text:
\begin{document}

\title{The Steinhaus property and Haar-null sets}
\author{Pandelis Dodos}
\address{Texas A$\&$M University, Department of Mathematics, 3368 TAMU, College Station,
TX 77843-3368.}
\email{pdodos@math.ntua.gr}

\footnotetext[1]{2000 \textit{Mathematics Subject Classification}: 54H11, 28C10.}
\footnotetext[2]{\textit{Key words}: Haar-null sets, Polish groups,
Steinhaus Theorem, Borel measures.}

\maketitle

\begin{abstract}
It is shown that if $G$ is an uncountable Polish group and $A\subseteq G$ is a universally
measurable set such that $A^{-1}A$ is meager, then the set $T_l(A)=\{\mu\in P(G): \mu(gA)=0
\text{ for all } g\in G\}$ is co-meager. In particular, if $A$ is analytic and not left
Haar-null, then $1\in\inte(A^{-1}AA^{-1}A)$.
\end{abstract}

\section{Introduction}

The purpose of this paper is to show that there exists a satisfactory extension of the classical
Steinhaus Theorem for an arbitrary Polish group. In order to get the extension one needs, first,
to isolate the appropriate $\sigma$-ideal on which the result will be applied. For the class of
abelian Polish groups this is the $\sigma$-ideal of Haar-null sets, defined by J. P. R.
Christensen \cite{Chr}. However, in non-abelian (and non-locally-compact) Polish groups this
$\sigma$-ideal is no longer well-behaved. Actually, by the results of S. Solecki in \cite{S2},
the Steinhaus property of Haar-null sets fails in ``most" non-abelian Polish groups. Notice also
that the conclusion of the Steinhaus Theorem is rather strong. If $A\subseteq\mathbb{R}$ is
of positive Lebesgue measure, then $A-A$ contains a neighborhood of $0$. If we relax the
conclusion to $A-A$ is not meager\footnote[3]{We recall that a subset $A$ of a topological
space $X$ is said to be \textit{meager} (or \textit{of first category}) if $A$ is covered
by a countable union of closed nowhere dense sets. The complement of a meager set is usually
referred as \textit{co-meager}.}, then this is valid in every abelian Polish group.

To state our result we need some definitions. Let $G$ be a Polish group and $A\subseteq G$ be
a universally measurable set. The set $A$ is said \textit{Haar-null} if there exists $\mu\in
P(G)$ (i.e. $\mu$ is a Borel probability measure on $G$) such that $\mu(g_1Ag_2)=0$ for all
$g_1, g_2\in G$. It is said to be \textit{left Haar-null} if there exists $\mu\in P(G)$ such
that $\mu(gA)=0$ for all $g\in G$. By the results in \cite{ST} and \cite{S2}, the notions of
Haar-null and left Haar-null set are distinct (however, they obviously agree on abelian groups).
We let
\[ T(A)=\{ \mu\in P(G): \mu(g_1Ag_2)=0 \text{ for all } g_1,g_2\in G\} \]
and
\[ T_l(A)=\{ \mu\in P(G): \mu(gA)=0 \text{ for all } g\in G\}. \]
It is easy to see that if $A$ is analytic\footnote[4]{We recall that a subset $A$ of a Polish
space $X$ is said to be \textit{analytic} if there exists a continuous map $f:\nn^\nn\to X$ with
$f(\nn^\nn)=A$. It is a classical result that every Borel subset of a Polish space is analytic.
It is also well-known that an analytic set which is not meager is actually co-meager in a
non-empty open set.}, then both $T(A)$ and $T_l(A)$ are faces (i.e. extreme convex subsets) of
$P(G)$ with the Baire property. It follows by \cite[Theorem 4]{D2} that the sets $T(A)$ and
$T_l(A)$ are either meager, or co-meager. A set $A$ is said to be \textit{generically Haar-null}
if $T(A)$ is co-meager. Respectively, the set $A$ is said to be \textit{generically left
Haar-null} if $T_l(A)$ is co-meager.

For every Polish group $G$ the class of generically left Haar-null subsets of $G$ forms a
$\sigma$-ideal. Notice that if $A$ is not generically left Haar-null, then $A$ should not be
considered as a small set (it is null only for a relatively small set of measures). This is
indeed true as the following theorem demonstrates.
\medskip

\noindent \textbf{Theorem A.} \textit{Let $G$ be an uncountable Polish group and $A$ be a
universally measurable subset of $G$. Assume that $A^{-1}A$ is meager. Then $T_l(A)$ is
co-meager. }

\textit{Thus, if $A$ is analytic and not generically left Haar-null (in particular, not left Haar-null), then $A^{-1}A$ is non-meager.}
\medskip

The locally-compact abelian case of Theorem A can be also derived by the results of M. Laczkovich
in \cite{La}, who proved that if $A$ is not covered by an $F_\sigma$ Haar-measure zero set, then
$A^{-1}A$ is co-meager in a neighborhood of the identity. To see that this implies Theorem A,
one invokes \cite[Proposition 5]{D1} which states that if $G$ is locally-compact and
$A\subseteq G$ is covered by an $F_\sigma$ Haar-null set, then $T_l(A)$ is co-meager. Both M.
Laczkovich's result as well as the result of J. P. R. Christensen \cite{Chr} that Haar-null sets
satisfy the Steinhaus property in abelian Polish groups, are heavily depended on the classical
Steinhaus Theorem. The proof of Theorem A follows quite different arguments. It is based on the
fact that if $\hhh$ is a dense $G_\delta$ and hereditary subset of $K(G)$, then this is witnessed
in the probabilities of $G$.

\subsection{Preliminaries} Our general notation and terminology follows \cite{Kechris}. By
$\nn=\{0,1,2,...\}$ we denote the natural numbers. For any Polish space $X$ by $K(X)$ we denote
the hyperspace of all compact subsets of $X$ with the Vietoris topology and by $P(X)$ the space
of all Borel probability measures on $X$ with the weak* topology. Both are Polish (see
\cite{Kechris}). If $d$ is a compatible complete metric of $X$, then by $d_H$ we denote the
Hausdorff metric on $K(X)$ associated to $d$, defined by
\[ d_H(K,C)=\inf\{ \ee>0: K\subseteq C_\ee \text{ and } C\subseteq K_\ee\} \]
where $A_\ee=\{x\in X:d(x,A)\leq\ee\}$ for every $A\subseteq X$. All balls in $K(X)$ are taken
with respect to $d_H$ and are denoted by $B_H$. In $P(X)$ we consider the so called L\'{e}vy
metric $\rho$, defined by
\begin{eqnarray*}
\rho(\mu,\nu)=
\inf\big\{ \ee>0 & : & \mu(A)\leq \nu(A_\ee)+\ee \text{ and } \nu(A)\leq\mu(A_\ee) +\ee \\
& & \text{for every compact (or Borel) subset } A \text{ of } X \big\}
\end{eqnarray*}
(see \cite{BL} for more details). All balls in $P(X)$ are taken with respect to $\rho$ and are
denoted by $B_P$. If $G$ is a Polish group and $\mu, \nu\in P(G)$, then by $\mu*\nu$ we denote
their convolution, defined by
\[ \mu*\nu(A)=\int_G \mu(Ax^{-1})d\nu(x). \]
A subset $\hhh$ of $K(X)$ is said to be \textit{hereditary} if for every $K\in\hhh$ and every
$C\in K(X)$ with $C\subseteq K$ we have that $C\in\hhh$. All the other pieces of notation we use
are standard.

\section{Hereditary, dense $G_\delta$ sets and measures}

Throughout this section $X$ will be a Polish space and $\hhh$ a hereditary, dense $G_\delta$
subset of $K(X)$. By $d$ we denote a compatible complete metric of $X$.
\begin{lem}
\label{l1} Let $X$ and $\hhh$ as above. Then there exists a sequence $(\uuu_n)$ of open, dense
and hereditary subsets of $K(X)$ such that $\hhh=\bigcap_n \uuu_n$.
\end{lem}
\begin{proof} Write $\hhh=\bigcap_n \vvv_n$ where each $\vvv_n$ is open and dense but not
necessarily hereditary. Fix $n$ and define
\[ \ccc_n=\{ K\in K(X): \exists C\subseteq K \text{ compact with } C\notin \vvv_n\}. \]
It is easy to check that $\ccc_n$ is closed and $\ccc_n \cap \hhh=\varnothing$. So if we set
$\uuu_n=K(X)\setminus \ccc_n$ we see that the sequence $(\uuu_n)$ has all the desired properties.
\end{proof}
In the sequel we will say that the sequence $(\uuu_n)$ obtained by Lemma \ref{l1}, is the
\textit{normal form} of $\hhh$. We need the following lemmas.
\begin{lem}
\label{l2} Let $\uuu\subseteq K(X)$ be open, dense and hereditary. Let also $x_0,...,x_n$ be
distinct points in $X$ and $r_1>0$. Then there exist $y_0,...,y_n$ distinct points in $X$ such
that $d(x_i,y_i)<r_1$ for all $i\in\{0,...,n\}$ and moreover $\{y_0,...,y_n\}\in \uuu$.
\end{lem}
\begin{proof}
We may assume that $B(x_i,r_1)\cap B(x_j,r_1)=\varnothing$ for all $i,j\in \{0,...,n\}$ with
$i\neq j$. Let
\[ \vvv=\big\{ K: K\subseteq \bigcup_{i=0}^n B(x_i,r_1)\text{ and }
K\cap B(x_i,r_1) \neq\varnothing \ \forall i=0,...,n \big\}.\]
Then $\vvv$ is open. As $\uuu$ is open and dense, there exists $K\in\vvv\cap\uuu$.
For every $i\in\{0,...,n\}$ we select $y_i\in K\cap B(x_i,r_1)$. As $\uuu$ is hereditary,
we see that $\{y_0,...,y_n\}\in\uuu$. Clearly $y_0,...,y_n$ are as desired.
\end{proof}
\begin{lem}
\label{l3} Let $\uuu\subseteq K(X)$ be open, dense and hereditary. Let also $\ee>0$. Then the set
\[ G_{\uuu,\ee}=\{\mu\in P(X): \exists K\in\uuu \text{ with } \mu(K)\geq 1-\ee \}\]
is co-meager in $P(X)$.
\end{lem}
\begin{proof} Fix $\uuu$ and $\ee>0$ as above. We will show that for every $V\subseteq P(X)$ open
there exists $W\subseteq V$ open such that $W\subseteq G_{\uuu,\ee}$. This will finish the proof
(actually it implies that $G_{\uuu,\ee}$ contains a dense open set). So let $V\subseteq P(X)$
open. As finitely supported measures are dense in $P(X)$, we may select
$\nu=\sum_{i=0}^n a_i \delta_{x_i}$ and $r>0$ such that
\begin{enumerate}
\item[(1)] $a_i>0$ for all $i\in\{0,...,n\}$ and $\sum_{i=0}^n a_i=1$,
\item[(2)] $B_P(\nu,r)\subseteq V$.
\end{enumerate}
By Lemma \ref{l2}, there exist $y_0,...,y_n$ distinct points in $X$ with $\{y_0,...,y_n\}\in \uuu$
and such that $d(x_i,y_i)<\frac{r}{2}$ for all $i\in\{0,...,n\}$. We set
$\mu=\sum_{i=0}^n a_i \delta_{y_i}$. Then it is easy to see that
\begin{enumerate}
\item[(3)] $\rho(\mu,\nu)\leq \frac{r}{2}$.
\end{enumerate}
Let $F=\{y_0,...,y_n\}$. As $\uuu$ is open and $F\in\uuu$ there exists $\theta>0$ such that
\begin{enumerate}
\item[(4)] $\theta<\min\{\frac{\ee}{3},\frac{r}{3}\}$ and
\item[(5)] $B_H(F,2\theta)\subseteq \uuu$.
\end{enumerate}
Then $W=B_P(\mu,\theta)$ is as desired. Indeed, by (2), (3) and (4) it is clear that $W$ is a
subset of $V$. We only need to check that $W$ is a subset of $G_{\uuu,\ee}$. Let $\lambda\in W$
arbitrary. Then $\rho(\lambda,\mu)<\theta$ and so
\[ 1=\mu(F)\leq \lambda(F_{\theta}) +\theta \]
which gives that $\lambda(F_\theta)\geq 1-\frac{\ee}{3}$ by the choice of $\theta$. By the inner
regularity of $\lambda$, there exists $C\subseteq F_\theta$ compact such that
$\lambda(C)\geq 1-\ee$. We set $K=C\cup F$. Then $d_H(K,F)\leq\theta$ and so, by (5), $K\in \uuu$.
Moreover, $\lambda(K)\geq \lambda(C)\geq 1-\ee$. This implies that $\lambda\in G_{\uuu,\ee}$ and
the proof is completed.
\end{proof}
Our goal in this section is to prove the following.
\begin{prop}
\label{p1} Let $\hhh$ be a hereditary, dense $G_\delta$ subset of $K(X)$. Then the set
\[ G_\hhh=\{ \mu\in P(X): \forall \ee>0 \ \exists K\in \hhh \text{ with } \mu(K)\geq 1-\ee \} \]
is co-meager in $P(X)$.
\end{prop}
\begin{proof}
Let $(\uuu_n)$ be the normal form of $\hhh$. For every $n,m\in \nn$ let
\[ G_{n,m}=\big\{ \mu\in P(X): \exists K\in \uuu_n \text{ with } \mu(K)\geq 1-\frac{1}{m+1}\big\}.\]
By Lemma \ref{l3}, we have that $G_{n,m}$ is co-meager. Hence, so is $\bigcap_{n,m} G_{n,m}$.
We claim that $G_\hhh=\bigcap_{n,m} G_{n,m}$. This will finish the proof. It is clear that
$G_\hhh\subseteq\bigcap_{n,m} G_{n,m}$. Conversely, fix $\mu\in \bigcap_{n,m} G_{n,m}$ and let
$\ee>0$ arbitrary. Pick a sequence $(\ee_n)$ of positive reals such that
\[ \sum_{n\in\nn} \ee_n<\frac{\ee}{2}. \]
Pick also a sequence $(m_n)$ of natural numbers with $\frac{1}{m_n+1}\leq\ee_n$ for every
$n\in\nn$. As
\[ \mu\in \bigcap_{n,m} G_{n,m} \subseteq \bigcap_n G_{n,{m_n}}\]
we may select a sequence $(K_n)$ in $K(X)$ such that
\begin{enumerate}
\item[(1)] $K_n\in \uuu_n$ and
\item[(2)] $\mu(K_n)\geq 1-\frac{1}{m_n+1}\geq 1-\ee_n$.
\end{enumerate}
For every $n\in\nn$ we let $F_n= \bigcap_{i=0}^n K_i$ and we set $F=\bigcap_n K_n$.
Then $F_n\downarrow F$. Notice that $F\in \uuu_n$ as $F\subseteq F_n\subseteq K_n\in\uuu_n$
and $\uuu_n$ is hereditary. Hence $F\in \bigcap_n \uuu_n=\hhh$. Moreover, by (2) above, we have
\[ \mu(F_n)=\mu(K_0\cap ... \cap K_n)\geq 1- \sum_{k=0}^n \ee_k. \]
As $F_n\downarrow F$ we get that
\[ \mu(F)=\lim_{n\in\nn} \mu(F_n)\geq 1-\sum_{n\in\nn} \ee_n \geq 1-\ee. \]
This shows that $\mu\in G_\hhh$, as desired.
\end{proof}

\section{Left Haar-null sets in Polish groups}

Our aim is to give the proof of Theorem A stated in the introduction.
\begin{proof}[Proof of Theorem A]
Let $G$ be an uncountable Polish group and $A$ be a universally measurable subset of $G$ such that
$A^{-1}A$ is meager. We select a sequence $(C_n)$ of closed, nowhere dense subsets of $G$ with the
following properties.
\begin{enumerate}
\item[(i)] $1\notin C_n$ for all $n\in\nn$.
\item[(ii)] $A^{-1}A\setminus\{1\} \subseteq \bigcup_n C_n$.
\end{enumerate}
For every $n\in\nn$ let
\[ \uuu_n=\{ K\in K(G): K^{-1}K \cap C_n=\varnothing \}. \]
Clearly every $\uuu_n$ is hereditary. Moreover, as the function $f:K(G)\to K(G)$
defined by $f(K)=K^{-1}K$ is continuous, we see that every $\uuu_n$ is open.
\begin{claim}
\label{cl1} For every $n\in\nn$ the set $\uuu_n$ is dense in $K(G)$.
\end{claim}
\begin{proof}[Proof of Claim \ref{cl1}]
As finite sets are dense in $K(G)$, it is enough to show that for every finite subset
$\{x_0,...,x_l\}$ of $G$ and every $r>0$ there exist $y_0,...,y_l$ distinct points in $G$ with
\[\big\{ y_i^{-1}y_j: i,j\in\{0,...,l\} \text{ with } i\neq j\big\} \cap C_n=\varnothing\]
and such that $d(x_i,y_i)\leq r$ for all $i\in\{0,...,l\}$ (here $d$ is simply a compatible
complete metric of $G$). The points $y_0,...,y_l$ will be chosen by recursion. We set $y_0=x_0$.
Assume that $y_0,...,y_k$ have been chosen for some $k<l$ so as $\big\{ y_i^{-1}y_j:
i,j\in\{0,...,k\} \text{ with } i\neq j\big\} \cap C_n=\varnothing$. For every $g\in G$ the
functions $x\mapsto gx^{-1}$ and $x\mapsto gx$ are homeomorphisms. It follows that the set
$F_k=\bigcup_{i=0}^k (y_iC_n^{-1}\cup y_iC_n)$ is a closed set with empty interior. Hence there
exists $y_{k+1}\in B(x_{k+1},r)$ such that $y_{k+1}\notin F_k\cup\{ y_0,...,y_k\}$. This implies
that for every $i\in\{0,...,k\}$ we have $y_{k+1}^{-1}y_i\notin C_n$ and $y_i^{-1}y_{k+1}\notin
C_n$. This completes the recursive selection and the proof of the claim is completed.
\end{proof}
It follows by the above claim that the set $\hhh=\bigcap_n \uuu_n$ is a hereditary, dense
$G_\delta$ subset of $K(G)$ and that $(\uuu_n)$ is a normal form of $\hhh$. Notice that if
$K\in \hhh$, then $K^{-1}K \cap A^{-1}A =\{1\}$. By Proposition \ref{p1}, we have that the set
\[ B_1=\{ \mu\in P(G): \forall \ee>0 \ \exists K\in\hhh \text{ with } \mu(K)\geq 1-\ee \} \]
is co-meager. Our assumption that $G$ is uncountable implies that the Polish group $G$ viewed
as a topological space is perfect. Hence, the set of all non-atomic Borel probability measures
on $G$ is co-meager in $P(G)$ (see \cite{Kn}, or \cite{PRV}). It follows that the set
\[ B_2 =\{ \mu\in P(G): \mu \text{ is non-atomic and } \mu\in B_1\}\]
is co-meager in $P(G)$. We will show that $B_2\subseteq T_l(A)$. This will finish the proof.
We need the following fact (its easy proof is left to the reader).
\begin{fact}
Let $\mu\in P(G)$. Then $\mu\in T_l(A)$ if and only if for every $\nu\in P(G)$ we have
$\nu*\mu(A)=0$.
\end{fact}
Fix $\mu\in B_2$. By the above fact, in order to verify that $\mu\in T_l(A)$ we have to show that
$\nu*\mu(A)=0$ for every $\nu\in P(G)$. So, let $\nu\in P(G)$ arbitrary. Let also $\ee>0$
arbitrary. As $\mu\in B_2\subseteq B_1$, there exists $K\in\hhh$ with $\mu(K)\geq 1-\ee$. Then
\begin{eqnarray*}
\nu*\mu(A) & = & \int_G \nu(Ay^{-1}) d\mu(y) \leq \int_K \nu(Ay^{-1})d\mu(y) + \mu(G\setminus K)\\
& \leq & \int_K \nu(Ay^{-1})d\mu(y) + \ee.
\end{eqnarray*}
We set $I=\{ y\in K: \nu(Ay^{-1})>0\}$.
\begin{claim}
\label{cl2} The set $I$ is countable.
\end{claim}
\begin{proof}[Proof of Claim \ref{cl2}]
Notice that if $y, z\in I$ with $y\neq z$, then $Ay^{-1}\cap Az^{-1}=\varnothing$. For if not,
we would have that $1\neq y^{-1}z \in K^{-1}K\cap A^{-1}A$, which contradicts the fact that
$K\in \hhh$. It follows that the family $\{ Ay^{-1}: y\in I\}$ is a family of pairwise disjoint
sets of positive $\nu$-measure. Hence $I$ is countable, as claimed.
\end{proof}
The measure $\mu$ is non-atomic as $\mu\in B_2$. Hence, by Claim \ref{cl2}, we see that
$\mu(I)=0$. It follows that
\[ \int_K \nu(Ay^{-1}) d\mu(y)= \int_I \nu(Ay^{-1}) d\mu(y)\leq\mu(I)=0\]
and so $\nu*\mu(A)\leq\ee$. Since $\ee$ was arbitrary, this implies that $\nu*\mu(A)=0$.
The proof of Theorem A is completed.
\end{proof}
Combining Theorem A with Pettis' Theorem (see \cite[Theorem 9.9]{Kechris}) we get the following
corollary.
\begin{cor}
\label{c1} Let $G$ be an uncountable Polish group and $A$ an analytic subset of $G$.
If $A$ is not generically left Haar-null (in particular, if $A$ is not left Haar-null),
then $1\in\inte(A^{-1}AA^{-1}A)$.
\end{cor}
Clearly Theorem A implies that in non-locally-compact groups, compact sets are generically left
Haar-null. Another application of this form concerns the size of analytic subgroups of Polish
groups. Specifically we have the following corollary which may be considered as the
non-locally-compact analogue of M. Laczkovich's Theorem \cite{La}.
\begin{cor}
\label{c2} Let $G$ be an uncountable Polish group and $H$ be an analytic subgroup of $G$ with
empty interior. Then $H$ is generically left Haar-null.
\end{cor}
\noindent \textbf{What about Haar-null sets?} We would like to remark on the possibility of
extending Theorem A to Haar-null sets instead of merely left Haar-null. As it has been shown
by S. Solecki in \cite{S2}, the Steinhaus property of the $\sigma$-ideal of Haar-null sets fails
in a large number of Polish groups (in a sense, it fails for most non-abelian Polish groups).
Precisely, by \cite[Theorem 6.1]{S2}, if $(H_n)$ is a sequence of countable groups such that
infinitely many of them are not FC (see \cite{S2} for the definition of FC groups), then one can
find a closed set $A\subseteq \prod_n H_n$ which is not Haar-null and $A^{-1}A$ is meager. So,
there is no analogue of Theorem A for Haar-null sets in arbitrary Polish groups. Yet there is one
if we further assume that the group $G$ satisfies the following non-singularity condition.
\begin{enumerate}
\item[(C)] For every analytic and meager subset $A$ of $G$, the conjugate saturation
$[A]=\{x: \exists g\in G \ \exists a\in A \text{ with } x=gag^{-1} \}$ of $A$ is meager.
\end{enumerate}
Clearly every abelian Polish group satisfies (C). Moreover we have the following.
\begin{prop}
\label{p2} Let $G_1$ and $G_2$ be Polish groups. If both $G_1$ and $G_2$ satisfy (C), then so
does $G_1\times G_2$.
\end{prop}
\begin{proof}
Let $A\subseteq G_1\times G_2$ be analytic and meager. By the Kuratowski-Ulam theorem (see
\cite[Theorem 8.41]{Kechris}), we have
\[ \forall^* x\in G_1 \text{ the section } A_x=\{y\in G_2: (x,y)\in A\} \text{ of } A
\text{ is meager.} \]
As $G_2$ satisfies (C), by another application of the Kuratowski-Ulam theorem we get that
\[ A_1=\{ (x,z): \exists g_2, y\in G_2 \text{ with } (x,y)\in A \text{ and } y=g_2zg_2^{-1} \}\]
is analytic and meager. With the same reasoning we see that the set
\[ A_2=\{ (w,z): \exists g_1, x\in G_1 \text{ with } (x,z)\in A_1 \text{ and } x=g_1wg_1^{-1} \}\]
is analytic and meager too. Noticing that $A_2=[A]$, the result follows.
\end{proof}
For groups that satisfy (C) we have the following strengthening of Theorem A.
\begin{prop}
\label{p3} Let $G$ be an uncountable Polish group that satisfies (C). If $A$ is an analytic
subset of $G$ such that $A^{-1}A$ is meager, then $T(A)$ is co-meager.
\end{prop}
\begin{proof}
The proof is similar to the proof of Theorem A, and so, we shall only indicate the necessary
changes. Let $A\subseteq G$ be analytic such that $A^{-1}A$ is meager. Notice that $A^{-1}A$
is analytic. The group $G$ satisfies (C). It follows that the set $[A^{-1}A]$ is meager too.
Arguing as in the proof of Theorem A this implies that there exists a co-meager set $B_2$ of
non-atomic Borel probability measures on $G$ such that for every $\mu\in B_2$ and every $\ee>0$
there exists $K\subseteq G$ compact with $\mu(K)\geq 1-\ee$ and $K^{-1}K\cap [A^{-1}A]=\{1\}$.
We claim that $B_2\subseteq T(A)$. To this end, it is enough to show that for every $\mu\in B_2$,
every $\nu\in P(G)$ and every $x\in G$ we have $\nu*\mu(Ax)=0$. Let $\ee>0$ arbitrary and pick
$K\subseteq G$ compact as described above. Then
\[ \nu*\mu(Ax)\leq \int_K \nu(Axy^{-1}) d\mu(y) + \ee. \]
We set $I=\{y\in K: \nu(Axy^{-1})>0\}$. Observe that if $y, z\in I$ with $y\neq z$,
then $(Axy^{-1})\cap (Axz^{-1})=\varnothing$ (for if not, we would have that
$1\neq y^{-1}z\in K^{-1}K \cap [A^{-1}A]$). By the countable chain condition of $\nu$, we get
that $I$ is countable and the result follows.
\end{proof}
\begin{rem}
The $\sigma$-ideal of generically left Haar-null sets is a quite satisfactory $\sigma$-ideal of
measure-theoretic small sets in arbitrary Polish groups. Beside Theorem A, this is also supported
by the results in \cite{D1} asserting that every analytic and generically left Haar-null subset
$A$ of $G$ can be covered by a \textit{Borel} set $B$ with the same property. The fact that this
ideal is well-behaved is also reflected in the complexity of the collection of all closed
generically left Haar-null sets (in the Effros-Borel structure). It is much better than the
one of closed Haar-null sets, at least in abelian Polish groups. Specifically, it follows by the
results of S. Solecki in \cite{S1}, that in non-locally-compact abelian Polish groups the
$\sigma$-ideal of closed generically Haar-null sets is $\PB^1_1$-complete. The corresponding
collection of closed Haar-null sets is much more complicated (it is both $\SB^1_1$ and
$\PB^1_1$-hard).
\end{rem}


text:
\section{Protocol}
\label{sec:protocol}

\subsection{Rateless Coding}

Puncturing and shortening are two suitable strategies able to adapt the rate of a channel code as we have already shown in a previous work~\cite{Elkouss10}. When $p$ punctured symbols of a codeword are removed, a $[n, k]$ code is converted into a $[n-p, k]$ code (see Fig.~\ref{fig:puncturing}). Whereas, when shortening, $s$ symbols are removed during the encoding process, and a $[n, k]$ code is converted into a $[n-s, k-s]$ code (see Fig.~\ref{fig:shortening}).

Supposing that $R_0$ is the original coding rate of a family of LDPC codes defined by:

\begin{equation}
R_0 = 1 - \frac{\displaystyle \sum_i \lambda_i / i}{\displaystyle \sum_j \rho_j / j}
\end{equation}

\noindent where $\lambda_i$ and $\rho_i$ are the coefficients of their generating polynomials. This rate can be modulated applying puncturing and shortening procedures as defined below. The modulated rate is then calculated as:

\begin{equation}
R = \frac{k - s}{n - p - s} = \frac{R_0 - \sigma}{1 - \pi - \sigma}
\end{equation}

\noindent where $\pi = p/n$ and $\sigma = s/n$ are the ratios of punctured symbols and shortened symbols respectively.

Both strategies, puncturing and shortening, may be applied in an isolated way in order to increase or decrease the coding rate respectively. However, we propose the use of both strategies simultaneously defining a new constant parameter, $\delta = \pi + \sigma$. This proposal is based primarily on two reasons:

\begin{enumerate}

\item Applying the same proportion of puncturing and shortening in every modulation. In consequence, regardless of the coding rate, the key length that can be corrected using the modulated code is known in advance.

\item As it is discussed in the next section, the use of puncturing and shortening simultaneously allow us to modify a previously modulated code in order to decrease the coding rate and to repeat an unsatisfactory correction process.

\end{enumerate}

The efficiency, defined in Eq.~\ref{eq:efficiency}, of the modulated code depends on the coding rate and the ratio $\delta$ of puncturing and shortening as shown in Fig.~\ref{fig:efficiency}.

\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{results/efficiency-th.ps}
\caption{Efficiency of \textit{Cascade}~\cite{Brassard94} and efficiency thresholds of LDPC codes with rate modulation for an error rate from 2\% to 10\%. The range is representative of good to very bad quantum channels for the QKD case. Efficiencies have been calculated, using the expression defined in Eq.~\ref{eq:efficiency}, for three different codes with rates $R_0$: $0.5$, $0.6$ and $0.7$. Two $\delta$ values, $0.1$ and $0.05$, have been used for the rate modulation of these codes. An additional $\delta = 0.5$ has been used with $R_0 = 0.5$ covering the entire error rate range. The curves show how the efficiency of the LDPC code depends on the ratio of puncturing and shortening, $\delta$, and the original coding rate, $R_0$. Higher $\delta$ values imply a bigger range of coding rates covered, unfortunately the efficiency drops for high values of $\delta$~\cite{Pishro-Nik07}. The efficiency drop increases for simultaneous use of high $\delta$ values with high coding rates.}
\label{fig:efficiency}
\end{figure}

\subsection{Interactive Reconciliation}

The process of adapting the coding rate of an LDPC code is usually done with a previous estimate of the error rate to be corrected, this estimation is traditionally carried out by exchanging a sample of the sequence on the public channel. We propose here a new protocol for information reconciliation with LDPC codes that does not require to estimate the error rate. The protocol is based on syndrome coding, but adding a new functionality: feedback information about the success of the decoding process (see Fig.~\ref{fig:interactive-decoding}). With this feedback the original one-way approach becomes an interactive protocol, as described below, with more flexibility in order to correct optimally a range of error rates. The protocol is blind in the sense that it is able to adapt to different channel configurations without a prior estimate and, the reconciliation is successful as long as the channel's characteristics are within a pre-established range.

\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{figures/interactive-decoding.eps}
\caption{Source coding with side information and feedback.}
\label{fig:interactive-decoding}
\end{figure}

The protocol is described by the following three steps:

\subsubsection*{Step 0) Raw Key Exchange}

Initially it is assumed that two sources, $X$ and $Y$, generate two correlated symbol sequences, $\mathbf{x}$ and $\mathbf{y}$ belonging to Alice and Bob respectively. Moreover, it is also supposed that the two symbol sequences have discrepancies within a bounded error rate range, $[e_0, e_1]$. From this hypothesis, Alice and Bob can choose an LDPC code with an information rate aimed to correct an intermediate point in the interval. Depending on the range and the efficiency target, the parties agree on a $\delta$ value to cover the entire range of required coding rates.

\begin{equation}
R_\mathrm{min} = \frac{R_0 - \delta}{1 - \delta} \leq R \leq \frac{R_0}{1 - \delta} = R_\mathrm{max}
\end{equation}

\noindent such that $R_\mathrm{min} \leq 1 - h_2(e_1)$ and $R_\mathrm{max} \geq 1 - h_2(e_0)$, where $h_2$ is the binary Shannon entropy.

As initial coding rate is chosen the highest value, $R = R_\mathrm{max}$, such that all symbols used to modulate the rate correspond to a punctured symbol, i.e. $\delta = \pi$ and $\sigma = 0$. In this case the protocol provides the minimum amount of information.

\subsubsection*{Step 1) Encoding}

Once it has been established a value for the coding rate, both parties compute the number of symbols to be punctured and shortened, $p$ and $s$ respectively:

\begin{equation}
\begin{split}
s & = \lceil \left( R_0 - R (1 - \delta) \right) n \rceil \\
p & = \lfloor \delta n \rfloor - s
\end{split}
\end{equation}

The first time this step is run, Alice randomly chooses the symbols to be punctured ---there are no shortened symbols in the first round---, and set them with random values. Once Alice knows which positions correspond with punctured symbols, and their values, she calculates the syndrome (compressed information), $\mathbf{z} = \mathbf{x} H^t$, and sends it to Bob along with their positions.

In subsequent runs of this step, Alice chooses randomly a preestablished proportion of punctured symbols that will be converted to shortened symbols and transmits to Bob their positions and values. The proportion of converted symbols in each round must be agreed by both parties at the beginning of the protocol, and it depends on the maximum number of rounds that is allowed and the desired efficiency.

\subsubsection*{Step 2) Decoding}

Bob uses his correlated sequence, $\mathbf{y}$, and the information provided by punctured and shortened symbols as starting point to find a sequence with a syndrome that matches the syndrome received from Alice, $\mathbf{z}$. The protocol is successfully concluded in this step when Bob decodes a word matching the syndrome received. Otherwise, if the decoding process is stopped because the maximum number of pre-set iterations has been reached, then Bob agrees with Alice a decrease in the coding rate, if possible, and they return to the previous Step~1.

The protocol fails if the coding rate takes its minimum value and decoding is unsuccessful, i.e. $R = (R_0 - \delta) / (1 - \delta)$, which happens for $\delta = \sigma$ and $\pi = 0$.

\begin{figure}
\centering
\includegraphics[width=0.6\linewidth]{figures/protocol.eps}
\caption{Example of a complete execution of the interactive protocol. The example shows how the symbols of an initial sequence are distributed in different positions of a codeword. The remaining positions of the codeword are initially marked as punctured symbols. In each round, the protocol replaces a proportion of punctured symbols with shortened symbols, thus reducing the coding rate.}
\label{fig:protocol}
\end{figure}

A graphic description of this protocol is shown in Fig.~\ref{fig:protocol}. The figure illustrates an example showing three executing rounds.

Different executions of the proposed protocol may conclude with different ratios for puncturing and shortening, $\pi$ and $\sigma$ respectively, i.e. different protocol executions reconcile the original sequences with different efficiencies. Efficiency for a single protocol execution is defined in Eq.~\ref{eq:efficiency}, the efficiency of this protocol can be measured by taking an average value. Puncturing and shortening ratios are then calculated by:

\begin{equation}
\hat{\pi} = \frac{1}{M} \sum_{i = 1}^{M} \pi^{(i)}
\; ; \quad
\hat{\sigma} = \frac{1}{M} \sum_{i = 1}^{M} \sigma^{(i)}
\end{equation}

\noindent where $M$ is the number of executions. The average efficiency is then calculated as:

\begin{equation}
\label{eq:average-efficiency}
\hat{f} = \frac{1}{h_2(e)} - \frac{R_0 - \hat{\sigma}}{(1 - \delta) h_2(e)} = \frac{1 - R_0 - \hat{\pi}}{(1 - \delta) h_2(e)}
\end{equation}

\noindent where $e$ is the crossover probability that has been corrected.

Let $Q$ be the maximum number of rounds, the efficiency value in an isolated execution is increased in each round by a constant factor, $\epsilon$, that depends on the proportion of new shortened symbols, $q = \delta / Q$ such that $\pi_{j + 1} = \pi_{j} - q$ and $\sigma_{j + 1} = \sigma_{j} + q$. The efficiency of an execution that concludes in the round $j$ can be also calculated by $f_{j} = f_{0} + j \epsilon$, where:

\begin{equation}
f_{0} = \frac{1 - R_0 - \delta}{(1 - \delta) h_2(e)}
\; ; \quad
\epsilon = \frac{q}{(1 - \delta) h_2(e)}
\end{equation}

config: arxiv


TITLE: Computing $DF$ when $F(x, y)=f(x, y, g(x, y))$
QUESTION [0 upvotes]: I am trying to compute the derivative of $F$ in terms of $Df$ and $Dg$ given the formula $F(x, y)=f(x, y, g(x, y))$. But I don't know how to apply chain rule to this.

REPLY [1 votes]: HINT: $$dF=\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy +\frac{\partial f}{\partial g(x,y)}d[g(x,y)]$$
and $$d[g(x,y)]=dg=\frac{\partial g}{\partial x}dx +\frac{\partial g}{\partial y}dy$$
Hence $$dF=\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy +\frac{\partial f}{\partial g}\cdot \left(\frac{\partial g}{\partial x}dx +\frac{\partial g}{\partial y}dy\right)$$
And the rest is for you.


TITLE: (Non?)-linearity of the consistency strength ordering in ZF
QUESTION [11 upvotes]: Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all sentences $\sigma,\tau\in Sent$, $\sigma$ has less consistency strength than $\tau$ (or $\sigma\leq_{cons}\tau$) iff $cons(ZF+\tau)\rightarrow cons(ZF+\sigma)$. My main question (which I believe should have a negative answer) is the following:
$$\mbox{ Is } \leq_{cons}\mbox{ linear? If not is there a natural counterexample?}$$
In fact I have several similar questions concerning $\leq_{cons}$ and I would be glad if someone could provide me with some references. Thanks!

REPLY [6 votes]: Unlike - seemingly - all the answers above, and since strangely he seems to have missed this question (as not so often), Pr. J.D. Hamkins claims he has natural statements that contradict the pre-linearity of this pre-ordering!
He will talk about them on 25 January at 7pm UTC:
http://jdh.hamkins.org/natural-instances-of-nonlinearity-in-the-hierarchy-of-consistency-strength-uwm-logic-seminar-january-2021/


TITLE: Is $\vec{v} = \frac{2\pi r}{T}$ Referring to Tangential Speed or Velocity?
QUESTION [0 upvotes]: I seem to be a bit pedantic when it comes to studying physics. Lately, I have been seeing the equation $\vec{v} = \frac{2\pi r}{T}$, which has been causing me trouble. Shouldn't the right expression give us a scalar value, hence giving us the tangential speed and not the tangential velocity? If $\vec{v}$ is always changing around a circle, how come we find the right expression to be a constant value? I was told that the velocity here referred to the instantaneous velocity, but I couldn't understand why we are using the whole circumference of the circle as the displacement. And what does it mean for displacement to be a circumference, shouldn't that be distance instead?
Hopefully I have been able to address the issue I had in mind.

REPLY [6 votes]: I assume $T$ is the period of rotation. $2\pi r/T$ is the magnitude of the tangential velocity, i.e. speed. The equation
$$\vec v = \frac{2\pi r}{T}$$
doesn't make much sense, because the right hand side is not a vector. $r$ on the right hand side is not the displacement vector: it is the magnitude of the displacement vector measured from the center of rotation, or in other words, the radius of the circle on which the body is rotating.
The equation $v = 2\pi r/T$ is easy to interpret. Every period $T$, the body traverses a distance of $2 \pi r$ (the circumference of the circle; again, this is not a displacement), so the speed is this distance divided by the period.
If you wish to express the instantaneous tangential velocity as a vector quantity, it can be written as
$$ \vec v = \vec \omega \times \vec r $$
where $\vec r$ is the displacement vector with respect to the center of rotation, and $\vec \omega$ is the angular velocity vector with magnitude $\omega=2\pi/T$.

REPLY [0 votes]: While the magnitude of $\vec{v}$ remains constant in an orbit, its direction is in a constant state of change, thus the use of the vector sign. The direction will always be tangent to the orbit at any given time, which is what gives rise to this 'instantaneous velocity': the constant speed and its direction at any instant.
Indeed, you don't need to compute this as a vector, and wikipedia has an entire article on 'orbital speed' . It all comes down to convention of notation, what you are doing, and who you're doing it for. The math stays the same.
Now, for the equation $$v=\frac{2\pi r}{T}$$
we can easily see that we are using the orbital circumference as a 'distance' and orbital period as our 'time', which is chosen for convenience as it is much easier to measure one period, and one circumference than it is to measure some fraction of that.
Now you can choose to express this orbital speed $v$ at any given instant, as an instantaneous velocity $\vec{v}$, which is equivalent in magnitude, but tack on whatever direction its going in that moment (a tangent to its orbital path). I normally choose not to, but maybe it becomes important when you consider scenarios like a ball on a string that detaches at some moment, and flies off at tangential velocity $\vec{v}$.


text:
\begin{document}

\title{Relativistic quantum motion of an electron in spinning cosmic string spacetime in the presence of uniform magnetic field and Aharonov-Bohm potential}

\author{M\'{a}rcio M. Cunha}
\email{marciomc05@gmail.com}
\affiliation{
        Departamento de F\'{i}sica,
        Universidade Federal do Maranh\~{a}o,
        65085-580, S\~{a}o Lu\'{i}s, Maranh\~{a}o, Brazil
      }
\author{Edilberto O. Silva}
\email{edilbertoo@gmail.com}
\affiliation{
        Departamento de F\'{i}sica,
        Universidade Federal do Maranh\~{a}o,
        65085-580, S\~{a}o Lu\'{i}s, Maranh\~{a}o, Brazil
      }
\date{\today }
\begin{abstract}
In this manuscript, we study the relativistic quantum mechanics of an electron in external fields in the spinning cosmic string spacetime. We obtain the Dirac equation, write the first and second-order equations from it, and then we solve these equations for bound states. We show that there are bound states solutions for the first-order equation Dirac. For the second-order equation, we show that its wave functions are given in terms of the Kummer functions and we determine the energies of the particle. We examine the behavior of the energies as a function of the physical parameters of the model, such as rotation, curvature, magnetic field, Aharonov-Bohm flux, and quantum numbers. Our study reveals that both curvature and rotation influence more intensely when these parameters have values smaller than 0.3. We also find that, depending on the values of these parameters, there are energy nonpermissible levels.
\end{abstract}

\pacs{03.65.Ge, 03.65.Pm, 04.62.+v, 71.15.Rf}
\maketitle

\section{Introduction}
\label{intro}

Symmetry is key-ingredient in the description of natural phenomena. The notion of symmetry is an essential feature in several areas of physics.  In this context, the well-known Noether's theorem \cite{lanczos2012variational} establishes a connection between symmetry and conservation laws of relevant physical quantities. In quantum mechanics, we often use symmetry to obtain crucial results concerning angular momenta operators \cite{sakurai2014modern}.
Likewise, symmetry is important in the topic of quantum information \cite{PhysRevB.95.045111}.
Symmetry is also important in the framework of relativity \cite{westman2009coordinates}, and for this reason, it is indispensable in research areas such as particle physics \cite{gibson1980symmetry} and cosmology \cite{preskill1991cosmology}.

A pertinent question in the research areas cited above refers to think about what happens when some symmetry is broken in a given physical system.
When a system suffers a phase transition, for example, it can lose some type of symmetry.
Another example of symmetry-breaking occurs in condensed matter systems: by employing the Volterra process \cite{solyom2007fundamentals}, it is possible to break some symmetry of the system due to the creation of a topological defect, like disclinations and dislocations \cite{RevModPhys.80.61,puntigam1997volterra}, for instance.

Topological defects can emerge in a large number of physical systems covering themes such as liquid crystals \cite{lavrentovich2012defects}, graphene physics \cite{alden2013strain}, magnetism \cite{kou2011tunable} and cosmology \cite{durrer1999topological}.
Recent studies also have reported the importance of topological defects in Life Sciences \cite{kawaguchi2017topological,saw2017topological}.
On the point of view of cosmology, defects in the spacetime topology can be viewed as a possible consequence of the evolution of the early universe, which has suffered phase transitions due to the temperature decreasing and the process of expansion \cite{hindmarsh1995cosmic,vilenkin2000cosmic}.

 In this contribution, we are particularly involved in studying the topological defect known as a cosmic string. A cosmic string is a linear defect, similar to a flux tube in type-II superconductors \cite{hindmarsh1995cosmic}. The spacetime around a cosmic string has a conical symmetry, identically to the case of a disclination \cite{moraes2000condensed}. The concept of a cosmic string it was introduced in the literature by Kibble \cite{kibble1976topology}. Since then, this topic has been investigated in diverse forms.  An intriguing facet in this subject refers to the quantum mechanical description of a particle in a region of the spacetime containing a cosmic string. It can be done both in the scenario of relativistic and nonrelativistic quantum mechanics. For instance, the hydrogen atom in a spacetime of a cosmic string it was analyzed in Ref. \cite{PhysRevD.66.105011}. In Ref. \cite{marques2005exact}, it was considered the problem of a relativistic electron in the presence of both Coulomb and scalar potentials in the cosmic string spacetime. Results about vacuum polarization in a cosmic string spacetime were reported in Ref. \cite{PhysRevD.74.025017}.
Again, the cosmic string spacetime it was considered as a background to examine relativistic oscillators \cite{PhysRevA.84.032109}, quantum phases \cite{bakke2016relativistic}, and fermionic currents \cite{bezerra2016induced}.

 A relevant issue in this context consists of taking into consideration the influence of electromagnetic fields in the quantum particle motion. Landau levels \cite{Book.2005.Griffiths} and the Aharonov-Bohm effect \cite{PR.1959.115.485,peshkin1989aharonov}, for instance, are essential ingredients in the investigation of quantum systems even in a flat spacetime. It can be explained because Landau levels are a quantum analog of classical cyclotron motion, while the Aharonov-Bohm effect reveals the significance of the vector potential in the quantum world.
Then, studying the contribution of magnetic fields to the quantum mechanical description of a system in spacetime having a topological defect is a natural development. Examples of studies dealing with Landau levels and the Aharonov-Bohm effect in the presence of topological defects can be accessed in Refs. \cite{de2001landau} and \cite{azevedo1998topological}, respectively. In particular, the inclusion of electromagnetic interactions in the case of a cosmic string background also has been considered. For instance, in Ref. \cite{medeiros2012relativistic}, it was analyzed the quantum dynamics of a charged particle in the presence of a magnetic field and scalar potential. In Ref. \cite{PhysRevD.93.043545}, various configurations of confined magnetic fields are examined and the existence of induced vacuum fermionic currents is investigated.

On the other hand, we can be interested in analyzing the behavior of a quantum system when noninertial effects turn on. These effects play a fundamental role in the description of systems governed by classical mechanics. Noninertial effects also can take place on quantum systems, providing novel theoretical predictions and feasible experimental developments.
For instance, the emergence of quantum phases in rotating systems, in analogy to the Aharonov-Bohm effect \cite{aharonov1973quantum,semon1982experimental} were investigated. In addition, a relation between the Hall effect and inertial forces it was established \cite{johnson2000inertial}.
Besides, if a given system it is put to rotate, it has consequences in diverse physical properties like spin transport \cite{PhysRevB.84.104410,chowdhury2014spin}, electronic structure \cite{garcia2017geometric}, and even can present magnetization due rotation, like in the Barnett effect \cite{PhysRevB.92.174424}. While a magnetic field produces a spin-field coupling, resulting in the anomalous
Zeeman effect \cite{nouredine2009quantum}, rotation produces an analog effect, due the spin-rotation coupling \cite{danner2020spin}.
Thus, rotation can contribute similarly to a magnetic field in the dynamics of a quantum system.
More, noninertial effects are an interesting issue in the situation in which spacetime contains topological defects. In this case, the noninertial effects and the presence of a topological defect can be included in the quantum mechanical description by employing the same tools: we can use a metric tensor to a spinning spacetime with a topological defect \cite{clement1990rotating}.
The spacetime of a spinning cosmic string has been considered as background for several problems involving quantum systems. For instance, the Schr\"{o}dinger equation in that spacetime it was solved in Ref. \cite{hassanabadi2015motion}. Bound states for neutral particles in a rotating frame of a cosmic string were analyzed in Ref. \cite{PhysRevD.82.084025}. Likewise, rotating effects on a Landau-Aharonov-Casher System in the spacetime of a cosmic string were investigated in Ref. \cite{bakke2015rotating}.

As we already have mentioned, in some cases rotation presents similarities within electromagnetic fields. This way, it is also an attractive question examining how the electromagnetic interactions affect the particle quantum motion of a rotating system in the presence of a topological defect. A recent example of studying dealing with both topological and noninertial effects in the presence of an Aharonov-Bohm potential can be accessed in Ref. \cite{oliveira2019topological}. In Ref. \cite{Wang_canadian_journal2017}, it was addressed the problem of a spinless relativistic particle subjected to a uniform magnetic field in the spinning cosmic string spacetime.
The Dirac oscillator in the spacetime of a cosmic string considering noninertial effects and the presence of the  Aharonov–Casher effect it was analyzed in Ref. \cite{EPJC_oliveira2019topological}.
In Ref. \cite{wang2018study}, it was analyzed the problem of a charged half-spin particle depicted by the Dirac equation in the presence of a uniform magnetic field in the rotating cosmic string spacetime.
A meaningful aspect in this context consists of analyzing how different configurations of magnetic fields affect the quantum particle motion.

In this paper, we study the relativistic quantum mechanics of an electron in the presence of both a uniform magnetic field and Aharonov-Bohm potential in the spinning cosmic string spacetime. In other words, we solve the Dirac equation in this scenario and investigate how the rotation, curvature and external magnetic fields affect the wave functions and energies of the electron.

The manuscript is organized as follows. In Section II, we present some algebraic elements necessary to construct the field equations in curved spacetime and write the Dirac equation describing the quantum motion of the electron in the presence of external magnetic fields in the spinning cosmic string background. In Section III, we deal with first-order solutions and study the existence of isolated solutions for the particular case of a particle at rest. In Section IV, we take our attention to the case when the energy of the particle is different from its rest energy. We map the Dirac equation problem in curved space with minimal coupling into a Sturm-Liouville problem for the upper component of the Dirac spinor and, using an appropriate ansatz, we derive the radial equation. We solve the radial equation and find the wave functions and energies of the particle. We make a detailed discussion of the results and also comparisons with other studies in the literature. In Section V, we present our conclusions. In our work, we use natural units, $\hbar = c = G = 1$.

\section{Dirac equation in the spinning cosmic string spacetime}
In this section, we briefly present the main tools needed to construct the Dirac equation in the conical spacetime in the presence of noninertial effects. The first step consists in take a look at the metric tensor characterizing this geometry. Next, we will choose an appropriate tetrad basis and implement the fields configuration involved through the performing of a minimal substitution.
The spacetime induced by a rotating cosmic string is described by the metric
\begin{equation}
ds^{2}=\left( dt+ad\varphi \right) ^{2}-dr^{2}-\alpha ^{2}r^{2}d\varphi
^{2}-dz^{2},  \label{metric}
\end{equation}
where $-\infty <z<\infty $, $r\geqslant 0$ and $0\leqslant \varphi \leqslant
2\pi$. The parameter $\alpha $ is related to the linear mass density $\mu$
of the cosmic string through the relation $\alpha =1-4\mu$ and it runs in
the interval $(0,1]$. The quantity $a=4J$ is the rotation parameter, with $J$ representing the angular momentum of the spinning cosmic string. The relativistic quantum dynamics of a spin-$1/2$ particle interacting with external magnetic fields in the rotating cosmic string spacetime is governed by the Dirac equation
\begin{equation}
\left[ i\gamma ^{\mu }\left( x\right) \left( \partial _{\mu }+\Gamma _{\mu
}\left( x\right) +ieA_{\mu }(x)\right) -M\right] \Psi \left( x\right) =0,
\label{diracsc}
\end{equation}
where $M$ is the mass of the particle and $\gamma ^{\mu }\left( x\right)$ are the Dirac matrices in the
rotating cosmic string spacetime, which are defined in terms of the tetrad fields $e_{a}^{\mu }$ and Dirac matrices in the flat space $\gamma ^{a}$ in the following way:
\begin{equation}
\gamma ^{\mu }\left( x\right) =e_{a}^{\mu }\left( x\right) \gamma ^{a},
\label{gmatrices}
\end{equation}
where
\begin{equation}
\gamma ^{a}=\left( \gamma ^{0},\gamma ^{i}\right),\, \text{with}\, \gamma ^{0}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right),\,\gamma ^{i}=\left(
\begin{array}{cc}
0 & \sigma ^{i} \\
-\sigma ^{i} & 0
\end{array}
\right),
\end{equation}
are the standard Dirac matrices
and $\sigma ^{i}=\left( \sigma ^{x},\sigma ^{y},\sigma ^{z}\right)$ are the usual Pauli matrices. The matrices (\ref{gmatrices}) satisfy the following relation:
\begin{equation}
\left\{ \gamma ^{\mu }\left( x\right) ,\gamma ^{\nu }\left( x\right)
\right\} =2g^{\mu \nu }\left( x\right).
\end{equation}
Also, in Eq. (\ref{diracsc}), $\Gamma _{\mu }\left( x\right)$ is the
spin affine connection given by
\begin{equation}
\Gamma _{\mu }\left( x\right) =\frac{1}{4}\gamma ^{a}\gamma ^{b}e_{a}^{\nu
}\left( x\right) \left[ \partial _{\mu }e_{b\nu }\left( x\right) -\Gamma
_{\mu \nu }^{\sigma }e_{b\sigma }\left( x\right) \right], \label{affine}
\end{equation}
where $\Gamma_{\mu \nu}^{\sigma}$ are the Christoffel symbols of the second kind and $e_{a}^{\mu }(x)$ is the tetrad field. The tetrad basis satisfies the relations
\begin{align}
&e_{\mu }^{a}\left( x\right) e_{\nu}^{b}\left(x\right) \eta_{ab}=g_{\mu \nu }\left( x\right),\\
&e_{\mu}^{a}\left(x\right) e_{\nu
}^{b}\left(x\right)=\delta_{a}^{b},\\
&e_{a}^{\mu}\left(x\right)
e_{\nu}^{a}\left(x\right) =\delta_{\mu}^{\nu}.
\end{align}
In Eq. (\ref{affine}), the Greek letters are used for tensor indices while the Latin letters are denoting Minkowski indices. We use the tetrad basis and its inverse defined as \cite{EPJC.2019.79.311}
\begin{eqnarray}
e_{\mu }^{a}\left( x\right)  &=&\left(
\begin{array}{cccc}
1 & 0 & a & 0 \\
0 & \cos \varphi  & -r\alpha \sin \varphi  & 0 \\
0 & \sin \varphi  & r\alpha \cos \varphi  & 0 \\
0 & 0 & 0 & 1
\end{array}
\right) , \\
e_{a}^{\mu }\left( x\right)  &=&\left(
\begin{array}{cccc}
1 & \frac{a\sin \varphi }{r\alpha } & -\frac{a\cos \varphi }{r\alpha } & 0
\\
0 & \cos \varphi  & \sin \varphi  & 0 \\
0 & -\frac{\sin \varphi }{r\alpha } & \frac{\cos \varphi }{r\alpha } & 0 \\
0 & 0 & 0 & 1
\end{array}
\right).\label{trdi}
\end{eqnarray}
For this choice, it can be shown that the non-vanishing affine connection is given by
\begin{equation}
\Gamma _{\mu }=\left( 0,0,\Gamma _{\varphi },0\right), \, \text{with} \; \Gamma _{\varphi }=\frac{i}{2}\left( 1-\alpha \right) \Sigma^{z},
\end{equation}
with
\begin{equation}
\Gamma _{\varphi }=\frac{i}{2}\left( 1-\alpha \right) \Sigma^{z},
\end{equation}
where
\begin{equation}
\Sigma ^{z}=\left(
\begin{array}{cc}
\sigma ^{z} & 0 \\
0 & \sigma ^{z}
\end{array}
\right), \;\; \sigma ^{z} =\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right).
\end{equation}
By using the tetrad basis (\ref{trdi}), the matrices (\ref{gmatrices}) can be written explicitly as
\begin{align}
&\gamma ^{t} =\gamma ^{0}-a\gamma ^{\varphi },  \\
&\gamma ^{r}=\left(
\begin{array}{cc}
0 & \sigma ^{r} \\
-\sigma ^{r} & 0
\end{array}
\right), \;\;\gamma ^{\varphi }=\left(
\begin{array}{cc}
0 & \sigma ^{\varphi } \\
-\sigma ^{\varphi } & 0
\end{array}
\right),
\end{align}
with
\begin{equation}
\sigma^{r} =\left(
\begin{array}{cc}
0 & e^{-i\varphi } \\
e^{+i\varphi } & 0
\end{array}
\right), \;\;\sigma ^{\varphi }=\frac{1}{r\alpha }\left(
\begin{array}{cc}
0 & -ie^{-i\varphi } \\
ie^{+i\varphi } & 0
\end{array}
\right)
\end{equation}
being the Pauli matrices in the curved spacetime.

Since we are first interested in studying the solutions of the Dirac equation in its present form (Eq. (\ref{diracsc})), we need to write the corresponding system of first order coupled differential equations. For this to be accomplished, let's assume the time-dependence of the wave functions together with the decomposition of the fermion field in the form
\begin{equation}
    \Psi \left( r,\varphi \right)=e^{-iEt}\left(
\begin{array}{c}
\psi _{1}\left( r,\varphi \right)  \\
\psi _{2}\left( r,\varphi \right)
\end{array}
\right),\label{S1}
\end{equation}
with
\begin{align}
&\psi _{1}\left( r,\varphi \right)=\left(
\begin{array}{c}
\psi _{a}\left( r,\varphi \right)  \\
\psi _{b}\left( r,\varphi \right)
\end{array}
\right) =\left(
\begin{array}{c}
e^{im\varphi }f_{+}\left( r\right)  \\
ie^{i\left( m+1\right) \varphi }f_{-}\left( r\right)
\end{array}
\right) ,\label{P1} \\
& \psi _{2}\left( r,\varphi \right) =\left(
\begin{array}{c}
\psi _{c}\left( r,\varphi \right)  \\
\psi _{d}\left( r,\varphi \right)
\end{array}
\right) =\left(
\begin{array}{c}
e^{im\varphi }g_{+}\left( r\right)  \\
ie^{i\left( m+1\right) \varphi }g_{-}\left( r\right)
\end{array}
\right). \label{P2}
\end{align}
The system we will analyze takes into account the particle is immersed in a region where there is a uniform magnetic field and also the potential due to a thin long solenoid along the z-axis. Having this field configuration in mind, we study
the physical implications due to noninertial effects and the Aharonov-Bohm potential on the relativistic Landau quantization. We also take into account the translational invariance of the system along the $z$-direction, which allows us to eliminate the third direction ($p_z=z=0$) and, consequently, we can consider only the planar motion \cite{PRL.1990.64.503,PRD.1994.50.7715,PRD.2012.85.041701,AoP.2013.339.510}. In this case, the
particle experiences a superposition of potential vectors written in the
Coulomb gauge as
\begin{equation}
\mathbf{A}=\left( 0,-\alpha rA_{\varphi }, 0\right) ,  \label{vectorA}
\end{equation}
with
\begin{align}
A_{\varphi } &=A_{\varphi ,1}+A_{\varphi ,2} , \label{potential} \\
A_{\varphi ,1} &=\frac{Br}{2},\;\;\; A_{\varphi ,2}=\frac{\phi }{\alpha r},\label{potentialv}
\end{align}
where $B$ is the magnetic field magnitude, $\phi =\Phi /\Phi _{0}$, $\Phi$ is the
magnetic flux and $\Phi _{0}=2\pi /e$ is the quantum of magnetic flux along the solenoid. This configuration also provides an superposition of magnetic fields in the
z-direction
\begin{equation}
B=B_{z,1}+B_{z,2},
\end{equation}
with
\begin{equation}
B_{1,z}=B,  \;\;\; B_{z,2}=\phi \frac{\delta (r)}{\alpha r},  \label{fields}
\end{equation}
Note that the particle only interacts with the magnetic field due to the
potential vector $A_{\varphi ,1}$. Here, we are focused on studying the
electron motion only in the $r\neq 0$ region, so that we can neglect the
point interaction $B_{z,2}$ and, consequently, consider only regular wave
functions.

Using the results above, the Dirac equation (\ref{diracsc}) can be written as
\begin{align}
(E-&M) \,\psi_{1}+\sigma ^{r}i\partial _{r}\psi _{2}\notag\\ &+\sigma
^{\varphi }\left( i\partial _{\varphi }+eA_{\varphi }-aE-\frac{s}{2}\left(
1-\alpha \right) \right) \psi_{2}=0, \label{Eqpsi}\\
(E+&M) \,\psi_{2}+\sigma ^{i}i\partial _{r}\psi _{1}\notag \\ &+\sigma
^{\varphi }\left( i\partial _{\varphi }+eA_{\varphi }-aE-\frac{s}{2}\left(
1-\alpha \right) \right) \psi_{1}=0. \label{Eqchi}
\end{align}
At this point, we are ready to solve the equations (\ref{Eqpsi}) and (\ref{Eqchi}) by considering two distinct circumstances:

(i) Take our attention to isolated solutions of the first order Dirac equation by imposing the condition $E= \pm M$;

(ii) By imposing the condition $E\neq \pm M$,  we looking for solutions of the second order Dirac equation.

We will show in the next two sections that there are bound state solutions for both cases and discuss their main physical properties. To distinguish each case in (i), in the next section we use the superscripts ($\pm$) to label the quantities corresponding to $E= \pm M$.

\section{Solution of the equation of motion to $E=\pm M$}

To study the existence of isolated solutions of the Dirac equation (\ref{diracsc}), we must set $E=\pm M$ in Eqs.  (\ref{Eqpsi}) and (\ref{Eqchi}). In literature, such solutions are known to be excluded from the Sturm-Liouville problem. The search for isolated solutions of the Dirac equation has been performed in different physical contexts \cite
{EPJC.2019.79.596,EPL.2014.108.30003,AoP.2013.338.278,JPA.2007.40.263,PLA.2006.351.379}. The bound state solution must satisfy the normalization condition
\begin{equation}
\int_{0}^{\infty }\left( |\psi_{1}(r)|^{2}+|\psi_{2}(r)|^{2}\right) rdr=1.
\label{norm}
\end{equation}
By making $E=+M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}) and using Eqs. (\ref{P1}) and (\ref{P2}), we get
\begin{align}
&\frac{dg_{+}^{(+)}(r)}{dr}-\frac{L_{m}^{(+)}}{r\alpha }g_{+}^{(+)}(r)+\frac{eBr}{2}g_{+}^{(+)}(r) =0,\label{eqg1}
\\
&\frac{dg_{-}^{(+)}(r)}{dr}+\frac{L_{m+1}^{(+)}}{r\alpha }g_{-}^{(+)}(r)-\frac{eBr}{2}g_{-}^{(+)}(r) =0,\label{eqg2}
\\
&\frac{df_{+}^{(+)}(r)}{dr}-\frac{L_{m}^{(+)}}{r\alpha }f_{+}^{(+)}(r)+\frac{eBr}{2}f_{+}^{(+)}(r)
=-2Mg_{-}^{(+)}(r), \label{eqg3}\\
&\frac{df_{-}^{(+)}(r)}{dr}+\frac{L_{m+1}^{(+)}}{r\alpha }f_{-}^{(+)}(r)-\frac{eBr}{2}f_{-}^{(+)}(r)
=2Mg_{+}^{(+)}(r),\label{eqg4}
\end{align}
with
\begin{align}
L_{m}^{(+)}& =m-\phi +aM+\frac{s}{2}\left( 1-\alpha \right),  \label{lmm1}\\
L_{m+1}^{(+)}& =m+1-\phi +aM+\frac{s}{2}\left( 1-\alpha \right).\label{lmm2}
\end{align}
The solution of the coupled linear differential equations system (\ref{eqg1})-(\ref{eqg4}) is given by
\begin{align}
f_{+}^{(+)}(r)& =e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }}\left[
a_{2}+a_{1}M\left( -\frac{eB}{2}\right) ^{\Omega _{a}}\Gamma _{a}^{(+)}\right],\label{sp1} \\
f_{-}^{(+)}(r)& =e^{\frac{1}{4}eBr^{2}}r^{-\frac{L_{m+1}^{(+)}}{\alpha }}\left[
b_{2}-b_{1}M\left( \frac{eB}{2}\right) ^{-\Omega _{b}}\Gamma _{b}^{(+)}\right] , \label{sp2} \\
g_{+}^{(+)}(r)& =b_{1}e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }}, \label{sp3} \\
g_{-}^{(+)}(r)& =a_{1}e^{\frac{1}{4}Ber^{2}}r^{-\frac{L_{m+1}^{(+)}}{\alpha }},\label{sp4}
\end{align}
with
\begin{align}
\Omega _{a}& =\frac{1}{2\alpha }\left( L_{m}^{(+)}+L_{m+1}^{(+)}-\alpha
\right) , \\
\Omega _{b}& =\frac{1}{2\alpha }\left( L_{m}^{(+)}+L_{m+1}^{(+)}+\alpha
\right) ,
\end{align}
where
\begin{align}
\Gamma _{a}^{(+)}& =\Gamma \left( -\Omega _{a},-\frac{1}{2}eBr^{2}\right) , \\
\Gamma _{b}^{(+)}& =\Gamma \left( \Omega _{b},\frac{1}{2}eBr^{2}\right) .
\end{align}
are upper incomplete Gamma functions \cite{Book.1972.Abramowitz}, and
$a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ are constants. Analyzing the solutions (\ref{sp1}) and (\ref{sp3}), we note that
$e^{-\frac{1}{4}eBr^{2}}$ dominates over $r^{\frac{L_{m}^{(+)}}{\alpha }}$
for any value of $L_{m}^{(+)}/\alpha $, in such way both solutions converge when $r\rightarrow 0$ and $r\rightarrow \infty $. This will not occur for the
function $e^{\frac{1}{4}eBr^{2}}$ in the solutions (\ref{sp2}) and (\ref{sp4}). Moreover, since the incomplete Gamma
functions $\Gamma _{a}^{(+)}$ and $\Gamma _{b}^{(+)}$ always diverge, then
the function $f_{+}^{(+)}(r)$ will only converges as $r\rightarrow 0$ if
$a_{1}=0$ while the function $f_{-}^{(+)}(r)$ will always diverge when $r\rightarrow \infty$ and,
therefore, will not be a square-integratable function. Thus, the only solution allowed for the equations system (\ref{eqg1})-(\ref{eqg4}) results
\begin{equation}
f_{+}^{(+)}(r)=a_{2}e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(+)}}{\alpha }},\, \text{with}\;\frac{L_{m}^{(+)}}{\alpha }\geqslant 0 ,\label{fnl}
\end{equation}
with $f_{-}^{(+)}(r)=g_{+}^{(+)}(r)=g_{-}^{(+)}(r)=0$. Solution (\ref{fnl})
satisfies equation (\ref{norm}) and constitutes a bound state solution for the case $
E=M$, i.e., an isolated solution to the Dirac equation (\ref{diracsc}) in the metric spacetime (\ref{metric}).

Proceeding in an analogous way, now we make $E=-M$ in Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). We find the system of equations
\begin{align}
&\frac{df_{+}^{(-)}(r)}{dr}-\frac{L_{m}^{(-)}}{r\alpha }f_{+}^{(-)}(r)+\frac{eBr}{2}f_{+}^{(-)}(r)=0,  \label{f1}
\\
&\frac{df_{-}^{(-)}(r)}{dr}+\frac{L_{m+1}^{(-)}}{r\alpha }f_{-}^{(-)}(r)-\frac{eBr}{2}f_{-}^{(-)}(r)=0,\label{f2}\\
&\frac{dg_{+}^{(-)}(r)}{dr}-\frac{L_{m}^{(-)}}{r\alpha }g_{+}^{(-)}(r)+\frac{eBr}{2}g_{+}^{(-)}(r)
=2Mf_{-}^{(-)}(r),  \label{g1}\\
&\frac{dg_{-}^{(-)}(r)}{dr}+\frac{L_{m+1}^{(-)}}{r\alpha }g_{-}^{(-)}(r)-\frac{eBr}{2}g_{-}^{(-)}(r)
=-2Mf_{+}^{(-)}(r). \label{g2}
\end{align}
with
\begin{eqnarray}
L_{m}^{(-)} &=&m-\phi -aM+\frac{s}{2}\left( 1-\alpha \right), \label{lpm1} \\
L_{m+1}^{(-)} &=&m+1-\phi -aM+\frac{s}{2}\left( 1-\alpha \right).\label{lpm2}
\end{eqnarray}
The solution of the coupled linear ordinary differential equations system (\ref{f1})-(\ref{g2}) is given by
\begin{align}
f_{+}^{(-)}(r)& =c_{1}e^{-\frac{1}{4}eBr^{2}}r^{\frac{L_{m}^{(-)}}{\alpha }},\label{sms1}
\\
f_{-}^{(-)}(r)& =d_{1}e^{\frac{1}{4}Ber^{2}}r^{-\frac{L_{m+1}^{(-)}}{\alpha }
}, \label{sms2}\\
g_{+}^{(-)}(r)& =e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(-)}}{\alpha}
}\left[-d_{1}M\left( -\frac{eB}{2}\right) ^{\Lambda _{c}}\Gamma
_{c}^{\left( -\right) }+d_{2}\right], \label{sms3}\\
g_{-}^{(-)}(r)& =e^{\frac{1}{4}eBr^{2}}r^{-\frac{L_{m+1}^{(-)}}{\alpha }}
\left[ c_{1}M\left( \frac{eB}{2}\right) ^{-\Lambda _{d}}\Gamma _{d}^{\left(
-\right) }+c_{2}\right] \label{sms4},
\end{align}
with
\begin{align}
\Lambda _{c}& =\frac{1}{2\alpha }\left( L_{m}^{(-)}+L_{m+1}^{(-)}-\alpha
\right) , \\
\Lambda _{d}& =\frac{1}{2\alpha }\left( L_{m}^{(-)}+L_{m+1}^{(-)}+\alpha
\right) ,
\end{align}
where
\begin{align}
\Gamma _{c}^{\left( -\right) }& =\Gamma \left( -\Lambda _{c},-\frac{1}{2}
Ber^{2}\right) , \\
\Gamma _{d}^{\left( -\right) }& =\Gamma \left( \Lambda _{d},\frac{1}{2}
eBr^{2}\right) .
\end{align}
By making the same analysis of the solutions as we have made for the case $E=M$, i.e., analyzing the behavior of the functions for $r\rightarrow \pm\, \infty$, we find that the only solution that admits bound state is (\ref{sms3}). Thus, the solution for the case $E=-M$ satisfying the normalization condition (\ref{norm}) is given by
\begin{equation}
g_{+}^{(-)}(r) =d_{2}e^{-\frac{1}{4}Ber^{2}}r^{\frac{L_{m}^{(-)}}{\alpha }},\, \text{with}\;\frac{L_{m}^{(-)}}{\alpha }\geqslant 0,\label{slem}
\end{equation}
with $f_{+}^{(-)}(r) =
f_{-}^{(-)}(r)=g_{-}^{(-)}(r)=0$.
Note that the solutions (\ref{fnl}) and (\ref{slem}) are affected by rotation through Eqs. (\ref{lmm1}) and (\ref{lpm1}), respectively.

\section{Solution of the equation of motion to $E\neq \pm M$}

In this section, we solve the second order equation to $\psi$ that we find from the Eqs. (\ref{Eqpsi}) and (\ref{Eqchi}). The solution of this equation is different from that one calculated in the previous section and allow us to obtain an expression for the particle energies. By isolating $\psi_{2}$ in Eq. (\ref{Eqchi}) and replacing in Eq. (\ref{Eqpsi}), we are able to write the second order differential equation for $\psi_{1}$ as
\begin{align}
&\left(E^{2}-M^{2}\right) \psi_{1} +\partial _{r}^{2}\psi_{1} +\frac{1}{r}\partial
_{r}\psi_{1} +\frac{1}{\alpha r}\sigma ^{z}e\left( \partial _{r}A_{\varphi
}\right) \psi_{1}  \notag \\
& +\frac{1}{\alpha ^{2}r^{2}}\left( \partial _{\varphi }-ieA_{\varphi }+i
\frac{1-\alpha }{2}\sigma ^{z}+iaE\right) ^{2}\psi_{1} =0.\label{eav}
\end{align}
Using the decomposition of the fermion field (\ref{P1}) (ignoring the subscript ($+$)) together with Eqs. (\ref{vectorA}), (\ref{potential}) and (\ref{potentialv}), we obtain the radial equation for $f(r)$
\begin{equation}
\left( \frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{L^{2}}{\alpha
^{2}r^{2}}-\frac{e^{2}B^{2}r^{2}}{4}+k^{2}\right) f\left( r\right) =0, \label{re}
\end{equation}
where
\begin{equation}
k^{2}=E^{2}-M^{2}+\frac{eB}{\alpha}L+seB,
\end{equation}
\begin{equation}
L=m-\phi +\frac{s\left( 1-\alpha \right) }{2}+aE. \label{am}
\end{equation}
\begin{figure}[!t]
	\centering
	\includegraphics[width=\columnwidth]{Fif2DEBa.pdf}
	\includegraphics[width=\columnwidth]{Fif2DEBb.pdf}
    \caption{Sketch of the energy levels $E_{n}^{(>)}$ (Eq. (\ref{Ep})) as a function of the magnetic field $B$ for different values of $n$. In panel (a), $s=+1$ and in panel (b), $s=-1$. The positive energies are represented by solid lines and the negative by dashed lines. We assume $e=1$ and $M=1.$}
	\label{fig2DEb}
\end{figure}
Note that there are three other equivalent equations and there is no need to solve them here because their respective energies would also be equivalent. Equation (\ref{re}) is the confluent hypergeometric equation
 and its solution is well known. Thus, it can be shown that the solution to $\psi_a$ is \begin{align}
&\psi_{a}\left( r,\varphi \right) =c_{nm}\left( \frac{eB}{2}\right) ^{
\frac{1}{2}\left(1+{\frac{\left\vert L\right\vert }{\alpha}}\right)}e^{im\varphi }{r}^{{
\frac{\left\vert L\right\vert }{\alpha }}}e{^{-\frac{1}{4}\,eB{r}^{2}}}
\notag \\
& \times {{_{1}F_{1}}\left(\frac{1}{2} \left(1+{\frac{\left\vert L\right\vert }{\alpha}}\right)+{\frac{{k}^{2}}{2eB}},\,1+{\frac{\left\vert L\right\vert }{\alpha
}},\,\frac{1}{2}eB{r}^{2}\right) }, \label{pphi}
\end{align}
where ${_{1}F_{1}}\left(a,b,z \right)$ denotes the confluent hypergeometric function of the first kind or Kummer's function $M(a,b,z)$ and $c_{nm}$ the normalization constant.
It can be shown that the hypergeometric function ${_{1}F_{1}}\left(a,b,z \right)$ has a divergent behavior for large values of $z$. Because of this, bound state solutions for Eq. (\ref{pphi}) are only possible if we impose that this function becomes a polynomial of degree $n$. For this to be accomplished, we require that $1/2+{\left\vert L\right\vert /2\alpha }+{{k}^{2}/2Be=-n}$, where $n\in\mathbb{Z}^{*}$, with $\mathbb{Z}^{*}$ denoting the set of the nonnegative integers.
\begin{figure}[!h]
	\centering
	\includegraphics[width=\columnwidth]{Fig3DEnm.pdf}
    \caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $n$ and $m$ for $a=0.5$, $\alpha =0.5$, $B=1$, $e=1$, $M=1$, $s=1$ and $\phi =1$.}
	\label{En3Dnm}
\end{figure}However, as we can see in Eq. (\ref{am}), the absolute value of the effective angular moment $L$ is defined in terms of the energy $E$. In this way, to obtain the energy eigenvalues from the above condition, we must consider $|L|>0$ and $|L|<0$, respectively, and then solve them for $E$. By making this, we get
\begin{align}
E_{n}^{(>)}& = \pm\sqrt{eB\left( 2n-s+1\right) +M^{2}}, \label{Ep}\\
E_{nm}^{(<)}& =-\frac{aeB}{\alpha } \pm \frac{1}{\alpha }\sqrt{
a^{2}e^{2}B^{2}+\alpha Q},\label{Em}
\end{align}
with the following requirement:
\begin{equation}
a^{2}e^{2}B^{2}+\alpha \,Q \geqslant 0, \label{cnd}
\end{equation}
where
\begin{equation*}
Q=\alpha eB\left( 2n-\frac{2}{\alpha }\left( m-\phi +\frac{s}{2}\right)
+1\right) +\alpha M^{2}.
\end{equation*}
\begin{figure}[!h]
	\centering
	\includegraphics[width=\columnwidth]{Fig3DEaa.pdf}
    \caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $\alpha$ and $a$ for $B = 4$, $M = 1$, $e=1$, $n=1$, $\phi=2$, $s = 1$ and $m = 1$.}
	\label{En3Daa}
\end{figure}
In Eqs. (\ref{Ep}) and (\ref{Em}), the superscripts ($>,<$) refer to the energies calculated for $\left\vert L\right\vert >0$ and $\left\vert L\right\vert <0$, respectively.
For a given choice of the element of spin $s$, the energy $E_{n}^{(>)}$ depends only on the quantum number $n$ and the magnetic field $B$. For a given value of $n$, the energy increases when the magnetic field is increased. In Fig. \ref{fig2DEb}, we show the profile of $E_{n}^{(>)}$ for the first four states for $s=1$. The energy levels for $s=-1$ (Fig. \ref{fig2DEb}(b)) are slightly larger than the profile for the case $s=1$ (Fig. \ref{fig2DEb}(a)).

The energies (\ref{Ep}) and (\ref{Em}) denote the relativistic Landau levels in the present context. These energies can be directly compared with those obtained for the relativistic oscillator (Dirac oscillator) addressed in Ref. \cite{EPJC.2019.79.311}. Although that scenario is different from the one we are exploring here, there are similarities between the profiles of the energy levels in both models. For example, for $s=1$, the energy (48) of the Ref. \cite{EPJC.2019.79.311} depends only on the frequency of the oscillator and the quantum number $n$. In our case, by defining the cyclotron frequency $\omega _{c}=eB/M$, Eq. (\ref{Ep}) results
\begin{equation}
\tilde{E}_{nm}^{(>)}=\pm \sqrt{2nM\omega _{c}+M^{2}},
\end{equation}
which makes such a similarity clear. Since the energies (\ref{Em}) are the only ones that depend on all the physical parameters involved in the current problem, we study them in more detail. For a given set of fixed parameters, for example, $a=0.5$, $\alpha =0.5$, $B=1.0$, $e=1.0$, $M=1$, $s=1$ and $\phi=1$, we have the profile of the energy levels as a function of $n$ and $m$ (Fig. \ref{En3Dnm}). We can clearly see that $|{E}_{nm}^{(<)}|$ increases with $n$ and $m$. The green solid bars denote the discrete energy values for a given $m$ and $n$.
\begin{figure}[!h]
	\centering
	\includegraphics[width=\columnwidth]{Fig3DEbf.pdf}
    \caption{Sketch of the energy (Eq. (\ref{Em})) as a function of $B$ and $\phi$ for $a=1$, $\alpha =0.5$, $e=1$, $m=1$, $M=1$, $n=1$ and $s=1$.}
	\label{En3Dbf}
\end{figure}
On the other hand, when we investigate the behavior of (\ref{Em}) as a function of $\alpha$ and $a$ for specific values of the other parameters, we see that the negative spectrum changes more rapidly when compared with the positive one (Fig. \ref{En3Daa}). In the positive spectrum, both rotation and curvature lead to a linear change, except in the region with $\alpha < 0.3$ and arbitrary $a$ . In the negative spectrum, we see that the curvature effects are more predominant in the region where alpha has values smaller than 0.2. In this region, any variation in the rotation parameter implies in an abrupt change in the energy spectrum. Modifications in the energies with $\alpha < 0.3$ is an expected manifestation in our analyses. Its physical implication is inherent in the metric (\ref{metric}) and is an immediate consequence of the topological cone, which becomes more singular for smaller $\alpha$ values. To complete our analysis, we investigate the profile of the energy (\ref{Em}) as a function of magnetic field $B$ and the magnetic flux through the solenoid, $\phi$. Similarly to Fig. \ref{En3Daa}, by fixing the other parameters, we see that the energy of the anti-particle varies more rapidly when compared to the energy of the particle (Fig. \ref{En3Dbf}). Clearly, we observe that the energy of the particle varies very slowly throughout the region of flux and magnetic field . As a final commentary, we clarify that the cases discussed in Figs. \ref{En3Dnm}, \ref{En3Daa} and \ref{En3Dbf} can be investigated for other fixed parameter values. In this way, it can be shown that there are forbidden energies, depending on the values of the parameters considered. In general, this occurs when both the $\alpha$ parameter and the rotation parameter $a$ are smaller than $0.3$ and the other parameters assuming higher values than those we use here.

\section{Conclusions}
\label{sec:conclusions}

In the present manuscript, we have addressed the problem of the relativistic quantum motion of an electron in the spinning cosmic string background considering the presence of a uniform magnetic field and the Aharonov-Bohm potential. We have shown that this combination of potentials allows bound states configurations in the scenario of first-order solutions as well as in the case of second-order solutions of the Dirac equation.  It is worth noting the role played by the two different terms in the vector potential. As already known in the literature, we have shown that the uniform field is responsible for a behavior analog to a harmonic oscillator, which leads to the relativistic Landau quantization while the Aharonov-Bohm flux contributes to the angular momentum of the particle. In the case of first order solutions, which were obtained by solving Eqs. (\ref{fnl}) and (\ref{slem}) for $E=+M$ and $E=-M$, respectively, the oscillator-like behavior provided by the uniform magnetic field guarantees the convergent first-order solutions and, consequently, the existence of bound states. The isolated solutions obtained (Eqs. (\ref{fnl}) and (\ref{slem})) are particular solutions of the Dirac equation (\ref{diracsc}).

We have also studied the more general problem by solving the second-order equation implied by equations (\ref{Eqpsi}) and (\ref{Eqchi}) for the upper component of the Dirac spinor for $E\neq \pm M$. Using appropriate solutions (Eq. (\ref{S1})) we have derived the radial equation and shown that its solution is given in terms of the Kummer functions from which we have extracted the expression for the energy levels of the particle (Eqs. (\ref{Ep}) and (\ref{Em})). For the field configuration considered, we have found that the effective angular momentum of the electron depends on its energy and the Aharonov-Bohm flux tube while the potential vector that generates the uniform field leads to a charged oscillator. This implies that such field superposition provides distinct effects on the motion of the particle. Additionally, in some cases, the rotation produces a combined effect with both the uniform magnetic field and the curvature (see Eq. (\ref{Em})). We have shown that the energy levels of the particle and antiparticle depend on the values of the physical parameters involved. In the case of energy (\ref{Em}), its validity is conditioned to Eq. (\ref{cnd}). Depending on the choice we make for the parameters, we can obtain forbidden energies. The sketches in Figs. \ref{En3Dnm}, \ref{En3Daa}, and \ref{En3Dbf}  illustrate the profiles of the particle and antiparticle energies and show that they belong to the same spectrum. The effects of curvature and rotation are more evident when $\alpha < 0.3$, being the antiparticle energy the most affected. As a final comment, we would like to emphasize that the model studied in this article generalizes others found in the literature, such as those of Refs. \cite{oliveira2019topological,wang2018study} for the case including a superposition of external magnetic fields and the investigation of isolated solutions of the Dirac equation. Furthermore, we present a detailed discussion on the energy levels of the particle which, in general, is not found in the literature.

\section*{Acknowledgments}
We would like to thanks E.R.B. Mello (Universidade Federal da Para\'{i}ba, PB, Brazil) for his remarks and comments.
This work was partially supported by the Brazilian agencies CAPES, CNPq and
FAPEMA. EOS acknowledges CNPq Grants 427214/2016-5 and 307203/2019-0, and
FAPEMA Grants 01852/14 and 01202/16. This study was financed in part by the Coordena\c{c}\~{a}o de
Aperfei\c{c}oamento de Pessoal de N\'{\i}vel Superior - Brasil (CAPES) -
Finance Code 001. MMC acknowledges CAPES
Grant 88887.358036/2019-00.

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TITLE: Crash course on algebraic geometry with view to applications in physics
QUESTION [28 upvotes]: Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?
What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references.

REPLY [1 votes]: Some algebraic geometry with the main purpose of understanding D-branes in the context of mirror symmetry is reviewed in Paul Aspinwall's 'D-branes on Calabi-Yau manifolds' (http://arxiv.org/abs/hep-th/0403166).
I have not begun reading it thoroughly myself, but it seems accessible to physicists, at least those with the basic math background for string theory.
I believe category theory is necessary in understanding algebraic geometry, and a very basic introduction to category theory with the objective of understanding how topological quantum field theories (TQFTs), their observables and their boundary conditions form categories can be found in Ketan Vyas' thesis 'Topics in topological and holomorphic quantum field theory' (http://thesis.library.caltech.edu/5894/). It should be easily readable by those familiar with TQFTs.


TITLE: Bounding the minimal maximum norm of a solution of a linear system.
QUESTION [5 upvotes]: I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear system $Ax = b$. What I want to know, is some general upper bound on the minimum $\ell_{\infty}$ norm of the solution.
In other words: I want to know that I can find a solution of the system with a norm not greater than...
Bound should be in terms of: the number of variables, the number of equations (we can assume that this number is much smaller if it helps) and of course of $A$ and $b$.
For example, if we have $n$ variables and one equation: $a_1x_1 + a_2x_2 + \ldots + a_nx_n = b$, then the minimal norm is given by $\frac{|b|}{|a_1|+|a_2| + \ldots + |a_n|}$. I suspect that for an arbitrary number of equations, smaller than $n$, there should exist some nice estimation of the norm. I have searched in the web for such a result on my own, but have only managed to find a papers concerning the problem of finding such solution algoritmically and no explicit bounds were mentioned.

REPLY [2 votes]: The question is easier with $L^2$ norm instead of $L^\infty$ norm -- obviously the former upper bounds the latter. Now, you are trying to minimize the norm of $x$ subject to the constraints that $x \cdot v_i = b_i.$ Using Lagrange multipliers, we see that we need $x = \sum \lambda_j v_j.$ Plugging into the original equation, we get $\mathbf{\lambda} V^t V = b.$ Since $V^t V$ is symmetric, it can be (orthogonally) diagonalized) to the matrix of the squares of singular values of $V.$ Let $\sigma_0$ be the smallest such singular value. This upper bounds all $\lambda$s by $b_{\max}/\sigma_0^2,$ and then we get an upper bound on the $L^2$ norm of $x$ in the obvious way.