# On the doubly stochastic realization of spectra

Issam Kaddoura
Department of Mathematics,
Lebanese International University,
Saida, Lebanon
<EMAIL_ADDRESS>
&Bassam Mourad
Department of Mathematics,
Faculty of Science, Lebanese University,
Beirut, Lebanon
<EMAIL_ADDRESS>

#  On a special class of block matrices and some applications

Issam Kaddoura
Department of Mathematics,
Lebanese International University,
Saida, Lebanon
<EMAIL_ADDRESS>
&Bassam Mourad
Department of Mathematics,
Faculty of Science, Lebanese University,
Beirut, Lebanon
<EMAIL_ADDRESS>

###### Abstract

In this paper, we study a particular class of block matrices placing an
emphasis on their spectral properties. Some related applications are then
presented.

_Keywords_ generalized permutation matrices, block generalized permutation
matrices, nonnegative matrices, doubly stochastic matrices, inverse eigenvalue
problem, matrix number theory

AMS CLASSIFICATION: 15A18, 15A51,15A24, 15A29, 12D10.

##  1 Introduction

Recall that every element $\pi$ in the permutation group $S_{n}$ corresponds
to a permutation matrix $P_{\pi}=(p_{ij})$ given by

$p_{ij}=\left\\{\begin{array}[c]{cc}1&\text{if}\ j=\pi(i)\\\ &\\\ 0&\ \text{
otherwise}.\end{array}\right.$

Let $\mathbb{C}$, $\mathbb{Z}$ and $\mathbb{N}$ denote the field of complex
numbers, the ring of integers and the set of natural numbers, respectively.
The $n\times n$ diagonal matrix whose $i$-th diagonal entry is $u_{i}$ will be
denoted by diag$(u_{1},...,u_{n})$. A square matrix having in each row and
column only one nonzero element is called a _generalized permutation_ (or
_monomial_) matrix.

For any vector $u=[u_{1},...,u_{n}]^{T}$ in $\mathbb{C}^{n}$, denote by
$D_{u}=$diag$(u_{1},...,u_{n})$ and for any permutation $\pi\in S_{n}$, let
$\pi(u)=[u_{\pi(1)},...,u_{\pi(n)}]^{T}\in\mathbb{C}^{n}$. If
$v=[v_{1},...,v_{n}]^{T}\in\mathbb{C}^{n}$, then the Hadamard product of $u$
and $v$ which is denoted by $u\odot v$, is defined in the normal way as
$u\odot v=[u_{1}v_{1},...,v_{n}v_{n}]^{T}$.

It is easy to see that an $n\times n$ matrix $U$ is a generalized permutation
matrix if and only if $U=D_{u}P_{\pi}$, in which $u\in\mathbb{C}^{n}$, $\pi\in
S_{n}$ and $D_{u}=$diag$(u_{1},...,u_{n})$. One may also check that

$P_{\pi}D_{u}=D_{\pi(u)}P_{\pi}.$ (1)

Next, recall that _the order_ of a permutation $\pi_{s},$ which is denoted by
$O(\pi_{s})$, is defined as the smallest positive integer $d$ such that
$\pi_{s}^{d}=\pi_{0}$ where $\pi_{0}$ is the identity in $S_{n}$. It is clear
that if $d$ is the order of a permutation $\pi_{s}$ and $P_{\pi_{s}}$ is the
permutation matrix corresponding to it, then $P_{\pi_{s}}^{d}=I_{n}$ where
$I_{n}$ is the $n\times n$ identity matrix.

The following lemma is needed for our purposes where its proof can be found in
[4].

###### Lemma 1.1

Let $s\in\\{0,1,2,3.....,n-1\\}$, $n\in\mathbb{N}$ and $\pi_{s}\in S_{n}$ such
that $\pi_{s}(i)\equiv i+s\operatorname{mod}(n).$ Then,

  1. 1.

$d=O(\pi_{s})=\frac{\operatorname{lcm}(n,s)}{s}=\frac{n}{\gcd(n,s)}.$

  2. 2.

$n$ is prime if only if $O(\pi_{s})=n$.

  3. 3.

If $\gcd(n,s)=1$, then $O(\pi_{s})=n$.

  4. 4.

If $O(\pi_{s})=\frac{\operatorname{lcm}(n,s)}{s}=d,$ then
$\pi_{s}^{qd+r}=\pi_{s}^{r}$ for any integer $q$ and $0\leq r<d.$

  5. 5.

If $O(\pi_{s})=d$, then $\pi_{s}^{r_{1}}=\pi_{s}^{r_{2}}$ if and only if
$r_{1}=r_{2}(\operatorname{mod}d).$

If $P=(p_{ij})$ is the permutation matrix corresponding to the permutation
$\pi_{1}(i)\equiv i+1\operatorname{mod}(n)$ i.e. $P$ is the $n\times n$
permutation matrix with 1 in the positions $(i,i+1)$ and $(n,1)$ for all
$i=1,\ldots n-1$, then a circulant matrix $C\in M_{n}(\mathbb{C})$ is an
element in the algebra generated by $P$ i.e. $C$ is of the form

$C=c_{0}I_{n}+c_{1}P+\ldots+c_{n-1}P^{n-1}$

for some complex numbers $c_{0},...,c_{n-1}$. As the eigenvalues and
eigenvectors of $P$ are well known, so are those of any circulant matrix $C$.

In [4], we generalized the class of circulant matrices to another class of
matrices which we called generalized weighted circulant matrices as follows.
Let $G_{w}$ be the class of all generalized permutation matrices corresponding
to the following family of permutations $\pi_{s}(i)\equiv
i+s\operatorname{mod}(n)$ where $s\in w:=\\{0,1,2,3.....,n-1\\}$ and with
$\pi_{0}(i)=\pi_{n}(i)\equiv i\operatorname{mod}(n)$ is the identity in
$S_{n}$. Then, a generalized weighted circulant matrix ia an element in the
algebra generated by an element $T$ in $G_{w}$. The main objective in [4] was
to find explicit expressions for the eigenvalues and eigenvectors of any $T\in
G_{w}$, and to present some applications based on these formulas.

In this work, we shall generalize the class of generalized weighted circulant
matrices to what we call generalized weighted circulant block matrices.
Roughly speaking, the idea is to start with any generalized permutation matrix
$X$ in $G_{w}$ and replace the nonzero entries in $X$ by square block matrices
of the same size to obtain a block generalized permutation matrix $M_{X}$ and
then a generalized weighted circulant block matrix is an element in the
algebra generated by $M_{X}$. Our main objective here is to study the spectral
properties of this new class of matrices and to present some related
applications as well.

We shall start by generalizing the notation introduced at the beginning to
block matrices. If $A=(A_{ij})$ is a matrix partitioned into blocks, then to
avoid any confusion we shall sometimes write $A(i,j)$ to refer to the block
$A_{ij}$ specifically when $i$ and $j$ are long expressions.

Recall that $A$ is said to be a _block generalized permutation_ matrix (BGPM)
if $A$ has only one nonzero block in each block row and block column. It is
easy to check that an $n\times n$ block matrix $A$ having each block of size
$k\times k$ is a block generalized permutation matrix with nonzero blocks
$A_{1.\pi(1)},\ldots,A_{n.\pi(n)}$ if and only if $A=D_{A}(P_{\pi}\otimes
I_{k})$ where
$D_{A}=\texttt{diag}(A_{1,\pi(1)},...,A_{n,\pi(n)})=\bigoplus\limits_{i=1}^{n}A_{i,\pi(i)}$
and $\otimes$ stands for the Kronecker product of matrices. Similar to (1), an
inspection shows that for any $\pi,\partial\in S_{n}$, the following equality
is valid

$\biggl{(}P_{\pi}\otimes
I_{k}\biggr{)}\bigoplus\limits_{i=1}^{n}A_{i,\partial(i)}=\bigoplus\limits_{i=1}^{n}A_{\pi(i),\partial\circ\pi(i)}\biggl{(}P_{\pi}\otimes
I_{k}\biggr{)}$ (2)

Next, we present the following useful lemma that exhibits a formula for
multiplying distinct block generalized permutation matrices.

###### Lemma 1.2

Suppose that
$U_{\pi}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i))\right)\biggl{(}P_{\pi}\otimes
I_{k}\biggr{)}$ and
$V_{\partial}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}V(i,\partial(i))\right)\biggl{(}P_{\partial}\otimes
I_{k}\biggr{)}$ are two block generalized $kn\times kn$ permutation matrices
corresponding to the permutations $\pi$ and $\partial$ respectively, then

$U_{\pi}V_{\partial}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U_{i,\pi(i)}V_{\pi(i),\partial\circ\pi(i)}\right)\biggl{(}P_{\partial\circ\pi}\otimes
I_{k}\biggr{)},$

and similarly
$V_{\partial}U_{\pi}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}V_{i,\partial(i)}U_{\partial(i),\pi\circ\partial(i)}\right)\biggl{(}P_{\pi\circ\partial}\otimes
I_{k}\biggr{)}.$

Proof. By making use of (2), we can write

$\displaystyle U_{\pi}V_{\partial}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i))\right)\biggl{(}P_{\pi}\otimes
I_{k}\biggr{)}\left({\displaystyle\bigoplus\limits_{i=1}^{n}}V(i,\partial(i))\right)\biggl{(}P_{\partial}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i))\right)\left({\displaystyle\bigoplus\limits_{i=1}^{n}}V(\pi(i),\partial\circ\pi(i))\right)\biggl{(}P_{\pi}\otimes
I_{k}\biggr{)}\biggl{(}P_{\partial}\otimes I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U_{i,\pi(i)}V_{\pi(i),\partial\circ\pi(i)}\right)\biggl{(}P_{\partial\circ\pi}\otimes
I_{k}\biggr{)}.$

Of course, the second equality is now obvious.

###### Lemma 1.3

Suppose that
$U_{\pi}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i))\right)\biggl{(}P_{\pi}\otimes
I_{k}\biggr{)}$ is a BGPM. Then for any positive integer $r$, we have that
$U_{\pi}^{r}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{r}}U(\pi^{j-1}(i),\pi^{j}(i))\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)}.$

Proof. We proceed by induction on $r$. Obviously, it is true for $r=1$.
Suppose now that the formula is true for $r-1$ i.e.

$U_{\pi}^{r-1}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{r-1}}U(\pi^{j-1}(i),\pi^{j}(i))\right)\biggl{(}P_{\pi^{r-1}}\otimes
I_{k}\biggr{)}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}B(i,\partial(i))\right)\biggl{(}P_{\partial}\otimes
I_{k}\biggr{)}$

where
$B(i,\partial(i))={\displaystyle\prod\limits_{j=1}^{r-1}}U(\pi^{j-1}(i),\pi^{j}(i))$
and $\partial=\pi^{r-1}.$ Now, making use of the previous lemma we obtain

$\displaystyle U_{\pi}^{r}$
$\displaystyle=U_{\pi}.U_{\pi}^{r-1}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U_{i,\pi(i)}B_{\pi(i),\partial\circ\pi(i)}\right)\biggl{(}P_{\partial\circ\pi}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U_{i,\pi(i)}B_{\pi(i),\pi^{r-1}\circ\pi(i)}\right)\biggl{(}P_{\pi^{r-1}\circ\pi}\otimes
I_{k}\biggr{)}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U_{i,\pi(i)}B_{\pi(i),\pi^{r}(i)}\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i)).{\displaystyle\prod\limits_{j=1}^{r-1}}U(\pi^{j-1}(\pi(i)),\pi^{j}(\pi(i)))\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i)).{\displaystyle\prod\limits_{j=1}^{r-1}}U(\pi^{j}(i),\pi^{j+1}(i))\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i)).{\displaystyle\prod\limits_{j-1=1}^{r-1}}U(\pi^{j-1}(i),\pi^{j}(i))\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi(i)).{\displaystyle\prod\limits_{j=2}^{r}}U(\pi^{j-1}(i),\pi^{j}(i))\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)}$
$\displaystyle=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{r}}U(\pi^{j-1}(i),\pi^{j}(i))\right)\biggl{(}P_{\pi^{r}}\otimes
I_{k}\biggr{)},$

and the proof is complete.

###### Corollary 1.4

If $P_{\pi_{s}}$ is a permutation matrix corresponding to $\pi_{s}(i)\equiv
i+s\operatorname{mod}(n)$ where $s\in\\{0,1,2,3.....,n-1\\}$, then

$\biggl{(}P_{\pi_{s}}\otimes I_{k}\biggr{)}^{O(\pi_{s})}=I.$

Proof. $\biggl{(}P_{\pi_{s}}\otimes
I_{k}\biggr{)}^{O(\pi_{s})}=\biggl{(}P_{\pi_{s}}\biggr{)}^{O(\pi_{s})}\otimes\biggl{(}I_{k}\biggr{)}^{O(\pi_{s})}=I_{n}\otimes
I_{k}=I_{nk}.$

Now, we are in a position to present formulaes for the eigenvalues and the
characteristic polynomial of any block generalized permutation matrix
corresponding to the family of the permutations $\pi_{s}(i)\equiv
i+s\operatorname{mod}(n),s\in\\{0,1,2,3.....,n-1\\}$ and $n\in\mathbb{N}.$

###### Theorem 1.5

Let
$U_{\pi_{s}}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi_{s}(i))\right)\biggl{(}P_{\pi_{s}}\otimes
I_{k}\biggr{)}$ be a $kn\times kn$ block generalized permutation matrix where
$s\in\\{1,2,3.....,n-1\\},$ $\pi_{s}(i)\equiv i+s\operatorname{mod}(n),$ and
$U(i,\pi_{s}(i))$ is a $k\times k$ matrix for each $i=1,2,3,..,n.$ Moreover,
let $d=O(\pi_{s})$ and $g=\gcd(n,s)$. Then,

$U_{\pi_{s}}^{d}={\displaystyle\bigoplus\limits_{i=1}^{dg}}\biggl{(}{\displaystyle\prod\limits_{j=1}^{d}}U(i+(j-1)s,i+js)\biggr{)}.$

Therefore, the characteristic polynomial $p(x)$ of$\ U_{\pi_{s}}^{d}$ is given
by

$\ \ \
p(x)={\displaystyle\prod\limits_{i=1}^{g}}\det\left({\displaystyle\prod\limits_{j=1}^{d}}U\bigl{(}\pi_{s}^{j-1}(i),\pi_{s}^{j}(i)\bigr{)}-xI_{k}\right)^{d},$

and hence the eigenvalues $\mu(U_{\pi_{s}})$ of $U_{\pi_{s}}$ are determined
by solving

$\mu^{d}(U_{\pi_{s}})=\lambda_{i}(U_{\pi_{s}}^{d}),\ \ \ i=1,2,..g.$

Proof. From the preceding lemma, we know that

$U_{\pi}^{d}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{d}}U(\pi_{s}^{j-1}(i),\pi_{s}^{j}(i))\right)\biggl{(}P_{\pi^{d}}\otimes
I_{k}\biggr{)},$

and from Lemma 1.1, we conclude that $n=gd$ and $P_{\pi_{s}}^{d}=I.$ This
implies that

$U_{\pi_{s}}^{d}={\displaystyle\bigoplus\limits_{i=1}^{dg}}\left({\displaystyle\prod\limits_{j=1}^{d}}U(i+(j-1)s,i+js)\right).$

It is well known that the diagonal blocks of $U_{\pi_{s}}^{d}$ which are
${\displaystyle\prod\limits_{j=1}^{d}}U(i+(j-1)s,i+js),$
${\displaystyle\prod\limits_{j=1}^{d}}U(i+g+(j-1)s,i+g+js),\ldots,{\displaystyle\prod\limits_{j=1}^{d}}U(i+(d-1)g+(j-1)s,i+(d-1)g+js)$
have the same eigenvalues. This in turn means that the following $d$
determinants are equal
$\det\left({\displaystyle\prod\limits_{j=1}^{d}}U(i+(j-1)s,i+js)-xI_{k}\right)=\det\left({\displaystyle\prod\limits_{j=1}^{d}}U(i+g+(j-1)s,i+g+js)-xI_{k}\right)$
$=\ldots=\det\left({\displaystyle\prod\limits_{j=1}^{d}}U(i+(d-1)g+(j-1)s,i+(d-1)g+js)-xI_{k}\right).$
Consequently, the characteristic polynomial of$\ U_{\pi_{s}}^{d}$ is given by

$\ \ \
p(x)={\displaystyle\prod\limits_{i=1}^{g}}\det\left({\displaystyle\prod\limits_{j=1}^{d}}A(\pi_{s}^{j-1}(i),\pi_{s}^{j}(i))-xI_{k}\right)^{d}.$

This completes the proof.

The particular case $s=1$ gives the following result which is Theorem 1 in
Mirsky[7] (see also [10]).

###### Corollary 1.6

Let

$\displaystyle U_{\pi_{1}}=\left(\begin{array}[]{ccccc}0&U_{1}&0&\ldots&0\\\
0&0&U_{2}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&\ddots&0\\\
0&0&\ldots&0&U_{n-1}\\\ U_{n}&0&\ldots&0&0\\\ \end{array}\right)$ (8)

where the $U_{i}$ are $k\times k$ matrices. If the eigenvalues of the $k\times
k$ matrix $U_{1}U_{2}...U_{n}$ are $w_{1},\ w_{2},\ldots,\ w_{k}$. Then, the
eigenvalues of $U_{\pi_{1}}$ are the $kn$ numbers consisting of all $n$-th
roots of each of the numbers $w_{1},\ w_{2},\ ,...,\ w_{k}$.

To illustrate our results, we start with the following simple example.

###### Example 1

Consider $n=4,\ k=2,\ s=2,$ and $\pi_{2}(i)\equiv i+2\operatorname{mod}4.$ Let
$U_{\pi_{2}}$ be given by
$U_{1,\pi_{2}(1)}=U(1,3)=\left(\begin{array}[c]{cc}a&0\\\
0&b\end{array}\right),\
U_{2,\pi_{2}(2)}=U(2,4)=\left(\begin{array}[c]{cc}c&0\\\
0&d\end{array}\right),$
$U_{3,\pi_{2}(3)}=U(3,1)=\left(\begin{array}[c]{cc}x&0\\\
0&y\end{array}\right),\
U_{4,\pi_{2}(4)}=U(4,2)=\left(\begin{array}[c]{cc}z&0\\\
0&t\end{array}\right),$ so that

$U_{\pi_{2}}=\left(\begin{array}[c]{cccc}\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{a}&{0}\\\
{0}&{b}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)\\\ \left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{c}&{0}\\\
{0}&{d}\end{array}\right)\\\ \left(\begin{array}[c]{cc}{x}&{0}\\\
{0}&{}y\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)\\\ \left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{z}&{0}\\\
{0}&{t}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)\end{array}\right).$

Clearly, the order of $\pi_{2}$ is
$d=O(\pi_{2})=\dfrac{\operatorname{lcm}(4,2)}{2}=2,$ and $g=gcd(4,2)=2,$ so
that in order to determine the spectrum of $U_{\pi_{2}},$ we first need to
determine the spectrum of the block generalized permutation matrix
$U_{\pi_{2}}^{2}.$ Obviously,

$\displaystyle U_{\pi_{2}}^{2}$
$\displaystyle={\displaystyle\bigoplus\limits_{i=1}^{2}}{\displaystyle\prod\limits_{j=1}^{2}}U(i+2(j-1),i+2j){\displaystyle\bigoplus\limits_{i=3}^{4}}{\displaystyle\prod\limits_{j=1}^{2}}U(i+2(j-1)s,i+2j)$
$\displaystyle={\displaystyle\bigoplus\limits_{i=1}^{2}}U(i,i+2)U(i+2,i){\displaystyle\bigoplus\limits_{i=3}^{4}}U(i,i+2)U(i+2,i)$
$\displaystyle=U(1,3)U(3,1){\displaystyle\bigoplus}U(2,4)U(4,2){\displaystyle\bigoplus}U(3,1)U(1,3){\displaystyle\bigoplus}U(4,2)U(2,4))$
$\displaystyle=\allowbreak\left(\begin{array}[c]{cc}{ax}&{0}\\\
{0}&{by}\end{array}\right){\displaystyle\bigoplus}\allowbreak\left(\begin{array}[c]{cc}{cz}&{0}\\\
{0}&{dt}\end{array}\right){\displaystyle\bigoplus}\allowbreak\left(\begin{array}[c]{cc}{ax}&{0}\\\
{0}&{by}\end{array}\right){\displaystyle\bigoplus}\left(\begin{array}[c]{cc}{cz}&{0}\\\
{0}&{dt}\end{array}\right)$

Therefore, the eigenvalues of $U_{\pi_{2}}$ are clearly given by $\sqrt{ax},\
-\sqrt{ax},\ \sqrt{by},\ -\sqrt{by},\ \sqrt{cz},\ -\sqrt{cz},\ \sqrt{dt},\
-\sqrt{dt}.$

Nest, we take a similar numeric example but with minor changes namely the
nonzero diagonal blocks are taken to be non-diagonal.

###### Example 2

Consider $n=4,\ k=2,\ s=2,$ and $\pi_{2}(i)\equiv i+2\operatorname{mod}4.$ Let
$U_{\pi_{2}}$ be given by
$U_{1,\pi_{2}(1)}=U(1,3)=\left(\begin{array}[c]{cc}1&2\\\
1&-1\end{array}\right),\
U_{2,\pi_{2}(2)}=U(2,4)=\left(\begin{array}[c]{cc}3&0\\\
2&-2\end{array}\right),$
$U_{3,\pi_{2}(3)}=U(3,1)=\left(\begin{array}[c]{cc}2&5\\\
0&0\end{array}\right),\
U_{4,\pi_{2}(4)}=U(4,2)=\left(\begin{array}[c]{cc}5&1\\\
-1&-1\end{array}\right),$ so that

$U_{\pi_{2}}=\left(\begin{array}[c]{cccc}\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{1}&{2}\\\
{1}&{-1}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)\\\ \left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{3}&{0}\\\
{2}&{-2}\end{array}\right)\\\ \left(\begin{array}[c]{cc}{2}&{5}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)\\\ \left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{5}&{1}\\\
{-1}&{-1}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)&\left(\begin{array}[c]{cc}{0}&{0}\\\
{0}&{0}\end{array}\right)\end{array}\right).$

Clearly, the order of $\pi_{2}$ is
$d=O(\pi_{2})=\dfrac{\operatorname{lcm}(4,2)}{2}=2,$ and $g=gcd(4,2)=2,$ so
that in order to determine the spectrum of $U_{\pi_{2}},$ we first need to
determine the spectrum of the block generalized permutation matrix
$U_{\pi_{2}}^{2}.$ Obviously,

$\displaystyle U_{\pi_{2}}^{2}$
$\displaystyle={\displaystyle\bigoplus\limits_{i=1}^{2}}{\displaystyle\prod\limits_{j=1}^{2}}U(i+2(j-1),i+2j){\displaystyle\bigoplus\limits_{i=3}^{4}}{\displaystyle\prod\limits_{j=1}^{2}}U(i+2(j-1)s,i+2j)$
$\displaystyle={\displaystyle\bigoplus\limits_{i=1}^{2}}U(i,i+2)U(i+2,i){\displaystyle\bigoplus\limits_{i=3}^{4}}U(i,i+2)U(i+2,i)$
$\displaystyle=U(1,3)U(3,1){\displaystyle\bigoplus}U(2,4)U(4,2){\displaystyle\bigoplus}U(3,1)U(1,3){\displaystyle\bigoplus}U(4,2)U(2,4))$
$\displaystyle=\allowbreak\left(\begin{array}[c]{cc}{2}&{5}\\\
{2}&{5}\end{array}\right){\displaystyle\bigoplus}\allowbreak\left(\begin{array}[c]{cc}{15}&{3}\\\
{12}&{4}\end{array}\right){\displaystyle\bigoplus}\allowbreak\left(\begin{array}[c]{cc}{7}&{-1}\\\
{0}&{0}\end{array}\right){\displaystyle\bigoplus}\left(\begin{array}[c]{cc}{17}&{-2}\\\
{-5}&{2}\end{array}\right)$

In other words,

$U_{\pi_{2}}^{2}=\left(\begin{array}[c]{cccccccc}{2}&{5}&{0}&{0}&{0}&{0}&{0}&{0}\\\
{2}&{5}&{0}&{0}&{0}&{0}&{0}&{0}\\\ {0}&{0}&{15}&{3}&{0}&{0}&{0}&{0}\\\
{0}&{0}&{12}&{4}&{0}&{0}&{0}&{0}\\\ {0}&{0}&{0}&{0}&{7}&{-1}&{0}&{0}\\\
{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\\ {0}&{0}&{0}&{0}&{0}&{0}&{17}&{-2}\\\
{0}&{0}&{0}&{0}&{0}&{0}&{-5}&{2}\end{array}\right),$

and a simple check shows that the characteristic polynomial of
$U_{\pi_{2}}^{2}$ is given by
$p(x)=x^{2}\left(x-7\right)^{2}\left(-19x+x^{2}+24\right)^{2}.$ Therefore, the
eigenvalues of $U_{\pi_{2}}^{2}$ can be exhibited as
$\frac{19}{2}+\frac{1}{2}\sqrt{265},\ \frac{19}{2}+\frac{1}{2}\sqrt{265},\
\frac{19}{2}-\frac{1}{2}\sqrt{265},\ \frac{19}{2}-\frac{1}{2}\sqrt{265},\ 0,\
0,\ 7,\ 7.$ However, this in turn means that the eigenvalues of $U_{\pi_{2}}$
are given by $\frac{1}{2}\sqrt{2}\sqrt{19-\sqrt{265}},\
-\frac{1}{2}\sqrt{2}\sqrt{19-\sqrt{265}},\
\frac{1}{2}\sqrt{2}\sqrt{\sqrt{265}+19},\
-\frac{1}{2}\sqrt{2}\sqrt{\sqrt{265}+19},\ 0,\ 0,\ \sqrt{7},\\\ -\sqrt{7}.$

###### Definition 1.7

Let
$U_{\pi_{s}}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi_{s}(i))\right)\biggl{(}P_{\pi_{s}}\otimes
I_{k}\biggr{)}$ be a $kn\times kn$ block generalized permutation matrix where
$s\in\\{1,2,3.....,n-1\\},$ $\pi_{s}(i)\equiv i+s\operatorname{mod}(n),$ and
$U(i,\pi_{s}(i))$ is a $k\times k$ matrix for each $i=1,2,3,..,n.$ Then, a
block generalized permutation circulant matrix $C$ is an element in the
algebra generated by $U_{\pi_{s}}$ i.e.

$C=\sum_{r=0}^{m}c_{r}(U_{\pi_{s}})^{r},$

where $m\in\mathbb{N}$ and the $c_{r}$ are complex constants.

Making use of the preceding theorem, we conclude the following.

###### Theorem 1.8

Let
$U_{\pi_{s}}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}U(i,\pi_{s}(i))\right)\biggl{(}P_{\pi_{s}}\otimes
I_{k}\biggr{)}$ be a $kn\times kn$ block generalized permutation matrix where
$s\in\\{1,2,3.....,n-1\\},$ $\pi_{s}(i)\equiv i+s\operatorname{mod}(n),$ and
$U(i,\pi_{s}(i))$ is a $k\times k$ matrix for each $i=1,2,3,..,n.$ Moreover,
let $d=O(\pi_{s})$ and $g=\gcd(n,s).$ If
$C=\sum_{r=0}^{m}c_{r}(U_{\pi_{s}})^{r}$ is a generalized permutation
circulant matrix generated by $U_{\pi_{s}}$, then the eigenvalues of $C$ are
given by

$\lambda_{t,r}(C)=\sum_{r=0}^{m}c_{r}\left(\sqrt[d]{\lambda_{t}}\right)^{r},\
t=1,2,\ldots,g.$

where $\lambda_{t}$ is an eigenvalue of
${\displaystyle\prod\limits_{j=1}^{d}}U(\pi_{s}^{j-1}(t),\pi_{s}^{j}(t))$ for
$t=1,2,...,g.$

###  1.1 Application 1

Recall that the inverse eigenvalue problem for special types of matrices is
concerned with constructing a matrix that maintains the required structure
from its set of eigenvalues. Constructing matrices for the inverse eigenvalue
problems arise in a remarkable variety of applications, including system and
control theory, geophysics, molecular spectroscopy, quantum mechanics,
particle physics, structure analysis, economics and operation research to name
a few (see, e.g. [2]).

An $n\times n$ matrix $A$ is said to be nonnegative  if all entries in $A$ are
nonnegative. An $n\times n$ nonnegative matrix $A$ whose each row and column
sum equals to 1 is said to be doubly stochastic. If $J_{n}$ denotes the
$n\times n$ matrix whose all entries are $\frac{1}{n}$, then clearly an
$n\times n$ nonnegative matrix $A$ is doubly stochastic if and only if
$AJ_{n}=J_{n}A=J_{n}.$ The nonnegative inverse eigenvalue problem (NIEP) (see,
e.g., [3]) is concerned with finding the necessary and sufficient conditions
on a set of $n$ complex numbers to be the spectrum of an $n\times n$
nonnegative matrix $A$. Similarly, the doubly stochastic inverse eigenvalue
problem (DIEP) deals with finding necessary and sufficient conditions on a set
of $n$ complex numbers in order to be the spectrum of an $n\times n$ doubly
stochastic matrix $A$.

Next, we shall show that Theorem 1.5 and Theorem 1.7 can serve as building
blocks for obtaining many new results concerning the preceding two problems.
More explicitly, one of the obvious applications of Theorem 1.5 is concerned
with generating new sufficient conditions from known ones for both the NIEP
and DIEP. More generally, any known results concerning these problems can lead
to new results as we shall illustrate below.

####  1.1.1 nonnegative matrices

Here, we shall show how Theorem 1.5 can be used to find new sufficient
conditions for the resolutions of the NIEP. In addition, we shall see how any
known sufficient conditions for the NIEP in low dimension can lead to new
sufficient conditions for the NIEP in higher dimension. We shall start by
proving that for any positive integer $n$, the $n$-th roots of the eigenvalues
of nonnegative matrices can occur as spectra of nonnegative matrices.

###### Theorem 1.9

Let $A$ be a $k\times k$ nonnegative matrix with eigenvalues $\lambda_{1},$
$\lambda_{2},$… $\lambda_{k}$. For each positive integer $n$ and for each
$i=1,...,k$, let $\mu_{1}^{(i)},$ $\mu_{2}^{(i)},$…,$\mu_{n}^{(i)}$ be the
$n$th roots of $\lambda_{i}$, then there exists a $kn\times kn$ nonnegative
matrix with eigenvalues

$\\{\mu_{1}^{(1)},\mu_{2}^{(1)},...,\mu_{n}^{(1)},\ldots,\mu_{1}^{(k)},\mu_{2}^{(k)},...,\mu_{n}^{(k)}\\}.$

Moreover, we recursively conclude that for any positive integer $s$, the $s$th
roots of the $n$th roots of $\\{\lambda_{1},$ $\lambda_{2},$… $\lambda_{k}\\}$
can occur as spectra of nonnegative matrices.

Proof. By Corollary1,6, it suffices to take

$\displaystyle
U_{\pi_{1}}(n)=\left(\begin{array}[]{ccccc}0&U_{1}&0&\ldots&0\\\
0&0&U_{2}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&\ddots&0\\\
0&0&\ldots&0&U_{n-1}\\\ U_{n}&0&\ldots&0&0\\\ \end{array}\right)$ (14)

where $U_{n}=A$ and $U_{i}=I_{k}$ for each $i=1,...,n-1$ and then we obtain
the nonnegative matrix
$T_{n}(A)=\left(\begin{array}[]{ccccc}0&I_{k}&0&\ldots&0\\\
0&0&I_{k}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&\ddots&0\\\
0&0&\ldots&0&I_{k}\\\ A&0&\ldots&0&0\\\ \end{array}\right)$ with the required
spectrum. In order to prove the second part, we now consider $U_{\pi_{1}}(s)$
and then we repeat the same process this time with $U_{s}=T_{n}(A)$ and
$U_{i}=I_{n}$ for each $i=1,...,s-1$ to obtain the required result.

Next, we shall see how any known sufficient conditions for the NIEP can serve
as building blocks for obtaining many new ones. Although, one can
systematically collect all the known sufficient conditions for the NIEP and
use it to obtain new ones, however this will not be done here and instead we
restrict ourselves only to two examples to illustrate the idea. Indeed, we
shall illustrate our results by considering the following theorem whose proof
can be found in [6].

###### Theorem 1.10

Let $\lambda_{1}>0\geq\lambda_{2}\geq...\geq\lambda_{k}.$ Then there exists a
$k\times k$ nonnegative matrix whose eigenvalues are
$\\{\lambda_{1},...,\lambda_{n}\\}$.

###### Corollary 1.11

Let $\lambda_{1}>0\geq\lambda_{2}\geq...\geq\lambda_{k}.$ Then for any
positive integer $n$, there exists a $kn\times kn$ matrix $A$ whose
eigenvalues are the $n$th roots of $\\{\lambda_{1},...,\lambda_{k}\\}.$ More
generally, for any positive integer $s$, the $s$th roots of the $n$th roots of
$\\{\lambda_{1},$ $\lambda_{2},$… $\lambda_{k}\\}$ can occur as spectra of
nonnegative matrices.

In [12] we have proved the following theorem.

###### Theorem 1.12

Let $\left(\lambda_{2},\ldots,\lambda_{k}\right)$ be any list of complex
numbers which is closed under complex conjugation and let the real and
imaginary parts of $\lambda_{i}$ be respectively denoted by
$x_{i}=\Re(\lambda_{i})$ and $y_{i}=\Im(\lambda_{i})$. Then, the following
holds.
1) Suppose that $\left(\lambda_{2},\ldots,\lambda_{k}\right)$ are such that
$\lambda_{2}\geq\ldots\geq\lambda_{p}$ are real for $2\leq p\leq k$ and
$\lambda_{p+1},\ldots,\lambda_{k}$ are non-real. Moreover, let

$w=\min\\{\pm\lambda_{2},\ldots,\pm\lambda_{p},\pm x_{p+1},\pm
y_{p+1},\ldots,\pm x_{\frac{k+p}{2}},\pm y_{\frac{k+p}{2}}\\}.$

Then, there exists a $k\times k$ nonnegative matrix with eigenvalues
$\left((k-1)|w|,\lambda_{2},\ldots,\lambda_{k}\right)$.
2) Suppose now that $\left(\lambda_{2},\ldots,\lambda_{k}\right)$ with
$k=2h-1$, is such that none of its components is real and let $w=\min\\{\pm
x_{2},\pm y_{2},\ldots,\pm x_{h},\pm y_{h}\\},$ Then, there exists a $k\times
k$ nonnegative matrix with eigenvalues
$\left(k|w|,\lambda_{2},\ldots,\lambda_{k}\right)$.

###### Corollary 1.13

Let $\left(\lambda_{2},\ldots,\lambda_{k}\right)$ be any list of complex
numbers which is closed under complex conjugation and let the real and
imaginary parts of $\lambda_{i}$ be respectively denoted by
$x_{i}=\Re(\lambda_{i})$ and $y_{i}=\Im(\lambda_{i})$. Then, the following
holds.
1) Suppose that $\left(\lambda_{2},\ldots,\lambda_{k}\right)$ are such that
$\lambda_{2}\geq\ldots\geq\lambda_{p}$ are real for $2\leq p\leq k$ and
$\lambda_{p+1},\ldots,\lambda_{k}$ are non-real. Moreover, let

$w=\min\\{\pm\lambda_{2},\ldots,\pm\lambda_{p},\pm x_{p+1},\pm
y_{p+1},\ldots,\pm x_{\frac{k+p}{2}},\pm y_{\frac{+kp}{2}}\\}.$

Then, for any positive integer $n$, there exists a $kn\times kn$ matrix $A$
whose eigenvalues are the $n$th roots of
$\left((k-1)|w|,\lambda_{2},\ldots,\lambda_{k}\right)$. More generally, for
any positive integer $s$, the $s$th roots of the $n$th roots of
$\left((k-1)|w|,\lambda_{2},\ldots,\lambda_{k}\right)$ can occur as spectra of
nonnegative matrices.
2) Suppose now that $\left(\lambda_{2},\ldots,\lambda_{k}\right)$ with
$k=2h-1$, is such that none of its components is real and let $w=\min\\{\pm
x_{2},\pm y_{2},\ldots,\pm x_{h},\pm y_{h}\\},$ Then, for any positive integer
$n$, there exists a $kn\times kn$ matrix $A$ whose eigenvalues are the $n$th
roots of $\left(k|w|,\lambda_{2},\ldots,\lambda_{k}\right)$. More generally,
for any positive integer $s$, the $s$th roots of the $n$th roots of
$\left(k|w|,\lambda_{2},\ldots,\lambda_{k}\right)$ can occur as spectra of
nonnegative matrices.

####  1.1.2 doubly stochastic matrices

We start here by noting that in the proof of Theorem 1.8 if $A$ happens to be
doubly stochastic then the matrix $T_{n}(A)$ is also doubly stochastic. Thus
we have the following.

###### Theorem 1.14

Let $A$ be a $k\times k$ doubly stochastic matrix with eigenvalues
$\lambda_{1},$ $\lambda_{2},$… $\lambda_{k}$. For each $i=1,...,k$, let
$\mu_{1}^{(i)},$ $\mu_{2}^{(i)},$…,$\mu_{n}^{(i)}$ be the $n$th roots of
$\lambda_{i}$, then there exists an $kn\times kn$ doubly stochastic matrix
with eigenvalues

$\\{\mu_{1}^{(1)},\mu_{2}^{(1)},...,\mu_{n}^{(1)},\ldots,\mu_{1}^{(k)},\mu_{2}^{(k)},...,\mu_{n}^{(k)}\\}.$

Moreover, we recursively conclude that for any positive integer $s$, the $s$th
roots of the $n$th roots of $A$ can occur as spectra of doubly stochastic
matrices.

In [8], we have proved the following theorem.

###### Theorem 1.15

For $n=2k$, then the $n$-list $(1,0,...,0,-1)$ is the spectrum of an $n\times
n$ doubly stochastic matrix, and for $n=2k+1$, the $n$-list $(1,0,...,0,-1)$
can not be the spectrum of an $n\times n$ doubly stochastic matrix,

Noting that $\\{1,-1\\}$ are the square roots of unity, we prove the following
generalization of the preceding theorem.

###### Theorem 1.16

Let $k>1$ be any positive integer and Let $w=e^{\frac{2\pi i}{k}}$ be a
primitive $k$th root of unity. Then, the list
$(1,0,...,0,w,w^{2},...,w^{k-1})$ is the spectrum of $kn\times kn$ doubly
stochastic matrix.

Proof. In view of Corollary1,6, it suffices to take

$\displaystyle
U_{\pi_{1}}(n)=\left(\begin{array}[]{ccccc}0&U_{1}&0&\ldots&0\\\
0&0&U_{2}&\ddots&\vdots\\\ \vdots&\vdots&\ddots&\ddots&0\\\
0&0&\ldots&0&U_{n-1}\\\ U_{n}&0&\ldots&0&0\\\ \end{array}\right)$ (20)

where $U_{n}=J_{k}$ and $U_{i}=I_{k}$ (or any $k\times k$ doubly stochastic
matrix as in this case $U_{1}.U_{2}...U_{n-1}J_{k}=J_{k}$) for each
$i=1,...,n-1$ and then we obtain a doubly stochastic matrix $M$ whose
eigenvalues are the $n$th roots of the eigenvalues of $J_{k}$ which are
$\\{1,0,...,0\\}$ and thus $M$ satisfies the requirements of the theorem.

In a similar fashion to the case of the NIEP, any known sufficient conditions
for the DIEP, will essentially lead to new results. We illustrate our
discussion in 3 situations. The first one solves the DIEP for the case $n=2.$

###### Lemma 1.17

There exists a $2\times 2$ symmetric doubly stochastic matrix with spectrum
$(1,\lambda)$ if and only if $-1\leq\lambda\leq 1.$

Proof. It suffices to note that for $-1\leq\lambda\leq 1,$ the matrix
$X=\left(\begin{array}[]{cc}\frac{1+\lambda}{2}&\frac{1-\lambda}{2}\\\
\frac{1-\lambda}{2}&\frac{1+\lambda}{2}\\\ \end{array}\right)$ is doubly
stochastic and has spectrum $(1,\lambda).$

As a conclusion, we have the following.

###### Corollary 1.18

Let $-1\leq\lambda\leq 1.$ Then, for positive integers $n$ and $s$, the $s$th
roots of the $n$tth roots of $\\{1,\lambda\\}$ occur as the spectra of doubly
stochastic matrices.

The second situation deals with the case $n=3$ (see, e.g. [9, 11]) and can be
stated as follows.

###### Theorem 1.19

There exists a symmetric $3\times 3$ doubly stochastic matrix with spectrum
$(1,\lambda,\mu)$ if and only if $-1\leq\lambda\leq 1,$ $-1\leq\mu\leq 1,$
$\lambda+3\mu+2\geq 0$ and $3\lambda+\mu+2\geq 0.$

Proof. Suppose that $\lambda\geq\mu,$ then the matrix
$X=\frac{1}{6}\left(\begin{array}[]{ccc}2+4\lambda&2-2\lambda&2-2\lambda\\\
2-2\lambda&2+\lambda+3\mu&2+\lambda-3\mu\\\
2-2\lambda&2+\lambda-3\mu&2+\lambda+3\mu\end{array}\right)$ is doubly
stochastic and has spectrum $(1,\lambda,\mu).$ Now if $\mu\geq\lambda,$ then
the proof can be completed by exchanging the roles of $\lambda$ and $\mu.$

As a result, we have the following.

###### Corollary 1.20

Suppose that $-1\leq\lambda\leq 1,$ $-1\leq\mu\leq 1,$ $\lambda+3\mu+2\geq 0$
and $3\lambda+\mu+2\geq 0.$ Then, for positive integers $n$ and $s$, the $s$th
roots of the $n$tth roots of $\\{1,\lambda,\mu\\}$ occur as the spectra of
doubly stochastic matrices.

Now if we let $\Pi_{3}$ denote the region of the complex plane specified as
the convex hull of the cubic roots of unity. Now our third illustration stems
from the following theorem which is due to [11] and the proof which we include
here appears in [6].

###### Theorem 1.21

[11] Let $z$ be a complex number and $\bar{z}$ be its complex conjugate. Then
$(1,z,\bar{z})$ is the spectrum of a $3\times 3$ doubly stochastic matrix if
and only if $z\in\Pi_{3}.$

Proof. It suffices to check that the circulant matrix

$X=\frac{1}{3}\left(\begin{array}[]{ccc}1+2r\cos\theta&1-2r\cos(\frac{\pi}{3}+\theta)&1-2r\cos(\frac{\pi}{3}-\theta)\\\
1-2r\cos(\frac{\pi}{3}-\theta)&1+2r\cos\theta&1-2r\cos(\frac{\pi}{3}+\theta)\\\
1-2r\cos(\frac{\pi}{3}+\theta)&1-2r\cos(\frac{\pi}{3}-\theta)&1+2r\cos\theta\end{array}\right)$

has spectrum $(1,z=re^{i\theta},\bar{z}=re^{-i\theta}),$ and $X$ is doubly
stochastic if and only if $z=re^{i\theta}\in\Pi_{3}.$

Consequently, we have the following.

###### Corollary 1.22

Let $z\in\Pi_{3}.$ Then, for positive integers $n$ and $s$, the $s$th roots of
the $n$tth roots of $\\{1,z,\bar{z}\\}$ occur as the spectra of doubly
stochastic matrices.

###### Remark 1.23

It is worthy to note that in our applications here, we have only used the
permutation $\pi_{1}\in S_{n}$ which is defined by $\pi_{1}(i)\equiv
i+1\operatorname{mod}(n).$ Of course, when using different permutations we can
clearly obtain new results by making use of the same techniques.

###  1.2 Application 2

Recall that Fermat’s last theorem states that the equation $\
x^{p}+y^{p}=z^{p}$ has no nontrivial solutions over the integers for all
$p>2$. More generally, Beal’s conjecture states that the equation
$x^{p}+y^{q}=z^{r}$ has no nontrivial solutions where $p,\ q$ and $r$ are
integers greater than 2 and $x,y$ and $z$ are coprime integers (see for
example [5, 13]).

Not much is known concerning integral matrix solutions to the equations
$X^{p}+Y^{p}=Z^{p}$ and $X^{p}+Y^{q}=Z^{r}$ where $X$, $Y$ and $Z$ are
$n\times n$ integer matrices with a particular structure. More generally, in
this subsection we shall present some results concerning finding integer
matrix solutions to the two equations $aX^{p}+bY^{p}=cZ^{p}$ and
$aX^{p}+bY^{q}=cZ^{r}$ for any integers $a,b$ and $c$. By making use of
certain properties of generalized permutation matrices, we are able to prove
the existence of an infinite number of solutions for these two problems such
that these solution matrices have all of their entries nonzero natural
numbers.

We shall begin with the following theorem.

###### Theorem 1.24

Let $a$, $b$ and $c$ be any integers. Then, for every positive integer $k$,
there is an infinite number of $n\times n$ square matrices $A,$ $B$ and $C$
with integer entries such that $aA^{k}+bB^{k}=cC^{k}.$ In particular, we can
select all entries of $A,$ $B$ and $C$ to be nonzero natural numbers.

Proof. As usual, let $\ \pi_{s}(i)\equiv i+s\operatorname{mod}n,\ s\in$
$\\{1,2,3..,n-1\\}.$ Now choose $s_{1},s_{2},s_{3}\in$ $\\{1,2,3.....,n-1\\}$
such that $ks_{1}\equiv ks_{2}\equiv ks_{3}\operatorname{mod}n$ (obviously our
choice is valid at least for $s_{1}=s_{2}=s_{3}),$ then $\ \pi_{s_{1}}^{k}=\
\pi_{s_{2}}^{k}=\ \pi_{s_{3}}^{k}$ and
$P_{\pi_{s_{1}}^{k}}=P_{\pi_{s_{2}}^{k}}=P_{\pi_{s_{3}}^{k}}.$ Next, consider
the following three generalized permutation block $mn\times mn$ matrices:
$A_{\pi_{s_{1}}}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}A(i,\pi_{s_{1}}(i))\right)\biggl{(}P_{\pi_{s_{1}}}\otimes
I_{m}\biggr{)}$,
$B_{\pi_{s_{2}}}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}B(i,\pi_{s_{2}}(i))\right)\biggl{(}P_{\pi_{s_{2}}}\otimes
I_{m}\biggr{)}$ and
$C_{\pi_{s_{3}}}=\left({\displaystyle\bigoplus\limits_{i=1}^{n}}C(i,\pi_{s_{3}}(i))\right)\biggl{(}P_{\pi_{s_{3}}}\otimes
I_{m}\biggr{)}$ where $A(i,\pi_{s_{1}}(i)),B(i,\pi_{s_{2}}(i))$ and
$C(i,\pi_{s_{3}}(i))$ are $m\times m$ nonzero matrices whose nonzero entries
are parameters for $i=1,...,n$. From Lemma 1.3, we get

$A_{\pi_{s_{1}}}^{k}=\biggl{(}{\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{k}}A(i+(j-1)s_{1},i+js_{1})\biggr{)}\biggl{(}P_{\pi_{s_{1}}^{k}}\otimes
I_{m}\biggr{)},$

$B_{\pi_{s_{2}}}^{k}=\biggl{(}{\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{k}}B(i+(j-1)s_{2},i+js_{2})\biggr{)}\biggl{(}P_{\pi_{s_{2}}^{k}}\otimes
I_{m}\biggr{)},$
$C_{\pi_{s_{3}}}^{k}=\biggl{(}{\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{k}}C(i+(j-1)s_{3},i+js_{3})\biggr{)}\biggl{(}P_{\pi_{s_{3}}^{k}}\otimes
I_{m}\biggr{)}.$

Since $\ \pi_{s_{1}}^{k}=\ \pi_{s_{2}}^{k}=\ \pi_{s_{3}}^{k}$, then solving
the following system of equations,

$\displaystyle
a\biggl{(}{\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{k}}A(i+(j-1)s_{1},i+js_{1})\biggr{)}+b\biggl{(}{\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{k}}B(i+(j-1)s_{2},i+js_{2})\biggr{)}$
$\displaystyle=\ \ \ \ \ \ \ \ \ \ \ \ \ $ $\displaystyle
c\biggl{(}{\displaystyle\bigoplus\limits_{i=1}^{n}}{\displaystyle\prod\limits_{j=1}^{k}}C(i+(j-1)s_{3},i+js_{3})\biggr{)}$

over $\mathbb{Z}$, yields the desired result
$aA_{\pi_{s_{1}}}^{k}+bB_{\pi_{s_{2}}}^{k}=cC_{\pi_{s_{3}}}^{k}.$

An inspection shows that we can always find some relations between the
parameters of these solution matrices to obtain an appropriate subfamily of
solutions $T,R$ and $W$ which are pairwise commuting matrices and all of whose
nonzero entries are positive integers. Now, let $D=p(T,R,W)$ be any polynomial
in $T,R$ and $W$ whose all entries are nonzero natural numbers. Obviously, $D$
commutes with the matrices $T,R$ and $W$. Clearly, the matrices $A=DT,$ $B=DR$
and $C=DW$ have all their entries as nonzero natural numbers and
$aA^{k}+bB^{k}-cC^{k}=$
$a\left(DT\right)^{k}+b\left(DR\right)^{k}-c\left(DW\right)^{k}=0$.

As a conclusion, we have the following.

###### Corollary 1.25

If $n$ a is prime, then for any integers $a$, $b$ and $c$ there is an infinite
number of $n\times n$ matrices $A,\ B$ and $C$ with integer entries such that
$aA^{n}+bB^{n}=cC^{n}.$

Another conclusion is given in the following.

###### Corollary 1.26

Let $a$, $b$ and $c$ be any integers. Then the equation $aA^{n}+bB^{p}=cC^{q}$
where $n,\ p$ and $q$ are arbitrary positive integers, has an infinite family
of matrix solutions over the square matrices with integer entries. In
particular, we can select all entries in the solution matrices to be natural
numbers.

Proof. Let $k=lcm(n,p,q)$, then there exists positive integers $r,\ s$ and $t$
such that $k=rn,\ k=sp$, and $k=tq$. From the previous theorem, we know that
there exists an infinite number of matrices such that $aA^{k}+bB^{k}=cC^{k}.$
But this in turn means that $aA^{rn}+bB^{sp}=cC^{tq}.$ Therefore,
$a\left(A^{r}\right)^{n}+b\left(B^{s}\right)^{p}=c\left(C^{t}\right)^{q},$ and
the proof is complete.

###### Example 3

In order to illustrate our results, we shall solve for positive integer
matrices the following equation

$4X^{5}=Y^{5}+3Z^{5}.$

We start with following generalized permutation matrices whose all nonzero
entries are considered as parameters

$T=\left(\begin{array}[c]{ccccc}0&m&0&0&0\\\ 0&0&n&0&0\\\ 0&0&0&p&0\\\
0&0&0&0&q\\\ r&0&0&0&0\end{array}\right),\
R=\left(\begin{array}[c]{ccccc}0&0&u&0&0\\\ 0&0&0&v&0\\\ 0&0&0&0&w\\\
s&0&0&0&0\\\ 0&t&0&0&0\end{array}\right),\
W=\left(\begin{array}[c]{ccccc}0&0&0&x&0\\\ 0&0&0&0&y\\\ z&0&0&0&0\\\
0&k&0&0&0\\\ 0&0&h&0&0\end{array}\right)$

We then find some relations among their entries so that $T,R$ and $W$ are
pairwise commuting matrices. In particular, we consider the following setting

$T=\left(\begin{array}[c]{ccccc}0&tuw&0&0&0\\\ 0&0&suv&0&0\\\ 0&0&0&tvw&0\\\
0&0&0&0&suw\\\
stv&0&0&0&0\end{array}\right),R=\left(\begin{array}[c]{ccccc}0&0&u&0&0\\\
0&0&0&v&0\\\ 0&0&0&0&w\\\ s&0&0&0&0\\\ 0&t&0&0&0\end{array}\right),\mbox{ and
}$ $W=\left(\begin{array}[c]{ccccc}0&0&0&tuvw&0\\\ 0&0&0&0&suvw\\\
stvw&0&0&0&0\\\ 0&stuw&0&0&0\\\ 0&0&stuv&0&0\end{array}\right).$

By solving the following system $4T^{5}-R^{5}-3W^{5}=0$ we obtain $stuvw-1=0$.
Taking $s=\frac{1}{tuvw}$ and defining

$A=tuvwT=\left(\begin{array}[c]{ccccc}0&t^{2}u^{2}vw^{2}&0&0&0\\\
0&0&uv&0&0\\\ 0&0&0&t^{2}uv^{2}w^{2}&0\\\ 0&0&0&0&uw\\\
tv&0&0&0&0\end{array}\right)$

$B=tuvwR=\left(\begin{array}[c]{ccccc}0&0&tu^{2}vw&0&0\\\ 0&0&0&tuv^{2}w&0\\\
0&0&0&0&tuvw^{2}\\\ 1&0&0&0&0\\\ 0&t^{2}uvw&0&0&0\end{array}\right)$

$C=tuvwW=\left(\begin{array}[c]{ccccc}0&0&0&t^{2}u^{2}v^{2}w^{2}&0\\\
0&0&0&0&uvw\\\ tvw&0&0&0&0\\\ 0&tuw&0&0&0\\\ 0&0&tuv&0&0\end{array}\right)$
where $t,u,v$ and $w$ are now restricted to be in $\mathbb{N}$, then we
clearly have $4A^{5}-B^{5}-3C^{5}=0$. Next, let $r,a,b,c$ and $d$ be any
positive integers, and define the matrix

$\displaystyle D$ $\displaystyle=rI_{5}+aA+bB+cC+dC^{3}$
$\displaystyle=\left(\begin{array}[c]{ccccc}r&at^{2}u^{2}vw^{2}&btu^{2}vw&ct^{2}u^{2}v^{2}w^{2}&dt^{3}u^{4}v^{3}w^{4}\\\
dt^{2}u^{2}v^{3}w^{2}&r&auv&btuv^{2}w&cuvw\\\
ctvw&dt^{4}u^{3}v^{3}w^{4}&r&at^{2}uv^{2}w^{2}&btuvw^{2}\\\
b&ctuw&dt^{2}u^{3}v^{2}w^{2}&r&auw\\\
atv&bt^{2}uvw&ctuv&dt^{4}u^{3}v^{4}w^{3}&r\end{array}\right)$

which obviously commutes with $A,\ B$ and $C$. Up to this point, let
$X=DA=\left(\begin{array}[c]{ccccc}dt^{4}u^{4}v^{4}w^{4}&rt^{2}u^{2}vw^{2}&at^{2}u^{3}v^{2}w^{2}&bt^{3}u^{3}v^{3}w^{3}&ct^{2}u^{3}v^{2}w^{3}\\\
ctuv^{2}w&dt^{4}u^{4}v^{4}w^{4}&ruv&at^{2}u^{2}v^{3}w^{2}&btu^{2}v^{2}w^{2}\\\
bt^{2}uv^{2}w^{2}&ct^{3}u^{2}v^{2}w^{3}&dt^{4}u^{4}v^{4}w^{4}&rt^{2}uv^{2}w^{2}&at^{2}u^{2}v^{2}w^{3}\\\
atuvw&bt^{2}u^{2}vw^{2}&ctu^{2}vw&dt^{4}u^{4}v^{4}w^{4}&ruw\\\
rtv&at^{3}u^{2}v^{2}w^{2}&bt^{2}u^{2}v^{2}w&ct^{3}u^{2}v^{3}w^{2}&dt^{4}u^{4}v^{4}w^{4}\end{array}\right),$
$Y=DB=\left(\begin{array}[c]{ccccc}ct^{2}u^{2}v^{2}w^{2}&dt^{5}u^{5}v^{4}w^{5}&rtu^{2}vw&at^{3}u^{3}v^{3}w^{3}&bt^{2}u^{3}v^{2}w^{3}\\\
btuv^{2}w&ct^{2}u^{2}v^{2}w^{2}&dt^{3}u^{4}v^{4}w^{3}&rtuv^{2}w&atu^{2}v^{2}w^{2}\\\
at^{2}uv^{2}w^{2}&bt^{3}u^{2}v^{2}w^{3}&ct^{2}u^{2}v^{2}w^{2}&dt^{5}u^{4}v^{5}w^{5}&rtuvw^{2}\\\
r&at^{2}u^{2}vw^{2}&btu^{2}vw&ct^{2}u^{2}v^{2}w^{2}&dt^{3}u^{4}v^{3}w^{4}\\\
dt^{4}u^{3}v^{4}w^{3}&rt^{2}uvw&at^{2}u^{2}v^{2}w&bt^{3}u^{2}v^{3}w^{2}&ct^{2}u^{2}v^{2}w^{2}\end{array}\right)$
and
$Z=DC=\left(\begin{array}[c]{ccccc}bt^{2}u^{2}v^{2}w^{2}&ct^{3}u^{3}v^{2}w^{3}&dt^{4}u^{5}v^{4}w^{4}&rt^{2}u^{2}v^{2}w^{2}&at^{2}u^{3}v^{2}w^{3}\\\
atuv^{2}w&bt^{2}u^{2}v^{2}w^{2}&ctu^{2}v^{2}w&dt^{4}u^{4}v^{5}w^{4}&ruvw\\\
rtvw&at^{3}u^{2}v^{2}w^{3}&bt^{2}u^{2}v^{2}w^{2}&ct^{3}u^{2}v^{3}w^{3}&dt^{4}u^{4}v^{4}w^{5}\\\
dt^{3}u^{3}v^{3}w^{3}&rtuw&atu^{2}vw&bt^{2}u^{2}v^{2}w^{2}&ctu^{2}vw^{2}\\\
ct^{2}uv^{2}w&dt^{5}u^{4}v^{4}w^{4}&rtuv&at^{3}u^{2}v^{3}w^{2}&bt^{2}u^{2}v^{2}w^{2}\end{array}\right),$
then it is easy to check that $4X^{5}-Y^{5}-3Z^{5}=0$.

## Acknowledgments

The first author is supported from LIU.

## References

  * [1] J. L. Brenner and J. de Pillis. Fermat’s equation $A^{p}+B^{p}=C^{p}$ for matrices of integers. Math. Mag., 45 (1972) pp. 12-15.
  * [2] M.T. Chu and G. H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Application, Oxford Science Publications, Oxford University Press, 2005.
  * [3] Charles R. Johnson, Carlos Marijuan, Pietro Paparella and Miriam Pisonero, The NIEP, Operator Theory, Operator Algebras, and Matrix Theory, 267, (2018) pp. 199–220.
  * [4] I. Kaddoura and B. Mourad, On a class of matrices generated by certain generalized permutation matrices and applications, Linear and Multilinear Algebra, 67 (2019) pp. 2117–2134
  * [5] R. D. Mauldin, A Generalization of Fermat’s Last Theorem: The Beal Conjecture and Prize Problem, Notices of the AMS 44 (1997) pp. 1436-1439.
  * [6] H. Minc, Nonnegative matrices Springer-Verlag, New York-Heidelberg, 1988.
  * [7] L. Mirsky. An inequality for characteristic roots and singular values of complex matrices. Monatsh. Math., 70 (1966) pp. 357–359.
  * [8] B. Mourad, An inverse problem for symmetric doubly stochastic matrices, Inverse Problems 19 (2003) pp. 821-831.
  * [9] B. Mourad, On a Lie-theoretic approach to generalized doubly stochastic matrices and applications, Linear and Multilin. Alg., 52, (2004) pp. 99-113.
  * [10] A. L. Nickerson, P. Paparella, Jordan Chains of $H$-Cyclic Matrices II, Electronic Journal of Linear Algebra, 38, (2022) pp. 486–93.
  * [11] H. Perfect and L. Mirsky. Spectral properties of doubly-stochastic matrices. Monatsh. Math., 69 (1965) 35–57.
  * [12] K. Rammal, B. Mourad, H. Abbas, et al. On the doubly stochastic realization of spectra. Banach J. Math. Anal. 16, 49 (2022). https://doi.org/10.1007/s43037-022-00203-8.
  * [13] P. Ribenboim, 13 lectures on Fermat’s last theorem. Springer-Verlag, New York-Heidelberg, 1979.


# From holography to the geometry of the spacetime

José A. S. Pelegrín
Departamento de Geometría y Topología,
Universidad de Granada, 18071 Granada, Spain
E-mail<EMAIL_ADDRESS>

###### Abstract

We show the geometric consequences that the holographic principle has on the
spacetime. Namely, we prove that complete spacelike hypersurfaces in a
spacetime that satisfies the holographic principle are non-parabolic. This has
important consequences on the behaviour of the Brownian motion in these
spacelike hypersurfaces, as well as provides a method for finding examples of
universes that lead to a violation of this principle.

MSC 2010: 53C80, 58J05, 53C50

Keywords: Holographic principle, transience, spacelike hypersurface.

##  1 Introduction

The recent revolution in string theory and black holes theory have changed the
way we understand the nature of the universe. Among the new unexpected results
that have been obtained, one of utmost importance is the holographic
principle, a mathematical conjecture that describes the relation between the
spacetime geometry and the number of quantum states of matter. Namely, the
degrees of freedom of a spatial region reside not in the interior but on its
boundary. This fact reduces the complexity of the physical systems and,
supported by the AdS/CFT duality [31], it has been argued that a fundamental
theory that aims to unify gravity and quantum mechanics should include this
counterintuitive result, i.e., it should be a holographic theory [6]. Our aim
in this article is to obtain some geometric consequences derived from the
assumption of this holographic principle in a relativistic universe.

Historically, the first step towards the holographic principle was due to
Bekenstein [3] in 1981, when guided by the second law of thermodynamics he
obtained that the entropy in a spherically symmetric spatial region of the
spacetime $S(V)$ satisfied

$S(V)\leq\frac{\mathrm{Area}(\partial V)}{4},$

being $\mathrm{Area}(\partial V)$ the area of the boundary of $V$ in Planck
units. This bound is saturated ,i.e., equality holds, by the Bekenstein-
Hawking entropy associated with a black hole horizon. As a result, no stable
spherical system can have a higher entropy than a black hole of equal size.

Thermodinamical entropy $S$ has a statistical interpretation as the logarith
of the number of independent quantum states compatible with the macroscopic
state of a physical system [7]. Therefore, the dimension of the system’s
Hilbert space is given by $dim(\mathcal{H})=e^{S}$. This enabled ’t Hooft [27]
and Susskind [26] to strengthen the interpretation of Bekenstein’s result,
stating that the number of independent degrees of freedom contained in a
spatial volume $\textit{N}_{dof}(V)$ (which is defined as the logarithm of the
dimension of the system’s Hilbert space) satisfies

$\textit{N}_{dof}(V)\leq\frac{\mathrm{Area}(\partial V)}{4}.$

Thus, a physical system can be completely determined by data stored on its
boundary without exceeding a density of one bit per Planck area [6].

In this article, by means of some potential theory results we are able to
obtain geometric properties of the spacelike hypersurfaces immersed in a
spacetime (i.e., a connected time-orientable Lorentzian manifold endowed with
a time-orientation) of arbitrary dimension where the holographic principle
holds. Given an $n$-dimensional manifold $M$, an immersion
$\psi:M\longrightarrow\overline{M}$ in an $(n+1)$-dimensional spacetime
$(\overline{M},\overline{g})$ is called a spacelike hypersurface if
$g=\psi^{*}\overline{g}$ is a Riemannian metric on $M$ [21]. Roughly speaking,
spacelike hypersurfaces describe the physical universe that some observers can
measure in a given instant of their proper time. These geometric objects are
of great importance in General Relativity both from a physical and
mathematical perspective. Among other reasons, we recall that the initial
value problem for the Einstein’s field equation is formulated in terms of a
spacelike hypersurface [24] and even in causality theory the existence of a
certain spacelike hypersurface can determine the causal properties of the
spacetime [13].

Hence, once we show the restrictions that the holographic principle imposes on
the geometry of the spacelike hypersurfaces, the search of models that lead to
a violation of this principle is reduced to finding spacetimes with spacelike
hypersurfaces where these geometric restrictions do not hold. In particular,
we obtain that this holographic principle is violated in the so called
spatially parabolic spacetimes, i.e., those that admit a parabolic spacelike
hypersurface [25]. Note that spatially parabolic spacetimes are a
generalization of spatially closed models (i.e., spacetimes that admit a
compact spacelike hypersurface). In fact, it was proved in [7] that for a
certain entropy distribution spatially closed spacetimes lead to a violation
of the holographic principle.

This article is organized as follows. In Section 2 we fix the formulation of
the holographic principle that we will use throughout this paper, namely, the
spacelike entropy bound. Moreover, we will also describe some properties of
the Brownian motion on a Riemannian manifold and how it is affected by the
manifold’s geometry. In Section 3 we obtain our main results concerning the
influence of the holographic principle on the geometry of complete spacelike
hypersurfaces in arbitrary spacetimes. Finally, we devote Section 4 to
particularize our previous results for Generalized Robertson-Walker spacetimes
that obey the holographic principle and obtain several results for the
behaviour of the Brownian motion in spacelike hypersurfaces in these ambient
spacetimes.

##  2 Preliminaries

In this section we will fix the formulation of the holographic principle that
we will later use in order to obtain our results as well as recall some
aspects of the study of the Brownian motion on Riemannian manifolds that will
appear along this article.

###  2.1 The spacelike entropy bound

Since Bekenstein’s bound only holds for spherically symmetric weakly
gravitating matter systems in assymptotically flat spacetimes [3], we will use
the following generalization, which is known as the spacelike entropy bound
(see [7] and [27]).

###### Definition 1

The spacelike entropy bound is said to hold in a spacetime $\overline{M}$ if
for every spacelike hypersurface $\psi:M\longrightarrow\overline{M}$ in
$\overline{M}$ the entropy of all matter systems contained in any geodesic
ball $B_{R}\subset M$, $S(B_{R})$, satisfies

$S(B_{R})\leq\frac{\mathrm{Area}(\partial B_{R})}{4},$

where $\mathrm{Area}(\partial B_{R})$ is the area of the boundary of $B_{R}$.

This formulation of the holographic principle has been criticized (although
some of the arguments against the spacelike entropy bound are weakened by the
results obtained in [28]), existing several alternative ones. For instance,
Fischler and Susskind [12] proposed to constrain only the entropy passing
through the past light cone. Later, continuing with these ideas of trying to
establish a lightlike formulation for the holographic principle, Bousso [6]
proposed a covariant entropy bound in terms of light sheets. It has also been
stated in terms of the cosmological apparent horizon [2] as well as taking
other cosmological considerations into account (see [8] and [11]).

However, every formulation of the holographic principle has received
criticism. In fact, the controversy generated is mostly due to the examples of
universes that violate each possible formulation (see [7], [8] and [19], for
instance). As we will show in this article, the validity of the holographic
principle has influence on the geometry of the spacetime. Therefore, the
search of counterexamples to each formulation is reduced to finding spacetimes
that do not satisfy the associated geometric restrictions.

###  2.2 Brownian motion on a Riemannian manifold

The irregular movement of microscopic particles suspended in a liquid has been
known since 1828, when Brown studied the erratic motion of pollen grains on
water. However, it was not until 1905 that Einstein explained this effect,
obtaining that it satisfies a diffusion equation. Later, Wiener [30] was able
to construct a rigorous continuous model of the Brownian motion.

Thanks to Wiener’s model, we can study the behaviour of a particle with a
Brownian motion on an arbitrary Riemannian manifold. In fact, the easiest way
to construct Brownian motion on a Riemannian manifold $(M,g)$ is by means of
the heat kernel $p(x,y,t)$, a function on $M\times M\times(0,+\infty)$ which
is the smallest positive fundamental solution of the heat equation

$\frac{\partial p}{\partial t}-\frac{1}{2}\Delta p=0\ \ \ \textnormal{on}\ M,$

in the variables $(x,t)$ (the point $y$ is fixed), with initial data

$p(\cdot,y,t)\rightarrow\delta_{y}\ \ \ \textnormal{for}\ t\rightarrow 0^{+}.$

Indeed, Dodziuk [10] proved the existence of the heat kernel on any arbitrary
Riemannian manifold regardless of geodesic completeness. The properties of the
heat kernel allow us to construct a (sub)Markov process $X_{t}$ on $M$ with
transition density $p$ (see [9]), which is a diffusion process known as
Brownian motion on $M$. If we denote by $\mathbb{P}_{x}$ corresponding measure
in the space of paths emanating from $x$, we can define the notions of
recurrence and trancience as in [15, Def. 2.1].

###### Definition 2

Brownian motion $X_{t}$ on a Riemannian manifold $M$ is recurrent if, for any
non-empty open set $\Omega$ and for any point $x\in M$,

$\mathbb{P}_{x}\\{\textnormal{there is a sequence}\ t_{k}\rightarrow+\infty\
\textnormal{such that}\ X_{t_{k}}\in\Omega\\}=1.$

Otherwise $X_{t}$ is transient.

Moreover, by means of geometric analysis’ techniques, it has been proved that
the recurrence of the Brownian motion on a manifold is equivalent to that
manifold being parabolic [15]. Therefore, the Brownian motion will be
transient on non-parabolic Riemannian manifolds. We recall that a complete
Riemannian manifold $M$ is parabolic if the only bounded from below,
superharmonic $C^{2}$-functions on $M$ are the constants. Hence, the
development of potential theory and the study of parabolicity has greatly
contributed to the better understanding of the Brownian motion’s recurrence
[20].

In fact, whereas in the Euclidean space $\mathbb{R}^{n}$ the behaviour of the
Brownian motion only depends on the dimension (being recurrent if $n\leq 2$
and transient otherwise), in an arbitrary Riemannian manifold recurrence is
related to some geometric properties such as curvature or volume growth of the
geodesic balls (see, for instance, [20] and [29]). Thus, in this article we
will be able to study the Brownian motion on complete spacelike hypersurfaces
once we obtain the geometric consequences that the holographic principle has
on these Riemannian manifolds.

##  3 Spacelike hypersurfaces in general ambient spacetimes

Once we have established the formulation of the holographic principle that
will be used along this paper as well as the relation between parabolicity of
a Riemannian manifold and recurrence of the Brownian motion, in this section
we will obtain several results concerning the geometric restrictions that the
holographic principle imposes on complete (non-compact) spacelike
hypersurfaces immersed in general ambient spacetimes of arbitrary dimension.
In particular, we have the following result.

###### Theorem 3

Let $\psi:M\longrightarrow\overline{M}$ be a complete spacelike hypersurface
with non-negative Ricci curvature in a spacetime that satisfies the spacelike
entropy bound. If the entropy density is bounded from below by a positive
constant $\sigma_{0}$, then the Brownian motion on $M$ is transient. In
particular, if $M$ is a simply connected spacelike surface, it is conformally
equivalent to $\mathbb{H}^{2}$.

  * Proof.

Let us suppose that the spacetime satisfies the spacelike entropy bound. If
the entropy density is bounded from below by a constant $\sigma_{0}>0$, we
know that the geodesic balls in $M$ satisfy

$\sigma_{0}\mathrm{Vol}(B_{R})\leq S(B_{R})\leq\frac{\mathrm{Area}(\partial
B_{R})}{4}.$ (1)

Hence, using the coarea formula in (1) we obtain

$4\sigma_{0}\leq\frac{d}{dR}\log(\mathrm{Vol}(B_{R})).$

Integrating this, we get

$\mathrm{Vol}(B_{R})\geq e^{4\sigma_{0}R}.$ (2)

Thus, we have

$\int_{1}^{+\infty}\frac{R\
dR}{\mathrm{Vol}(B_{R})}\leq\int_{1}^{+\infty}\frac{R\
dR}{e^{4\sigma_{0}R}}<+\infty.$ (3)

It was proved in [29] that for Riemannian manifolds of non-negative Ricci
curvature the finiteness of the previous integral is equivalent to the
transience of the Brownian motion. The last statement of the theorem follows
from [15, Thm. 5.1]. $\square$

###### Remark 4

Note that the converse does not hold in general. For instance, let us consider
the spacelike hyperplanes $\\{t_{0}\\}\times\mathbb{R}^{3},\
t_{0}\in\mathbb{R}$, in the Lorentz-Minkowski spacetime $\mathbb{L}^{4}$.
Clearly, the Brownian motion is transient on these spacelike hyperplanes.
However, as it was shown in [12] we obtain a violation of the spacelike
entropy bound by considering a constant entropy density $\sigma_{0}$ in
$\\{t_{0}\\}\times\mathbb{R}^{3},\ t_{0}\in\mathbb{R}$ and a geodesic ball of
radius $R>\frac{3}{4\sigma_{0}}$.

Indeed, we can extend our result to the case of spacelike hypersurfaces with
negative Ricci curvature with a controlled radial decay immersed in a
spacetime that obeys the spacelike entropy bound with a more general entropy
distribution. Namely, if the inverse of the entropy distribution is in
$L^{1}([0,+\infty))$ (from now on, we will denote this function space by
$L^{1}(+\infty)$) we obtain the next theorem.

###### Theorem 5

Let $\psi:M\longrightarrow\overline{M}$ be a complete spacelike hypersurface
in a spacetime that satisfies the spacelike entropy bound for an entropy
distribution that satisfies

$\frac{1}{S(B_{R})}\in L^{1}(+\infty).$ (4)

Assume that there exists a constant $C_{1}>0$ such that the Ricci curvature of
$M$ satisfies

$\mathrm{Ric}(p)\geq-C_{1}\rho^{-2}(p),$ (5)

for all $p\in M$, being $\rho(p)$ the distance function to a fixed point $x\in
M$. If there exist $q\in M$ and $C_{2}>0$ such that the volume comparison
condition

$\mathrm{Vol}(B_{R}(q))\leq C_{2}\mathrm{Vol}(B_{\frac{R}{2}}(p)),$ (6)

holds for all $p\in\partial B_{R}(q)$, then the Brownian motion on $M$ is
transient.

  * Proof.

From the spacelike entropy bound, we obtain that (4) yields to

$\frac{1}{\mathrm{Area}(\partial B_{R})}\in L^{1}(+\infty).$ (7)

Now, it is an easy calculus exercise to show that (7) implies the next volume
growth condition (see [4, Lemma 5.10]).

$\frac{R}{\mathrm{Vol}(B_{R})}\in L^{1}(+\infty).$ (8)

By means of [20, Thm. 1.9] we obtain that under our assumptions (8) is
equivalent to the non-parabolicity of $M$. $\square$

###### Remark 6

Since the property of existence of positive Green’s functions is quasi-
isometric invariant (see [15, Cor. 5.3]), we can also prove the transience of
the Brownian motion on a spacelike hypersurface $M$ in these ambient
spacetimes if $M$ is quasi-isometric to a manifold satisfying the assumptions
of Theorem 5. In fact, although the volume comparison condition (6) may seem
restrictive, it is actually satisfied on any complete non-compact Riemannian
manifold with non-negative Ricci curvature due to the Bishop comparison
theorem [5].

Moreover, the validity of the holographic principle for an entropy
distribution satisfying (4) also implies the transience of the Brownian motion
in the particular case where the spacelike hypersurfaces are model manifolds
in the sense of Greene and Wu [14, Def. 2.11]. Namely, we have

###### Theorem 7

Let $\psi:M\longrightarrow\overline{M}$ be a spacelike hypersurface which is a
model manifold in a spacetime $\overline{M}$ that satisfies the spacelike
entropy bound for an entropy distribution that satisfies (4). Then, the
Brownian motion on $M$ is transient.

  * Proof.

If the spacetime satisfies the spacelike entropy bound, from (4) we can deduce

$\frac{1}{\mathrm{Area}(\partial B_{R})}\in L^{1}(+\infty).$ (9)

The result follows from the fact that for a model manifold $M$ condition (9)
is equivalent to non-parabolicity (see [18, Prop. 1.1]). $\square$

By means of these results we can obtain spatially open spacetimes where the
spacelike entropy bound does not hold. In particular, the spacelike entropy
bound is violated in those spacetimes that admit a parabolic spacelike
hypersurface, known as spatially parabolic spacetimes [25]. Indeed, from
Theorem 3 we have the following corollary.

###### Corollary 8

Let $\psi:M\longrightarrow\overline{M}$ be a parabolic spacelike hypersurface
with non-negative Ricci curvature. Then, the spacelike entropy bound does not
hold on $M$ for an entropy density which is bounded from below by a positive
constant.

  * Proof.

If $M$ satisfies the spacelike entropy bound for an entropy density
$\sigma\geq\sigma_{0}>0$, Theorem 3 yields to the non-parabolicity of $M$,
reaching a contradiction. $\square$

##  4 Spacelike hypersurfaces in GRW spacetimes

In this section we will obtain more geometric consequences of the holographic
principle in the family of models known as Generalized Robertson-Walker (GRW)
spacetimes. By a GRW spacetime we mean a product manifold
$\overline{M}=I\times F$ of an open interval $I$ of the real line $\mathbb{R}$
and an $n(\geq 2)$-dimensional (connected) Riemannian manifold $(F,g_{F})$,
endowed with the Lorentzian metric

$\overline{g}=-\pi^{*}_{I}(dt^{2})+f(\pi_{I})^{2}\,\pi_{F}^{*}(g_{F}),$

where $\pi_{I}$ and $\pi_{F}$ denote the projections onto $I$ and $F$,
respectively.

The Lorentzian manifold $(\overline{M},\overline{g})$ is a warped product (in
the sense of [21, Chap. 7]) with base $(I,-dt^{2})$, fiber $(F,g_{F})$ and
warping function $f$. These spacetimes were introduced in [1] to extend the
classical notion of Robertson-Walker spacetime to the case where the fiber
does not necessarily have constant sectional curvature and has arbitrary
dimension. Therefore, they include some well-known spacetimes such as Lorentz-
Minkowski spacetime, de Sitter spacetime and Einstein-de Sitter spacetime.

For a spacelike hypersurface $\psi:M\longrightarrow\overline{M}$ in a GRW
spacetime, we will denote by $\tau:=\pi_{I}\circ\psi$ the restriction of
$\pi_{I}$ along $\psi$. Moreover, the time-orientation of $\overline{M}$
allows to take a unique unitary future-pointing vector field
$N\in\mathfrak{X}^{\bot}(M)$ globally defined on $M$. If we denote by $A$ the
shape operator associated to $N$, the corresponding mean curvature function is
given by $H:=-\frac{1}{n}\mathrm{trace}(A)$.

In these ambient spacetimes, reasoning as in [22, Lemma 3], we can obtain the
following bound for the Ricci curvature of spacelike hypersurfaces.

###### Lemma 9

Let $\psi:M\longrightarrow\overline{M}$ be an $n$-dimensional spacelike
hypersurface in a GRW spacetime whose warping function satisfies $(\log
f)^{\prime\prime}\leq 0$ and whose fiber has non-negative sectional curvature.
If the mean curvature function of $M$ satisfies

$H^{2}\leq\frac{4(n-1)}{n^{2}}\frac{f^{\prime}(\tau)^{2}}{f(\tau)^{2}},$ (10)

then, the Ricci curvature of $M$ is non-negative.

  * Proof.

Given $p\in M$, let us take a local orthonormal frame
$\left\\{E_{1},\ldots,E_{n}\right\\}$ around $p$. From the Gauss equation we
get that the Ricci curvature of $M$, $\mathrm{Ric}$, satisfies

$\displaystyle\mathrm{Ric}(Y,Y)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\overline{g}(\overline{\mathrm{R}}(Y,E_{i})E_{i},Y)+nHg(AY,Y)+g(A^{2}Y,Y)$
(11) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\overline{g}(\overline{\mathrm{R}}(Y,E_{i})E_{i},Y)+\left|AY+\frac{nH}{2}Y\right|^{2}-\frac{n^{2}H^{2}}{4}|Y|^{2},$

for all $Y\in\mathfrak{X}(M)$. Now, from [21, Prop. 7.42], we have

$\displaystyle\sum_{i=1}^{n}\overline{g}(\overline{\mathrm{R}}(Y,E_{i})E_{i},Y)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\overline{g}(\overline{\mathrm{R}}(Y^{F},E_{i}^{F})E_{i}^{F},Y^{F})+(n-1)\frac{f^{\prime}(\tau)^{2}}{f(\tau)^{2}}|Y|^{2}$
(12) $\displaystyle-(n-2)(\log f)^{\prime\prime}(\tau)\,g(Y,\nabla\tau)^{2}$
$\displaystyle-(\log f)^{\prime\prime}(\tau)|\nabla\tau|^{2}|Y|^{2},$

where $Y^{F}$ and $E_{i}^{F}$ denote the projections on the fiber $F$ of $Y$
and $E_{i}$, respectively, and $\nabla\tau$ is the gradient of $\tau$ on $M$.
Therefore, introducing (12) in (11) and taking our assumptions into account we
obtain that the Ricci curvature of $M$ is non-negative. $\square$

###### Remark 10

The assumptions made in this lemma on the GRW spacetime of having a fiber with
non-negative sectional curvature and satisfying $(\log f)^{\prime\prime}\leq
0$ imply that the Ricci tensor of the spacetime satisfies

$\overline{\mathrm{Ric}}(z,z)\geq 0,$

for all lightlike vectors $z\in T\overline{M}$. This is known as the null
convergence condition, which is an algebraic consequence of the Einstein’s
field equation closely related with the weak energy condition [16, Sec. 4.3].

We are now in a position to obtain the next result for spacelike hypersurfaces
in spatially open GRW spacetimes that satisfy the spacelike entropy bound.

###### Theorem 11

Let $\psi:M\longrightarrow\overline{M}$ be an $n(\geq 3)$-dimensional complete
spacelike hypersurface in a GRW spacetime whose warping function satisfies
$(\log f)^{\prime\prime}\leq 0$ and whose fiber has non-negative sectional
curvature. If $\overline{M}$ satisfies the spacelike entropy bound for an
entropy density which is bounded from below by a positive constant and the
mean curvature function of $M$ satisfies (10), then the Brownian motion on $M$
is transient.

  * Proof.

Under our assumptions, we can use Lemma 9 to ensure that the Ricci curvature
of $M$ is non-negative. This and the fact that the spacelike entropy bound
holds for an entropy density $\sigma\geq\sigma_{0}>0$ allow us to apply
Theorem 3 to deduce the transience of the Brownian motion on $M$. $\square$

Note that in Theorem 11 we have omitted the $2$-dimensional case. This is due
to the fact that Lemma 9 enables us to find $3$-dimensional GRW spacetimes
that violate the spacelike entropy bound. Namely, we get

###### Proposition 12

Let $\psi:M\longrightarrow\overline{M}$ be a complete spacelike surface in a
$3$-dimensional GRW spacetime whose warping function satisfies $(\log
f)^{\prime\prime}\leq 0$ and whose fiber has non-negative sectional curvature.
If the mean curvature function of $M$ satisfies

$H^{2}\leq\frac{f^{\prime}(\tau)^{2}}{f(\tau)^{2}},$ (13)

then, the spacelike entropy bound for an entropy density
$\sigma\geq\sigma_{0}>0$ does not hold on $M$.

  * Proof.

From Lemma 9 we obtain that the Gaussian curvature of $M$ is non-negative.
Since $M$ is complete, a classical result by Ahlfors and Blanc-Fiala-Huber for
Riemannian surfaces ensures its parabolicity (see [17]). If the spacelike
entropy bound for an entropy density $\sigma\geq\sigma_{0}>0$ holded on $M$,
Theorem 3 would yield to the non-parabolicity of $M$, reaching a
contradiction. $\square$

###### Remark 13

Notice that inequalities like (10) and (13) that relate the mean curvature
function of a spacelike hypersurface with the Hubble function $f^{\prime}/f$
have been previously used to characterize the spacelike slices in GRW
spacetimes (see [23] and references therein). Therefore, in this article we
find new applications for this type of inequalities.

## Acknowledgements

The author would like to thank the anonymous referees for their deep reading
and valuable suggestions to improve this article. The author is supported by
Spanish MINECO and ERDF project MTM2016-78807-C2-1-P.

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# Dense Lineability and algebrability of $\ell^{\infty}\setminus c_{0}$

Dimitris Papathanasiou  Dimitris Papathanasiou, Euler International
Mathematical Institute, Pesochnaya Naberezhnaya 10,Saint Petersburg, Russia,
197376<EMAIL_ADDRESS>
###### Abstract.

We show that the set $\ell^{\infty}\setminus c_{0}$ is maximal dense-lineable
and densely strongly $\mathfrak{c}$-algebrable answering a question posed by
Nestoridis and complementing a result by García-Pacheco, Martín and Seoane-
Sepúlveda.

###### Key words and phrases:

Lineability, algebrability, sequence spaces

###### 2020 Mathematics Subject Classification:

15A03, 46B25, 46B87

The research is supported by “Native towns”, a social investment program of
PJSC “Gazprom Neft”

##  1\. Introduction

Let $\ell^{\infty}$ be the Banach space of real bounded sequences normed with
the uniform norm, and $c_{0}$ be the subspace of $\ell^{\infty}$ of sequences
converging to zero. Since $c_{0}$ is a closed subspace of $\ell^{\infty}$, the
set $\ell^{\infty}\setminus c_{0}$ is a dense $G_{\delta}$ subset of
$\ell^{\infty}$. Rosenthal [5] showed that $c_{0}$ is quasi-complemented in
$\ell^{\infty}$ from which it immediately follows that the set
$(\ell^{\infty}\setminus c_{0})\cup\\{0\\}$ contains a closed and infinite
dimensional subspace. In the same direction, García-Pacheco, Martín and
Seoane-Sepúlveda [3] showed that if we consider $\ell^{\infty}$ as a Banach
algebra, endowed with the coordinatewise product, then the set
$(\ell^{\infty}\setminus c_{0})\cup\\{0\\}$ even contains a closed infinitely
generated subalgebra.

In the recent paper [4], Nestoridis proved that if $0<p<q<\infty$, the set
$(\ell^{q}\setminus\ell^{p})\cup\\{0\\}$ contains a dense linear subspace and
posed the following question:

######  Question 1.1 (Nestoridis).

Does the set $(\ell^{\infty}\setminus c_{0})\cup\\{0\\}$ contain a dense
linear subspace?

Nestoridis also pointed out that the fact that $\ell^{\infty}$ is non-
separable could add difficulties in trying to answer the above mentioned
question. Indeed, the algebraic dimension of a dense linear subspace of
$\ell^{\infty}$ must be $\mathfrak{c}$ (see Proposition 1.5) which makes it
difficult to control that all its non-zero elements lie outside $c_{0}$. On
the other hand, one could argue that the fact that $\ell^{\infty}$ is non-
separable, in contrast to $c_{0}$ provides enough space in the set
$(\ell^{\infty}\setminus c_{0})\cup\\{0\\}$ to find a dense linear subspace.

We will provide a positive answer to Nestoridis’ question by proving the
following more detailed result. Borrowing the terminology from Aron et al. [1]
we will call a subset $M$ of a topological vector space $X$ maximal dense-
lineable provided $M\cup\\{0\\}$ contains a dense linear subspace of $X$ of
dimension $\dim(X)$.

######  Theorem 1.2.

The set $\ell^{\infty}\setminus c_{0}$ is maximal dense-lineable.

Motivated by the result of García-Pacheco, Martín and Seoane-Sepúlveda, we
also considered the question of whether the set $(\ell^{\infty}\setminus
c_{0})\cup\\{0\\}$ contains a dense subalgebra, where as before, we endow
$\ell^{\infty}$ with the coordinatewise product. Following Bartoszewicz and
Gła̧b [2] we will call a subset $M$ of a Banach algebra $X$ densely strongly
$\mathfrak{c}$-algebrable provided $M\cup\\{0\\}$ contains a dense subalgebra
of $X$ generated by an algebraically independent set of cardinality
$\mathfrak{c}$.

######  Theorem 1.3.

The set $\ell^{\infty}\setminus c_{0}$ is densely strongly
$\mathfrak{c}$-algebrable.

Theorems 1.2 and 1.3 immediately yield the following corollary which fills a
gap in the literature of lineability and algebrability.

######  Corollary 1.4.

If $0<p<\infty$ then the following hold.

  1. (i)

The set $(\ell^{\infty}\setminus\ell^{p})\cup\\{0\\}$ is maximal dense-
lineable.

  2. (ii)

If we endow $\ell^{\infty}$ with the coordinatewise product, the set
$(\ell^{\infty}\setminus\ell^{p})\cup\\{0\\}$ is densely strongly
$\mathfrak{c}$-algebrable.

The idea behind the proofs of Theorems 1.2 and 1.3 is common. For
$A\subset\mathbb{N}$ we denote by $\chi_{A}:\mathbb{N}\rightarrow\mathbb{R}$
the characteristic sequence of $A$. The set
$\\{\chi_{A}:A\subset\mathbb{N\\}}$ generates a dense linear subspace of
$\ell^{\infty}$. We want to conveniently perturb those characteristic
sequences in such a way that the subspace or the subalgebra generated by the
modified sequences intersects $c_{0}$ trivially.

Theorem 1.3 is clearly stronger than Theorem 1.2 however, we provide two
separate proofs for those results since we feel that the proof of Theorem 1.2
is more intuitive than the proof of Theorem 1.3. In this way, one does not
have to go through the more technical proof of Theorem 1.3 if interested in
mere dense-lineability.

We also observe that due to the following simple fact, in our setting the
maximality part of Theorem 1.2 and the cardinality part of Theorem 1.3 follow
for free.

######  Proposition 1.5.

  1. (i)

If $E$ is a dense subspace of $\ell^{\infty}$ then $\dim(E)=\mathfrak{c}$.

  2. (ii)

If $A$ is a dense subalgebra of $\ell^{\infty}$ (endowed with the
coordinatewise product) then each set of generators of $A$ has cardinality
$\mathfrak{c}$.

###### Proof.

To prove (i), we observe that since $\overline{E}=\ell^{\infty}$, $E$ has to
be infinite-dimensional. Now, considering linear combinations with rational
coefficients of a Hamel basis of $E$, and taking into account that any dense
subset of $\ell^{\infty}$ has cardinality $\mathfrak{c}$, it follows that
$\dim(E)=\mathfrak{c}$.

Concerning (ii), let $G$ be any set generating $A$. That means that if
$\tilde{G}$ is the set of all elements of the form

$g_{i_{1}}^{n_{1}}\dots g_{i_{k}}^{n_{k}}$

where $k,n_{1},\dots,n_{k}\in\mathbb{N}$ and $\\{g_{i_{1}},\dots,g_{i_{k}}\\}$
is a finite subset of $G$, then,

$A=\text{span}(\tilde{G}).$

Now, $G$ has to be infinite since otherwise, $\dim(A)\leq\aleph_{0}$ which by
(i), contradicts the fact that $A$ is dense in $\ell^{\infty}$. For the same
reason we get that

$\mathfrak{c}=\operatorname{card}(\tilde{G})\leq\operatorname{card}(G)\aleph_{0}=\operatorname{card}(G)$

from which it follows that $\operatorname{card}(G)=\mathfrak{c}$. ∎

##  2\. Proof of Theorem 1.2

We will need two lemmas, the first is a well known fact that appears in many
places in literature, but we state it and prove it here for the sake of self
containment.

######  Lemma 2.1.

There are $\mathfrak{c}$ infinite subsets of $\mathbb{N}$ pairwise finitely
intersected.

###### Proof.

Let $(q_{n})_{n=1}^{\infty}$ be an enumeration of $\mathbb{Q}\cap[0,1]$. For
each $r\in[0,1]\setminus\mathbb{Q}$ we may find an increasing sequnce of
natural numbers $(n_{k}(r))_{k=1}^{\infty}$ such that $q_{n_{k}(r)}\rightarrow
r$, as $k\rightarrow\infty$. The sets $(n_{k}(r))_{k=1}^{\infty}$, for
$r\in[0,1]\setminus\mathbb{Q}$ have the desired properties. ∎

The next simple fact provides our main tool in perturbing sequences with
finitely many values to sequences that lie outside $c_{0}$ and will also be
used in the proof of Theorem 1.3.

######  Lemma 2.2.

Let $(X,\|\cdot\|)$ be a (real or complex) normed space, $a=(a_{n})$ a
sequence in $X$ with finite range, and $b=(b_{n})$ a sequence somewhere dense
in $X$. Then the sequence $a+b$ is also somewhere dense in $X$.

###### Proof.

Let $\\{c_{1},\dots,c_{k}\\}$, for some $k\in\mathbb{N}$ be the range of $a$
and $\\{N_{i}:i=1,\dots,k\\}$ a partition of $\mathbb{N}$ such that

$a_{n}=c_{i},\quad\text{for}\quad n\in N_{i}.$

Now, since

$\\{b_{n}:n\in\mathbb{N}\\}=\bigcup_{i=1}^{k}\\{b_{n}:n\in N_{i}\\}$

there is $j\in\\{1,\dots,k\\}$ such that

$\\{b_{n}:n\in N_{j}\\}$

is somewhere dense. Since

$\\{a_{n}+b_{n}:n\in N_{j}\\}=c_{j}+\\{b_{n}:n\in N_{j}\\}$

the latter set is a somewhere dense subset in $X$. ∎

We may now proceed with the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

If $A\subset\mathbb{N}$, we want to associate to $\chi_{A}$ a sequence of
suitable elements from $\ell^{\infty}\setminus c_{0}$ converging to
$\chi_{A}$. To this end, we let by Lemma 2.1

$\\{(a_{(A,j)}(k))_{k=1}^{\infty}:(A,j)\in\mathcal{P}(\mathbb{N})\times\mathbb{N}\\}$

be a family of infinite, pairwise finitely intersected subsets of
$\mathbb{N}$, indexed by $\mathcal{P}(\mathbb{N})\times\mathbb{N}$ and
enumerated as increasing sequences. Note that we are using the fact that the
cardinality of the set $\mathcal{P}(\mathbb{N})\times\mathbb{N}$ is
$\mathfrak{c}$. Let also $x=(x_{n})_{n=1}^{\infty}\in\ell^{\infty}$ be a
sequence which is somewhere dense in $\mathbb{R}$ (an enumeration of the
rationals of $[0,1]$ would do the job). For $A\subset\mathbb{N}$ and
$j\in\mathbb{N}$, we define the following elements of $\ell^{\infty}$,

$f_{(A,j)}=\chi_{A}+\frac{1}{j}\sum_{k=1}^{\infty}x_{k}\chi_{\\{a_{(A,j)}(k)\\}}.$

Clearly,

$f_{(A,j)}\rightarrow\chi_{A}\quad\text{as}\quad j\rightarrow\infty$

hence, the set

$Y=\text{span}\\{f_{(A,j)}:A\subset\mathbb{N},j\in\mathbb{N}\\}$

is dense in $\ell^{\infty}$. We want to show that

$Y\cap c_{0}=\\{0\\}.$

For $p\geq 1$, let $c_{1},\dots,c_{p}\in\mathbb{R}$ be such that $c_{i}\neq
0$, for some $i\in\\{1,\dots,p\\}$ and $(A_{1},j_{1}),\dots,(A_{p},j_{p})$ be
pairwise distinct elements of $\mathcal{P}(\mathbb{N})\times\mathbb{N}$. We
consider $k_{0}\in\mathbb{N}$ satisfying that

$\\{a_{(A_{i},j_{i})}(k):k\geq k_{0}\\}\cap\\{a_{(A_{l},j_{l})}(k):k\geq
k_{0}\\}=\emptyset$

for all $l\neq i$, with $l\in\\{1,\dots p\\}$. For $k\geq k_{0}$, we then have
that

$\displaystyle
c_{1}f_{(A_{1},j_{1})}(a_{(A_{i},j_{i})}(k))+\dots+c_{p}f_{(A_{p},j_{p})}(a_{(A_{i},j_{i})}(k))=$
$\displaystyle
c_{1}\chi_{A_{1}}(a_{(A_{i},j_{i})}(k))+\dots+c_{p}\chi_{A_{p}}(a_{(A_{i},j_{i})}(k))+\frac{1}{j_{i}}c_{i}x_{k}.$

This, together with Lemma 2.2 give that the sequence

$(c_{1}f_{(A_{1},j_{1})}(a_{(A_{i},j_{i})}(k))+\dots+c_{p}f_{(A_{p},j_{p})}(a_{(A_{i},j_{i})}(k)))_{k=1}^{\infty}$

is somewhere dense in $\mathbb{R}$. In particular,

$c_{1}f_{(A_{1},j_{1})}+\dots+c_{p}f_{(A_{p},j_{p})}\notin c_{0}$

which shows that $\ell^{\infty}\setminus c_{0}$ contains a dense linear
subspace. An application of Proposition 1.5 (i) concludes the proof. ∎

##  3\. Proof of Theorem 1.3

The way we perturbed the characteristic sequences in the proof of Theorem 1.2
does not work for proving Theorem 1.3. The problem is that the product of two
sequences supported on finitely intersected subsets of $\mathbb{N}$ will be a
finitely supported sequence thus it will belong to $c_{0}$. Obviously, we
cannot hope on separating totally an uncountable family of subsets of
$\mathbb{N}$. Thus, we establish the following result that allows us to
overcome this obstacle. We thank the anonymous referee for indicating to us
this short, topological argument.

######  Lemma 3.1.

There is a family of cardinality $\mathfrak{c}$

$(f_{\gamma})_{\gamma<\mathfrak{c}}\subset[0,1]^{\mathbb{N}}$

such that for each finite subset
$\\{\gamma_{1},\dots,\gamma_{k}\\}\subset\mathfrak{c}$, where
$k\in\mathbb{N}$, the set

$\\{(f_{\gamma_{1}}(n),\dots,f_{\gamma_{k}}(n)):n\in\mathbb{N}\\}$

is dense in $[0,1]^{k}$.

###### Proof.

The proof is an immediate consequence of the fact that the Tychonoff cube
$[0,1]^{\mathfrak{c}}$ endowed with the product topology, is separable (see
for instance [6, Theorem 16.4 (c)]). Indeed, let

$\\{x_{n}:n\in\mathbb{N}\\}$

be a dense subset of $[0,1]^{\mathfrak{c}}$. For each $\gamma<\mathfrak{c}$,
we define the element $f_{\gamma}\in[0,1]^{\mathbb{N}}$ by

$f_{\gamma}(n)=x_{n}(\gamma),\quad n\in\mathbb{N}.$

The fact that the family $(f_{\gamma})_{\gamma<\mathfrak{c}}$ satisfies the
claim follows by the definition of the product topology. ∎

###### Proof of Theorem 1.3.

By Lemma 3.1 and the fact that the cardinality of
$\mathcal{P}(\mathbb{N})\times\mathbb{N}$ is $\mathfrak{c}$, for
$A\subset\mathbb{N}$ and $j\in\mathbb{N}$, we get sequences

$f_{(A,j)}\in\ell^{\infty}$

with $\|f_{(A,j)}\|\leq\frac{1}{j}$ and satisfying that if

$\\{(A_{1},j_{1}),\dots,(A_{k},j_{k})\\}$

with $k\in\mathbb{N}$, is a finite subset of
$\mathcal{P}(\mathbb{N})\times\mathbb{N}$, then

$\\{(f_{(A_{1},j_{1})}(n),\dots,f_{(A_{k},j_{k})}(n)):n\in\mathbb{N}\\}$

is somewhere dense in $\mathbb{R}^{k}$. Now, let us set

$h_{(A,j)}=\chi_{A}+f_{(A,j)},\quad(A,j)\in\mathcal{P}(\mathbb{N})\times\mathbb{N}.$

Clearly,

$h_{(A,j)}\rightarrow\chi_{A}\quad\text{as}\quad j\rightarrow\infty$

hence, the set

$\\{h_{(A,j)}:A\subset\mathbb{N},j\in\mathbb{N}\\}$

spans a dense subspace of $\ell^{\infty}$. We will show that if $B$ is the
algebra generated by $\\{h_{(A,j)}:A\subset\mathbb{N},j\in\mathbb{N}\\}$, then

$B\cap c_{0}=\\{0\\}.$

Let for $k\in\mathbb{N}$, $P\in\mathbb{R}[x_{1},\dots x_{k}]$ be a non-zero
polynomial vanishing at the origin. The zero set of $P$ is nowhere dense in
$\mathbb{R}^{k}$. If $(A_{1},j_{1}),\dots,(A_{k},j_{k})$ are distinct elements
of $\mathcal{P}(\mathbb{N})\times\mathbb{N}$, since the range of the sequence

$(\chi_{A_{1}}(n),\dots,\chi_{A_{k}}(n))_{n=1}^{\infty}$

is finite, Lemma 2.2 ensures that

$\\{(h_{(A_{1},j_{1})}(n),\dots,h_{(A_{k},j_{k})}(n)):n\in\mathbb{N}\\}$

is somewhere dense in $\mathbb{R}^{k}$. Hence, there is $\delta>0$ and a
subsequence $(n_{l})$ of natural numbers, such that

$|P(h_{(A_{1},j_{1})}(n_{l}),\dots,h_{(A_{k},j_{k})}(n_{l}))|\geq\delta.$

In particular, since the underlying product is the coordinatewise, we get that

$|P(h_{(A_{1},j_{1})},\dots,h_{(A_{k},j_{k})})(n_{l})|=|P(h_{(A_{1},j_{1})}(n_{l}),\dots,h_{(A_{k},j_{k})}(n_{l}))|\geq\delta$

which yields that

$P(h_{(A_{1},j_{1})},\dots,h_{(A_{k},j_{k})})\notin c_{0}$

hence, $B\cap c_{0}=\\{0\\}$ and the family
$\\{h_{(A,j)}:A\subset\mathbb{N},j\in\mathbb{N}\\}$ is algebraically
independent. Now, either by noticing that

$\operatorname{card}\\{h_{(A,j)}:A\subset\mathbb{N},j\in\mathbb{N}\\}=\mathfrak{c}$

or by an immediate application of Proposition 1.5 (ii) the proof is complete.
∎

######  Remark 3.2.

The proofs of Theorem 1.2 and Theorem 1.3 can be easily modified to provide
the same results in the complex case.

######  Remark 3.3.

The proofs of Theorem 1.2 and Theorem 1.3 actually show that the set

$\ell^{\infty}\setminus c$

is maximal dense-lineable and densely strongly $\mathfrak{c}$-algebrable,
where $c$ is the space of real convergent sequences. For the analogue of
Theorem 1.3 one would need to use the fact that the image under any non-
constant polynomial of several variables of an open set has non-empty
interior.

Acknowledgement. The author would like to thank the anonymous referee for
his/her insightful comments which substantially improved the paper.

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