# On metric spaces with given transfinite asymptotic dimensions

Yan Wu∗ Jingming Zhu∗∗ Taras Radul ∗∗∗ 111 ∗ College of Data Science, Jiaxing
University, Jiaxing , 314001, P.R.China.
E-mail<EMAIL_ADDRESS>
${}^{\ast}\ast$ College of Data Science, Jiaxing University, Jiaxing , 314001,
P.R.China.
E-mail<EMAIL_ADDRESS>
∗∗∗ Institute of Mathematics, Casimirus the Great University of Bydgoszcz,
Poland;
Department of Mechanics and Mathematics, Ivan Franko National University of
Lviv, Universytetska st., 1.79000 Lviv, Ukraine.
E-mail<EMAIL_ADDRESS>
Abstract. For every countable ordinal number $\xi$, we construct a metric
space $X_{\xi}$ whose transfinite asymptotic dimension and complementary-
finite asymptotic dimension are both $\xi$.

Keywords  Asymptotic dimension, Transfinite asymptotic dimension,
Complementary-finite asymptotic dimension;

222 This research was supported by the National Natural Science Foundation of
China under Grant (No.12071183,11871342)

##  1 Introduction

In coarse geometry, asymptotic dimension of a metric space is an important
concept which was defined by Gromov for studying asymptotic invariants of
discrete groups [1]. This dimension can be considered as an asymptotic
analogue of the Lebesgue covering dimension. As a large scale analogue of W.E.
Haver’s property $C$ in dimension theory, A. Dranishnikov introduced the
notion of asymptotic property $C$ in [2]. It is well known that every metric
space with finite asymptotic dimension has asymptotic property $C$. But the
inverse is not true, which means that there exists some metric space $X$ with
infinite asymptotic dimension and asymptotic property $C$. Therefore how to
classify the metric spaces with infinite asymptotic dimension into smaller
categories becomes an interesting problem.

In [3], T. Radul defined the transfinite asymptotic dimension (trasdim) which
can be viewed as a transfinite extension of the asymptotic dimension and
proved that for a metric space $X$, trasdim$(X)<\infty$ if and only if $X$ has
asymptotic property $C$. He also gave examples of metric spaces with
trasdim$=\infty$ and with trasdim$=\omega$, where $\omega$ is the smallest
infinite ordinal number (see [3]). But whether there is a metric space $X$
with $\omega<$trasdim$(X)<\infty$ (stated as“omega conjecture”in [4] by M.
Satkiewicz) was unknown until recently. A metric space $X$ with
trasdim$X=\omega+1$ was constructed in [5]. Some examples of metric spaces
with higher trasdim were constructed in [6]. Let us remark that trasdim takes
only countable values [3]. In this paper, for every countable ordinal number
$\xi$, we construct a metric space $X_{\xi}$ with trasdim$(X_{\xi})=\xi$ and
coasdim$(X_{\xi})=\xi$, which generalized the results in [5] and [6]. Let us
remark that this result combined with a recent result in [7] gives an answer
to a problem stated by [8].

The paper is organized as follows: In Section 2, we recall some definitions
and properties of transfinite asymptotic dimension and complementary-finite
asymptotic dimension. In Section 3, we introduce a concrete metric space
$X_{\xi}$ with trasdim$(X_{\xi})=\xi$ and coasdim$(X_{\xi})=\xi$ for every
countable ordinal number $\xi$.

##  2 Preliminaries

Our terminology concerning the asymptotic dimension follows from [9] and for
undefined terminology we refer to [3] and [6].

Let $(X,d)$ be a metric space and $U,V\subseteq X$, let

$\text{diam}~{}U=\text{sup}\\{d(x,y)~{}|~{}x,y\in U\\}\text{ and
}d(U,V)=\text{inf}\\{d(x,y)~{}|~{}x\in U,y\in V\\}.$

Let $R>0$ and $\mathcal{U}$ be a family of subsets of $X$. $\mathcal{U}$ is
said to be _$R$ -bounded_ if

$\text{diam}~{}\mathcal{U}\doteq\text{sup}\\{\text{diam}~{}U~{}|~{}U\in\mathcal{U}\\}\leq
R.$

In this case, $\mathcal{U}$ is said to be _uniformly bounded_. Let $r>0$, a
family $\mathcal{U}$ is said to be _ $r$-disjoint_ if

$d(U,V)\geq r~{}\text{for every}~{}U,V\in\mathcal{U}\text{~{}with~{}}U\neq V.$

In this paper, we denote $\bigcup\\{U~{}|~{}U\in\mathcal{U}\\}$ by
$\bigcup\mathcal{U}$, denote
$\\{U~{}|~{}U\in\mathcal{U}_{1}\text{~{}or~{}}~{}U\in\mathcal{U}_{2}\\}$ by
$\mathcal{U}_{1}\cup\mathcal{U}_{2}$. Let $A$ be a subset of a metric space
$X$ and $\epsilon>0$. We denote $\\{x\in X~{}|~{}d(x,A)<\epsilon\\}$ by
$N_{\epsilon}(A)$ and denote $\\{x\in X~{}|~{}d(x,A)\leq\epsilon\\}$ by
$\overline{N_{\epsilon}(A)}$. We denote
$\\{N_{\epsilon}(U)~{}|~{}U\in\mathcal{U}\\}$ by $N_{\epsilon}(\mathcal{U})$
and denote $\\{\overline{N_{\epsilon}(U)}~{}|~{}U\in\mathcal{U}\\}$ by
$\overline{N_{\epsilon}(\mathcal{U})}$.

Definition 2.1. ([1]) A metric space $X$ is said to have _finite asymptotic
dimension_ if there exists $n\in\mathbb{N}$, such that for every $r>0$, there
exists a sequence of uniformly bounded families
$\\{\mathcal{U}_{i}\\}_{i=0}^{n}$ of subsets of $X$ such that the family
$\bigcup_{i=0}^{n}\mathcal{U}_{i}$ covers $X$ and each $\mathcal{U}_{i}$ is
$r$-disjoint for $i=0,1,\cdots,n$. In this case, we say that the _asymptotic
dimension_ of $X$ less than or equal to $n$, which is denoted by asdim$(X)\leq
n$.

We say that asdim$(X)=n$ if asdim$(X)\leq n$ and asdim$(X)\leq n-1$ is not
true.

T. Radul generalized asymptotic dimension of a metric space $X$ to transfinite
asymptotic dimension which is denoted by trasdim$(X)$ (see [3]). We will need
the set-theoretical function Ord from [11] which classifies families of finite
non-empty subsets of a set $T$. We will consider only the particular case when
$T=\mathbb{N}$.

Definition 2.2. ([11]) Let $Fin\mathbb{N}$ denote the collection of all
finite, nonempty subsets of $\mathbb{N}$ and let $M\subseteq Fin\mathbb{N}$.
For $\sigma\in\\{\varnothing\\}\cup Fin\mathbb{N}$, let

$M^{\sigma}=\\{\tau\in Fin\mathbb{N}~{}|~{}\tau\cup\sigma\in M\text{ and
}\tau\cap\sigma=\varnothing\\}.$

Let $M^{a}$ abbreviate $M^{\\{a\\}}$ for $a\in\mathbb{N}$. Define _the ordinal
number_ Ord$M$ inductively as follows:

$\displaystyle\text{Ord}M=0$ $\displaystyle\Leftrightarrow$ $\displaystyle
M=\varnothing,$ $\displaystyle\text{Ord}M\leq\alpha$
$\displaystyle\Leftrightarrow$
$\displaystyle\forall~{}a\in\mathbb{N},~{}\text{Ord}M^{a}<\alpha,$
$\displaystyle\text{Ord}M=\alpha$ $\displaystyle\Leftrightarrow$
$\displaystyle\text{Ord}M\leq\alpha\text{ and }\text{Ord}M<\alpha\text{ is not
true},$ $\displaystyle\text{Ord}M=\infty$ $\displaystyle\Leftrightarrow$
$\displaystyle\text{Ord}M\leq\alpha\text{ is not true for every ordinal number
}\alpha.$

We call a family $M\subseteq Fin\mathbb{N}$ _inclusive_ if and only if for
each $\sigma,\tau\in Fin\mathbb{N}$ such that $\tau\subseteq\sigma$ and
$\sigma\in M$, we have $\tau\in M$. We consider only inclusive families in the
following.

The following lemmas are particular cases of the corresponding lemmas from
[11].

######  Lemma 2.1.

([11]) Let $M\subseteq Fin\mathbb{N}$ and $n\in\mathbb{N}$. Then

Ord$M\leq n$ if and only if $|\sigma|\leq n$ for every $\sigma\in M$.

######  Lemma 2.2.

([11]) If $M,N\subseteq Fin\mathbb{N}$ and $M\subseteq N$, then
$\text{Ord}M\leq\text{Ord}N.$

######  Lemma 2.3.

([11]) If $M,N\subseteq Fin\mathbb{N}$, then $\text{Ord}(M\cup
N)\leq\text{max}\\{\text{Ord}M,\text{Ord}N\\}.$

######  Lemma 2.4.

([11]) Let $L$ and $L^{\prime}$ be sets. Let $FinL$ and $FinL^{\prime}$ denote
collections of all finite, nonempty subsets of $L$ and $L^{\prime}$,
respectively. Let $M\subseteq FinL$, $M^{\prime}\subseteq FinL^{\prime}$ and
$\phi:L\to L^{\prime}$ be a function such that for every $\sigma\in M$, we
have $\phi(\sigma)\in M^{\prime}$ and $|\phi(\sigma)|=|\sigma|$. Then
Ord$M\leq$Ord $M^{\prime}$.

Let $M\subseteq Fin\mathbb{N}$ and $K$ be an infinite subset of $\mathbb{N}$.
Then there is a standard bijection $\varphi$ from $\mathbb{N}$ to $K$ which
keeps the order. We define
$M[K]=\Big{\\{}\\{\varphi(k_{1}),\varphi(k_{2}),\cdots,\varphi(k_{m})\\}~{}\Big{|}~{}\\{k_{1},k_{2},\cdots,k_{m}\\}\in
M\Big{\\}}$. Note that $M=M[\mathbb{N}]$. By Lemma 2.4, we obtain the
following result.

######  Corollary 2.1.

Let $M\subseteq Fin\mathbb{N}$ and $K$ is a infinite subset of $\mathbb{N}$,
then Ord$M=$Ord$M[K]$.

Definition 2.3. ([3]) Given a metric space $X$, define the following
collection:

$\begin{split}A(X,d)=\\{\sigma\in Fin\mathbb{N}~{}|~{}&\text{ there are no
uniformly bounded families }\mathcal{U}_{i}\text{ for }i\in\sigma\\\ &\text{
such that each }\mathcal{U}_{i}\text{ is }i\text{-disjoint and
}\bigcup_{i\in\sigma}\mathcal{U}_{i}\text{~{}covers~{}}X\\}\end{split}$

and

$\begin{split}A_{2}(X,d)=\\{\sigma\in Fin\mathbb{N}~{}|~{}&\text{ there are no
uniformly bounded families }\mathcal{V}_{i}\text{ for }i\in\sigma\\\ &\text{
such that each }\mathcal{V}_{i}\text{ is }2^{i}\text{-disjoint and
}\bigcup_{i\in\sigma}\mathcal{V}_{i}\text{ covers }X\\}.\end{split}$

The _transfinite asymptotic dimension_ of $X$ is defined as trasdim$(X)=$
Ord$A(X,d)$ .

######  Lemma 2.5.

Ord$A(X,d)=$Ord$A_{2}(X,d)$.

###### Proof.

Since $A(X,d)\subseteq A_{2}(X,d)$, Ord$A(X,d)\leq$Ord$A_{2}(X,d)$ by Lemma
2.2. The inverse inequality follows from Lemma 2.4 by considering the
exponential function $\phi:\mathbb{N}\to\mathbb{N}$ defined by
$\phi(n)=2^{n}$. ∎

Definition 2.4. ([12]) Every ordinal number $\gamma$ can be represented as
$\gamma=\lambda(\gamma)+n(\gamma)$, where $\lambda(\gamma)$ is the limit
ordinal or $0$ and $n(\gamma)\in\mathbb{N}\cup\\{0\\}$. Let $X$ be a metric
space, we define _complementary-finite asymptotic dimension_ of $X$
(coasdim$(X)$) inductively as follows:

  * •

coasdim$(X)=-1$ $\Leftrightarrow$ $X=\emptyset$,

  * •

coasdim$(X)\leq\lambda(\gamma)+n(\gamma)$ $\Leftrightarrow$ for every $r>0$,
there exist $r$-disjoint uniformly bounded families
$\mathcal{U}_{0},\cdots,\mathcal{U}_{n(\gamma)}$ of subsets of $X$ such that
coasdim$(X\setminus\bigcup\bigcup_{i=0}^{n(\gamma)}\mathcal{U}_{i})<\lambda(\gamma)$,

  * •

coasdim$(X)=\gamma$ $\Leftrightarrow$ coasdim$(X)\leq\gamma$ and
coasdim$(X)\leq\beta$ is not true for any $\beta<\gamma$,

  * •

coasdim$(X)=\infty$ $\Leftrightarrow$ coasdim$(X)\leq\gamma$ is not true for
any ordinal number $\gamma$.

$X$ is said to have _complementary-finite asymptotic dimension_ if
coasdim$(X)\leq\gamma$ for some ordinal number $\gamma$.

Remark 2.1. It is easy to see that for every $n\in\mathbb{N}\cup\\{0\\}$,
coasdim$(X)\leq n$ if and only if asdim$(X)\leq n$.

######  Lemma 2.6.

([12]) Let $X$ be a metric space with $X_{1},X_{2}\subseteq X$. Then

coasdim$(X_{1}\cup
X_{2})\leq\text{max}\\{\text{coasdim}(X_{1}),\text{coasdim}(X_{2})\\}$.

##  3 Main result

###  3.1 Families with given Ord

For each $\tau=\\{k_{0},k_{1},\cdots,k_{s}\\}\in Fin\mathbb{N}$, we choose an
indexation such that $k_{0}<k_{1}<\cdots<k_{s}$. For every
$n\in\mathbb{N}\cup\\{0\\}$, let

$K(\tau,n)=\\{k_{n+1},k_{n+2}\cdots,k_{s}\\}$ if $n<s$ and
$K(\tau,n)=\varnothing$ if $n\geq s$,

and let

$i(\tau,n)=\text{max}\\{k_{0},k_{1}\cdots,k_{n}\\}=k_{n}$ if $n\leq s$ and
$i(\tau,n)=k_{s}$ if $n>s$.

For each limit ordinal $\alpha$, we fix an increasing sequence
$\\{\zeta_{i}(\alpha)+i\\}$ of ordinals such that

each $\zeta_{i}(\alpha)$ is a limit ordinal or 0 and
$\alpha=\text{sup}_{i}(\zeta_{i}(\alpha)+i)$.

When $\alpha=\text{sup}_{i}(\beta+i)$ for some limit ordinal $\beta$, we put
$\zeta_{i}(\alpha)=\beta$ for each $i\in\mathbb{N}$.

For each countable ordinal number $\xi$, we write $\xi=\gamma(\xi)+n(\xi)$,
where $\gamma(\xi)$ is a limit ordinal or 0 and
$n(\xi)\in\mathbb{N}\cup\\{0\\}$. We build a family $S_{\xi}\subseteq
Fin\mathbb{N}$ by induction.

Definition 3.5.  Let $n\in\mathbb{N}$ and let $\xi$ be a countable ordinal
number , we define

  * •

$S_{n}=\\{\sigma\in Fin\mathbb{N}~{}|~{}|\sigma|\leq n\\}$.

  * •

$S_{\xi}=S_{\gamma(\xi)+n(\xi)}=\Big{\\{}\sigma\in
Fin\mathbb{N}~{}\Big{|}~{}K(\sigma,n(\xi))\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing~{}\text{~{}for~{}some
}l\in\\{1,2,\cdots,i(\sigma,n(\xi))\\}\Big{\\}}.$

Remark 3.2.

  * •

It follows from the definition of the family $S_{\xi}$ that $\sigma\in
S_{\xi}$ for each $\sigma\in Fin\mathbb{N}$ with $|\sigma|\leq n(\xi)+1$.

  * •

Note that if $\sigma=\\{k_{0},\cdots,k_{s}\\}\in S_{\xi}$, then
$\\{l,k_{0},\cdots,k_{s}\\}\in S_{\xi+1}$ for every $l<k_{0}$ with
$l\in\mathbb{N}$.

######  Lemma 3.1.

Let $\xi=\gamma(\xi)+n(\xi)$ be an infinite ordinal number and
$\tau=\\{k_{0},\cdots,k_{s}\\}\in S_{\xi}$. Then we have

$\tau\in S_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}\text{ for some
}l\in\\{1,2,\cdots,i(\tau,n(\xi))\\}.$

###### Proof.

Since $\tau=\\{k_{0},\cdots,k_{s}\\}\in S_{\xi}$, we have $K(\tau,n(\xi))\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing~{}\text{~{}for~{}some
}l\in\\{1,2,\cdots,i(\tau,n(\xi))\\}$.

  * •

If $K(\tau,n(\xi))\in S_{\zeta_{l}(\gamma(\xi))+l},$ then
$\tau=\\{k_{0},\cdots,k_{n(\xi)},k_{n(\xi)+1},\cdots,k_{s}\\}\in
S_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ by Remark 3.2.

  * •

If $K(\tau,n(\xi))=\varnothing$, then $K(\tau,l+n(\xi)+1)\subseteq
K(\tau,n(\xi))$ implies $K(\tau,l+n(\xi)+1)=\varnothing$. So $\tau\in
S_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$.

∎

######  Lemma 3.2.

The family $S_{\xi}$ is inclusive for each countable ordinal $\xi$.

###### Proof.

  * •

It follows from the definition of $S_{\xi}$ that the result is true when $\xi$
is finite.

  * •

Assume that the result holds for every infinite ordinal number $\xi<\alpha$.
Now for $\xi=\alpha$, let
$\tau=\\{k_{j_{0}},k_{j_{1}},\cdots,k_{j_{t}}\\}\subseteq\sigma=\\{k_{0},\cdots,k_{s}\\}\in
S_{\xi}$. Note that $t\leq s$ and
$K(\tau,n(\xi))\subseteq~{}K(\sigma,n(\xi))$. If $t\leq n(\xi)$, then we have
$\tau\in S_{\xi}$ by definition. Consider the case $t>n(\xi)$. Then we have
$s>n(\xi)$ and

$K(\sigma,n(\xi))\in S_{\zeta_{l}(\gamma(\xi))+l}~{}\text{~{}for~{}some }1\leq
l\leq i(\sigma,n(\xi))=k_{n(\xi)}\leq k_{j_{n(\xi)}}=i(\tau,n(\xi)).$

By inductive assumption, we have $K(\tau,n(\xi))\in
S_{\zeta_{l}(\gamma(\xi))+l}$. So $\tau\in S_{\gamma(\xi)+n(\xi)}$.

∎

######  Lemma 3.3.

$S_{\gamma+n}\subseteq S_{\gamma+m}$ , where $\gamma$ is a limit ordinal and
$m\in\mathbb{N},n\in\mathbb{N}\cup\\{0\\}$ such that $n<m$.

###### Proof.

For every $\sigma=\\{k_{0},\cdots,k_{s}\\}\in S_{\gamma+n}$.

  * •

If $s\leq m$, then we have $\sigma\in S_{\gamma+m}$ by definition.

  * •

If $s>m$, then $s>n$. It follows that

$K(\sigma,n)\in S_{\zeta_{l}(\gamma)+l}~{}\text{~{}for~{}some }1\leq l\leq
i(\sigma,n)=k_{n}<k_{m}=i(\sigma,m).$

Since $n<m$, we have $K(\sigma,m)\subseteq K(\sigma,n).$ Since
$S_{\zeta_{l}(\gamma)+l}$ is inclusive by Lemma 3.2, we have

$K(\sigma,m)\in S_{\zeta_{l}(\gamma)+l}\text{~{}for~{}some
}l\in\\{1,2,\cdots,i(\sigma,m)\\}.$

Hence $\sigma\in S_{\gamma+m}.$

∎

######  Lemma 3.4.

For each countable ordinals $\xi$, Ord$S_{\xi}\leq\xi$.

###### Proof.

By Lemma 2.1, the result is true when $\xi$ is finite. Assume that the result
holds for every $\xi<\alpha$. Now $\xi=\alpha$. For every $a\in\mathbb{N}$ and
$\sigma\in S_{\xi}^{a}$, $\\{a\\}\sqcup\sigma\in S_{\xi}.$ It follows that

$K(\\{a\\}\sqcup\sigma,n(\xi))\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing~{}\text{for~{}some
}l\in\\{1,2,\cdots,i(\\{a\\}\sqcup\sigma,n(\xi))\\}.$

  * •

When $n(\xi)\in\mathbb{N}$.

    * –

If $i(\\{a\\}\sqcup\sigma,n(\xi))>a$, then
$K(\sigma,n(\xi)-1)=K(\\{a\\}\sqcup\sigma,n(\xi))\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing$. Note that

$1\leq l\leq i(\\{a\\}\sqcup\sigma,n(\xi))=i(\sigma,n(\xi)-1)\text{~{}which
implies that~{}}\sigma\in S_{\gamma(\xi)+n(\xi)-1}.$

    * –

If $i(\\{a\\}\sqcup\sigma,n(\xi))\leq a$, then $l\leq a\text{ and
}K(\sigma,n(\xi))\subseteq K(\\{a\\}\sqcup\sigma,n(\xi)).$ Since
$S_{\zeta_{l}(\gamma(\xi))+l}$ is inclusive, $K(\sigma,n(\xi))\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing$. So by Remark 3.2, $\sigma\in
S_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ for some $l\in\\{1,2,\cdots,a\\}$.

Therefore,

$S_{\xi}^{a}\subseteq
S_{\gamma(\xi)+n(\xi)-1}\cup(\bigcup_{l=1}^{a}S_{\zeta_{l}(\gamma(\xi))+n(\xi)+l+1}).$

Then by Lemma 2.3 and inductive assumption,

$\text{Ord}(S_{\xi}^{a})\leq\gamma(\xi)+n(\xi)-1<\xi.$

  * •

When $n(\xi)=0$.

    * –

If $i(\\{a\\}\sqcup\sigma,0)=a$, then $\sigma=K(\\{a\\}\sqcup\sigma,0)\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing$, for some
$l\in\\{1,2,\cdots,a\\}$.

    * –

If $i(\\{a\\}\sqcup\sigma,0)<a$, then

$K(\sigma,0)\subseteq K(\\{a\\}\sqcup\sigma,0)\in
S_{\zeta_{l}(\gamma(\xi))+l}\cup\varnothing~{}\text{for~{}some
}l\in\\{1,2,\cdots,i(\\{a\\}\sqcup\sigma,0)\\}$,

which implies $\sigma\in S_{\zeta_{l}(\gamma(\xi))+l+1}$ for some $l<a$, by
Lemma 3.2.

So by Lemma 3.3,

$S_{\xi}^{a}\subseteq\bigcup_{l=1}^{a}S_{\zeta_{l}(\gamma(\xi))+l+1}.$

Then by Lemma 2.3 and inductive assumption,

$\text{Ord}(S_{\xi}^{a})\leq\zeta_{a}(\gamma(\xi))+a+1<\xi.$

So in both cases, we have Ord$S_{\xi}\leq\xi$.

∎

Let $\sigma=\\{k_{0},\cdots,k_{s}\\}\in Fin\mathbb{N}$ and
$\tau=\\{p_{0},\cdots,p_{s}\\}\in Fin\mathbb{N}$ . We denote $\sigma\leq\tau$
if for each $i\in\\{0,1,\cdots,s\\}$, $k_{i}\leq p_{i}$.

######  Lemma 3.5.

Let $\xi$ be an countable ordinal number and $\sigma\in S_{\xi}$. Then for
each $\tau\in\text{Fin}\mathbb{N}$ such that $\sigma\leq\tau$, we have
$\tau\in S_{\xi}$.

###### Proof.

It is easy to see the result is true when $\xi$ is finite. Assume that the
result holds for every countable ordinal number $\xi<\alpha$. For
$\xi=\alpha$, let $\sigma=\\{k_{0},\cdots,k_{s}\\}\in S_{\xi}$ and
$\tau=\\{p_{0},\cdots,p_{s}\\}$ such that $k_{i}\leq p_{i}$ for
$i\in\\{0,1,\cdots,s\\}$.

  * •

If $s\leq n(\xi)$, then $\tau\in S_{\xi}$ by Remark 3.2.

  * •

If $s>n(\xi)$, then $K(\sigma,n(\xi))\in S_{\zeta_{l}(\xi)+l}$ for some $1\leq
l\leq i(\sigma,n(\xi))=k_{n(\xi)}\leq p_{n(\xi)}=i(\tau,n(\xi))$. Note that
$K(\sigma,n(\xi))\leq K(\tau,n(\xi))$. By inductive assumption,

$K(\tau,n(\xi))\in S_{\zeta_{l}(\xi)+l}$ for some
$l\in\\{1,2,\cdots,i(\tau,n(\xi))\\}$.

So $\tau\in S_{\xi}$.

∎

######  Proposition 3.1.

Ord$S_{\xi}=\xi$ for each countable ordinal number $\xi$.

###### Proof.

By Lemma 2.1, it is easy to obtain that the result is true when $\xi$ is
finite. Assume that the result holds for every countable ordinal number
$\xi<\alpha$. For $\xi=\alpha$. By Lemma 3.4, it suffices to show that
Ord$S_{\xi}\geq\xi$.

  * •

If $n(\xi)=0$, then it means that $\xi$ is a limit ordinal. For every
$k\in\mathbb{N}$, let

$S_{\zeta_{k}(\xi)+k}(k)=\\{\tau\in S_{\zeta_{k}(\xi)+k}\mid t>k$ for each
$t\in\tau\\}$.

Then for each $\tau\in S_{\zeta_{k}(\xi)+k}(k)$, $\\{k\\}\sqcup\tau\in
S_{\xi}$ by definition. Since $S_{\xi}$ is inclusive by Lemma 3.2, $\tau\in
S_{\xi}$. So $S_{\zeta_{k}(\xi)+k}(k)\subseteq S_{\xi}$.

Note that

$\text{Ord}S_{\zeta_{k}(\xi)+k}=\text{Ord}S_{\zeta_{k}(\xi)+k}(k).$

Indeed, by Lemma 2.2, $S_{\zeta_{k}(\xi)+k}(k)\subseteq S_{\zeta_{k}(\xi)+k}$
implies $\text{Ord}S_{\zeta_{k}(\xi)+k}(k)\leq\text{Ord}S_{\zeta_{k}(\xi)+k}.$
There is a function $\phi:\mathbb{N}\rightarrow\mathbb{N}$ defined by
$\phi(i)=i+k$. Then by Lemma 3.5,

$\text{ $\phi(\sigma)\in S_{\zeta_{k}(\xi)+k}(k)$ whenever $\sigma\in
S_{\zeta_{k}(\xi)+k}$ and $|\phi(\sigma)|=|\sigma|$}.$

By Lemma 2.4,
$\text{Ord}S_{\zeta_{k}(\xi)+k}\leq\text{Ord}S_{\zeta_{k}(\xi)+k}(k).$ So
$\text{Ord}S_{\zeta_{k}(\xi)+k}=\text{Ord}S_{\zeta_{k}(\xi)+k}(k).$

By Lemma 2.2 and inductive assumption,

$\text{Ord}S_{\xi}\geq\text{Ord}S_{\zeta_{k}(\xi)+k}(k)=\text{Ord}S_{\zeta_{k}(\xi)+k}=\zeta_{k}(\xi)+k\text{~{}for
each~{}}k\in\mathbb{N}.$

Hence $\text{Ord}S_{\xi}\geq\xi$.

  * •

If $n(\xi)\in\mathbb{N}$, then for every $k\in\mathbb{N}$, let

$S_{\xi-1}(k)=\\{\tau\in S_{\xi-1}\mid t>k$ for each $t\in\tau$}.

Then for each $\tau\in S_{\xi-1}(k)$, by Remark 3.2, $\\{k\\}\sqcup\tau\in
S_{\xi}$ which implies $\tau\in S_{\xi}^{k}$. Then $S_{\xi-1}(k)\subseteq
S_{\xi}^{k}$. By the similar argument and inductive assumption,

$\text{Ord}S_{\xi}^{k}\geq\text{Ord}S_{\xi-1}(k)=\text{Ord}S_{\xi-1}=\xi-1.$

Hence $\text{Ord}S_{\xi}\geq\xi$.

∎

###  3.2 Construction of spaces $X_{\xi}$

Definition 3.6. ([13]) Let $X$ be a metric space and let $A,B$ be a pair of
disjoint subsets of $X$. We say that a subset $L\subseteq X$ is a _partition_
of $X$ between $A$ and $B$ if there exist open sets $U,W\subseteq X$
satisfying the following conditions

$A\subseteq U,B\subseteq W\text{ and }X=U\sqcup L\sqcup W.$

Definition 3.7. ([6]) Let $X$ be a metric space and let $A,B$ be a pair of
disjoint subsets of $X$. For every $\epsilon>0$, we say that a subset
$L\subseteq X$ is an _$\epsilon$ -partition_ of $X$ between $A$ and $B$ if
there exist open sets $U,W\subseteq X$ satisfying the following conditions

$A\subseteq U,B\subseteq W,X=U\sqcup L\sqcup W,d(L,A)>\epsilon\text{ and
}d(L,B)>\epsilon$

Clearly, an $\epsilon$-partition $L$ of $X$ between $A$ and $B$ is a partition
of $X$ between $A$ and $B$.

######  Lemma 3.6.

([13], Lemma 1.8.14) Let $F_{i}^{+}$, $F_{i}^{-}$, where $i=1,2,\ldots,n$, be
the pairs of opposite faces of $I^{n}\doteq[0,1]^{n}$. If $I^{n}=L_{0}\supset
L_{1}\supset\ldots\supset L_{n}$ is a decreasing sequence of closed sets such
that $L_{i+1}$ is a partition of $L_{i}$ between $L_{i}\cap F_{i+1}^{+}$ and
$L_{i}\cap F_{i+1}^{-}$ for $i\in\\{0,1,2,\ldots,n-1\\}$, then
$L_{n}\neq\varnothing$.

######  Lemma 3.7.

Let $L_{0}\doteq[0,B]^{n}$ for some $B>0$, $F_{i}^{+}$, $F_{i}^{-}$, where
$i=1,2,\cdots,n$, be the pairs of opposite faces of $L_{0}$ and let
$0<\epsilon<\frac{1}{6}B$. For $k=1,2,\cdots,n$, let $\mathcal{U}_{k}$ be an
$\epsilon$-disjoint and $\frac{1}{3}B$-bounded family of subsets of
$[0,B]^{n}$. Then there exists an $\epsilon$-partition $L_{k+1}$ of $L_{k}$
between $F_{k+1}^{+}\cap L_{k}$ and $F_{k+1}^{-}\cap L_{k}$ such that
$L_{k+1}\subseteq L_{k}\cap(\bigcup\mathcal{U}_{k+1})^{c}$ for
$k=0,1,2,\cdots,n-1$. Moreover, $L_{n}\neq\varnothing$.

###### Proof.

For $k=1,2,\cdots,n$, let
$\mathcal{A}_{k}\doteq\\{U\in\mathcal{U}_{k}~{}|~{}d(U,F_{k}^{+})\leq
2\epsilon\\}$ and
$\mathcal{B}_{k}\doteq\\{U\in\mathcal{U}_{k}~{}|~{}d(U,F_{k}^{+})>2\epsilon\\}$.
Note that $\mathcal{A}_{k}\cup\mathcal{B}_{k}=\mathcal{U}_{k}$. Let

$A_{k}=\bigcup\\{N_{\frac{\epsilon}{3}}(U)~{}|~{}U\in\mathcal{A}_{k}\\}\text{
and }B_{k}=\bigcup\\{N_{\frac{\epsilon}{3}}(U)~{}|~{}U\in\mathcal{B}_{k}\\}.$

Then
$d(A_{k},F_{k}^{-})>B-\frac{1}{3}B-2\epsilon-\frac{\epsilon}{3}>\frac{4}{3}\epsilon$
and $d(B_{k},F_{k}^{+})>2\epsilon-\frac{\epsilon}{3}>\frac{4}{3}\epsilon$. It
follows that

$(A_{k}\cup N_{\frac{4}{3}\epsilon}(F_{k}^{+}))\cap(B_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{-}))=\varnothing.$

Let $L_{k+1}\doteq L_{k}\setminus((A_{k+1}\cup
N_{\frac{4}{3}\epsilon}(F_{k+1}^{+}))\cup(B_{k+1}\cup
N_{\frac{4}{3}\epsilon}(F_{k+1}^{-})))$ for $k=0,1,2,\cdots,n-1$.

Note that $F_{k+1}^{+}\cap L_{k}\neq\varnothing$ and $F_{k+1}^{-}\cap
L_{k}\neq\varnothing$ for $k=0,1,\cdots,n-1.$ Indeed,

  * •

Clearly, $F_{1}^{+}\cap L_{0}\neq\varnothing$ and $F_{1}^{-}\cap
L_{0}\neq\varnothing$.

  * •

For $k=1,2,\cdots,n-1$,

$\begin{split}L_{k}&=L_{k-1}\setminus((A_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{+}))\cup(B_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{-})))\\\ &=L_{k-2}\setminus((A_{k-1}\cup
N_{\frac{4}{3}\epsilon}(F_{k-1}^{+}))\cup(B_{k-1}\cup
N_{\frac{4}{3}\epsilon}(F_{k-1}^{-}))\cup(A_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{+}))\cup(B_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{-})))\\\ &=\cdots\\\
&=L_{0}\setminus((A_{1}\cup N_{\frac{4}{3}\epsilon}(F_{1}^{+}))\cup(B_{1}\cup
N_{\frac{4}{3}\epsilon}(F_{1}^{-}))\cup\cdots\cup(A_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{+}))\cup(B_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{-}))).\end{split}$

Note that $F_{k+1}^{+}\setminus(N_{\frac{4}{3}\epsilon}(F_{1}^{+}))\cup
N_{\frac{4}{3}\epsilon}(F_{1}^{-})\cup\cdots\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{+})\cup N_{\frac{4}{3}\epsilon}(F_{k}^{-}))$ is
a nonempty connected set with diameter greater than $\frac{5}{9}B$ and
$(A_{1}\cup B_{1}\cup\cdots A_{k}\cup B_{k})$ is a union set of a
$\frac{\epsilon}{3}$-disjoint and $\frac{4}{9}B$-bounded family. Then

$F_{k+1}^{+}\setminus((A_{1}\cup
N_{\frac{4}{3}\epsilon}(F_{1}^{+}))\cup(B_{1}\cup
N_{\frac{4}{3}\epsilon}(F_{1}^{-}))\cup\cdots\cup(A_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{+}))\cup(B_{k}\cup
N_{\frac{4}{3}\epsilon}(F_{k}^{-})))\neq\varnothing.$

It follows that $F_{k+1}^{+}\cap L_{k}\neq\varnothing$. Similarly,
$F_{k+1}^{-}\cap L_{k}\neq\varnothing$.

Therefore, $L_{k+1}$ is an $\epsilon$-partition of $L_{k}$ between
$F_{k+1}^{+}\cap L_{k}$ and $F_{k+1}^{-}\cap L_{k}$ such that $L_{k+1}\subset
L_{k}\cap(\bigcup\mathcal{U}_{k+1})^{c}$. By Lemma 3.6,
$L_{n}\neq\varnothing$.

∎

Let $L=\\{n+2~{}|~{}n\in\mathbb{N}\\}$ and
$S_{\xi}[L]=\Big{\\{}\\{k_{0}+2,k_{1}+2,\cdots,k_{s}+2\\}~{}\Big{|}~{}\\{k_{0},k_{1},\cdots,k_{s}\\}\in
S_{\xi}\Big{\\}}$. Then by Corollary 2.1, Ord$S_{\xi}[L]$=Ord$S_{\xi}=\xi$ for
each countable ordinal number $\xi$.

Definition 3.8. For $\tau=\\{k_{0}+2,k_{1}+2,\cdots,k_{m}+2\\}\in S_{\xi}[L]$,
we define

$X_{\tau}=\Big{\\{}(x_{i})_{i=0}^{m}\in(2^{k_{0}}\mathbb{Z})^{m+1}~{}\Big{|}~{}|\\{j~{}|~{}x_{j}\notin
2^{k_{p}}\mathbb{Z}\\}|\leq p\text{ for every
}p\in\\{0,1,\cdots,m\\}\Big{\\}}$.

We consider $X_{\tau}$ with sup-metric.

######  Proposition 3.2.

$\tau\in A_{2}(X_{\tau},d)$ for every
$\tau=\\{k_{0}+2,k_{1}+2,\cdots,k_{m}+2\\}\in S_{\xi}[L]$.

###### Proof.

Suppose that $\tau=\\{k_{0}+2,k_{1}+2,\cdots,k_{m}+2\\}\notin
A_{2}(X_{\tau},d)$, then there are $B$-bounded families
$\mathcal{U}_{0},\mathcal{U}_{1},\cdots,\mathcal{U}_{m}$ such that
$\mathcal{U}_{j}$ is $2^{k_{j}+2}$-disjoint for every $j\in\\{0,1,\cdots,m\\}$
and $\bigcup_{i=0}^{m}\mathcal{U}_{i}$ covers $[0,8B]^{m+1}\cap X_{\tau}$ for
some $B>2^{k_{m}+1}$.

Assume that $p_{1}=\frac{8B}{2^{k_{m}}}\in\mathbb{N}$. Take a bijection
$\psi:\\{1,2,\cdots,p_{1}^{m+1}\\}\to\\{0,1,2,\cdots,p_{1}-1\\}^{m+1}$. Let

$Q(t)=\prod_{j=1}^{m+1}[2^{k_{m}}\psi(t)_{j},2^{k_{m}}(\psi(t)_{j}+1)],\text{where
}\psi(t)_{j}\text{ is the $j$th coordinate of }\psi(t).$

Let $\mathcal{Q}_{1}=\\{Q(t)~{}|~{}t\in\\{1,2,\cdots,p_{1}^{m+1}\\}\\}$, then
$[0,8B]^{m+1}=\bigcup_{Q\in\mathcal{Q}_{1}}Q$. Note that

$[0,8B]^{m+1}\cap
X_{\tau}\subseteq\bigcup_{Q\in\mathcal{Q}_{1}}\partial_{m}Q$, where
$\partial_{m}Q$ is the $m$-dimensional skeleton of $Q$.

Let $L_{0}=[0,8B]^{m+1}$. Since $\overline{N_{2^{k_{m}}}(\mathcal{U}_{m})}$ is
$2^{k_{m}+1}$-disjoint and $(2^{k_{m}+1}+B)$-bounded, by Lemma 3.7, there
exists a $2^{k_{m}+1}$-partition $L_{1}$ of $[0,8B]^{m+1}$ between $F_{1}^{+}$
and $F_{1}^{-}$ such that

$L_{1}\subseteq(\bigcup\overline{N_{2^{k_{m}}}(\mathcal{U}_{m})})^{c}\cap[0,8B]^{m+1}$
and $d(L_{1},F_{1}^{+/-})>2^{k_{m}+1}$,

where $F_{1}^{+/-}$ is a pair of opposite facets of $[0,8B]^{m+1}$. Since
$L_{1}$ is a partition of $[0,8B]^{m+1}$ between $F_{1}^{+}$ and $F_{1}^{-}$,
$[0,8B]^{m+1}=L_{1}\sqcup A_{1}\sqcup B_{1}$ such that $A_{1}$, $B_{1}$ are
open in $[0,8B]^{m+1}$ and $A_{1}$, $B_{1}$ contain two opposite facets
$F_{1}^{-}$, $F_{1}^{+}$ respectively.

Let $\mathcal{M}_{1}=\\{Q\in\mathcal{Q}_{1}~{}|~{}Q\cap
L_{1}\neq\varnothing\\}$ and $M_{1}=\bigcup\mathcal{M}_{1}$. Since $L_{1}$ is
a $2^{k_{m}+1}$-partition of $[0,8B]^{m+1}$ between $F_{1}^{+}$ and
$F_{1}^{-}$, then $M_{1}$ is a partition of $[0,8B]^{m+1}$ between $F_{1}^{+}$
and $F_{1}^{-}$, i.e., $[0,8B]^{m+1}=M_{1}\sqcup A^{\prime}_{1}\sqcup
B^{\prime}_{1}$ such that $A^{\prime}_{1}$, $B^{\prime}_{1}$ are open in
$[0,8B]^{m+1}$ and $A^{\prime}_{1}$, $B^{\prime}_{1}$ contain two opposite
facets $F_{1}^{-}$, $F_{1}^{+}$ respectively. Let
$L^{\prime}_{1}=\partial_{m}M_{1}=\bigcup\\{\partial_{m}Q|Q\in\mathcal{M}_{1}\\}$,
then $[0,8B]^{m+1}\setminus(L^{\prime}_{1}\sqcup A^{\prime}_{1}\sqcup
B^{\prime}_{1})$ is the union of some disjoint open $m+1$-dimensional cubes
with length of edge $=2^{k_{m}}$. So $L_{1}^{\prime}$ is a partition of
$[0,8B]^{m+1}$ between $F_{1}^{+}$ and $F_{1}^{-}$ and
$L^{\prime}_{1}\subseteq(\bigcup\mathcal{U}_{m})^{c}\cap[0,8B]^{m+1}$.

Since $\overline{N_{2^{k_{m-1}}}(\mathcal{U}_{m-1})}$ is
$2^{k_{m-1}+1}$-disjoint and $(2^{k_{m-1}+1}+B)$-bounded, there exists a
$2^{k_{m-1}+1}$-partition $L_{2}$ of $L^{\prime}_{1}$ between $F_{2}^{+}\cap
L^{\prime}_{1}$ and $F_{2}^{-}\cap L^{\prime}_{1}$ such that

$L_{2}\subseteq(\bigcup\overline{N_{2^{k_{m-1}}}(\mathcal{U}_{m-1})})^{c}\cap
L^{\prime}_{1}$ and $d(L_{2},F_{2}^{+/-})>2^{k_{m-1}+1}$.

Since $L_{2}$ is a partition of $L^{\prime}_{1}$ between $F_{2}^{+}\cap
L^{\prime}_{1}$ and $F_{2}^{-}\cap L^{\prime}_{1}$,
$L^{\prime}_{1}=L_{2}\sqcup A_{2}\sqcup B_{2}$ such that $A_{2}$, $B_{2}$ are
open in $L^{\prime}_{1}$ and $A_{2}$, $B_{2}$ contain two opposite facets
$F_{2}^{-}\cap L^{\prime}_{1}$, $F_{2}^{+}\cap L^{\prime}_{1}$ respectively.

Assume that $p_{2}=\frac{8B}{2^{k_{m-1}}}\in\mathbb{N}$. Take a bijection
$\psi:\\{1,2,\cdots,p_{2}^{m+1}\\}\to\\{0,1,2,\cdots,p_{2}-1\\}^{m+1}$. Let

$Q^{\prime}(t,l)=\prod_{j=1}^{l}(2^{k_{m-1}}\psi(t)_{j},2^{k_{m-1}}(\psi(t)_{j}+1))\times
2^{k_{m-1}}\psi(t)_{l}\times\prod_{j=l+1}^{m+1}(2^{k_{m-1}}\psi(t)_{j},2^{k_{m-1}}(\psi(t)_{j}+1)),$

where $\psi(t)_{j}$ is the $j$-th coordinate of $\psi(t)$. Let

$\mathcal{Q}_{2}=\\{\overline{Q^{\prime}(t,l)\cap
L_{1}^{\prime}}~{}~{}|~{}~{}t\in\\{1,2,\cdots,p_{2}^{m+1}\\},l\in\\{1,\cdots,m+1\\}\\}$

be a family of $m$-dimensional cubes with length of edges=$2^{k_{m-1}}$, then
$L^{\prime}_{1}=\bigcup_{Q\in\mathcal{Q}_{2}}Q$. Let

$\mathcal{M}_{2}=\\{Q^{\prime}\in\mathcal{Q}_{2}~{}|~{}Q^{\prime}\cap
L_{2}\neq\varnothing\\}$ and $M_{2}=\bigcup\mathcal{M}_{2}$.

Since $L_{2}$ is a $2^{k_{m-1}+1}$-partition of $L^{\prime}_{1}$ between
$F_{2}^{+}\cap L^{\prime}_{1}$ and $F_{2}^{-}\cap L^{\prime}_{1}$, then
$M_{2}$ is a partition of $L^{\prime}_{1}$ between $F_{2}^{+}\cap
L^{\prime}_{1}$ and $F_{2}^{-}\cap L^{\prime}_{1}$, i.e.,
$L^{\prime}_{1}=M_{2}\sqcup A^{\prime}_{2}\sqcup B^{\prime}_{2}$ such that
$A^{\prime}_{2}$, $B^{\prime}_{2}$ are open in $L^{\prime}_{1}$ and
$A^{\prime}_{2}$, $B^{\prime}_{2}$ contain two opposite facets $F_{2}^{-}\cap
L^{\prime}_{1}$, $F_{2}^{+}\cap L^{\prime}_{1}$ respectively. Let
$L^{\prime}_{2}=\partial_{m-1}M_{2}=\bigcup\\{\partial_{m-1}Q^{\prime}~{}|~{}Q^{\prime}\in\mathcal{M}_{2}\\}$.
Then $L^{\prime}_{1}\setminus(L^{\prime}_{2}\sqcup A^{\prime}_{2}\sqcup
B^{\prime}_{2})$ is the union of some disjoint open $m$-dimensional cubes with
length of edge $=2^{k_{m-1}}$. So $L^{\prime}_{2}$ is a partition of
$L^{\prime}_{1}$ between $F_{2}^{+}\cap L^{\prime}_{1}$ and $F_{2}^{-}\cap
L^{\prime}_{1}$ and
$L^{\prime}_{2}\subseteq(\bigcup(\mathcal{U}_{m-1}\cup\mathcal{U}_{m}))^{c}\cap[0,8B]^{m+1}$.

After $m+1$ steps, we obtain $L^{\prime}_{m+1}$ to be a partition of
$L^{\prime}_{m}$ between $F_{m+1}^{+}\cap L^{\prime}_{m}$ and $F_{m+1}^{-}\cap
L^{\prime}_{m}$ and
$L^{\prime}_{m+1}\subseteq(\bigcup(\bigcup_{i=0}^{m}\mathcal{U}_{i}))^{c}\cap[0,8B]^{m+1}$.
Note that $L^{\prime}_{m+1}$ is $0$-dimensional cubes with length of edge
$=2^{k_{0}}$. By the construction, $L^{\prime}_{m+1}\subseteq X_{\tau}$. Since
$\bigcup_{i=0}^{m}\mathcal{U}_{i}$ covers $[0,8B]^{m+1}\cap X_{\tau}$,
$L^{\prime}_{m+1}=\varnothing$ which is a contradiction with Lemma 3.7.

∎

Definition 3.9.  Let

$\bigoplus\mathbb{Z}=\\{(x_{i})~{}|~{}x_{i}\in\mathbb{Z}\text{ and there
exists }k\in\mathbb{N}\text{ such that }x_{j}=0\text{ for each }j\geq k\\}$

with the sup-metric $\rho$. For every
$\tau=\\{k_{0}+2,k_{1}+2,\cdots,k_{m}+2\\}\in S_{\xi}[L]$, we define an
isometric embedding $i_{\tau}:X_{\tau}\to\bigoplus\mathbb{Z}$ by

$i_{\tau}(x)_{i}=x_{i}~{}~{}\forall~{}i\in\\{0,1,\cdots,m\\}\text{ and
}i_{\tau}(x)_{i}=0~{}~{}\forall~{}i\notin\\{0,1,\cdots,m\\},$

where $i_{\tau}(x)_{i}$ is the $i$-th coordinate of $i_{\tau}(x)$ and
$x=(x_{0},x_{1},\cdots,x_{m})\in X_{\tau}$.

For any countable ordinal number $\xi$, we define $X_{\xi}$ as the disjoint
union of $X_{\tau}$ with $\tau\in S_{\xi}[L]$, i.e.,

$X_{\xi}=\bigsqcup_{\tau\in S_{\xi}[L]}X_{\tau}$

with the metric $d_{\xi}$ which is defined as

$\displaystyle d_{\xi}(x,y)=\begin{cases}\rho(i_{\tau}(x),i_{\tau}(y))&\text{
if }x,y\in
X_{\tau},\cr\max\\{s(\tau_{1}),s(\tau_{2}),\rho(i_{\tau_{1}}(x),i_{\tau_{2}}(y))\\}&\text{
if }x\in X_{\tau_{1}},y\in X_{\tau_{2}}\text{ and
}\tau_{1}\neq\tau_{2},\end{cases}$

where $s(\sigma)=2^{\max\sigma}$ for any $\sigma\in S_{\xi}[L]$.

######  Theorem 3.1.

trasdim$(X_{\xi})\geq\xi$.

###### Proof.

By Proposition 3.2, for every $\tau\in S_{\xi}[L]$, $\tau\in
A_{2}(X_{\tau},d)\subseteq A_{2}(X_{\xi},d)$. i.e., $S_{\xi}[L]\subseteq
A_{2}(X_{\xi},d).$ So

$\text{trasdim}X_{\xi}=\text{Ord}A(X_{\xi},d)=\text{Ord}A_{2}(X_{\xi},d)\geq\text{Ord}S_{\xi}[L]=\xi.$

∎

######  Theorem 3.2.

coasdim$(X_{\xi})\leq\xi$ for each countable ordinal number $\xi$.

###### Proof.

For finite $\xi$ it follows from Theorem 25 in [10]. We proceed by transfinite
induction. For $k\in\mathbb{N}$ we put $T_{k}=\\{\tau\in
S_{\xi}[L]\mid|\tau|\geq n(\xi)+2$ and $i(\tau,n(\xi))\geq k+2\\}$ and
$K_{k}=S_{\xi}[L]\setminus T_{k}$.

Since
$X_{\tau}\subset\\{(x_{1},\dots,x_{|\tau|})\in\mathbb{Z}^{|\tau|}\mid|\\{j~{}|~{}x_{j}\notin
2^{k}\mathbb{Z}\\}|\leq n(\xi)\\}$ for each $\tau\in T_{k}$, by Lemma 3.8 in
[6], for any $r\in\mathbb{N}$, there exists $p\geq 2^{r}$ such that for each
$\tau\in T_{p}$ there exist uniformly bounded $2^{r}$-disjoint families
$\mathcal{U}_{0},\dots,\mathcal{U}_{n(\xi)}$ which cover $X_{\tau}$. Since the
distance between $X_{\tau}$ and $X_{\sigma}$ is greater or equal then $2^{p}$
for $\tau$, $\sigma\in T_{p}$, there exist uniformly bounded $2^{r}-$disjoint
families which are still called $\mathcal{U}_{0},\dots,\mathcal{U}_{n(\xi)}$
such that $\bigcup_{i=0}^{n(\xi)}\mathcal{U}_{i}$ covers the set
$Y=\coprod\\{X_{\tau}\mid\tau\in T_{p}\\}$.

So, we need to prove that coasdim$(X_{\xi}\setminus
Y)=$coasdim$(\coprod_{\tau\in K_{p}}X_{\tau})<\gamma(\xi)$. For each $\tau\in
K_{p}$ we can consider $X_{\tau}$ as a subset of
$X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ for some $l\in\\{1,\dots,p+1\\}$ by
Lemma 3.1 and Remark 3.2. Moreover, we have that the natural embedding
$e_{\tau}:X_{\tau}\to X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ is isometrical by
the definition of the metrics on $X_{\tau}$ and on
$X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$. Let us define a metric space $(Z,d)$.
Put $Z=\coprod_{l\in\\{1,\dots,p+1\\}}X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$.
Define the metric $d$ as follows. On each
$X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ we consider its metric
$d_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$. If $x\in
X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ and $y\in
X_{\zeta_{s}(\gamma(\xi))+s+n(\xi)+1}$ with $s\neq l$, we have that $x\in
X_{\tau}$ and $y\in X_{\sigma}$ for some $\tau\in
S_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1}$ and $\sigma\in
S_{\zeta_{s}(\gamma(\xi))+s+n(\xi)+1}$. Then we put
$d(x,y)=\max\\{s(\tau),s(\sigma),\rho(i_{\tau}(x),i_{\sigma}(y))\\}$ where
$\rho$ is the sup-metric in $\bigoplus\mathbb{Z}$. By Lemma 2.6 we have

$\text{coasdim}(\coprod_{l=1}^{p+1}X_{\zeta_{l}(\gamma(\xi))+l+n(\xi)+1})\leq\max_{l\in\\{1,\dots,p+1\\}}\zeta_{p+1}(\gamma(\xi))+l+n(\xi)+1\leq\zeta_{p+1}(\gamma(\xi))+p+n(\xi)+2<\gamma(\xi).$

Define the map $e:\coprod_{\tau\in K_{p}}X_{\tau}\to Z$ by the conditions
$e(x)=e_{\tau}(x)$ for each $x\in X_{\tau}$. We have that $e$ is an
isometrical embedding. Hence coasdim$(\coprod_{\tau\in
K_{p}}X_{\tau})\leq$coasdim$Z<\gamma(\xi)$.

∎

######  Lemma 3.8.

([6]) Let $X$ be a metric space, if $X$ has complementary-finite asymptotic
dimension, then trasdim$(X)\leq\text{coasdim}(X)$.

######  Theorem 3.3.

coasdim$(X_{\xi})=\xi$ and trasdim$(X_{\xi})=\xi$ for every countable ordinal
$\xi$.

###### Proof.

By Theorem 3.2 and Lemma 3.8, trasdim$(X_{\xi})\leq\xi$. Then
trasdim$(X_{\xi})=\xi$ by Theorem 3.1. By Lemma 3.8,
coasdim$(X_{\xi})\geq$trasdim$(X_{\xi})=\xi$. Then coasdim$(X_{\xi})=\xi$ by
Theorem 3.2.

∎

##  4 Acknowledgements

The authors wish to thank the reviewers for careful reading and valuable
comments. This work was supported by NSFC grant of P.R. China (No. 12071183,
11871342). And the authors (Yan Wu and Jingming Zhu) want to thank V.M.
Manuilov and Benyin Fu for helpful discussions. The third author (Taras Radul)
is grateful to Taras Banakh for valuable and stimulating discussions.

## References

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  * [4] M. Satkiewicz, _Transfinite Asymptotic Dimension._ arXiv:1310.1258v1, 2013\.
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††2020 Mathematics Subject Classification: Primary 11J20 ; Secondary 11J83,
28A78

# On a Kurzweil type theorem via ubiquity

Taehyeong Kim  Taehyeong Kim. Department of Mathematics, Technion,
<EMAIL_ADDRESS>
###### Abstract.

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.

##  1\. 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
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ and an $m\times n$ matrix $A\in
M_{m,n}(\mathbb{R})$, we say that $\mathbf{b}\in\mathbb{R}^{m}$ is
$\psi$-approximable for $A$ if there exist infinitely many solutions
$\mathbf{q}\in\mathbb{Z}^{n}$ to the following inequality

${\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}}<\psi(\|\mathbf{q}\|).}$

Denote by $W_{A}(\psi)$ the set of such vectors in the unit cube $[0,1]^{m}$.
Here and hereafter, $\|\mathbf{x}\|=\max_{1\leq i\leq m}|x_{i}|$ and
$\|\mathbf{x}\|_{\mathbb{Z}}=\min_{\mathbf{n}\in\mathbb{Z}^{m}}\|\mathbf{x}-\mathbf{n}\|$
for $\mathbf{x}\in\mathbb{R}^{m}$. We say that $A\in M_{m,n}(\mathbb{R})$ is
badly approximable if

$\liminf_{\|\mathbf{q}\|\to\infty}\|\mathbf{q}\|^{\frac{n}{m}}\|A\mathbf{q}\|_{\mathbb{Z}}>0.$

Kurzweil proved the following zero-one law for $W_{A}(\psi)$.

######  Theorem 1.1.

[Kur55] If $A\in M_{m,n}(\mathbb{R})$ is badly approximable, then for any
decreasing $\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ we have

${|W_{A}(\psi)|=\begin{dcases}0&\quad\text{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}<\infty,\\\ 1&\quad\text{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{m}=\infty.\end{dcases}}$

Here and hereafter, $|\cdot|$ stands for the Lebesgue measure on
$\mathbb{R}^{m}$.

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.

In this article, we will consider similar results when the badly approximable
condition is relaxed. We say that $A\in M_{m,n}(\mathbb{R})$ is singular if
for any $\epsilon>0$ for all large enough $X\geq 1$ there exists
$\mathbf{q}\in\mathbb{Z}^{n}$ such that

$\|A\mathbf{q}\|_{\mathbb{Z}}<\epsilon X^{-\frac{n}{m}}\quad\text{and}\quad
0<\|\mathbf{q}\|<X.$

Otherwise we call it non-singular (or regular following [Cas57]). One can
check that $A\in M_{m,n}(\mathbb{R})$ is singular if and only if for any
$\epsilon>0$ for all large enough $\ell\in\mathbb{Z}_{\geq 1}$ there exists
$\mathbf{q}\in\mathbb{Z}^{n}$ such that

(1.1) ${\|A\mathbf{q}\|_{\mathbb{Z}}<\epsilon
2^{-\frac{n}{m}\ell}\quad\text{and}\quad 0<\|\mathbf{q}\|<2^{\ell}.}$

Hence $A\in M_{m,n}(\mathbb{R})$ is non-singular if and only if there exists
$\epsilon>0$ such that the set

$L(\epsilon):=\\{\ell\in\mathbb{Z}_{\geq 1}:\text{ there is no solution
}\mathbf{q}\in\mathbb{Z}^{n}\text{ to }\eqref{eq_sing}\text{ with }\ell\\}$

is unbounded. We call $L(\epsilon)$ $\epsilon$-return sequence for $A$.

######  Remark 1.2.

  1. (1)

Note that $A\in M_{m,n}(\mathbb{R})$ is badly approximable if and only if
there exist $\epsilon>0$ and $\ell\geq 1$ such that
$L(\epsilon)\supset\mathbb{Z}_{\geq\ell}$.

  2. (2)

In a dynamical point of view as in [Dan85], the set $L(\epsilon)$ corresponds
to return times to a compact set related to $\epsilon$ of a certain diagonal
flow in the space of lattices.

The following is the main theorem of this article.

######  Theorem 1.3.

Let $A\in M_{m,n}(\mathbb{R})$ be non-singular with $\epsilon$-return sequence
$L(\epsilon)=\\{\ell_{i}\\}_{i\geq 1}$. For any decreasing
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ and $0\leq s\leq m$, the
$s$-dimensional Hausdorff measure of $W_{A}(\psi)$ is given by

$\mathcal{H}^{s}(W_{A}(\psi))=\mathcal{H}^{s}([0,1]^{m})\quad\text{if}\quad\sum_{i=1}^{\infty}2^{\ell_{i}n}\psi(2^{\ell_{i}})^{s}=\infty$

For $\delta>0$, let $\psi_{\delta}(q)=\delta q^{-\frac{n}{m}}$. Denote
$\mathbf{Bad}_{A}(\delta)=[0,1]^{m}\setminus W_{A}(\psi_{\delta})$ and
$\mathbf{Bad}_{A}=\bigcup_{\delta>0}\mathbf{Bad}_{A}(\delta)$. Theorem 1.3
with $\psi=\psi_{\delta}$ and $s=m$ directly implies the following corollary.

######  Corollary 1.4.

If $A\in M_{m,n}(\mathbb{R})$ is non-singular, then for any $\delta>0$, the
set $\mathbf{Bad}_{A}(\delta)$ has Lebesgue measure zero, hence,
$\mathbf{Bad}_{A}$ has Lebesgue measure zero.

######  Remark 1.5.

  1. (1)

There are some historical remarks about Corollary 1.4. The one-dimensional
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. (2)

A zero-one law for Lebesgue measure of $W_{A}(\psi)$ 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 $W_{A}(\psi)$
in multidimensional case.

  3. (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 $A\in M_{m,n}(\mathbb{R})$ is badly approximable, then for any decreasing
$\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ and $0\leq s\leq m$, we have

${\mathcal{H}^{s}(W_{A}(\psi))=\begin{dcases}0&\quad\text{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{s}<\infty,\\\
\mathcal{H}^{s}([0,1]^{m})&\quad\text{if
}\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{s}=\infty.\end{dcases}}$

Moreover, the convergent part holds for every $A\in M_{m,n}(\mathbb{R})$.

###### Proof.

The convergent part will be proved in Section 2. Since the divergence and
convergence of the sums

${\sum_{\ell=1}^{\infty}2^{\ell
n}\psi(2^{\ell})^{s}\quad\text{and}\quad\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{s}\quad\text{coincide},}$

the divergent part follows from Theorem 1.3 and Remark 1.2 (1). ∎

We explore some counterparts of Kurzweil’s theory. We denote by
$w(A,\mathbf{b})$ the supremum of the real numbers $w$ for which, for
arbitrarily large $X$, the inequalities

$\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}}<X^{-w}\quad\text{and}\quad\|\mathbf{q}\|<X$

have an integral solution $\mathbf{q}\in\mathbb{Z}^{n}$. We also denote by
$\widehat{w}(A)$ the supremum of the real numbers $w$ for which, for all
sufficiently large $X$, the inequalities

$\|A\mathbf{q}\|_{\mathbb{Z}}<X^{-w}\quad\text{and}\quad\|\mathbf{q}\|<X$

have an non-zero integral solution $\mathbf{q}\in\mathbb{Z}^{n}$. If
$\widehat{w}({{}^{t}A})>\frac{m}{n}$, then by [BL05, Theorem], for almost all
$\mathbf{b}\in\mathbb{R}^{m}$,

$w(A,\mathbf{b})=\frac{1}{\widehat{w}({{}^{t}A})}<\frac{n}{m}.$

Thus for any $\delta>0$, the Lebesgue measure of $\mathbf{Bad}_{A}(\delta)$ is
full. This is opposite to Corollary 1.4. Note that if
$\widehat{w}({{}^{t}A})>\frac{m}{n}$, then ${}^{t}A$ is singular, hence $A$ is
singular. So, they do not conflict with each other.

If we consider the case $\widehat{w}({{}^{t}A})=\frac{m}{n}$ and ${}^{t}A$ is
singular, then we cannot deduce from [BL05, Theorem] that
$\mathbf{Bad}_{A}(\delta)$ is of full Lebesgue measure for any $\delta>0$. We
will give a certain sufficient condition for $\mathbf{Bad}_{A}(\delta)$ being
of full Lebesgue measure for any $\delta>0$.

If $\mathrm{rk\,}_{\mathbb{Z}}({{}^{t}A}\mathbb{Z}^{m}+\mathbb{Z}^{n})<m+n$,
then $\widehat{w}({{}^{t}A})=\infty$, hence we may assume that
$\mathrm{rk\,}_{\mathbb{Z}}({{}^{t}A}\mathbb{Z}^{m}+\mathbb{Z}^{n})=m+n$. Then
following [BL05, Section 3], there exists a sequence of best approximations
$(\mathbf{y}_{k})_{k\geq 1}$ in $\mathbb{Z}^{m}$ for ${}^{t}A$. Denote
$Y_{k}=\|\mathbf{y}_{k}\|$ and
$M_{k}=\|{{}^{t}A}\mathbf{y}_{k}\|_{\mathbb{Z}}$.

######  Theorem 1.7.

If

$\sum_{k\geq
1}\left(Y_{k+1}^{\frac{m}{n}}M_{k}\right)^{\frac{n}{m+n}}<\infty,$

then for any $\delta>0$, the Lebesgue measure of $\mathbf{Bad}_{A}(\delta)$ is
full.

######  Remark 1.8.

  1. (1)

The summability assumption implies that $Y_{k+1}^{\frac{m}{n}}M_{k}\to 0$ as
$k\to\infty$, hence ${}^{t}A$ is singular.

  2. (2)

This theorem is stronger than the previous observation because if
$\widehat{w}({{}^{t}A})>\frac{m}{n}$, then there is $\gamma>0$ such that
$Y_{k+1}^{\frac{m}{n}+\gamma}M_{k}<1$ for all sufficiently large $k\geq 1$.
Hence,

$\sum_{k\geq
1}\left(Y_{k+1}^{\frac{m}{n}}M_{k}\right)^{\frac{n}{m+n}}<\sum_{k\geq
1}\left(Y_{k+1}^{-\gamma}\right)^{\frac{n}{m+n}}<\infty$

since $Y_{k}$ increases at least geometrically (see [BL05, Lemma 1]).

  3. (3)

It was proved in [BKLR21, KKL] that $A$ is singular on average if and only if
there exists $\delta>0$ such that $\mathbf{Bad}_{A}(\delta)$ has full
Hausdorff dimension. Thus it seems very interesting to obtain an equivalent
Diophantine property of $A$ for $\mathbf{Bad}_{A}(\delta)$ being of full
Lebesgue measure.

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.

Acknowledgments. I would like to thank Victor Beresnevich and Nikolay
Moshchevitin for providing helpful comments.

##  2\. 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].

######  Lemma 2.1 (Hausdorff-Cantelli).

Let $\\{B_{i}\\}_{i\geq 1}$ be a sequence of measurable sets in
$\mathbb{R}^{m}$. For any $0\leq s\leq k$,

${\mathcal{H}^{s}(\limsup_{i\to\infty}B_{i})=0\quad\text{if}\quad\sum_{i}\textup{diam}(B_{i})^{s}<{\infty}.}$

Note that

$W_{A}(\psi)=\limsup_{\|\mathbf{q}\|\to\infty}B(A\mathbf{q},\psi(\|\mathbf{q}\|)),$

where $B(A\mathbf{q},\psi(\|\mathbf{q})\|)$ denotes the ball in
$\mathbb{R}^{m}$ of radius $\psi(\|\mathbf{q}\|)$ centered at $A\mathbf{q}$
modulo $1$, and

$\sum_{\mathbf{q}\in\mathbb{Z}^{n}}\operatorname{diam}(B(A\mathbf{q},\psi(\|\mathbf{q}\|)))^{s}<\infty\iff\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{s}<\infty.$

Hence, using Hausdorff-Cantelli lemma, we have that for any $0\leq s\leq m$,

${\mathcal{H}^{s}(W_{A}(\psi))=0\quad\text{if}\quad\sum_{q=1}^{\infty}q^{n-1}\psi(q)^{s}<\infty.}$

This prove the convergent part of Corollary 1.6.

##  3\. 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 $\mathbb{T}^{m}$ with the supremum norm $\|\cdot\|$. With notation
in [BDV06] we set up the following:

$J=\\{\mathbf{q}\in\mathbb{Z}^{n}\\},\quad
R_{\mathbf{q}}=A\mathbf{q}\in\mathbb{T}^{m},\quad\mathcal{R}=\\{R_{\mathbf{q}}:\mathbf{q}\in
J\\},\quad\beta_{\mathbf{q}}=\|\mathbf{q}\|.$

Let $l=\\{l_{i}\\}$ and $u=\\{u_{i}\\}$ be positive increasing sequences such
that

$l_{i}<u_{i}\quad\text{and}\quad\lim_{i\to\infty}l_{i}=\infty.$

For a decreasing function $\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$, we define

$\Delta_{l}^{u}(\psi,i):=\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:l_{i}<\|\mathbf{q}\|\leq
u_{i}}B(A\mathbf{q},\psi(\|\mathbf{q}\|)).$

It follows that

$W_{A}(\psi)=\limsup_{i\to\infty}\Delta_{l}^{u}(\psi,i):=\bigcap_{M=1}^{\infty}\bigcup_{i=M}^{\infty}\Delta_{l}^{u}(\psi,i).$

Throughout, $\rho:\mathbb{R}^{+}\to\mathbb{R}^{+}$ will denote a function
satisfying $\lim_{r\to\infty}\rho(r)=0$ and is referred to as the ubiquitous
function. Let

${\Delta_{l}^{u}(\rho,i):=\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:l_{i}<\|\mathbf{q}\|\leq
u_{i}}B(A\mathbf{q},\rho(u_{i})).}$

######  Definition 3.1 (Local ubiquity).

Let $B$ be an arbitrary ball in $\mathbb{T}^{m}$. Suppose that there exist a
ubiquitous function $\rho$ and an absolute constant $\kappa>0$ such that

(3.1) ${|B\cap\Delta_{l}^{u}(\rho,i)|\geq\kappa|B|\quad\text{for }i\geq
i_{0}(B).}$

Then the pair $(\mathcal{R},\beta)$ is said to be a locally ubiquitous system
relative to $(\rho,l,u)$.

Finally, a function $h$ is said to be $u$-regular if there exists a positive
constant $\lambda<1$ such that for $i$ sufficiently large

${h(u_{i+1})\leq\lambda h(u_{i}).}$

With notation in [BDV06], the Lebesgue measure on $\mathbb{T}^{m}$ is of type
(M2) with $\delta=m$ and the intersection conditions are also satisfied with
$\gamma=0$. These conditions are not stated here but these extra conditions
exist and need to be established for the more abstract ubiquity.

Combining [BDV06, Corollary 2 and Corollary 4], we have the following theorem.

######  Theorem 3.2.

[BDV06] Suppose that $(\mathcal{R},\beta)$ is a local ubiquitous system
relative to $(\rho,l,u)$ and assume further that $\rho$ is $u$-regular. Then
for any $0\leq s\leq m$

${\mathcal{H}^{s}(W_{A}(\psi))=\mathcal{H}^{s}(\mathbb{T}^{m})\quad\text{if}\quad\sum_{i=1}^{\infty}\frac{\psi(u_{i})^{s}}{\rho(u_{i})^{m}}=\infty.}$

###  3.2. 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
$\mathbf{q}\in\mathbb{Z}^{n}\setminus\\{0\\}$ such that

$\|A\mathbf{q}\|_{\mathbb{Z}}<C\quad\text{and}\quad\|\mathbf{q}\|<X.$

Then for any $\mathbf{b}\in\mathbb{R}^{m}$, there exists
$\mathbf{q}\in\mathbb{Z}^{n}$ such that

$\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}}\leq
C_{1}\quad\text{and}\quad\|\mathbf{q}\|\leq X_{1},$

where

$C_{1}=\frac{1}{2}(h+1)C,\quad X_{1}=\frac{1}{2}(h+1)X,\quad h=X^{-n}C^{-m}.$

This principle implies the following corollary.

######  Corollary 3.4.

Let $A\in M_{m,n}(\mathbb{R})$ be non-singular and let
$L(\epsilon)=\\{\ell_{i}\\}_{i\geq 1}$ be the $\epsilon$-return sequence for
$A$. Then for any $\mathbf{b}\in\mathbb{R}^{m}$, there exists
$\mathbf{q}\in\mathbb{Z}^{n}$ such that

$\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}}\leq\frac{1}{2}(\epsilon^{-m}+1)\epsilon
2^{-\frac{n}{m}\ell_{i}}\quad\text{and}\quad\|\mathbf{q}\|\leq\frac{1}{2}(\epsilon^{-m}+1)2^{\ell_{i}}.$

###### Proof.

It follows directly from Theorem 3.3 with $C=\epsilon
2^{-\frac{n}{m}\ell_{i}}$ and $X=2^{\ell_{i}}$. ∎

###  3.3. Weyl type uniform distribution

In this subsection, we will show Weyl type uniform distribution result for the
sequence
$\left\\{A\mathbf{q}\right\\}_{\mathbf{q}\in\mathbb{Z}^{n}}\subset\mathbb{T}^{m}$.
For $A\in M_{m,n}(\mathbb{R})$, Kronecker’s theorem (see e.g. [Cas57, Chapter
iii@, Theorem iv@]) asserts that the sequence
$\left\\{A\mathbf{q}\right\\}_{\mathbf{q}\in\mathbb{Z}^{n}}$ is dense in
$\mathbb{T}^{m}$ if and only if the subgroup

$G({{}^{t}A}):={{}^{t}A}\mathbb{Z}^{m}+\mathbb{Z}^{n}\subset\mathbb{R}^{n}$

has maximal rank $m+n$ over $\mathbb{Z}$. If $A$ is non-singular, then
${{}^{t}}A$ is non-singular, hence $G({{}^{t}A})$ has maximal rank $m+n$ over
$\mathbb{Z}$. By Kronecker’s theorem, the sequence
$\left\\{A\mathbf{q}\right\\}_{\mathbf{q}\in\mathbb{Z}^{n}}$ is dense in
$\mathbb{T}^{m}$.

But the dense result is not enough for our purpose. We need the following Weyl
type uniform distribution result.

######  Proposition 3.5.

If $G({{}^{t}A})$ has maximal rank $m+n$ over $\mathbb{Z}$, then the sequence
$\left\\{A\mathbf{q}\right\\}_{\mathbf{q}\in\mathbb{Z}^{n}}$ is uniformly
distributed in the following sense: for any ball $B\subset\mathbb{T}^{m}$

$\frac{\\#\\{A\mathbf{q}\in B:\|\mathbf{q}\|\leq
N\\}}{\\#\\{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
N\\}}\to|B|\quad\text{as}\quad N\to\infty.$

###### Proof.

We first claim that for any $\mathbf{c}\in\mathbb{Z}^{m}\setminus\\{0\\}$

$\frac{1}{\\#\\{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
N\\}}\sum_{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq N}e^{2\pi
i(\mathbf{c}\cdot A\mathbf{q})}\to 0\quad\text{as}\quad N\to\infty.$

Indeed, by the maximal rank assumption, we have
${{}^{t}}A\mathbf{c}\in\mathbb{R}^{n}\setminus\mathbb{Q}^{n}$. Without loss of
generality, we may assume that the first coordinate, say $\alpha$, of
${{}^{t}}A\mathbf{c}$ is irrational. It follows from $\mathbf{c}\cdot
A\mathbf{q}={{}^{t}}A\mathbf{c}\cdot\mathbf{q}$ that

$\begin{split}\frac{1}{N^{n}}\left|\sum_{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
N}e^{2\pi i(\mathbf{c}\cdot
A\mathbf{q})}\right|&=\frac{1}{N^{n}}\left|\sum_{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
N}e^{2\pi i({{{}^{t}}A}\mathbf{c}\cdot\mathbf{q})}\right|\\\
&\ll\frac{1}{N^{n}}N^{n-1}\sum_{q_{1}=-N}^{N}e^{2\pi i\alpha q_{1}}\\\
&\leq\frac{1}{N}\frac{2}{|e^{2\pi i\alpha}-1|}.\end{split}$

Since $\\#\\{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq N\\}\asymp N^{n}$,
this prove the claim.

Following the proof of classical Weyl’s criterion (see e.g. [KN74, Theorem
2.1]), we can deduce that for any ball $B\subset\mathbb{T}^{m}$ we have

$\frac{\\#\\{A\mathbf{q}\in B:\|\mathbf{q}\|\leq
N\\}}{\\#\\{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
N\\}}\to|B|\quad\text{as}\quad N\to\infty.$

∎

######  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.

##  4\. Proof of Theorem 1.3

Let $A$ be non-singular and let $L(\epsilon)=\\{\ell_{i}\\}_{i\geq 1}$ be the
$\epsilon$-return sequence. With the notations in Subsection 3.1, we take
sequences $l=l(\epsilon)=\\{l_{i}\\}$ and $u=u(\epsilon)=\\{u_{i}\\}$ as
follows:

$u_{i}=\frac{1}{2}(\epsilon^{-m}+1)2^{\ell_{i}}\quad\text{and}\quad
l_{i}=c_{1}u_{i},$

with some positive constant $c_{1}=c_{1}(\epsilon)<1$, 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 $(\mathcal{R},\beta)$ is a locally ubiquitous system relative to
$\left(\rho(r)=c_{2}r^{-\frac{n}{m}},l,u\right)$ with the constant
$c_{2}=\epsilon\left(\frac{1}{2}(\epsilon^{-m}+1)\right)^{1+\frac{n}{m}}$.

###### Proof.

Fix any ball $B=B(x,r_{0})$ in $\mathbb{T}^{m}$. By Corollary 3.4, we have
that

$B=B\cap\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
u_{i}}B\left(A\mathbf{q},\frac{1}{2}(\epsilon^{-m}+1)\epsilon
2^{-\frac{n}{m}\ell_{i}}\right).$

Since

$\frac{1}{2}(\epsilon^{-m}+1)\epsilon
2^{-\frac{n}{m}\ell_{i}}=\epsilon\left(\frac{1}{2}(\epsilon^{-m}+1)\right)^{1+\frac{n}{m}}u_{i}^{-\frac{n}{m}}=\rho(u_{i}),$

it follows that

(4.1)
${\begin{split}|B|\leq&\left|B\cap\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
l_{i}}B(A\mathbf{q},\rho(u_{i}))\right|\\\
&+\left|B\cap\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:l_{i}<\|\mathbf{q}\|\leq
u_{i}}B(A\mathbf{q},\rho(u_{i}))\right|\end{split}}$

Using Proposition 3.5 with $2B=B(x,2r_{0})$, there is an absolute constant
$C>0$ independent of the choice of $B$ such that for all large enough $i\geq
1$

$\\#\\{A\mathbf{q}\in 2B:\|\mathbf{q}\|\leq l_{i}\\}\leq Cl_{i}^{n}|B|.$

Thus for all large enough $i\geq 1$ so that $\rho(u_{i})<r_{0}$, we have

$\left|B\cap\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:\|\mathbf{q}\|\leq
l_{i}}B(A\mathbf{q},\rho(u_{i}))\right|\leq
Cl_{i}^{n}|B|(2\rho(u_{i}))^{m}=(2c_{2})^{m}Cc_{1}^{n}|B|.$

By taking $0<c_{1}<1$ so that $(2c_{2})^{m}Cc_{1}^{n}<\frac{1}{2}$, which
depends only on $\epsilon$, it follows from (4.1) that for all large enough
$i\geq 1$

${\begin{split}|B\cap\Delta_{l}^{u}(\rho,i)|&=\left|B\cap\bigcup_{\mathbf{q}\in\mathbb{Z}^{n}:l_{i}<\|\mathbf{q}\|\leq
u_{i}}B(A\mathbf{q},\rho(u_{i}))\right|\\\
&\geq|B|-(2c_{2})^{m}Cc_{1}^{n}|B|>\frac{1}{2}|B|.\end{split}}$

∎

###### Proof of Theorem 1.3.

It follows from $\ell_{i+1}\geq\ell_{i}+1$ that for any $i\geq 1$

$\rho(u_{i+1})=\frac{1}{2}(\epsilon^{-m}+1)\epsilon
2^{-\frac{n}{m}\ell_{i+1}}\leq
2^{-\frac{n}{m}}\frac{1}{2}(\epsilon^{-m}+1)\epsilon
2^{-\frac{n}{m}\ell_{i}}=2^{-\frac{n}{m}}\rho(u_{i}),$

hence $\rho$ is $u$-regular. Since the divergence and convergence of the sums

${\sum_{i=1}^{\infty}\frac{\psi(u_{i})^{s}}{\rho(u_{i})^{m}}\quad\text{and}\quad\sum_{i=1}^{\infty}\frac{\psi(2^{\ell_{i}})^{s}}{\rho(2^{\ell_{i}})^{m}}\quad\text{coincide},}$

Theorem 3.2 and Theorem 4.1 imply Theorem 1.3. ∎

##  5\. Proof of Theorem 1.7

The proof of Theorem 1.7 relies on the following Transference principle.

######  Theorem 5.1.

[Cas57, Chapter v@, Theorem xvii@ A] For any $\mathbf{b}\in\mathbb{R}^{m}$,
$C>0$, and $X>1$, if there exists $\mathbf{q}\in\mathbb{Z}^{n}$ such that

$\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}}\leq
C\quad\text{and}\quad\|\mathbf{q}\|\leq X,$

then for any $\mathbf{y}\in\mathbb{Z}^{m}$

$\|\mathbf{b}\cdot\mathbf{y}\|_{\mathbb{Z}}\leq(m+n)\max\left(X\|{{}^{t}A}\mathbf{y}\|_{\mathbb{Z}},C\|\mathbf{y}\|\right).$

As in the introduction, let $(\mathbf{y}_{k})_{k\geq 1}$ be a sequence of best
approximations for ${}^{t}A$. Let $Y_{k}=\|\mathbf{y}_{k}\|$,
$M_{k}=\|{{}^{t}A}\mathbf{y}_{k}\|_{\mathbb{Z}}$, and

$\gamma_{k}=\left(Y_{k+1}^{\frac{m}{n}}M_{k}\right)^{\frac{n}{m+n}}$

for each $k\geq 1$. For any $\delta>0$, consider the set

$B_{\delta}(\\{\gamma_{k}\\})=\\{\mathbf{b}\in[0,1]^{m}:\|\mathbf{b}\cdot\mathbf{y}_{k}\|_{\mathbb{Z}}>(m+n)\delta^{\frac{m}{m+n}}\gamma_{k}\text{
for all large enough }k\geq 1\\}.$

It follows from the assumption $\sum_{k\geq 1}\gamma_{k}<\infty$ that
$|B_{\delta}(\\{\gamma_{k}\\})|=1$. Thus it is enough to show the following
claim.
Claim. $B_{\delta}(\\{\gamma_{k}\\})\subset\mathbf{Bad}_{A}(\delta)$.

###### Proof.

Fix any $\mathbf{b}\in B_{\delta}(\\{\gamma_{k}\\})$. For all large enough
$k\geq 1$,

$\begin{split}\|\mathbf{b}\cdot\mathbf{y}_{k}\|_{\mathbb{Z}}&>(m+n)\delta^{\frac{m}{m+n}}\left(Y_{k+1}^{\frac{m}{n}}M_{k}\right)^{\frac{n}{m+n}}\\\
&>(m+n)\delta^{\frac{m}{m+n}}\left(Y_{k}^{\frac{m}{n}}M_{k}\right)^{\frac{n}{m+n}}\\\
&=(m+n)\max\left(X_{k}M_{k},\delta
X_{k}^{-\frac{n}{m}}Y_{k}\right),\end{split}$

for $X_{k}=\left(\delta Y_{k}M_{k}^{-1}\right)^{\frac{m}{m+n}}$. Using Theorem
5.1 with $X=X_{k}$ and $C=\delta X_{k}^{-\frac{n}{m}}$, it follows that for
all $\mathbf{q}\in\mathbb{Z}^{n}$ with $\|\mathbf{q}\|\leq X_{k}$,

$\|A\mathbf{q}-\mathbf{b}\|_{\mathbb{Z}}>\delta
X_{k}^{-\frac{n}{m}}\geq\delta\|\mathbf{q}\|^{-\frac{n}{m}}.$

Since $X_{k}\to\infty$ as $k\to\infty$, this implies that
$\mathbf{b}\in\mathbf{Bad}_{A}(\delta)$. ∎

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# Global deceleration and inward movements of X-ray knots and rims of RCW 103

Hiromasa Suzuki Department of Physics, Faculty of Science and Engineering,
Konan University, 8-9-1 Okamoto, Higashinada, Kobe, Hyogo 658-8501, Japan
Takaaki Tanaka Department of Physics, Faculty of Science and Engineering,
Konan University, 8-9-1 Okamoto, Higashinada, Kobe, Hyogo 658-8501, Japan
Tsuyoshi Inoue Department of Physics, Faculty of Science and Engineering,
Konan University, 8-9-1 Okamoto, Higashinada, Kobe, Hyogo 658-8501, Japan
Hiroyuki Uchida Department of Physics, Kyoto University, Kitashirakawa Oiwake-
cho, Sakyo, Kyoto 606-8502, Japan  Takuto Narita Department of Physics, Kyoto
University, Kitashirakawa Oiwake-cho, Sakyo, Kyoto 606-8502, Japan

###### Abstract

Kinematics of shocks, ejecta knots, and the compact remnant of a supernova
remnant gives an insight into the nature of the progenitor and surrounding
environment. We report on a proper motion measurement of X-ray knots and rims
of the magnetar-hosting supernova remnant RCW 103. Chandra data obtained in
three epochs, 1999, 2010, and 2016 are used. We find a global deceleration of
12 knots and rims both in northern and southern regions within the last $\sim
24$ yrs, even though its age is thought to be larger than 2 kyr. Some of them
even changed their moving directions from outward ($\sim 1,000$ km s-1) to
inward ($\sim-2,000$ km s-1). Our findings can be explained with a collision
with a high-density medium both in the northern and southern edges of the
remnant, although the remnant may still be expanding in the wind-blown cavity.
The proper motion of the associated magnetar 1E 161348$-$5055 is possibly
detected with a velocity of $\approx 500$ km s-1.

Supernova remnants (1667); X-ray sources (1822); Shocks (2086); Circumstellar
matter (241); Magnetars (992)

††journal: ApJ††facilities: Chandra ††software: HEASoft (v6.30.1; HEASARC
2014), CIAO (v4.15; Fruscione et al. 2006)

##  1 Introduction

RCW 103 is a young or middle-aged supernova remnant (SNR) hosting the compact
object 1E 161348$-$5055 (Tuohy & Garmire, 1980). Its age, i.e., elapsed time
after the supernova explosion, is estimated to be 2.0–4.4 kyr (Carter et al.,
1997; Braun et al., 2019). It has a nearly circular shape with a spatial
extent of $\sim 10\arcmin$ or $\sim 9$ pc at the estimated distance of 3.1 kpc
(Reynoso et al., 2004). Interestingly, its morphologies are similar among
radio, infrared, optical, and X-rays. All show bright emissions in the
southern large area and northern small part. Radio continuum observations
revealed a smooth structure without clear shells (Dickel et al., 1996). Paron
et al. (2006) found an interacting 12CO cloud in the southern area. Infrared
observations also found interacting H2 gas and other elements (Oliva et al.,
1990, 1999; Rho et al., 2001; Reach et al., 2006; Pinheiro Gonçalves et al.,
2011). A 1720 MHz OH maser detection from the southern area also supports the
cloud interaction (Frail et al., 1996). Carter et al. (1997) detected
H$\alpha$ filaments from both south and north, with the northern filament
being much fainter. They estimated the age to be $\sim 2$ kyr based on optical
proper motions of $\sim 1,100$ km s-1.

The compact object 1E 161348$-$5055 has been known as an extraordinary compact
object with a very long periodicity $\sim 6.67$ h (De Luca et al., 2006). In
2016, it exhibited a bursting activity and began to be recognized as a
magnetar (D’Aì et al., 2016; Rea et al., 2016; Tendulkar et al., 2017).
Previous X-ray observations shed light on the relation between the progenitor
and magnetar (Nugent et al., 1984; Frank et al., 2015; Braun et al., 2019;
Zhou et al., 2019). A common conclusion is that the supernova explosion was
less energetic (with an explosion energy of $10^{49}$–$10^{50}$ erg) and the
progenitor was not very massive ($\lesssim 13$ M⊙). Most recently, Narita et
al. (2023) identified X-ray emission from shock-heated circumstellar medium
(CSM) near the edges of RCW 103. They found an enhanced N/O abundance ratio
($\sim 4$) of the CSM, and suggested that the progenitor rotation was not
rapid ($\lesssim 100$ km s-1) and a magnetar formation by dynamo effects in
massive stars ($>35$ M⊙) is unlikely.

From another aspect, constraining the X-ray kinematic properties including
movements of forward shocks, ejecta knots, and the associated magnetar is of
great importance as well to understand the nature of the progenitor and
magnetar. In this paper, we report on proper motion measurements of X-ray
bright knots and rims, and the associated magnetar. Out original purpose was
to determine the explosion center and obtain tight constraints on the age and
kinematics. However, we find a global deceleration and inward movements of the
X-ray knots and rims. In Section 2, we summarize the observation log and data
reduction processes. Our proper motion analysis and results are described in
Section 3. We discuss the origin of the deceleration and inward movements in
Section 4, and conclude in Section 5.

##  2 Observation and data reduction

We use five Chandra ACIS-I (Garmire, 1997) observations of the RCW 103 region
listed in Table 1, which consist of three epochs (1999, 2010, and 2016). The
baselines for the proper motion study are $\approx 11$ yr and $\approx 6$ yr,
for the first and second intervals, respectively. The observation log is
summarized in Table 1.

We use the analysis software CIAO (v4.15; Fruscione et al. 2006) and
calibration database v4.10.2 for the data reduction and analysis. We process
the raw data using the standard data reduction method (chandra_repro).

Table 1: Chandra observation log  ObsID | R.A. (2000) | Dec. (2000) | Date | Exposure (ks) |  $\Delta x$ (pixel)aaCoordinate transformation parameters with respect to those of ObsID 11823. Transformation directions $+\Delta x$ and $+\Delta y$ correspond to $-$R.A. and $+$Dec., respectively. |  $\Delta y$ (pixel)aaCoordinate transformation parameters with respect to those of ObsID 11823. Transformation directions $+\Delta x$ and $+\Delta y$ correspond to $-$R.A. and $+$Dec., respectively.
---|---|---|---|---|---|---
123 | 244$\fdg$40550 | 51$\fdg$02022 | 1999 Sep. 26 | 13.4 | 0.51 | 0.39
11823 | 244$\fdg$41094 | 51$\fdg$02284 | 2010 Jun. 01 | 62.5 | $\cdots$ | $\cdots$
12224 | 244$\fdg$40770 | 51$\fdg$02419 | 2010 Jun. 27 | 17.8 | $-0.10$ | 0.13
18459 | 244$\fdg$41579 | 51$\fdg$04894 | 2016 May 23 | 25.8 |  $\cdots$ bbNo correction is performed due to the low quality of point-like sources. |  $\cdots$ bbNo correction is performed due to the low quality of point-like sources.
18854 | 244$\fdg$41333 | 51$\fdg$04969 | 2016 May 30 | 13.0 | $-0.64$ | 0.29

##  3 Analysis and results

We perform a proper motion study on RCW 103. The procedures and results are
presented in this section. In our analysis, we use the software HEASoft
(v6.30.1; HEASARC 2014), XSPEC (v12.12.1; Arnaud 1996), and AtomDB 3.0.9.
Throughout the paper, uncertainties in the text, figures, and tables indicate
$1\sigma$ confidence intervals.

Figure 1 presents an X-ray image of the whole remnant with our analysis
regions. We choose bright knots and sharp edges as our analysis regions. The
profile extraction directions are determined by eye to roughly correspond to
the directions perpendicular to the boundaries or toward the geometric center
of the remnant.

Figure 1: Exposure-corrected Chandra image of RCW 103 in the energy band of
0.5–5.0 keV. The radial-profile extraction regions are indicated with the
green boxes.

###  3.1 Aspect correction

To maximize the reliability and accuracy of the proper motion measurement, we
perform the aspect correction to individual observations. As a large fraction
of the central region of the field of view is covered by the bright target
source, we only find 7–9 point-like sources with off-axis angles of
$>4\arcmin$. We perform the correction with the CIAO tool wcs_match and
wcs_update. Considering the small number of detected point-like sources and
their off-axis positions, we perform coordinate transformations without
rotation and scaling. The resultant transformation parameters are listed in
Table 1. The relative offsets of 0.1–0.6 pixels, which correspond to
0$\farcs$05–0$\farcs$3, are reasonable according to the pointing accuracy of
Chandra ACIS-I, $\approx 0\farcs$67.111The reference for the pointing accuracy
is https://cxc.harvard.edu/cal/ASPECT/celmon/.

###  3.2 Proper motions of X-ray knots and rims

We measure proper motions using radial profiles extracted from the regions
indicated in Figure 1. The two observations in 2010 are merged after the
aspect correction. The same process is done for the observations in 2016.
Thus, hereafter, we use three images obtained in 1999, 2010, and 2016. Flux
profiles are extracted from the exposure-corrected images in the 0.5–5.0 keV
energy range. Two examples are shown in Figure 2 (a-1) and (b-1).

We use the same method as that taken in Tanaka et al. (2021) and Suzuki et al.
(2022) in calculating the velocities. Two profiles obtained from two epochs
are used. We artificially shift the second profile by $\Delta x$ and evaluate
the difference against the first one with $\chi^{2}(\Delta x)$, which is
defined as

$\chi^{2}(\Delta x)=\sum_{i}\frac{(f_{i}-g_{i}(\Delta x))^{2}}{\Delta
f_{i}^{2}+\Delta g_{i}(\Delta x)^{2}},$ (1)

where $f_{i}$ and $\Delta f_{i}$ indicate the flux and error of the bin number
$i$ in the first observation, and $g_{i}$ and $\Delta g_{i}$ indicate those of
the shifted second profile. This calculation is repeated with various values
of $\Delta x$ and we obtain $\chi^{2}$ as a function of $\Delta x$. The
minimum $\chi^{2}$ value ($\chi^{2}_{\rm min}$) and corresponding shift
($\Delta x_{\rm min}$) are determined by fitting the $\chi^{2}$–$\Delta x$
plot with a parabola function. We calculate proper motion velocity from the
best-fit $\Delta x_{\rm min}$ and known baselines. The profile shift is not
limited to an integer multiple of the bin width. We re-bin the shifted profile
$g(\Delta x)$ with the same bin arrangement as $f$ with an assumption of a
uniform probability distribution inside each bin. Then, the profile-shift
ranges that give $\chi^{2}(\Delta x)=\chi^{2}_{\rm min}+1$ are calculated from
the best-fit parabola functions. These ranges are considered to be 1$\sigma$
confidence ranges of the profiles shifts.

The $\chi^{2}$–$\Delta x$ plots of Reg. 5 and Reg. 6 are shown in Figure 2
(a-2) and (b-2), respectively. The $\Delta x_{\rm min}$ values in the second
interval (2010 to 2016) are negative, whereas those in the first (1999 to
2010) are positive. These indicate that their movements were outward before
2010 but they changed their moving directions to inward after 2010. We show
the calculated velocities of all the analysis regions in the two intervals in
Table 2 and Figure 3. One can see a global deceleration from the first to
second interval. Among them, Regs. 5–8 are firmly found to be moving inward in
the second interval.

We here evaluate possible systematic uncertainties. The pointing accuracy of
Chandra has to be considered, which would be $<0\farcs 67$ after the aspect
correction. Even if we assume that the astrometry offsets between the images
are significant, some of the knots or rims still should be moving inward in
the second interval, because the analysis regions include both northern and
southern edges. The profile extraction directions are determined by eye, and
thus the measured velocities will have some uncertainties due to deviations
from the true moving directions. We evaluate this uncertainty by slightly
changing the extraction direction ($\pm 10$ deg) of Reg. 6, finding a
$\lesssim 100$ km s-1 variation in velocities. This uncertainty is not
significant because the statistical uncertainties are much larger. We also
repeat the analysis procedure with 1) an alternative aspect correction with an
optical source catalog and 2) different extraction energy ranges of 1.0–5.0
keV and 1.5–5.0 keV. The measured velocities are largely consistent with the
ones obtained above with the same tendency (See Appendix A and B).

Figure 2: Radial profiles and derived $\chi^{2}$–$\Delta x$ (shift) plots extracted from Reg. 5 (a) and from Reg. 6 (b). In the radial profiles, negative directions correspond to outer regions. In the $\chi^{2}$–$\Delta x$ plots, positive values indicate outward movements.  Figure 3: Proper motion velocities of our analysis regions. Positive velocities indicate outward movements. A distance of 3.1 kpc is assumed in calculation of the velocities.  Table 2: Proper motion velocitiesaaA distance of 3.1 kpc is assumed. Minus velocities indicate inward movements (Same for Table 3 and 4). Region | Velocity (km s-1) | Velocity (km s-1)
---|---|---
| 1999–2010 | 2010–2016
Reg.1 | $1000\pm 600$ | $-1000\pm 1000$
Reg.2 | $1400\pm 300$ | $-600\pm 500$
Reg.3 | $1600\pm 600$ | $-1100\pm 800$
Reg.4 | $990\pm 400$ | $-500\pm 400$
Reg.5 | $1600\pm 600$ | $-2100\pm 700$
Reg.6 | $1100\pm 300$ | $-1300\pm 500$
Reg.7 | $-10\pm 500$ | $-1500\pm 800$
Reg.8 | $1100\pm 300$ | $-1500\pm 500$
Reg.9 | $60\pm 400$ | $-100\pm 500$
Reg.10 | $880\pm 400$ | $-600\pm 600$
Reg.11 | $1400\pm 500$ | $-70\pm 700$
Reg.12 | $1200\pm 400$ | $-1200\pm 800$

###  3.3 Proper motion of the associated magnetar 1E 161348$-$5055

Using the aspect-corrected, 0.5–5.0 keV images, we measure the proper motion
of the associated magnetar, 1E 161348$-$5055\. For individual observations, we
determine the positions of the magnetar and their statistical errors using the
CIAO tool wavdetect. The determined locations are presented in Figure 4. The
angular displacement between 1999 and 2016 (ObsID 18854) is measured to be
$0\farcs 585\pm 0\farcs 025$, which is converted to $501\pm 21$ km s-1 at a
distance of 3.1 kpc (Reynoso et al., 2004). We note that this displacement
might be insignificant if we consider systematic uncertainties due to the
pointing accuracy.

Figure 4: Locations of the magnetar 1E 161348$-$5055 in 1999, 2010, and 2016.
The central positions and radii of the ellipses indicate the estimated
positions of the magnetar at different times and their statistical errors.

##  4 Discussion

We find a global deceleration of the X-ray knots and rims in RCW 103 in the
last $\sim 24$ yrs, even though its age is thought to be larger than 2 kyr.
Some of them were even moving inward in the second interval, from 2010 to
2016. We here discuss the origin of this sudden deceleration.

Narita et al. (2023) proposed that X-ray emitting plasma near the outer edges
are CSM dominated. They also suggested that the remnant is still expanding in
the wind-blown bubble based on the derived progenitor properties. The X-ray
bright southern and northern edges, on which this work focuses, coincide with
the locations of H$\alpha$ emission (Carter et al., 1997). The southern edge
is thought to be interacting with a molecular cloud (Dickel et al., 1996).
Considering these facts and suggestions, we propose a scenario that both
northern and southern regions interact with molecular or atomic clouds,
although the remnant is still expanding in the wind-blown bubble. We assume
that the northern part is also interacting with a high-density medium but it
is yet to be detected. The weaker H$\alpha$ emission and slower deceleration
in the northern part support an interpretation that the interacting medium
there has a lower density than the southern part.

Regs. 5–8 are found to have decelerated from $\sim+1,000$ km s-1 (outward) to
$\sim-2,000$ km s-1 (inward). This can be interpreted as a reflection of the
shocks due to a collision with a high-density medium. The shock reflection by
an interaction with a high density cloud is studied analytically by Miesch &
Zweibel (1994) and Inoue et al. (2012). For a high Mach number incident shock,
like an SNR blast wave shock, the relation between the incident/reflection
shock velocities and density jump at a cloud surface is given by eq. (A3) of
Inoue et al. (2012). In the case of an incident shock velocity $\sim 1,000$ km
s-1 and a reflection shock velocity $\sim-2,000$ km s-1, the required density
jump is calculated to be $\sim 36$. This is consistent with a typical density
jump between a diffuse ISM and HI clouds. We note that a shock wave can be
reflected whereas shock-heated plasma will not be. The reflected shock
enhances thermal X-rays while moving inward, which can be observed as an
inward movement if the newly enhanced emission is bright enough. If the
observed X-ray radial profiles originate from mixtures of outward- and inward-
moving plasma, actual reflected-shock velocities in the observer’s frame might
be larger than the measured velocities.

Inward moving filaments were found in a few other young SNRs, such as
Cassiopeia A (Sato et al., 2018) and RCW 86 (Suzuki et al., 2022). The inward
filaments are interpreted as reverse shocks for Cassiopeia A and reflection
shocks for RCW 86. The case of RCW 103 is similar to RCW 86. A large
difference is the locations where the inward movements are observed: in the
present case, they are at the outer edges of the X-ray emission, whereas they
are well behind outermost filaments in the case of RCW 86. This is consistent
with our interpretation, a very recent collision in RCW 103.

To summarize, the global deceleration can be understood as a result of a
collision of the shocks with a high-density medium (molecular or atomic
cloud), although the X-ray emitting plasma may still be expanding in the wind-
blown bubble.

##  5 Conclusion

We examined proper motions of X-ray knots and rims in the southern and
northern edges of RCW 103. We found a global deceleration of them within the
last $\sim 24$ yrs, even though its age is thought to be larger than 2 kyr.
Among them, Regs. 5–8 were found to have changed the moving directions from
$\sim+1,000$ km s-1 (outward) to $\sim-2,000$ km s-1 (inward). We confirmed
that the deceleration and inward movements are robust to the uncertainties in
the moving directions and the pointing accuracy of Chandra. The inward
movements can be understood as a shock reflection due to a collision with a
high-density medium. As a conclusion, the global deceleration can be explained
as due to a collision with a high-density medium both in the northern and
southern regions, although the X-ray emitting plasma may still be expanding in
the wind-blown bubble.

We appreciate a fruitful discussion with H. Sano about the surrounding medium.
This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI : 10.26093/cds/vizier). The original description of
the VizieR service was published in 2000, A&AS 143, 23. This work was
partially supported by JSPS/MEXT grant Nos. JP21J00031 (HS), JP19H01936,
JP21H04493 (TT), and JP22H01265 (HU).

##  Appendix A Aspect correction with the NOMAD-1 optical source catalog

In order to evaluate systematic uncertainties associated with the aspect
correction, we here apply another aspect correction. We use the NOMAD-1
optical source catalog (Zacharias et al., 2004) available via the VizieR
service222https://doi.org/10.26093/cds/vizier (Ochsenbein et al., 2000) to
register the point-like sources in the Chandra images. We find 4–6 X-ray
sources (depending on observations) which match the catalog sources. After
correcting all the images, we measure the proper motions in the same way as in
Section 3. The resultant velocities are listed in Table 3. Overall, the
velocities are consistent with the ones obtained in Section 3, suggesting that
the systematic uncertaintes due to the aspect correction are small compared to
the statistical errors.

Table 3: Proper motion velocities in the alternative aspect correction case Region | Velocity (km s-1) | Velocity (km s-1)
---|---|---
| 1999–2010 | 2010–2016
Reg.1 | $1300\pm 500$ | $-3000\pm 1000$
Reg.2 | $1100\pm 300$ | $-1500\pm 500$
Reg.3 | $1100\pm 600$ | $-2100\pm 800$
Reg.4 | $900\pm 400$ | $-1600\pm 500$
Reg.5 | $1700\pm 600$ | $-2400\pm 700$
Reg.6 | $1300\pm 300$ | $-900\pm 500$
Reg.7 | $400\pm 500$ | $-1300\pm 700$
Reg.8 | $1400\pm 300$ | $-400\pm 500$
Reg.9 | $600\pm 400$ | $1400\pm 600$
Reg.10 | $900\pm 400$ | $-100\pm 700$
Reg.11 | $800\pm 500$ | $400\pm 700$
Reg.12 | $1500\pm 400$ | $-900\pm 800$
Reg.13 | $100\pm 800$ | $-2900\pm 1200$

##  Appendix B Proper motions in different energy ranges

We here check for systematic uncertainties of the proper motions due to the
extraction energy range. Because the detector calibration might be less
reliable at low energies due to the contamination on the sensor surface
(Marshall et al., 2004; O’Dell et al., 2015; Plucinsky et al., 2018), we test
two additional cases where we use the 1.0–5.0 keV and 1.5–5.0 keV energy
ranges. The measured velocities are listed in Table 4. In the latter case,
velocities are constrained only for Regs. 6, 7, and 8, due to the limited
statistics. In both cases, one can see that the results are mostly consistent
with the ones in Section 3, showing a global deceleration and a change in
moving directions.

Table 4: Proper motion velocities in the different energy-selection cases Region | Velocity (km s-1) | Velocity (km s-1)
---|---|---
| 1999–2010 | 2010–2016
1.0–5.0 keV case |
Reg.1 | $600\pm 1100$ | $-2000\pm 1100$
Reg.2 | $2200\pm 500$ | $-1000\pm 600$
Reg.3 | $800\pm 900$ | $-300\pm 1000$
Reg.4 | $-500\pm 900$ | $-600\pm 600$
Reg.5 | $1400\pm 900$ | $-1400\pm 900$
Reg.6 | $1700\pm 400$ | $-1309\pm 600$
Reg.7 | $100\pm 600$ | $-1600\pm 900$
Reg.8 | $1200\pm 400$ | $-1600\pm 500$
Reg.9 | $400\pm 500$ | $-400\pm 600$
Reg.10 | $800\pm 500$ | $-500\pm 800$
Reg.11 | $1700\pm 600$ | $100\pm 800$
Reg.12 | $1000\pm 600$ | $-500\pm 900$
Reg.13 | $1400\pm 1100$ | $-3200\pm 1700$
1.5–5.0 keV case |
Reg.6 | $2100\pm 900$ | $-3000\pm 1100$
Reg.7 | $1700\pm 2000$ | $-2300\pm 1500$
Reg.8 | $900\pm 1000$ | $-1100\pm 900$

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\frac{\log~p }{N\submT}\|\bthetahat\supmeta- \bthetabar\|_1^2 \precsim \frac{\log~p }{N\submT} \frac{\log~ d}{n\submS +n\submT}s_\btheta ^2,
\]

\[
\frac{\log~p }{N\submT} \frac{\log~ d}{n\submS +n\submT}s_\btheta ^2 \precsim   \sqrt{\frac{\log~ d}{n\submT}}\|\bdel_0\|_1
\]

under the assumption that $s_\btheta \precsim n\submT/\log ~d$;

\[
\sqrt{\frac{\log~ d}{n\submS +n\submT}}s_\btheta  \precsim   \|\bdel_0\|_1,
\]

we have

\[
\|\bthetahat\supmeta- \bthetabar \|_2^2 \precsim   \sqrt{\frac{\log~ d}{n\submT}}\|\bdel_0\|_1,
\]

\[
\frac{\log~p }{N\submT}\|\bthetahat\supmeta- \bthetabar\|_1^2 \precsim \frac{\log~p }{N\submT}\|\bdel_0\|_1^2,
\]

\[
\frac{\log~p }{N\submT}\|\bdel_0\|_1^2 \precsim \sqrt{\frac{\log~ d}{n\submT}}\|\bdel_0\|_1
\]

under the assumption $\|\bdel_0\|_1 \precsim  \sqrt{n\submT/\log~d}$.

\[
\|\bthetahat\supmeta- \bthetabar \|_2^2 + \frac{\log~ p}{N\submT}\|\bthetahat\supmeta- \bthetabar \|_1^2  \precsim \frac{\log~ d}{n\submS +n\submT}s_\btheta + \sqrt{\frac{\log~ d}{n\submT}}\|\bdel_0\|_1.
\]

Similarly, we have

\[
\|\wh \btheta\supmT - \bthetabar \|_2^2 + \frac{\log~ p}{N\submT}\|\wh \btheta\supmT - \bthetabar \|_1^2  \precsim \frac{\log~ d}{n\submT}s_\btheta.
\]

Hence, by Theorem <ref>,
we obtain
_1 ≾√(log  p /N+ n) s_+ √(N+ n/log p ){( log d/n+ ns_+ √(log d/n) _0_1) ∧( log  d/ns_+ n^-1/2)},
_2^2 ≾log  p /N+ n s_+ ( log d/n+ ns_+ √(log d/n) _0_1) ∧( log  d/ns_+ n^-1/2).

§.§ Supplementary Lemmas

Let $X_1, \ldots, X_n$ be independent mean $0$ sub-exponential random
variables and $K = \max_i\|X_i\|_{\psi_1}$.
Then for any $t>0$, we have
(1/n|∑_i=1^n X_i|≥t
) ≤2 exp{ -C^''
min(t^2/K^2, t/K)n },
where $C^{\prime\prime}>0$ is a universal constant.

For more details see Lemma K2 in the supplementary materials to <cit.>.

A random variable $X$ is sub-Gaussian if and only if $X^2$ is sub-exponential. Moreover,
X_ψ_2^2 ≤X^2_ψ_1 ≤2X_ψ_2^2.

Consider any bounded random variable $X$, thus $|X|\leq M$ almost
surely for some $M$. Then $X$ is a sub-Gaussian random variable with
$\|X\|_{\psi_2} \leq M$.
That is, $\|X\|_{\psi_2} \leq \|X\|_{\infty}$.

Assume $X$ and $Y$ are sub-Gaussian random variables. Then $\|XY\|_{\psi_1} \leq 2\|X\|_{\psi_2}\|Y\|_{\psi_2}$.

§.§ Real Data Application: Results of Estimation of $\bb_0$

SUPSource: Selected predictors and corresponding estimates of the regression coefficients

Intercept     Age         Gender                                  rs264011      rs1367117     rs2030746     rs13078807     rs1801725

-1.9279       0.6290         0.1599                                0.0951        0.0778        0.0666        0.0662         0.0552

rs2200733     rs17013203     rs6882076                             rs2814982     rs998584      rs6908357     rs12670798     rs4917014

-0.0505       -0.0483        -0.0344                               0.0333        0.0323        0.0317        -0.0208        0.0196

rs1883025     rs7903146      rs7130440     rs7941030     rs838880      rs3751413     rs3751812      rs2287019

-0.0168       0.0153        -0.0152       -0.0140       -0.0093       -0.0072       -0.0060        -0.0040

SUP: Selected predictors and corresponding estimates of the regression coefficients

Intercept     Age         rs7903146      rs1532085    rs2642442

-0.9353       0.6761      0.1107         0.1089       -0.0627

rs10767664     rs4650994     rs13355303     rs1260326     rs12603905

0.0613     -0.0492     -0.0219     0.0212     0.0092

CS: Selected predictors and corresponding estimates of the regression coefficients

Intercept     Age         Gender      rs17013203     rs7903146
rs4917014     rs2814982     rs3751812

-1.8716      0.7091      0.1555      0.1015       0.0997
0.0829      -0.0690     0.0676

rs6908357     rs3751413     rs7130440     rs2200733
rs10757278     rs13078807     rs1532085     rs7941030

0.0654       -0.0556     -0.0529     0.0484
-0.0471     0.0442     0.0429     -0.0413

rs2030746     rs264011     rs13355303     rs2287019
rs1801725     rs2708027     rs5219      rs6882076

-0.0393     -0.0389     -0.0353     -0.0352
0.0323     -0.0275     -0.0239     0.0221

rs1883025     rs13265604     rs2596565     rs3177928
rs1367117     rs4299376     rs478607     rs987237

-0.0206     0.0202     0.0154     0.0147
-0.0099     -0.0084     -0.0083     -0.0080

rs2606736     rs838880     rs1559474     rs11246602
rs12670798     rs386000     rs9816226

0.0074     -0.0074     -0.0048     0.0046
0.0031     -0.0029     -0.0018

SUPTrans: Selected predictors and corresponding estimates of the regression coefficients

Intercept     Age     Gender     rs7903146     rs3751812     rs4917014     rs2814982

-0.9976      0.6847     0.1154     0.0615     0.0357     0.0244     -0.0231

rs17013203     rs6908357     rs3751413     rs13078807     rs7130440     rs264011

0.0147     0.0129     -0.0119     0.0082     -0.0051     -0.0047

SAS: Selected predictors and corresponding estimates of the regression coefficients

Intercept     Age     rs1260326     rs7903146
rs2602836     rs174546     rs2081687     rs2642442

-1.0063     0.7255     0.1646     0.0808
0.0575     -0.0482     -0.0388     -0.0360

rs10128711     rs964184     rs1965629     rs11181609
rs8077889     rs7960253     rs9383938     rs10968576

0.0354     0.0348     0.0333     -0.0307
-0.0285     -0.0276     0.0266     0.0250

rs3198697     rs4253772     rs708272     rs2972146
rs1077514     rs1475219     rs1559474     rs2479409

-0.0244     -0.0241     -0.0220     -0.0218
0.0213     -0.0181     -0.0179     0.0163

rs4650994     rs2131925     rs492602     rs2606736
rs12603905    rs10102164    rs1229470    rs5756931

-0.0154     0.0151     0.0142     0.0138
-0.0125     0.0124     -0.0117     -0.0107

rs2871865     rs10060696     rs2287623     rs7074609
rs11723806     rs4142995     rs5112     rs499974

0.0091     0.0088     0.0084     0.0083
0.0077     0.0076     -0.0067     0.0060
rs13078807     rs13128910     rs11065987     rs7565375
Gender     rs581080     rs2710642     rs17013203

-0.0049     -0.0048     -0.0045     0.0044
0.0039     0.0038     -0.0037     -0.0034

rs6805251     rs6725887     rs1047891     rs514230

0.0023     -0.0022     -0.0022     -0.0018

STRIFLE: Selected predictors and corresponding estimates of the regression coefficients

Intercept     Age     rs174546     rs7903146
rs964184     rs10968576     rs1559474     rs581080

-1.2207     0.7973     -0.1042     0.1001
0.0367     0.0336    -0.0325     0.0319

rs2081687     rs2602836    rs1965629     rs4650994
rs4142995     rs7248104     rs10128711     rs11065987

-0.0313     0.0309     0.0297    -0.0278
0.0273     0.0268     0.0261     -0.0256

rs9383938     rs12145743     rs11181609     rs8077889
rs3198697     rs2241423     rs1122608     rs138777

0.0246     -0.0233     -0.0223     -0.0190
-0.0170     0.0164     -0.0160     0.0160

rs2479409     rs2112347     rs5112     rs5756931
rs2642442     rs4253772     rs12328675     rs708272

0.0153     0.0133     -0.0126     -0.0123
-0.0117     -0.0116     -0.0109     -0.0095
rs5219     rs2871865     rs10904908     rs7515577
rs2078267     rs7565375     rs206936     rs6725887

-0.0092     0.0086     -0.0076     0.0066
-0.0062     0.0047     0.0043     -0.0023
rs2131925     rs2972146    rs4942486     rs2287623
rs314253     rs12678919     rs10811662

0.0010     -0.0008     0.0007     0.0006
-0.0006     0.0005     -0.0003