Upper triangular operator matrices and stability of their various spectra

Nikola Sarajlija111e-mail address<EMAIL_ADDRESS>
University of Novi Sad, Faculty of Sciences, Novi Sad, 21000, Serbia

August 29, 2024

###### Abstract

Denote by $T_{n}^{d}(A)$ an upper triangular operator matrix of dimension $n$
whose diagonal entries $D_{i}$ are known, where $A=(A_{ij})_{1\leq i<j\leq n}$
is an unknown tuple of operators. This article is aimed at investigation of
defect spectrum
$\mathcal{D}^{\sigma_{*}}=\bigcup\limits_{i=1}^{n}\sigma_{*}(D_{i})\setminus\sigma_{*}(T_{n}^{d}(A))$
, where $\sigma_{*}$ is a spectrum corresponding to various types of
invertibility: (left, right) invertibility, (left, right) Fredholm
invertibility, left Weyl invertibility, right Weyl invertibility. We give
characterizations for each of the previous types, and provide some sufficent
conditions for the stability of certain spectrum (case
$\mathcal{D}^{\sigma_{*}}=\emptyset$). Our main results hold for an arbitrary
dimension $n\geq 2$ in arbitrary Hilbert or Banach spaces without assuming
separability, thus generalizing results from [11], [12]. Hence, we complete a
trilogy to previous work [9], [10] of the same author, whose goal was to
explore basic invertibility properties of $T_{n}^{d}(A)$ that are studied in
Fredholm theory. We also retrieve a result from [1] in the case $n=2$, and we
provide a precise form of the well known ’filling in holes’ result from [8].

Keywords: defect spectrum, invertibility, stability, $n\times n$, Banach
spaces
MSC 2010: 47A10 47A53

##  1 Introduction. Basic concepts

Throughout this text, let $X,Y,X_{1},...,X_{n}$ be arbitrary Banach spaces.
Collection of linear and bounded operators from $X$ to $Y$ is denoted by
$\mathcal{B}(X,Y)$, where $\mathcal{B}(X):=\mathcal{B}(X,X)$, and if
$T\in\mathcal{B}(X,Y)$ its null and range space are denoted by
$\mathcal{N}(T)$ and $\mathcal{R}(T)$, respectively. $\mathcal{N}(T)$ is
closed subspace of $X$, and we denote its linear dimension by $\alpha(T)$.
$\mathcal{R}(T)$ is a subspace of $Y$, and we denote its codimension by
$\beta(T)$. If $\alpha(T)$ ($\beta(T)$) is not finite we write
$\alpha(T)=\infty$ ($\beta(T)=\infty$). We define
$\mathrm{ind}(T):=\alpha(T)-\beta(T)$ as the index of $T$, if at least one of
the quantities $\alpha(T),\beta(T)$ is finite. Notice that
$\mathrm{ind}(T)\in\mathds{Z}\cup\\{+\infty,-\infty\\}.$

Let $T\in\mathcal{B}(X)$. Size $\alpha(T)$ ($\beta(T)$) introduced in the
previous paragraph measures how close $T$ is to a left (right) invertible
operator. If $\alpha(T)=0$ and $\mathcal{R}(T)$ is closed and complemented,
then $T$ is left invertible, write $T\in\mathcal{G}_{l}(X)$. If $\beta(T)=0$
and $\mathcal{N}(T)$ is complemented, then $T$ is right invertible, write
$T\in\mathcal{G}_{r}(X)$ [3, p. 468]. If $\alpha(T)=\beta(T)=0$ then $T$ is
invertible, write
$T\in\mathcal{G}(X)=\mathcal{G}_{l}(X)\cap\mathcal{G}_{r}(X)$. Sizes $\alpha$
and $\beta$ are systematically studied in Fredholm theory, which basic
notation we introduce right now [14]. Families of left and right Fredholm
operators, respectively, are defined as

$\displaystyle\Phi_{l}(X)=\\{T\in\mathcal{B}(X):\ \alpha(T)<\infty\ and\
\mathcal{R}(T)\ is\ closed\ and\ complemented\\}$

and

$\displaystyle\Phi_{r}(X)=\\{T\in\mathcal{B}(X):\beta(T)<\infty\ and\
\mathcal{N}(T)\ is\ complemented\\}.$

The set of Fredholm operators is

$\Phi(X)=\Phi_{l}(X)\cap\Phi_{r}(X)=\\{T\in\mathcal{B}(X):\alpha(T)<\infty\
and\ \beta(T)<\infty\\}.$

Families of left and right Weyl operators, respectively, are defined as

$\Phi_{l}^{-}(X)=\\{T\in\Phi_{l}(X):\ \mathrm{ind}(T)\leq 0\\}$

and

$\Phi_{r}^{+}(X)=\\{T\in\Phi_{r}(X):\ \mathrm{ind}(T)\geq 0\\}.$

Corresponding spectra of an operator $T\in\mathcal{B}(X)$ are defined as
follows:
-the left spectrum: $\sigma_{l}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\mathcal{G}_{l}(X)\\}$;
-the right spectrum: $\sigma_{r}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\mathcal{G}_{r}(X)\\}$;
-the spectrum: $\sigma(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\mathcal{G}(X)\\}$;
-the left essential spectrum: $\sigma_{le}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\Phi_{l}(X)\\}$;
-the right essential spectrum: $\sigma_{re}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\Phi_{r}(X)\\}$;
-the essential spectrum: $\sigma_{e}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\Phi(X)\\}$;
-the left Weyl spectrum: $\sigma_{lw}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\Phi_{l}^{-}(X)\\}$;
-the right Weyl spectrum: $\sigma_{rw}(T)=\\{\lambda\in\mathds{C}:\lambda-T\not\in\Phi_{r}^{+}(X)\\}$;
All of these spectra are compact nonempty subsets of the complex plane.
Their complements are denoted by
$\rho_{l}(T),\rho_{r}(T),\rho(T),\rho_{le}(T),\rho_{re}(T),\rho_{e}(T),\rho_{lw}(T),\rho_{rw}(T)$,
respectively.

In this article we investigate spectral properties of upper triangular block
operators whose diagonal entries are known while the others are not. Denote by
$T_{n}^{d}(A)$ such an operator, where subscript $n$ denotes the matrix
dimension, superscript $d$ (’down’) means that operators below the main
diagonal are all zero, and $A=(A_{ij})_{1\leq i<j\leq n}$ is a tuple of
unknown operators above the main diagonal. In other words,

$T_{n}^{d}(A)=\begin{bmatrix}D_{1}&A_{12}&A_{13}&...&A_{1,n-1}&A_{1n}\\\
0&D_{2}&A_{23}&...&A_{2,n-1}&A_{2n}\\\ 0&0&D_{3}&...&A_{3,n-1}&A_{3n}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&...&D_{n-1}&A_{n-1,n}\\\
0&0&0&...&0&D_{n}\end{bmatrix}\in\mathcal{B}(X_{1}\oplus
X_{2}\oplus\cdots\oplus X_{n}),$ (1.1)

For convenience, we denote by $\mathcal{B}_{n}$ the collection of all
described tuples $A=(A_{ij})$.

Investigation of spectral properties of block operators began with $2\times 2$
block operators (see [3], [7], [8] and many others). General $n\times n$ block
operators were first studied by Benhida et al. in 2005 [2], but afterwards
they were mainly neglected until a few years ago ([9], [10], [11], [12]). This
article has aim to complete investigation of basic spectral properties of
$T_{n}^{d}(A)$ that appear in Fredholm theory. Our main task will be to study
defect spectrum
$\mathcal{D}^{\sigma_{*}}=\bigcup\limits_{i=1}^{n}\sigma_{*}(D_{i})\setminus\sigma_{*}(T_{n}^{d}(A))$,
where $\sigma_{*}$ is one of the spectra introduced above. In case of
$\mathcal{D}^{\sigma_{*}}=\emptyset$ we say that spectrum $\sigma_{*}$ is
stable [13]. Stability of spectrum has usually been explored in a connection
with single valued extension property or SVEP for short ([5], [6], [13]). Here
we use a different method. It turns out that a relation between
$\bigcup_{k=1}^{n}\sigma_{*}(D_{k})$ and $\sigma_{*}(T_{n}^{d}(A))$ can be
deduced using some characterization statements expressed for appropriate types
of invertibility. Thus, we exploit results from [9], [10] of the present
author in order to gain information about defect spectrum.

Article is organized as follows. For the end of this section we give a useful
lemma that we use in the sequel. Its proof for $n=2$ that is conducted in [3]
extends to an arbitrary dimension $n\geq 2$ as well. In the next section we
give results related to left/right Weyl, (left,right) Fredholm and
(left,right) defect spectrum, respectively.

###### Lemma 1.1

Let $T_{n}^{d}(A)\in\mathcal{B}(X_{1}\oplus\cdots\oplus X_{n}).$ Then:

  * (i)

$\sigma_{le}(D_{1})\subseteq\sigma_{le}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})$;

  * (ii)

$\sigma_{re}(D_{n})\subseteq\sigma_{re}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{re}(D_{k});$

  * (iii)

$\sigma_{le}(D_{1})\cup\sigma_{re}(D_{n})\subseteq\sigma_{e}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})$;

  * (iv)

$\sigma_{lw}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})$;

  * (v)

$\sigma_{rw}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{rw}(D_{k})$;

  * (vi)

$\sigma_{l}(D_{1})\subseteq\sigma_{l}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})$;

  * (vii)

$\sigma_{r}(D_{n})\subseteq\sigma_{r}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{r}(D_{k})$;

  * (viii)

$\sigma_{l}(D_{1})\cup\sigma_{r}(D_{n})\subseteq\sigma(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma(D_{k})$.

##  2 Defect spectra and stability results

###  2.1 Left and right Weyl spectrum

In this subsection we generalize results from [12, Section 3] to arbitrary
Hilbert spaces. We report that Corollaries 3.3 and 3.8 in [12] do not hold
with the equivalence: ’only if’ part is not valid. The reason for this is that
their proofs summon [12, Theorems 2.5, 2.6] which do not hold with the
equality (see [9, Corollaries 3.4, 3.10] for corrected versions). Corollaries
4, 8 and 12 from [11, Section 3] are not valid for analogous reasons. In the
sequel we provide correct forms of the previous.

###### Theorem 2.1

([9, Corollary 3.4]) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces, then

$\sigma_{le}(D_{1})\cup\Big{(}\bigcup\limits_{k=2}^{n+1}\delta_{k}\Big{)}\subseteq\\\
\bigcap\limits_{A\in\mathcal{B}_{n}}\sigma_{lw}(T_{n}^{d}(A)),$ (2.1)

where

$\delta_{k}:=\Big{\\{}\lambda\in\mathds{C}:\ \alpha(D_{k}-\lambda)=\infty\
and\ \sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)<\infty\Big{\\}},\ 2\leq k\leq
n,$ $\delta_{n+1}:=\Big{\\{}\lambda\in\mathds{C}:\
\sum\limits_{s=1}^{n}\beta(D_{s}-\lambda)<\sum\limits_{s=1}^{n}\alpha(D_{s}-\lambda)\Big{\\}}.$

Using the previous theorem we will prove the following generalization to
arbitrary Hilbert spaces of [12, Theorem 3.1].

###### Theorem 2.2

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces, then

$\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})=\sigma_{lw}(T_{n}^{d}(A))\cup\Delta_{1}\cup\Delta_{2}$
(2.2)

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta_{1}=\bigcup\limits_{k=2}^{n}\\{\lambda\in\mathds{C}:\
\alpha(D_{k}-\lambda)=\infty,\ \alpha(D_{s}-\lambda)<\infty\ for\ 2\leq s\leq
k-1\ and$
$\displaystyle\sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)=\infty\\}\cap\rho_{le}(D_{1})\cap\\{\lambda\in\mathds{C}:\
\sum\limits_{s=1}^{n}\beta(D_{s}-\lambda)\geq\sum\limits_{s=1}^{n}\alpha(D_{s}-\lambda)\\},$
$\displaystyle\Delta_{2}=\bigcup\limits_{k=1}^{n}\Big{\\{}\lambda\in\mathds{C}:\
\alpha(D_{s}-\lambda)<\infty\ for\ all\ 1\leq s\leq
n,\sum\limits_{s=1}^{n}\beta(D_{s}-\lambda)\geq$
$\displaystyle\sum\limits_{s=1}^{n}\alpha(D_{s}-\lambda)\ and\
\Big{(}\alpha(D_{k}-\lambda)>\beta(D_{k}-\lambda)\ or\
\mathcal{R}(D_{k}-\lambda)\ is\ not\ closed\Big{)}\Big{\\}}.$

###### Remark 2.3

For $k=2$ condition ’$\alpha(D_{s}-\lambda)<\infty\ for\ 2\leq s\leq k-1$’ in
$\Delta_{1}$ is omitted.

Proof. Obviously,
$\Delta_{1}\cup\Delta_{2}\subseteq\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})$,
and
$\sigma_{lw}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})$
according to Lemma 1.1. Assume that
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})\setminus\sigma_{lw}(T_{n}^{d}(A))$.
Then by Theorem 2.1 we get that $\lambda$ does not belong to the left side of
(2.1), which together with observation
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})$ easily gives
$\lambda\in\Delta_{1}\cup\Delta_{2}$. $\square$

Now, we can give a sufficient condition for the stability of the left Weyl
spectrum.

###### Corollary 2.4

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces, then

$\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})=\sigma_{lw}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.2.

If we summon the separability condition, the previous corollary along with one
additional assumption provides a characterization result for the stability of
the left Weyl spectrum.

###### Corollary 2.5

([12, Theorem 3.1], corrected version) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional separable Hilbert spaces and
$\mathcal{R}(D_{s}-\lambda),\ 2\leq s\leq n,\ \lambda\in\mathds{C}$ are
closed, then

$\bigcup\limits_{k=1}^{n}\sigma_{lw}(D_{k})=\sigma_{lw}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if and only if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.2.

Proof. Sufficiency is clear, and necessity follows from Theorem 2.2 and the
second inclusion in [9, Corollary 3.4]. $\square$

By duality, we obtain results related to the stability of the right Weyl
spectrum. We begin with the following generalization of [12, Theorem 3.6] to
arbitrary Hilbert spaces.

###### Theorem 2.6

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces, then

$\bigcup\limits_{k=1}^{n}\sigma_{rw}(D_{k})=\sigma_{rw}(T_{n}^{d}(A))\cup\Delta_{1}\cup\Delta_{2}$

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta_{1}=\bigcup\limits_{k=1}^{n-1}\\{\lambda\in\mathds{C}:\
\beta(D_{k}-\lambda)=\infty,\ \beta(D_{s}-\lambda)<\infty\ for\ k+1\leq s\leq
n-1\ and$
$\displaystyle\sum\limits_{s=k+1}^{n}\alpha(D_{s}-\lambda)=\infty\\}\cap\rho_{re}(D_{n})\cap\\{\lambda\in\mathds{C}:\
\sum\limits_{s=1}^{n}\alpha(D_{s}-\lambda)\geq\sum\limits_{s=1}^{n}\beta(D_{s}-\lambda)\\},$
$\displaystyle\Delta_{2}=\bigcup\limits_{k=1}^{n}\\{\lambda\in\mathds{C}:\
\beta(D_{s}-\lambda)<\infty\ for\ all\ 1\leq s\leq n,\
\sum\limits_{s=1}^{n}\alpha(D_{s}-\lambda)\geq$
$\displaystyle\sum\limits_{s=1}^{n}\beta(D_{s}-\lambda)\ and\
\beta(D_{k}-\lambda)>\alpha(D_{k}-\lambda)\\}.$

###### Remark 2.7

For $k=n-1$ condition ’$\beta(D_{s}-\lambda)<\infty\ for\ k+1\leq s\leq n-1$’
in $\Delta_{1}$ is omitted.

Now, we can give a sufficient condition for the stability of the right Weyl
spectrum.

###### Corollary 2.8

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces, then

$\bigcup\limits_{k=1}^{n}\sigma_{rw}(D_{k})=\sigma_{rw}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.6.

If we include the separability assumption, we obtain characterization for the
stability of the right Weyl spectrum.

###### Corollary 2.9

([12, Theorem 3.6], corrected version) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional separable Hilbert spaces and
$\mathcal{R}(D_{s}-\lambda),\ 1\leq s\leq n-1,\ \lambda\in\mathds{C}$ are
closed, then

$\bigcup\limits_{k=1}^{n}\sigma_{rw}(D_{k})=\sigma_{rw}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if and only if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.6.

###  2.2 Fredholm spectrum

In this subsection we generalize results from [11, Section 3] to arbitrary
Banach spaces. All of the statements in the sequel will be expressed using a
notion of the complements condition introduced in [10].

###### Definition 2.10

We say that an operator $T\in\mathcal{B}(X,Y)$ satisfies the complements
condition if and only if:
$(i)$ $\mathcal{N}(T)$ is complemented;
$(ii)$ $\overline{\mathcal{R}(T)}$ is complemented.

Hereafter, term ’complemented’ always stands for the topological complement.

Hence, if $T\in\mathcal{B}(X,Y)$ satisfies the complements condition, we have
decompositions

$X=\mathcal{N}(T)\oplus\mathcal{N}(T)^{\prime}\quad
Y=\overline{\mathcal{R}(T)}\oplus\overline{\mathcal{R}(T)}^{\prime}$

where ’ sign denotes a complement.

For example, class of inner regular operators satisfies the complements
condition ([4, 1.1.5 Corollary]). Therefore, all classes of operators
introduced in Section 1 satisfy the complements condition.

###### Theorem 2.11

([10, Corollary 3.41]) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\sigma_{le}(D_{1})\cup\Big{(}\bigcup\limits_{k=2}^{n}\delta_{k}\Big{)}\subseteq\bigcap\limits_{A\in\mathcal{B}_{n}}\sigma_{le}(T_{n}^{d}(A)),$
(2.3)

where

$\delta_{k}:=\Big{\\{}\lambda\in\mathds{C}:\ \alpha(D_{k}-\lambda)=\infty\
and\ \sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)<\infty\Big{\\}},\ 2\leq k\leq
n.$

Now, we generalize [11, Theorem 4] to arbitrary Banach spaces.

###### Theorem 2.12

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})=\sigma_{le}(T_{n}^{d}(A))\cup\Delta_{1}\cup\Delta_{2}$
(2.4)

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta_{1}=\bigcup\limits_{k=2}^{n}\\{\lambda\in\mathds{C}:\
\alpha(D_{k}-\lambda)=\infty,\ \alpha(D_{s}-\lambda)<\infty\ for\ 2\leq s\leq
k-1$ $\displaystyle and\
\sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)=\infty\\}\cap\rho_{le}(D_{1}),$
$\displaystyle\Delta_{2}=\bigcup\limits_{k=1}^{n}\\{\lambda\in\mathds{C}:\
\alpha(D_{s}-\lambda)<\infty\ for\ all\ s=1,...,n\ and$
$\displaystyle\mathcal{R}(D_{k}-\lambda)\ is\ not\ complemented\\}.$

###### Remark 2.13

For $k=2$ condition ’$\alpha(D_{s}-\lambda)<\infty\ for\ 2\leq s\leq k-1$ in
$\Delta_{1}$’ is omitted.

Proof. Obviously,
$\Delta_{1}\cup\Delta_{2}\subseteq\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})$,
and
$\sigma_{le}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})$
according to Lemma 1.1. Assume that
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})\setminus\sigma_{le}(T_{n}^{d}(A))$.
Then by Theorem 2.11 we get that $\lambda$ does not belong to the left side of
(2.3), which together with observation
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})$ easily gives
$\lambda\in\Delta_{1}\cup\Delta_{2}$. $\square$

Now, we can give sufficient condition for the stability of the left Fredholm
spectrum.

###### Corollary 2.14

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})=\sigma_{le}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.12.

###### Corollary 2.15

([11, Corollary 4], corrected version) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. If
$X_{1},...,X_{n}$ are infinite dimensional separable Hilbert spaces and
$\mathcal{R}(D_{s}-\lambda),\ 2\leq s\leq n,\ \lambda\in\mathds{C}$ are
closed, then

$\bigcup\limits_{k=1}^{n}\sigma_{le}(D_{k})=\sigma_{le}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if and only if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.12.

Proof. Sufficiency is obvious, and necessity follows from Theorem 2.12 and the
second inclusion in [9, Corollary 3.17]. $\square$

For the right Fredholm spectrum we provide the following results. First we
generalize [11, Theorem 5] to arbitrary Banach spaces.

###### Theorem 2.16

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\bigcup\limits_{k=1}^{n}\sigma_{re}(D_{k})=\sigma_{re}(T_{n}^{d}(A))\cup\Delta_{1}\cup\Delta_{2}$

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta_{1}=\bigcup\limits_{k=1}^{n-1}\\{\lambda\in\mathds{C}:\
\beta(D_{k}-\lambda)=\infty,\ \beta(D_{s}-\lambda)<\infty\ for\ k+1\leq s\leq
n-1\ and$
$\displaystyle\sum\limits_{s=k+1}^{n}\alpha(D_{s}-\lambda)=\infty\\}\cap\rho_{re}(D_{n}),$
$\displaystyle\Delta_{2}=\bigcup\limits_{k=1}^{n}\\{\lambda\in\mathds{C}:\
\beta(D_{k}-\lambda)<\infty\ for\ all\ 1\leq k\leq n\ and$
$\displaystyle\mathcal{N}(D_{k}-\lambda)\ is\ not\ complemented\\}.$

###### Remark 2.17

For $k=n-1$ condition ’$\beta(D_{s}-\lambda)<\infty\ for\ k+1\leq s\leq n-1$’
in $\Delta_{1}$ is omitted.

Sufficient condition for the stability of the right Fredholm spectrum follows.

###### Corollary 2.18

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\bigcup\limits_{k=1}^{n}\sigma_{re}(D_{k})=\sigma_{re}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.16.

Let us summon separability and go down to Hilbert spaces.

###### Corollary 2.19

([11, Corollary 8], corrected version) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional separable Hilbert spaces and
$\mathcal{R}(D_{s}-\lambda),\ 1\leq s\leq n-1,\ \lambda\in\mathds{C}$ are
closed. Then

$\bigcup\limits_{k=1}^{n}\sigma_{re}(D_{k})=\sigma_{re}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if and only if

$\Delta_{1}\cup\Delta_{2}=\emptyset,$

where $\Delta_{1},\Delta_{2}$ are defined as in Theorem 2.16.

For the end of this section, we provide statements dealing with the Fredholm
spectrum. We begin with

###### Theorem 2.20

([10, Corollary 3.55]) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\sigma_{le}(D_{1})\cup\sigma_{re}(D_{n})\cup\Big{(}\bigcup\limits_{k=2}^{n-1}\delta_{k}\Big{)}\cup\delta_{n}\subseteq\\\
\bigcap_{A\in\mathcal{B}_{n}}\sigma_{e}(T_{n}^{d}(A))$ (2.5)

where

$\displaystyle\delta_{k}=\Big{\\{}\lambda\in\mathds{C}:\
\alpha(D_{k}-\lambda)=\infty\ and\
\sum_{s=1}^{k-1}\beta(D_{s}-\lambda)<\infty\Big{\\}}\cup$
$\displaystyle\Big{\\{}\lambda\in\mathds{C}:\ \beta(D_{k}-\lambda)=\infty\
and\ \sum_{s=k+1}^{n}\alpha(D_{s}-\lambda)<\infty\Big{\\}},\quad 2\leq k\leq
n-1,$ $\displaystyle\delta_{n}=\Big{\\{}\lambda\in\mathds{C}:\
\alpha(D_{n}-\lambda)=\infty\ and\
\sum_{s=1}^{n-1}\beta(D_{s}-\lambda)<\infty\Big{\\}}\cup$
$\displaystyle\Big{\\{}\lambda\in\mathds{C}:\ \beta(D_{1}-\lambda)=\infty\
and\ \sum_{s=2}^{n}\alpha(D_{s}-\lambda)<\infty\Big{\\}},$

###### Theorem 2.21

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})=\sigma_{e}(T_{n}^{d}(A))\cup\Delta$

holds for every $A\in\mathcal{B}_{n}$, where

$\Delta=(\Delta_{1}\cup\Delta_{2})\cap\rho_{le}(D_{1})\cap\rho_{re}(D_{n}),$
$\displaystyle\Delta_{1}=\bigcup\limits_{k=2}^{n-1}\Big{\\{}\lambda\in\mathds{C}:\
\Big{(}\alpha(D_{k}-\lambda)=\sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)=\infty\
and\ \alpha(D_{s}-\lambda)<\infty$ $\displaystyle for\ 2\leq s\leq k-1\Big{)}\
or\
\Big{(}\beta(D_{k}-\lambda)=\sum\limits_{s=k+1}^{n}\alpha(D_{s}-\lambda)=\infty\
and\ \beta(D_{s}-\lambda)<\infty$ $\displaystyle for\ k+1\leq s\leq
n-1\Big{)}\Big{\\}},$ $\displaystyle\Delta_{2}=\Big{\\{}\lambda\in\mathds{C}:\
\Big{(}\alpha(D_{n}-\lambda)=\sum\limits_{s=1}^{n-1}\beta(D_{s}-\lambda)=\infty\
and\ \alpha(D_{s}-\lambda)<\infty$ $\displaystyle for\ 2\leq s\leq n-1\Big{)}\
or\
\Big{(}\beta(D_{1}-\lambda)=\sum\limits_{s=2}^{n}\alpha(D_{s}-\lambda)=\infty\
for\ 2\leq s\leq n-1$ $\displaystyle and\ \beta(D_{s}-\lambda)<\infty\ for\
2\leq s\leq n-1\Big{)}\Big{\\}}.$

Proof. Obviously, $\Delta\subseteq\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})$,
and
$\sigma_{e}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})$
according to Lemma 1.1. Assume that
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})\setminus\sigma_{e}(T_{n}^{d}(A))$.
Then by Theorem 2.20 we get that $\lambda$ does not belong to the left side of
(2.5), which together with observation
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})$ easily gives
$\lambda\in\Delta$. $\square$

Now we can give sufficient condition for the stability of the Fredholm
spectrum.

###### Corollary 2.22

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$D_{s}-\lambda,\ 2\leq s\leq n-1,\ \lambda\in\mathds{C}$ satisfy the
complements condition. Then

$\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})=\sigma_{e}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta=\emptyset,$

where $\Delta$ is defined as in Theorem 2.21.

###### Corollary 2.23

([11, Corollary 12], corrected version) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional separable Hilbert spaces and
$\mathcal{R}(D_{s}-\lambda),\ 2\leq s\leq n,\ \lambda\in\mathds{C}$ are
closed. Then

$\bigcup\limits_{k=1}^{n}\sigma_{e}(D_{k})=\sigma_{e}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if and only if

$\Delta=\emptyset,$

where $\Delta$ is defined as in Theorem 2.21.

Proof. Sufficiency is obvious, and necessity follows from Theorem 2.21 and the
second inclusion in [9, Corollary 3.29]. $\square$

Notice that for $n=2$ statements related to the Fredholm spectrum become
especially elegant.

###### Theorem 2.24

([1, Corollary 3.2 ]) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2})$. Then

$\displaystyle\sigma_{e}(D_{1})\cup\sigma_{e}(D_{2})=\sigma_{e}(T_{n}^{d}(A))\cup\Big{(}\\{\lambda\in\mathds{C}:\
\beta(D_{1}-\lambda)=\alpha(D_{2}-\lambda)=\infty\\}\cap$
$\displaystyle\rho_{le}(D_{1})\cap\rho_{re}(D_{2})\Big{)}$

holds for every $A\in\mathcal{B}_{2}$.

###### Corollary 2.25

Let $D_{1}\in\mathcal{B}(X_{1}),\ D_{2}\in\mathcal{B}(X_{2})$. Then

$\sigma_{e}(D_{1})\cup\sigma_{e}(D_{2})=\sigma_{e}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{2}$ if

$\\{\lambda\in\mathds{C}:\
\beta(D_{1}-\lambda)=\alpha(D_{2}-\lambda)=\infty\\}\cap\\\
\rho_{le}(D_{1})\cap\rho_{re}(D_{2})=\emptyset.$

Thus, we recover Remark 3 from [11, Section 3].

##  3 Spectrum

We begin with the left and the right spectrum, and afterwards conclude with
the spectrum.

###### Theorem 3.1

([10, Corollary 2.3]) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\sigma_{l}(D_{1})\cup\Big{(}\bigcup\limits_{k=2}^{n}\Delta_{k}\Big{)}\subseteq\\\
\bigcap\limits_{A\in\mathcal{B}_{n}}\sigma_{l}(T_{n}^{d}(A))$ (3.1)

where

$\Delta_{k}:=\Big{\\{}\lambda\in\mathds{C}:\
\alpha(D_{k}-\lambda)>\sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)\Big{\\}},\
2\leq k\leq n,$

###### Theorem 3.2

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})=\sigma_{l}(T_{n}^{d}(A))\cup\Delta$
(3.2)

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta=\bigcup\limits_{k=2}^{n}\\{\lambda\in\sigma_{l}(D_{k}):\
\alpha(D_{k}-\lambda)\leq\sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda)\\}.$

Proof. Obviously, $\Delta\subseteq\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})$,
and
$\sigma_{l}(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})$
according to Lemma 1.1. Assume that
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})\setminus\sigma_{l}(T_{n}^{d}(A))$.
Then by Theorem 3.1 we get that $\lambda$ does not belong to the left side of
(3.1), which together with observation
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})$ easily gives
$\lambda\in\Delta$. $\square$

###### Corollary 3.3

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\bigcup\limits_{k=1}^{n}\sigma_{l}(D_{k})=\sigma_{l}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta=\emptyset,$

where $\Delta$ is defined as in Theorem 3.2.

By duality we obtain results related to the right spectrum.

###### Theorem 3.4

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\bigcup\limits_{k=1}^{n}\sigma_{r}(D_{k})=\sigma_{r}(T_{n}^{d}(A))$ (3.3)

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta=\bigcup\limits_{k=1}^{n-1}\\{\lambda\in\sigma_{r}(D_{k}):\
\beta(D_{k}-\lambda)\leq\sum\limits_{s=k+1}^{n}\alpha(D_{s}-\lambda)\\},$

###### Corollary 3.5

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\bigcup\limits_{k=1}^{n}\sigma_{r}(D_{k})=\sigma_{r}(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta=\emptyset,$

where $\Delta$ is defined as in Theorem 3.4.

###### Theorem 3.6

([10, Corollary 2.15]) Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\sigma_{l}(D_{1})\cup\sigma_{r}(D_{n})\cup\Big{(}\bigcup\limits_{k=2}^{n-1}\Delta_{k}\Big{)}\cup\Delta_{n}\subseteq\\\
\bigcap_{A\in\mathcal{B}_{n}}\sigma(T_{n}^{d}(A)),$ (3.4)

where

$\displaystyle\Delta_{k}=\Big{\\{}\lambda\in\mathds{C}:\
\alpha(D_{k}-\lambda)>\sum_{s=1}^{k-1}\beta(D_{s}-\lambda)\Big{\\}}\cup$
$\displaystyle\Big{\\{}\lambda\in\mathds{C}:\
\beta(D_{k}-\lambda)>\sum_{s=k+1}^{n}\alpha(D_{s}-\lambda)\Big{\\}},\quad
2\leq k\leq n-1,$ $\displaystyle\Delta_{n}=\Big{\\{}\lambda\in\mathds{C}:\
\alpha(D_{n}-\lambda)>\sum_{s=1}^{n-1}\beta(D_{s}-\lambda)\Big{\\}}\cup$
$\displaystyle\Big{\\{}\lambda\in\mathds{C}:\
\beta(D_{1}-\lambda)>\sum_{s=2}^{n}\alpha(D_{s}-\lambda)\Big{\\}}.$

###### Theorem 3.7

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\bigcup\limits_{k=1}^{n}\sigma(D_{k})=\sigma(T_{n}^{d}(A))\cup\Delta$

holds for every $A\in\mathcal{B}_{n}$, where

$\displaystyle\Delta=\bigcup\limits_{k=2}^{n-1}\Big{\\{}\lambda\in\mathds{C}:\
\Big{(}\beta(D_{k}-\lambda)\leq\sum\limits_{s=k+1}^{n}\alpha(D_{s}-\lambda)\
and\ \lambda\in\sigma_{r}(D_{k})\Big{)}$ $\displaystyle or\
\Big{(}\alpha(D_{k}-\lambda)\leq\
\sum\limits_{s=1}^{k-1}\beta(D_{s}-\lambda\Big{)}\ and\
\lambda\in\sigma_{l}(D_{k})\Big{)}\Big{\\}}\bigcup$
$\displaystyle\Big{\\{}\lambda\in\mathds{C}:\
0<\beta(D_{1}-\lambda)\leq\sum\limits_{s=2}^{n}\alpha(D_{s}-\lambda)\ or\
0<\alpha(D_{n}-\lambda)\leq\sum_{s=1}^{n-1}\beta(D_{s}-\lambda)\Big{\\}}.$

Proof. Obviously, $\Delta\subseteq\bigcup\limits_{k=1}^{n}\sigma(D_{k})$ and
$\sigma(T_{n}^{d}(A))\subseteq\bigcup\limits_{k=1}^{n}\sigma(D_{k})$ according
to Lemma 1.1. Assume that
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma(D_{k})\setminus\sigma(T_{n}^{d}(A))$.
Then by Theorem 3.6 we get that $\lambda$ does not belong to the left side of
(3.4), which together with observation
$\lambda\in\bigcup\limits_{k=1}^{n}\sigma(D_{k})$ easily gives
$\lambda\in\Delta$. $\square$

###### Corollary 3.8

Let $D_{1}\in\mathcal{B}(X_{1}),\
D_{2}\in\mathcal{B}(X_{2}),...,D_{n}\in\mathcal{B}(X_{n})$. Assume that
$X_{1},...,X_{n}$ are infinite dimensional Hilbert spaces. Then

$\bigcup\limits_{k=1}^{n}\sigma(D_{k})=\sigma(T_{n}^{d}(A))$

holds for every $A\in\mathcal{B}_{n}$ if

$\Delta=\emptyset,$

where $\Delta$ is defined as in Theorem 3.7.

Notice that for $n=2$ statements related to the spectrum become especially
elegant.

###### Theorem 3.9

Let $D_{1}\in\mathcal{B}(X_{1}),\ D_{2}\in\mathcal{B}(X_{2})$. Assume that
$X_{1},X_{2}$ are infinite dimensional Hilbert spaces. Then

$\displaystyle\sigma(D_{1})\cup\sigma(D_{2})=\sigma(T_{2}^{d}(A))\cup\Big{\\{}\lambda\in\mathds{C}:\
0<\beta(D_{1}-\lambda)\leq\alpha(D_{2}-\lambda)\ or$ $\displaystyle
0<\alpha(D_{2}-\lambda)\leq\beta(D_{1}-\lambda)\Big{\\}}.$

holds for every $A\in\mathcal{B}_{2}$.

Observe that Theorem 3.9 is a more precise version of [8, Corollary 7] in the
Hilbert space setting. Namely, in the notation of [8], we have specified the
form of holes in $\sigma(M_{C})$.

###### Corollary 3.10

Let $D_{1}\in\mathcal{B}(X_{1}),\ D_{2}\in\mathcal{B}(X_{2})$. Assume that
$X_{1},X_{2}$ are infinite dimensional Hilbert spaces. Then

$\sigma(D_{1})\cup\sigma(D_{2})=\sigma(T_{2}^{d}(A))$

holds for every $A\in\mathcal{B}_{2}$ if

$\displaystyle\Big{\\{}\lambda\in\mathds{C}:\
0<\beta(D_{1}-\lambda)\leq\alpha(D_{2}-\lambda)\ or$ $\displaystyle
0<\alpha(D_{2}-\lambda)\leq\beta(D_{1}-\lambda)\Big{\\}}=\emptyset.$

Acknowledgments
This work was supported by the Ministry of Education, Science and
Technological Development of the Republic of Serbia under Grant No.
451-03-9/2021-14/ 200125
I wish to express my gratitude to professor Dragan S. Djordjević for
introduction to this topic and for useful comments that greatly improved the
form of this paper.

## References

  * [1] Bai Q. & Huang J. & Chen A., Weyl type theorems of 2×2 upper triangular operator matrices. J. Math. Anal. Appl. 434 (2016), no. 2, 1065–1076.
  * [2] Benhida C. & Zerouali E. H. & Zguitti H. Spectral properties of upper triangular block operators. Acta Sci. Math. (Szeged) 71 (2005), no. 3-4, 681–690.
  * [3] Djordjević D. S., Perturbations of spectra of operator matrices, J. Oper. Theory. 48(3), 467-486 (2002)
  * [4] Djordjević D. S. & Rakočević V., Lectures on generalized inverses, University of Niš, Faculty of Sciences and Mathematics, Niš, 2008
  * [5] Djordjević S. V. & Han Y. M., A note on Weyl’s theorem for operator matrices. Proc. Amer. Math. Soc. 131 (2003), no. 8, 2543–2547.
  * [6] Djordjević S. V. & Zguitti H., Essential point spectra of operator matrices through local spectral theory. J. Math. Anal. Appl. 338 (2008), no. 1, 285–291.
  * [7] Du H. K. & Jin P., Perturbation of spectrums of 2×2 operator matrices, Proc. Amer. Math. Soc. 121 (1994), no. 3, 761–766
  * [8] Han J. K. & Lee H. Y. & Lee W. Y., Invertible completions of 2×2 upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (2000), no. 1, 119–123
  * [9] Sarajlija N. Fredholmness and Weylness of block operator matrices, submitted to Complex Anal. Oper. Theory222available at http://arxiv.org/abs/2108.12425
  * [10] Sarajlija N. Perturbing the spectrum of $T_{n}^{d}(A)$ and some related results, submitted to J. Korean Math. Soc.333available at https://arxiv.org/abs/2108.12153
  * [11] Wu X. & Huang J., Essential spectrum of upper triangular operator matrices, Ann. Funct. Anal, 11 (2020), no. 3, 780–798
  * [12] Wu X. & Huang J., Weyl spectrum of upper triangular operator matrices, Acta Math. Sin. 36(7), 783-796 (2020)
  * [13] Zerouali E. H. & Zguitti H. Perturbation of spectra of operator matrices and local spectral theory. J. Math. Anal. Appl. 324 (2006), no. 2, 992–1005.
  * [14] Živković Zlatanović S. Č. & Rakočević V. & Djordjević D. S., Fredholm theory, to appear


https://perso.pages.math.cnrs.fr/users/claude.sabbah

# Kodaira-Saito vanishing for the irregular Hodge filtration

Claude Sabbah CMLS, CNRS, École polytechnique, Institut Polytechnique de
Paris, 91128 Palaiseau cedex, France<EMAIL_ADDRESS>
###### Abstract.

After making correct, and then improving, our definition of the category of
irregular mixed Hodge modules thanks to Mochizuki’s recent results, we show
how these results allow us to obtain Kodaira-Saito-type vanishing theorems for
the irregular Hodge filtration of irregular mixed Hodge modules.

###### 1991 Mathematics Subject Classification:

14F10, 32S40, 34M40


# A reciprocity theorem for the Cohen-Ramanujan sums and its application to
Cohen-Ramanujan expansions in the second variable

K Vishnu Namboothiri  Department of Mathematics, Baby John Memorial Government
College, Chavara, Sankaramangalam, Kollam, Kerala - 691583, India  Department
of Collegiate Education, Government of Kerala, India<EMAIL_ADDRESS>and  Vinod Sivadasan  Department of Mathematics, University College,
Thiruvananthapuram (Research Centre of the University of Kerala), Kerala -
695034, India  Department of Mathematics, College of Engineering Trivandrum,
Thiruvananthapuram, Kerala - 695034, India  Department of Collegiate
Education, Government of Kerala, India<EMAIL_ADDRESS>
###### Abstract.

For an arithmetical function $f$, its Ramanujan expansion is a series
expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k)c_{k}(n)$ where
$a(k)$ are complex numbers and
$c_{k}(n):=\sum\limits_{\begin{subarray}{c}m=1\\\
(m,k)=1\end{subarray}}^{k}e^{\frac{2\pi imn}{k}}$ is the Ramanujan sum. Here
we prove a reciprocity result on Cohen-Ramanujan sums
$c_{k}^{s}(n):=\sum\limits_{\begin{subarray}{c}{h=1}\\\
(h,k^{s})_{s}=1\end{subarray}}^{k^{s}}e^{\frac{2\pi inh}{k^{s}}}$ to change
the position of $k$ and $n$ in a twisted function and use it to prove that for
certain arithmetical functions $f$, Cohen-Ramanujan series expansions in the
form $\sum\limits_{k=1}^{\infty}a(k)c_{k}^{(s)}(n)$ exist if and only if
expansions in the form $\sum\limits_{k=1}^{\infty}b(k/n)c_{n}^{(s)}(k)$ exist.

###### Key words and phrases:

Ramanujan sum, Ramanujan expansions, Cohen-Ramanujan sum, Cohen-Ramanujan
expansions, Jordan totient function, Klee’s function, Möbius inversion
formula, reciprocity theorem

###### 2010 Mathematics Subject Classification:

11A25, 11L03

##  1\. Introduction

For an arithmetical function $f$, its _Ramanujan expansion_ is a series
expansion in the form $f(n)=\sum\limits_{k=1}^{\infty}a(k)c_{k}(n)$ where
$a(k)$ are complex numbers and $c_{k}(n)$ is the _Ramanujan sum_ defined as

$\displaystyle c_{k}(n):=\sum\limits_{\begin{subarray}{c}{m=1}\\\
(m,k)=1\end{subarray}}^{k}e^{\frac{2\pi imn}{k}}.$

provided that the series on the right of $f(n)$ converges. Srinivasa Ramanujan
gave such expansions for some arithmetical functions in [12] which were
pointwise convergent. Such expansions and necessary and sufficient conditions
for such expansions to exist became a topic of discussion in many papers. A.
Wintner [13] gave a condition for the existence of Ramanujan expansions which
was later improved by H. Delange in [4]. L. Lucht gave some other conditions
in [9] and [10] for such expansions to exist. The paper [11] by Ram Murty
discusses some new and old results related to these expansions.

Ramanujan sum was generalized by many later. E. Cohen gave a generalization of
the Ramanujan sum in [2] defining

(1) $\displaystyle c_{k}^{s}(n):=\sum\limits_{\begin{subarray}{c}{h=1}\\\
(h,k^{s})_{s}=1\end{subarray}}^{k^{s}}e^{\frac{2\pi inh}{k^{s}}},$

where $(a,b)_{s}$ is the generalized gcd of $a$ and $b$. A. Chandran and K. V.
Namboothiri derived some conditions for the existence of Ramanujan series like
expansions for certain arithmetical functions in terms of the generalized
Ramanujan sum (1) (henceforth called as the _Cohen-Ramanujan sum_) in [1]. The
method of arguments they used were those employed by L. Lucht in [9] and [10].

Let us call $k$ to be the first variable and $n$ the second in $c_{k}(n)$ and
$c_{k}^{s}(n)$. The Ramanujan expansions and Cohen-Ramanujan expansions
mentioned in the papers above are all in the first variable $k$ of the
(Cohen-)Ramanujan sum. K. R. Johnson took a different route and proved that
for certain arithmetical functions $f$ having Ramanujan expansions (which he
called as the $C$-series expansions), there exist expansions in the second
variable ([8, Theorem 7]). Precisely, he proved that

_A function f has an absolutely convergent C-series representation
$f(n)=\sum\limits_{k=1}^{\infty}a(k)c_{k}(n)$ with $a(k)=0$ if $k$ is not
square-free, if and only if there exists an absolutely convergent
$C^{\prime}$-series representation
$f(n)n^{*}\mu(\overline{n})=\sum\limits_{k=1}^{\infty}b(k/n^{*})c_{n}(k)$._

Here $\overline{n}$ is the largest square-free divisor of $n$, called the
_core_ of $n$ and $n^{*}=n/\overline{n}$. He proved it with the help of a
_reciprocity result_ ([8, Theorem 6]) which enabled him to interchange the
positions of $k$ and $n$ in the Ramanujan sum twisted by the Möbius function
$\mu$. Here we derive an analogous reciprocity result (Theorem 3.8) involving
Cohen-Ramanujan sums and consequently show that for certain arithmetical
functions $f$, an expansion in the form
$\sum\limits_{k=1}^{\infty}a(k)c_{k}^{(s)}(n^{s})$ exists if and only if an
expansion in the form
$\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{k=1}^{\infty}\xi_{n^{*}}\,(k)b(k/n^{*})c_{n}^{(s)}(k^{s})$
exists (Theorem 3.9). Here $\xi_{d}(k)=d$ if $d|k$ and 0 otherwise.

##  2\. Notations and basic results

By $(a,b)_{s}$, we mean the _generalized GCD_ of $a$ and $b$ defined to be the
largest $d^{s}\in\mathbb{N}$ ($d\in\mathbb{N}$) such that $d^{s}|a$ and
$d^{s}|b$. Hence $(a,b)_{1}$ is the usual GCD $(a,b)$. By $J_{s}$ we mean the
_Jordan totient function_ defined as
$J_{s}(n):=n^{s}\prod\limits_{\begin{subarray}{c}{p\mid
n}\end{subarray}}(1-\frac{1}{p^{s}})$. Note that $J_{1}(n)=\varphi(n)$ where
$\varphi$ is the Euler totient function. $\omega(k)$ will be used to denote
the number of distinct prime divisors of $k$. Unless otherwise stated, $p$
will be used to represent primes.

_The Cohen-Ramanujan sum_ (or _generalized Ramanujan sum_) defined by E. Cohen
in [2] is the sum given in equation (1). By ([2, Theorem 1]), $c_{k}^{(s)}(n)$
is multiplicative in $k$. When $k$ is a prime power, the values of the Cohen-
Ramanujan sum are as folows ([2, Theorem 3]).

(2) $\displaystyle
c_{p^{j}}^{(s)}(n)=\begin{cases}p^{sj}-p^{s(j-1)}&\quad\text{if }p^{sj}|n\\\
-p^{s(j-1)}&\quad\text{if }p^{s(j-1)}||n\\\ 0&\quad\text{if }p^{s(j-1)}\nmid
n.\end{cases}$

Cohen-Ramanujan sum satisfies the following relation with the Möbius function
$\mu$ ([2, Theorem 3]):

(3) $\displaystyle c_{k}^{(s)}(n)=\sum\limits_{\begin{subarray}{c}d|k\\\
d^{s}|n\end{subarray}}\mu(k/d)d^{s}.$

Also, with the Jordan totient function, it holds the relation ([3, Theorem 2])

(4) $\displaystyle c_{k}^{(s)}(n)=\frac{J_{s}(n)\mu(m)}{J_{s}(m)}\quad\text{
where }m=\frac{n}{(k,n)}.$

Sometimes we club the conditions $d|k$ and $d^{s}|n$ into the single condition
$d^{s}|(n,k^{s})_{s}$.

By the notation $d^{m}||n$ we mean that $d^{m}|n$ and $d^{m+1}\nmid n$. For a
prime $p$, the exponent of $p$ in the prime factorization of $k$ will be
denoted by $e_{p}(k)$. Hence $p^{e_{p}(k)}||k$. Analogously, we write
$a:=e_{p}^{(s)}(n)$ if $p^{as}|n$ and $p^{(a+1)s}\nmid n$. In this case, we
also use the notation $p^{as}||_{s}n$. We use the Möbius inversion formula and
a variant of it many times in this paper. For two arithmetical functions $f$
and $g$, it states that

(5) $\displaystyle
g(k)=(\mu*f)(k):=\sum\limits_{d|k}\mu(d)f\left(\frac{k}{d}\right)\Longleftrightarrow
f(k)=(g*u)(k)=\sum\limits_{d|k}g(d)$

where $*$ is the Dirichlet product, and $u\equiv 1$ is the unit function
defined by $u(n)=1$ for all $n\in\mathbb{N}$.

A more general inversion formula was given by Hardy and Wright ([7, Theorem
270]) which states that
$g(k)=\sum\limits_{\begin{subarray}{c}d=1\end{subarray}}^{\infty}\mu(d)f(kd)$
if and only if
$f(k)=\sum\limits_{\begin{subarray}{c}d=1\end{subarray}}^{\infty}g(kd)$.

##  3\. Main Results and proofs

We have already stated in the previous section that Cohen-Ramanujan sum
$c_{k}^{(s)}(n)$ is multiplicative in $k$. However, it is not multiplicative
in $n$. It was proved by Donovan and Rearick in [5] that though $c_{k}(n)$ is
not multiplicative in $n$, the twisted function $\mu(k)c_{k}(n)$ is
multiplicative in $n$ when $k$ is sqaure-free. We will prove an analogous
result here for the Cohen-Ramanujan sum.

######  Theorem 3.1.

For positive integers $m,n$ with $(m,n)=1$, and a square-free positive integer
$k$, we have
$\mu(k)c_{k}^{(s)}(mn)=\mu(k)c_{k}^{(s)}(m)\times\mu(k)c_{k}^{(s)}(n)$.

###### Proof.

Write $k=k_{1}k_{2}$ with $(k_{1},n)=(k_{2},m)=1$. Then

$\displaystyle\mu(k_{1}k_{2})c_{k_{1}k_{2}}^{(s)}(mn)=\mu(k_{1})c_{k_{1}}^{(s)}(mn)\times\mu(k_{2})c_{k_{2}}^{(s)}(mn).$

Now use identity (3) to proceed further to get the above to be equal to

$\displaystyle\mu(k_{1})\sum\limits_{\begin{subarray}{c}d|k_{1}\\\
d^{s}|mn\end{subarray}}d^{s}\mu(k_{1}/d)\times\mu(k_{2})\sum\limits_{\begin{subarray}{c}d|k_{2}\\\
d^{s}|mn\end{subarray}}d^{s}\mu(k_{2}/d).$

By our assumption on $k_{1},k_{2},m$, and $n$, we can see that
$\\{d:d|k_{1}\text{ and }d^{s}|mn\\}=\\{d:d|k_{1}\text{ and
}d^{s}|m\\}=\\{d:d|k\text{ and }d^{s}|m\\}$. Similarly, $\\{d:d|k_{2}\text{
and }d^{s}|mn\\}=\\{d:d|k\text{ and }d^{s}|n\\}$. So

$\displaystyle\mu(k_{1}k_{2})c_{k_{1}k_{2}}^{(s)}(mn)$
$\displaystyle=\mu(k_{2})\sum\limits_{\begin{subarray}{c}d|k\\\
d^{s}|m\end{subarray}}d^{s}\mu(k_{1}/d)\times\mu(k_{1})\sum\limits_{\begin{subarray}{c}d|k\\\
d^{s}|n\end{subarray}}d^{s}\mu(k_{2}/d)$
$\displaystyle=\sum\limits_{\begin{subarray}{c}d|k\\\
d^{s}|m\end{subarray}}d^{s}\mu(k/d)\times\sum\limits_{\begin{subarray}{c}d|k\\\
d^{s}|n\end{subarray}}d^{s}\mu(k/d)$
$\displaystyle=c_{k}^{(s)}(m)c_{k}^{(s)}(n)$
$\displaystyle=\mu(k)c_{k}^{(s)}(m)\times\mu(k)c_{k}^{(s)}(n).$

∎

K. R. Johnson used the the multiplicative property of $\mu(k)c_{k}(.)$ to show
that $\mu(k)c_{k}(n)=\sum\limits_{d|(k,n)}d\mu(d)$ for any square-free $k$
([8, Lemma 1]). He did not give details for proving it, but just stated that
the multiplicative nature of both sides of this equality can be used to prove
this quickly. We will derive an analogous result giving a detailed proof.

######  Theorem 3.2.

For square-free positive integer $k$, we have

$\displaystyle\mu(k)c^{(s)}_{k}(n)=\sum\limits_{d^{s}|(k^{s},n)_{s}}d^{s}\mu(d)$

###### Proof.

From Theorem 3.1, $\mu(k)c^{(s)}_{k}(n)$ is multiplicative in $n$. So if we
show that $\sum\limits_{d^{s}|(k^{s},n)_{n}}d^{s}\mu(d)$ is also
multiplicative in $n$, we can complete the proof by verifying the identity in
the statement for prime powers.

Let $m,n$ be positive integers such that $(m,n)=1$. Now $d^{s}|(k^{s},mn)_{s}$
implies that $d^{s}|k^{s}$ and $d^{s}|mn$. Since $(m,n)=1$, it must be true
that $d=d_{1}d_{2}$ with $(d_{1},d_{2})=1$ and $(d_{1},n)=1=(d_{2},m)$. So as
we saw previously, $d^{s}|(k^{s},mn)_{s}$ if and only if
$d_{1}^{s}|k^{s},d_{1}^{s}|m,d_{2}^{s}|k^{s}$, and $d_{2}^{s}|n$. Hence

$\displaystyle\sum\limits_{d^{s}|(k^{s},n)_{n}}d^{s}\mu(d)=\sum\limits_{d_{1}^{s}|(k^{s},m)_{s}}d_{1}^{s}\mu(d_{1})\times\sum\limits_{d_{2}^{s}|(k^{s},n)_{s}}d_{2}^{s}\mu(d_{2}).$

Now we will show that for every prime $p$ and positive integer $a$, we have
$\mu(k)c^{(s)}_{k}(p^{a})=\sum\limits_{d^{s}|(k^{s},p^{a})_{s}}d^{s}\mu(d)$.
If $p|k$, then since $k$ is square-free, $p^{s}||_{s}k^{s}$. In addition, if
$a\geq s$, then $d^{s}|(k^{s},p^{a})_{s}\Rightarrow d^{s}=1$ or $d^{s}=p^{s}$.
Hence if $p|k$ and $a\geq s$, then
$\sum\limits_{d^{s}|(k^{s},p^{a})_{s}}d^{s}\mu(d)=\mu(1)+p^{s}\mu(p)=1-p^{s}.$
Now
$\mu(k)c^{(s)}_{k}(p^{a})=\mu(p)c^{(s)}_{p}(p^{a})\times\mu(k/p)c^{(s)}_{k/p}(p^{a})$.
From equation 2, $c_{p}^{(s)}(p^{a})=p^{s}-1$ and
$c^{(s)}_{k/p}(p^{a})=\prod\limits_{\begin{subarray}{c}i=1\\\
p_{i}|k,\,p_{i}\neq
p\end{subarray}}^{r}c^{(s)}_{p_{i}}(p^{a})=\prod\limits_{\begin{subarray}{c}i=1\\\
p_{i}\neq p\end{subarray}}^{r}(-p^{s(1-1)})=(-1)^{\omega(k/p)}$. Also, since
$k$ is square-free, $\mu(k/p)=(-1)^{\omega(k/p)}$. Therefore
$\mu(k)c^{(s)}_{k}(p^{a})=(-1)(p^{s}-1)=1-p^{s}$.

Now if $p\nmid k$ or $a<s$, then $d^{s}|(k^{s},p^{a})_{s}\Rightarrow d^{s}=1$.
So $\sum\limits_{d^{s}|(k^{s},p^{a})_{s}}d^{s}\mu(d)=1$. Now
$\mu(k)c^{(s)}_{k}(p^{a})=\prod\limits_{p_{i}|k}\mu(p_{i})c^{(s)}_{p_{i}}(p^{a})=\prod\limits_{p_{i}|k}(-1)(-1)=1$.
So in all the cases, we get
$\mu(k)c^{(s)}_{k}(p^{a})=\sum\limits_{d^{s}|(k^{s},p^{a})_{s}}d^{s}\mu(d)$.
This completes the proof of the theorem. ∎

Recall that for a positive integer $k$, the _core_ of $k$ is defined to be its
largest square-free divisor denoted by $\overline{k}$ and
$k^{*}=k/\overline{k}$. Hardy proved in [6] that $c_{k}(n)=0$ if $k^{*}\nmid
n$. This is equivalent to the statement that $c_{p^{r}}(n)=0$ unless
$p^{r-1}|n$. An analogous statement in terms of the Cohen-Ramanujan sum is
appearing in equation (2). We use is to reformulate Hardy’s result as follows
and skip the proof of it as it follows easily from equation (2).

######  Lemma 3.3.

$c_{k}^{(s)}(n)=0$ if $(k^{*})^{s}\nmid n$.

In [8, Lemma 2], K. R. Johnson stated that
$c_{k}(nk^{*})=k^{*}c_{\overline{k}}(n)$. He did not provide any details on
how to arrive at this result, but just mentioned that one may use Hardy’s
result (which is the above lemma with $s=1$) to verify it. We give a
generalization of this result in terms of the Cohen-Ramanujan sum and prove
it.

######  Lemma 3.4.

$c_{k}^{(s)}(n(k^{*})^{s})=(k^{*})^{s}c_{\overline{k}}^{(s)}(n).$

###### Proof.

$\displaystyle c_{k}^{(s)}(n(k^{*})^{s})$
$\displaystyle=\prod\limits_{p^{r}||k}c_{p^{r}}^{(s)}(n(k^{*})^{s})$
$\displaystyle=\prod\limits_{\begin{subarray}{c}p^{r}||k\\\
p^{s}|n\end{subarray}}c_{p^{r}}^{(s)}(n(k^{*})^{s})\times\prod\limits_{\begin{subarray}{c}p^{r}||k\\\
p^{s}\nmid n\end{subarray}}c_{p^{r}}^{(s)}(n(k^{*})^{s})$
$\displaystyle=\prod\limits_{\begin{subarray}{c}p^{r}||k\\\
p^{s}|n\end{subarray}}(p^{rs}-p^{(r-1)s})\prod\limits_{\begin{subarray}{c}p^{r}||k\\\
p^{s}\nmid n\end{subarray}}(-p^{(r-1)s}).$

On the other hand,

$\displaystyle(k^{*})^{s}c_{\overline{k}}^{(s)}(n)$
$\displaystyle=\prod\limits_{\begin{subarray}{c}p^{r}||k\end{subarray}}p^{(r-1)s}\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}|n\end{subarray}}c_{p}^{(s)}(n)\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}\nmid n\end{subarray}}c_{p}^{(s)}(n)$
$\displaystyle=\prod\limits_{\begin{subarray}{c}p^{r}||k\\\
p^{s}|n\end{subarray}}p^{(r-1)s}\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}|n\end{subarray}}(p^{s}-1)\prod\limits_{\begin{subarray}{c}p^{r}||k\\\
p^{s}\nmid n\end{subarray}}p^{(r-1)s}\prod\limits_{\begin{subarray}{c}p|k\\\
p^{s}\nmid n\end{subarray}}(-1)$ $\displaystyle=c_{k}^{(s)}(n(k^{*})^{s}).$

∎

Now we give a sufficient condition for the existence of Cohen-Ramanujan
expansions of certain arithmetical functions in the second variable. We call
the Cohen-Ramanujan expansion in the first variable as the CR-series expansion
and the expansion in the second variable as the CR’-series expansion. This
result is analogous to [8, Theorem 4].

######  Theorem 3.5.

Let $f$ be an arithmetical function. Let $a(k)$ be a function such that
$a(k)=0$ if $k\neq(\overline{k})^{s}$. For other values of $k$, $a(k)$ is
defined recursively using the relation

(6) $\displaystyle f(k)=\sum\limits_{d|k}d^{s}\mu(d)a(d^{s}).$

Suppose

(7)
$\displaystyle\sum\limits_{k=1}^{\infty}\sum\limits_{n=1}^{\infty}|a(n^{s}k^{s})|<\infty.$

Define

(8) $\displaystyle b(k^{s})=\sum\limits_{n=1}^{\infty}\mu(n)a(n^{s}k^{s}),$

with $b$ defined to be 0 on positive integers that are not $s^{\text{th}}$
powers. Then

(9)
$\displaystyle\mu(k)f(k)=\sum\limits_{n=1}^{\infty}b(n^{s})c_{k}^{(s)}(n^{s})$

for all square-free $k$. Also, the right-hand side series in equation (9)
converges absolutely.

###### Proof.

By the inversion formula we stated previously, we have
$b(k^{s})=\sum\limits_{n=1}^{\infty}\mu(n)a(n^{s}k^{s})$ if and only if
$a(k^{s})=\sum\limits_{n=1}^{\infty}b(n^{s}k^{s})$. So from equation (6), we
get

$\displaystyle f(k)$ $\displaystyle=\sum\limits_{d|k}d^{s}\mu(d)a(d^{s})$
$\displaystyle=\sum\limits_{d|k}d^{s}\mu(d)\sum\limits_{n=1}^{\infty}b(n^{s}d^{s})$
$\displaystyle=\sum\limits_{d|k}d^{s}\mu(d)\sum\limits_{\begin{subarray}{c}n=1\\\
d|n\end{subarray}}^{\infty}b(n^{s})$
$\displaystyle=\sum\limits_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}b(n^{s})\sum\limits_{d^{s}|(k^{s},n^{s})_{s}}d^{s}\mu(d)$

which is equal to
$\sum\limits_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}b(n^{s})c_{k}^{(s)}(n^{s})\mu(k)$
by Theorem 3.2.

Now
$\sum\limits_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}|b(n^{s})|\,|c_{k}^{(s)}(n^{s})|\leq
k\sum\limits_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}\sum\limits_{\begin{subarray}{c}m=1\end{subarray}}^{\infty}|a(n^{s}m^{s})|$
implying the absolute convergence.

∎

Now we give a converse of the above theorem.

######  Theorem 3.6.

Suppose that $f$ has a series representation as in (9) with
$\sum\limits_{\begin{subarray}{c}k=1\end{subarray}}^{\infty}b(k^{s})<\infty$
and $b(n)=0$ if $n\neq(\overline{n})^{s}$. Then $b(k^{s})$ is given by (8).

###### Proof.

We will more or less work backward the proof of the previous theorem. For
square-free $k$, from equation (9), we have

$\displaystyle f(k)$
$\displaystyle=\sum\limits_{n=1}^{\infty}b(n^{s})c_{k}^{(s)}(n^{s})\mu(k)$
$\displaystyle=\sum\limits_{\begin{subarray}{c}n=1\end{subarray}}^{\infty}b(n^{s})\sum\limits_{d^{s}|(k^{s},n^{s})_{s}}d^{s}\mu(d)$
$\displaystyle=\sum\limits_{d|k}d^{s}\mu(d)\sum\limits_{\begin{subarray}{c}n=1\\\
d|n\end{subarray}}^{\infty}b(n^{s})$
$\displaystyle=\sum\limits_{d|k}d^{s}\mu(d)\sum\limits_{n=1}^{\infty}b(n^{s}d^{s}).$

Thus $a(d^{s})=\sum\limits_{n=1}^{\infty}b(n^{s}d^{s})$ which implies that
$b(d^{s})=\sum\limits_{n=1}^{\infty}\mu(n)a(n^{s}d^{s})$. ∎

######  Lemma 3.7.

Let $k$ and $n$ be square-free integers. Then
$\mu(k)c_{k}^{(s)}(n^{s})=\mu(n)c_{n}^{(s)}(k^{s})$.

###### Proof.

Note that from Theorem 3.2, we get an expression for
$\mu(k)c_{k}^{(s)}(n^{s})$ which is symmetric in $k$ and $n$ thus permitting
us interchanging the positions of $k$ and $n$. ∎

Now we prove a _reciprocity theorem_.

######  Theorem 3.8.

For square-free integers $k$ and $n$, we have

(10)
$\displaystyle\frac{\mu(\overline{k})}{(k^{*})^{s}}c_{k}^{(s)}(n^{s}(k^{*})^{s})=\frac{\mu(\overline{n})}{(n^{*})^{s}}c_{n}^{(s)}(k^{s}(n^{*})^{s}).$

###### Proof.

We have

$\displaystyle\mu(\overline{k})c_{\overline{k}}^{(s)}(\overline{n}^{s})$
$\displaystyle=\mu(\overline{k})\sum\limits_{d^{s}|(\overline{k}^{s},\overline{n}^{s})}d^{s}\mu(d)$
$\displaystyle=\mu(\overline{k})\sum\limits_{d^{s}|(\overline{k}^{s},n^{s})}d^{s}\mu(d)$
$\displaystyle=\mu(\overline{k})c_{\overline{k}}^{(s)}(n^{s})$
$\displaystyle=\mu(\overline{k})\frac{c_{k}^{(s)}(n^{s}(k^{*})^{s})}{(k^{*})^{s}}$

where we used Lemma 3.4 to arrive at the last step above. On the other hand,
using Lemma 3.7, we get

$\displaystyle\mu(\overline{k})c_{\overline{k}}^{(s)}(\overline{n}^{s})$
$\displaystyle=\mu(\overline{n})c_{\overline{n}}^{(s)}(\overline{k}^{s})$
$\displaystyle=\mu(\overline{n})c_{\overline{n}}^{(s)}(k^{s})$
$\displaystyle=\mu(\overline{n})\frac{c_{n}^{(s)}(k^{s}(n^{*})^{s})}{(n^{*})^{s}}.$

∎

We are going to show now that the existence of a $CR$-series expansion is
equivalent to the existence of $CR^{\prime}$-series expansion for certain
functions.

######  Theorem 3.9.

A function $f$ has an absolutely convergent Cohen-Ramanujan series expansion
in the first variable $f(n)=\sum\limits_{k=1}^{\infty}a(k)c_{k}^{(s)}(n^{s})$
where $a(k)=0$ if $k$ is not square-free if and only if $f$ has an absolutely
convergent series expansion in the second variable in the form
$f(n)=\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{k=1}^{\infty}\xi_{n^{*}}(k)\,b(k/n^{*})c_{n}^{(s)}(k^{s})$.

###### Proof.

We mainly use the reciprocity theorem Theorem 3.8 in the proof. Suppose that
$f$ has a absolutely convergent $CR$-series expansion with coefficients $a(k)$
as given in the statement. Then

$\displaystyle f(n)$
$\displaystyle=\sum\limits_{k=1}^{\infty}a(k)c_{k}^{(s)}(n^{s})$
$\displaystyle=\sum\limits_{k=1}^{\infty}\mu(k)a(k)\mu(k)c_{k}^{(s)}(n^{s})$
$\displaystyle=\sum\limits_{k=1}^{\infty}\mu(k)a(k)\mu(\overline{k})c_{\overline{k}}^{(s)}(n^{s})\quad\text{since
$\overline{k}=k$ when $k$ is square-free}$
$\displaystyle=\sum\limits_{k=1}^{\infty}\mu(k)a(k)\mu(\overline{k})\frac{c_{k}^{(s)}(n^{s}(k^{*})^{s})}{(k^{*})^{s}}\quad\text{by
Lemma \ref{th:ck-ck-core}}.$

Now by the reciprocity theorem, the above is equal to

$\displaystyle\sum\limits_{k=1}^{\infty}\mu(k)a(k)\frac{\mu(\overline{n})}{(n^{*})^{s}}c_{n}^{(s)}(k^{s}(n^{*})^{s})$
$\displaystyle=\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{\begin{subarray}{c}k=1\\\
n^{*}|k\end{subarray}}^{\infty}\mu(k/n^{*})a(k/n^{*})c_{n}^{(s)}(k^{s})$
$\displaystyle=\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{k=1}^{\infty}\xi_{n^{*}}(k)\,b(k/n^{*})c_{n}^{(s)}(k^{s})$

where $b(k/n^{*})=\mu(k/n^{*})a(k/n^{*})$ which is 0 if $k/n^{*}$ is not
square-free.

Conversely suppose that
$f(n^{s})=\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{k=1}^{\infty}\xi_{n^{*}}(k)\,b(k/n^{*})c_{n}^{(s)}(k^{s})$
where $b(m)=0$ if $m$ is not square-free. Then

$\displaystyle f(n)$
$\displaystyle=\sum\limits_{k=1}^{\infty}\frac{\mu(\overline{n})}{(n^{*})^{s}}b(k)c_{n}^{(s)}(k^{s}(n^{*})^{s})$
$\displaystyle=\sum\limits_{k=1}^{\infty}b(k)\frac{\mu(\overline{k})}{(k^{*})^{s}}c_{k}^{(s)}(n^{s}(k^{*})^{s})\quad\text{by
Theorem \ref{th:mu-c-reciprocity}}$
$\displaystyle=\sum\limits_{k=1}^{\infty}b(k)\frac{\mu(\overline{k})}{(k^{*})^{s}}\times(k^{*})^{s}c_{\overline{k}}^{(s)}(n^{s})\quad\text{by
Lemma \ref{th:ck-ck-core}}$

Write $\alpha(k)=b(k)\frac{\mu(\overline{k})}{(k^{*})^{s}}$. For a fixed
positive integer $k$, note that there can be many positive integers $m$ such
that $\overline{m}=k=\overline{k}$. Denote
$H(k)=\\{m\in\mathbb{N}:\overline{m}=k\\}$. Then

$\displaystyle f(n^{s})=\sum\limits_{k=1}^{\infty}\left(\sum\limits_{m\in
H(k)}\alpha(m)(m^{*})^{s}\right)c_{k}^{(s)}(n^{s})=\sum\limits_{k=1}^{\infty}a(k)c_{k}^{(s)}(n^{s})$

where $a(k)=\sum\limits_{m\in H(k)}\alpha(m)(m^{*})^{s}$ for each square-free
positive integer $k$. ∎

######  Example 3.10.

The construction of this example follows the methods used by Ramanujan in
Sections 2, 3 and 9 in [12]. Consider the Klee’s function defined by
$\Phi_{s}(n)=n\prod\limits_{p^{s}|n}\left(1-\frac{1}{p^{s}}\right)$.
$\Phi_{s}$ can be expanded as

$\displaystyle\Phi_{s}(n)$
$\displaystyle=n\prod\limits_{p^{s}|n}\left(1+\frac{\mu(p)}{p^{s}}\right)$
$\displaystyle=n\sum\limits_{d^{s}|n}\frac{\mu(d)}{d^{s}}.$

Let $\xi^{(s)}$ be the function such that $\xi^{(s)}_{d}(n)=d^{s}$ if
$d^{s}|n$ and 0 otherwise. Hence equation (3) can be rewritten as
$c_{k}^{(s)}(n)=\sum\limits_{d|k}\xi^{(s)}_{d}(n)\frac{\mu(d)}{d^{s}}$ which
by the Möbius inversion formula implies that
$\xi^{(s)}_{k}(n)=\sum\limits_{d|k}c_{d}^{(s)}(n)$. Hence for $t\geq n$, we
have

$\displaystyle\Phi_{s}(n)$
$\displaystyle=\sum\limits_{d=1}^{t}\frac{1}{d^{s}}\xi^{(s)}_{d}(n)\mu(d)\frac{n}{d^{s}}$
$\displaystyle=\sum\limits_{d=1}^{t}\frac{1}{d^{s}}\sum\limits_{v|d}c_{v}^{(s)}(n)\mu(d)\frac{n}{d^{s}}.$

Let us collect the coefficients of $c_{1}^{(s)}(n),c_{2}^{(s)}(n),\ldots$.
Since $1|d$ for every $d$, the coefficient of $c_{1}^{(s)}(n)$ is
$\sum\limits_{d=1}^{t}\frac{1}{d^{s}}\mu(d)\frac{n}{d^{s}}$. Now $2|d$ for
$d=2,4,\ldots,2[t/2]$ where for any real number $x$, $[x]$ represents the
greatest integer not exceeding $x$. Hence instead of $d$, we may take $2d$ for
$d$ in the range $1,2,\ldots,[t/2]$ in the coefficient so that the coefficient
of $c_{2}^{(s)}(n)$ will be
$\sum\limits_{d=1}^{[t/2]}\frac{1}{(2d)^{s}}\mu(2d)\frac{n}{(2d)^{s}}$.
Continuing like this, we get

$\displaystyle\Phi_{s}(n)$
$\displaystyle=c_{1}^{(s)}(n)\sum\limits_{d=1}^{t}\mu(d)\frac{n}{d^{2s}}+\ldots+c_{k}^{(s)}(n)\sum\limits_{d=1}^{[t/k]}\mu(kd)\frac{n}{(kd)^{2s}}+\ldots.$

Letting $t\rightarrow\infty$ and noting that
$\sum\limits_{k=1}^{\infty}\frac{\mu(kd)}{d^{2s}}=\frac{\mu(k)k^{2s}}{J_{2s}(k)\zeta(2s)}$
([12, Equation 9.5]), we get

$\displaystyle\frac{\Phi_{s}(n)\zeta(2s)}{n}$
$\displaystyle=\frac{\mu(1)}{J_{2s}(1)}c_{1}^{(s)}(n)+\ldots+\frac{\mu(k)}{J_{2s}(k)}c_{k}^{(s)}(n)+\ldots.$

If we use the relation $\Phi_{s}(n^{s})=J_{s}(n)$ between the Jordan totient
function and Klee’s function, then we get

(11) $\displaystyle\frac{\Phi_{s}(n)\zeta(2s)}{n}$
$\displaystyle=\frac{\mu(1)}{\Phi_{2s}(1)}c_{1}^{(s)}(n)+\ldots+\frac{\mu(k)}{\Phi_{2s}(k^{s})}c_{k}^{(s)}(n)+\ldots,$

or, equivalently

$\displaystyle\frac{J_{s}(n^{s})\zeta(2s)}{n}$
$\displaystyle=\frac{\mu(1)}{J_{2s}(1)}c_{1}^{(s)}(n)+\ldots+\frac{\mu(k)}{J_{2s}(k)}c_{k}^{(s)}(n)+\ldots.$

It is interesting to compare the above with a similar expression equation 9.6
obtained by Ramanujan in [12]. To avoid any confusion, note that Ramanujan
used the symbol $\phi_{s}$ to denote the Jordan totient function in [12]. Now
the series expression equation (11) satisfies the requirement in Theorem 3.9
with $a(k)=\frac{\mu(k)}{\Phi_{2s}(k^{s})}$. Hence the $CR^{\prime}$-series
expansion for $\frac{\Phi_{s}(n)\zeta(2s)}{n}$ will be

$\displaystyle\frac{\Phi_{s}(n)\zeta(2s)}{n}$
$\displaystyle=\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{k=1}^{\infty}\xi_{n^{*}}(k)\,\mu(k/n^{*})a(k/n^{*})c_{n}^{(s)}(k^{s})$
$\displaystyle=\frac{\mu(\overline{n})}{(n^{*})^{s}}\sum\limits_{k=1}^{\infty}\frac{\xi_{n^{*}}(k)}{\Phi_{2s}(k^{s})}\,c_{n}^{(s)}(k^{s}).$

## References

  * [1] Arya Chandran and K Vishnu Namboothiri.  On a Ramanujan type expansion of arithmetical functions.  The Ramanujan Journal, 62:177–188, 2023.
  * [2] Eckford Cohen.  An extension of Ramanujan’s sum.  Duke Mathematical Journal, 16(2):85–90, 1949.
  * [3] Eckford Cohen.  Trigonometric sums in elementary number theory.  The American Mathematical Monthly, 66(2):105–117, 1959.
  * [4] Hubert Delange.  On Ramanujan expansions of certain arithmetical functions.  Acta Arith, 31(3):259–270, 1976.
  * [5] George S Donovan and David Rearick.  On Ramanujan’s sum.  Det Kgl. Norske Vidensk Selsk. Fordhandlinger, 39:1–2, 1966.
  * [6] G H Hardy.  Note on Ramanujan’s trigonometrical function $c_{q}(n)$, and certain series of arithmetical functions.  Proceedings of the Cambridge Philosophical Society, 20:263–271, 1921.
  * [7] Godfrey Harold Hardy and Edward Maitland Wright.  An Introduction to the Theory of Numbers.  Oxford University Press, London, 1979.
  * [8] Kenneth R Johnson.  A reciprocity law for Ramanujan sums.  Pacific Journal of Mathematics, 39(1):99–105, 1982.
  * [9] Lutz Lucht.  Ramanujan expansions revisited.  Archiv der Mathematik, 64(2):121–128, 1995.
  * [10] Lutz G Lucht.  A survey of Ramanujan expansions.  International Journal of Number Theory, 6(08):1785–1799, 2010.
  * [11] M Ram Murty.  Ramanujan series for arithmetical functions.  Hardy-Ramanujan Journal, 36, 2013.
  * [12] Srinivasa Ramanujan.  On certain trigonometrical sums and their applications in the theory of numbers.  Trans. Cambridge Philos. Soc, 22(13):259–276, 1918.
  * [13] Aurel Wintner.  Eratosthenian averages.  Waverly Press, Baltimore, MD, 1943.


# Intersecting the Twin dragon with rational lines

Shigeki Akiyama, Paul Großkopf, Benoît Loridant and Wolfgang Steiner  Tsukuba
University, Institute of Mathematics
Tennodai-1-1-1
Tsukuba 350-8571, Japan<EMAIL_ADDRESS>Université libre de
Bruxelles
Boulevard du Triomphe
1050 Bruxelles, Belgium<EMAIL_ADDRESS>Leoben University
Franz Josefstrasse 18
8700 Leoben, Austria<EMAIL_ADDRESS>Université de Paris,
CNRS, IRIF, F–75006 Paris, France<EMAIL_ADDRESS>Dedicated to Professor Jörg
Thuswaldner on the occasion of his $50^{th}$ birthday

###### Abstract.

The Knuth Twin Dragon is a compact subset of the plane with fractal boundary
of Hausdorff dimension $s=(\log\lambda)/(\log\sqrt{2})$,
$\lambda^{3}=\lambda^{2}+2$. Although the intersection with a generic line has
Hausdorff dimension $s-1$, we prove that this does not occur for lines with
rational parameters. We further describe the intersection of the Twin Dragon
with the two diagonals as well as with various axis parallel lines.

###### Key words and phrases:

Number system, Hausdorff dimension

##  1\. Introduction

We investigate the intersections of the Knuth Twin Dragon with rational lines.
Let $\alpha=-1+i$, then

$\mathcal{K}=\left\\{\sum_{k=1}^{\infty}\frac{d_{k}}{\alpha^{k}}\,:\,d_{k}\in\\{0,1\\}\right\\}$

is the Knuth Twin Dragon. The Hausdorff dimension of its boundary $\partial K$
is $\mathfrak{s}=\frac{\log\lambda}{\log\sqrt{2}}\approx 1.5236$, where
$\lambda$ is the real number satisfying $\lambda^{3}=\lambda^{2}+2$. For lines

(1.1) $\Delta_{p,q,r}=\\{x+iy\in\mathbb{C}\,:\,px+qy=r\\}$

with $p,q,r\in\mathbb{Z}$, we show that the $\alpha$-expansions of
$\mathcal{K}\cap\Delta_{p,q,r}$ are recognized by a finite automaton.

By a result of John Marstrand [5], the intersection of $\partial\mathcal{K}$
with Lebesgue almost all lines going through $\mathcal{K}$ has Hausdorff
dimension $\mathfrak{s}-1$, meaning that in the set of all parameter triples
$(p,q,r)\in\mathbb{R}^{3}$ for which
$\Delta_{p,q,r}\cap\mathcal{K}\neq\emptyset$, the exceptional cases form a
Lebesgue null set. We obtain here that the Hausdorff dimension of the
intersection of the boundary of the Twin Dragon with rational lines is never
equal to $\mathfrak{s}-1$.

Further we revisit results by Shigeki Akiyama and Klaus Scheicher [1] and add
uncountably many examples of horizontal, vertical, and diagonal lines.

We mention that similar results were obtained in [4] for lines intersecting
the Sierpinski carpet $F$. The set $F$ has Hausdorff dimension $\frac{\log
8}{\log 3}$. Manning and Simon showed that, given a slope
$\alpha\in\mathbb{Q}$, the intersection of $F$ with the line $y=\alpha
x+\beta$ is strictly less than $\frac{\log 8}{\log 3}-1$ for Lebesgue almost
every $\beta$.

##  2\. Main statement and proof

We first recall the notions of a canonical number system and its fundamental
domain. Let $\beta$ be an algebraic integer and
$\mathcal{N}=\\{0,1,\dots,|N(\beta)|-1\\}$, where $N(x)$ denotes the norm of
$x$ over $\mathbb{Q}(\beta)/\mathbb{Q}$. The pair $(\beta,\mathcal{N})$ is
called a _canonical number system_ (CNS) if each $\gamma\in\mathbb{Z}[\beta]$
admits a representation of the form

(2.1) $\gamma=\sum_{k=0}^{n}d_{k}\beta^{k},\quad d_{k}\in\mathcal{N}.$

We call $\beta$ the _radix_ or _base_ and $\mathcal{N}$ the set of _digits_.
The representation (2.1) is unique up to leading zeros.

The Knuth Twin Dragon $\mathcal{K}$ appears as the _fundamental domain_ of the
CNS $(\alpha,\mathcal{N})$, where $\alpha=-1+i$ is the root of the polynomial
$x^{2}-2x-2$ and $\mathcal{N}=\\{0,1\\}$. The fundamental domain of a CNS is
the set of all numbers that can be expressed with purely negative exponents.
Since $\alpha^{4}=-4$, it is often useful to consider groups of four digits:

$\sum_{k=1}^{\infty}\frac{d_{k}}{\alpha^{k}}=\sum_{k=1}^{\infty}\frac{\sum_{j=0}^{3}d_{4k-j}\alpha^{j}}{\alpha^{4k}}=\sum_{k=1}^{\infty}\frac{b_{k}}{(-4)^{k}},$

with the possibilities for $b_{k}=\sum_{j=0}^{3}d_{4k-j}\alpha^{j}$ being

$\begin{array}[]{llll}[0000]_{\alpha}=0,&[0001]_{\alpha}=1,&[0010]_{\alpha}=-1{+}i,&[0011]_{\alpha}=i,\\\
{}[0100]_{\alpha}=-2i,&[0101]_{\alpha}=1{-}2i,&[0110]_{\alpha}=-1{-}i,&[0111]_{\alpha}=-i,\\\
{}[1000]_{\alpha}=2{+}2i,&[1001]_{\alpha}=3{+}2i,&[1010]_{\alpha}=1{+}3i,&[1011]_{\alpha}=2{+}3i,\\\
{}[1100]_{\alpha}=2,&[1101]_{\alpha}=3,&[1110]_{\alpha}=1{+}i,&[1111]_{\alpha}=2{+}i.\end{array}$

In other words, we have

$\mathcal{K}=\left\\{\sum_{k=1}^{\infty}\frac{b_{k}}{(-4)^{k}}\,:\,b_{k}\in\mathcal{D}\right\\},$

with

$\mathcal{D}=\\{-1{-}i,-1{+}i,-2i,-i,0,i,1{-}2i,1,1{+}i,1{+}3i,2,2{+}i,2{+}2i,2{+}3i,3,3{+}2i\\}$

Points in the intersection of $\mathcal{K}$ with lines
$\Delta_{p,q,r}=\\{x+iy\,:\,px+qy=r\\}$ can now be characterized by their
digit expansion in the following way.

######  Lemma 2.1.

We have $z\in\mathcal{K}\cap\Delta_{p,q,r}$ if and only if there is a digit
sequence $b_{1}b_{2}\cdots\in\mathcal{D}^{\mathbb{N}}$ with

$z=\sum_{k=1}^{\infty}\frac{b_{k}}{(-4)^{k}}\quad\mbox{and}\quad
r=\sum_{k=1}^{\infty}\frac{p\,\mathfrak{R}(b_{k})+q\,\mathfrak{I}(b_{k})}{(-4)^{k}}.$

Here, $\mathfrak{R}(b)$ denotes the real part and $\mathfrak{I}(b)$ denotes
the imaginary part of $b\in\mathbb{C}$.

We will show that we can characterize the digit expansion of the points in the
intersection $\Delta_{p,q,r}\cap\mathcal{K}$ via a Büchi automaton, that is a
finite automaton that accepts infinite paths. Using this representation we
will be able to calculate the Hausdorff dimension of the intersection
$\mathcal{K}\cap\Delta_{p,q,r}$ as well as the Hausdorff dimension of
$\partial K\cap\Delta_{p,q,r}$.

######  Definition 2.2.

A _Büchi automaton_ is a $5$-tuple $(Q,A,E,I,T)$, where
$Q=\\{q_{1},\dots,q_{N}\\}$ is a finite set of _states_ , $A$ is a finite
_alphabet_ , $E\subset Q\times A\times Q$ is a set of _edges_ and $I,T\subset
Q$ the set of _initial_ and _terminal states_. Let $A^{*}$ denote the set of
all (finite) words and $A^{\omega}$ denote the set of all (right) infinite
words. A word $w\in A^{*}$, $w=w_{1}\cdots w_{n}$, is _accepted_ by the
automaton if and only if there are states $q_{i_{0}},\dots,q_{i_{n}}$ such
that $q_{i_{0}}\in I$, $q_{i_{n}}\in T$ and $(q_{i_{k-1}},w_{k},q_{i_{k}})\in
E$ for all $k$. We call such a finite path _successful_ , and we call an
infinite path successful if and only if infinitely many subpaths are
successful. An infinite word $w\in A^{\omega}$ is accepted by the automaton if
there exists an infinite successful path with _label_ $w$. The set of all
$w\in A^{\omega}$ that are accepted by the automaton is called its
$\omega$-_language_.

Büchi automata are really helpful to describe self-similar sets. The automaton
in Figure 1 characterizes all infinite sequences of digits $0,1$ in base
$\alpha$ that give rise to boundary points in $\partial\mathcal{K}$; see [3,
7].

$g_{1}$$g_{2}$$g_{3}$$g_{4}$$g_{5}$$g_{6}$100,1010,101

Figure 1. An automaton characterizing $\partial\mathcal{K}$ (in base
$\alpha$).

Let $L_{1},L_{2}$ two $\omega$-languages in the same alphabet accepted by
$\mathcal{A}$ respectively $\mathcal{B}$. It can be necessary to create
automata accepting the union of the languages or their intersection. The union
is not difficult: one just uses the union of states and edges, as well as the
union of terminal and initial states. The intersection generally requires
heavy computations, especially in the non-deterministic case, where a larger
framework than Büchi automata needs to be used. But it becomes easy in some
cases. We prove one particular case that will be useful to prove our main
statements.

######  Lemma 2.3.

Let $L_{1},L_{2}$ be two $\omega$-languages on the same alphabet $A$ accepted
by Büchi automata. If one of the automata has only terminal states, then there
is a Büchi automaton accepting $L_{1}\cap L_{2}$.

###### Proof.

Define $\mathcal{A}\times\mathcal{B}=(Q_{\mathcal{A}}\times
Q_{\mathcal{B}},A,E,I_{\mathcal{A}}\times
I_{\mathcal{B}},T_{\mathcal{A}}\times T_{\mathcal{B}})$, where $E$ consists of
the edges $(a,b)\overset{d}{\rightarrow}(a^{\prime},b^{\prime})$ with
$a\overset{d}{\rightarrow}a^{\prime}$ and
$b\overset{d}{\rightarrow}b^{\prime}$. Let $w\in A^{\omega}$ be a word that is
accepted by $\mathcal{A}\times\mathcal{B}$. Then there exists an infinite path
in the automaton. Projecting to the first coordinate gives an infinite path
through $\mathcal{A}$. Therefore, we have $w\in L_{1}$ and with the same
reasoning $w\in L_{2}$. Now let $w\in L_{1}\cap L_{2}$. There exists a path
$a_{0}a_{1}\cdots$ through $\mathcal{A}$ and a path $b_{0}b_{1}\cdots$ through
$\mathcal{B}$. Then $(a_{0},b_{0})(a_{1},b_{1})\cdots$ is a path in the
product automaton. Assume w.l.o.g. that all states of $\mathcal{A}$ are
terminal. Then, for every finite subpath $b_{0}b_{1}\cdots b_{k}$ accepted by
$\mathcal{B}$, the corresponding path $a_{0}a_{1}\cdots a_{k}$ in
$\mathcal{A}$ is also accepted, hence $(a_{0},b_{0})(a_{1},b_{1})\cdots$ is
successful. ∎

In general, if $\Delta_{p,q,r}\cap\mathcal{K}$ is described by a Büchi
automaton $\mathcal{A}$ and the boundary $\partial\mathcal{K}$ by a Büchi
automaton $\mathcal{G}$, then $\partial\mathcal{K}\cap\Delta_{p,q,r}$ is
described by the product automaton $\mathcal{A}\times\mathcal{G}$.
Interpreting this Büchi automaton as a graph directed construction for
$\partial\mathcal{K}\cap\Delta_{p,q,r}$, we will have a way to compute the
Hausdorff dimension of this set via results of Mauldin and Williams [6]. Let
us state and prove our main statements.

######  Theorem 2.4.

Let $p,q,r\in\mathbb{Z}$, $\Delta_{p,q,r}$ as in (1.1) and $\mathcal{K}$ the
Knuth Twin Dragon. Then the intersection $\mathcal{K}\cap\Delta_{p,q,r}$ can
be described by a Büchi automaton.

###### Proof.

For $s,s^{\prime}\in\mathbb{Z}$ we define an edge relation by

(2.2) $s\overset{b}{\rightarrow}s^{\prime}\Longleftrightarrow
s^{\prime}=p\,\mathfrak{R}(b)+q\,\mathfrak{I}(b)-4s.$

Now consider a path
$-r=s_{0}\overset{b_{1}}{\rightarrow}s_{1}\overset{b_{2}}{\rightarrow}\cdots\overset{b_{n}}{\rightarrow}s_{n}$.
Then

$s_{n}=(-4)^{n}(-r)+\sum_{k=1}^{n}(-4)^{n-k}\big{(}p\,\mathfrak{R}(b_{k})+q\,\mathfrak{I}(b_{k})\big{)},$

i.e.,

$\frac{s_{n}}{(-4)^{n}}=-r+\sum_{k=1}^{n}\frac{p\,\mathfrak{R}(b_{k})+q\,\mathfrak{I}(b_{k})}{(-4)^{k}}.$

Using Lemma 2.1, we immediately get that

$(x,y)=[0.b_{1}b_{2}b_{3}\cdots]_{-4}\in\mathcal{K}\cap\Delta_{p,q,r}\quad\mbox{if
and only if}\quad\lim_{n\to\infty}\frac{s_{n}}{(-4)^{n}}=0.$

We now show that the elements $s_{n}$ lying on paths starting with $s_{0}=-r$
and $\lim_{n\to\infty}\frac{s_{n}}{(-4)^{n}}=0$ are bounded by a constant
$c(p,q)$. Indeed, we have

$\frac{s_{n}}{(-4)^{n}}=-r+\sum_{k=1}^{n}\frac{p\,\mathfrak{R}(b_{k})+q\,\mathfrak{I}(b_{k})}{(-4)^{k}}=-\sum_{k=n+1}^{\infty}\frac{p\,\mathfrak{R}(b_{k})+q\,\mathfrak{I}(b_{k})}{(-4)^{k}},$

and therefore

$|s_{n}|=4^{n}\left|\sum_{k=n+1}^{\infty}\frac{p\,\mathfrak{R}(b_{k})+q\,\mathfrak{I}(b_{k})}{(-4)^{k}}\right|\leq\frac{\max\\{|p\,\mathfrak{R}(b)+q\,\mathfrak{I}(b)|:b\in\mathcal{D}\\}}{3}=c(p,q).$

Defining the set of states $Q=\\{s\in\mathbb{Z}:|s|\leq
c(p,q)\\}\cup\\{-r\\}$, $I=\\{-r\\}$, $T=Q$ and edges as in 2.2, gives us the
desired Büchi automaton. ∎

######  Theorem 2.5.

Let $p,q,r\in\mathbb{Z}$, $\Delta_{p,q,r}$ as in (1.1) and $\mathcal{K}$ the
Knuth Twin Dragon. Then the Hausdorff dimension of the intersection
$\partial\mathcal{K}\cap\Delta_{p,q,r}$ is never $\mathfrak{s}-1$, where
$\mathfrak{s}$ is the Hausdorff dimension of $\partial\mathcal{K}$.

###### Proof.

The Büchi automaton of Theorem 2.4 gives rise to a description of the
intersection $\mathcal{K}\cap\Delta_{p,q,r}$ as the attractor of a graph
directed construction (_GIFS_):

$\mathcal{K}\cap\Delta_{p,q,r}=K_{-r},\quad\mbox{with}\quad
K_{s}=\bigcup_{s\overset{b}{\rightarrow}s^{\prime}\in\mathcal{A}}\frac{K_{s^{\prime}}+b}{-4}\qquad(s\in
Q).$

As mentioned above, $\partial\mathcal{K}$ is also the attractor of a GIFS:

$\partial\mathcal{K}=\bigcup_{g\in Q^{\prime}}K_{g},\quad\mbox{with}\quad
K_{g}=\bigcup_{g\overset{b}{\rightarrow}g^{\prime}\in\mathcal{G}}\frac{K_{g^{\prime}}+b}{-4}\qquad(g\in
Q^{\prime}),$

where $\mathcal{G}$ is the automaton characterizing $\partial\mathcal{K}$ in
base $-4$. The automaton $\mathcal{G}$ can be obtained from the automaton
$\mathcal{G}^{\prime}$ of Figure 1 as follows.

  * •

The set of states $Q^{\prime}$ is the same as for $\mathcal{G}^{\prime}$; all
states are initial and terminal.

  * •

There is an edge from $g$ to $g^{\prime}$ in $\mathcal{G}$ whenever there is a
path of length $4$ from $g$ to $g^{\prime}$ in $\mathcal{G}^{\prime}$. The
label of this edge in $\mathcal{G}$ is the digit vector
$[d_{1}d_{2}d_{3}d_{4}]_{\alpha}$ corresponding to the labels
$d_{1},d_{2},d_{3},d_{4}$ in $\mathcal{G}^{\prime}$ along the path of length
$4$.

In that way, $\mathcal{A}$ and $\mathcal{G}$ are built on the same alphabet.
By Lemma 2.3, the intersection $\mathcal{A}\times\mathcal{G}$ is a Büchi
automaton describing the intersection $\Delta_{p,q,r}\cap\partial\mathcal{K}$.
By Mauldin and Williams [6], the Hausdorff dimension of a GIFS attractor can
be computed from the spectral radius $\beta$ of the incidence matrix of a
strongly connected component of the associated automaton; see further details
in Remark 2.6. In particular, in our case,

$\mathrm{dim}_{H}(\partial\mathcal{K}\cap\Delta_{p,q,r})=\frac{\log\beta}{\log{4}},$

where the involved number $\beta$ is an algebraic integer.

Now, the dimension of the boundary of the Twin Dragon is
$\mathfrak{s}=\frac{\log\lambda}{\log\sqrt{2}}$, with
$\lambda^{3}=\lambda^{2}+2$. To have
$\frac{\log\beta}{\log{4}}=\mathfrak{s}-1$, we need
$\beta=\frac{\lambda^{4}}{4}$. However, the minimal polynomial of
$\frac{\lambda^{4}}{4}$ is $4x^{3}-9x^{2}+2x-1$, thus $\frac{\lambda^{4}}{4}$
is not an algebraic integer. ∎

######  Remark 2.6.

We shortly explain why the results of Mauldin and Williams [6] indeed apply to
our setting. All the similarities in our graphs are contractions of the form
$T(x)=\frac{x+b}{-4}$, with the same ratio $-\frac{1}{4}$. Therefore, if $G$
denotes any of our graphs, we only need to check the existence of
nonoverlapping compact sets $J_{1},\ldots,J_{n}$ (one for each node
$1,\ldots,n$ of $G$) with the property

$\forall\,i\in\\{1,\ldots,n\\},\;J_{i}\supset\bigcup_{i\xrightarrow{T}j\in
G}T(J_{j}),$

each union being nonoverlapping.

For the graph $G=\mathcal{G}$ of our paper (with states $g\in Q^{\prime}$),
the intersections of $\mathcal{K}$ with its six neighboring tiles in the plane
tiling generated by $\mathcal{K}$ are compact sets playing the role of the
$J_{i}$’s, that is, satisfying the above nonoverlapping conditions; see for
example [2]. These intersections are exactly the sets $K_{g}$ defined in the
proof of Theorem 2.5.

Now, the graph $G=\mathcal{A}\times\mathcal{G}$ of our paper can be
interpreted as a _subgraph_ of $\mathcal{G}$: taking the product of
$\mathcal{A}$ and $\mathcal{G}$ means to select paths of $\mathcal{G}$. The
states of $\mathcal{A}\times\mathcal{G}$ are of the form $(r,g)$, for some
integers $r$ and $g\in Q^{\prime}$. Defining

$K_{r,g}:=\Delta_{p,q,-r}\cap K_{g},$

we obtain compact sets fulfilling the nonoverlapping requirements mentioned
above.

##  3\. Further results of intersections of the Twin Dragon with rational
lines

In this section, we want to extend the work of [1], where the intersections
with the $x$-and the $y$-axis are calculated. The intersections of these lines
with $\partial K$ are significatively different from the expected result for
intersections of fractals and lines, as they consist only of two points.
First, we show that their result extends to uncountably many axis-parallel
lines (where we do not have finite automata), and using the self-similar
structure, to diagonal lines. Then we give one example of a more complicated
intersection.

######  Theorem 3.1.

Let $a_{1}a_{2}\cdots$ be a sequence in $\\{0,1\\}^{\omega}$ not ending in
$(01)^{\omega}$, and

$r=\sum_{k=1}^{\infty}\frac{2a_{k}}{(-4)^{k}}.$

Then

$\partial\mathcal{K}\cap\Delta_{1,0,r}=\left\\{r+\left(r-\tfrac{2}{5}\right)i,\,r+\left(r+\tfrac{3}{5}\right)i\right\\},$

and $\mathcal{K}\cap\Delta_{1,0,r}$ is the closed line segment
$r+\left[r-\frac{2}{5},r+\frac{3}{5}\right]i$.

Figure 2. The Knuth Twin Dragon $\mathcal{K}$ and its intersection with
$\Delta_{1,0,r}$ for some $r$ as in Theorem 3.1 (red) and with
$\Delta_{1,0,-1/5}$ (blue).

###### Proof.

We first use Lemma 2.1 to describe $\mathcal{K}\cap\Delta_{1,0,r}$, i.e., we
determine the sequences $b_{1}b_{2}\cdots\in\mathcal{D}$ such that
$\mathfrak{R}\left(\sum_{k=1}^{\infty}b_{k}(-4)^{-k}\right)=r$, i.e.,

$\sum_{k=1}^{\infty}\frac{2a_{k}-\mathfrak{R}(b_{k})}{(-4)^{k}}=0.$

Since $\mathfrak{R}(b_{k})\in\\{-1,0,1,2,3\\}$, we have
$2a_{k}-\mathfrak{R}(b_{k})\in\\{-3,-2,\dots,2,3\\}$ and thus

$\left|\sum_{k=n+1}^{\infty}\frac{2a_{k}-\mathfrak{R}(b_{k})}{(-4)^{k}}\right|\leq\frac{1}{4^{n}}\quad\mbox{for
all}\ n\geq 0.$

Moreover, equality holds if and only if $2a_{k}-\mathfrak{R}(b_{k})$ is
alternately $3$ and $-3$, which implies that $a_{k}$ is alternately $1$ and
$0$, which we have excluded. This gives that

$\left|\sum_{k=n+1}^{\infty}\frac{2a_{k}-\mathfrak{R}(b_{k})}{(-4)^{k}}\right|<\frac{1}{4^{n}}\quad\mbox{and}\quad\sum_{k=n+1}^{\infty}\frac{2a_{k}-\mathfrak{R}(b_{k})}{(-4)^{k}}=\sum_{k=1}^{n}\frac{\mathfrak{R}(b_{k})-2a_{k}}{(-4)^{k}}\in\frac{\mathbb{Z}}{4^{n}}$

for all $n\geq 1$, hence $\mathfrak{R}(b_{k})=2a_{k}$ for all $k\geq 1$. For
the corresponding sequences $d_{1}d_{2}\cdots$ (with
$\sum_{j=0}^{3}d_{4k-j}\alpha^{j}=b_{k}$) this implies that

(3.1)
$d_{4k-3}d_{4k-2}d_{4k-1}d_{4k}\in\\{a_{k}000,a_{k}011,a_{k}100,a_{k}111\\}\quad\mbox{for
all}\ k\geq 1.$

Now consider sequences $d_{1}d_{2}\cdots$ of the form (3.1) in the boundary
automaton $\mathcal{G}$ given in Figure 1. The only paths labeled by $abcc$,
$a,b,c\in\\{0,1\\}$, starting from $g_{1}$, $g_{2}$, $g_{5}$ and $g_{6}$
respectively are

$g_{1}\overset{0000}{\longrightarrow}g_{6},g_{1}\overset{0011}{\longrightarrow}g_{2},g_{2}\overset{1000}{\longrightarrow}g_{5},g_{2}\overset{1011}{\longrightarrow}g_{1},g_{5}\overset{0100}{\longrightarrow}g_{6},g_{5}\overset{0111}{\longrightarrow}g_{2},g_{6}\overset{1100}{\longrightarrow}g_{5},g_{6}\overset{1111}{\longrightarrow}g_{1}.$

Therefore, for an infinite successful path of the form (3.1) starting from
$g_{1}$, $g_{2}$, $g_{5}$ or $g_{6}$, the sequence $a_{1}a_{2}\cdots$ is
alternately $0$ and $1$, which we have excluded. Hence, it suffices to
consider paths that are in $g_{3}$ and $g_{4}$ after $4k$ steps for all $k\geq
0$. From

$g_{3}\overset{a100}{\longrightarrow}g_{4}\quad\mbox{and}\quad
g_{4}\overset{a011}{\longrightarrow}g_{3}\qquad(a\in\\{0,1\\}),$

we see that the only points in $\partial\mathcal{K}\cap\Delta_{1,0,r}$ are

$\begin{split}\sum_{k=1}^{\infty}\frac{a_{k}\alpha^{3}}{(-4)^{k}}+\sum_{k=1}^{\infty}\frac{\alpha^{6}+\alpha+1}{16^{k}}&=r\,(1+i)+\frac{3i}{5},\\\
\sum_{k=1}^{\infty}\frac{a_{k}\alpha^{3}}{(-4)^{k}}+\sum_{k=1}^{\infty}\frac{\alpha^{5}+\alpha^{4}+\alpha^{2}}{16^{k}}&=r\,(1+i)-\frac{2i}{5}.\end{split}$

Since $r\,(1+i)\in\mathcal{K}$, $\mathcal{K}\cap\Delta_{1,0,r}$ is the line
segment between these points. ∎

######  Theorem 3.2.

For $-\frac{8}{15}<r<\frac{2}{15}$, we have

$\begin{split}-2i\,(\mathcal{K}\cap\Delta_{0,1,r/2})&=(\mathcal{K}\cap\Delta_{1,0,r})+\\{0,i\\},\\\
(-1+i)\,(\mathcal{K}\cap\Delta_{1,1,-r})&=\mathcal{K}\cap\Delta_{1,0,r}\,,\\\
(-1+i)\,(\mathcal{K}\cap\Delta_{1,-1,r/2})&=(\mathcal{K}\cap\Delta_{0,1,r/2})+\\{0,1\\},\\\
2\,(1+i)\,(\mathcal{K}\cap\Delta_{1,-1,r/2})&=(\mathcal{K}\cap\Delta_{1,0,r})+\\{-2i,-i,0,i\\}.\end{split}$

In particular, for $r$ as in Theorem 3.1, the sets
$\mathcal{K}\cap\Delta_{0,1,r/2}$, $\mathcal{K}\cap\Delta_{1,1,-r}$ and
$\mathcal{K}\cap\Delta_{1,-1,r/2}$ are closed line segments with endpoints

$\begin{split}\partial\mathcal{K}\cap\Delta_{0,1,r/2}&=\partial(\mathcal{K}\cap\Delta_{0,1,r/2})=\left\\{-\tfrac{4}{5}-\tfrac{r}{2}+\tfrac{r}{2}\,i,\tfrac{1}{5}-\tfrac{r}{2}+\tfrac{r}{2}\,i\right\\},\\\
\partial\mathcal{K}\cap\Delta_{1,1,-r}&=\partial(\mathcal{K}\cap\Delta_{1,1,-r})=\left\\{-\tfrac{1}{5}+\left(\tfrac{1}{5}-r\right)i,\tfrac{3}{10}-\left(\tfrac{3}{10}+r\right)i\right\\},\\\
\partial\mathcal{K}\cap\Delta_{1,-1,r/2}&=\partial(\mathcal{K}\cap\Delta_{1,-1,r/2})=\left\\{-\tfrac{3}{5}+\tfrac{r}{2}-\tfrac{3}{5}\,i,\tfrac{2}{5}+\tfrac{r}{2}+\tfrac{2}{5}\,i\right\\}.\end{split}$

Figure 3. The intersection of
$\mathcal{K}=\alpha^{-1}\left(\mathcal{K}\cup(\mathcal{K}+1)\right)$ with
lines $\Delta_{0,1,r/2}$, $\Delta_{1,1,-r}$, and $\Delta_{1,-1,r/2}$ for some
$r$ as in Theorem 3.1.

###### Proof.

Note that $\alpha\,\mathcal{K}=\mathcal{K}\cup(\mathcal{K}+1)$ and

$\alpha\,\Delta_{1,1,-r}=\Delta_{1,0,-r},\quad\alpha\,\Delta_{0,1,r/2}=\Delta_{1,1,-r},\quad\alpha\,\Delta_{1,-1,r/2}=\Delta_{0,1,r/2}.$

Moreover, we have

$(\mathcal{K}+1)\cap\Delta_{1,0,r}=\emptyset=(\mathcal{K}-1)\cap\Delta_{1,0,r}=(\mathcal{K}+\alpha)\cap\Delta_{1,0,r}$

since $-\frac{8}{15}<r<\frac{2}{15}$ and

$\begin{split}\min\\{x\,:\,x+iy\in\mathcal{K}\\}&=\sum_{k=1}^{\infty}\left(\frac{3}{(-4)^{2k-1}}+\frac{-1}{(-4)^{2k}}\right)=-\sum_{k=1}^{\infty}\frac{13}{16^{k}}=-\frac{13}{15},\\\
\max\\{x\,:\,x+iy\in\mathcal{K}\\}&=\sum_{k=1}^{\infty}\left(\frac{-1}{(-4)^{2k-1}}+\frac{3}{(-4)^{2k}}\right)=\sum_{k=1}^{\infty}\frac{7}{16^{k}}=\frac{7}{15}.\end{split}$

Using these geometric properties, we obtain that

$\begin{split}\alpha\,(\mathcal{K}\cap\Delta_{1,1,-r})&=(\mathcal{K}\cup(\mathcal{K}+1))\cap\Delta_{1,0,r}=\mathcal{K}\cap\Delta_{1,0,r\,},\\\
\alpha^{2}\,(\mathcal{K}\cap\Delta_{0,1,r/2})&=\big{(}\mathcal{K}\cup(\mathcal{K}+1)\cup(\mathcal{K}+\alpha)\cup(\mathcal{K}+\alpha+1)\big{)}\cap\Delta_{1,0,r}\\\
&=(\mathcal{K}\cap\Delta_{1,0,r})\cup\big{(}(\mathcal{K}+i)\cap\Delta_{1,0,r})=(\mathcal{K}\cap\Delta_{1,0,r})+\\{0,i\\},\\\
\alpha\,(\mathcal{K}\cap\Delta_{1,-1,r/2})&=\big{(}\mathcal{K}\cup(\mathcal{K}+1)\big{)}\cap\Delta_{0,1,r/2}=(\mathcal{K}\cap\Delta_{0,1,r/2})+\\{0,1\\},\\\
\alpha^{3}\,(\mathcal{K}\cap\Delta_{1,-1,r/2})&=\alpha^{2}\,(\mathcal{K}\cap\Delta_{0,1,r/2})-\\{0,2i\\}=(\mathcal{K}\cap\Delta_{1,0,r})+\\{-2i,-i,0,i\\}.\end{split}$

For $r$ as in Theorem 3.1, we have $-\frac{8}{15}<r<\frac{2}{15}$ since

$\begin{split}\min\left\\{\sum_{k=1}^{\infty}\frac{2a_{k}}{(-4)^{k}}\,:\,a_{1}a_{2}\cdots\in\\{0,1\\}^{\omega}\right\\}&=-\sum_{k=1}^{\infty}\frac{8}{16^{k}}=-\frac{8}{15},\\\
\max\left\\{\sum_{k=1}^{\infty}\frac{2a_{k}}{(-4)^{k}}\,:\,a_{1}a_{2}\cdots\in\\{0,1\\}^{\omega}\right\\}&=\sum_{k=1}^{\infty}\frac{2}{16^{k}}=\frac{2}{15},\end{split}$

and the minimum and maximum are attained only for the sequences
$(10)^{\omega}$ and $(01)^{\omega}$, which we have excluded. Therefore,
Theorem 3.1 and the formulae above give that

$\begin{split}\mathcal{K}\cap\Delta_{1,1,-r}&=-\tfrac{1+i}{2}\left(r\,(1+i)+\left[-\tfrac{2}{5},\tfrac{3}{5}\right]i\right)=-r\,i+\left[-\tfrac{1}{5},\tfrac{3}{10}\right](1-i),\\\
\mathcal{K}\cap\Delta_{0,1,r/2}&=\tfrac{i}{2}\left(r\,(1+i)+\left[-\tfrac{2}{5},\tfrac{8}{5}\right]\,i\right)=r\,\tfrac{-1+i}{2}+\left[-\tfrac{4}{5},\tfrac{1}{5}\right],\\\
\mathcal{K}\cap\Delta_{1,-1,r/2}&=\tfrac{1-i}{4}\left(r\,(1+i)+\left[-\tfrac{12}{5},\tfrac{8}{5}\right]i\right)=\tfrac{r}{2}+\left[-\tfrac{3}{5},\tfrac{2}{5}\right](1+i),\end{split}$

which proves the statements for the intersection of $\mathcal{K}$ with lines.
For the intersections of $\partial\mathcal{K}$ with lines, it only remains to
check that the points in

$\alpha^{-2}\big{(}(\mathcal{K}\cap\Delta_{1,0,r})\cap((\mathcal{K}\cap\Delta_{1,0,r})+i)\big{)}=\left\\{\tfrac{1}{\alpha^{2}}\left(r\,(1+i)+\tfrac{3i}{5}\right)\right\\}$

and

$\alpha^{-1}\big{(}(\mathcal{K}\cap\Delta_{0,1,r/2})\cap((\mathcal{K}\cap\Delta_{0,1,r/2})+1)\big{)}=\left\\{\tfrac{1}{\alpha^{3}}\left(r\,(1+i)-\tfrac{2}{5}\,i\right)\right\\}$

are not in $\partial K$. By the proof of Theorem 3.1, the digit expansion

$[.a_{1}100a_{2}011a_{3}100a_{4}011\cdots]_{\alpha}=r\,(1+i)+\tfrac{3}{5}\,i$

is given by a path starting only from $g_{3}$ in the boundary automaton
$\mathcal{G}$. Dividing by $\alpha^{2}$ adds $00$ in front of the expansion,
but $g_{3}$ cannot be reached by $00$, hence
$\frac{1}{\alpha^{2}}\left(r\,(1+i)+\frac{3i}{5}\right)$ is not on the
boundary of $K$. Similarly, the digit expansion

$[.a_{1}011a_{2}100a_{3}011a_{4}100\cdots]_{\alpha}=r\,(1+i)-\tfrac{2}{5}\,i$

is given by a path starting from $g_{4}$ in the boundary automaton
$\mathcal{G}$, and $g_{4}$ cannot be reached by $000$, thus
$\frac{1}{\alpha^{3}}\left(r\,(1+i)-\frac{2}{5}\,i\right)$ is not on the
boundary of $K$. This proves that all intersections of $\mathcal{K}$ with the
given lines are line segments. ∎

We can use this method to find a vertical line with a more interesting
intersection. For example, if we look at $\Delta_{1,0,-1/4}$, we see that the
only expansion $\sum_{k=1}^{\infty}\frac{b_{k}}{(-4)^{k}}$ with
$b_{k}\in\mathcal{D}$ having real part $-1/4$ is $b_{1}b_{2}\cdots=100\cdots$.
In base $\alpha$, we must therefore have
$d_{1}d_{2}d_{3}d_{4}\in\\{0001,0101,1010,1110\\}$, which correspond to the
digits $1,1-2i,1+3i,1+i\in\mathcal{D}$. The remaining digit sequences
$d_{5}d_{6}\cdots$ give points in
$\frac{1}{\alpha^{4}}(\mathcal{K}\cap\Delta_{1,0,0})$, thus

$\mathcal{K}\cap\Delta_{1,0,-1/4}=-\tfrac{1}{4}+\left(\left[-\tfrac{9}{10},-\tfrac{13}{20}\right]\cup\left[-\tfrac{2}{5},\tfrac{1}{10}\right]\cup\left[\tfrac{7}{20},\tfrac{3}{5}\right]\right)\,i.$

We go on with $\Delta_{1,0,-1/4+1/16}$ and see that points in the intersection
have imaginary part with an expansion in base $-4$ starting with two digits in
$\\{-2,0,1,3\\}$ and ending with digits in $\\{-1,0,1,2\\}$. For the limit
$\Delta_{1,0,-1/5}$ of lines of this form, we obtain the following
intersection with $\mathcal{K}$, see Figure 2.

######  Theorem 3.3.

We have

$\mathcal{K}\cap\Delta_{1,0,-1/5}=\left\\{-\frac{1}{5}+\sum_{k=1}^{\infty}\frac{d_{k}}{(-4)^{k}}\,i\,:\,d_{k}\in\\{-2,0,1,3\\}\
\mbox{for all}\ k\geq 1\right\\},$

and a point is in $\partial\mathcal{K}\cap\Delta_{1,0,-1/5}$ if and only if it
is of the form $-\tfrac{1}{5}+\sum_{k=1}^{\infty}d_{k}(-4)^{-k}\,i$, where
$d_{1}d_{2}\cdots$ is a path in the automaton in Figure 4.

###### Proof.

Since $-\frac{1}{5}=\sum_{k=1}^{\infty}(-4)^{-k}$, we obtain in the same way
as in the proof of Theorem 3.1 that
$\mathfrak{R}\big{(}\sum_{k=1}^{\infty}b_{k}(-4)^{-k}\big{)}=-\frac{1}{5}$
with $b_{k}\in\mathcal{D}$ if and only if $\mathfrak{R}(b_{k})=1$ for all
$k\geq 1$, i.e., $b_{k}\in\\{1{-}2i,1,1{+}i,1{+}3i\\}$. The corresponding
4-digit blocks in base $\alpha$ are $0101$, $0001$, $1110$, and $1010$. This
proves the characterization of $\mathcal{K}\cap\Delta_{1,0,-1/5}$.

In the boundary automaton, the digit blocks $0101$, $0001$, $1110$, and $1010$
are accepted only from $g_{3}$ and $g_{4}$, and we have the transitions

$g_{3}\overset{0101}{\longrightarrow}g_{3},\
g_{3}\overset{0001}{\longrightarrow}g_{4},\
g_{3}\overset{0101}{\longrightarrow}g_{4},\
g_{4}\overset{1010}{\longrightarrow}g_{4},\
g_{4}\overset{1010}{\longrightarrow}g_{3},\
g_{4}\overset{1110}{\longrightarrow}g_{3}.$

Taking imaginary parts of the corresponding numbers in $\mathcal{D}$ gives the
automaton in Figure 4. ∎

$g_{3}$$g_{4}$-2-2,031,3

Figure 4. Automaton recognizing the imaginary parts of points in
$\partial\mathcal{K}\cap\Delta_{1,0,-1/5}$ in base $-4$.

######  Theorem 3.4.

The Hausdorff dimension of $\mathcal{K}\cap\Delta_{1,0,-1/5}$ is 1 and

$\dim_{H}(\partial\mathcal{K}\cap\Delta_{1,0,-1/5})=\tfrac{\log 3}{\log
4}\approx 0.7925>\mathfrak{s}-1.$

###### Proof.

We can interpret the intersection with $\Delta_{1,0,-1/5}$ as the self-similar
digit tile in $\mathbb{R}$ with $A=-4$ and $D=\\{-2,0,1,3\\}$. Since $D$ is a
complete residue system modulo 4, this tile has non empty interior and
therefore is of dimension 1.

For the boundary, we have $\partial\mathcal{K}\cap\Delta_{1,0,-1/5}=K_{3}\cup
K_{4}$, with

$-4K_{3}=(K_{3}-2)\cup(K_{4}-2)\cup
K_{4},\quad-4K_{4}=(K_{3}+1)\cup(K_{3}+3)\cup(K_{4}+3).$

Therefore, by [6], the Hausdorff dimension of
$\partial\mathcal{K}\cap\Delta_{1,0,-1/5}$ is $\log\beta/\log 4$, where
$\beta$ is the Perron-Frobenius eigenvalue of the matrix $\binom{1\ 2}{2\ 1}$,
i.e., $\beta=3$. ∎

### Acknowledgments.

The authors were supported by the project I3346 of the Japan Society for the
Promotion of Science (JSPS) and the FWF, the project FR 07/2019 of the
Austrian Agency for International Cooperation in Education and Research
(OeAD), the project PHC Amadeus 42314NC, and the project ANR-18-CE40-0007
CODYS of the Agence Nationale de la Recherche (ANR).

## References

  * [1] S. Akiyama and K. Scheicher, Intersecting two-dimensional fractals with lines, Acta Sci. Math. (Szeged), 71 (2005), pp. 555–580.
  * [2] S. Akiyama and J. M. Thuswaldner, The topological structure of fractal tilings generated by quadratic number systems, Comput. Math. Appl., 49 (2005), pp. 1439–1485.
  * [3] P. J. Grabner, P. Kirschenhofer, and H. Prodinger, The sum-of-digits function for complex bases, J. London Math. Soc. (2), 57 (1998), pp. 20–40.
  * [4] A. Manning and K. Simon, Dimension of slices through the Sierpinski carpet, Trans. Amer. Math. Soc., 365 (2013), pp. 213–250.
  * [5] J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3), 4 (1954), pp. 257–302.
  * [6] R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), pp. 811–829.
  * [7] K. Scheicher and J. M. Thuswaldner, Neighbours of self-affine tiles in lattice tilings, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 241–262.