\section{Dynamical systems in number theory}
In the last decades dynamical systems became very important for the
development of modern number theory. The present paper focuses on
Furstenberg's refinements of Poincaré's recurrence theorem and
applications of these ideas to Diophantine problems.

A (measure-theoretic) dynamical system is formally given as a
quadruple $(X,\mathfrak{B}, \mu, T)$, where $(X,\mathfrak{B},\mu)$
is a probability space with $\sigma$-algebra $\mathfrak{B}$ of
measurable sets and $\mu$ a probability measure; $T\colon
X\rightarrow X$ is a measure-preserving transformation on this
space, \textit{i.e.} $\mu(T^{-1}A)= \mu(A)$ for all measurable
sets $A\in\mathfrak{B}$. In the theory of dynamical systems,
properties of the iterations of the transformation $T$ are of
particular interest. For this purpose we only consider invertible
transformations and call such dynamical systems invertible.

The first property, we consider, originates from Poncaré's famous
recurrence theorem (see Theorem 1.4 of
\cite{walters1982:introduction_to_ergodic} or Theorem 2.11 of
\cite{einsiedler_ward2011:ergodic_theory_with}) saying that
starting from a set $A$ of positive measure $\mu(A)>0$ and
iterating $T$ yields infinitely many returns to $A$. More
generally, we call a subset $\mathcal{R}\subset\mathbb{N}$ of the
positive integers a set of recurrence if for all invertible
dynamical systems and all measurable sets $A$ of positive measure
$\mu (A)>0$ there exists $n\in\mathcal{R}$ such that $\mu(A\cap
T^{-n}A)>0$. Then Poincaré's recurrence theorem means that $\N$
is a set of recurrence.

A second important theorem for dynamical systems is Birkhoff's ergodic
theorem (see Theorem 1.14 of
\cite{walters1982:introduction_to_ergodic} or Theorem 2.30 of
\cite{einsiedler_ward2011:ergodic_theory_with}). We call $T$ ergodic
if the only invariant sets under $T$ are sets of measure $0$ or of
measure $1$, \textit{i.e.} $T^{-1}A=A$ implies $\mu(A)=0$ or
$\mu(A)=1$. Then Birkhoff's ergodic theorem connects average in time
with average in space, \textit{i.e.}
\[\lim_{N\rightarrow\infty}\frac{1}{N}\sum^{N-1}_{n=0}f \circ
T^{n}(x)= \int_{X}f(x)d\mu(x)\]
for all $f\in L^{1}(X,\mu)$ and $\mu$-almost all $x \in X$.

Let us explain an important application of this theorem to number
theory. For $q\geq2$ a positive integer, consider
$T\colon[0,1)\rightarrow [0,1)$ defined by $T(x)=\{qx\},$ where
$\{t\}=t-\lfloor t\rfloor$ denotes the fractional part of $t$.
If $x\in\R$ is given by its $q$-ary digit expansion
$x=\lfloor x\rfloor+ \sum^{\infty}_{j=1}a_{j}(x)q^{-j}$, then the
digits $a_{j}(x)$ can be computed by iterating this transformation
$T$: $a_{j}(x)=i$ if $T^{j-1}x\in\left[\tfrac iq,\frac{i+1}q\right)$
with $i\in\{0,1,\ldots,q-1\}$. Moreover, since $a_j(Tx)=a_{j+1}(x)$
for $j\geq1$ the transformation $T$ can be seen as a left shift of the
expansion.

Now we call a real number $x$ simply normal in base $q$ if
\[\lim_{\mathbb{N}\rightarrow\infty}\frac{1}{N}\#\{j\leq N\colon a_{j}=d\}= \frac{1}{q}\]
for all $d=0, \ldots, q-1$, \textit{i.e.} all digits $d$ appear
asymptotically with equal frequencies $1/q.$ A number $x$ is
called $q$-normal if it is simply normal with respect to all bases
$q, q^{2},q^{3},\ldots$. This is equivalent to the fact that the
sequence $(\{q^{n}x\})_{n\in\mathbb{N}}$ is uniformly distributed
modulo $1$ (for short: u.d.~mod $1$), which also means that all
blocks $d_{1},d_{2},\ldots, d_{L}$ of subsequent digits appear in
the expansion of $x$ asymptotically with the same frequency
$q^{-L}$ (\textit{cf.} \cite{bugeaud2012:distribution_modulo_one,
  drmota_tichy1997:sequences_discrepancies_and,
  kuipers_niederreiter1974:uniform_distribution_sequences}). For
completeness, let us give here one possible definition of u.d.
sequences $(x_{n})$: a sequence of real numbers $x_{n}$ is called
u.d.~mod $1$ if for all continuous functions $f: [0,1]\rightarrow
\mathbb{R}$

\begin{equation}\label{ud}
\lim_{N\rightarrow\infty}\frac{1}{N}\sum^{N}_{n=1}f(x_{n})=
\int^{1}_{0}f(x)dx.
\end{equation}

Note, that by Weyl's criterion the class of continuous functions
can be replaced by trigonometric functions $e(hx)=e^{2\pi ihx}$,
$h\in \mathbb{N}$ or by characteristic functions $1_{I}(x)$ of
intervals $I=[a,b)$. Applying Birkhoff's ergodic theorem, shows
that Lebesgue almost all real numbers are $q$-normal in any base
$q\geq 2$. Defining a real number to be absolutely normal if it is
$q$-normal for all bases $q\geq 2$, this immediately yields that
almost all real numbers are absolutely normal.

In particular, this shows the existence of absolutely normal
numbers. However, it is a different story to find constructions of
(absolutely) normal numbers. It is a well-known difficult open problem
to show that important numbers like $\sqrt{2}$, $\ln 2$, $e$, $\pi$
etc. are simply normal with respect to some given base $q\geq 2$. A
much easier task is to give constructions of $q$-normal numbers for
fixed base $q$. Champernowne
\cite{champernowne1933:construction_decimals_normal} proved that

$$0.1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\ldots$$

is normal to base $10$ and later this type of constructions was
analysed in detail. So, for instance, for arbitrary base $q\geq 2$

\[0.\langle\lfloor g(1)\rfloor\rangle_{q}\; \langle\lfloor g(2)\rfloor\rangle_{q}\ldots\]

is $q$-normal, where $g(x)$ is a non-constant polynomial with real
coefficients and the $q-$normal number is constructed by concatenating
the $q-$ary digit expansions $\langle\lfloor g(n)\rfloor\rangle_{q}$
of the integer parts of the values $g(n)$ for $n=1,2,\ldots$. These
constructions were extended to more general classes of functions $g$
(replacing the polynomials) (see
\cite{nakai_shiokawa1992:discrepancy_estimates_class,
  nakai_shiokawa1990:class_normal_numbers,
  madritsch_thuswaldner_tichy2008:normality_numbers_generated,
  madritsch_tichy2013:construction_normal_numbers,
  davenport_erdoes1952:note_on_normal,
  schiffer1986:discrepancy_normal_numbers}) and the concatenation of
$\langle[g(p)]\rangle_{q}$ along prime numbers instead of the positive
integers (see \cite{nakai_shiokawa1997:normality_numbers_generated,
  madritsch2014:construction_normal_numbers,
  copeland_erdoes1946:note_on_normal,
  madritsch_tichy2013:construction_normal_numbers}).

All such constructions depend on the choice of the base number
$q\geq 2$, and thus they are not suitable for constructing
absolutely normal numbers. A first attempt to construct absolutely
normal numbers is due to Sierpinski
\cite{sierpinski1917:demonstration_elementaire_du}. However,
Turing \cite{turing1992:note_on_normal} observed that Sierpinski's
``construction'' does not yield a computable number, thus it is
not based on a recursive algorithm. Furthermore, Turing gave an
algorithm for a construction of an absolutely normal number. This
algorithm is very slow and, in particular, not polynomially in
time. It is very remarkable that Becher \text{et al.}
\cite{becher_heiber_slaman2013:polynomial_time_algorithm}
established a polynomial time algorithm for the construction of
absolutely normal numbers. However, there remain various questions
concerning the analysis of these algorithms. The discrepancy of
the corresponding sequences is not studied and the order of
convergence of the expansion is very slow and should be
investigated in detail. Furthermore, digital expansions with
respect to linear recurring base sequences seam appropriate to be
included in the study of absolute normality from a computational
point of view.

Let us now return to Poincaré's recurrence theorem which shortly states
that the set $\mathbb{N}$ of positive integers is a recurrence set. In
the 1960s various stronger concepts were introduced:
\begin{enumerate}[(i)]
\item $\mathcal{R}\subseteq \mathbb{N}$ is called a nice recurrence
  set if for all invertible dynamical systems and all measurable sets
  $A$ of positive measure $\mu(A)>0$ and all $\varepsilon > 0,$ there
  exist infinitely many $n\in \mathcal{R}$ such that $$\mu(A\cup
  T^{-n}A)>\mu(A)^{2}- \varepsilon.$$
\item $\mathcal{H}\subseteq\mathbb{N}$ is called a van der Corput set
  (for short: vdC set) if for all $h\in\mathcal{H}$ the following
  implications holds:
  \[(x_{n+h}-x_{n})_{n\in\mathbb{N}}\; \text{is u.d.~mod $1$}\Longrightarrow
  (x_{n})_{n\in\mathbb{N}}\; \text{is u.d.~mod $1$.}\]
\end{enumerate}

Clearly, any nice recurrence set is a recurrence set. By van der
Corput's difference theorem (see
\cite{kuipers_niederreiter1974:uniform_distribution_sequences,
  drmota_tichy1997:sequences_discrepancies_and}) the set
$\mathcal{H}= \N$ of positive integers is a vdC set. Kamae and
Mend\`es-France \cite{kamae_mendes1978:van_der_corputs} proved that
any vdC set is a nice recurrence set. Ruzsa
\cite{ruzsa1984:connections_between_uniform} conjectured that any
recurrence set is also vdC. An important tool in the analysis of
recurrence sets is their equivalence with intersective (or difference)
sets established by Bertrand-Mathis
\cite{bertrand-mathis1986:ensembles_intersectifs_et}. We call a set
$\mathcal{I}$ intersective if for each subset $E\subseteq \mathbb{N}$
of positive (upper) density, there exists $n\in\mathcal{I}$ such that
$n=x-y$ for some $x,y\in E$. Here the upper density of $E$ is defined
as usual by
\[\overline{d}(E)=\limsup_{N\to\infty}\frac{\#(E\cap[1,N])}{N}.\]
Bourgain \cite{bourgain1987:ruzsas_problem_on} gave an example of an
intersective set which is not a vdC set, hence contradicting the above
mentioned conjecture of Ruzsa.

Furstenberg \cite{furstenberg1977:ergodic_behavior_diagonal}
proved that the values $g(n)$ of a polynomial $g\in \mathbb{Z}[x]$
with $g(0)=0$ form an intersective set and later it was shown by
Kamae and Mend\`es-France \cite{kamae_mendes1978:van_der_corputs}
that this is a vdC set, too. It is also known, that for fixed
$h\in \mathbb{Z}$ the set of shifted primes $\{p\pm h\colon
p\text{ prime}\}$ is a vdC set if and only if $h=\pm 1.$
(\cite[Corollary 10]{montgomery1994:ten_lectures_on}). This leads
to interesting applications to additive number theory, for
instance to new proofs and variants of theorems of S\'ark\H{o}zy
\cite{sarkozy1978:difference_sets_sequences1,
  sarkoezy1978:difference_sets_sequences3,
  sarkoezy1978:difference_sets_sequences2}. A general result
concerning intersective sets related to polynomials along primes is
due to Nair \cite{nair1992:certain_solutions_diophantine}.

In the present paper we want to extend the concept of recurrence sets,
nice recurrence sets and vdC sets to subsets of $\Z^{k},$ following
the program of Bergelson and Lesigne
\cite{bergelson_lesigne2008:van_der_corput} and our earlier
paper~\cite{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}. In
section 2 we summarize basic facts concerning these concepts,
including general relations between them and counter examples. Section
3 is devoted to a sufficient condition for establishing the vdC
property. In the final section 4 we collect various examples and give
some new applications.

\section{Van der Corput sets}

In this section we provide various equivalent definitions of van der
Corput sets in $\Z^k$. In particular, we give four different
definitions, which are $k$-dimensional variants of the one dimensional
definitions, whose equivalence is due to Ruzsa
\cite{ruzsa1984:connections_between_uniform}. These generalizations
were established by Bergelson and Lesigne
\cite{bergelson_lesigne2008:van_der_corput}.  Then we present a set,
which is not a vdC set in order to give some insight into the
structure of vdC sets. Finally, we define the
higher-dimensional variant of nice recurrence sets.

\subsection{Characterization via uniform distribution}

Similarly to above we first define a van der Corput set (vdC set for
short) in $\Z^k$ via
uniform distribution.
\begin{defi}
  A subset $\mathcal{H}\subset\Z^k\setminus\{0\}$ is a vdC set if any
  family $(x_{\mathbf{n}})_{\mathbf{n}\in\N^k}$ of real numbers is u.d.~mod $1$
  provided that it has
  the property that for all $\mathbf{h}\in \mathcal{H}$ the family
  $(x_{\mathbf{n}+\mathbf{h}}-x_{\mathbf{n}})_{\mathbf{n}\in\N^k}$ is
  u.d.~mod $1$.

  Here the property of u.d.~mod $1$ for the multi-indexed family
  $(x_{\mathbf{n}})_{\mathbf{n}\in\N^k}$ is defined via a natural
  extension of \ref{ud}:
\end{defi}

\begin{equation}\label{ud1}
\lim_{N_1,N_2,\ldots,N_k\to+\infty}\frac1{N_1N_2\cdots N_k}
  \sum_{0\leq\mathbf{n}<(N_1,N_2,\ldots,N_k)}f(x_{\mathbf n})=
\int^{1}_{0}f(x)dx
\end{equation}

for all continuous functions $f:[0,1]\rightarrow\mathbb{R}.$ Here
in the limit $N_1, N_2,\ldots, N_k$ are tending to infinity
independently and $<$ is defined componentwise.

Using the $k$-dimensional variant of van der Corput's inequality
we could equivalently define a vdC set as follows:

\begin{defi}
  A subset $\mathcal{H}\subset\Z^k\setminus\{0\}$ is a van der Corput
  set if for any family $(u_\mathbf n)_{n\in\Z^k}$ of complex numbers of
  modulus $1$ such that \[\forall\mathbf{h}\in \mathcal{H},\quad
  \lim_{N_1,N_2,\ldots,N_k\to+\infty}\frac1{N_1N_2\cdots N_k}
  \sum_{0\leq\mathbf{n}<(N_1,N_2,\ldots,N_k)}u_{\mathbf{n}+\mathbf{h}}\overline{u_{\mathbf{n}}}=0\]
the relation
\[\lim_{N_1,N_2,\ldots,N_k\to+\infty}\frac1{N_1N_2\cdots
    N_k}
  \sum_{0\leq\mathbf{n}<(N_1,N_2,\ldots,N_k)}u_{\mathbf{n}}=0\]
  holds.
\end{defi}

\subsection{Trigonometric polynomials and spectral characterization}

The first two definitions are not very useful for proving or
disproving that a set $\mathcal{H}$ is a vdC set. Similar to the one
dimensional case the following spectral characterization involving
trigonometric polynomials is a better tool.

\begin{thm}[{\cite[Proposition
    1.18]{bergelson_lesigne2008:van_der_corput}}]
  A subset $\mathcal{H}\subset\Z^k\setminus\{0\}$ is a van der Corput set if and
  only if for all $\varepsilon>0$, there exists a real trigonometric
  polynomial $P$ on the $k$-torus $\T^k$ whose spectrum is contained
  in $\mathcal{H}$ and which satisfies $P(0)=1$, $P\geq-\varepsilon$.
\end{thm}

The set of polynomials fulfilling the last theorem for a given
$\varepsilon$ forms a convex set. Moreover the conditions may be
interpreted as some infimum. Therefore we might expect some dual
problem, which is actually provided by the following theorem. For
details see Bergelson and Lesigne
\cite{bergelson_lesigne2008:van_der_corput} or Montgomery
\cite{montgomery1994:ten_lectures_on}.

\begin{thm}[{\cite[Theorem
    1.8]{bergelson_lesigne2008:van_der_corput}}]
  Let $\mathcal{H}\subset\Z^k\setminus\{0\}$. Then $\mathcal{H}$ is a
  van der Corput set if and only if for any positive measure $\sigma$
  on the $k$-torus $\T^k$ such that, for all $\mathbf{h}\in \mathcal{H}$,
  $\widehat{\sigma}(\mathbf{h})=0$, this implies
  $\sigma(\{(0,0,\ldots,0)\})=0$.
\end{thm}

\subsection{Examples}

The structure of vdC sets is better understood by first giving a
counter example. The following lemma shows to be very useful in
the construction of counter examples.

\begin{lem}\label{mt:infinite_intersection}
Let $\mathcal{H}\subset\N$. If there exists $q\in\N$ such that the
set $\mathcal{H}\cap q\N$ is finite, then the set $\mathcal{H}$ is
not a vdC set.
\end{lem}

\begin{proof}
The proof is a combination of the following two observations of Ruzsa
\cite{ruzsa1984:connections_between_uniform} (see Theorem 2
and Corollary 3 of \cite{montgomery1994:ten_lectures_on}):
\begin{enumerate}
\item Let $m\in\N$. The sets $\{1,\ldots,m\}$ and $\{n\in\N\colon
  m\nmid n\}$ are both not vdC sets.
\item Let $\mathcal{H}=\mathcal{H}_1\cup\mathcal{H}_2\subset\N$. If
  $\mathcal{H}$ is a vdC set, then $\mathcal{H}_1$ or $\mathcal{H}_2$
  also has to be a vdC set.
\end{enumerate}
Suppose there exists a $q\in\N$ such that $\mathcal{H}\cap q\N$ is
finite. Then we may split $\mathcal{H}$ into the sets
$\mathcal{H}\cap q\N$ and $\mathcal{H}\setminus q\N$. The first
one is finite and the second one contains no multiples of $q$.
Therefore both are not vdC sets and hence $\mathcal{H}$ is not a
vdC set.
\end{proof}

The first counter example deals with arithmetic progressions.
\begin{lem}\label{lem:arithmetic_prog}
  Let $a,b\in\N$. If the set $\{an+b\colon n\in\N\}$ is a vdC set,
  then $a\mid b$.
\end{lem}

\begin{proof}
Let $b\in\N$ and $\mathcal{H}=\{an+b\colon n\in\N\}$ be a
vdC set. Then by Lemma \ref{mt:infinite_intersection} we must have
\[an+b\equiv b\equiv 0\bmod a\quad\text{infinitely often.}\]
This implies that $a\mid b$.
\end{proof}

The sufficiency (and also the necessity) of the requirement $a\mid b$
follows from the following result of Kamae and Mend\`es-France
\cite{kamae_mendes1978:van_der_corputs} (\textit{cf.} Corollary 9 of
\cite{montgomery1994:ten_lectures_on}).
\begin{lem}
Let $P(z)\in\Z[z]$ and suppose that $P(z)\to+\infty$ as
$z\to+\infty$. Then $\mathcal{H}=\{P(n)>0\colon n\in\N\}$ is a vdC set
if and only if for every positive integer $q$ the congruence
$P(z)\equiv 0\pmod q$ has a root.
\end{lem}

Now we want to establish a similar result for sets of the form
$\{ap+b\colon p\text{ prime}\}$. In this case the following result is
due to Bergelson and Lesigne
\cite{bergelson_lesigne2008:van_der_corput} which is a generalization of
the case $f(x)=x$ due to Kamae and Mend\`es-France
\cite{kamae_mendes1978:van_der_corputs}.
\begin{lem}[{\cite[Proposition
    1.22]{bergelson_lesigne2008:van_der_corput}}]\label{bl:prop1.22}
Let $f$ be a (non zero) polynomial with integer coefficients and
zero constant term. Then the sets $\{f(p-1)\colon
p\in\mathbb{P}\}$ and $\{f(p+1)\colon p\in\mathbb{P}\}$ are vdC
sets in $\Z$.
\end{lem}

We show the converse direction.
\begin{lem}
Let $a$ and $b$ be non-zero integers. Then the set $\{ap+b\colon
p\in\mathbb{P}\}$ is a vdC set if and only if $\lvert a\rvert=\lvert
b\rvert$, \textit{i.e.} $ap+b=a(p\pm 1)$.
\end{lem}

\begin{proof}
It is clear from Lemma \ref{bl:prop1.22} that $\{ap+b\colon
p\in\mathbb{P}\}$ is a vdC set if $\lvert a\rvert=\lvert b\rvert$.

On the contrary a combination of Lemma \ref{mt:infinite_intersection}
and Lemma \ref{lem:arithmetic_prog} yields that $a\mid b$. Now we
consider the sequence modulo $b$. Then by Lemma
\ref{mt:infinite_intersection} we get that
\[ap+b\equiv ap\equiv 0\bmod b\quad\text{infinitely often.}\] Since $(p,b)>1$ only
holds for finitely many primes $p$ we must have $b\mid a$. Combining
these two requirements yields $\lvert a\rvert=\lvert b\rvert$.
\end{proof}

\section{A sufficient condition}

In this section we want to formulate a general sufficient
condition which provides us with a tool to show for plenty of
different examples that they generate a vdC set. This is a
generalization of the conditions of Kamae and Mend\`es-France
\cite{kamae_mendes1978:van_der_corputs} and Bergelson and Lesigne
\cite{bergelson_lesigne2008:van_der_corput}. Before stating the
condition we need an auxiliary lemma.

\begin{lem}[{\cite[Corollary
    1.15]{bergelson_lesigne2008:van_der_corput}}]
\label{bl:lem_linear_algebra}
  Let $d$ and $e$ be positive integers, and let $L$ be a linear
  transformation from $\Z^d$ into $\Z^e$ (represented by an $e\times
  d$ matrix with integer entries). Then the following assertions hold:
\begin{enumerate}
\item If $D$ is a vdC set in $\Z^d$ and if $0\not\in L(D)$, then
  $L(D)$ is a vdC set in $\Z^e$.
\item Let $D\in\Z^d$. If the linear map $L$ is one-to-one, and if
  $L(D)$ is a vdC set in $\Z^e$, then $D$ is a vdC set in $\Z^d$.
\end{enumerate}
\end{lem}

Our main tool is the following general result. Applications
are given in the next section.

\begin{prop}\label{mt:sufficient_condition}
Let $g_1,\ldots,g_k\colon\N\to\Z$ be arithmetic functions. Suppose
that $g_{i_1},\ldots,g_{i_m}$ is a basis of the $\Q$-vector space
$\mathrm{span}(g_1,\ldots,g_k)$. For each $q\in\N$, we introduce
\[D_q:=\left\{(g_{i_1}(n),\ldots,g_{i_m}(n)\colon n\in\N\text{ and $q!\mid
  g_{i_j}(n)$ for all $j=1,\ldots,m$}\right\}.\]
Suppose further that, for every $q$, there exists a sequence
$(h^{(q)}_n)_{n\in\N}$ in $D_q$ such that, for all
$\mathbf{x}=(x_1,\ldots,x_m)\in\R^m\setminus\Q^m$, the sequence
$(h^{(q)}_n\cdot\mathbf{x})_{n\in\N}$ is uniformly distributed mod
$1$. Then
\[\widetilde{D}:=\{(g_1(n),\ldots,g_k(n))\colon n\in\N\}\in\Z^k\]
is a vdC set.
\end{prop}

\begin{proof}
We first show that the set
\[D:=\{(g_{i_1}(n),\ldots,g_{i_m}(n))\colon n\in\N\}\]
is a vdC set in $\Z^m$. For $q,N\in\N$ we define a family of trigonometric
polynomials
\[P_{q,N}:=\frac1N\sum_{n=1}^Ne\left(h_n^{(q)}\cdot\mathbf{x}\right).\]
By hypothesis, $\lim_{N\to\infty}P_{q,N}(x)=0$ for
$x\not\in\Q^m$. For fixed $q$ there exists a subsequence $(P_{q,N'})$
which converges pointwise to a function $g_q$. Since $g_q(x)=1$
(for $x\in\Q^m$ and $q$ sufficiently large) and $g_q(x)=0$ (for
$x\not\in\Q^m$), the sequence $(g_q)$ is pointwise convergent to the indicator
function of $\Q^m$. For a positive measure $\sigma$ on the
$m$-dimensional torus with vanishing Fourier transform
$\widehat{\sigma}$ on $D$, we have $\int P_{q,N}\mathrm{d}\sigma=0$
for all $q,N$. Thus $\sigma(\Q^m)=0$ follows from the dominating
convergence theorem, obviously $\sigma(\{0,0,\ldots,0\})=0$, and thus $D$ is a vdC
set.

In order to prove that $\widetilde{D}$ is a vdC set we apply Lemma
\ref{bl:lem_linear_algebra} twice. Since $g_{i_1},\ldots,g_{i_m}$ is a
base of $\mathrm{span}(g_1,\ldots,g_k)$, we can write each $g_j$ as a
linear combination (with rational coefficients) of
$g_{i_1},\ldots,g_{i_m}$. Multiplying with the common denominator of
the coefficients yields
\[a_jg_j=b_{j,1}g_{i_1}+\cdots+b_{j,m}g_{i_m}\]
for $j=1,\ldots,k$ and certain $a_j,b_{j,\ell}\in\Z$. Considering the
transformation $L\colon\Z^m\to\Z^k$ given by the matrix $(b_{j,\ell})$
and applying part (1) of Lemma \ref{bl:lem_linear_algebra} shows that
\[\{(a_1g_1(n),\ldots,a_kg_k(n))\colon n\in\N\}\]
is a vdC set for certain integers $a_1,\ldots,a_k$.

Now consider the transformation $\widetilde{L}\colon\Z^k\to\Z^k$ given by
the $k\times k$ diagonal matrix with entries $a_1,\ldots,a_k$ in the
diagonal. Then by part (2) of Lemma \ref{bl:lem_linear_algebra} also
$\widetilde{D}$ is a vdC set and the proposition is proved.
\end{proof}

\section{Various examples and applications to additive problems}

In this section we consider multidimensional variants of prime
powers, entire functions and $x^\alpha\log^\beta x$ sequences.

\subsection{Prime powers}

In a recent paper the authors together with Bergelson, Kolesnik and
Son
\cite{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}
consider sets of the form
\[\{(\alpha_1(p_n\pm1)^{\theta_1},\ldots,\alpha_k(p_n\pm1)^{\theta_k})\colon
n\in\N\},\] where $\alpha_i,\beta_i\in\R$ and $p_n\in\mathcal{P} $
runs over all prime numbers. These sets are vdC, however, we
missed the treatment of a special case in the proof. In
particular, if for some $i\neq j$ the exponents satisfy
$\theta_i=\theta_j=:\theta$, then the vector
$(p_n^{\theta},p_n^{\theta})$ is not uniformly distributed mod 1.

Here we close this gap.
\begin{thm}
If $\alpha_i$ are positive integers and $\beta_i$ are positive and non-integers, then
$$ D_1 = \{ \left(  (p-1)^{\alpha_1}, \cdots , (p-1)^{\alpha_k}, [(p-1)^{\beta_1}], \cdots , [(p-1)^{\beta_\ell}]  \right) | \, p \in\mathcal{P} \},$$
and
$$ D_2 = \{ \left(  (p+1)^{\alpha_1}, \cdots , (p+1)^{\alpha_k}, [(p+1)^{\beta_1}], \cdots , [(p+1)^{\beta_\ell}]  \right) | \, p \in \mathcal{P} \}$$
are vdC sets in $\Z^{k+\ell}$.
\end{thm}

\begin{proof}
Since $x^{\theta_1}$ and $x^{\theta_2}$ are $\Q$-linear dependent
for all $x\in\Z$ if and only if $\theta_1=\theta_2$, an
application of Proposition \ref{mt:sufficient_condition} yields
that it suffices to consider the case where all exponents are
different. However, this follows by the same arguments as in the
proof of Theorem 4.1 in
\cite{bergelson_kolesnik_madritsch+2014:uniform_distribution_prime}.
\end{proof}

\subsection{Entire functions}

In this section we consider entire functions of bounded logarithmic
order. We fix a transcendental entire function $f$ and denote by
$S(r):=\max_{\lvert z\rvert\leq r}\lvert f(z)\rvert$. Then we call
$\lambda$ the logarithmic order of $f$ if
\[\limsup_{r\to\infty}\frac{\log S(r)}{\log r}=\lambda.\]

The central tool is the following result of Baker
\cite{baker1984:entire_functions_and}.

\begin{thm}[{\cite[Theorem 2]{baker1984:entire_functions_and}}]\label{baker:uniform_distribution_of_entire_functions}
Let $f$ be a transcendental entire function of logarithmic order
$1<\lambda<\frac43$. Then the sequence
\[\left(f(p_n)\right)_{n\geq1}\]
is uniformly distributed mod $1$.
\end{thm}

Our second example of a class of vdC sets is the following.
\begin{thm}\label{thm:entire_functions_vdC}
Let $f_1,\ldots,f_k$ be entire functions with distinct logarithmic orders
$1<\lambda_1,\lambda_2,\ldots,\lambda_k<\frac43$, respectively.
Then the set
\[D:=\{(\lfloor f_1(p_n)\rfloor,\ldots,\lfloor f_k(p_n)\rfloor)\colon
n\in\N\}\]
is a vdC set.
\end{thm}

\begin{proof}
  We enumerate $D=(\mathbf{d}_n)_{n\geq1}$, where
  \[\mathbf{d}_n:=\left(\lfloor f_1(p_n)\rfloor,\ldots,\lfloor f_k(p_n)\rfloor\right).\]

  First we show, that for every $q\in\N$ the set
  \[D^{(q)}:=\{(d_1,\ldots,d_k)\in D\colon q\mid d_i\}\] has positive
  relative density in $D$. We note that if
  $0\leq\left\{\frac{f_i(p_n)}{q}\right\}<\frac1q$ for $1\leq i\leq k$,
  then $\mathbf{d}_n\in D^{(q)}$. By Theorem
  \ref{baker:uniform_distribution_of_entire_functions} the
  sequence \[\left(\left(\frac{f_1(p_n)}{q},\ldots,\frac{f_k(p_n)}q\right)\right)_{n\geq1}\]
  is uniformly distributed and thus $D^{(q)}$ has positive density in
  $D$.

  For each $q\in\N$ we enumerate the elements of
  $D^{(q!)}=(\mathbf{d}^{(q!)}_n)_{n\geq1}$, such that
  $\lvert\mathbf{d}^{(q!)}_n\rvert$ is increasing. Since the
  logarithmic orders are distinct we immediately get that the
  functions $f_i$ are $\Q$-linearly independent. Thus by Proposition
  \ref{mt:sufficient_condition} it is sufficient to show that for all
  $q\in\N$ and all $\mathbf{x}=(x_1,\ldots,x_k)\in\R^k\setminus\Q^k$
  the sequence $(\mathbf{d}^{(q!)}_n\cdot\mathbf{x})_{n\geq1}$ is
  u.d. mod 1.

  Using the orthogonality relations for additive characters we get for
  any non-zero integer $h$, that
  \begin{multline*}
  \frac1{\lvert\{n\leq N\colon\mathbf{d}_n\in D^{(q!)}\}\rvert}
    \sum_{n\leq N}e\left(h\left(d^{(q!)}_n\cdot\mathbf{x}\right)\right)\\
  =\frac1{\lvert\{n\leq N\colon\mathbf{d}_n\in D^{(q!)}\}\rvert}
    \frac1{(q!)^k}\sum_{j_1=1}^{q!}\cdots\sum_{j_k=1}^{q!}
    \frac1N\sum_{n\leq
    N}e\left(d_n\cdot\left(h\mathbf{x}+\left(\frac{j_1}{q!},\ldots,\frac{j_k}{q!}\right)\right)\right).
  \end{multline*}
  The innermost sum is of the form
  \[\sum_{n\leq N}e(g(p_n)),\]
  with $g(x)=\sum_{i=1}^k\alpha_i\lfloor f_i(x)\rfloor$ for a certain
  $(\alpha_1,\ldots,\alpha_k)\in\R^k\setminus\Q^k$.

  By relabeling the terms we may suppose that there exists an $\ell$
  such that $\alpha_1,\ldots,\alpha_\ell\not\in\Q$ and
  $\alpha_{\ell+1},\ldots,\alpha_k\in\Q$. Furthermore we may write
  $\alpha_j=\frac{a_j}q$ for $\ell+1\leq j\leq m$. Then
  \[e(g(p_n))=e\left(\sum_{i=1}^k\alpha_k\lfloor
    f_i(p_n)\rfloor\right)= \prod_{j=1}^\ell
  s_j(\alpha_jf_j(p_n),f_j(p_n))\prod_{j=\ell+1}^kt_j(\lfloor
  f_j(p_n)\rfloor),\] where $s_j(x,y)=e(x-\{y\}\alpha_j)$ ($1\leq
  j\leq\ell$) and $t_j(z)=e\left(a_j\frac zq\right)$ ($\ell+1\leq
  j\leq k$).

  Since $s_j(x,y)$ is Riemann-integrable on $\T^2$ for
  $j=1,\ldots,\ell$ \; and $t_j(z)$ is continuous on $\Z_q=\Z/q\Z$, the
  function $\prod_{j=1}^\ell s_j\prod_{j=\ell+1}^kt_j$ is
  Riemann-integrable on $\T^{2\ell}\times\Z_q^{k-\ell}$.

  Now an application of Theorem
  \ref{baker:uniform_distribution_of_entire_functions} yields that for
  any $u\in\N$ the sequence
\[\left(\alpha_1f_1(p_n),f_1(p_n),\ldots,\alpha_\ell
  f_\ell(p_n),f_\ell(p_n),\frac{f_{\ell+1}(p_n)}u,\ldots,\frac{f_k(p_n)}u\right)_{n\geq1}\]
is u.d. in $\T^{2\ell}\times\T^{k-\ell}$. Since $\lfloor
x\rfloor\equiv a\pmod q$ is equivalent to $\frac xq\in[\frac
aq,\frac{a+1}q]$, we deduce that
\[\left(\alpha_1f_1(p_n),f_1(p_n),\ldots,\alpha_\ell
  f_\ell(p_n),f_\ell(p_n),\lfloor
  f_{\ell+1}(p_n)\rfloor,\ldots,\lfloor
  f_k(p_n)\rfloor\right)_{n\geq1}\]
is u.d. in $\T^{2\ell}\times\Z_q^{k-\ell}$, and Weyl's criterion
implies that
\[\lim_{N\to\infty}\frac1N\sum_{n\leq
  N}e\left(\sum_{i=1}^k\alpha_i\lfloor f_i(p_n)\rfloor\right)=0,\]
proving the theorem.
\end{proof}

\subsection{Functions of the form $x^\alpha\log^\beta x$}

In the one-dimensional case Boshernitzan \textit{et al.}
\cite{boshernitzan_kolesnik_quas+2005:ergodic_averaging_sequences}
showed, among other things, that these sets are vdC sets. Our aim
is to show an extended result for the $k$-dimensional case.
Therefore we use the following general criterion, which is a
combination of Fejer's theorem and van der Corput's difference
theorem.
\begin{thm}[{\cite[Theorem 3.5]{kuipers_niederreiter1974:uniform_distribution_sequences}}]
  Let $f(x)$ be a function defined for $x > 1$ that is $k$-times
  differentiable for $x > x_0$. If $f^{(k)}(x)$ tends monotonically to
  $0$ as $x\to\infty$ and if \; $\lim_{x\to\infty}x\lvert
  f^{(k)}(x)\rvert=\infty$, then the sequence $(f(n))_{n\geq1}$ is
  u.d. mod 1.
\end{thm}

Applying this theorem we get the following
\begin{cor}\label{cor:uniform_distribution_of_n_log_powers}
  Let $\alpha\neq0$ and
  \begin{itemize}
  \item either $\sigma>0$ not an integer and $\tau\in\R$ arbitrary
  \item or $\sigma>0$ an integer and $\tau\in\R\setminus[0,1]$.
  \end{itemize}
  Then the sequence $(\alpha n^\sigma\log^\tau n)_{n\geq2}$ is
  u.d. mod 1.
\end{cor}

Our third example is the following class of vdC sets.
\begin{thm}
Let $\alpha_1,\ldots,\alpha_k>0$ and $\beta_1,\ldots,\beta_k\in\R$,
such that $\beta_i\not\in[0,1]$ whenever $\alpha_i\in\Z$ for
$i=1,\ldots,k$. Then the set
\[D:=\{(\lfloor n^{\alpha_1}\log^{\beta_1}n\rfloor,\ldots,\lfloor
n^{\alpha_k}\log^{\beta_k}n\rfloor)\colon n\in\N\}\]
is a vdC set.
\end{thm}

\begin{proof}
Following the same arguments as is the proof of Theorem
\ref{thm:entire_functions_vdC} and replacing the uniform distribution
result for entire functions (Theorem \ref{thm:entire_functions_vdC})
by the corresponding result for $n^\alpha\log^\beta n$ sequences (Corollary
\ref{cor:uniform_distribution_of_n_log_powers}) yields the proof.
\end{proof}

\section*{Acknowledgment}

This research work was done when the first author was a visiting
lecturer at the Department of Analysis and Computational Number Theory
at Graz University of Technology. The author thanks the institution
for its hospitality.

The second author acknowledges support of the project F 5510-N26
within the special research area ``Quasi Monte-Carlo Methods and
applications'' founded by the Austrian Science Fund.

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\section{Introduction}
Observational evidences for multiple stellar populations in globular
clusters (GCs) challenge the traditional view of GCs hosting simple
stellar populations, which are formed in one generation from a cloud
of uniform composition (single-generation cluster). One important
evidence is the presence of star-to-star abundance variations. The
most famous example is the Na-O anticorrelation that oxygen-depleted
stars have higher sodium abundances, which is exhibited among the
main sequence (MS) turnoff (TO) stars, sub-giant branch (SGB) stars
and red-giant branch (RGB) stars in some GCs
\citep{Carretta2004,Gratton2004}. Another important evidence is that
some GCs show multiple MSs \citep{Piotto2007}, multiple SGBs
\citep{Milone2008,Marino2009}, or anomalously wide RGB
\citep{Yong2008,Lee2009} in the Hertzsprung-Russell (HR) diagram.

Several scenarios have been proposed to explain the formation of
multiple stellar populations \citep{Gratton2012}, and they can be
classified into two contrasting hypotheses: \emph{Primordial}
Scenario vs. \emph{Evolutionary} Scenario
\citep{Kraft1994,Carretta2004}. In the \emph{Primordial} Scenario,
multiple stellar populations in GCs are formed in the different
materials that are primordial chemical inhomogeneities, for example,
a merger of two separate GCs \citep{Lee1999,Mackey2007} or the
self-pollution of the intra-cluster gas occurring in the early
evolution of clusters
\citep{D'Antona2002,Decressin2007,deMink2009a}. However, a merger
event would not be expected to occur in the halo of the Milky Way
due to the very large relative velocities of GCs
\citep{Gratton2012}. For the the self-enrichment of GCs, it is
difficult to explain the high fraction of the second-generation
formed from the gas polluted by the first-generation stars
\citep{Bastian2009}, and the nature of possible polluters remains
unclear \citep{Sills2010,Gratton2012}.

In the \emph{Evolutionary} Scenario, initial GCs are believed to be
single-generation clusters, and multiple stellar populations in GCs
are attributed to the evolution of stars affected by some physical
processes, such as the first dredge-up, Sweigart-Mengel Mixing
\citep{Sweigart1979}, primordial rotation \citep{Bastian2009}.
However, it is necessary to explain why only some of stars in GCs
are affected by these physical processes. Moreover,
\citet{Decressin2007} suggested that the MS TO stars in GCs are not
hot enough for producing the observed Na-O anti-correlation, because
the central temperature (about 25 $\times$ $10^6$\,K) of a
0.85\,M$_{\rm \odot}$ TO stars is higher than the required
temperature of CNO cycle (about 20 $\times$ $10^6$\,K), but is lower
than that of the production of $^{23}$Na from proton-capture on
$^{20}$Ne (about 35 $\times$ $10^6$\,K).

Binary interactions have been used to explain the formation of
multiple stellar populations as a kind of \emph{Primordial}
Scenario. Chemically enriched material ejected from massive binaries
have been proposed to form a second population of stars
\citep{deMink2009a}, or to be accreted by low-mass pre-main-sequence
stars \citep{Bastian2013}. However, binary interactions are not
considered as a kind of \emph{Evolutionary} Scenario. They are not
investigated to produce the abundance anomalous stars
\emph{directly}, although they have been proposed as an explanation
for extended MS in the HR diagrams of intermediate age clusters
\citep{Yang2011,Li2012}.

Binary systems can produce stars more massive than normal single
stars in the same evolutionary stage by binary interactions, e.g.
the merged stars or the accretor stars. These stars are not hot
enough for the production of $^{23}$Na from proton-capture on
$^{20}$Ne, but they are hot enough for proton-captures on $^{16}$O
and $^{22}$Ne, which respectively lead to the destruction of
$^{16}$O and to the production of $^{23}$Na at temperatures $>$20
$\times$ $10^6$\,K \citep{Prantzos2006}. Furthermore, these stars
produced by binary interactions will be rapid rotating because
binary interactions can convert orbital angular momentum into their
spin angular momentum \citep{deMink2013}. Rotationally induced
mixing can bring part of the nuclear-processed products from the
core to the surface \citep{Decressin2007}, and stellar rotation was
used to explain the observed nitrogen enrichment found in several
massive MS stars \citep{Brott2009}. Therefore, these rapidly
rotating stars reproduce the properties of the abundance anomalous
stars observed in GCs (e.g. the Na-O anticorrelation), and binary
interactions will be a possible scenario for the formation of
multiple stellar populations in GCs as a kind of \emph{Evolutionary}
Scenario.

In this paper, we adopt a simple model for this scenario to
investigate the formation of multiple stellar populations. In
Section 2, we explore the formation of rapidly rotating stars from
binary interactions by performing binary evolution calculations, and
calculated the subsequent evolution of these rapidly rotating stars
to obtain their parameters (e.g. effective temperatures,
luminosities and surface abundances). Then, we estimate the
distributions of the binary population by performing the binary
population synthesis study. In Section 3, we give discussion and
conclusions.

\section{THE MODEL AND SIMULATION}
In our scenario, the anomalous stars observed in GCs are the merged
stars and the accretor stars produced by binary interactions, and
these binary products are rapidly rotating and more massive than the
normal stars. These binary products have different effective
temperatures, luminosities and surface abundances from single stars
in the same evolutionary stage due to their own evolution. Thus, the
stellar population with binaries (the binary population) can
reproduce the observations of multiple stellar populations, because
the binary population includes two populations: (1) normal single
stars with normal abundance (initial abundance of GCs); (2) the
merged stars and the accretor stars (produced by binary
interactions) that have abnormal surface abundances by their own
evolution.

\subsection{Binary evolution calculations}

We considered three evolutionary channels for the formation of
rapidly rotating stars from binary interactions: (1) a binary
evolves into contact and then merges into a rapidly rotating star
(contact merger channel for the merged stars)
\citep{Jiang2012,deMink2013}; (2) a binary evolves into Roche lobe
overflow (RLOF) and experiences unstable mass transfer while both
components are still MS stars, and then merges into a rapidly
rotating star (unstable RLOF merger channel for the merged stars)
\citep{Jiang2012}; (3) a binary evolves into RLOF and experiences
stable mass transfer, and the secondary obtains the mass and angular
momentum transferred from the primary, and then becomes a rapidly
rotating star (stable RLOF channel for the accretor stars)
\citep{deMink2013}.

In these channels, a binary needs to evolve into contact phase or
unstable RLOF phase while both components are still MS stars, or
evolve into stable RLOF phase while the secondary is still a MS star
\citep{Jiang2012,deMink2013}. This is because for MS stars, the
binding energy of the envelope might be large enough to form rapidly
rotating stars. To determine whether the binary evolves into these
phase, it is necessary to perform detailed binary evolution
calculations. We use Eggleton's stellar evolution code
\citep{Eggleton1971,Eggleton1972,Eggleton1973} that has been updated
with the latest input physics during the last four decades
\citep{Han1994,Pols1995,Pols1998,Nelson2001,Eggleton2002}. In our
calculations, we used the metallicity $Z = 0.001$ (corresponding to
typical GCs) and considered the nonconservative effects, includes a
model of dynamo-driven mass loss, magnetic braking and tidal
friction for the evolution of stars with cool convective envelopes
\citep{Eggleton2002}.

Altogether, we have calculated the evolution of 5200 binary systems,
thus obtained a large, dense model grid that covers the following
ranges of initial primary mass ($M_{10}$), initial mass ratio ($q_0
= M_{20}/M_{10}$) and initial orbital period ($P_0$):
log\,$M_{10}=-0.3, -0.25, . . . , 0.2$;
log\,$q_0=$log\,$(M_{20}/M_{10})=0.05, 0.10, . . . , 0.5$; and
log\,$(P_0/P_{\rm ZAMS})= 0.05, 0.10, . . . $, where $P_{\rm ZAMS}$
is the orbital period at which the initially more massive component
would just fill its Roche lobe on the zero-age MS. These ranges of
initial parameters cover the systems that can form rapidly rotating
stars in old stellar population by Case A binary evolution.
According to these binary models, the parameters of rapidly rotating
stars are established when the binary evolves into contact, unstable
RLOF, or the end of stable RLOF. For simplicity, in the contact
merger channel and the unstable RLOF merger channel, we assume that
there are 10\% mass loss during the merger phase
\citep{Jiang2012,deMink2013} and the binary products rotate at 80\%
of break-up velocity. For the stable RLOF channel, we calculate the
angular momentum obtained by the secondary, and then calculate its
rotational velocity. We only consider the secondaries that can get
enough angular momentum to rotate at 80\% of break-up velocity.

\subsection{Rotation evolution calculations}
To calculate the subsequent evolution of rapidly rotating stars
produced by binary interactions, we have used the Modules for
Stellar Experiments in Astrophysics code (MESA version 4631)
\citep{Paxton2011}. This code implements the effects of rotation in
the transport of angular momentum and chemical mixing, and it has
been used to study the effects of rotational mixing
\citep{Chatzopoulos2012a,Chatzopoulos2012b}. We do not consider the
effect of the helium-rich core of the original components on the
subsequent evolution of rapidly rotating stars, and we assume that
these rapidly rotating stars have abundances similar to zero age MS
stars. In addition, we consider that the effect of rotation on
low-mass stars with convective envelopes might be weaker due to
magnetic braking \citep{Hurley2000,Bastian2009}. We calculate the
stellar evolutionary models with $M = 0.6, 0.7 ... 1.6$\,M$_{\rm
\odot}$ for $Z = 0.001$ that cover rapidly rotating stars produced
by binary interactions in old stellar populations. For simplicity,
these models run for two different degrees of rotation: non-rotating
and 80\% of break-up velocity, which are used to investigate the
evolution of surface abundances of single stars and rapidly rotating
stars produced by binary interactions. All evolutionary tracks have
been computed up to the point of core helium ignition.

To illustrate the evolutionary tracks of these stars, we show four
evolution examples of surface mass fractions (sodium versus oxygen
abundances) in Fig. 1. During the evolution of rapidly rotating star
with $M=1.3$\,M$_{\rm \odot}$ (dotted line), the surface mass
fraction of sodium ($X_{\rm Na23}$) increases from $1.22\times
10^{-6}$ to $5.95\times 10^{-6}$ , and that of oxygen ($X_{\rm
O16}$) decreases from $4.67\times 10^{-4}$ to $9.54\times 10^{-5}$.
This is because the destruction of O and the production of $^{23}$Na
in the central region result from proton-captures on $^{16}$O and
$^{22}$Ne \citep{Langer1993,Prantzos2006,Decressin2007}, and they
are brought to the surface by rotational mixing. In other examples,
rapidly rotating stars with $M=1.0, 1.1, 1.2$\,M$_{\rm \odot}$
evolve in a similar way as in the previous example, although the
variation ranges of surface abundances decrease with decreasing
mass. Hence, these rapidly rotating stars can be oxygen depletion
and sodium enhancement. For single stars with no rotation (open
circles in Fig. 1), the surface mass fractions of sodium and oxygen
do not show significant variations.

\subsection{Binary population synthesis}
In the binary population synthesis study, we have performed a Monte
Carlo simulation. We adopt the following input for the simulation
(see Han et al. 1995). (1) The initial mass function (IMF) of
\citet{Miller1979} is adopted. (2) The mass-ratio distribution is
taken to be constant. (3) We take the distribution of separations to
be constant in log\,$a$ for wide binaries, where $a$ is the orbital
separation. Our adopted distribution implies that $\sim$50 per cent
of stellar systems are binary systems with orbital periods less than
100\,yr. We follow the evolution of a million sample binaries
according to grids of binary models and the evolution models of
rapidly rotating stars described above.

The simulation gives the distributions of many properties of the
binary population, e.g. the masses, the effective temperatures, the
luminosities, the surface abundances, etc. Fig. 2 are selected
distributions of the binary population at 10 Gyr that may be helpful
for understanding the scenario of binary interactions for the
formation of multiple stellar populations. Fig. 2(a) shows the
distribution of single stars and binary products in the Na-O plane.
Single stars (open circles) are concentrated at the bottom right
corner and do not show significant variations. However, binary
products have large star-to-star variations. The distribution of
binary products first extends upward to high sodium abundance region
from the original abundance location, and then turns left to the
oxygen depletion region. Thus, the binary population with binary
interactions can show a Na-O anticorrelation.

To investigate the distributions of binary products with different
abundances in HR diagram, we classify them into four groups based on
the surface mass fraction of Na and O, which are denoted as pluses
with different colours. The distribution of the binary population
with binary interactions in HR diagram is shown in Fig. 2(b). It is
obvious that this population has at least two MSs, three TOs, three
SGBs and extended RGB, which are produced by single stars (open
circles) and binary products (pluses). More importantly, these
sequences have a large spread in sodium and oxygen abundances. This
is consistent with the fact that the Na-O anticorrelation is
observed among MS TO stars, SGB stars and RGB stars in some GCs
\citep{Carretta2004,Gratton2004}. Observed red giants in 17 GCs
given by \citet{Carretta2009} are shown in Fig. 2(c), and they have
a distribution similar to that of our simulated binary populations.
If we assume that the binary populations (black line and blue lines)
with different initial abundances, such as in different GCs, have
the same variation ranges of surface abundances, then we would find
that these binary populations in the simulation are in good
agreement with the observed stars in 17 GCs as shown in Fig. 2(c).

We noted that single stars and binary products in the binary
population have different mass distributions as shown in Fig. 2(d),
and binary products are more massive than single stars. For the
binary population with log $L/L_{\rm \odot}
> 0$, the mass range of single stars is from 0.75 to 0.9\,M$_{\rm \odot}$,
while the mass range of binary products is from 0.8\,M$_{\rm \odot}$
to 1.35\,M$_{\rm \odot}$, although these single stars and binary
products have the same range of evolutionary stages. The maximum
mass of binary products is larger than that of single stars by
$\sim0.45$\,M$_{\rm \odot}$. We do not compare stars with log
$L/L_{\rm \odot} < 0$ because the sample of very low-mass binary
products is not complete in our calculations. Although the effect of
rotation on low-mass stars might be weaker due to magnetic braking
\citep{Hurley2000,Bastian2009}, massive binary products could have
significantly different surface abundances from single stars. Binary
products are more massive than single stars in the same evolutionary
stage, and they rotate rapidly, i.e. close to break-up velocity. The
destruction of $^{16}$O and the production of $^{23}$Na in the
central region result from proton-captures on $^{16}$O and
$^{22}$Ne, and the products of nuclear burning are brought to the
surface by rotational mixing. This can explain why the abundance
variations take place in these stars, and not in single stars at the
same evolutionary stage.

\section{Discussion and conclusions}
In this paper, we presented binary interactions as a possible
scenario for the formation of multiple stellar populations in GCs.
In this scenario, binary interactions can convert orbital angular
momentum into spin angular momentum and produce rapidly rotating
stars. These stars might have different properties from single
stars, such as surface abundances, temperatures and luminosities,
because they experienced binary interactions and their evolution is
affected by rotationally induced mixing. The existence of binary
products and single stars results in multiple stellar populations in
GCs, although initial GCs are single-generation clusters. In our
simple model, we carried out binary evolutionary calculations and
constructed the evolutionary models of rapidly rotating stars formed
by binary interactions. By performing a Monte Carlo simulation, we
obtained the distributions of the binary population and investigated
the formation of multiple stellar populations from binary
interactions.

Stellar rotation have been shown that can explain the broad MS and
spread TO observed in intermediate age stellar clusters
\citep{Bastian2009, Li2012}. However, \citet{Bastian2009} suggested
that this is unlikely to cause multiple stellar populations observed
in old GCs, because the stars in GCs are low mass and are not
expected to be rapidly rotating stars. We show that old binary
population could show multiple sequences in HR diagram as shown in
Fig. 2(b), because binary interactions could produce rapidly
rotating stars. Binary interactions, including mass transfer
\citep{Pols1991,deMink2013} or merger \citep{Tylenda2011,Jiang2013},
have been used to investigate the formation of rapidly rotating
stars. Moreover, observed evidences have shown that binary
interactions are very common in GCs. Many contact binaries have been
observed in GCs \citep{Rucinski2000}, and these binaries are
believed to merge into a rapidly rotating star (contact merger
channel). \citet{Mathieu2009} show that many of blue stragglers are
rotating faster than normal MS stars , which might be formed by mass
transfer \citep{Geller2011}. Therefore, binary interactions could
explain multiple sequences in HR diagram, which is important
evidence of multiple stellar populations.

Other important evidence of multiple stellar populations in GCs is
the presence of star-to-star abundance variations, and the most
famous example is the Na-O anticorrelation. We show that old binary
population can reproduce a Na-O anticorrelation as a result of the
formation of rapidly rotating stars from binary interactions. If the
binary populations with different initial abundances, such as in
different GCs, are assumed to have the same variation ranges of
surface abundances, we found that these binary populations in the
simulation are in good agreement with the observed stars in 17 GCs
given by \citet{Carretta2009}. In our scenario, this anticorrelation
could be found in TO stars, SGB stars and RGB stars, which is in
agreement with the observation that show the Na-O anticorrelation in
these evolutionary stages \citep{Carretta2004,Gratton2004}.
Moreover, our model predicts that these abundance anomalous stars
should be more massive than MS TO stars, and they should be blue
stragglers or blue straggler progeny. Therefore, binary interactions
could explain the existence of the Na-O anticorrelation in GCs. This
suggests that binary interactions may be a possible scenario for the
formation of multiple stellar populations, and the traditional view
of single-generation cluster could explain the photometric and
spectroscopic observations of multiple stellar populations by
considering the effects of binary interactions.

Observed multiple stellar populations show the cluster-to-cluster
variations \citep[e.g.][]{Carretta2009}, and these variations also
need to be explained \citep{Sills2010}. This cluster-to-cluster
variations might be due to the dependence of binary interactions on
the cluster properties, such as age, initial abundance, initial
fraction of binaries and initial orbital separation distribution.
These parameters determine the nature of binary interactions and the
subsequent evolution of binary products. When the initial fraction
of binaries is very low, the binary population would have few binary
products and be similar to a simple, single stellar population. The
investigation of the properties of binaries in 13 low-density GCs
revealed that the fraction of binaries ranges from 0.1 to 0.5 in the
core depending on the cluster \citep{Sollima2007}. Moreover, the
stellar density in GCs is sufficiently high to affect the binaries
that can be destroyed, created or modified by dynamical interaction
\citep[e.g.][]{Hut1992}. Higher stellar densities could result in
more efficient binary dissolution and thus a lower binary fraction
\citep{Marks2011}. Consequently, the binary population might have
different distributions and different fractions of binary products
in various clusters. A more detailed study of the binary population
with binary interactions might help to understand the dependence of
binary interactions on the cluster properties in the further
studies.

It is clear that our model is still quite simple and does not
investigate in much detail so far. In our investigation, we do not
consider the effect of the helium-rich core of the original
components on the subsequent evolution of binary products. Our
simulation shows that 36 per cent of all rapidly rotating stars are
formed by binary merger (including contact merger channel and
unstable RLOF merger channel). These rapidly rotating stars formed
by binary merger might contain the helium-rich core of the original
primary, and their remaining relative lifetimes are smaller than
those of single stars with the same masses
\citep[e.g.][]{Sills1997,Glebbeek2008,deMink2013}. For rapidly
rotating stars produced through stable RLOF, this effect need not to
be considered because they are the less-massive components of
binaries and their cores are still hydrogen rich at the onset of
mass transfer \citep[e.g.][]{Sills1997,Glebbeek2008,deMink2013}. For
rotating velocity, we simply assume that all binary products rotate
initially at 80\% of break-up velocity. A lower initial rotating
velocity or a faster spin-down will lead to a weaker variation of
surface abundances. These effects need more detailed investigation
in further study. In addition, the observations suggested that the
fraction of binaries in GCs may be lower than that in the field and
open clusters \citep{Rubenstein1997}. This might be because GCs are
much older than the field and open clusters, and consequently more
binaries in GCs are not expected to be recognized as binaries due to
their large luminosity ratio and their large orbital periods, or
they may not even be binaries due to merger \citep{deMink2013}.

\acknowledgments

We thank an anonymous referee for his/her valuable comments. This
work was supported by the Natural Science Foundation of China (Nos
11073049, 11033008, 11103073, 11390374 and 11373063), Science and
Technology Innovation Talent Programme of the Yunnnan Province (No.
2013HA005) and the Western Light Youth Project.


\section{Introduction}
Since the experimental realization of Bose-Einstein condensation in optical traps,
much effort has been devoted to the study of spinor
superfluid \cite{hall1,hall2,matthews,stenger,ho,ohmi,review}. With the
degree of freedom of spin, spinor superfluid has much richer phases than scalar
superfluid as it has both superfluid order and spin textures.
However, most spinor superfluids studied till now carry both spin current and
mass current. It will be interesting to see whether these two currents can
decouple and further whether a  pure spin current which has no mass current
can flow frictionlessly.  Our motivation also originates
from condensed matter physics, in which the concept of spin
superconductor \cite{sun2,sun3},
formed by the Cooper-like pairs of electrons and holes and carrying spin
super-current, is proposed. It is interesting to have these ideas realized
in the field of cold atoms.

In this work we focus on the unpolarized spin-1 Bose-Einstein condensate (BEC), where the pure spin current
can be generated by applying a small magnetic gradient.  It was found
in Ref. \cite{fujimoto} that  a planar pure spin current  in such
a system is always unstable as the $m=1,-1$ components can collide
into the $m=0$ component destroying the spin current. We find that
the pure spin current can be stabilized with the quadratic Zeeman effect
and become a super-flow.  Furthermore, we propose a scheme to create pure
spin current at the ground state, thus free from the issue of instability. Our scheme
utilizes the spin-orbit coupling. Specifically, we study a spin-1 BEC
with Rashba spin-orbit coupling confined in a two-dimensional harmonic trap,
and numerically find the ground state of the system. For antiferromagnetic
interactions, opposite vortices appear in the $m=1,-1$ components with equal
amplitude when the spin-orbit coupling is weak. Such a state carries
pure spin current and no mass current. This spin current is a super-flow as it is
the ground state and must be stable.

We note that there has been a lot of theoretical and experimental work on the counterflow of two species BEC \cite{law,kuklov,yukalov,takeuchi,engels,hamner,fujimoto,ishino,jltp,abad,kurn,cherng}.
For two miscible BECs with counterflow, it is found that there is a critical relative speed between the two species, beyond which the state is dynamically unstable \cite{law,kuklov,yukalov,takeuchi,engels,hamner,ishino,jltp,abad}. It is shown that
the instability can lead to proliferation of solitons \cite{engels,hamner} and
quantum turbulence \cite{takeuchi}.  This kind of counterflow is very similar
to a spin current but it is not for two reasons: (1) Theoretically, if we regard
the two species as two components of a pseudo-spin, this  pseudo-spin has no SU(2) rotational
symmetry as the number of bosons in each species is conserved. (2) Experimentally,
it is hard to control the number of bosons in each component to create a spin current
that has no mass current.

The paper is organized as follows. In Sec. \ref{planar}, we first study the stability
of a spin-1 planar counterflow. We identify the mechanisms associated with
the instabilities, and find that the quadratic Zeeman effect  can stabilize such a planar
flow. We then study the similar situation in the circular geometry
in Sec. \ref{circular}. The pure spin current consists of a vortex and
anti-vortex in the $m=-1,1$ components. The experimental schemes
to realize the stable pure spin current is discussed in Sec. \ref{expe}.
Finally, we briefly summarize our main results in Sec. \ref{conclusion}.

\section{Planar flow}\label{planar}
The dynamics of a spin-1 BEC in free space is governed by the
mean field Gross-Pitaevskii (GP) equation \cite{review},
\begin{equation}
\label{ }
i\hbar\frac{\partial}{\partial t}\psi_m=-\frac{\hbar^2\nabla^2}{2M}\psi_m+c_0 \rho\psi_m+c_2\sum_{n=-1}
^1\mathbf{s}\cdot\mathbf{S}_{mn}\psi_n,
\end{equation}
where $\psi_m$ ($m=1,0,-1$) are the components of the macroscopic wave function.
$\rho=\sum_{m=-1}^1|\psi_m|^2$ is the total density, ${\bf s}_i=\sum_{mn}\psi_m^*(S_i)_{mn}\psi_n$
is the spin density vector and ${\bf S}=(S_x, S_y, S_z)$ is the spin operator vector with $S_i$ ($i=x,y,z$)
being the three Pauli matrices in the spin-1 representation.
The collisional interactions include a spin-independent
part $c_{0}=4\pi\hbar^{2}(a_{0}+2a_{2})/3M$ and a spin-dependent
part $c_{2}=4\pi\hbar^{2}(a_{2}-a_{0})/3M$, with $a_{f}(f=0,2)$
being the $s$-wave scattering length for spin-1 atoms in the
symmetric channel of total spin $f$.

We consider a spin current state of the above GP equation with the form
\begin{equation}
\label{scurrent}
\psi=\sqrt{\frac{n}{2}}\left(
\begin{array}{c}
e^{i\mathbf{k}_1\cdot{\bf r}}\\  0\\
e^{i\mathbf{k}_2\cdot{\bf r}}
\end{array}
\right)\,,
\end{equation}
where $n$ is the density of the uniform BEC.
The requirement of equal chemical potential leads to $|\mathbf{k}_1|=|\mathbf{k}_2|$.
In the case where $\mathbf{k}_1=-\mathbf{k}_2$,  this state carries a pure spin current:
the total mass current is zero as  it has equal mass counterflow while the spin current is nonzero.

It is instructive to first consider the special case when there is no counterflow, i.e., ${\bf k}_1={\bf k}_2=0$.
The excitation spectra are found to be $\epsilon^0=\sqrt{2c_2n\epsilon_q+\epsilon_q^2}$ and $\epsilon_1^{\pm1}=\sqrt{2c_0n\epsilon_q+\epsilon_q^2}$, $\epsilon_2^{\pm1}=\sqrt{2c_2n\epsilon_q+\epsilon_q^2}$, respectively,
with $\epsilon_q=\hbar^2q^2/2M$.
So for antiferromagnetic interaction ($c_0>0, c_2>0$), all branches of the spectra are real
and  there is a double degeneracy in one branch of the spectra. The phonon excitations give two sound velocities, $\sqrt{n c_i/M}$ ($i=0,2$), corresponding to the speeds of density wave and spin wave, respectively. However, the existence of phonon excitation
does not mean that  the pure spin current (${\bf k}_1={\bf k}_2\neq 0$) is a super-flow
as we can not obtain the current with ${\bf k}_1={\bf k}_2\neq 0$ from the state
with ${\bf k}_1={\bf k}_2= 0$ by a Galilean transformation.

The stability of the spin current  has been studied
in Ref. \cite{fujimoto} for the  case $\mathbf{k}_1=-\mathbf{k}_2\neq 0$.
It is found that, for  the antiferromagnetic interaction case ($c_0>0, c_2>0$), the excitation
spectrum of the $m=0$ component always has nonzero imaginary part
in the long wavelength limit as long as there is counterflow between the two components, and the imaginary excitations in the $m=1,-1$ components only appear for a large enough relative velocity $v_1=2\sqrt{n c_2/M}$. For the ferromagnetic
interaction case ($c_0>0, c_2<0$), both excitation spectra of the $m=0$ and $m=1,-1$ components have nonzero imaginary parts for any relative velocity. This means that the pure spin current cannot be stable in any cases.

For the general non-collinear case (${\bf k}=\frac{{\bf k}_1+{\bf k}_2}{2}\neq0$) and antiferromagnetic interaction, the excitation spectrum for the $m=0$ component is found to be
\begin{equation}
\label{ }
\epsilon^0=\sqrt{\left(\epsilon_q+\frac{\hbar^2}{2M}\left(|\mathbf{k}|^2-|\mathbf{k}_1|^2\right)+c_2n\right)^2-c_2^2n^2}.
\end{equation}
We see here that as long as the momenta of the two components are not exactly parallel, i.e., $\mathbf{k}_1$ is not exactly equal to $\mathbf{k}_2$, then $|\mathbf{k}|<|\mathbf{k}_1|$, and
there is always dynamical instability for the long wavelength excitations.

Therefore, the spin current in Eq. (\ref{scurrent}) is generally unstable and not a super-flow.
This instability  originates from the interaction process described by
$\psi_0^{\dagger}\psi_0^{\dagger}\psi_1\psi_{-1}$ in the second
quantized Hamiltonian. This energetically favored process converts two
particles in the $m=1,-1$ components, respectively,  into two stationary particles
in the $m=0$ component. To suppress such a process and achieve a stable
pure spin current, one can utilize the quadratic Zeeman effect.
With the quadratic Zeeman effect of negative coefficient, the Hamiltonian
adopts an additional term $\lambda m^2$ ($\lambda<0$ and $m=1,0,-1$).
This term does not change the energy of the $m=0$
component, but lowers the energy of the other two components $m=1,-1$.
As a result, there arises a barrier for two atoms in the $m=1,-1$ components scattering to the $m=0$ component, and the scattering process  is thus suppressed.

The above intuitive argument can be made more rigorous and quantitative.
Consider the case $\mathbf{k}_1=-\mathbf{k}_2$. With the quadratic Zeeman term,
the  excitation spectrum for the $m=0$ component changes to
\begin{equation}
\label{ }
\epsilon^0=\sqrt{\left(\epsilon_q-\frac{\hbar^2 |\mathbf{k}_1|^2}{2M}+c_2n-\lambda\right)^2-c_2^2n^2}.
\end{equation}
So as long as $-\lambda-\hbar^2 |\mathbf{k}_1|^2/2M>0$, long wavelength excitations will be stable for the $m=0$ component. From the excitation
of the $m=0$ component, one can obtain a critical relative velocity of the spin current,
$v_0=2\sqrt{-2\lambda/M}$.  There is  another nonzero critical velocity $v_1=2\sqrt{n c_2/M}$ determined by the
excitations of the $m=1,-1$ components.  The overall critical velocity of the system is the smaller
one of $v_0$ and $v_1$.  Therefore, below the critical relative velocity $v_{\rm c}={\rm min}\{v_0,v_1\}$, the pure spin current is stable and a super-flow. The experimental scheme
to realize such a Zeeman effect will be discussed  in Sec. \ref{expe}.

\section{Circular flow}\label{circular}
In the cylindrical geometry, we consider a pure spin current formed by two vortices
 with opposite circulation
in the $m=1,-1$ components. From similar arguments, one can expect that interaction will make
such a current unstable.  Inspired by the quadratic Zeeman effect method above,
we propose to use spin-orbit coupling to stabilize it. The spin-orbit coupling can be
viewed as a momentum-dependent effective magnetic field that exerts only on
the $m=1,-1$ components. Therefore, it is possible that spin-orbit coupling lowers
the energy of $m=1,-1$ components, and consequently suppresses the interaction process
leading to the instability.

The model of spin-1 BEC subject to Rashba spin-orbit coupling
can be described by the following energy functional,
\begin{align}
\label{energy}
\mathcal{E}\left[\psi_{\alpha}\right]= & \int d\mathbf{r}\Bigg\{\sum_{\alpha}\frac{\hbar^{2}|\nabla\psi_{\alpha}|^{2}}{2M}+\rho V(r)+\frac{c_{0}}{2}\rho^{2}
+\frac{c_{2}}{2}\mathbf{s}^{2} \nonumber \\
& +\gamma\langle S_xp_y-S_yp_x\rangle\Bigg\},
\end{align}
where $\rho$ is the density, $V(r)=\frac{1}{2}M\omega^2(x^2+y^2)$ is the trapping potential, and $\gamma$ is the strength of spin-orbit coupling. $\langle\cdots\rangle$ is the expectation value taken with respect to the three component wave function
$\psi=(\psi_1,\psi_0,\psi_{-1})^T$. The strength of the spin-orbit coupling $\gamma$ defines a characteristic length
$a_{\rm soc}=\hbar/M\gamma$, and can be rescaled to be dimensionless with respect to the harmonic oscillator length
$a_{\rm h}=\sqrt{\hbar/M\omega}$. Then we characterize the strength of spin-orbit coupling with the dimensionless
quantity $\kappa=a_{\rm h}/a_{\rm soc}=\gamma\sqrt{M/\hbar\omega}$.
The spin-orbit coupling of Rashba type here can be generated in various ways, which will be discussed in the next section.

The above model can describe a spin-1 BEC of $^{23}$Na confined in a
two-dimensional harmonic trap. Assume the atom number is about $10^6$.
Using the estimate of scattering lengths $a_0=50a_B$, $a_2=55a_B$ \cite{epjd},
with $a_B$ being the Bohr radius, the ground state of spin-1 $^{23}$Na should be
antiferromagnetic because $c_0>0, c_2>0$ \cite{ho}.
Previous studies of spin-1 BEC with Rashba spin-orbit coupling mostly focus on
the strong spin-orbit coupling regime, where the ground
state is found to be the plane wave phase or the stripe phase, for ferromagnetic interaction and antiferromagnetic interaction, respectively \cite{zhai}.
Here we are interested in the antiferromagnetic interaction case and the
weak spin-orbit coupling regime ($\kappa\ll 1$), and calculate the ground
state wave function of the energy functional with the method of imaginary
time evolution.

\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{wf}
\caption{(color online) Amplitudes (a1,b1,c1) and phase angles (a2,b2,c2) of the three component wave function
$\psi=(\psi_1,\psi_0,\psi_{-1})^T$ at the ground state of Hamiltonian (\ref{energy}) for a BEC of $^{23}$Na confined
in a 2D harmonic trap. The particle number is $10^6$, the frequency of the trap is $2\pi\times42$ Hz, and the dimensionless spin-orbit coupling strength is $\kappa=0.04$. The units of the $X$ and $Y$ axes are $a_{\rm h}$.}
\label{wf}
\end{center}
\end{figure}

We find that when the spin-orbit coupling is weak ($\kappa\ll 1$), the ground state wave function
has the form
\begin{equation}\label{ground}
\psi=\left(\begin{array}{c}
\chi_{1}(r)e^{-i\phi}\\
\chi_{0}(r)\\
\chi_{-1}(r)e^{i\phi}
\end{array}\right),
\end{equation}
with $\chi_{1}(r)=-\chi_{-1}(r)$ and all $\chi_i$ real. The ground state is
shown in Fig. \ref{wf}. Such a ground state consists of an anti-vortex
in the $m=1$ component and a vortex in the $m=-1$ component.
The $m=0$ component does not carry angular momentum. Since $|\psi_1|=|\psi_{-1}|$,
the net mass current vanishes.

The wave function in Eq. (\ref{ground}) can be understood in the single particle level.
In terms of the ladder operators of spin and angular momentum, the spin-orbit coupling term reads
\begin{equation}
\mathcal{H}_{\rm soc}=\frac{\gamma\sqrt{M\hbar\omega}}{2}\left[S_+\left(\hat{a}_R-\hat{a}_L^{\dagger}
\right)+S_-\left(\hat{a}_R^{\dagger}-\hat{a}_L\right)\right],
\end{equation}
where $S_{\pm}$ is the ladder operator of spin, and $\hat{a}_{L(R)}^{\dagger}$ is the creation operator of the
left (right) circular quanta \cite{cohen}. When the spin-orbit coupling is very weak ($\kappa\ll 1$), its effect can be accounted
for in a perturbative way. From the ground state $\Psi^{(0)}=|0,0\rangle$, the first order correction to the
wave function for small $\gamma$ is given by
\begin{align}
\Psi^{(1)}=&\frac{\gamma\sqrt{M\hbar\omega}}{2\hbar\omega}\left(-S_+\hat{a}_L^{\dagger}
+S_-\hat{a}_R^{\dagger}\right)|0,0\rangle \nonumber \\
=&\frac{\kappa}{2}\left(-|1,-1\rangle+|-1,1\rangle\right),
\end{align}
where $|m_s,m_o\rangle$ denotes a state with spin quantum number $m_s$ and orbital magnetic quantum number
$m_o$. One immediately sees that $\psi_1$ has angular momentum $-\hbar$ and $\psi_{-1}$ has angular momentum $\hbar$.
Besides, the amplitudes of both $\psi_1$ and $\psi_{-1}$ are proportional to $\kappa$.

There exits a continuity equation for spin density and spin current, which  is
\begin{equation}
\frac{d}{d t}\left(\psi^{\dagger}{\bf S}_\mu\psi\right)+\nabla\cdot\mathbf{J}_{\mu}^{s}=0.
\end{equation}
The spin current density tensor $\mathbf{J}_{\mu}^{s}$ ($\mu=x, y, z$ denotes the spin component) is defined as \cite{aref1,aref2}
\begin{align}
\label{def}
\mathbf{J}_{\mu}^{s}=&\frac{1}{2}\left\{\psi^{\dagger}S_\mu{\bf v}\psi+{\rm c.c.}\right\} \nonumber \\
 =&\frac{1}{2}\left\{\sum_{m,n,l}\psi_m^*\left(S_\mu\right)_{mn}{\bf v}_{nl}\psi_l+{\rm c.c.}\right\},
\end{align}
where
\begin{equation}
{\bf v}_{nl}=\frac{{\bf p}}{M}+\gamma\left(\hat{z}\times{\bf S}_{nl}\right),
\end{equation}
and c.c. means the complex conjugate. The second part in ${\bf v}_{nl}$  is
induced by the spin-orbit coupling.

By the definition in Eq. (\ref{def}), the spin current density carried by
the ground state (\ref{ground}) is
\begin{eqnarray}
\label{spincurrent}
\mathbf{J}_{x}^{s}=&\gamma\sin2\phi|\psi_1|^2\hat{x}+\gamma\left(|\psi_0|^2+2|\psi_1|^2\sin^2\phi\right)\hat{y}, \nonumber\\
\mathbf{J}_{y}^{s}=&-\gamma\left(|\psi_0|^2+2|\psi_1|^2\cos^2\phi\right)\hat{x}-\gamma\sin2\phi|\psi_1|^2\hat{y}, \nonumber\\
\mathbf{J}_{z}^{s}
=&\left(-\frac{2\hbar|\psi_{1}|^{2}}{Mr}+\sqrt{2}\gamma|\psi_1\psi_0|\right)\hat{\phi}.
\end{eqnarray}
From both analytical and numerical results of the wave function, $|\psi_1|\ll|\psi_0|$, so $\mathbf{J}_{x}^{s}$
roughly points in the $y$ direction, while $\mathbf{J}_{y}^{s}$ almost points in the $-x$ direction.
$\mathbf{J}_{z}^{s}$
represents a flow whose amplitude has rotational symmetry.
 From the numerical results shown in
Fig. \ref{current}, we see that $\mathbf{J}_{z}^{s}$ is a counter-clockwise flow.
The amplitudes of $\mathbf{J}_{x}^{s}$ and $\mathbf{J}_{y}^{s}$ are of the same order, both proportional to $\kappa$,
while that of $\mathbf{J}_{z}^{s}$, proportional to $\kappa^2$, is much smaller.
It is evident that the state in Eq. (\ref{ground}) carries no mass current and only pure spin current. Since the spin current is in the ground state, it
must be stable.  In this way,  we have realized a superfluid of pure spin current,
or a pure spin super-current.

\begin{figure}
\begin{center}
\includegraphics[width=8.5cm]{currentnew}
\caption{(color online) Distribution of the spin current densities $\mathbf{J}_{x}^{s}$ (blue arrow), $\mathbf{J}_{y}^{s}$ (red arrow) and
$\mathbf{J}_z^s$ (black arrow)
of the ground state shown in Fig. \ref{wf}. The length of the arrows represents the
strength of the spin current. The arrow length of different colors is not to scale. $\kappa=0.04$.
The units of the $X$ and $Y$ axes are $a_{\rm h}$.}
\label{current}
\end{center}
\end{figure}

\section{experimental schemes}\label{expe}
In this section, we propose the experimental schemes to generate and detect
the pure spin currents discussed in Sec. \ref{planar} and Sec. \ref{circular}.

The planar pure spin current can be easily generated. By  applying a magnetic field gradient,
the two components $m=1,-1$  will be accelerated in opposite directions and a pure
spin current is generated as done in Refs. \cite{engels,hamner}.  To stabilize this
spin current, one needs to generate the quadratic  Zeeman effect. We apply an  oscillating magnetic field $B\sin\omega t$ with the frequency $\omega$ being much larger than
the characteristic frequency of the condensate, e.g., the chemical potential $\mu$.
The time averaging removes the linear Zeeman effect; only the quadratic Zeeman effect
remains.  The coefficient of the quadratic  Zeeman effect from the second-order
perturbation theory is given by $\lambda=\left(g\mu_{\rm B} B\right)^2/\Delta E_{\rm hf}$, where $g$ is the Land\'e $g$-factor of the atom, $\mu_{\rm B}$ is the Bohr
magneton, and $\Delta E_{\rm hf}$ is the hyperfine energy splitting \cite{ueda}. For the $F=2$ manifold of $^{87}$Rb, $\Delta E_{\rm hf}<0$, so the coefficient of the quadratic
Zeeman effect is negative.

The circular flow  in Sec. \ref{circular} may find prospective realizations in
two different systems: cold atoms and exciton BEC. In cold atoms, we consider a
system consisting of a BEC of $^{23}$Na confined in a pancake trap, where the
confinement in the $z$ direction is so tight  that one can treat the system effectively
as two dimensional. The spin-orbit coupling can be induced by two different methods.
One is by the exertion of a strong external electric field ${\bf E}$ in the $z$ direction. Due to the relativistic effect,
the magnetic moment of the atom will experience a weak spin-orbit coupling,
where the strength $\gamma=g\mu_B|{\bf E}|/Mc^2$. Here $M$ is the atomic mass
and $c$ is the speed of light.
For weak spin-orbit coupling (small $\gamma$), the fraction of atoms in the $m=1,-1$ components
is proportional to $\gamma^2$.
For an experimentally observable fraction of atoms, e.g., $0.1\%$ of $10^6$ atoms,
using the typical parameters of $^{23}$Na BEC,
the estimated electric field is of the same order of magnitude as the vacuum breakdown field.
For atoms with smaller mass or larger magnetic moment, the required electric field can be lowered.
Another method of realizing spin-orbit coupling
is to exploit the atom laser interaction, where strong spin-orbit coupling can be
created in principle \cite{socreview}.
In exciton BEC systems, as the effective mass of exciton is much smaller than that of atom, the required
electric field is four to five orders of magnitude smaller, which is quite feasible in experiments \cite{exciton1,exciton2,exciton3,exciton4}.

The vortex and anti-vortex in the $m=1,-1$ components can be detected by the method
of time of flight.
First one can split the three spin components with the Stern-Gerlach effect.
The appearance of vortex or anti-vortex in the $m=1,-1$ components is
signaled by a ring structure in the time of flight image. After a sufficiently long
time of expansion, the ring structure should be clearly visible.

\section{conclusion}\label{conclusion}
In summary, we have studied the stability of a pure spin current of a spin-1 BEC.
In the planar flow, the system always suffers from dynamical instability. The origin of the
instability is the interaction process that
converts two  particles in the $m=1,-1$ components into the $m=0$ component. Based on this,
we propose a method to stabilize the pure spin current by utilizing the quadratic  Zeeman effect.
In the circular flow, we have proposed to use spin-orbit coupling to make the pure spin current stable.
For weak spin-orbit coupling,  we have found that the ground state of the system is a
superfluid of pure spin current. The experimental schemes to
realize and detect these pure spin currents have been discussed.

\section*{Acknowledgements} This work is supported by the NBRP of China (2013CB921903,2012CB921300) and the NSF of China (11274024,11274364,11334001,11429402).


\section{Introduction}
\hspace{4mm} For real numbers $a,b$ and $c$ with $c\neq
0,-1,-2,\cdots$, the Gaussian hypergeometric function is defined by
\begin{equation}
F(a,b;c;x)={}_{2}F_{1}(a,b;c;x)=\sum_
{n=0}^{\infty}\frac{(a,n)(b,n)}{(c,n)}\frac{x^n}{n!},\ |x|<1.
\end{equation}
Here $(a,0)=1$ for $a\neq0$, and
$(a,n)=a(a+1)(a+2)(a+3)\cdots(a+n-1)$ is the shifted factorial
function for $n=1,2,\cdots$. In connection with the Gaussian
hypergeometric function, the well-known complete elliptic integrals
$\mathcal{K}(r)$ and $\mathcal{E}(r)(0<r<1)$ of the first and second
kinds [1, 2] are defined by
\begin{equation}
\left\{\begin{array}{ll}
{\mathcal{K}}(r)={\pi}F({1}/{2},{1}/{2};1;r^2)/2=\int_{0}^{{\pi}/{2}}(1-r^2\sin^2\theta)^{-1/2}d \theta,\\
{\mathcal{K}}(0)={\pi}/2,\qquad{\mathcal{K}}(1)=\infty
\end{array}\right.
\end{equation}
and
\begin{equation}
\left\{\begin{array}{ll}
{\mathcal{E}}(r)={\pi}F(-{1}/{2},{1}/{2};1;r^2)/2=\int_{0}^{{\pi}/{2}}(1-r^2\sin^2\theta)^{1/2} d\theta,\\
{\mathcal{E}}(0)={\pi}/2,\qquad{\mathcal{E}}(1)=1,
\end{array}\right.
\end{equation}
respectively. The following formulas for $\mathcal{K}(r)$ were
presented in [3]:
\begin{equation}
\frac{d{\mathcal{K}(r)}}{dr}=\frac{{\mathcal{E}(r)}-{(1-r^2)}{\mathcal{K}(r)}}{r(1-r^2)},
\end{equation}
\begin{equation}
\mathcal{K}\left(\frac{2\sqrt{r}}{1+r}\right)=(1+r)\mathcal{K}(r).
\end{equation}

Let $H(a,b)=2ab/(a+b)$, $G(a,b)=\sqrt{ab}$,
$L(a,b)=(b-a)/(\log{a}-\log{b})$ and $A(a,b)=(a+b)/2$ be the
classical harmonic, geometric, logarithmic and arithmetic means of
two distinct positive real numbers $a$ and $b$, respectively. Then
it is well known that the inequalities $H(a,b)<G(a,b)<L(a,b)<A(a,b)$
hold for all $a,b>0$ with $a\neq b$.

The classical arithmetic-geometric mean $AG(a,b)$ of two positive
number $a$ and $b$ is defined as the common limit of sequences
$\{a_{n}\}$ and $\{b_{n}\}$, which are given by
\begin{align*}
&a_{0}=a,\qquad b_{0}=b,\\
&a_{n+1}=(a_{n}+b_{n})/2=A(a_{n},b_{n}),\qquad
b_{n+1}=\sqrt{a_{n}b_{n}}=G(a_{n},b_{n}).
\end{align*}

It is well known that inequalities
\begin{equation}
G(a,b)<\sqrt{A(a,b)G(a,b)}<AG(a,b)<A(a,b)
\end{equation}
hold for all $a,b>0$ with $a\neq b$.

Recently, the harmonic, geometric, logarithmic, arithmetic-geometric
and arithmetic means have been the subject of intensive research. In
particular, many remarkable inequalities for these means can be
found in the literature [4-13].

The Gaussian identity [3] shows that
\begin{equation}
AG(1,r)\mathcal{K}(\sqrt{1-r^2})=\frac{\pi}{2}
\end{equation}
for all $r\in(0,1)$.

Carlson and Vuorinen [14], and Brackenn [15] proved that
\begin{equation*}
L(a,b)<AG(a,b)
\end{equation*}
for all $a,b>0$ with $a\neq b$. Vamanamurthy and Vuorinen [16]
established that
\begin{equation*}
AG(a,b)<\frac{\pi}{2}L(a,b)
\end{equation*}
for all $a,b>0$ with $a\neq b$.

For $t_{1},t_{2},t_{3},t_{4}\in(0,1/2)$, very recently Chu et al.
[17, 18] proved that the inequalities
\begin{equation}
G(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)>AG(a,b),
\end{equation}
\begin{equation}
H(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)>AG(a,b),
\end{equation}
\begin{equation}
G(t_{3}a+(1-t_{3})b,t_{3}b+(1-t_{3})a)>L(a,b),
\end{equation}
and
\begin{equation}
H(t_{4}a+(1-t_{4})b,t_{4}b+(1-t_{4})a)>L(a,b)
\end{equation}
hold for all $a,b>0$ with $a\neq b$ if and only if $t_{1}\geq
1/2-\sqrt{2}/4$, $t_{2}\geq 1/4$, $t_{3}\geq 1/2-\sqrt{6}/6$ and
$t_{4}\geq 1/2-\sqrt{3}/6$.

Let $t\in(0,1/2)$, $s\geq 1$ and
\begin{equation}
Q_{t,s}(a,b)=G^{s}(ta+(1-t)b, tb+(1-t)a)A^{1-s}(a,b).
\end{equation}
Then it is not difficult to verify that
\begin{equation*}
Q_{t,1}(a,b)=G(ta+(1-t)b, tb+(1-t)a),
\end{equation*}
\begin{equation*}
Q_{t,2}(a,b)=H(ta+(1-t)b, tb+(1-t)a)
\end{equation*}
and $Q_{t,s}(a,b)$ is strictly increasing with respect to
$t\in(0,1/2)$ for fixed $a,b>0$ with $a\neq b$.

It is natural to ask what are the least values $t_{1}=t_{1}(s)$ and
$t_{2}=t_{2}(s)$ in $(0,1/2)$ such that inequalities
$Q_{t_{1},s}(a,b)>AG(a,b)$ and $Q_{t_{2},s}(a,b)>L(a,b)$ hold for
all $a,b>0$ with $a\neq b$ and $s\geq 1$. The aim of this paper is
to answer these questions, our main results are the following
Theorems 1.1 and 1.2.

\medskip
{\bf Theorem 1.1.} If $t\in(0,1/2)$ and $s\geq 1$ then the
inequality
\begin{equation}
Q_{t,s}(a,b)>AG(a,b)
\end{equation}
holds for all $a,b>0$ with $a\neq b$ if and only if $t\geq
1/2-\sqrt{2s}/(4s)$.

\medskip
{\bf Theorem 1.2.} If $t\in(0,1/2)$ and $s\geq 1$ then the
inequality
\begin{equation}
Q_{t,s}(a,b)>L(a,b)
\end{equation}
holds for all $a,b>0$ with $a\neq b$ if and only if $t\geq
1/2-\sqrt{6s}/(6s)$.

\medskip
{\bf Remark 1.1.} Let $s=1,2$ in Theorem 1.1, then inequality (1.13)
becomes inequalities (1.8) and (1.9), respectively.

\medskip
{\bf Remark 1.2.} Let $s=1,2$ in Theorem 1.2, then inequality (1.14)
becomes inequalities (1.10) and (1.11), respectively.

\section{Lemmas}
\hspace{4mm} \setcounter{equation}{0}

In order to prove Theorems 1.1 and 1.2 we need two lemmas, which we
present in this section.

\medskip
{\bf Lemma 2.1.} Let $u\in[0,1]$, $s\geq 1$ and
\begin{equation}
f_{u,s}(x)=\frac{s}{2}\log(1-ux^2)-\log\left(\frac{\pi}{2\mathcal{K}(x)}\right).
\end{equation}
Then $f_{u,s}>0$ for all $x\in(0,1)$ if and only if $2su\leq 1$.

\medskip
{\bf\em Proof.} From (1.4) and (2.1) one has
\begin{equation}
f_{u,s}'(x)=-\frac{usx}{1-ux^2}+\frac{\mathcal{E}(x)-(1-x^2)\mathcal{K}(x)}{x(1-x^2)\mathcal{K}(x)}=\frac{F_{u,s}(x)}{x(1-x^2)(1-ux^2)\mathcal{K}(x)},
\end{equation}
where
\begin{equation}
F_{u,s}(x)=-sux^2(1-x^2)\mathcal{K}(x)+(1-ux^2)[\mathcal{E}(x)-(1-x^2)\mathcal{K}(x)].
\end{equation}

It follows from (1.1)-(1.3) and (2.3) together with elaborated
computations that
\begin{align*}
&\mathcal{E}(x)-(1-x^2)\mathcal{K}(x)\\
=&\frac{\pi}{2}\left[\sum\limits_{n=0}^{\infty}\frac{(-1/2,n)(1/2,n)}{(n!)^2}x^{2n}-(1-x^2)\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{(n!)^2}x^{2n}\right]\\
=&\frac{\pi}{2}\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{2n!(n+1)!}x^{2n+2},
\end{align*}
\begin{align}
\frac{2}{\pi}F_{u,s}(x)=&-sux^2(1-x^2)\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{(n!)^2}x^{2n}+(1-ux^2)\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{2n!(n+1)!}x^{2n+2}\nonumber\\
=&-su\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{(n!)^2}x^{2n+2}+su\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{(n!)^2}x^{2n+4}\nonumber\\
&+\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{2n!(n+1)!}x^{2n+2}-u\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{2n!(n+1)!}x^{2n+4}\nonumber\\
=&-sux^2-su\sum\limits_{n=0}^{\infty}\frac{(1/2,n+1)^2}{[(n+1)!]^2}x^{2n+4}+su\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{(n!)^2}x^{2n+4}\nonumber\\
&+\frac{x^2}{2}+\sum\limits_{n=0}^{\infty}\frac{(1/2,n+1)^2}{2(n+1)!(n+2)!}x^{2n+4}-u\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2}{2n!(n+1)!}x^{2n+4}\nonumber\\
=&x^2\left[\frac{1}{2}-su+\sum\limits_{n=0}^{\infty}\frac{(1/2,n)^2A_{n}}{2(n+1)!(n+2)!}x^{2n+2}\right],
\end{align}
where
\begin{equation}
A_{n}=su(n+2)(2n+\frac{3}{2})+(n+\frac{1}{2})^2-u(n+1)(n+2)>0.
\end{equation}

We divide the proof into two cases.

{\bf\em Case 1.1.} $2su\leq 1$. Then (2.2)-(2.5) lead to conclusion
that $f_{u,s}(x)$ is strictly increasing in $(0,1)$. Therefore,
$f_{u,s}(x)>f_{u,s}(0^+)=0$ for all $x\in(0,1)$ follows from (1.2)
and (2.1) together with the monotonicity of $f_{u,s}(x)$ in $(0,1)$.

{\bf\em Case 1.2.} $2su>1$. Then (2.2)-(2.4) lead to conclusion that
there exists $\delta_{1}\in(0,1)$ such that $f_{u,s}(x)$ is strictly
decreasing in $(0,\delta_{1})$. Therefore,
$f_{u,s}(x)<f_{u,s}(0^+)=0$ for all $x\in(0,\delta_{1})$ follows
from (1.2) and (2.1) together with the monotonicity of $f_{u,s}(x)$
in $(0,\delta_{1})$.

\medskip
{\bf Lemma 2.2.} Let $u\in [0,1]$, $s\geq 1$,
$\arth(x)=\frac{1}{2}\log\left(\frac{1+x}{1-x}\right)$ be the
inverse hyperbolic tangent function, and
\begin{equation}
g_{u,s}(x)=\frac{s}{2}\log(1-ux^2)+\log\left(\frac{\arth(x)}{x}\right).
\end{equation}
Then $g_{u,s}(x)>0$ for all $x\in(0,1)$ if and only if $3su\leq 2$.

\medskip
{\bf\em Proof.} From (2.6) one has
\begin{equation}
g_{u,s}'(x)=-\frac{sux}{1-ux^2}+\frac{x-(1-x^2)\arth{(x)}}{x(1-x^2)\arth{(x)}}=\frac{G_{u,s}(x)}{x(1-x^2)(1-ux^2)\arth(x)},
\end{equation}
where
\begin{equation}
G_{u,s}(x)=-sux^2(1-x^2)\arth(x)+(1-ux^2)[x-(1-x^2)\arth(x)].
\end{equation}

Making use of series expansion and (2.8) we have
\begin{align}
G_{u,s}(x)=&-sux^2(1-x^2)\sum\limits_{n=0}^{\infty}\frac{x^{2n+1}}{2n+1}+(1-ux^2)\left[x-(1-x^2)\sum\limits_{n=0}^{\infty}\frac{x^{2n+1}}{2n+1}\right]\nonumber\\
=&-su\sum\limits_{n=0}^{\infty}\frac{x^{2n+3}}{2n+1}+su\sum\limits_{n=0}^{\infty}\frac{x^{2n+5}}{2n+1}+(1-ux^2)
\sum\limits_{n=0}^{\infty}\frac{2x^{2n+3}}{(2n+1)(2n+3)}\nonumber\\
=&x^3\left[\frac{2}{3}-su+\sum\limits_{n=0}^{\infty}\frac{B_{n}x^{2n+2}}{(2n+1)(2n+3)(2n+5)}\right],
\end{align}
where
\begin{equation}
B_{n}=2u(s-1)(2n+5)+2(2n+1)>0.
\end{equation}

We divide the proof into two cases.

{\bf\em Case 1.1.} $3su\leq 2$. Then (2.7)-(2.10) lead to conclusion
that $g_{u,s}(x)$ is strictly increasing in $(0,1)$. Therefore,
$g_{u,s}(x)>g_{u,s}(0^+)=0$ for all $x\in(0,1)$ follows from (2.6)
together with the monotonicity of $g_{u,s}(x)$ in $(0,1)$.

{\bf\em Case 1.2.} $3su>2$. Then (2.7)-(2.9) lead to conclusion that
there exists $\delta_{2}\in(0,1)$ such that $g_{u,s}(x)$ is strictly
decreasing in $(0,\delta_{2})$. Therefore,
$g_{u,s}(x)<g_{u,s}(0^+)=0$ for all $x\in(0,\delta_{2})$ follows
from (2.6) and the monotonicity of $g_{u,s}(x)$ in $(0,\delta_{2})$.

\section{Proof of Theorems 1.1 and 1.2}
\hspace{4mm} \setcounter{equation}{0} {\bf Proof of Theorem 1.1.}
Since both $Q_{t,s}(a,b)$ and $AG(a,b)$ are symmetric and
homogeneous of degree 1. Without loss of generality, we assume that
$a>b$. Let $x=(a-b)/(a+b)\in(0,1)$. Then from (1.5) and (1.7)
together with $b/a=(1-x)/(1+x)$ we have
\begin{align}
\frac{AG(a,b)}{A(a,b)}=&\frac{AG(1,b/a)}{A(1,b/a)}=\frac{\pi}{\mathcal{K}\sqrt{1-(b/a)^2}(1+b/a)}\nonumber\\
=&\frac{\pi(1+x)}{2\mathcal{K}(2\sqrt{x}/(1+x))}=\frac{\pi}{2\mathcal{K}(x)}.
\end{align}

It follow from (1.12) and (3.1) that
\begin{align}
&\log\left(\frac{Q_{t,s}(a,b)}{AG(a,b)}\right)=\log\left(\frac{Q_{t,s}(a,b)}{A(a,b)}\right)-\log\left(\frac{AG(a,b)}{A(a,b)}\right)\nonumber\\
&=\frac{s}{2}\log\left[1-(1-2t)^2x^2\right]-\log\left[\frac{\pi}{2\mathcal{K}(x)}\right].
\end{align}

Therefore, Theorem 1.1  follows from Lemma 2.1 and (3.2).

\medskip
{\bf Proof of Theorem 1.2.} Since both $Q_{t,s}(a,b)$ and $L(a,b)$
are symmetric and homogeneous of degree 1. Without loss of
generality, we assume that $a>b$. Let $x=(a-b)/(a+b)\in(0,1)$. Then
(1.12) leads to
\begin{align}
&\log\left(\frac{Q_{t,s}(a,b)}{L(a,b)}\right)=\log\left(\frac{Q_{t,s}(a,b)}{A(a,b)}\right)-\log\left(\frac{L(a,b)}{A(a,b)}\right)\nonumber\\
&=\frac{s}{2}\log\left[1-(1-2t)^2x^2\right]+\log\left(\frac{\arth(x)}{x}\right).
\end{align}

Therefore, Theorem 1.2 follows from Lemma 2.2 and (3.3).\\

{\bf Acknowledgement:} This work was supported by the Natural
Science Foundation of China (Grant Nos. 11071059, 11071069,
11171307), and the Innovation Team Foundation of the Department of
Education of Zhejiang Province (Grant no. T200924).