text:
\begin{document}

\title{Quantification of the Banach-Saks property}
\author{Hana Bendov\'a, Ond\v{r}ej F.K. Kalenda and Ji\v{r}\'{\i} Spurn\'y}

\address{Department of Mathematical Analysis \\
Faculty of Mathematics and Physic\\ Charles University\\
Sokolovsk\'{a} 83, 186 \ 75\\Praha 8, Czech Republic}
\email{haaanja@gmail.com}
\email{kalenda@karlin.mff.cuni.cz}
\email{spurny@karlin.mff.cuni.cz}

\subjclass[2010]{46B20}
\keywords{Banach-Saks set; weak Banach-Saks set; uniformly weakly convergent sequence; $\ell_1$-spreading model; quantitative versions; measures of weak non-compactness}

\thanks{Our research was supported in part by the grant GA\v{C}R P201/12/0290}

\begin{abstract}
We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property.
We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish
a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence
and $\ell_1$-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an
analogue of the James distortion theorem for $\ell_1$-spreading models.
\end{abstract}
\maketitle

\section{Introduction}

A bounded subset $A$ of a Banach space $X$ is said to be a \emph{Banach-Saks set} if each sequence in $A$ has a Ces\`aro convergent subsequence. A Banach space $X$ is said to have the \emph{Banach-Saks property} if each bounded sequence in $X$ has a Ces\`aro convergent subsequence, i.e., if its closed unit ball $B_X$ is a Banach-Saks set.

This property goes back to S.~Banach and S.~Saks who proved in \cite{BS} that, in the modern terminology, the spaces $L^p(0,1)$ for $p\in(1,+\infty)$ enjoy the Banach-Saks property. Any Banach space with the Banach-Saks property is reflexive \cite{NiWa} and there are reflexive spaces without the Banach-Saks property \cite{baer}. On the other hand, superreflexive spaces enjoy the Banach-Saks property
(S.~Kakutani showed in \cite{Kaku} that uniformly convex spaces have the Banach-Saks property and by \cite{enflo} any superreflexive space admits a uniformly convex renorming).

A localized version of the mentioned result of \cite{NiWa} says that any Banach-Saks set is relatively weakly compact (see \cite[Proposition 2.3]{LA}). This inspires the definition of the \emph{weak Banach-Saks property} -- a Banach space $X$ is said to have this property if any weakly compact subset of $X$ is a Banach-Saks set, i.e., if any weakly convergent sequence in $X$ admits a Ces\`aro convergent subsequence. There are nonreflexive spaces enjoying the weak Banach-Saks property, for example $c_0$ or $L^1(\mu)$ \cite{farnum,szlenk}.

In the present paper we investigate possibilities of quantifying the Banach-Saks property. This is inspired by a large number of recent results on quantitative versions of various theorems and properties of Banach spaces, see, e.g., \cite{AC-meas,AC-jmaa,CKS,CMR,f-krein,Gr-krein,kks-adv,bendova}. Another approach to quantification of the Banach-Saks property and some related properties was followed by A.~Kryczka in a recent series of papers \cite{kr08a,kr08s,kr12,kr13}. More precisely, in the quoted papers mainly quantitative versions of Banach-Saks and weak Banach-Saks operators are investigated.

The quantification means, roughly speaking, to replace implications between some notions by inequalities between certain quantities. Let us now introduce the basic quantities we will use.

Let $(x_k)$ be a bounded sequence in a Banach space. Following \cite{kks-adv} we set
\begin{eqnarray}
\ca{x_k}&=&\inf_{n\in\en} \sup\{\norm{x_k-x_l}\colon k,l\ge n\},\\
\wca{x_k}&=&\inf \{\ca{x_{k_n}}\colon (x_{k_n})\text{ is a subsequence of }(x_k)\}.
\end{eqnarray}
The first quantity measures how far the sequence is from being norm Cauchy. Clearly, $\ca{x_k}=0$ if and only if the sequence $(x_k)$ is norm Cauchy (hence norm convergent).

Since we are interested in Ces\`aro convergence of sequences, it is natural to define
\begin{eqnarray}
\cca{x_k}&=&\ca{\frac1k(\sum_{i=1}^k x_i)},\\
\tcca{x_k}&=&\inf\{\cca{x_{k_n})}\colon (x_{k_n})\text{ is a subsequence of }(x_k)\}.
\end{eqnarray}

Let us remark that the quantities $\cca{}$ and $\tcca{}$ behave differently than the quantities $\ca{}$ and $\wca{}$. More precisely, the quantity $\ca{}$ decreases when passing to a subsequence but it is not the case of $\cca{}$. Indeed, a subsequence of a Ces\`aro convergent sequence need not be Ces\`aro convergent, in fact, any bounded sequence is a subsequence of a Ces\`aro convergent sequence.

For a  bounded set $A$ in a Banach space $X$ we introduce the following two quantities:
\begin{eqnarray}
\bs{A}&=&\sup\{\tcca{x_k}\colon (x_k)\subset A\},\\
\wbs{A}&=&\sup\{\tcca{x_k}\colon (x_k)\subset A\text{ is  weakly convergent}\}.
\end{eqnarray}

The first one measures how far is $A$ from being Banach-Saks. Indeed, $\bs{A}=0$ if and only if $A$ is Banach-Saks by Corollary~\ref{c:kvalit} below. Further, the same statement yields that $\wbs A=0$ if and only if any weakly convergent sequence in $A$ has a Ces\`aro convergent subsequence (let us stress that the limit could be outside $A$). The sets with the latter property will be called \emph{weak Banach-Saks sets}.

\section{Preliminaries}

We use mostly a standard notation. If $X$ is a Banach space, $B_X$ denotes its closed unit ball. If $A$ is any set, we denote by $\# A$ the cardinality of $A$. (We use this notation mainly for finite sets).

We investigate, among others, quantifications of the relationship of the Banach-Saks property to compactness and weak compactness.
To formulate such results we need some quantities measuring non-compactness and weak non-compactness.
There are several ways how to do it. We will use the notation from \cite{kks-adv}. Let us recall the basic quantities.

If $A$ and $B$ are two nonempty subsets of a Banach space $X$, we set
\begin{eqnarray*}
	\dd(A,B)&=&\inf\{\|a-b\|: a\in A, b\in B\}, \\
	\dh(A,B)&=&\sup\{\dd(a,B) : a\in A\}.
\end{eqnarray*}
Hence, $\dd(A,B)$ is the ordinary distance of the sets $A$ and $B$ and $\dh(A,B)$ is the non-symmetrized Hausdorff distance (note that the Hausdorff distance of $A$ and $B$ is equal to $\max\{\dh(A,B),\dh(B,A)\}$).

Let $A$ be a bounded subset of a Banach space $X$. Then the Hausdorff measure of non-compactness
of $A$ is defined by
\begin{equation*}
	\chi(A)=\inf\{\dh(A,F): \emptyset\ne F\subset X\mbox{ finite}\} =
	\inf\{\dh(A,K): \emptyset\ne K\subset X\mbox{ compact}\}.
\end{equation*}

This is the basic measure of non-compactness. We will need one more such measure:
\begin{equation*}
 \beta(A)=\sup\{\wca{x_k} : (x_k)\mbox{ is a sequence in }A\}.
\end{equation*}
It is easy to check that for any bounded set $A$ we have
\begin{equation*}

		\chi(A)\le\beta(A)\le 2\chi(A),
\end{equation*}
thus these two measures are equivalent. (And, of course, they equal zero if and only if the respective set is relatively compact.)

An analogue of the Hausdorff measure of non-compactness for measuring weak non-compactness is
the de Blasi measure of weak non-compactness
\begin{equation*}
	\omega(A)=\inf\{\dh(A,K) : \emptyset\ne K\subset X\mbox{ is weakly compact}\}.
\end{equation*}
Then $\omega(A)=0$ if and only if $A$ is relatively weakly compact. Indeed, the `if' part is obvious and the `only if' part follows from \cite[Lemma 1]{deblasi}.

There is another set of quantities measuring weak non-compactness. Let us mention two of them:
\begin{eqnarray*}
\wk{A}&=&\dh(\wscl{A},X),\\
\wck{X}{A}&=&\sup\{\ \dd(\clu{X}{x_k},X) : (x_k)\mbox{ is a sequence in }A\}.
\end{eqnarray*}
By $\wscl{A}$ we mean the weak$^*$ closure of $A$ in $X^{**}$ (the space $X$ is canonically embedded in $X^{**}$) and $\clu{X}{x_k}$ is the set of all weak$^*$ cluster points in $X^{**}$ of  the sequence $(x_k)$.
It follows from \cite[Theorem 2.3]{AC-meas} that for any bounded subset $A$ of a Banach space $X$
we have
\begin{gather*}

\wck X{A}\le\wk{A}\le 2\wck X{A}, \\
\wk{A}\le \omega(A).
\end{gather*}

So, putting together these inequalities with the measures of norm non-compactness we obtain the
following diagram:
\begin{equation*}

	\begin{array}{cccccccc}
	&&\chi(A)&\le&\beta(A)&\le&2\chi(A) \\&&\upleq&&&&& \\
	&&\omega(A)&&&&& \\ &&\upleq&&&&& \\
	\wck{X}{A}&\le&\wk{A}&\le& 2\wck{X}{A}&&
\end{array}
\end{equation*}

Let us remark that the inequality $\omega(A)\le\chi(A)$ is obvious and that the quantities $\omega(\cdot)$ and $\wk{\cdot}$ are not equivalent, see \cite{tylli-cambridge,AC-meas}.

Some quantities related to the Banach-Saks property were defined and used in \cite{kr12}. Let us recall them.
If $(x_k)$ is a bounded sequence in $X$, we define the \emph{arithmetic separation} of $(x_k)$ by
$$\asep(x_k)=\inf\left\{\norm{\frac1{\#F}\left(\sum_{n\in F} x_n-\sum_{n\in H} x_n\right)} :
F,H\subset \en, \#F=\#H, \max F<\min H\right\}.$$
Further, for any bounded set $A\subset X$ define
$$\begin{aligned}
\varphi(A)&=\sup\{\asep(x_k):(x_k)\mbox{ is a sequence in }A\},\\
\varphi'(A)&=\sup\{\asep(x_k):(x_k)\mbox{ is a weakly convergent sequence in }A\}.
\end{aligned}$$
The quantities $\asep$ and $\varphi$ are from \cite{kr12}, the quantity $\varphi'$ is an obvious modification.

\section{Quantitative relation to compactness and weak compactness}

Since any convergent sequence is also Ces\`aro convergent, relatively norm compact sets are Banach-Saks. Further, a set is Banach-Saks if and only if it is weakly Banach-Saks and relatively weakly compact. In this section we investigate quantitative versions of these relationships. Positive results are summed up in the following theorem.

\begin{thm}
\label{P:wbs<bs<beta}
Let $A$ be a bounded subset of a Banach space $X$. Then
\begin{equation}
\label{E:wbs<bs<beta}
\max\{\wck XA,\wbs{A}\}\le \bs{A}\le \beta(A).
\end{equation}
\end{thm}

The second inequality in the theorem quantifies the implication

\begin{center}
\textit{$A$ is relatively norm compact $\Rightarrow$ $A$ is Banach-Saks,}
\end{center}

the first one quantifies the implication

\begin{center}
\textit{$A$ is Banach-Saks $\Rightarrow$ $A$ is weakly Banach-Saks and relatively weakly compact.}
\end{center}

We point out that the latter implication cannot be quantified by using the de Blasi measure $\omega$ and that the converse implication is purely qualitative. This is illustrated by the following two examples.

\begin{example}
\label{EX:omega,bs}
There exists a separable Banach space $X$ such that
\[
\forall \ep>0\ \exists A\subset B_X\colon \bs{A}<\ep\ \&\ \omega(A)>\frac12.
\]
\end{example}

\begin{example}
\label{EX:be+ell1}
There exists a separable Banach space $X$ such that
\[
\forall \ep>0\ \exists A\subset B_X\colon \bs{A}=2\ \&\ \wbs{A}=0\ \&\ \omega(A)<\ep.
\]
\end{example}

The rest of this section will be devoted to the proofs of these results.
The proof of Example~\ref{EX:be+ell1} will be postponed to the next section since we will
make use of Theorem~\ref{P:wbs<2wu<4sm}.

\begin{proof}[Proof of Theorem~\ref{P:wbs<bs<beta}]

We start by the second inequality. It immediately follows from the following lemma which is a quantitative version of the well-known fact that any convergent sequence is Ces\`aro convergent.

\begin{lemma} Let $(x_k)$ be a bounded sequence in a Banach space $X$. Then
$$\cca{x_k}\le\ca{x_k}.$$
\end{lemma}

\begin{proof} Set $M=\sup_k\|x_k\|$ and fix any $c>\ca{x_k}$. Further, set
$y_m=\frac1m\sum_{i=1}^m x_{i}$ for $m\in\en$.

We find $n_0\in\en$ such that $\norm{x_{n}-x_{m}}<c$ for each $n,m\ge n_0$.
Let $\ep>0$ be given. We find $n_1\ge n_0$ such that $\frac{Mn_0}{n_1}<\ep$.
Then we have for $n_1\le n\le m$ inequalities
\[
\begin{aligned}
\norm{y_m-y_n}&= \norm{\frac1m\sum_{i=1}^ m x_{i}-\frac1n\sum_{j=1}^nx_{j}}
=\norm{\sum_{i=1}^m\sum_{j=1}^n\frac{x_i-x_j}{mn}}\\
&\le
\sum_{i=1}^m\sum_{j=1}^{n_0}\frac{\norm{x_i-x_j}}{mn}
+\sum_{i=1}^{n_0}\sum_{j=n_0+1}^{n}\frac{\norm{x_i-x_j}}{mn}
+\sum_{i=n_0+1}^{m}\sum_{j=n_0+1}^{n}\frac{\norm{x_i-x_j}}{mn} \\
&\le \frac{2M m n_0}{mn}+\frac{2Mn_0(n-n_0)}{mn}+ \frac{(m-n_0)(n-n_0)c}{mn}\\
&\le \frac{2M n_0}{n_1}+\frac{2Mn_0}{n_1} + c<c+4\ep.
\end{aligned}
\]
Since $\ep$ is arbitrary, $\cca{x_{k}}=\ca{y_m}\le c$. \end{proof}

Since the inequality $\bs A\ge \wbs A$ is obvious, it remains to prove $\bs A\ge \wck XA$. To do that we first prove the following lemma using an auxiliary quantity $\gamma_0$ defined by the formula

\begin{eqnarray*}\gamma_0(A)&=&\sup\{\ |\lim_m\lim_n x^*_m(x_n)|:      \\ && \qquad\qquad (x^*_m)\mbox{ is a sequence in } B_{X^*} \mbox{ weak* converging to }0,  \\ && \qquad\qquad (x_n) \mbox{ is a sequence in } A  \\ && \qquad\qquad\mbox{ and all the involved limits exist}\}.
\end{eqnarray*}

\begin{lemma}
\label{L:gamma0<bs}
Let $A$ be a bounded set in a Banach space $X$. Then
\begin{equation*}

\gamma_0(A)\le \bs{A}.
\end{equation*}
\end{lemma}

\begin{proof}
Let $\gamma_0(A)>c$. Then there exists a sequence $(x_k)$ in $A$ and a sequence $(x_j^*)$ in $B_{X^*}$ weak* converging to $0$ such that $\lim_j\lim_k x_j^*(x_k)>c$.
Without loss of generality we may assume that
\[
\forall j\in\en\colon \lim_k x_j^*(x_k)>c.
\]
Let $y_k=\frac1k(x_1+\cdots +x_k)$, $k\in\en$. Then
\[
\forall j\in\en\colon \lim_k x_j^*(y_k)=\lim_k x_j^*(x_k)>c.
\]
Let $\ep>0$ be arbitrary. Fix $k\in\en$. Since $(x^*_j)$ weak* converges to zero, there exists $j\in\en$ such that $x^*_j(y_k)<\ep$. Then we find $l>k$ such that $x_j^*(y_l)>c$. Then
\[
\norm{y_l-y_k}\ge x_j^*(y_l-y_k)>c-\ep.
\]
Hence $\cca{x_k}=\ca{y_k}\ge c$. Further, any subsequence of $(x_k)$ has the same properties, hence $\tcca{x_k}\ge c$. Therefore $\bs{A}\ge c$ and the proof is complete.
\end{proof}

Let us now complete the proof of the remaining inequality.

Assume first that $X$ is separable. Then $(B_{X^*},w^*)$ is metrizable, and thus angelic.
By \cite[Theorem 6.1]{CKS}, $\gamma_0(A)=\wck{X}{A}$, and thus $\wck{X}{A}\le \bs A$ by the previous lemma.

Assume now that $X$ is arbitrary and $\wck{X}{A}>c$ for some $c>0$. Let $(x_k)$ be a sequence in $A$ with $\dd(\clu{X}{x_k},X)>c$.
Set $Y=\ov{\span}\{x_k\colon k\in\en\}$. Then $Y$ is a separable subspace of $X$ and $\dd(\clu{Y}{x_k},Y)>c$.

Indeed, let $y^{**}\in \clu{Y}{x_k}$ be arbitrary. Set
$x^{**}(x^*)=y^{**}(x^*|_Y)$, $x^*\in X^*$. Then $x^{**}\in\clu{X}{x_k}$ and for each $y\in Y$ we have due to the Hahn-Banach theorem
\[
\begin{aligned}
\norm{y^{**}-y}_{Y}&=\sup_{y^*\in B_{Y^*}} \abs{y^{**}(y^*)-y^*(y)}\\
&=\sup_{x^*\in B_{X^*}}\abs{(y^{**}(x^*|_Y)-x^*(y)}\\
&=\norm{x^{**}-y}_X.
\end{aligned}
\]
Therefore
$$\dd_{Y^{**}}(y^{**},Y)=\dd_{X^{**}}(x^{**},Y)\ge
\dd_{X^{**}}(x^{**},X),$$
hence
$$\dd(\clu{Y}{x_k},Y)\ge \dd(\clu{X}{x_k},X)>c.$$
Hence $\wck Y{A\cap Y}>c$, by the separable case we get
\[
\bs{A}\ge \bs{A\cap Y}\ge \wck{Y}{A\cap Y}>c,
\]
which concludes the proof of Theorem~\ref{P:wbs<bs<beta}.
\end{proof}

\begin{proof}[Proof of Example~\ref{EX:omega,bs}]
Let us fix $\alpha>0$ and consider an equivalent norm $|\cdot|_\alpha$ on $\ell_1$
given by the formula
$$|x|_\alpha=\max\{\alpha\|x\|_1,\|x\|_\infty\}$$
and let
$$X=\bigg(\oplus_{n=1}^\infty (\ell_1,|\cdot|_{1/n})\bigg)_{\ell_1}.$$
It is clear that $X$ is a separable Banach space. Further, $X$ has the Schur property as it is an $\ell_1$-sum of spaces with the Schur property
(this follows by a straightforward modification of the proof that $\ell_1$ has the Schur property, see \cite[Theorem~5.19]{fhhmpz}).

Further, let us define the following elements of $X$:
 $${x}^n_k=(0,\dots,0,\stackrel{n\text{-th}}{e_k},0,\dots),\qquad n,k\in\en,$$
where $e_k$ is the $k$-th canonical basic vector in $\ell_1$. Fix $n\in\en$ and set
$$A_n=\{{x}^n_k\colon k\in\en\}.$$
Since $\|{x}^n_k\|=1$ for  $k\in\en$, we get $A_n\subset B_X$. Further, $\|{x}^n_k-{x}^n_{k'}\|\ge 1$ whenever $k\ne k'$, so $\beta(A_n)\ge 1$ and hence $\chi(A_n)\ge \frac12$. Since $X$ has the Schur property, we get $\omega(A_n)=\chi(A_n)\ge\frac12$.

Let $(z_k)$ be an arbitrary sequence in $A_n$. If it has a constant subsequence, then $\tcca{z_k}=0$. Otherwise there is a one-to-one subsequence $(z_{k_i})$. It is clear that
$$\norm{\frac1m(z_{k_1}+\dots+z_{k_m})}=\norm{\frac1m({x}^n_1+\dots+{x}^n_m)}=\max\left\{\frac1n,\frac1m\right\} \mbox{\quad for }m\in\en,$$
hence $\cca{z_{k_i}}\le \frac2n$. So $\bs{A_n}\le\frac2n$.

This completes the proof.
\end{proof}

\section{Quantitative characterization of weak Banach-Saks sets}

It follows from the results summarized in \cite[Section 2]{LA} that the following assertions are equivalent
for a subset $A$ of a Banach space:
\begin{itemize}
	\item $A$ is a weak Banach-Saks set.
	\item Each weakly convergent sequence in $A$ admits a uniformly weakly convergent subsequence.
	\item No weakly convergent sequence in $A$ generates an $\ell_1$-spreading model.
\end{itemize}

More precisely, in the quoted paper the authors formulate characterizations of Banach-Saks sets, adding to the other assertions the assumption that $A$ is relatively weakly compact. In this section we will prove a quantitative version of these characterizations. To formulate it, we need to introduce some natural quantities related to the above-mentioned properties:

Let $(x_k)$ be a sequence which weakly converges to some $x\in X$. This sequence is said to be \emph{uniformly weakly convergent} if
for each $\ep>0$
$$\exists n\in\en\ \forall x^*\in B_{X^*}\colon \#\{k\in\en\colon \abs{x^*(x_k-x)}>\ep\}\le n.$$
The quantity $\wu{x_k}$ is then defined to be the infimum of all $\ep>0$ satisfying this condition. Further, we set
\begin{eqnarray*}

\twu{x_k}&=&\inf\{\wu{x_{k_n}}\colon (x_{k_n})\text{ is a subsequence of }x_k\}.
\end{eqnarray*}
Finally, for a bounded set $A$ we define
\begin{eqnarray*}
\swu{A}&=&\sup\{\twu{x_k}\colon (x_k)\subset A\text{ is weakly convergent}\}.
\end{eqnarray*}

We continue by a definition related to spreading models.
Let $(x_k)$ be a bounded sequence. We say that it \emph{generates an $\ell_1$-spreading model with $\delta>0$} if
\[
\forall F\subset \en, \#F\le\min F\; \forall (\alpha_i)_{i\in F}\in\er^F\colon \norm{\sum_{i\in F}\alpha_ix_i}\ge \delta\sum_{i\in F} \abs{\alpha_i}.
\]
The sequence $(x_k)$ \emph{generates an $\ell_1$-spreading model} if it generates an $\ell_1$-spreading model with some $\delta>0$.

For a bounded set $A$ we set
\[
 \begin{aligned}
\sm{A}=\sup\{\delta>0\colon
&\exists (x_k)\subset A, x_k\overset{w}{\rightarrow} x,\\
& (x_k-x)\text{ generates an $\ell_1$-spreading model with $\delta$} \}.
\end{aligned}
\]

Now we are ready to formulate the promised quantitative characterizations.

\begin{thm}
\label{P:wbs<2wu<4sm}
Let $A$ be a bounded set in a Banach space $X$. Then
\begin{equation}\label{E:wbs<wu<sm}
\sm{A}\le \frac12\wbs{A}\le \swu{A}\le 2\sm{A}.
\end{equation}
\end{thm}

\begin{remark} For any bounded set $A\subset X$  have
\begin{equation}\label{eq:phi}
\sm{A}\le \frac12\varphi'(A)\le \wbs{A}.
\end{equation}
Indeed, it is easy to check that $\cca{x_k}\ge\frac12\asep(x_k)$ for any bounded sequence $(x_k)$. Since the quantity $\asep$ cannot decrease when we pass to a subsequence, we get $\tcca{x_k}\ge\frac12\asep(x_k)$, hence the second inequality in \eqref{eq:phi} follows. The first inequality follows from Lemma~\ref{l:phi} below.

Hence, by \eqref{eq:phi} and Theorem~\ref{P:wbs<2wu<4sm} the quantity $\varphi'$ inspired by \cite{kr12}
is equivalent to our quantities. Further, since clearly $\varphi'\le\varphi$, we get
$$\sm{A}\le\frac12\varphi'(A)\le \frac12\varphi(A)\le \bs{A}.$$
The last inequality follows from the previous paragraph. It seems not to be clear whether the quantity $\varphi$ is equivalent to $\bs{ }$ also for sets which are not relatively weakly compact.
\end{remark}

As a consequence of Theorem~\ref{P:wbs<2wu<4sm} we get that the introduced quantities $\wbs{}$ and $\bs{}$ really measure the failure
of the weak Banach-Saks (Banach-Saks, respectively) property of a set.

\begin{cor}\label{c:kvalit}
Let $A$ be a bounded set in a Banach space $X$.
\begin{itemize}
	\item If $\wbs{A}=0$, then $A$ is a weak Banach-Saks set.
	\item If $\bs{A}=0$, then $A$ is a Banach-Saks set.
\end{itemize}
\end{cor}

To prove the corollary we will use the following lemma, which also proves the inequality $\wbs{A}\le 2\swu{A}$ from Theorem~\ref{P:wbs<2wu<4sm}.

\begin{lemma}\label{L:ccawu}
Let $(x_k)$ be a sequence in a Banach space $X$ which weakly converges to some $x\in X$. Then $\cca{x_k}\le 2\wu{x_k}$.
\end{lemma}

\begin{proof} Let $M=\sup\limits_k \norm{x_k}$. Fix an arbitrary $c>\wu{x_k}$. Let $N\in\en$ be such that
\begin{equation}\label{E:N}
\forall x^*\in B_{X^*}\colon \#\{k\in\en\colon \abs{x^*(x_k-x)}>c\}\le N.
\end{equation}
Set $z_k=\frac1k(x_1+\cdots+x_k)$, $k\in\en$. Given $\ep>0$, we find $n_0\in\en$ such that $\frac{2MN}{n_0}<\ep$.

Now, for any couple of indices $n_0\le n<m$ and $x^*\in B_{X^*}$ we obtain from \eqref{E:N}
\[
\begin{aligned}
\abs{x^*(z_m-z_n)}&=\abs{x^*(z_m-x)+x^*(x-z_n)}\\
&=\abs{\left(\frac1m-\frac1n\right)\left(x^*(x_1-x)+\cdots +x^*(x_n-x)\right)\right.\\ &\qquad\left.+ \frac1m\left(x^*(x_{n+1}-x)+\cdots+x^*(x_m-x)\right)}\\
&\le \frac{2NM}{n}+c\left(n\left(\frac1n-\frac1m\right)+\frac{m-n}{m}  \right)\\
&\le \ep+c\left(2-2\frac{n}{m}   \right)\\
&\le 2c+\ep.
\end{aligned}
\]
Since $x^*\in B_{X^*}$ is arbitrary,
\[
\norm{z_m-z_n}\le 2c+\ep
\]
for each $n_0\le n<m$. Thus $\cca{x_k}=\ca{z_k}\le 2c$, which completes the proof.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{c:kvalit}.]
Suppose that $\wbs{A}=0$. By Theorem~\ref{P:wbs<2wu<4sm} we deduce $\swu{A}=0$. Let $(x_k)$ be a weakly convergent sequence in $A$.
Then we can construct by induction sequences $(y_k^n)_{k=1}^\infty$ for $n\in\en$ such that
\begin{itemize}
	\item $y_k^1=x_k$, $k\in\en$;
	\item $(y_k^{n+1})$ is a subsequence of $(y_k^n)$;
	\item $\wu{y_k^{n+1}}<\frac1{n+1}$.
\end{itemize}
Consider the diagonal sequence $(z_k)=(y_k^k)$. Then $(z_k)$ is a subsequence of $(x_k)$, $\wu{z_k}=0$ and hence $\cca{z_k}=0$ (by Lemma~\ref{L:ccawu}),
so $(z_k)$ is Ces\`aro convergent. This completes the proof that $A$ is a weak Banach-Saks set.

For the second part, suppose that $\bs{A}=0$. Hence $\wbs{A}=0$, so by the first part, $A$ is a weak Banach-Saks set. Further, by Theorem~\ref{P:wbs<bs<beta} we get $\wck XA=0$, hence $A$ is relatively weakly compact. Therefore $A$ is a Banach-Saks set.
\end{proof}

We continue with the inequality $2\sm{A}\le \wbs{A}$. It follows immediately from the following lemma.

\begin{lemma}\label{L:sm} Let $(x_k)$ be a bounded sequence in a Banach space $X$ and $x\in X$. Suppose that  $(x_k-x)$ generates an $\ell_1$-spreading model with a constant $c$. Then $\tcca{x_{k^3}}\ge 2c$.
\end{lemma}

\begin{proof}
Since $\cca{x_k-x}=\cca{x_k}$, we may without loss of generality suppose that $x=0$.
Let $M=\sup_k\|x_k\|$.
Let $(z_k)$ be any subsequence of $(x_{k^3})$. We will show that $\cca{z_k}\ge 2c$. To this end we notice that, for $F\subset \en$ satisfying $\#F\le (\min  F)^3$, we have $\norm{\sum_{i\in F}\alpha_iz_i}\ge c\sum_{i\in F}\abs{\alpha_i}$ whenever $(\alpha_i)$ are  arbitrary scalars.

Let $N\in\en$ be given. We set $n=N+N^2$ and $m=N+N^3$. Then the set $F=\{N+1,\dots, m\}$ satisfies $\#F\le (\min F)^3$, which implies
\[
\begin{aligned}
\left\|\frac1m\left(z_1\right.\right.&\left.\left.+\cdots+z_m\right)-\frac1n\left(z_1+\cdots+z_n\right)\right\|\\
&=\norm{\left(\frac1m-\frac1n\right)(z_1+\cdots+z_n)+\frac1m(z_{n+1}+\cdots +z_m)}\\
&\ge \norm{\left(\frac1m-\frac1n\right)(z_{N+1}+\cdots+z_n)+\frac1m(z_{n+1}+\cdots+z_m)}
\\&\qquad\qquad-\norm{\left(\frac1m-\frac1n\right)(z_1+\cdots+z_N)}\\
&\ge c\left(\frac1n-\frac1m\right)(n-N)+\frac{c}{m}(m-n)-MN\left(\frac1n-\frac1m\right)\\
&=c\frac{N^2(N^3-N^2)}{(N+N^3)(N+N^2)}+c\frac{N^3-N^2}{N^3+N}-M\frac{N(N^3-N^2)}{(N+N^3)(N+N^2)}.
\end{aligned}
\]
Since the last term converges to $2c$ as $N$ tends to infinity, $\cca{z_k}\ge 2c$, which completes the proof.
\end{proof}

The following lemma provides the promised proof of the first inequality in \eqref{eq:phi}. It is an easier variant of the previous lemma.

\begin{lemma}\label{l:phi} Let $(x_k)$ be a bounded sequence in a Banach space $X$ and $x\in X$. Suppose that  $(x_k-x)$ generates an $\ell_1$-spreading model with a constant strictly greater than $c$. Then there is $n\in\en$ such that
$\asep((x_{k^2})_{k\ge n})>2c$.
\end{lemma}

\begin{proof} Fix $d>c$ such that  $(x_k-x)$ generates an $\ell_1$-spreading model with the constant $d$. Let $M=\sup_k\norm{x_k}$.
Fix $n\in\en$ such that $n\ge2$ and
$$2d-\sqrt{\frac2n}(d+2M)>2c.$$
If we show that
$$\asep((x_{k^2})_{k\ge n})\ge 2d-\sqrt{\frac2n}(d+2M),$$
the proof will be completed. To do that fix any $m\in\en$ and indices
$$n\le p_1<p_2<\dots<p_m<q_1<q_2<\dots< q_m.$$
If $m\le\frac12n^2$, then $2m\le n^2$ and therefore
$$\norm{\frac1m\left(\sum_{i=1}^m x_{p_i^2}-\sum_{i=1}^m x_{q_i^2}\right)}
=\norm{\frac1m\left(\sum_{i=1}^m (x_{p_i^2}-x)-\sum_{i=1}^m (x_{q_i^2}-x)\right)}
\ge d\cdot \frac1m\cdot 2m =2d.$$
Finally, suppose that $m>\frac12n^2$. Let $j\in\{1,\dots,m\}$ be the smallest number satisfying
$p_j^2\ge 2m$. Such a number exists since $p_m^2\ge m^2\ge 2m$ (as $m>\frac12n^2\ge2$). Moreover, necessarily $j\le \sqrt{2m}+1$. Indeed, if $p_i^2<2m$, then $i^2\le p_i^2<2m$ and hence $i<\sqrt{2m}$. We have
$$\begin{aligned}
\left\|\frac1m\left(\sum_{i=1}^m x_{p_i^2}\right.\right.&-\left.\left.\sum_{i=1}^m x_{q_i^2}\right)\right\|
=\norm{\frac1m\left(\sum_{i=1}^m (x_{p_i^2}-x)-\sum_{i=1}^m (x_{q_i^2}-x)\right)}\\
&\ge\norm{\frac1m\left(\sum_{i=j}^m (x_{p_i^2}-x)-\sum_{i=1}^m (x_{q_i^2}-x)\right)}-
\norm{\frac1m\sum_{i=1}^{j-1} (x_{p_i^2}-x)}\\
&\ge d\cdot \frac1m\cdot(2m-\sqrt{2m})-\frac1m\cdot 2M \cdot\sqrt{2m}
\\&=2d- \sqrt{\frac2m}(d+2M)\ge 2d- \sqrt{\frac2n}(d+2M).
\end{aligned}$$
This completes the proof.
\end{proof}

Finally we will show $\swu{A}\le 2\sm{A}$. This follows from the following lemma. The lemma essentially follows from \cite[Theorem 2.1]{LA}. However, since this theorem is not proved in \cite{LA} and we were not able to completely
recover it from the references therein and since we, moreover, need to know precise constants, we decided to give
here a complete proof of the lemma using the results of \cite{AMT,LA-T}.

\begin{lemma}\label{L:ramsey} Let $c>0$ and let $(x_k)$ be a weakly null sequence with $\twu{x_k}>c$. Then there is a subsequence of $(x_k)$ generating an $\ell_1$-spreading model with the  constant $\frac c2$. \end{lemma}

\begin{proof} Fix $\delta\in(0,1)$ such that $\twu{x_k}>c+3\delta$. We begin by showing that without loss of generality (up to passing to a subsequence)
we may suppose that for any finite set $F\subset\en$ the following holds:

\begin{equation}\label{eq:AMT}
\begin{aligned}
\Big(
\exists x^*\in B_{X^*}&\;\forall k\in F\colon x^*(x_k)\ge c+3\delta \Big)
\\ \implies &\exists y^*\in B_{X^*}
\Big(\forall k\in F: y^*(x_k)>c+2\delta\mbox{ and }\sum_{k\in \en\setminus F} \abs{y^*(x_k)}<\delta\Big).
\end{aligned}
\end{equation}

To this end we will use \cite[Lemma 2.4.7]{AMT}. The set
$$D=\{ (x^*(x_k))_{k=1}^\infty \colon x^*\in B_{X^*} \}$$
is a convex symmetric weakly compact subset of $c_0$. (Indeed, the mapping $x^*\mapsto (x^*(x_k))_{k=1}^\infty$ is a mapping of $B_{X^*}$ onto $D$ which is continuous from the weak$^*$ topology to the weak topology of $c_0$.) Further, fix $\ep\in(0,1)$ such that
$$(1-\ep)(c+3\delta)>c+2\delta\mbox{ and }\ep(c+3\delta)<\delta.$$
Then the quoted lemma applied to the set $D$, the constant $c+3\delta$ in place of $\delta$ and $\ep$
yields an infinite set $M\subset \en$ such that for any finite set $F\subset M$ \eqref{eq:AMT} is satisfied with $M$ in place of $\en$.
Up to passing to a subsequence we may suppose that $M=\en$, therefore without loss of generality \eqref{eq:AMT} holds for any finite set $F\subset\en$.

Further, set
$$\K_0=\{ \{k\in\en\colon x^*(x_k)\ge c+3\delta\} \colon x^*\in B_{X^*} \},$$
let $\K$ be the family of those subsets of $\en$ which are contained in an element of $\K_0$.

To complete the proof we will use the following lemma which is an easier variant of \cite[Theorem 2]{LA-T} and was suggested to us by the referee:

\begin{lemma}\label{LLL} Let $\F$ be a nonempty hereditary family of finite subsets of $\en$.
Then there is an infinite set $M\subset \en$ such that one of the following conditions is satisfied:
\begin{itemize}
	\item[(a)] There is some $d\in\en\cup\{0\}$ such that $\F\cap \P(M)=[M]^{\le d}$.
	\item[(b)] There is a strictly increasing mapping $f\colon M\to \en$ such that $\{F\subset M:\#F\le f(\min F)\}\subset \F$.
\end{itemize}
\end{lemma}

(A family $\F$ is heredirary if $B\in\F$ whever $B\subset A$ and $A\in F$. Furhter,
by $\P(M)$ we denote the power set of $M$, by $[M]^{\le d}$ the family of all subsets of $M$ of cardinality at most $d$,
by $[M]^{d}$ the family of all subsets of $M$ of cardinality exactly $d$. )

\begin{proof} Suppose that there is an infinite set $M\subset \en$ and $d\in\en$ such that $\F\cap[M]^{d}=\emptyset$.
Let $d_0\in\en$ be the minimal number with this property. If $d_0=1$, then $\F\cap\P(M)=\{\emptyset\}=[M]^{\le 0}$. Suppose that $d_0>1$. By the classical Ramsey theorem there is an infinite set $N\subset M$ such that either $\F\cap[N]^{d_0-1}=\emptyset$ or $[N]^{d_0-1}\subset \F$. The first possibility cannot occur due to the minimality of $d_0$.
The second one implies $\F\cap \P(N)=[N]^{\le d_0-1}$ (as $\F$ is hereditary and $\F\cap [N]^{d_0}=\emptyset$). Hence, the condition (a) is satisfied.

Next suppose that such an infinite set $M\subset\en$ and $d\in\en$ do not exist. It means that $\F\cap[M]^d\ne\emptyset$ for each inifinite $M\subset \en$ and each $d\in\en$. Using the classical Ramsey theorem we deduce that for any infinite $M\subset\en$ and any $d\in\en$ there is an infinite set $N\subset M$ such that $[N]^d\subset\F$. Since $\F$ is hereditary,
automatically $[N]^{\le d}\subset F$. Therefore we can by induction construct a sequence $(M_n)$ with the following properties.
\begin{itemize}
	\item $\en\supset M_1\supset M_2\supset\cdots$
	\item $M_n$ is infinite for each $n\in\en$.
	\item $[M_n]^{\le n}\subset \F$ for each $n\in\en$.
\end{itemize}
Choose a strictly increasing sequence $(m_n)$ of natural numbers such that $m_n\in M_n$ for each $n\in\en$. Set $M=\{m_n:n\in\en\}$ and define $f\colon M\to \en$ by $f(m_n)=n$. Then $f$ witnesses that (b) is satisfied for $M$.
\end{proof}

Now we continue the proof of Lemma~\ref{L:ramsey}. We apply the preceding lemma to the family $\K$. We observe that the case (a) cannot occur. Indeed, suppose that $M\subset \en$ is infinite and $d\in\en$ such that $\K\cap\P(M)=[M]^{\le d}$.
Then, in particular, $\K_0\cap\P(M)\subset [M]^{\le d}$, hence
for each $x^*\in B_{X^*}$ we have
\begin{multline*}\#\{k\in M:\abs{x^*(x_k)}\ge c+3\delta\} = \#\{k\in M:{x^*(x_k)}\ge c+3\delta\}\\ + \#\{k\in M:{(-x^*)(x_k)}\ge c+3\delta\}\le 2d,\end{multline*}
hence $\twu{x_k}\le c+3\delta$, a contradiction with the choice of $\delta$.

Thus the case (b) must occur. Fix the relevant set $M$ and function $f$. Up to passing to a subsequence we may suppose that $M=\en$. Now we are going to check that the sequence $(x_{k})$ (which was made from the original one by passing twice to a subsequence) generates an $\ell_1$-spreading model with the constant $\frac c2$. To do that let $F\subset\en$ be a subset satisfying $\#F\le\min F$ and $(\alpha_i)_{i\in F}$ be any choice of scalars. Set $F^+=\{i\in F: \alpha_i>0\}$ and  $F^-=\{i\in F: \alpha_i<0\}$. Without loss of generality we may suppose that
$\sum_{i\in F^+}\alpha_i\ge \sum_{i\in F^-}(-\alpha_i)$ (otherwise we can pass to $(-\alpha_i)$). Since $\#F^+\le\#F\le \min F\le f({\min F})\le f(\min F^+)$, we get $F^+\in\K$. Therefore using \eqref{eq:AMT} we can find $x^*\in B_{X^*}$ such that
$$x^*(x_{i})>c+2\delta\mbox{ for }i\in F^+\mbox{\qquad and\qquad} \sum_{i\in\en\setminus F^+}|x^*(x_{i})|<\delta.$$
Then
$$\begin{aligned}\norm{\sum_{i\in F} \alpha_i x_{i}}&\ge \sum_{i\in F} \alpha_i x^*(x_{i}) \ge \sum_{i\in F^+} \alpha_i(c+2\delta)-\sum_{i\in F\setminus F^+}|\alpha_i||x^*(x_{i})|\\&\ge\frac12(c+2\delta)\sum_{i\in F} \abs{\alpha_i} - \delta\sum_{i\in F} \abs{\alpha_i} =\frac c2\sum_{i\in F} \abs{\alpha_i},\end{aligned}$$
which completes the argument.
\end{proof}

\begin{remark} The proof of Theorem~\ref{P:wbs<2wu<4sm} is inspired by the proof of \cite[Theorem 2.4]{LA}. More precisely, Lemma~\ref{L:ccawu} is straightforward. Lemma~\ref{L:sm} is a more elementary and more precise version of the argument in \cite[p. 2256, second paragraph]{LA}. The quoted approach would yield $\wbs{A}\ge\frac14\sm{A}$, with a little more care  $\wbs{A}\ge\frac12\sm{A}$. Our approach is more elementary, we use just the triangle inequality and not the possibility to extract a basic subsequence, and we obtain the best possible inequality.
Finally, Lemma~\ref{L:ramsey} is a more precise version of the proof of \cite[Theorem 2.4(a)$\implies$(b)]{LA}.
\end{remark}

We finish this section by giving the last missing proof from the previous section.

\begin{proof}[Proof of Example~\ref{EX:be+ell1}]
Let $B$ be the Baernstein space, i.e., the separable reflexive space constructed in \cite{baer} and let $(b_n)$ denotes its canonical basis. Let $X=B\oplus_{\infty} \ell_1$ and $(e_n)$ be the standard basis of $\ell_1$. For $\ep\in[0,1]$ set $A_\ep=\{(b_n,\ep e_n)\colon n\in\en\}$.
Then $A_\ep\subset B_X$. Since $(b_n)$ converges weakly to zero, the set $A_0$ is weakly compact and hence $\omega(A_\ep)\le \dh(A_\ep,A_0)\le\ep$.

Fix $\ep\in(0,1]$. It is clear that the sequence $(b_n,\ep e_n)$ is equivalent to the $\ell_1$ basis and hence $A_\ep$ contains
no nontrivial weakly convergent sequences, and thus trivially $\wbs{A_\ep}=0$.
Finally, by the very definitions we get $\bs {A_\ep}\ge \bs{A_0}$. By the construction of $B$ in \cite{baer} we know that $(b_n)$ is weakly null and generates an $\ell_1$-spreading model with $\delta=1$. Thus $\bs {A_0}\ge2$ by Theorem~\ref{P:wbs<2wu<4sm} and hence $\bs{A_\ep}\ge 2$. This completes the proof.
\end{proof}

\section{Quantities applied to the unit ball}

In this section we investigate possible values of the quantities $\bs{}$ and $\wbs{}$ when applied to the unit ball of a Banach space. There are two main results in this section. The first one is a dichotomy for the quantity $\wbs{}$, the second one deals with the quantity $\bs{}$ in nonreflexive spaces.

\begin{thm}
\label{T:dichotomie}
Let $X$ be a Banach space. Then
$$\wbs{B_X}=\begin{cases}0 & \mbox{ if $X$ has the weak Banach-Saks property},\\ 2 &
\mbox{ otherwise}.\end{cases}$$
In particular:
\begin{itemize}
	\item There is a separable reflexive Banach space $X$ with $\bs{B_X}=\wbs{B_X}=2$.
	\item There is a nonreflexive Banach space $X$ with separable dual with $\bs{B_X}=\wbs{B_X}=2$.
	\item If $X=C[0,1]$, then $\bs{B_X}=\wbs{B_X}=2$.
\end{itemize}
\end{thm}

\begin{thm}
\label{T:nr} \
\begin{enumerate}
	\item Let $X$ be a Banach space containing an isomorphic copy of $\ell_1$. Then $\bs{B_X}=2$. In particular, $\bs{B_{\ell_1}}=2$ and $\wbs{B_{\ell_1}}=0$.
	\item Let $X$ be a nonreflexive Banach space containing no isomorphic copy of $\ell_1$. Then $\bs{B_X}\in[1,2]$. In particular,
	$\bs{B_{c_0}}=1$, $\bs{B_{c}}=2$ and $\wbs{B_{c_0}}=\wbs{B_c}=0$, where $c$ denotes the space of convergent sequences equipped with the supremum norm.
\end{enumerate}
\end{thm}

The key ingredient of the proof of Theorem~\ref{T:dichotomie} is the following lemma which can be viewed as a variant of the James distortion theorem for spreading models.

\begin{lemma}
\label{l:dichotomie}
Let $X$ be a Banach space. Then $\sm{B_X}\in \{0,1\}$.
\end{lemma}

\begin{proof}
Assume that $\sm{B_X}>0$. Then there is a sequence $(x_k)$ in $B_X$ weakly converging to some $x\in B_X$ and $\delta>0$ such that the sequence  $(x_k-x)$ generates an $\ell_1$-spreading model with $\delta>0$, i.e.,
\[
\forall F\subset \en, \#F\le \min F\; \forall (\alpha_i)\colon \norm{\sum_{i\in F} \alpha_i (x_i-x)}\ge \delta\sum_{i\in F} \abs{\alpha_i}.
\] \
We will show that, for any $\omega>0$, there exists a weakly null sequence $(y_k)$ in $B_X$ which generates an $\ell_1$-spreading model with  $1-\omega$.

The first step is to replace the sequence $(x_k-x)$ by a normalized weakly null sequence generating an $\ell_1$-spreading model.
Since $(x_k-x)$ generates an $\ell_1$-spreading model, no subsequence is norm-convergent and hence $\inf\limits_k\norm{x_k-x}>0$. Thus,
if we set $u_k=\frac{x_k-x}{\norm{x_k-x}}$, then $(u_k)$ is a normalized weakly null sequence. Moreover, since $\|x_k-x\|\le2$ for each $k$,
the sequence $(u_k)$ generates an $\ell_1$-spreading model with $\frac\delta2$. By \cite[Proposition 1.5.4]{albiac-kalton} we can suppose (up to passing to a subsequence) that $(u_k)$ is a basic sequence. Moreover, by \cite[Theorem~6.6]{f-banach} we may assume (by passing to a further subsequence if necessary) that there exists a Banach space $(Y,\abs{\cdot})$ with a (subsymmetric) basis $(e_k)$ such that
\begin{equation}\label{E:dich}
\begin{aligned}
\forall \ep>0 &\;\forall N\in\en\; \exists m(\ep,N)\in \en\colon m(\ep,N)\le k_1<k_2<\cdots<k_N\implies \\
&\forall (\alpha_i)\colon(1-\ep)\abs{\sum_{i=1}^N \alpha_ie_i}\le \norm{\sum_{i=1}^N \alpha_i u_{k_i}}\le (1+\ep)\abs{\sum_{i=1}^N \alpha_ie_i}.
\end{aligned}
\end{equation}
Since $(u_k)$ generates an $\ell_1$-spreading model, the sequence $(e_k)$ is equivalent to the $\ell_1$-basis, so we may suppose that $Y$ is the space $\ell_1$ with an equivalent norm $\abs{\cdot}$ and $(e_k)$ is the canonical basis. Hence, if we set
\[
\beta=\inf\left\{\abs{\sum_{i=1}^\infty \alpha_ie_i}\colon \sum_{i=1}^\infty \abs{\alpha_i}=1\right\},
\]
then $\beta>0$ and
\[
\forall (\alpha_i)\in\ell_1\colon \abs{\sum_{i=1}^\infty \alpha_ie_i}\ge \beta\sum_{i=1}^\infty \abs{\alpha_i}.
\]

To complete the proof choose an arbitrary $\omega>0$. We can fix $\eta>0$ such that $\frac{1-\eta}{(1+\eta)^2}>1-\omega$.
Using the density of $c_{00}$ in $\ell_1$ we find  $n\in\en$ and scalars $(\alpha_i)_{i=1}^n$ such that $\sum_{i=1}^n \abs{\alpha_i}=1$ and $\abs{\sum_{i=1}^n \alpha_ie_i}<(1+\eta)\beta$.
Set $m_0=m(\eta,n)$. For every $k\in\en$  we set
\[
y_k=\frac{1}{(1+\eta)^2\beta}\sum_{i=1}^n\alpha_i u_{m_0+kn+i}.
\]
From \eqref{E:dich} and the choice of $(\alpha_i)_{i=1}^n$ we obtain
\[
\norm{y_k}\le \frac{1}{(1+\eta)^2\beta}(1+\eta)\abs{\sum_{i=1}^n \alpha_ie_i}\le \frac{(1+\eta)^2\beta}{(1+\eta)^2\beta}=1,
\]
hence $y_k$ are elements of $B_X$. Further, the sequence $(y_k)$ weakly converges to zero.

Let $N\in\en$ be fixed. Let $k_1<k_2<\cdots<k_N$ be indices, where $k_1n\ge m(\eta, nN)$. If $(\beta_j)_{j=1}^N$ are scalars,
then we have from \eqref{E:dich} and from the definition of $\beta$ estimates
\[
\begin{aligned}
\norm{\sum_{j=1}^n \beta_j y_{k_j}}&=\frac{1}{(1+\eta)^2\beta}\norm{\sum_{j=1}^N\beta_j\left(\sum_{i=1}^n \alpha_i u_{m_0+k_jn+i} \right)}\\
&\ge \frac{1-\eta}{(1+\eta)^2\beta}\abs{\sum_{j=1}^N \beta_j\left(\sum_{i=1}^n \alpha_i e_{(j-1)n+i}\right)}\\
&\ge \frac{1-\eta}{(1+\eta)^2}\sum_{j=1}^N \abs{\beta_j}\left(\sum_{i=1}^n\abs{\alpha_i}\right)\\
&=\frac{1-\eta}{(1+\eta)^2}\sum_{j=1}^N \abs{\beta_j} \ge (1-\omega)\sum_{j=1}^N \abs{\beta_j}.
\end{aligned}
\]
Hence we have shown the following statement for the sequence $(y_k)$:

\begin{multline*}
\forall N\in\en\colon

 \left(\frac{m(\eta,nN)}n\le k_1<k_2<\dots<k_N \right.\\\left.\implies

\forall (\beta_j)\colon \norm{\sum_{j=1}^N \beta_j y_{k_j}}\ge (1-\omega)\sum_{j=1}^N\abs{\beta_j}\right).
\end{multline*}

To finish the proof it is enough to extract a further subsequence from $(y_k)$ satisfying the definition of the  $\ell_1$-spreading model with $1-\omega$.
To this end, we choose an increasing sequence $(n_j)$ of indices satisfying $n_j\ge \frac{m(\eta,nj)}n$ and set $z_j=y_{n_j}$, $j\in\en$.
Let now $N\in\en$ and let $N\le k_1<k_2<\cdots< k_N$ be indices and $(\alpha_i)$ be scalars. Then
\[
\norm{\sum_{i=1}^N\alpha_j z_{k_j}}=\norm{\sum_{i=1}^N \alpha_j y_{n_{k_j}}}\ge (1-\omega)\sum_{i=1}^N \abs{\alpha_i},
\]
because $\frac{m(\eta,nN)}n\le n_N\le n_{k_1}<n_{k_2}<\cdots <n_{k_N}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{T:dichotomie}.]
The equality follows from Lemma~\ref{l:dichotomie} and Theorem~\ref{P:wbs<2wu<4sm}.

The first example of a separable reflexive space without the Banach-Saks property is constructed in \cite{baer}.

As a nonreflexive space with separable dual which fails the weak Banach-Saks property one can take
the Schreier space described for example in \cite[Construction 0.2]{CaSh}. It is not reflexive since it contains a copy of $c_0$
by \cite[Proposition 0.7]{CaSh}, it has separable dual since it has an unconditional basis and does not contain a copy of $\ell_1$
\cite[Proposition 0.4(iii) and Proposition 0.5]{CaSh}. It fails the weak Banach-Saks property since the basis is weakly null and generates an $\ell_1$-spreading model due to \cite[Proposition 0.4(iv)]{CaSh}.

The space $C[0,1]$ fails the weak Banach-Saks property since it contains any separable space as a subspace. A direct proof is contained already
in \cite{schreier}.

\end{proof}

\begin{proof}[Proof of Theorem~\ref{T:nr}.]
(1) Let $(x_k)$ be a bounded sequence and $\delta>0$ such that
\begin{equation}\label{eq:ell1}
\norm{\sum_{i=1}^n\alpha_i x_i}\ge\delta\sum_{i=1}^n\abs{\alpha_i},\qquad n\in\en, \alpha_1,\dots,\alpha_n\in\er.
\end{equation}
Then $\cca{x_k}\ge 2\delta$. Indeed, let $m\ge n$. Then
$$\begin{aligned}
\norm{\frac 1m\sum_{i=1}^m x_i - \frac 1n\sum_{i=1}^n x_i}
&=\norm{\left(\frac1m-\frac1n\right)\sum_{i=1}^n x_i + \frac 1m\sum_{i=n+1}^m x_i}
\\ &\ge \delta\left(n\left(\frac1n-\frac1m\right)+\frac1m(m-n)\right)
=2\delta\left(1-\frac nm\right).
\end{aligned}$$
For any fixed $n$ the latter expression has limit $2\delta$ when $m\to+\infty$. This shows that $\cca{x_k}\ge 2\delta$.
Since any subsequence of $(x_k)$ satisfies \eqref{eq:ell1} as well, we get even $\tcca{x_k}\ge 2\delta$.

Let $X$ be a Banach space containing an isomorphic copy of $\ell_1$. By the James distortion theorem there is, given $\ep\in(0,1)$, a normalized sequence $(x_k)$ in $X$ which satisfies \eqref{eq:ell1} with the constant $1-\ep$ in place of $\delta$. It follows that $\bs{B_X}\ge \tcca{x_k}\ge 2(1-\ep)$. Since $\ep\in(0,1)$ is arbitrary, $\bs{B_X}\ge 2$. The converse inequality is obvious.

Finally, the equality $\wbs{B_{\ell_1}}=0$ follows from the Schur property of $\ell_1$.

(2) The inequality $\bs{B_X}\le 2$ is trivial. The inequality $\bs{B_X}\ge 1$ follows from Theorem~\ref{P:wbs<bs<beta} since for a nonreflexive space $X$ one has $\wck X{B_X}=1$ (this follows for example from \cite[Theorem 1]{Gr-James} and \cite[Proposition 2.2]{CKS}).

The spaces $c_0$ and $c$ are isomorphic and, moreover, they enjoy the weak Banach-Saks property by \cite{farnum}. Therefore $\wbs{B_{c_0}}=\wbs{B_c}=0$.

To show that $\bs{B_c}=2$ define a sequence $(x_k)$ in $B_c$ by the formula
$$x_k(i)=\begin{cases} 1, & 1\le i\le k,\\ -1, & i\ge k+1. \end{cases}$$
Let $(x_{k_n})$ be any subsequence of $(x_k)$. Denote by $y_n=\frac1n\sum_{j=1}^n x_{k_j}$ for $n\in\en$. Let $m<n$ be two natural numbers. Then
$$y_m(k_m+1)=-1 \mbox{\quad and\quad} y_n(k_m+1)=\frac1n(m\cdot(-1) +(n-m)\cdot 1)=1-\frac{2m}n.$$
Hence $\norm{y_m-y_n}\ge 2-\frac{2m}n$. For any fixed $m$ this value has limit $2$ for $n\to\infty$, thus $\cca{x_{k_n}}\ge 2$. It follows that $\tcca{x_k}\ge 2$, so $\bs{B_c}\ge 2$.

Finally, it remains to show that $\bs{B_{c_0}}\le 1$. To do this let us fix a sequence $(x_k)$ in $B_{c_0}$. Up to passing to a subsequence we may suppose that the sequence $(x_k)$ pointwise converges to some $x\in B_{\ell^\infty}$. Fix an arbitrary $\ep\in(0,1)$. We will construct
increasing sequences of natural numbers $(k_n)$ and $(p_n)$ using the following inductive procedure:
\begin{itemize}
	\item $k_1=1$;
	\item $\abs{x_k(i)}<\ep$ for $k\le k_n$ and $i\ge p_n$;
	\item $\abs{x_k(i)-x(i)}<\ep$ for $i\le p_n$ and $k\ge k_{n+1}$.
\end{itemize}
It is clear that this construction can be performed, using the facts that the points $x_k$ belong to $c_0$ and that the sequence $(x_k)$ pointwise converges to $x$. Let us consider the sequence $(x_{k_n})$ and set $y_n=\frac1n(x_{k_1}+\dots+ x_{k_n})$ for $n\in\en$. Fix an arbitrary $i\in\en$.
Let $m\in\en$ be the minimal number such that $i\le p_m$. By the construction we have $\abs{x_{k_n}(i)}<\ep$ for $n<m$ and  $\abs{x_{k_n}(i)-x(i)}<\ep$ for $n>m$, thus
$$ \abs{x_{k_n}(i)-\frac{x(i)}2}<\frac12\abs{x(i)}+\ep \mbox{ for }n\in\en\setminus \{m\}.$$
It follows that for any $N\in\en$ we have
$$\begin{aligned} \abs{y_N(i)-\frac{x(i)}2}&\le \frac1N\sum_{n=1}^N\abs{x_{k_n}(i)-\frac{x(i)}2}\le \frac1N\left((N-1)\left(\frac12\abs{x(i)}+\ep\right)+\frac32\right)
\\&\le\frac12\abs{x(i)}+\ep +\frac3{2N}. \end{aligned} $$

Therefore for any $M,N\in\en$ we have
$$\norm{y_N-y_M}\le \|x\|+2\ep+\frac3{2N}+\frac3{2M},$$
so clearly $\cca{x_{n_k}}=\ca{y_k}\le \|x\|+2\ep$. Since $\ep\in(0,1)$ is arbitrary, we get $\tcca{x_k}\le \|x\|\le 1$. This completes the proof.
\end{proof}

\section{Final remarks and open problems}

It is natural to ask whether the inequalities in our results are optimal and which inequalities may become strict.

Let us start by Theorem~\ref{P:wbs<bs<beta}.
\begin{itemize}
  \item If $X=C[0,1]$ and $A=B_X$, then $\wbs{A}=\bs{A}=\beta(A)=2$, hence we have equalities. Indeed, $\wbs{A}=2$ by Theorem~\ref{T:dichotomie} and obviously $\beta(A)\le 2$.
	\item If $X=\ell_1$ and $A=B_X$, then $\wbs{A}=0$ by the Schur property of $\ell_1$, obviously $\wck X{A}\le 1$ (in fact, $\wck X{A}=1$ since $\ell_1$ is not reflexive) and $\bs{A}=\beta(A)=2$ by Theorem~\ref{T:nr}, hence the first inequality is strict, the second one becomes equality.
	\item If $X=c_0$ and $A=B_X$, then $\wbs{A}=0$ by \cite{farnum}, $\wck X{A}=1$ since $c_0$ is not reflexive (in this concrete case the equality can
	be verified directly by the use of the sequence $(x_k)$ where $x_k=e_1+\dots+e_k$), $\bs{A}=1$ by Theorem~\ref{T:nr} and $\beta(A)=2$ (the constant $2$ is attained by the sequence $x_k=e_1+\dots+e_k-e_{k+1}$), hence the first inequality becomes equality and the second one is strict.
\end{itemize}

So, it seems that Theorem~\ref{P:wbs<bs<beta} is optimal.

Further, let us focus on Theorem~\ref{P:wbs<2wu<4sm}. The first inequality may become equality -- it is the case if $A=B_X$ by Theorem~\ref{T:dichotomie} and Lemma~\ref{l:dichotomie}. However, we know no example when the first inequality is strict.
The second inequality may become equality, for example if $X$ is the Baernstein space from \cite{baer} and $A$ is the canonical basis of $X$.
In this case the last inequality is strict. We do not know any example when the second inequality is strict. So, we can ask the following
question.

\begin{question} Let $X$ be a Banach space and $A\subset X$ a bounded set. Is it necessarily true that
$$\wbs{A}=2\sm{A}=2\swu{A}\quad ?$$
\end{question}

\section*{Acknowledgement}

We are grateful to the referee for suggesting to us Lemma~\ref{LLL} which substantially simplified the original proof of Lemma~\ref{L:ramsey}.

\def\cprime{$'$} \def\cprime{$'$}


TITLE: Probability of the number of winners
QUESTION [1 upvotes]: I have a theoretical $7$ sided die with an equal probability of each result($1$-$7$). If I roll the die $30$ times and count for the winning number(s) what is the probability of having $1$ winning number? $2$ winning numbers? $3$? ... $6$ winning numbers? Winning number is defined as the number(s) that get rolled the most out of the $30$ rolls. i.e. if $1$ and $2$ are both rolled $7$ times and all other numbers are rolled less than $7$ times then there are $2$ winners ($1$ & $2$).
Thank you in advance for the help!

REPLY [0 votes]: I think there are essentially $1824$ non-equally probable patterns for the dice throws, corresponding to the partitions of $30$ into no more than $7$ parts.
One example pattern could be $8$ of one side, $8$ of a second, $8$ of a third, $4$ of a fourth, $1$ of a fifth, $1$ of a sixth and $0$ of the seventh, with $8+8+8+4+1+1+0=30$.  This would correspond to $3$ winners as it has $3$ sides appearing eight times, $1$ side appearing four times, $2$ sides appearing once and $1$ side not appearing, with $3+1+2+1=7$.  The number of possible equally-probable ways this pattern could appear is $\dfrac{30!}{(8!)^3(4!)^1(1!)^2(0!)^1}\times \dfrac{7!}{3!1!2!1!} \approx 7 \times{10^{19}}$ so to turn this into a probability you need to divide by $7^{30}$ to give a probability for this pattern of about $3\times 10^{-6}$.
There are $39$ partition patterns giving $3$ winners, and it would seem sensible to do the calculations across all the $1824$ partitions with a program.  For example the following R code
library(partitions)
rolls <- 30
sides <- 7
partitions <- restrictedparts(rolls, sides, include.zero=TRUE, decreasing=TRUE)
probwinners <- numeric(sides)
names(probwinners) <- 1:sides
for (n in 1:ncol(partitions)){
   part <- partitions[,n]
   numberwinners <- sum(part == part[1])
   probwinners[numberwinners] <- probwinners[numberwinners] + exp(
      lfactorial(rolls) - sum(lfactorial(part)) +
      lfactorial(sides) - sum(lfactorial(table(part))) -
      rolls * log(sides) )
   }
probwinners

gives the calculated probabilities
           1            2            3            4            5            6            7
7.525078e-01 1.872654e-01 4.409657e-02 1.361835e-02 2.484250e-03 2.758859e-05 0.000000e+00

which add up to $1$.  My earlier simulated values were close to these, providing some validation


TITLE: Complex differentiability equivalent to linear approximation
QUESTION [1 upvotes]: Let $G \subset \mathbb C$ be an open set and $f: G \to \mathbb C$ a complex function on $G$. Prove that the function $f$ is complex differentiable at a point $z \in G$ if and only if there exists a constant $A \in \mathbb C$ such that
  \begin{align*} f(z+h) -f(z) =
Ah+\omega(h) \end{align*}
  where $\omega : G_z \to \mathbb C$ satisfies
  \begin{align*} \lim_{|h| \to 0} |\omega(h)|/|h| = 0, \qquad h \ne 0,
\, z+h \in G,
\end{align*}
  where $G_z = \{h \in \mathbb C: h+z \in G\}$.

Proof:
"$\Rightarrow$": Let $f$ be complex differentiable at $z$. Then the limit
\begin{align*}
f'(z) = \lim_{h \to 0} \frac{f(z+h) - f(z)}{h}
\end{align*}
exists and is finite. Set $A := f'(z)$ and
\begin{align*}
\omega(h) := f(z+h) - f(z)
\end{align*}
Then
\begin{align*}
\lim_{|h| \to 0} \frac{|\omega(h)|}{|h|} = \lim_{|h| \to 0} \frac{|f(z+h) - f(z)|}{|h|} = 0 ???
\end{align*}
and
\begin{align*}
Ah + \omega(h) &= f'(z)h + f(z+h) - f(z) \\
&= \lim_{h \to 0} \frac{f(z+h)-f(z)}{h} h + f(z+h) - f(z) \\
&= \lim_{h \to 0} f(z+h)-f(z)  + f(z+h) - f(z) \\
&= f(z)-f(z) + f(z+h)-f(z)
= f(z+h)-f(z).
\end{align*}
Question: I have still some troubles with $\omega$ as I don't see that it satisfies the condition.
"$\Leftarrow$": Suppose there exists $A$ with the stated properties. Then
\begin{align*}
\frac{f(z+h)-f(z)}{h}= \frac{Ah+\omega(h)}{h} = A + \omega(h)/h \xrightarrow{h \to 0} A,
\end{align*}
and so $f$ is differentiable.

REPLY [1 votes]: In the first part of the proof, the definition of $\omega$ should be changed to
$$\begin{align*}
\omega(h) := f(z+h) - f(z) - A h
\end{align*}$$
With this definition,
$$
\lim_{ |h|  \to 0} \frac{ \omega(h) }{ h } = \lim_{|h| \to 0} \left(\frac{ f(z+h) - f(z) }{ h } -A\right) =   A-A=0
 $$
hence
$$ \lim_{ |h|  \to 0} \frac{ |\omega(h)| }{ |h |}=0$$
(A function tends to zero if and only if its absolute value does.)


TITLE: Need help in proving that $4$ divides $n$.
QUESTION [1 upvotes]: I'm pretty sure this question has a duplicate, but since I can't seem to find it, I'm asking it again.
If $a_{1},a_{2}...a_{n}$ are $1$ or $-1$ and if,$$a_{1}a_{2}a_{3}a_{4}+a_{2}a_{3}a_{4}a_{5}+...+a_{n}a_{1}a_{2}a_{3}=0$$prove that $n$ is divisible by $4$.
What I've done so far is manage to prove that $n$ is even, which is pretty obvious. I haven't come up with any workable leads so far, so I'd be much obliged if you could give me some pointers.

REPLY [4 votes]: Here's a nice way to do this problem, which relies on the following invariant: if one of the numbers, say $a_1$, is replaced by $-a_1$, then the sum remains the same modulo $4$ (namely $0$ modulo $4$).
Given that this invariant is correct, replace each $+1$ with a $-1$ in the list of the $a_i$. Then the sum is $n$, which equals $0$ modulo $4$ by the invariant property.
Why does the invariant work?
When we replace $a_1$ by $-a_1$ four of the terms of the sum change. There are three cases.

If two of these four terms are positive and two negative then the sum does not change.
If three of these four terms have the same sign then the sum changes by $\pm 4$.
If four of these four terms have the same sign then the sum changes by $\pm 8$.

In every case the sum remains invariant modulo $4$.


TITLE: If $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$, what does $2p+5q+10r+s$ equal?
QUESTION [3 upvotes]: Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?
I tried messing with both statements. But I usually just end up stuck or hit a dead end.
(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice)

REPLY [2 votes]: Suppose $(a+b\sqrt{2})^2=5$, with $a,b\in\mathbb{Q}$. Then
$$
a^2+2ab\sqrt{2}+2b^2=5
$$
If $ab\ne0$, we get
$$
\sqrt{2}=\frac{5-a^2-2b^2}{2ab}\in\mathbb{Q}
$$
which is a contradiction. Thus either $a=0$ or $b=0$. If $a=0$, then $2b^2=5$, a contradiction; if $b=0$, then $a^2=5$, a contradiction.
Since the degree of $\sqrt{5}$ over $\mathbb{Q}(\sqrt{2})=2$, because the minimal polynomial is $x^2-5$, we conclude that every element of $\mathbb{Q}(\sqrt{2},\sqrt{5})$ can be written in a unique way as
$$
x+y\sqrt{5}
$$
with $x,y\in\mathbb{Q}(\sqrt{2})$. Now
$$
0=p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=(s+p\sqrt{2})+(q+r\sqrt{2})\sqrt{5}
$$
implies $s+p\sqrt{2}=0$ and $q+r\sqrt{2}=0$. Therefore $s=p=q=r=0$, because an element in $\mathbb{Q}(\sqrt{2})$ can be written in a unique way as $a+b\sqrt{2}$ with $a,b\in\mathbb{Q}$.

The key result is that if $t$ is algebraic over the field $F$ and its degree is $2$, then every element of $F(t)$ can be written in a unique way as $a+bt$, with $a,b\in F$.


TITLE: Convergence in a finite metric space
QUESTION [2 upvotes]: Let's look at a metric space $(X,d)$
So we know that a set $A$ is closed iff for each convergent sequence $x_n\rightarrow x$ from $A$,
we have $x\in A$
However, if $X$ is finite, how can we look at such defenition?

REPLY [2 votes]: A finite metric space is discrete (every set is open), so every set is closed.
This means the only convergent sequences are those that are eventually constant. Since all $x_n\in A$, then $x\in A$ because $x_n=x$ for all sufficiently large $n$.


TITLE: Orthonormal coordinate coframe in arbitrary Riemannian manifold - question on published paper
QUESTION [2 upvotes]: In Miranda Jr, Pallara, Paronetto and Preunkert's Heat semigroup and functions of bounded variation on Riemannian manifolds (p.106, property (4)), the authors state that for an $n\ge 2$-dimensional Riemannian manifold $M$ without boundary, one can construct a collection of coordinate charts $(U_i^\eta,\psi_i)$ such that $T^*(U_i^\eta)\simeq U_i^\eta \times \mathbb{R}^n$, and $d\psi_i$ is an isometry between $T^*_x(M)$ and $\mathbb{R}^n$ for every $x \in U_i^\eta$.
Why is this possible when $M$ is not flat? The coordinate system $(U_i^\eta, \psi_i)$ seems to have an orthonormal coordinate frame, which is only possible if $M$ has a flat metric on $M$ (by e.g. Theorem 13.14 in Lee's Introduction to Smooth Manifolds, or this question).
Miranda, M. jun.; Pallara, D.; Paronetto, F.; Preunkert, M., Heat semigroup and functions of bounded variation on Riemannian manifolds, J. Reine Angew. Math. 613, 99-119 (2007). ZBL1141.58014.

REPLY [0 votes]: The existence of a local isometry implies flatness, since the curvature tensor is a local invariant. So this condition implies $(M, g)$ is flat on the neighborhood of interest.
One can arrange for coordinate charts to be an isometry at one point and have all first derivatives of the metric vanish at that point - these are called exponential coordinates - but no better is possible in general. It is likely the case (but one should check) that only this or something similar which can be arranged for using previously assumed hypotheses is actually used in the proofs involving this condition.


text:
\section{Introduction and Historical Background}\label{SEC_intro}
This paper is concerned with the ``\textit{almost existence}" phenomenon
  for periodic orbits of Hamiltonian dynamical systems.
We shall describe some of the background of this  phenomenon,
  and we relate it to the new feral curve theory, \cite{FH2}, which was
  recently initiated by the authors.
The new approach to the  ``\textit{almost existence}" phenomenon suggests
  a larger context which also features questions around the
  \(C^{\infty}\)-closing lemma in the following sense.
Consider a compact symplectic manifold \((W,\Omega)\) equipped
  with a smooth Hamiltonian \(H\colon W\rightarrow {\mathbb R}\).
Given a regular energy surface $H^{-1}(E)$, one can ask the question:
  Is it the case that  after a small smooth perturbation of $H$ the new
  $H'^{-1}(E)$ has the property that the set of periodic orbits are dense?
This is a classical question, and the $C^1$-closing lemma shows that the
  above assertion is true for a $C^2$-small perturbation of the Hamiltonian,
  i.e. for a $C^1$-close Hamiltonian vector field.
Whereas the $C^1$-closing lemma holds in general, it is known that the
  $C^{\infty}$-closing lemma does not; see \cite{Herman:1991}.
We believe that the analysis below contributes to a growing body of
  evidence which suggests that the validity of the Hamiltonian
  \(C^\infty\)-closing lemma is intimately connected to the existence of a
  sufficiently rich Gromov-Witten theory of the ambient space.
We touch on this speculation in Section \ref{SEC_speculation}.

The main goal of this paper is to set up a body of results to utilize the
  Feral Curve theory to study questions around almost existence and the
  closing lemma.
Our results show that given a compact pile of Hamiltonian energy surfaces,
  a sufficient supply of pseudoholomorphic curves associated to this pile
  implies ``almost existence"; see Theorem \ref{THM_existence_and_almost_existence}.
This is always attainable provided the pile of energy surfaces can be
  viewed as lying in a compact symplectic manifold with a sufficient supply
  of pseudoholomorphic curves.
As an exercise the reader might enjoy using our more local results to
  reprove the almost existence result for regular compact energy surfaces in
  ${\mathbb R}^{2n}$ by  using the large supply of pseudoholomorphic curves
  obtained as a deformation of holomorphic curves in ${\mathbb C\mathbb
  P}^n$.
Building on previous work in \cite{FH2}, we also show that under suitable
  topological constraints, the almost existence result can be improved to
  the following: Every energy level contains a non-trivial closed invariant
  subset, and for almost all of these energy levels this set is a periodic
  orbit;  see Theorem \ref{THM_existence_and_almost_existence}.
For the convenience of the reader we recall the necessary background.

\subsection{The Weinstein Conjecture}
As a starting point for our discussion, we consider two seminal papers by P.
  Rabinowitz, namely \cite{Rab:78} and \cite{Rab}, which are concerned
  with the existence of a periodic orbit on a given compact regular
  energy surface
  \begin{align*}
    M=H^{-1}(E)
    \end{align*}
  for an autonomous Hamiltonian system defined on the standard
  phase space ${\mathbb R}^{2n}$.
Rabinowitz showed  the existence of periodic orbits whenever suitable
  geometric conditions are met.
We refer the reader to the introduction of \cite{Abbas-Hofer} for the
  interesting broader historical perspective of Rabinowitz' results.

A. Weinstein analyzed these results, particularly \cite{Rab},  and
  proposed the far-reaching Weinstein Conjecture in \cite{Weinstein:79},
  which we shall describe momentarily.
First though, we provide some definitions.
Consider an odd-dimensional smooth manifold \(M^{2n+1}\),
  with a one-form \(\lambda\) for which \(\lambda\wedge (d\lambda)^n\) is
  a volume form.
In this case, we call \(\lambda\) a contact form for \(M\), which then
  uniquely determines a vector field \(X\) by the following equations:
  \begin{align*}
    \lambda(X)\equiv 1\qquad\text{and}\qquad i_X d\lambda \equiv 0.
    \end{align*}
In this case, we call \(X\) the \textit{Reeb vector field} associated to the
  contact form \(\lambda\).
In a modified form, the Weinstein Conjecture states the following.\\

\noindent{\textbf{Weinstein Conjecture (1978):}} \\
\noindent \emph{
Let \(M\) be a smooth closed odd-dimensional manifold equipped with a
  contact form and an associated Reeb vector field \(X\).
Then the dynamical system given by
  \begin{align*}
    \dot{x}=X(x),
    \end{align*}
  has a nontrivial periodic orbit.\\
}

The first breakthrough concerning the Weinstein Conjecture was Viterbo's
  celebrated result in \cite{Viterbo:1987}, which showed that a regular
  compact contact-type energy surface of a Hamiltonian system in ${\mathbb
  R}^{2n}$ carries a nontrivial periodic orbit.
In 1993 Hofer, in \cite{Hofer:1993}, showed that for a contact form
  $\lambda$ on a closed three-manifold $M$ the Weinstein conjecture is true
  provided at least one of the following holds.
Either $M=S^3$, or $\pi_2(M)\neq 0$, or $\lambda$ is overtwisted.
In 2007 Taubes proved in \cite{Taubes:2007} that the Weinstein conjecture in
  dimension three is true for all $(M,\lambda)$.

A natural question then becomes whether or not a generalization of this
  result holds for more general compact energy surfaces in
  \(\mathbb{R}^{2n}\).
However, if $n\geq 3$ results in \cite{Ginzburg:1995} and \cite{Her0} show
  that periodic orbits might not exist on a given energy surface.
There is also a result for $n=2$, see V. Ginzburg and B. G{\"u}rel,
  \cite{GG}, and thus it becomes interesting to study, in some sense, how
  often and how the generalization fails.
This is the content of Section \ref{SEC_almost_existence}.

\subsection{Almost Existence}\label{SEC_almost_existence}
By analyzing Viterbo's paper, the second author and E. Zehnder established
  in \cite{HZ:87} that many compact Hamiltonian energy levels contain
  periodic orbits.
Indeed, after some refinement by Rabinowitz in \cite{Rab:87} and Struwe in
  \cite{Struwe:90}, this phenomenon became referred to as ``\textit{almost
  existence}''.
Working in a context in which the almost existence phenomenon holds, the
  actual existence question
  for periodic orbits of Hamiltonian systems quite often can be phrased as
  ``a priori estimates imply existence", see \cite{BHR}.
Finally the phenomenon was explained in \cite{HZ} in terms of
  differentiability properties of the so-called Hofer-Zehnder capacity, see also
  \cite{HZ:1990}.

For our explicit purpose of connecting this topic to feral curve theory,
  we approach the subject in a particular way.
In a first definition we give an abstraction of a regular, smooth and
  compact Hamiltonian energy surface in a symplectic manifold, which we call
  a ``\textit{framed Hamiltonian manifold}''.
By forgetting some of the information carried by a framed Hamiltonian
  manifold we obtain what is called an ``\textit{odd-symplectic manifold}''.

\begin{definition}[framed Hamiltonian manifold and odd-symplectic]
  \label{DEF_framed}
  \hfill\\
A framed Hamiltonian manifold $(M,\lambda,\omega)$ consists of a
  smooth closed odd-dimen\-sional manifold $M=M^{2n+1}$, a two-form $\omega$
  and a one-form $\lambda$ such that $\lambda\wedge \omega^{n}$ is a volume
  form.
When such a \(\lambda\) exists but we only specify \((M, \omega)\), we
  call the pair an odd-symplectic manifold.
\end{definition}

A framed Hamiltonian manifold  $(M,\lambda,\omega)$ defines a dynamical
  system.
Namely there exists a non-singular vector field $X$ on $M$ uniquely
  characterized by the equations
\[
i_X\lambda\equiv 1\ \ \textrm{and}\ \ \ i_X\omega\equiv 0.
\]
As in the more special contact case we shall refer to $X$ as the
  \textit{Reeb vector field} (associated to $(M,\lambda,\omega)$).

Assume that we are given a symplectic manifold $(W,\Omega)$ without
  boundary and  consider a compact, smooth, regular and co-oriented
  hypersurface $M$ in \(W\).
Denoting by $\iota:M\rightarrow W$ the inclusion, we abbreviate $\omega=
  \iota^{\ast}\Omega$ and obtain the odd-symplectic manifold $(M,\omega)$.
If  $H:W\rightarrow {\mathbb R}$ is a smooth Hamiltonian and $H^{-1}(E)=M$
  for some number $E\in {\mathbb R}$ and $dH(m)\neq 0$ for $m\in M$ we can
  take a one-form $\lambda$ on $M$ such that
  $\lambda(X_H(m))=1$ for $m\in M$.
Then $\lambda\wedge\omega^{n}$ is a volume form on $M$.
Hence we obtain a framed Hamiltonian manifold $(M,\lambda,\omega)$.
One easily verifies that the Reeb vector field $X$ satisfies
\[
X_H(m)=X(m)\ \ \textrm{for all}\ m\in M,
\]
  where $X_H$ is the Hamiltonian vector field associated to $H$ and
  defined by $i_{X_H}\Omega=-dH$.

In order to study the almost existence phenomenon we also need to consider
  neighborhoods of a smooth, regular and compact energy surface.
Given a co-orientable, compact, smooth and regular hypersurface $M$
  contained in $W$, where $(W,\widetilde{\Omega})$ is a symplectic
  manifold, we obtain an odd-symplectic manifold $(M,\omega)$ as previously
  described.
Namely we take the inclusion $i:M\rightarrow W$ and define
  $\omega:=i^{\ast}\widetilde{\Omega}$.

We fix a one-form $\lambda$ such that $\lambda\wedge \omega^n$ is a volume
  form on $M$.
We can define on ${\mathbb R}\times M$ with coordinates $(t,m)$ a 2-form
  $\Omega$ by
\[
\Omega =d(t\lambda)+\omega.
\]
It is a trivial exercise that there exists an open neighborhood $U$ of
  $M\equiv \{0\}\times M$ such that $\Omega|U$ is a symplectic form.
Moreover, if we take $U$ small enough we find an embedding
  $\psi:U\rightarrow W$ onto an open neighborhood of $M\subset W$
  such that $\psi^{\ast}\widetilde{\Omega}= \Omega|U$ and in addition
  $\psi(0,m)=m$ for all $m\in M$.
Note that this also implies that given $\lambda_1$ and $\lambda_2$ so that
  $\lambda_i\wedge \omega^n$ are volume forms we find for the corresponding
  $\Omega_i$ defined by
\[
\Omega_i =d(t\lambda_i)+\omega,
\]
  open neighborhoods $U_1$ and $U_2$ of $M\subset {\mathbb R}\times M$
  such that $\Omega_i|U_i$ are symplectic and there exists a symplectic
  diffeomorphism $\psi:U_1\rightarrow U_2$, which is the identity on
  $M=\{0\}\times M$.
Depending on the case, whether or not $\lambda_1\wedge \omega^n$ or
  $\lambda_2\wedge \omega^n$ define the same orientation on $M$, we must
  have that $\psi$ maps $(s,m)$ for some $s>0$ to some $(s',m')$ with $\pm
  s'>0$.

From the previous discussion it follows that instead of working with a
  suitable open neighborhood $U$ of $M$ we may assume that $U={\mathbb
  R}\times M$ equipped with a symplectic form $\Omega$ such that
  $\omega=\iota^{\ast}\Omega$, where $\iota(m)=(0,m)$ for $m\in M$.

\begin{definition}[compatible $\Omega$]
  \label{DEF_compatible_Omega}
  \hfill\\
Given an odd-symplectic $(M,\omega)$ we call a symplectic form $\Omega$ on
  ${\mathbb R}\times M$ \textit{compatible} provided
  $\iota^{\ast}\Omega=\omega$, where $\iota:M\rightarrow {\mathbb R}\times
  M$ is defined by $\iota(m)=(0,m)$ for $m\in M$.
\end{definition}

The notion of \textit{``almost existence''} will be associated to the
  behavior of small neighborhoods of $M=\{0\}\times M$ in ${\mathbb R}\times
  M$ for a symplectic form which restricts to $\omega$.

\begin{definition}[almost existence property - odd symplectic]
  \label{DEF_almost_existence_odd_symplectic}
  \hfill\\
Consider a smooth, compact odd-symplectic manifold $(M,\omega)$, denote by
  $\Omega$ a compatible symplectic form  on ${\mathbb R}\times M$ and
  identify $M\equiv \{0\}\times M$.
We say that $(M,\omega)$ has the \textit{almost existence property}
  provided there exists an open neighborhood $U$ of $M$ with the following
  property.
Given any  proper, smooth and surjective Hamiltonian $H:V\rightarrow
  (-\varepsilon,\varepsilon)$ for some $\varepsilon>0$,  where $V$ is an
  open neighborhood of $M$ contained in $U$  such that $H^{-1}(0)=\iota(M)$
  and $dH(s,m)\neq 0$ for $m\in M$ and $s\in (-\varepsilon,\varepsilon)$
  define the  set $\Sigma_H$ by
\[
\Sigma_H=\{E\in (-\varepsilon,\varepsilon)\ \vert\ \dot{x}=X_H(x)\
\textrm{has a periodic orbit with}\ H(x)=E\}.
\]
We say that $(M,\omega)$ has the \textit{almost existence property}
  provided for suitable $U$ it holds for all $H$ as described above that
  $\textrm{measure}(\Sigma_H)=2\cdot\varepsilon$.
\end{definition}

Finally we can introduce a special class of symplectic manifolds.

\begin{definition}[almost existence property -- symplectic manifold]
  \label{DEF_almost_existence_property_symplectic}
  \hfill\\
A symplectic manifold $(W,\Omega)$ without boundary has the almost
  existence property provided for every regular, compact, and co-oriented
  hypersurface $M$ the pair $(M,\iota^{\ast}\Omega)$ has the almost existence
  property, where  $\iota:M\rightarrow W$ is the inclusion.
\end{definition}

We know that the standard symplectic vector space ${\mathbb R}^{2n}$ has
  the almost existence property, see \cite{HZ:1990}.
We also know that $T^4$ can be be equipped with a symplectic form so that
  $(T^4,\Omega)$ does not have the almost existence property, see also
  \cite{HZ:1990}.
In other words, the almost existence property is nontrivial.
We state the following  theorem for the convenience of the reader.
It is based on some known facts which we identify as a local property.

\begin{theorem}[local almost existence property]
  \label{THM_SOMETHING}\label{th1}
  \hfill\\
Every symplectic manifold $(W,\Omega)$ without boundary has the following
  property.
Given a point $w\in W$ there exists an open neighborhood $U(w)$ so that
  for every closed regular hypersurface  $M\subset U$ the pair
  $(M,i^{\ast}\Omega)$ has the almost existence property.
\end{theorem}

In other words, every symplectic manifold without boundary has the "local
  almost existence property".
It then becomes an interesting question what kind of more global compact,
  regular hypersurfaces in $(W,\Omega)$ have the almost existence property.
We formulate a more precise question next.

\begin{question}
  \label{Q_SOMETHING}
Assume that $(W,\Omega)$ is a symplectic manifold without boundary.
Suppose that $M\subset W$ is a smooth, compact, regular  hypersurface
  without boundary so that the inclusion $\iota: M\rightarrow W$ is isotopic
  to a small hypersurface, i.e. contained in some $U(w)$, see Theorem
  \ref{th1}.
Does then $M$ have the almost existence property?
If this is not always true, then for which class of symplectic manifolds
  (other than \(\mathbb{R}^{2n}\)) is it true?
\end{question}

We note that the literature suggests that compact symplectic manifolds
  with a sufficiently rich Gromov-Witten theory have the almost existence
  property.
The first paper,  predating Gromov-Witten theory, where such an idea  is
  used is \cite{HOFV}.
It shows that having a suitable moduli space of pseudoholomorphic curves
  associated to a symplectic manifold implies that the Weinstein
  conjecture holds for suitable energy surfaces.
The method in \cite{HOFV} was then used
  in \cite{LiuT} and combined with Gromov-Witten
  theory for a more convenient packaging of properties of moduli spaces.
This Gromov-Witten style approach has also been used to prove almost
  existence results in certain cases; see for example \cite{Lu}, Theorem 1.10.

\subsection{Statement of the Main Result}
Here we state the main result of the article.
The terms used below are standard, however they are provided explicitly in
  Section \ref{SEC_background} below.
For example, the notion of an \(\Omega\)-tame almost complex structure is
  provided in Definition \ref{DEF_almost_complex_structures}; the notion of
  a pseudoholomorphic map is provided in Definition
  \ref{DEF_pseudoholomorphic_map}; and the notion of the genus of such a map
  is provided in Definition \ref{DEF_genus}.

\setcounter{CounterSectionMain}{\value{section}}
\setcounter{CounterTheoremMain}{\value{theorem}}

\begin{theorem}[Main Result]
  \label{THM_main_maybe}
  \hfill\\
Let \((W, \Omega)\) be a symplectic manifold without boundary, and let
  \(H\colon W\to \mathbb{R}\) be a smooth proper\footnote{By proper here, we
  mean that for each compact set \(\mathcal{K}\subset \mathbb{R}\), the set
  \(H^{-1}(\mathcal{K})\) is compact.} Hamiltonian.
Fix \(E_-, E_+ \in H(W) \subset \mathbb{R}\) with \(E_- < E_+\), as well
  as positive constants, \(C_g>0\), and \(C_\Omega>0\).
Suppose that for each \(\Omega\)-tame almost complex structure \(J\) on
  \(W\) there exists a proper pseudoholomorphic map
  \begin{align*}
    u:(S,j)\to \{p\in W : E_- < H(p) < E_+ \}
    \end{align*}
  without boundary, which also satisfies the following conditions:
  \vspace{5pt}
\begin{enumerate}
  \item \emph{({genus and area bounds})}
  The following inequalities hold:
  \begin{align*}
    {\rm Genus}(S) \leq C_g\qquad\text{and}\qquad \int_{S} u^*\Omega
    \leq C_\Omega.
    \end{align*}
  \item \emph{({energy surjectivity})}
  The map \(H\circ u : S\to (E_- , E_+)\) is
  surjective.\vspace{5pt}
  \end{enumerate}
Then there is a periodic Hamiltonian orbit on almost every energy level in
  range \((E_-, E_+)\).
That is, if we let \(\mathcal{I}\subset (E_-, E_+)\) denote the energy
  levels of \(H\) which contain a Hamiltonian periodic orbit, then
  \(\mathcal{I}\) has full measure:
  \begin{align*}
    \mu(\mathcal{I}) = \mu\big((E_-, E_+)\big) = E_+ - E_-.
    \end{align*}
\end{theorem}

At this point there are a few points worth making.
The first is that the pseudoholomorphic maps in question here need not be
  compact -- in fact, a careful inspection of the requirements reveals
  that they cannot be compact.

A second point is that the pseudoholomorphic maps allowed by the above
  hypotheses may have domains \((S, j)\) which are diffeomorphic to an open   annulus, but sometimes the domains will be much worse.
For example, our assumptions allow for the possibility that a domain of a
  pseudoholomorphic map may be an infinitely punctured open disk with
  infinitely many closed disks removed.
Or worse: the open disk with the Cantor set removed.
Or any closed Riemann surface with any closed set removed.
To be clear, such assumptions are highly unusual in the standard theory of
  pseudoholomorphic curves, however they are standard in feral curve theory.
This is because feral curves in general are much wilder.
We illustrate this with a plausible example.

Suppose \((W,\Omega)\) is a closed symplectic manifold, with an
  \(\Omega\)-compatible almost complex structure \(J\), a smooth Hamiltonian
  \(H\colon W\to \mathbb{R}\) for which \(0\) is a regular value, and a
  pseudoholomorphic map \(\bar{u}\colon \overline{S}\to W \) where
  \(\overline{S}\) is a closed Riemann surface such as a sphere or torus,
  etc.
For \(\epsilon>0\), what structure does the following set have?
  \begin{align}\label{EQ_example_S}
    S := \big\{z\in \overline{S} : -\epsilon < H\big(\bar{u}(z)\big) <
    \epsilon\big\}
    \end{align}
As it need not be the case that \(\pm \epsilon\) are regular values of
  \(H\circ u\), there is no reason that \(S\) should necessarily admit a
  smooth compactification to a compact Riemann surface with smooth
  boundary.
Indeed, all one can really say is that it has the structure of a closed
  Riemann surface with some closed set removed, which is exactly the sort of
  domain surface that Theorem \ref{THM_main_maybe} allows.
Conceptually, it may be easier to think of the maps we allow as arising
  from restrictions like \(u:=\bar{u}\big|_{S}\), with \(S\) as in equation
  (\ref{EQ_example_S}), however it is worth noting that the existence of
  such an extension (or lack thereof) plays no role in our proof.

With such unusual freedom allowed for the domains of our pseudoholomorphic
  maps, the attentive reader may be concerned about the precise notion of
  genus.
This is made rigorous in Definition \ref{DEF_genus} below, although
  it amounts to exhausting \(S\) by compact two-dimensional manifolds with
  boundary and taking the limit of associated the genera.

A third point is that the hypothesis in Theorem \ref{THM_main_maybe}
  regarding the existence of a suitable pseudoholomorphic map for
  \emph{each} \(\Omega\)-tame almost complex structure seems very
  restrictive, however in practice this is not the case.
For example, Gromov-Witten invariants are invariants of a closed
  symplectic manifold obtained by algebraically counting pseudoholomorphic
  maps of specified genus, homology class, and incidence conditions.
In particular, Gromov-Witten invariants are independent of the choice of
  almost complex structure (provided that it is \(\Omega\)-tame), and thus
  a sufficiently rich Gromov-Witten theory for a closed symplectic
  manifold \((W, \Omega)\) is sufficient to guarantee the hypotheses of
  Theorem \ref{THM_main_maybe} are satisfied in many cases.

We will contrast the feral curve techniques used below with the methods
  used in \cite{LiuT} and \cite{Lu}  momentarily, however at present we
  point out that the the proof of Theorem \ref{THM_main_maybe} will show
  that its conclusion holds under weaker assumptions regarding the
  pseudoholomorphic curves.
Specifically, one only needs the existence of pseudoholomorphic curves for
  a particular sequence of adiabatically degenerating almost complex
  structures.
The notion of this adiabatic degeneration is rather technical and is made
  precise in Definition
  \ref{DEF_adiabatically_degenerating_almost_complex_structures}, however
  the idea is to degenerate the almost complex structure so as to
  geometrically ``stretch the neck'' along a continuum of energy levels.
With such an concept internalized, we direct the reader's attention to
  Theorem \ref{THM_localized_almost_existence} in Section
  \ref{SEC_main_proof} below, which is a localized version of Theorem
  \ref{THM_main_maybe} with the assumptions stripped to
  the absolute essentials.

With the exception of Struwe's results in \cite{Struwe}, which predate
  modern symplectic methods, all proofs of almost existence follow a similar
  pattern: prove the Hofer-Zehnder capacity of a domain containing the
  energy surface is finite, and the desired result follows from \cite{HZ}.
For example, in \cite{Lu} Lu takes cues from \cite{LiuT} to use the
  Gromov-Witten invariants to define a \emph{pseudo-}capacity which is
  finite and bounds the Hofer-Zehnder capacity; the almost existence result
  is then immediate.
In contrast, the proof of Theorem \ref{THM_main_maybe} makes no use of
  capacities at all, and only makes use of pseudhoholomorphic curves --
  specifically feral curve theory.
The idea is to stretch the neck along a continuum of energy levels, and
  analyze some basic properties in the limit.
For those familiar with methods from Symplectic Field Theory, it is worth
  pointing out that the key obstacle to overcome is that there are no Hofer
  energy bounds in this case, and hence infinite energy pseudoholomorphic
  curves (i.e feral curves) can appear in the limit ``building.''
The picture that emerges from this analysis is rather interesting.
It seems that as one takes this adiabatic degeneration, curves which cross
  the region of degeneration are inexorably drawn to collapse onto
  families of periodic orbits.
Or more precisely, such a collapse to families of orbits occurs
  \emph{almost} everywhere, and on the complementary measure zero set the
  curves are allowed to jump between such families, or even jump to a more
  general closed invariant subset.
Further analysis of such limiting curves seems well poised to illuminate
  additional dynamical features.

We close this introductory section with an application which appears to be
  inaccessible to methods relying on the finiteness of the Hofer-Zehnder
  capacity.
We state this as Theorem \ref{THM_existence_and_almost_existence} below,
  but first provide a definition.

\begin{definition}[positive contact type]
  \label{DEF_positive_contact_type_boundary}
  \hfill\\
Let \((W, \Omega)\) be a compact symplectic manifold with boundary.
Let \(M^+\) be a union of connected components of \(\partial W\).
We say \(M^+ \subset \partial W\) is  of \emph{positive contact type} provided
  there exists an outward pointing nowhere vanishing vector field \(Y\)
  defined in a neighborhood of \(M^+\) in \(W\) for which \(\mathcal{L}_Y
  \Omega = \Omega\); here \(\mathcal{L}\) denotes the Lie derivative.
In this case \(\lambda := i_Y \Omega \big|_{M^+}\) is a contact form on
  \(M^+\).
\end{definition}

\setcounter{CounterSectionIntertwine}{\value{section}}
\setcounter{CounterTheoremIntertwine}{\value{theorem}}

\begin{theorem}[intertwining existence and almost existence]
  \label{THM_existence_and_almost_existence}
  \hfill\\
Let \((W, \Omega)\) be a four-dimensional compact connected exact symplectic
  manifold with boundary \(\partial W = M^+ \cup M^-\).
Suppose \(M^+\) is positive contact type in the sense of Definition
  \ref{DEF_positive_contact_type_boundary}, and suppose that one of the
  following three conditions holds:
\begin{enumerate}
  \item
  \(M^+\) has a connected component diffeomorphic to \(S^3\),
  \item
  there exists an embedded \(S^2\subset M^+\) which is homotopically
  nontrivial in \(W\),
  \item
  \((M^+, \lambda)\) has a connected component which is overtwisted.
  \end{enumerate}
Then for each Hamiltonian \(H\in C^\infty(W,{\mathbb R})\) for which
  \(H^{-1}(\pm 1) = M^\pm\), the following is true.
For each \(s\in [-1,1]\) the energy level \(H^{-1}(s)\) contains a closed
  non-empty set other than the energy level \(H^{-1}(s)\) itself which is
  invariant under the Hamiltonian flow of \(X_H\); moreover for almost every
  \(s\in [-1, 1]\) this closed invariant subset is a periodic orbit.
\end{theorem}

Note that the image of $H$ is not required to lie in $[-1,1]$.

The proof of Theorem \ref{THM_existence_and_almost_existence} is
  provided in Section \ref{SEC_main_proof} below.

\subsection{Speculation on the
  \texorpdfstring{$C^\infty$}{C-infinity}-Closing Property}
  \label{SEC_speculation}

Before moving on to the proofs of Theorem \ref{THM_main_maybe} and Theorem
  \ref{THM_existence_and_almost_existence}, we wish to draw some
  connections to the \(C^\infty\)-closing lemma.
We also aim to pose some speculative questions which we believe the
  feral curve techniques below seem well poised to eventually answer.

We begin with an odd-symplectic manifold $(M,\omega)$, and take a compatible
  symplectic form $\Omega$ on ${\mathbb R}\times M$.
We consider the Fr\'echet space $C^{\infty}(M,{\mathbb R})$ and observe
  that every element $f$ defines a hypersurface in ${\mathbb R}\times M$ by
  setting
  \[
  M_f=\{(f(m),m)\in \mathbb{R}\times M :\ m\in M\}.
  \]
Considering the hypersurface $M_f\subset ({\mathbb R}\times M,\Omega)$ we
  obtain the distinguished line bundle ${\mathcal L}_{f}\rightarrow M_f$,
  given by
  \[
   {\mathcal L}_f:= \ker(\omega_f)\subset TM_f,
   \]
  where $\omega_f$ is the pull-back of $\Omega$ by the inclusion
  $M_f\rightarrow {\mathbb R}\times M$.
Since ${\mathcal L}_f\subset TM_f$ is a dimension one sub-bundle, it is an
  integrable distribution and we are interested in the closed leaves.
We denote by ${\mathcal C}_f$ the union of all points in $M_f$ which lie
  on a closed leaf.
We say that the periodic orbits are dense on $M_f$ provided
  $\textrm{cl}({\mathcal C}_f)=M_f$.

\begin{definition}[$C^\infty$-closing property]
  \label{DEF_closing_property_framed_hamiltonian}
  \hfill\\
We say that the odd-symplectic manifold $(M,\omega)$ has the
  $C^{\infty}$-\textit{closing property} provided that there exists a
  compatible $\Omega$ on ${\mathbb R}\times M$ so that for a Baire subset
  $\Sigma$ of $C^{\infty}(M,{\mathbb R})$  the following holds:
  \[
  \textrm{\emph{cl}}({\mathcal C}_f) = M_f\ \ \textrm{for all}\ f\in\Sigma.
  \]
  \end{definition}

Again one can use the closing property to define a particular class of
  symplectic manifolds.

\begin{definition}[$C^\infty$-closing property -- symplectic
  manifolds]
  \label{DEF_closing_property_symplectic}
  \hfill\\
We say a symplectic manifold $(W,\Omega)$ has the
  $C^{\infty}$-\textit{closing property} provided for every regular compact
  co-oriented hypersurface $M$ in $W$ the induced $(M,\omega)$ has the
  $C^{\infty}$-closing property.
\end{definition}

One can play around with the above definition by allowing only
  hypersurfaces isotopic to small ones or those carrying a suitable
  topology.
Alternatively, one might choose to only allow contact-type hypersurfaces.
We leave it to the reader to explore these ideas and we only mention the
  following conjecture.

\begin{conjecture}[local $C^{\infty}$-closing property]
  \label{CON_closing}
  \hfill\\
The standard symplectic vector space $({\mathbb
  R}^{2n},\Omega_{\textrm{standard}})$, $n\geq 2$,  has the
  $C^{\infty}$-closing property.
In particular all symplectic manifolds without boundary have the local
  $C^{\infty}$-closing property.
\end{conjecture}

The conjecture is open for all $n\geq 2$ and basically nothing is known for $n\geq 3$.
In the case of $n=2$ one knows a partial result, namely that compact,  regular hypersurfaces
in ${\mathbb R}^4$ of contact type have the $C^{\infty}$-closing property. However, nothing is known
for general compact regular hypersurfaces in ${\mathbb R}^4$.
Indeed, by a result of Irie, \cite{Irie:2015}, every $(M,\lambda,d\lambda)$, where
  $M$ is a closed three-manifold equipped with a contact form $\lambda$, has
  the $C^{\infty}$-closing property.
In particular every compact regular hypersurface of contact type in
  ${\mathbb R}^4$ has the $C^{\infty}$-closing property.
In the background of Irie's result and a follow-up result, \cite{ AI}, is
  the important volume formula by  Cristofaro-Gardiner,  Hutchings,  and
  Ramos, \cite{CHR}.
At present the proof of \cite{Irie:2015} based on \cite{CHR}  only works in the
  three-dimensional case.
Due to the use of Seiberg-Witten-Floer Theory it will need some new ideas
  to attack the higher-dimensional cases -- perhaps feral curves.

The germ of an idea goes as follows.
The key upshot of feral curves, is that one can stretch the neck along any
  hypersurface.
The downside is that one may find limit sets which are much more
  complicated than a finite set of periodic orbits.
However, Theorem \ref{THM_main_maybe} strongly suggests that generically
  (in the right sense) one can stretch the neck while following a
  \emph{single} curve for each almost complex structure, and pass to the
  limit to find a periodic orbit.
That is, at least one periodic orbit is (generically) found by tracking
  just one curve.
What if there are many curves to track?
High dimensional families of curves, for example.
Here the proposed richness of the Gromov-Witten invariants comes into
  play.
For example, consider \(\mathbb{R}^{2n} \simeq \mathbb{C}^n =
  \mathbb{C}P^n\setminus \mathbb{C}P^{n-1}\), and suppose we consider
  stretching the neck along some generic hypersurface \(M\subset
  \mathbb{R}^{2n}\).
By considering curves of high degree, one obtains high dimensional
  families of curves which stretch and break along periodic orbits.
This raises a question: Which orbits can be found by stretching the neck
  and tracking curves of any fixed (but arbitrarily large) degree?
All orbits, or just a subset?
A dense subset?
If it turns out that neck stretching can find (nearly) every orbit
  then feral curves seem well poised to recover the
  \(C^\infty\)-closing lemma for arbitrary regular, compact hypersurfaces in \(\mathbb{R}^{2n}\).
 Currently the known results about the contact-type case in ${\mathbb R}^4$ depend on Seiberg-Witten theory.
  The feral curve theory should be important in removing the contact-type hypothesis. In order to prove the results in higher dimensions
 one way to succeed seems  to be the development of suitable techniques to use higher-dimensional moduli spaces.

config: arxiv


TITLE: The number of e-th power residues $\bmod m$
QUESTION [2 upvotes]: For each positive integer $m$ define $\Bbb{Z}_m = {0, 1, 2, \ldots , m − 1}$, the set of all residues modulo $m$, and define
  $$C(m,e) = \{ k ∈ \Bbb{Z}_m \mid 0 \ne k ≡ a^e \pmod m \text{ for some }a \in \Bbb{Z} \}$$
  So $C(m,e)$ is the set of all non-zero e-th power residues mod m.

For a specific $e$ such that $e\mid m-1$ and $m$ is prime, I know that you can show that the number of elements in $C(m,e)$ is $(m-1)/e$. What about the cases when $e\nmid (m-1)$?
I computed $C(m,e)$ numerically using arbitrary values Matlab and I am pretty sure that when m is prime and $e\nmid (m-1)$, $C(m,e)$ has $m-1$ elements i.e. is a full set. How can we prove this?
And then what if m isn't prime but instead $m = pq$ and $e\mid(p-1)$ and $e\mid (q-1)$? Again I plugged in values in Matlab to find that integers with these properties gave the size of $C(m,e)$ to be$$\left(\frac{p-1}{e}+1 \right)\left(\frac{q-1}{e}+1\right)-1$$
But again I do not know how to show this analytically. Any hints or insights would be greatly appreciated.

REPLY [1 votes]: 1. Let $p$ be prime, and suppose that $e$ and $p-1$ are relatively prime. We will show that any $a$ not divisible by $p$, there is an $x$ such that $x^e\equiv a\pmod{p}$.
Since $e$ and $p-1$ are relatively prime, by the "Bezout Identity" there exist integers $s$ and $t$ such that $es+(p-1)t=1$. Then
$$a=a^1=a^{es+(p-1)t}=(a^s)^e (a^{p-1})^t\equiv (a^s)^e\pmod{p}.$$
So we have found an $x$, namely $a^s$, such that $x^e\equiv a\pmod{p}$.
Note that it is not enough to assume that $e$ does not divide $p-1$. For example, $4$ does not divide $7-1$, but $6$ is not a fourth power of anything modulo $7$.
2. Now we look at your assertion about products $pq$ of distinct primes, where $e\mid p-1$ and $e\mod q-1$. We show that, counting $0$, there are $$\left(\frac{p-1}{e}+1\right)\left(\frac{q-1}{e}+1\right)\tag{1}$$ distinct $e$-th powers, modulo $pq$.
Note first that any $e$-th power modulo $pq$ is an $e$-th power modulo $p$ and modulo $q$. Now let $a$ be an $e$-th power modulo $p$, and let $b$ be an $e$-th power modulo $q$. By the Chinese Remainder Theorem, there is a unique $c$ modulo $pq$ such that $c\equiv a\pmod{p}$ and $c\equiv b\pmod{q}$. We show that $c$ is an $e$-th power modulo $pq$.
Let $x^e\equiv a\pmod{p}$, and let $y^e\equiv b\pmod{q}$. Let $w$ be the solution of the system of congruences $w\equiv x\pmod{p}$, $w\equiv y\pmod{q}$.
Then $w^e\equiv a\equiv c \pmod{p}$ and  $w^e\equiv c\pmod{q}$, and therefore $c$ is an $e$-th power modulo $pq$.
It follows that the number of $e$-th powers modulo $pq$ is given by Formula 1. The formula and proof generalize to products of $k$ distinct primes.