker's theorem, the sequence tAqu qPZ n is dense in T m .But the dense result is not enough for our purpose.We need the following Weyl type uniform distribution result.Proposition 3.5.If Gp t Aq has maximal rank m `n over Z, then the sequence tAqu qPZ n is uniformly distributed in the following sense: for any ball B Ă T m #tAq P B : }q} ď N u #tq P Z n : }q} ď N u Ñ |B| as N Ñ 8.Indeed, by the maximal rank assumption, we have t Ac P R n zQ n .Without loss of generality, we may assume that the first coordinate, say α, of t Ac is irrational.It follows from c ¨Aq " t Ac ¨q that