text:
\section{Introduction}

The celebrated Szemer\'edi theorem \cite{Szem} from 1975 asserts that every subset of the natural numbers with positive upper density
contains arithmetic progressions of arbitrary length. This means that as soon as a set does not vanish asymptotically it has certain structure inside.
In 1977 Furstenberg \cite{Fu77} presented his groundbreaking ergodic theoretic proof of Szemer\'edi's theorem  based on multiple recurrence
which has had enormous impact.
In particular, various related results in additive number theory were proven using ergodic theoretic methods such as the Furstenberg-Katznelson multidimensional Szemer\'edi theorem \cite{FuKa78}, the Bergelson-Leibman polynomial Szemer\'edi theorem  \cite{BL96}, the Green-Tao theorem on the existence of arithmetic progressions in the primes \cite{GT08,GT10}, the Tao-Ziegler polynomial Green-Tao theorem \cite{TZ08,TZ13,TZ18} and the multidimensional Green-Tao theorem \cite{TaZi15,CMT,FoZh} (where \cite{CMT} does not use ergodic theory).  For some related works see also \cite{BM00,FHK07,BLL08,FrWi09,WoZi12,FHK13,Frantz15,MR19,BMR20}.

In this note we present a proof of multiple recurrence for ergodic systems (and hence of Szemer\'edi's theorem) being a mixture of three known proofs, namely Tao's modification \cite{Tao-ET} of the classical Furstenberg-Katznelson proof \cite{FuKa78}, the ori\-ginal Furstenberg proof \cite{Fu77} and (the beginning of)
the proof of multiple convergence by Host and Kra \cite{HK}.
For a finitary quantitative proof of Szemer\'edi's theorem with similar philosophy see Tao \cite{Tao-szem}.

Like the original Furstenberg proof \cite{Fu77} (and in contrast to the classical Furstenberg-Katznelson proof), our proof
does not use transfinite induction  and deals for multiple recurrence of order $k$ with a tower of  $k-1$ factors, namely Furstenberg's distal factors
of order
less than or equal to $k-1$.
We give an alternative proof of Furstenberg's result  \cite{Fu77} that the distal factor of order $k-1$ is characteristic for $k$-term multiple ergodic averages by showing that this factor is an extension of the Host-Kra-Ziegler factor of order $k-1$ (via working with the corresponding subspaces of functions).
 Thereby we do not need any information on the Host-Kra-Ziegler factors other than
the description of the corresponding subspaces via the Gowers-Host-Kra seminorms, not even the fact that they are factors. Note that the original proof of Furstenberg of the characteristic property of the distal factors of finite order is based on a description of conditional eigen\-functions\footnote{in Furstenberg's terminology, generalized eigenfunctions} of fibered product systems (see \cite[Theorem 7.1]{Fu77}) and analysis of diagonal measures, with a subsequent simplification by Frantzikinakis \cite[Theorem 5.2]{Frantz04}, still based on \cite[Theorem 7.1]{Fu77}, using the van der Corput trick.

Two decompositions will play a central role, namely a conditional version of the \jdlgd\ and the Host-Kra decomposition.
The first one, decomposing the $\Ell^2$-space into conditionally almost periodic and conditionally weakly mixing functions, is a stronger version due to Tao \cite[Chapter 2]{Tao-ET} of the dichotomy between almost periodic and weakly mixing extensions due to Furstenberg, Katznelson \cite{FuKa78}.
The Host-Kra decomposition decomposes, for every $k$, the $\Ell^2$-space into a part where the Gowers-Host-Kra seminorm of order $k+1$ vanishes and the orthogonal complement known as the Host-Kra-Ziegler factor of order $k$.
Then properties of the Gowers-Host-Kra seminorms and the conditional Jacobs-de Leeuw-Glicksberg decomposition quickly imply
 that the distal factors of finite order are extensions of the Host-Kra-Ziegler factors of the corresponding order,
see Proposition \ref{prop:Z_k-ap} below.
(Host, Kra \cite[Lemma 6.2]{HK}, \cite[Lemma 18.2]{HK-book} and Ziegler \cite[Thm.~6.1]{Ziegler} showed\footnote{Host and Kra showed that the Host-Kra-Ziegler factors are isometric extensions (as defined by Furstenberg \cite{Fu77}) of each other. See, e.g., Glasner \cite[Theorem 9.14]{Glasner-book} for the characterization of isometric extensions as those being generated by conditional eigenfunctions  implying almost periodicity of such extensions.} a stronger property of the Host-Kra-Ziegler factors, namely that they are compact extensions of each other. For an alternative proof using the conditional Jacobs-de Leeuw-Glicksberg decomposition see Zorin-Kranich \cite[Lemma 10.1]{Pavel-ET}.)

The rest of the argument is standard. By definition, the distal factors of finite order are maximal compact extensions of each other. By induction, starting with the fixed (one-point) factor and using the fact that compact extensions preserve multiple recurrence (Proposition \ref{prop:ap-ext-MR} below), one has multiple recurrence for the distal factors of all orders. Since the distal factor of order $k-1$ is an extension of the Host-Kra-Ziegler factor of order $k-1$, it
is characteristic for $k$-term multiple ergodic averages by the generalized von Neumann theorem (Proposition \ref{prop:ghk-control-mult-ave} and Corollary \ref{cor:ghk-control-mult-ave} below),
completing the proof.

As indicated above, the structure of the proof
 is essentially implicit in Host, Kra \cite{HK}, Ziegler \cite{Ziegler} as well as in Furstenberg \cite{Fu77} combined with Furstenberg, Katznelson, Ornstein \cite{FuKaOr82}, cf.~Tao \cite[Remark 4.7]{Tao-szem}. We present here a view based on the conditional Jacobs-de Leeuw-Glicksberg decomposition and properties of the Gowers-Host-Kra seminorms.

\textbf{Acknowledgements.} The author is very grateful to B\'alint Farkas and Asgar Jamneshan for helpful discussions, comments and references.

\section{Preliminaries}

Throughout this article we call a triple $(X,\mu,\varphi)$ a \emph{measure-preserving system}\ if $(X,\mu)$ is a
probability space and $\varphi$ is a measurable invertible transformation on $X$ with measurable inverse which is $\mu$-preserving, i.e., $\mu(\varphi^{-1}A)=\mu(A)$ holds for every measurable set $A\subset X$.\footnote{In this case $\varphi^{-1}$ is clearly $\mu$-preserving as well. The invertibility assumption is technical and will assure that the conditional expectation commutes with the transformation. Since in our context one can work with invertible systems without loss of generality, we assume all systems to be invertible.}
For a \mps\ $(X,\mu,\varphi)$ we call the linear invertible isometry $T:\Ell^1(X,\mu)\to \Ell^1(X,\mu)$ defined by $Tf:=f\circ \varphi$ the \emph{Koopman operator} of the system.
The Koopman operator acts as  an invertible isometry on $\Ell^p(X,\mu)$ for every $p\in [1,\infty]$ and is unitary on $\Ell^2(X,\mu)$. Since we mostly work with the Koopman operator $T$ instead of the transformation $\varphi$, we often write $(X,\mu,T)$ instead of $(X,\mu,\varphi)$.

A measure-preserving system  $(X,\mu,\varphi)$ is called \emph{ergodic} if it has no non-trivial invariant sets, i.e., if every measurable set satisfying $\varphi^{-1}A\subset A$ up to a null set (which is equivalent to $\varphi^{-1}A=A$ and hence, by the invertibility of $\varphi$, to $A=\varphi(A)$ up to a null set)
has measure zero or one. For the corresponding Koopman operator $T$ on every $\Ell^p(X,\mu)$ this is equivalent to $\Fix T=\C\Eins$ (i.e., $T$  has only constant invariant functions).

We now define multiple recurrence and multiple convergence. Thereby,  $f>0$ means $f\geq 0$ and $f\neq 0$ (as element of $\Ell^\infty(X,\mu)$).

\begin{definition}
Let $(X,\mu,T)$ be a measure-preserving system and $k\in \N$. We say that $(X,\mu,T)$ satisfies
\begin{itemize}
\item \emph{multiple recurrence of order $k$} (shortly \emph{MR$_k$}) if for every $0<f\in\Ell^\infty(X,\mu)$
\begin{equation}\label{eq:mult-rec}
\liminf_{N\to\infty}\aveN \int_X f\cdot T^nf\cdot T^{2n}f \cdots T^{kn}f\,d\mu>0.
\end{equation}
\item \emph{multiple convergence of order $k$}
if for every $f_1,\ldots,f_k\in\Ell^\infty(X,\mu)$ the limit of multiple ergodic averages
\begin{equation}\label{eq:mult-erg-ave}
\aveN T^nf_1\cdot T^{2n}f_2 \cdots T^{kn}f_k
\end{equation}
exists in $\Ell^2(X,\mu)$.
\end{itemize}
We say that $(X,\mu,T)$ satisfies \emph{multiple recurrence} (shortly \emph{MR}) if it satisfies MR$_k$ for all $k\in\N$ and define analogously \emph{multiple convergence}.
\end{definition}
Note that, given multiple convergence of order $k$, the property MR$_k$ means that every function $f\in \Ell^\infty(X,\mu)$ with $f>0$ correlates with the limit of the averages $\aveN T^nf\cdots T^{kn}f$. Moreover, for functions of the form $f=\Eins_A$ for some $A\subset X$ with $\mu(A)>0$ property \eqref{eq:mult-rec} implies for the underlying transformation $\varphi$ that $\mu(A\cap \varphi^{-n}A\cap\ldots\cap \varphi^{-kn}A)>0$ holds for many $n\in\N$ explaining the name ``multiple recurrence''.

We now state Szemer\'edi's theorem which inspired the study of multiple recurrence and multiple convergence.

\begin{thm}[Szemer\'edi \cite{Szem}]
Let $C\subset \N$ have \emph{positive upper density}, i.e., satisfy $\ol{d}(C):=
\limsup_{N\to\infty} \frac{|C\cap\{1,\ldots,N\}|}N>0$. Then $C$ contains arbitrarily long arithmetic progressions, i.e., for every $k\in\N$ there exist
$a,n\in \N$ with $a,a+n,\ldots,a+kn\in C$.
\end{thm}

Furstenberg  \cite{Fu77} divided his ergodic theoretic proof of Szemer\'edi's theorem into two parts. The difficult part was to show multiple recurrence for ergodic systems.

\begin{thm}[Furstenberg, multiple recurrence for ergodic systems]\label{thm:mr}
Every
ergodic\footnote{For our purposes ergodic systems  suffice by Proposition \ref{prop:corr-principle}. Note that multiple recurrence holds for general (also not necessarily invertible) \mpss, see, e.g., Einsiedler, Ward \cite[pp.~177--178]{EW}.}
\mps\
satisfies MR.\footnote{Furstenberg  \cite{Fu77} showed a stronger property than MR where the lower Ces\'aro limit $\liminf_{N\to\infty}\aveN$ in \eqref{eq:mult-rec} is replaced by the lower Banach limit $\liminf_{N-M\to\infty}\sum_{n=M+1}^N$.}
\end{thm}

The easy part was to build the following bridge between number theory and ergodic theory, see, e.g., Furstenberg \cite[p.~77]{Fu-book}, Kra \cite[Section 2.2]{Kra} or \cite[Section 20.1]{EFHN} for details.

\begin{proposition}[Furstenberg, correspondence principle]\label{prop:corr-principle}
Multiple recurrence for ergodic systems implies the assertion of Szemer\'edi's theorem. More precisely,
for every $C\subset \N$ there exists an ergodic \mps\ $(X,\mu,\varphi)$ and a measurable set $A\subset X$ with $\mu(A)\geq \ol{d}(C)$\footnote{By the same argument one can easily replace the upper density $\ol{d}(C)$ by the upper Banach density $d^*(C):=
\limsup_{N-M\to\infty}\frac{|C\cap \{M+1,\ldots,N\}|}{N-M}$ leading to the corresponding stronger version of Szemer\'edi's theorem proved by Furstenberg \cite{Fu77}.} such that, for every $k\in\N$, the set $C$ contains an arithmetic progression of length $k+1$ if and only if there exists $n\in\N$ such that the set
$A\cap \varphi^{-n}A\cap \ldots\cap \varphi^{-kn}A$ has positive measure.
\end{proposition}

In particular, it suffices to prove
that for every $f\in\Ell^\infty(X,\mu)$ with $f>0$ there exists $n\in\N$ such that $\int_X f\cdot T^nf\cdot T^{2n}f \cdots T^{kn}f\,d\mu>0$ which is a formally weaker property but turns out to be equivalent to MR$_k$. For a discussion and a deduction of multiple recurrence back from Szemer\'edi's theorem see, e.g., \cite[Section 20.2]{EFHN}.

Multiple convergence remained open
 until it was proved by Host, Kra \cite{HK} in 2005, with a subsequent alternative proof by Ziegler  \cite{Ziegler}.

\begin{thm}[Multiple convergence, Host, Kra \cite{HK}, Ziegler \cite{Ziegler}]\label{thm:mr}
Every
measure-preserving system
satisfies multiple convergence.
\end{thm}

\medskip

We now collect some definitions.

We do not need to assume systems to be separable (i.e., the underlying $\sigma$-algebra to be countably generated up to null sets).
So we work with the following abstract notion of a factor. A \mps\ $(Y,\nu,S)$ is called a \emph{(Markov) factor} of $(X,\mu,T)$ if there exists a Markov homomorphism\footnote{An operator $J:\Ell^1(Y,\nu)\to \Ell^1(X,\mu)$ is called \emph{Markov} if it is positive (i.e., $f\geq 0$ implies $Jf\geq 0$) and satisfies $J\Eins=\Eins$ as well as $ J'\Eins=\Eins$ (i.e., $\int_X Jf\,d\mu=\int_Yf\,d\nu$ holds for every $f\in \Ell^1(Y,\nu)$). A Markov operator is called \emph{Markov homomorphism} (or \emph{Markov embedding}) if it preserves the absolut value or, equivalently, is multiplicative on $\Ell^\infty(Y,\nu)$, see, e.g., \cite[Theorem 7.23]{EFHN}.}  $J:\Ell^1(Y,\nu)\to \Ell^1(X,\mu)$ which respects the Koopman operators, i.e., satisfies $JS=TJ$. In this case $(X,\mu,T)$ is called a  \emph{(Markov) extension} of $(Y,\nu,S)$ and $J$ is called the \emph{(Markov) factor map}.

\begin{remark}\label{rem:factors}
\begin{itemize}
\item[(a)] For a (Markov) factor $(Y,\nu,S)$ of a measure-preserving system $(X,\mu,T)$ with  factor map $J:\Ell^1(Y,\nu)\to \Ell^1(X,\nu)$ the set $J\Ell^\infty(Y,\nu)$ is a conjugation invariant, $T$- and $T^{-1}$-invariant subalgebra of $\Ell^\infty(X,\mu)$ containing $\Eins$, and there is a (up to isomorphism one-to-one) correspondence between  (Markov) factors of $(X,\mu,T)$ and  conjugation invariant, $T$- and $T^{-1}$-invariant subalgebras of $\Ell^\infty(X,\mu)$ containing $\Eins$. Every such subalgebra $\A$ of functions further corresponds to a $\varphi$- and $\varphi^{-1}$-invariant sub-$\sigma$-algebra of the original $\sigma$-algebra on $X$ for the underlying transformation $\varphi$, namely the smallest $\varphi$- and $\varphi^{-1}$-invariant sub-$\sigma$-algebra such that all functions from $\A$ are measurable with respect to it. We will work with the above characterization of factors via subalgebras of functions.
\item[(b)]
For separable systems every (Markov) factor map is induced by a \emph{point factor map}, i.e., $J$ is of the form $Jf=f\circ \pi$ for a  morphism $\pi:X\to Y$, i.e., a measure-preserving map such that $\pi\circ \varphi=\psi \circ \pi$ holds $\mu$-a.e.~for the underlying transformations $\varphi$ and $\psi$ on $X$ and $Y$, respectively.
For details see, e.g., \cite[Chapter 7]{EF-book}. The readers who prefer point factors to Markov factors can assume without loss of generality that the system is separable by passing to the $\varphi$- and $\varphi^{-1}$-invariant sub-$\sigma$-algebra generated by the function $f$ for multiple recurrence and by the functions $f_1,\ldots,f_k$ for multiple convergence of order $k$, respectively.
\end{itemize}
\end{remark}

In particular, for a \mps\ $(X,\mu,T)$ with factor $(Y,\nu,S)$ the corresponding factor map $J:\Ell^1(Y,\mu)\to \Ell^1(X,\nu)$
 acts as a contraction w.r.t.~the $\Ell^p$-norm for every $p\in[1,\infty]$.
The adjoint operator $J'$ extends to a Markov operator
$$
\EY:=J':\Ell^1(X,\mu)\to\Ell^1(Y,\nu)$$
called the \emph{conditional expectation operator}. It also acts as a contraction w.r.t.~the $\Ell^p$-norm for every $p\in[1,\infty]$.  Recall that the conditional expectation operator satisfies
\begin{equation}\label{eq:EY}
\int_Y \EY f\,d\nu=\int_X f\,d\mu
\quad
\EY(Jg\cdot f)=g \EY f\quad \forall f\in \Ell^1(X,\mu)\ \forall g\in\Ell^\infty(Y,\nu).
\end{equation}
Since we assume factors to be invertible,
one also has
\begin{equation}\label{eq:ET=SE}
\EY T=S\EY.
\end{equation}
The operator
$$
P_Y:=J\EY:\Ell^2(X,\mu)\to\Ell^2(X,\mu)
$$
is the orthogonal projection onto the subspace $J\Ell^2(Y,\nu)$. We call $P_Y$  the
\emph{projection onto the factor} $Y$. It extends to a Markov operator acting as a contraction w.r.t.~every $\Ell^p$-norm, $p\in[1,\infty]$. Another important property of $P_Y$ is
\begin{equation}\label{eq:PYf>0}
P_Yf>0\quad \text{for every }f>0.
\end{equation}
(Indeed, the Markov property of $J$ and $\EY$ implies $P_Yf\geq 0$
and $\int_XP_Yf\,d\mu=\int_Xf\,d\mu>0$ ensuring $P_Yf\neq 0$.)

We say that a subspace $\A$ of $\Ell^2(X,\mu)$ is \emph{induced by a  factor} $(Y,\nu,S)$ of $(X,\mu,T)$  if $\A=J\Ell^2(Y,\nu)$ for the corresponding Markov embedding $J$ or, equivalently,
$\A=\Ell^2(X,\Sigma',\mu)$ for the $\varphi$-invariant sub-$\sigma$-algebra $\Sigma'$ corresponding to $(Y,\nu,S)$ for the underlying transformation $\varphi$ on $X$.
 By Remark \ref{rem:factors}(a),  this is the case if and only if $\A\cap \Ell^\infty(X,\mu)$ is a conjugation invariant, $T$- and $T^{-1}$-invariant subalgebra of $\Ell^\infty(X,\mu)$ containing $\Eins$.

\section{Single recurrence and von Neumann's decomposition}

We recall how the mean ergodic theorem and the property MR$_1$ rely on the von Neumann decomposition
\begin{equation}\label{eq:mul-vN-dec}
H=\fix T\oplus \ol{\rg(I-T)}
\end{equation}
for contractions $T$ on a Hilbert space $H$, see, e.g., \cite[Theorem 8.6]{EFHN}.
Let $(X,\mu,T)$ be a \mps\ and $f\in\Ell^2(X,\mu)=:H$.\footnote{For $k=1$ one does not need $f$ to be bounded.} By the telescopic sum argument the second part of \eqref{eq:mul-vN-dec} does not contribute to the limit of the ergodic averages $\aveN T^nf$ and  $T$ acts as the identity operator on the first part of  \eqref{eq:mul-vN-dec}.  Thus if $f>0$ then
$$
\limaveN \int_X f\cdot T^nf\,d\mu=
\left\la\limaveN T^nf,f\right\ra
\|P_{\Fix T} f\|_{\Ell^2(X,\mu)}^2>0.
$$
The last inequality follows from the measure preserving property of $T$ (which implies $\int_X P_{\Fix T}f\,d\mu=\int_X f\,d\mu\neq 0$) and reflects the fact that $\Fix T$ is induced by a factor, namely the fixed factor.

In other words, the second part in the von Neumann decomposition \eqref{eq:mul-vN-dec} does not contribute to the limit of $\aveN T^nf$
and we need $P_{\Fix T}f>0$
to restrict to the case $\Ell^2(X)=\Fix T$ where the assertion is trivial due to the very structured behavior of the orbits (being one point).

\section{Double recurrence and the classical \jdlgd}

To illustrate the argument for higher $k$ we now briefly discuss the case $k=2$ which is analogous to the case $k=1$. For a detailed exposition see, e.g., \cite[Section 8.3]{EF-book}.

\begin{definition}
Let $T$ be a contraction on a Hilbert space $H$. We call $f\in H$ \emph{weakly mixing} if $\limaveN |\la T^nf,f\ra|^2=0$.
\end{definition}

\begin{remark}\label{rem:wmix}
By the Koopman-von Neumann lemma, see, e.g., \cite[Lemma 9.16]{EFHN}, for a contraction $T$ on a Hilbert space $H$, $f\in H$ is weakly mixing if and only if there exists a set $J\subset \N$ with density $1$ (i.e., satisfying $d(J):=
\lim_{N\to\infty} \frac{|J\cap\{1,\ldots,N\}|}N=1$) such that $\lim_{n\to\infty,n\in J}\la T^nf,f\ra=0$.
\end{remark}

Double convergence and double recurrence rely on the following decomposition replacing the von Neumann decomposition, where $\T$ denotes the unit circle.

\begin{thm}[Jacobs-de Leeuw-Glicksberg decomposition]\label{thm:JdLG}
Let $T\in \LLL(H)$ be a contraction on a Hilbert space $H$. Then one has the orthogonal decomposition
\begin{equation}\label{eq:jdlg-dec}
H=\clin\{f\in H:\, Tf=\lambda f\text{ for some }\lambda\in\T\}\oplus \{f\in H:\, f\text{ is weakly mixing}\}.
\end{equation}
\end{thm}

Let now $(X,\mu,T)$ be a \mps\ and $f,g\in \Ell^\infty(X,\mu)$. One shows via the so-called van der Corput trick that the double averages
$$
\aveN T^nf \cdot T^ng
$$
converge to zero whenever $f$ or $g$ is weakly mixing. Since the first part of the decomposition \eqref{eq:jdlg-dec} is induced by a factor, namely the Kronecker factor, we have that $P_\kr f,P_\kr g\in\Ell^\infty(X,\mu)$ for the corresponding projection $P_\kr$. Thus
double convergence reduces to showing it for eigenfunctions for which it is immediate. For double recurrence let $g:=f>0$. Then $P_\kr f>0$
by \eqref{eq:PYf>0} and we can assume by \eqref{eq:jdlg-dec} and the above observation that $(X,\mu,T)$ coincides with its Kronecker factor. In this case the orbit of $f$ is compact and one uses almost periodicity of $f$ to show that the limit in \eqref{eq:mult-rec} (for $k=2$) is greater than zero.

To summarize, the second part in the decomposition \eqref{eq:jdlg-dec} does not contribute to the limit and, using the fact that the first part is induced by a factor, we can assume that this factor coincides with the original system. Then one uses the very structured behavior of the orbits.

Next we consider two generalizations of \eqref{eq:jdlg-dec}, the  conditional Jacobs-de Leeuw-Glicksberg decomposition and the Host-Kra decomposition, see \eqref{eq:rel-jdlg-a+w} and \eqref{eq:hk-dec} below.

\section{Conditional \jdlgd

We now present the abstract conditional setting introduced by Tao \cite[Chapter 2]{Tao-ET}. To simplify the notation we will shorten
$\Ell^p(X,\mu)$ to $\Ell^p(X)$ for a probability space $(X,\mu)$.
Moreover, by writting $cf$ for $c\in\Ell^\infty(Y)$ and $f\in \Ell^1(X)$ we identify $c$ with its image under $J:\Ell^1(Y)\to\Ell^1(X)$.

\begin{definition}
Let $(X,\mu,T)$ be a \mps\ with
factor $(Y,\nu,S)$.
For $f,g\in \Ell^2(X)$ we call the function
$$
\la f,g\ra_{\Ell^2(X|Y)}:=\EY(f\cdot\ol{g}
\in\Ell^1(Y)
$$
the \emph{conditional scalar product} of $f$ and $g$ w.r.t.~$Y$ and the function
$$
\|f\|_{\Ell^2(X|Y)}:=(\la f,f\ra_{\Ell^2(X|Y)})^{\frac12}=(\EY(|f|^2))^{\frac12}\in\Ell^2(Y
$$
the \emph{conditional
norm} of $f$ w.r.t.~$Y$. Moreover, the \emph{conditional $\Ell^2$-space} is defined by
$$
\Ell^2(X|Y):=\{f\in\Ell^2(X):\, \|f\|_{\Ell^2(X|Y)}\in\Ell^\infty (Y)\}.
$$
A function $f\in \Ell^2(X|Y)$ is called
\emph{conditionally almost periodic} if for every $\veps>0$ there exists a \emph{finitely generated module zonotope} $Z$ (i.e., a set of the form
$$
\{c_1f_1+\ldots+c_mf_m:\,\|c_1\|_{\Ell^\infty(Y)},\ldots,\|c_m\|_{\Ell^\infty(Y)}\leq 1\}
$$
 for some $m\in\N$, $f_1,\ldots,f_m\in\Ell^2(X|Y)$) such that for every $n\in\Z$ there exists  $g\in Z$ with
$$
\|\|T^nf-g\|_{\Ell^2(X|Y)}\|_{\Ell^\infty(Y)}<\veps.
$$
We denote the space of all conditionally almost periodic functions by $A(X|Y)$.

A function $f\in \Ell^2(X)$ is called  \emph{conditionally weakly mixing}\footnote{We follow here Zorin-Kranich \cite[Section 8]{Pavel-ET}. Tao \cite[Section 2.14.1]{Tao-ET} calls a function $f\in\Ell^2(X|Y)$ conditionally weakly mixing if $\limaveN
\|\la T^nf,f\ra_{\Ell^2(X|Y)}\|_{\Ell^2(Y)}^2=0$. It is easy to see that these two definitions are equivalent on $\Ell^2(X|Y)$.} if
$$
\limaveN
\|\la T^nf,f\ra_{\Ell^2(X|Y)}\|_{\Ell^1(Y)}
=0.
$$
We denote the set of all such functions by $W(X|Y)$.
\end{definition}
To shorten the notation we abbreviate
$$
\|f\|_{\Ell^{2,\infty}(X|Y)}:=\|\|f\|_{\Ell^2(X|Y)}\|_{\Ell^\infty(Y)}
$$
for $f\in \Ell^2(X|Y)$. It is easy to see, by establishing a Cauchy-Schwarz inequality for the conditional scalar product, that $\|\cdot\|_{\Ell^{2,\infty}(X|Y)}$ is a norm on $\Ell^2(X|Y)$.
\begin{remark}\label{rem:AandW}
\begin{itemize}
\item[(a)] It follows directly from the definition that
$$
\|fg\|_{\Ell^{2,\infty}(X|Y)}\leq \|f\|_{\Ell^\infty(X)}\|g\|_{\Ell^{2,\infty}(X|Y)}
$$
holds for every $f\in\Ell^\infty(X)$ and $g\in\Ell^{2,\infty}(X|Y)$.
\item[(b)]
The space $\Ell^2(X|Y)$ is clearly an $\Ell^\infty(Y)$-module (i.e., a linear subspace closed under multiplication by $\Ell^\infty(Y)$-functions) with
$$
\Ell^\infty(X)\subset \Ell^2(X|Y)\subset \Ell^2(X).
$$
Moreover, $\Ell^2(X|Y)$ is $T$-invariant by \eqref{eq:ET=SE} with $T$ acting as a contraction w.r.t.~the norm $\|\cdot\|_{\Ell^{2,\infty}(X|Y)}$.
\item[(c)] It is easy to see that $A(X|Y)$ is a $T$- and $T^{-1}$-invariant sub-$\Ell^\infty(Y)$-module.  Moreover, it is closed in $\Ell^2(X|Y)$ w.r.t.~the norm $\|\cdot\|_{\Ell^{2,\infty}(X|Y)}$.
\item[(d)] It is easy but somewhat longer to show that $W(X|Y)$ is a closed linear subspace of $\Ell^2(X)$.\footnote{See Zorin-Kranich \cite[Lemma 8.1]{Pavel-ET} based on Tao \cite[Exercises 2.14.1, 2.14.2]{Tao-ET}.}
\item[(e)] Analogously to Remark \ref{rem:wmix},
 $f\in \Ell^2(X|Y)$ is conditionally weakly mixing if and only if there exists a set $J\subset \N$ with density $1$ such that $\lim_{n\to\infty,n\in J}\|\la T^nf,f\ra_{\Ell^2(X|Y)}\|_{\Ell^1(Y)}=0$.
\item[(f)] For the trivial one-point factor $(Y,\nu,S)$ the conditional definitions coincide with the unconditional ones. In particular, in this case $f\in W(X|Y)$ if and only if $f$ is weakly mixing.
\end{itemize}
\end{remark}

\begin{remark}[Connection to conditional almost periodicity in measure]\label{rem:capm}
A function $f\in \Ell^2(X|Y)$ is called  \emph{conditionally almost periodic in measure} if for every $\veps>0$ there exists a measurable set $M\subset Y$ with $\nu(M)>1-\veps$ such that $\Eins_Mf\in A(X|Y)$, see Tao \cite[Def.~2.13.7]{Tao-ET}, cf.~Furstenberg, Katznelson \cite[Section 2]{FuKa78} and Furstenberg \cite[Section 6.3]{Fu-book}. Denote the set of all such functions by $AM(X|Y)$. This set satisfies
$$
AM(X|Y)=\ol{A(X|Y)}\cap \Ell^2(X|Y),
$$
where the closure is taken in $\Ell^2(X)$, see, e.g., Zorin-Kranich \cite{Pavel}, cf.~Furstenberg, Katznelson \cite[Thm.~2.1]{FuKa78} and Furstenberg \cite[Thm.~6.13]{Fu-book}. For the reader's convenience we present here the short argument. The inclusion ``$\subset$'' follows directly from the definition of conditional almost periodicity. For the converse inclusion let $f\in\Ell^2(X|Y)$ such that $f=\lim_{n\to\infty} f_n$ in $\Ell^2(X)$ for some sequence $(f_n)\subset A(X|Y)$. By \eqref{eq:EY} we have
$$
\int_Y \EY(|f_n-f|^2)\,d\nu=\|f_n-f\|_{\Ell^2(X)}^2\to 0\quad \text{as }n\to\infty.
$$
Hence we can assume without loss of generality (passing to a subsequence if ne\-cessary) that $\EY(|f_n-f|^2)\to 0$ as $n\to\infty$ holds $\nu$-a.e. By Egorov's theorem there exists a measurable set $M\subset Y$ with $\nu(M)>1-\veps$ such that $\|\Eins_M\EY(|f_n-f|^2)\|_{\Ell^\infty(Y)}\to 0$, or, equivalently by \eqref{eq:EY},
$$
\|\Eins_Mf_n-\Eins_Mf\|_{\Ell^{2,\infty}(X|Y)}\to 0\quad \text{as }n\to\infty.
$$
Since for every $n\in\N$ the function $\Eins_M f_n$ is conditionally almost periodic (by using  the same module zonotopes as for $f_n$), so is $\Eins_M f$ by Remark \ref{rem:AandW}(c).
\end{remark}

The following conditional version of Theorem \ref{thm:JdLG} is due to Tao \cite[Section 2.14.2]{Tao-ET}, cf.~Furstenberg, Katznelson \cite[Prop.~2.2]{FuKa78}
and Furstenberg, Katznelson, Ornstein \cite[Thm.~10.1]{FuKaOr82}.
Here and later we mean the closure in $\Ell^2(X)$ if not specified otherwise.

\begin{thm}
Conditional Jacobs-de Leeuw-Glicksberg decomposition]\label{mr:higher-JdLG}
Let $(X,\mu,T)$ be a
\mps\ with
factor $(Y,\nu,S)$. Then
one has the orthogonal decomposition
\begin{equation}\label{eq:rel-jdlg-a+w}
\Ell^2(X)=\ol{A(X|Y)}\oplus W(X|Y).
\end{equation}
\end{thm}

\begin{remark}\label{rem:a=e}
Zorin-Kranich \cite{Pavel} showed that for regular
systems with ergodic factor
\begin{equation}\label{eq:A=E}
\ol{A(X|Y)}=\ol{E(X|Y)}
\end{equation}
holds for the set $E(X|Y)$ of conditional eigenfunctions, see  Jamneshan \cite[Theorem 4.1]{Ja} for the general case in the context of arbitrary group actions.
Here  $f\in \Ell^2(X|Y)$ is called a \emph{conditional eigenfunction} if
its orbit is contained in a $T$-invariant \emph{finitely generated sub-$\Ell^\infty(Y)$-module} (i.e., a set of the form
$$
\{c_1f_1+\ldots+c_mf_m:\,c_1,\ldots,c_m\in\Ell^\infty(Y)\}
$$ for some $m\in\N$, $f_1,\ldots,f_m\in\Ell^2(X|Y)$).\footnote{This  definition due to Furstenberg, Weiss \cite{FW96}, see also Zimmer
\cite[Section 7]{Zi}, is equivalent to the one of generalized eigenfunctions introduced by Furstenberg \cite{Fu77} for ergodic factors $(Y,\nu,S)$.} Thus we also have the orthogonal decomposition
\begin{equation}\label{eq:rel-jdlg-e+w}
\Ell^2(X)=\ol{E(X|Y)}\oplus W(X|Y),
\end{equation}
cf.~Zimmer \cite[Cor.~7.10]{Zi}.
For two different recent abstract approaches to
\eqref{eq:rel-jdlg-e+w} in the context of general group actions
see Jamneshan \cite{Ja} and Edeko, Haase, Kreidler \cite{EHK}.
\end{remark}

The following lemma is crucial,  see Tao \cite[Exercise 2.13.6]{Tao-ET} combined with Remark \ref{rem:capm}.
We assume ergodicity of the factor in order to simplify the argument and follow the proof by Zorin-Kranich \cite{Pavel}.

\begin{lemma}\label{lem:mul-E-factor}
Let $(X,\mu,T)$ be a \mps\ with an ergodic factor $(Y,\nu,S)$. Then
$\ol{A(X|Y)}$ is induced by a factor.
\end{lemma}

\begin{proof}
We proceed in three steps.

\underline{Step 1.} We first show that for every $f\in A(X|Y)$ one can take \emph{bounded} generators of the module zopotopes
in the definition of conditional almost periodicity.  Let $\veps>0$ and let $f_1,\ldots,f_m\in\Ell^2(X|Y)$ satisfy
$$
\orb(f):=\{T^nf:\,n\in\Z\}\subset U_\veps(Z(f_1,\ldots,f_m)),
$$
where we denote by $Z(f_1,\ldots,f_m)$ the module zonotope generated by $f_1,\ldots,f_m$ and by $U_\veps(Z)$ the $\veps$-neighborhood of a set $Z\subset\Ell^2(X|Y)$ with respect to the norm $\|\cdot\|_{\Ell^{2,\infty}(X|Y)}$.

Fix $j\in\{1,\ldots,m\}$ and consider for every $k\in \N$ the function $f_{j,k}:=f_j\Eins_{\{|f_j|\leq k\}}$ which is bounded by $k$. Since  the functions $f_{j,k}$ approximate $f_j$ in $\Ell^2(X)$, $\EY (|f_j-f_{j,k}|^2)$ converge to zero in $\Ell^1(Y)$ as $k\to\infty$. By Egorov's theorem (passing first to a subsequence if necessary), we see that there exists $k\in \N$ and a set $A_j\subset Y$ with $\nu(A_j)>1-\frac{\veps}m$ such that $\EY(|f_j-f_{j,k}|^2)<\frac{\veps}m$ on $A_j$. Write now $f_j=g_j+b_j+r_j$
with $g_j:=f_{j,k}$ being bounded by $k$, $b_j:=(f_j-f_{j,k})\Eins_{Y\setminus A_j}$ and the rest $r_j:=(f_j-f_{j,k})\Eins_{A_j}$. Since $\|r_j\|_{\Ell^{2,\infty}(X|Y)}\leq \frac{\veps}m$ by the definition of $k$, we have
\begin{equation}\label{eq:U2eps}
\orb(f)\subset U_{2\veps} (Z(g_1,\ldots,g_m,b_1,\ldots,b_m)).
\end{equation}
Denote by $\varphi$ and $\psi$ the underlying transformations on $X$ and $Y$, respectively.
Consider the ``good'' set  $B:=\cap_{j=1}^mA_j$ satisfying $\nu(B)>1-\veps$. By \eqref{eq:U2eps} and since the functions $b_j$  vanish on $B$, for every $l\in\N_{0}$ there exist $s_l\in Z(g_1,\ldots,g_m)$ (which is then bounded by $M:=km$) such that
$$
\|1_B T^lf-s_l\|_{\Ell^{2,\infty}(X|Y)}<2\veps
$$
which we can rewrite as
\begin{equation}\label{eq:S^lBf}
\|1_{\psi^lB} f-T^{-l}s_l\|_{\Ell^{2,\infty}(X|Y)}<2\veps.
\end{equation}
Consider now the function
$$
\tilde{f}= \sum_{l=0}^\infty \Eins_{\psi^l(B)}T^{-l}s_l \Eins_{Y\setminus(\cup_{l'=0}^{l-1}\psi^l(B))}.
$$
Clearly, $\tilde{f}$ is bounded by $M$. Moreover, since  $\cup_{l=0}^\infty \psi^l(B)=Y$ up to a null set by the ergodicity of $(Y,\nu,S)$, \eqref{eq:S^lBf} implies
\begin{equation}\label{eq:f-tildef}
\|f-\tilde{f}\|_{\Ell^{2,\infty}(X|Y)}\leq 2\veps.
\end{equation}
In particular, $\tilde{f}$ satisfies by \eqref{eq:U2eps}
\begin{equation}\label{eq:U4eps}
\orb(\tilde{f})\subset U_{4\veps} (Z(g_1,\ldots,g_m,b_1,\ldots,b_m)).
\end{equation}
We now truncate the
new generators as follows. Let $N:=\frac{M}{\veps}\sum_{j=1}^m \|b_j\|_{\Ell^{2,\infty}(X|Y)}$ and set $D:=\cap_{j=1}^m \{|b_j|\leq N\}\subset X$. By $\Eins_{X\setminus D}\leq \sum_{j=1}^m\frac{|b_j|}N$ we have
\begin{equation}\label{eq:norm-x-d}
\| \Eins_{X\setminus D}\|_{\Ell^{2,\infty}(X|Y)}\leq \frac1N \sum_{j=1}^m \|b_j\|_{\Ell^{2,\infty}(X|Y)}=\frac{\veps}M.
\end{equation}
By \eqref{eq:U4eps} for every $j\in\{1,\ldots,m\}$ and $n\in\Z$  there exist $c_{j,n},d_{j,n}\in \Ell^\infty(Y)$ bounded by $1$ such that
$$
\left\|T^n\tilde{f} - \sum_{j=1}^mc_{j,n}g_j- \sum_{j=1}^md_{j,n}b_j\right\|_{\Ell^{2,\infty}(X|Y)}<4\veps.
$$
It follows, by adding and subtracting $\Eins_DT^n\tilde{f}$ and using \eqref{eq:norm-x-d} and Remark \ref{rem:AandW}(a), that
\begin{eqnarray*}
&\ &\left\|T^n\tilde{f} - \sum_{j=1}^mc_{j,n}g_j\Eins_D- \sum_{j=1}^md_{j,n}b_j\Eins_D\right\|_{\Ell^{2,\infty}(X|Y)}\\
&\ &\quad \leq \left\|T^n\tilde{f} - \sum_{j=1}^mc_{j,n}g_j- \sum_{j=1}^md_{j,n}b_j\right\|_{\Ell^{2,\infty}(X|Y)}+\|T^n\tilde{f}\|_{\Ell^\infty(X)}\| \Eins_{X\setminus D}\|_{\Ell^{2,\infty}(X|Y)}\\
&\ &\quad < 4\veps + M\frac{\veps}M=5\veps.
\end{eqnarray*}
This implies by \eqref{eq:f-tildef}
$$
\orb(f)\subset U_{7\veps}(Z(g_1\Eins_D,\ldots,g_m\Eins_D,b_1\Eins_D,\ldots,b_m\Eins_D)),
$$
with every generator of this zonotope being bounded by $\max\{k,N\}$. This completes the proof of Step 1.

\underline{Step 2.} We now prove that $A(X|Y)\cap \Ell^\infty(X)$ is dense in $\ol{A(X|Y)}$.
To do this it is enough to show that for every $f\in A(X|Y)$  the function  $f^+:=\max\{0,f\}$ belongs to $A(X|Y)$ which by shifting, multiplying with $-1$ and applying the argument again implies conditional almost periodicity of the function $f\Eins_{\{|f|\leq N\}}$ for every $N\in \N$.
Let $f\in A(X|Y)$ and $\veps>0$. By Step 1 we can assume that the orbit of $f$ belongs to an $\veps$-neighborhood (w.r.t.~$\|\cdot\|_{\Ell^{2,\infty}(X|Y)}$) of a module zonotope with bounded generators $f_1,\ldots,f_m$. Using $\|\cdot\|_{\Ell^{2,\infty}(X|Y)}\leq \|\cdot\|_{\Ell^\infty(X)}$, by changing to a $2\veps$-neighborhood we can assume
that each $f_j$ is a finite linear combination of characteristic functions. Moreover, by
enlarging $m$ we can further assume that each $f_j$ is a positive multiple of a characteristic function with supports being disjoint (up to null sets). After these modifications, by
$$
\left\|T^n f^+ - \sum_{j=1}^m c_j^+f_j\right\|_{\Ell^{2,\infty}(X|Y)}\leq \left\|T^n f - \sum_{j=1}^m c_jf_j\right\|_{\Ell^{2,\infty}(X|Y)}
$$
we see that the orbit of $f^+$ belongs to the same neighborhood of the same module zonotope as the orbit of $f$ finishing the argument.

\underline{Step 3.} It is clear that the set $A(X|Y)\cap \Ell^\infty(X)$ is conjugation invariant, contains $\Eins$ and is invariant under both $T$ and $T^{-1}$. Moreover, by Step 2 it is dense in $\ol{A(X|Y)}$. It thus remains to show that it is a subalgebra of $\Ell^\infty(X)$. Let $f,g\in A(X|Y)\cap \Ell^\infty(X)$ and assume without loss of generality that $\|g\|_\infty\leq 1$. Let further $\veps>0$ and  take $f_1,\ldots, f_m\in\Ell^2(X|Y)$ with $\orb(f)\in U_\frac{\veps}{2}(Z(f_1,\ldots,f_m))$. By Step 1 we can assume that $f_1,\ldots,f_m\in \Ell^\infty(X)$ and denote $M:=\max\{\|f_1\|_\infty,\ldots,\|f_m\|_\infty\}$. Let further $g_1,\ldots,g_l\in\Ell^2(X|Y)$ satisfy $\orb(g)\in U_{\frac{\veps}{2mM}}(Z(g_1,\ldots,g_l))$. Then for every $n\in\Z$ and appropriate $a_n\in Z(f_1,\ldots,f_m)$ and $b_n\in Z(g_1,\ldots,g_l)$ we have by the triangle inequality and Remark \ref{rem:AandW}(a)
\begin{eqnarray*}
&\ &\|T^n(fg)-a_nb_n
\|_{\Ell^{2,\infty}(X|Y)}\leq \|T^nf-a_n\|_{\Ell^{2,\infty}(X|Y)}\|g\|_{\Ell^\infty(X)}\\
&\ &\quad +\|a_n\|_{\Ell^\infty(X)}\|T^ng-b_n \|_{\Ell^{2,\infty}(X|Y)}<\frac{\veps}2+mM\frac{\veps}{2mM}=\veps.
\end{eqnarray*}
Since each $a_nb_n$ belongs to the module zonotope generated by $f_jg_k$, $j\in\{1,\ldots,m\}$, $k\in\{1,\ldots,l\}$, this shows $fg\in A(X|Y)\cap \Ell^\infty(X)$. The proof is complete.
\end{proof}

\begin{remark}
The factor inducing $\ol{A(X|Y)}$ is the maximal compact extension
of $Y$ in $X$ in the sense of Definition \ref{def:comp-ext} below. Moreover, by \eqref{eq:A=E} it coincides with the maximal isometric extension of $Y$ in $X$  introduced by Furstenberg, Weiss \cite{FW96} for ergodic systems.
\end{remark}

As a corollary we obtain the following property of $W(X|Y)$.
\begin{lemma}\label{lem:mul-Wbdd}
Let $(X,\mu,T)$ be a
\mps\ with
factor $(Y,\nu,S)$. Then $W(X|Y)\cap\Ell^\infty(X)$ is dense in $W(X|Y)$ w.r.t.~the $\Ell^2$-norm.
\end{lemma}
\begin{proof}
Let $f\in W(X|Y)$ and take a sequence $(f_n)\subset \Ell^\infty(X)$ converging to $f$ in $\Ell^2(X)$. Consider the sequence $(P_{W(X|Y)}f_n)$
where $P_{W(X|Y)}$ denotes the orthogonal projection onto $W(X|Y)$. This sequence clearly belongs to $W(X|Y)$ and converges to $f$ by the Pythagoras theorem. Moreover, for every $n\in\N$ we have
$$
P_{W(X|Y)}f_n=f_n-P_{\ol{A(X|Y)}}f_n\in \Ell^\infty(X)
$$
by \eqref{eq:rel-jdlg-a+w} and Lemma \ref{lem:mul-E-factor}. (Recall that
the projection onto a factor is
contractive w.r.t.~the  $\Ell^\infty$-norm.)
\end{proof}

We now introduce the distal factors of finite order defined by Furstenberg \cite{Fu77} using conditional eigenfunctions instead of conditionally almost periodic functions (cf.~Remark \ref{rem:a=e}).

\begin{definition}\label{def:mul-n}
Let $(X,\mu,T)$ be an ergodic
\mps. We construct the sequence of factors inductively as follows. Start with $D_0:= \fix T=\C\Eins$ and the corresponding fixed (one-point) factor $\mathcal{D}_0$
and for every $k\in\N$ denote by $\mathcal{D}_{k}$ the factor
inducing $D_k:=\ol{A(X|\mathcal{D}_{k-1})}$
(see Lemma \ref{lem:mul-E-factor}). We call $\mathcal{D}_k$ the \emph{distal factor of order $k$} of $(X,\mu,T)$.
\end{definition}

Furstenberg \cite{Fu77} showed that for regular, ergodic systems  for each $k\in\N$ the factor $\mathcal{D}_k$ (defined via conditional eigenfunctions) is \emph{characteristic for $k$-term multiple ergodic averages \eqref{eq:mult-erg-ave}} in the sense that both convergence and the limit of the averages \eqref{eq:mult-erg-ave} remain unchanged if we replace every function by its projection
onto this factor. See also Frantzikinakis \cite[Theorem 5.2]{Frantz04} who deduced it from Furstenberg \cite[Theorem 7.1]{Fu77} using the van der Corput trick. We will give an alternative proof of this fact here, see Propositions \ref{prop:ghk-control-mult-ave} and  \ref{prop:Z_k-ap} below.

\section{Gowers-Host-Kra seminorms}\label{sec:GHK-seminorms}

The uniformity seminorms were introduced by Gowers \cite{Gowers} in his proof of Szemer\'edi's theorem via higher order Fourier analysis for rotations on cyclic gro\-ups $\Z_N$
and were extended by Host and Kra \cite{HK}\footnote{Initially Host, Kra \cite{HK} defined the seminorms using cube measure spaces. The two definitions are easily shown to be equivalent, see, e.g., Kra \cite[Lemma 7.4]{Kra} or \cite[Section 14.2]{EF-book}.} to arbitrary ergodic measure-preserving systems in their proof of multiple convergence.

\begin{definition}[Gowers-Host-Kra (uniformity) seminorms]
\label{def:GHK-1}
Let $(X,\mu,T)$ be an ergodic
 \mps \ and $f\in L^\infty(X,\mu)$. The \emph{Gowers-Host-Kra} (or \emph{uniformity}) \emph{seminorms} are defined inductively by
\begin{eqnarray}
\|f\|_{U_1}&:=&
\left| \int_X f\,d\mu\right|
,\label{eq:u1}\\
\|f\|_{U_{l+1}}^{2^{l+1}}&:=&\limsup_{N\to\infty}\aveN \|T^nf\cdot \ol{f}\|_{U_l}^{2^l},\quad l\in\N.\label{eq:ghk-def-seminorms}
\end{eqnarray}
\end{definition}

\begin{remark}[Second uniformity seminorm]\label{rem:semin-erg-12}
For an ergodic measure-preserving system $(X,\mu,T)$ the second uniformity seminorm satisfies
\begin{eqnarray*
\|f\|_{U_2}^4&=&\limsup_{N\to\infty}\aveN |\la T^nf,f\ra|^{2}.
\end{eqnarray*}
In particular,  $f\in\Ell^\infty(X)$ is weakly mixing if and only if $\|f\|_{U^2}=0$.
\end{remark}

It is easy to see that the  Gowers-Host-Kra  seminorms are increasing and satisfy
\begin{equation}\label{eq:U-Linfty}
\|\cdot\|_{U^k}\leq \|\cdot\|_{\Ell^\infty(X)}\quad\forall k\in\N.
\end{equation}
We need the following stronger property of these seminorms, see, e.g., \cite[Equation (12)]{ET}.

\begin{lemma}\label{lem:mul-uk-lp}
Let $(X,\mu,T)$ be an ergodic \mps\ and $k\in \N$. Then for $p_k:=\frac{2^k}{k+1}$ one has $\|f\|_{U^k}\leq \|f\|_{\Ell^{p_k}(X)}$ for every $f\in\Ell^\infty(X)$.
\end{lemma}

Let $(X,\mu,T)$ be an ergodic \mps.
Consider for every $k\in \N_{0}$ the orthogonal decomposition
\begin{equation}\label{eq:hk-dec}
\Ell^2(X)=Z_k\oplus \ol{\{f\in\Ell^\infty(X):\ \|f\|_{U^{k+1}}=0\}}
\end{equation}
known as the \emph{Host-Kra decomposition} of order $k$, where $Z_k$ is at first defined as the orthogonal complement of the second part. In particular, we have $Z_0=\C\Eins=\Fix T$ by ergodicity and the decomposition \eqref{eq:hk-dec} for $k=0$ coincides with the von Neumann decomposition \eqref{eq:mul-vN-dec}.
Moreover, for $k=1$ \eqref{eq:hk-dec} coincides with the
Jacobs-de Leeuw-Glicksberg decomposition \eqref{eq:jdlg-dec} by
Remark \ref{rem:semin-erg-12}
and Lemma \ref{lem:mul-Wbdd} applied to the one-point factor.

The following property shows the relevance of the uniformity seminorms for multiple convergence and recurrence. It is an easy consequence of the definition of the Gowers-Host-Kra seminorms and the van der Corput inequality, see, e.g., \cite[Section 14.1]{EF-book}.

\begin{proposition}[Generalized von Neumann theorem]
\label{prop:ghk-control-mult-ave}
Let $(X,\mu,T)$ be an ergodic measure-preserving system, $k\in\N$ and $f_1,\ldots,f_k\in\Ell^\infty(X)$. Then
\begin{equation}\label{eq:gener-vN-ineq}
\limN
\left\|\aveN T^nf_1\cdot T^{2n}f_2\cdots T^{kn}f_k\right\|_2\leq \min_{j=1,\ldots,k}j\|f_j\|_{U^k}.
\end{equation}
\end{proposition}

\begin{corollary}[The subspace $Z_{k-1}$ is characteristic for $k$-term multiple averages]\label{cor:ghk-control-mult-ave}
Let $(X,\mu,T)$ be an ergodic measure-preserving system, $k\in\N$ and $f_1,\ldots,f_k\in\Ell^\infty(X)$. Then
$$
\limaveN T^nf_1\cdot T^{2n}f_2\cdots T^{kn}f_k=0 \quad \text{in }\Ell^2(X)
$$
holds whenever $f_j\perp Z_{k-1}$ for some $j\in\{1,\ldots,k\}$.
\end{corollary}
\begin{proof}
Assume without loss of generality that $\|f_1\|_\infty\leq 1,\ldots,\|f_k\|_\infty\leq 1$ and let $f_j\perp Z_{k-1}$ for some $j\in\{1,\ldots,k\}$. By the Host-Kra decomposition \eqref{eq:hk-dec} there exists a sequence $(g_m)\subset \Ell^\infty(X)$ satisfying $\|g_m\|_{U^k}=0$ for all $m\in\N$ with $\lim_{m\to\infty}\|f_j-g_m\|_2=0$. Thus for every $m\in\N$, by decomposing $f_j=(f_j-g_m)+g_m$, we have by \eqref{eq:gener-vN-ineq} and the triangle inequality
\begin{eqnarray*}
\limsup_{N\to\infty}\left\|\aveN T^nf_1\cdot T^{2n}f_2\cdots T^{kn}f_k\right\|_2\leq \|f_j-g_m\|_{\Ell^2(X)}.
\end{eqnarray*}
Letting $m\to\infty$ proves the assertion.
\end{proof}

The following property of the subspaces $Z_k$ can be shown via constructing the factors using cube measure spaces, see Host, Kra \cite{HK},  \cite[Chapter 9]{HK-book} and, e.g., \cite[Section 14.4]{EF-book}. We will not need this fact and include it here for completeness.

\begin{proposition}\label{prop:HKZ-factors}
For every $k\in\N_{0}$ the subspace $Z_k$ is induced by a factor
 $\mathcal{Z}_k$, called the \emph{Host-Kra-Ziegler factor of order $k$}.
\end{proposition}

In particular, $\mathcal{Z}_0$ is the one-point factor and $\mathcal{Z}_1$ is the Kronecker factor.
The Host-Kra-Ziegler factors
were introduced by Host and Kra \cite{HK} and subsequently independently by Ziegler \cite{Ziegler} (see Leibman \cite{Le} for the equality of the two constructions) in their proofs for multiple convergence.
We refer to Host, Kra \cite{HK,HK-book} for a detailed analysis of these factors and the deep structure theorem which states that for ergodic regular systems each $\mathcal{Z}_k$ is an inverse limit of nilsystems of step $k$. Again, we will not need anything from this theory
here.

\begin{remark}[Multiple recurrence for weakly mixing systems]
For weakly mixing systems\footnote{A \mps\ $(X,\mu,\varphi)$ is called \emph{weakly mixing} if $\limaveN |\mu(A\cap \varphi^{-n}B)-\mu(A)\mu(B)|=0$ for every measurable $A,B\subset X$. This is equivalent to the orthogonal decomposition $\Ell^2(X)=\C\Eins\oplus \{\text{weakly mixing functions}\}$.} one easily shows using Remark \ref{rem:semin-erg-12} and induction  that  all Gowers-Host-Kra seminorms are equal to the first seminorm (see, e.g., Kra \cite[Section 7.3]{Kra}). Thus in this case $Z_k=\C\Eins$ holds for every $k\in\N$
and both multiple convergence and multiple recurrence follow from Corollary \ref{cor:ghk-control-mult-ave}
with
\begin{equation*}
\lim_{N\to\infty}\aveN \int_X f_0\cdot T^nf_1\cdot T^{2n}f_2 \cdots T^{kn}f_k\,d\mu=\prod_{j=0}^k\int_Xf_j\,d\mu
\end{equation*}
recovering a result of Furstenberg \cite[Equation (1)]{Fu77}.
\end{remark}

\section{Host-Kra-Ziegler factors versus distal factors \\
and end of the proof

We now connect the Host-Kra-Ziegler factors to distal factors of the corresponding order using the conditional Jacobs-de Leeuw-Glicksberg decomposition (Theorem \ref{mr:higher-JdLG}) and Lemma \ref{lem:mul-uk-lp}.

\begin{proposition}
Distal factors are extensions of Host-Kra-Ziegler factors]\label{prop:Z_k-ap}
Let $(X,\mu,T)$ be an ergodic \mps.
Then
$Z_{k}\subset D_{k}$ for every $k\in \N$.
\end{proposition}
\begin{proof}
Let $k\in \N_{0}$.
By the orthogonal decomposition
\begin{equation}\label{eq:mul-dec-dk+1}
\Ell^2(X)=D_{k+1}\oplus W(X|\DD_{k})
\end{equation}
 from the conditional \jdlg\ Theorem \ref{mr:higher-JdLG} and the Host-Kra decomposition \eqref{eq:hk-dec}
 for $k+1$ we need to show
$$
W(X|\DD_{k})\subset \ol{\{f\in\Ell^\infty(X):\, \|f\|_{U^{k+2}}=0\}}.
$$
By Lemma \ref{lem:mul-Wbdd} it suffices to show
\begin{equation}\label{eq:mul-uk+2=0}
\text{$\|f\|_{U^{k+2}}=0$  for every
$f\in W(X|\DD_{k})\cap \Ell^\infty(X)$. }
\end{equation}
We show \eqref{eq:mul-uk+2=0} for every $k\in\N_{0}$ by induction on $k$. For $k=0$ let $f\in W(X|\mathcal{D}_0)\cap \Ell^\infty(X)$, where $\mathcal{D}_0$ is the one-point factor. Then $f$ is weakly mixing by Remark \ref{rem:AandW}(f) and hence $\|f\|_{U^2}=0$ by Remark \ref{rem:semin-erg-12}.
Assume now that $k\in \N$ and that \eqref{eq:mul-uk+2=0} holds for $k-1$.
 Let $f\in W(X|\DD_{k})$ be bounded and assume without loss of generality $\|f\|_\infty\leq 1$. Take $n\in \N$. By the triangle inequality  and the decomposition \eqref{eq:mul-dec-dk+1}
for $k-1$
\begin{equation}\label{eq:mul-uk-triangle}
\|T^nf\cdot\ol{f}\|_{U^{k+1}}\leq \|P_{D_{k}}(T^nf\cdot\ol{f})\|_{U^{k+1}} + \|P_{W(X|\DD_{k-1})}(T^nf\cdot\ol{f})\|_{U^{k+1}}
\end{equation}
for the orthogonal projections $P_{D_{k}}$ and $P_{W(X|\DD_{k-1})}$ onto the factor $\DD_{k}$ and the subspace $W(X|\DD_{k-1})$, respectively.
(Recall that the first projection and hence also the second, complementary, projection
maps bounded functions to bounded functions.)
The last summand in \eqref{eq:mul-uk-triangle} equals zero by the induction hypothesis. Moreover, recall that $P_{D_k}=JE_{\DD_k}$ for the corresponding Markov factor map $J$ and that both $J$ and $E_{\DD_k}$ act as contractions w.r.t.~the $\Ell^p$-norm for every $p\in [1,\infty]$.
So we have by \eqref{eq:mul-uk-triangle}, Lemma \ref{lem:mul-uk-lp} and $\|f\|_\infty\leq 1$ denoting $p_{k+1}:=\frac{2^{k+1}}{k+2}$
\begin{eqnarray*}
&\ &
\|T^nf\cdot\ol{f}\|_{U^{k+1}}
\leq
\|P_{D_{k}}(T^nf\cdot\ol{f})\|_{U^{k+1}}
\leq
\|P_{D_{k}}(T^nf\cdot\ol{f})\|_{\Ell^{p_{k+1}}(X)}
\\
&\ &\quad \leq
\|E_{\DD_{k}}(T^nf\cdot\ol{f})\|_{\Ell^{p_{k+1}}(Y)}
\leq \|E_{\DD_{k}}(T^nf\cdot\ol{f})\|_{\Ell^1(Y)}^{1/p_{k+1}}.
\end{eqnarray*}
Since $\|T^nf\cdot\ol{f}\|_{U^{k+1}}\leq 1$ by $\|f\|_\infty\leq 1$ and \eqref{eq:U-Linfty},  this implies
$$
\|T^nf\cdot\ol{f}\|_{U^{k+1}}^{2^{k+1}}\leq \|T^nf\cdot\ol{f}\|_{U^{k+1}}^{p_{k+1}}\leq \|E_{\DD_{k}}(T^nf\cdot\ol{f})\|_{\Ell^1(Y)}
$$
and, by $f\in W(X|\DD_k)$,
$$
\|f\|_{U^{k+2}}^{2^{k+2}}=\limsup_{N\to\infty}\aveN \|T^nf\cdot\ol{f}\|_{U^{k+1}}^{2^{k+1}}=0
$$
follows.
This proves \eqref{eq:mul-uk+2=0} and completes the proof.
\end{proof}

\begin{remark}\label{rem:Fu-orig}
As mentioned in the introduction, as a corollary of  Proposition \ref{prop:ghk-control-mult-ave} and Proposition \ref{prop:Z_k-ap} we obtain an alternative proof of the fact proved by Furstenberg \cite{Fu77}, see also Frantzikinakis \cite[Theorem 5.2]{Frantz04} based on Furstenberg \cite[Theorem 7.1]{Fu77}, that the distal factor $\DD_{k-1}$ of order $k-1$ is characteristic for $k$-term multiple recurrence.
\end{remark}

We now define compact extensions, cf.~Furstenberg, Katznelson \cite[Def.~3.1]{FuKa78} and Furstenberg, Katznelson, Ornstein \cite[Section 9]{FuKaOr82}.
\begin{definition}\label{def:comp-ext}
A \mps\ $(X,\mu,T)$ is called a \emph{compact extension}\footnote{Tao \cite[Def.~2.13.7]{Tao-ET} calls an extension $(X,\mu,T)$ of $(Y,\nu,S)$ compact if every function $f\in\Ell^2(X|Y)$ is conditionally almost periodic in measure, cf.~Furstenberg, Katznelson \cite[Section 2]{FuKa78}, Furstenberg \cite[Section 6.3]{Fu-book}. By Remark \ref{rem:capm} the two definitions are equivalent.} of its factor $(Y,\nu,S)$ if $\ol{A(X|Y)}=\Ell^2(X)$.
\end{definition}
In particular, for an ergodic \mps\  $(X,\mu,T)$ and every $k\in \N_{0}$ the distal factor $\mathcal{D}_{k+1}$ is by definition
a compact extension of $\mathcal{D}_{k}$, namely
 the maximal compact extension of $\mathcal{D}_{k}$ in $(X,\mu,T)$.

The last ingredient of the proof of Theorem \ref{thm:mr} is the following property of compact extensions, see Tao \cite[Theorem 2.13.11]{Tao-ET} combined with Remark \ref{rem:capm}, cf.~Furstenberg, Katznelson \cite[Prop.~3.5]{FuKa78},
Furstenberg, Katznelson, Ornstein \cite[Thm.~9.1]{FuKaOr82} and Einsiedler, Ward \cite[Section 7.9]{EW}.

\begin{proposition}[Compact extensions preserve MR]\label{prop:ap-ext-MR}
Let $(X,\mu,T)$ be a compact extension of $(Y,\nu,S)$.
If $(Y,\nu,S)$ satisfies MR then so does $(X,\mu,T)$.
\end{proposition}

\begin{proof}[Proof of multiple recurrence for ergodic systems (Theorem \ref{thm:mr})]
Let $(X,\mu,T)$ be an ergodic \mps\ and $k\in\N$.

Denoting by $P_{D_{k-1}}$ the orthogonal projection onto $D_{k-1}$, for $f\in\Ell^\infty(X)$ with $f>0$ the function $P_{D_{k-1}}f$ is also bounded and satisfies $P_{D_{k-1}}f>0$  by Lemma \ref{lem:mul-E-factor} and \eqref{eq:PYf>0}.
Therefore Proposition \ref{prop:Z_k-ap} and Corollary \ref{cor:ghk-control-mult-ave} imply
\begin{equation}\label{eq:mul-mea-char-fac}
\limN \left\|\aveN T^nf\cdots T^{kn}f-\aveN T^nP_{D_{k-1}}f\cdots T^{kn}P_{D_{k-1}}f
\right\|_{\Ell^2(X)}=0
\end{equation}
and we can assume without loss of generality that $\Ell^2(X)=D_{k-1}$.
So we have the chain of factors
$$
X=
\DD_{k-1} \to \DD_{k-2} \to \ldots \to \DD_1 \to \DD_0=\{\cdot\}
$$
where $\{\cdot\}$ denotes the one-point factor,
and every factor in this chain is a compact extension of the next one by definition. Finally, the one-point factor clearly satisfies MR. Thus every $\DD_j$, and hence also $X=\DD_{k-1}$,  satisfies MR by Proposition \ref{prop:ap-ext-MR}.
The proof is complete.
\end{proof}

redpajama_set_name: RedPajamaArXiv


1) Just looking at the content of your wardrobe is enough to make you feel hot and drip into a puddle of sweat.
2) You of course have nothing to wear, since even a light cotton kurta feels as suffocating as thick wool sweater.
3) Ice cream suddenly feel like an appropriate answer to "What's for dinner".
4) Revise that thought, ice cream is appropriate at every meal.
5) Getting dressed in the morning feels like running a marathon, you loose our weight in sweat, and you hit the wall with your t-shirt not even fully lowered on your chest thinking "I can't do it".
8) Putting together a meal of canned baked beans and heating pre-cooked paratha on the stove, makes you feel like you pulled out a gourmet meal.
9) Your kids agree with the above and for once do not fuss at the dinning table.
11) You then give the finger to another health and wellbeing article that says you need to exercise daily.
12) Walking to the fridge to go take the water bottle out to drink glass number 22 IS exercise.
13) You think twice about going anywhere outside your home…ummm make that thrice.
14) Your dream vacation involves a ski resort, with lots of snow.
15) You cut down to one cup of hot tea a day, and supplement you caffeine needs with Diet coke.
16) If it can't be eaten as a salad, it's not worth buying it.
17) The clutter that bothered you a few weeks ago and needed to be organized no longer bother you.
18) Reading a book requires more brain cells than you have left in a functional state.
19) Googling rain invoking dances and rituals doesn't seem silly.
20) Sleeping really becomes the only cure to go through the afternoons.
Lol, I love it! I feel bad for you with all of that heat. I do not miss it!
I can imagine not missing it :) What makes it really bad these days is the nasty increase in humidity. One month until the monsoon should start reaching us, if it comes on time that year.
Seriously with us using AC in two rooms at night only we are already at 5000 rupees a month for electricity and it's already having my hyperventilate :-) this past week I needed a little AC at nap time too so my bill is probably going to be even higher this month. 24 hours a day is totally out of question. I try to spend as little time in the kitchen as possible.


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text:
Playwrights & Practitioners
Context & Criticism
Theatre Craft
Continuum Renaissance Drama Guides
Sort by: Alphabetical Cast Size Length Sort by Ascending Descending
1 Henry IV
This Critical Guide demonstrates why Shakespeare's drama about the education of a prince remains such a "renewable resource" for both schools and universities. Its impressive contributors show there is a perfect match between 1 Henry IV and the educational aims of the Continuum Renaissance Drama series. In a chain of ingeniously linked essays, they reveal how the play poses questions about multiculturalism, role play, history, generational conflict, and the control of the media that are as pressing today as in 1597. So this sophisticated Guide illustrates how Shakespeare does not simply serve modern education, but in deep, sometimes disturbing ways, teaches the teachers.
Doctor Faustus, is Christopher Marlowe's most popular play and is often seen as one of the overwhelming triumphs of the English Renaissance. It has had a rich and varied critical history often arousing violent critical controversy.
This guide offers students an introduction to its critical and performance history, surveying notable stage productions from its initial performance in 1594 to the present and including TV, audio and cinematic versions. It includes a keynote chapter outlining major areas of current research on the play and four new critical essays. Finally, a guide to critical, web-based and production-related resources and an annotated biography provide a basis for further individual research.
"This volume represents much of the star power in Marlowe Studies over recent decades, including three past presidents of the Marlowe Society of America along with two current officers. Readers may be assured that the scholarly assessments in this volume are well informed, judicious, insightful, and forward-looking." - Roslyn L. Knutson, Professor Emerita, University of Arkansas at Little Rock, USA; President, Marlowe Society of America
"This wide-ranging book provides food for thought for readers of all levels: it helps orient the beginner with useful surveys of the critical territory and it caters to more advanced readers with stimulating scholarly essays on topics such as postcolonialism, print culture and the occult. Andrew Duxfield approaches Renaissance hermeticism in an admirably clear and user-friendly way; Georgia Brown imaginatively examines writing/printing as both a context for and a theme of the play; and Andrew Hartley's experience of directing Dr Faustus provides brilliant and pithy insights about the play's theatricality and applicability to audiences and readers today. There are extensive (and helpfully annotated) reading lists to guide all readers through the critical maze. Every chapter is full of useful information and observations. This is a valuable addition to the library of every school or university - and to the bookcase of every student." - Laurie Maguire, Professor of English, University of Oxford, UK
"Doctor Faustus: A Critical Guide offers meticulously detailed and lucid overviews of the vexed textual, critical, and performance history of Marlowe's most widely read and performed play, devoting an entire chapter to recent trends in scholarship and interpretation. For these reasons, and because of its deliberate emphasis on the relatively neglected area of performance studies of Doctor Faustus, it is an exceptionally ambitious and valuable work." - Constance Brown Kuriyama, Professor of English, Texas Tech University, USA
"Scholars of English literature introduce students to some features of the play written by British playwright Christopher Marlowe (1564-93) in 1588, the year of the Spanish Armada. They cover the critical backstory, the performance history, current critical research, Renaissance hermeticism, imperialism as devilry in a postcolonial reading, staging Faustus on campus, and the other black arts of printing and writing. A final chapter surveys resources." -Eithne O'Leyne, BOOK NEWS, Inc.
This guide offers a clear picture of the critical landscape of past and current scholarly research on Doctor Faustus, both textual and performative... the volume is a useful resource for students and for scholars who are either just beginning their research or interested in surveying the various rudimentary critical scholarship of the day.
'[This book] aiming to be useful to teachers and students, views the play from varied perspectives. Starting with a useful "Timeline," which begins with Marlowe's life and continues to the present, cataloguing significant stage and publication history, it includes an introduction and eight chapters.'-The Shakespeare Newsletter
The guide serves a dual purpose, offering on the one hand survey chapters detailing the critical and
performance heritage, current critical trends, and available resources, and on the other, four 'New Directions' chapters which engage in original critical and contextual analysis of the play. Deats's introduction, as well as providing an overview of the volume's contents, offers the reader a grounding in the play's historical and literary sources and a broad account of the critical divide between orthodox moralistic and tragic heroic interpretations of the play. Years Work in English Studies, vol 91, no 1, 2012
The Duchess of Malfi
John Webster's classic revenge tragedy The Duchess of Malfi was first performed in 1613 and published in 1623. This guide offers students an introduction to its critical and performance history, including recent versions on stage and screen. It includes a keynote chapter outlining major areas of current research on the play and four new critical essays presenting new critical positions that offer divergent perspectives on Webster's religio-political allegiances and the politics and gendering of secrecy in the play. Finally, a guide to critical, web-based and production-related resources and an annotated bibliography provide a basis for further individual research.
'This book will be invaluable to anyone teaching this extraordinary play. The essays in the volume furnish detailed investigations of historical contexts and illuminating readings of the play while guiding the reader toward other relevant scholarship. I will certainly have it to hand the next time I teach The Duchess.'
Luckyj's extremely comprehensive and varied collection of essays will ensure that no teacher of the play need feel that they are going into a wilderness.
This new collection offers a dense, yet surprisingly easy-to-digest survey of literature that I would recommend to any person teaching or researching Webster. Sixteenth Century Journal
'Tis Pity She's A Whore
John Ford's tragedy 'Tis Pity She's A Whore was first performed between 1629 and 1633 and since then its themes of incest, love versus duty and forbidden passion have made it a widely studied and performed, if controversial, play.
This guide offers students an introduction to its critical and performance history, including TV and film adaptations. It includes a keynote chapter outlining major areas of current research on the play and four new critical essays. Finally, a guide to critical, web-based and production-related resources and an annotated bibliography provide a basis for further individual research.
"A collection of many fine critical insights, these new essays should fuel discussion of 'Tis Pity for years to come." - Richard Abrams, Department of English, University of Southern Maine, USA
"Lisa Hopkins has long been an outstanding authority on John Ford. In this excellent critical guide she provides an accessible introduction, which lays out the complexities of classifying both Ford and his plays, a critical backstory of 'Tis Pity, which includes the reworking of aspects of the play in fiction and film, and an engaging essay on the social and civic use of Ford's stage. She has also collected a range of stimulating essays by accomplished academics that tackle some of the dominant issues of the period in original ways: incest, anatomy, and the survival of prohibited Catholic beliefs and practices in Protestant England. The volume also effectively summarises scholarship of the play since C17, its performance history, and recent critical research and theoretical approaches. The collection is particularly innovative in its inclusion of teaching resources and the variety of ways 'Tis Pity has been incorporated into university syllabi throughout the world. This is an essential text for anyone studying or teaching 'Tis Pity She's a Whore." - Professor Robyn Bolam, St Mary's University College, UK
"A noted authority on dramatist John Ford, editor Hopkins has assembled an outstanding collection of essays dealing with Ford's famous tragedy 'Tis Pity She's A Whore. Contributors summarize the performance history of the play since it was first preformed almost 400 years ago, and deal with issues the play raises, including incest, anatomy, and the survival of banned Catholic beliefs and practices in Protestant England. The editor herself contributes a critical backstory, which looks at how aspects of the play have been reworked in fiction and film, as well as an engaging essay on the social and civic use of Ford's stage." -Eithne O'Leyne, BOOK NEWS, Inc.
Continuum Renaissance Drama offers practical and accessible introductions to the critical and performative contexts of key Elizabethan and Jacobean plays. Each guide introduces the text's critical and performance history but also provides students with an invaluable insight into the landscape of current scholarly research through a keynote essay on the state of the art and newly commissioned essays of fresh research from different critical perspectives.
Copyright © 2020 Bloomsbury Publishing Plc
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text:
YAL Wins Trench Fight in Student Senate! YAL Wins Majority Vote!
By Scott Lietzow
The beginning of the semester looked bright for the freshly forming Young Americans for Liberty at Harper College. We had members of our club as voting members on student senate and we thought we really could accomplish things on campus;.
But due to the rules at Harper that allow the Student Senate President and her executive board to appoint voting members and not have elections for these student representatives, she was able to strip one of our members of their voting rights and restrict other members of our club from obtaining voting rights. At first we were discouraged but then became outraged. Then we decided as a club that we needed to change things on campus to allow students to elect their representatives in the Senate.
So for 5 weeks we planned, lobbied the student senators, and brushed up on Roberts Rules of order. Then, on Friday Oct 19, we carried out our plan during the Student Senate meeting. We made sure to bring one of our club members who happens to be a writer for the school newspaper along as well. We immediately put our plan to suspend the rules in place, which led to all out trench fight between the senators and the executive board.
The executive board was trying to appoint new voting members that day, which according to the Constitution was not allowed because appointments can only happen before the fifth week of school. The senators, on the other hand, were trying to make the representatives (non-voting members) become senators and be given the right to vote. These representatives happened to be members of YAL and were not allowed beforehand to vote because of our views. The floor fight lasted for over an hour, and ended with a recess.
The recess took place because the senators demanded that the executive board review everything and come back with a decision to allow our previous representative to become senators. After the fifteen minute recess the executive board came back and granted all of YAL's members on Student Senate their voting rights and even gave voting rights back to the Senator who was stripped of his rights at the beginning of the year. When these senators were appointed it became clear to us that Young Americans for Liberty now has the majority vote in our Senate!
Keeping with the great libertarian conspiracy that we want to take over, so we can leave you alone, our first plan of action will be to petition the students to implement open voting for Student Senators, and with our majority vote on the Senate we will ensure that it gets passed. I want to thank our members Kristie Constabileo, Sahand Tsay, and Devarshi Patel for playing such a pivotal part in making this success happen!
Campus Activism ReportFall 2012Student Government


At ages 89 and 85 respectively, both Morris and Ettie are excited and keen to pack up their belongings and to start the next phase and new adventure of their lives.
Morris, as young spirited as ever, has stated that he is not coming to Israel to retire, but is looking for employment, even if he becomes a bus or taxi driver!!
Ettie, on the other hand, is looking forward to a new circle of friends and to joining a kaluki group where she has spent many years practicing the skills of card playing.
Alan, Diane and family made Aliya 17 years ago: their home in Raanana is as open today as it was years ago in Johannesburg, and their table is full of guests on Shabbatot and Chaggim. They look very forward to honouring Morris and Ettie in their home and at their table.
Alan flew to Johannesburg to assist in the packing and to attend to all the logistical work and admin and will fly back with them to Israel.
Both Morris and Ettie have said that it is a huge adjustment in their lives, but it is definitely worthwhile and certainly the correct thing for them to do.
Morris and Ettie have a daughter Beverley & Roy Lomnitz , grandchildren and two great grandchildren in Johannesburg as well as a son, Colin and Janice, grandchildren and great grandchildren in Canada.
"The years of travelling to see and be with family in Australia, Canada and other places are over for us and now that we will be in the center of the world, they will all be able to visit us in our new home in Israel", say Morris and Ettie.
Grandaughter Gabi Baker, has said she will miss them terribly and is motivated for her and husband Mark to think about joining them in Israel as soon as possible.
Morris and Ettie will be missed by the Johannesburg community and by the members of their Shul for over 70 years, Sydenham Highlands North. Their loss will most certainly be Israels gain.
We all wish Morris and Ettie Silber the very best of health, strength, peace and happiness in their mitzvah of making Aliya. Kol Hakavod to the both of them and let their Aliya be a source of inspiration and motivation to all Jews across the globe, to make Aliya, no matter from where or at what age.


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text:
import argparse
import logging
import os
from pathlib import Path
import shutil
import torchaudio

import soundfile as sf
from tqdm import tqdm
import pandas as pd

from examples.speech_synthesis.data_utils import extract_logmel_spectrogram
from examples.speech_to_speech.preprocessing.data_utils import gen_config_yaml
from examples.speech_to_text.data_utils import create_zip, get_zip_manifest, save_df_to_tsv
from fairseq.data.audio.audio_utils import convert_waveform

logger = logging.getLogger(__name__)

MANIFEST_COLUMNS = ["id", "src_audio", "src_n_frames", "tgt_audio", "tgt_n_frames"]

def prepare_target_data(args, tgt_audios):
    feature_name = "logmelspec80"
    zip_path = args.output_root / f"{feature_name}.zip"
    if zip_path.exists():
        print(f"{zip_path} exists.")
        return zip_path

    feature_root = args.output_root / feature_name
    feature_root.mkdir(exist_ok=True)

    print("Extracting Mel spectrogram features...")
    for tgt_audio in tqdm(tgt_audios):
        sample_id = tgt_audio.stem
        waveform, sample_rate = torchaudio.load(tgt_audio.as_posix())
        waveform, sample_rate = convert_waveform(
            waveform, sample_rate, normalize_volume=args.normalize_volume,
            to_sample_rate=args.sample_rate
        )
        extract_logmel_spectrogram(
            waveform, sample_rate, feature_root / f"{sample_id}.npy",
            win_length=args.win_length, hop_length=args.hop_length,
            n_fft=args.n_fft, n_mels=args.n_mels, f_min=args.f_min,
            f_max=args.f_max
        )
    print("ZIPing features...")
    create_zip(feature_root, zip_path)
    shutil.rmtree(feature_root)

    return zip_path

def process(args):
    os.makedirs(args.output_root, exist_ok=True)

    manifest = {}
    tgt_audios = []
    for split in args.data_split:
        print(f"Processing {split}...")

        manifest[split] = {c: [] for c in MANIFEST_COLUMNS}
        missing_tgt_audios = []
        src_audios = list(args.source_dir.glob(f"{split}/*.wav"))
        for src_audio in tqdm(src_audios):
            sample_id = src_audio.stem

            tgt_audio = args.target_dir / split / f"{sample_id}.wav"
            if not tgt_audio.is_file():
                missing_tgt_audios.append(sample_id)
                continue

            tgt_audios.append(tgt_audio)

            src_n_frames = sf.info(src_audio.as_posix()).frames
            manifest[split]["id"].append(sample_id)
            manifest[split]["src_audio"].append(src_audio.as_posix())
            manifest[split]["src_n_frames"].append(
                src_n_frames // 160
            )  # estimation of 10-ms frame for 16kHz audio

        print(f"Processed {len(manifest[split]['id'])} samples")
        if len(missing_tgt_audios) > 0:
            print(
                f"{len(missing_tgt_audios)} with missing target data (first 3 examples: {', '.join(missing_tgt_audios[:3])})"
            )

    # Extract features and pack features into ZIP
    zip_path = prepare_target_data(args, tgt_audios)

    print("Fetching ZIP manifest...")
    tgt_audio_paths, tgt_audio_lengths = get_zip_manifest(zip_path)

    print("Generating manifest...")
    for split in args.data_split:
        print(f"Processing {split}...")

        for sample_id in tqdm(manifest[split]["id"]):
            manifest[split]["tgt_audio"].append(tgt_audio_paths[sample_id])
            manifest[split]["tgt_n_frames"].append(tgt_audio_lengths[sample_id])

        out_manifest = args.output_root / f"{split}.tsv"
        print(f"Writing manifest to {out_manifest}...")
        save_df_to_tsv(pd.DataFrame.from_dict(manifest[split]), out_manifest)

    # Generate config YAML
    win_len_t = args.win_length / args.sample_rate
    hop_len_t = args.hop_length / args.sample_rate
    extra = {
        "features": {
            "type": "spectrogram+melscale+log",
            "sample_rate": args.sample_rate,
            "eps": 1e-5, "n_mels": args.n_mels, "n_fft": args.n_fft,
            "window_fn": "hann", "win_length": args.win_length,
            "hop_length": args.hop_length,
            "win_len_t": win_len_t, "hop_len_t": hop_len_t,
            "f_min": args.f_min, "f_max": args.f_max,
            "n_stft": args.n_fft // 2 + 1
        }
    }
    gen_config_yaml(
        args.output_root,
        audio_root=args.output_root.as_posix(),
        specaugment_policy="lb",
        feature_transform=["utterance_cmvn", "delta_deltas"],
        extra=extra,
    )

def main():
    parser = argparse.ArgumentParser()
    parser.add_argument(
        "--source-dir", required=True, type=Path, help="source audio directory"
    )
    parser.add_argument(
        "--target-dir", required=True, type=Path, help="target audio directory"
    )
    parser.add_argument(
        "--data-split",
        default=["train", "valid", "test"],
        nargs="+",
        help="data split names",
    )
    parser.add_argument(
        "--output-root", required=True, type=Path, help="output directory"
    )
    # target feature related
    parser.add_argument("--win-length", type=int, default=1024)
    parser.add_argument("--hop-length", type=int, default=256)
    parser.add_argument("--n-fft", type=int, default=1024)
    parser.add_argument("--n-mels", type=int, default=80)
    parser.add_argument("--f-min", type=int, default=20)
    parser.add_argument("--f-max", type=int, default=8000)
    parser.add_argument("--sample-rate", type=int, default=22050)
    parser.add_argument("--normalize-volume", "-n", action="store_true")

    args = parser.parse_args()

    process(args)

if __name__ == "__main__":
    main()

redpajama_set_name: RedPajamaGithub


"While Africa has made significant strides in economic growth and in poverty reduction, much remains to be done. The goal is clear - to eradicate poverty, create more inclusive socio-economic development and to integrate the continent.
"I am particularly delighted that the Bank is working closely with this administration in addressing the challenges faced by communities and people emerging from the insurgency in the North East and some of the militancy in the Niger Delta Region.
Address by His Excellency, Prof Yemi Osinbajo, SAN, The Vice President, Federal Republic of Nigeria at The Commissioning of the AfDB Nigeria Country Office, Abuja on Thursday, 18th January, 2018.
I am delighted to be here today, to celebrate with the African Development Bank the commissioning of its Nigeria Country Office Complex. This building here in Abuja is, among other things, an important symbol of the special relationship between the AfDB and Nigeria - a founding member and the largest shareholder of the Bank from inception to date. And since 1971, the Nigeria Country Office has served as an important convening point between the Bank, the Federal and State Governments, and the private sector.
Nigeria has been important in the growth of the institution as a major shareholder, donor and borrower all at the same time. And we have through the years shared the bank's vision and objectives. Just to underscore the love between the AFDB and Nigeria, as the President aptly described it as a marriage, we have loaned our own Dr. Akinwunmi Adesina to the bank, as he is to our great pride, Nigeria's first President of the Bank.
Since its founding in 1964, the African Development Bank, has evolved into one of the most efficient vehicles for the economic development and integration of the African continent. The bank has faithfully played its role as a trusted advisor and an honest broker in our region. And has earned its place as the voice of Africa on development issues.
As the continent's premier financial institution, it is at the center of Africa's transformation to bring real and lasting progress to the lives of African people.
While Africa has made significant strides in economic growth and in poverty reduction, much remains to be done. The goal is clear - to eradicate poverty, create more inclusive socio-economic development and to integrate the continent.
To contribute to this agenda, the African Development Bank, through the High-5s - unveiled by Dr. Adesina when he took office in 2015 - is in our view, right on track by investing more in infrastructure, agriculture, education, healthcare, and increased access to, in particular, affordable energy and water.
More importantly, the Bank is to be commended for its ongoing efforts to close the gender gap, empower women and youths, and to ultimately strengthen and expand social safety nets for our most vulnerable populations.
With its large market of over 185 million people, Nigeria will continue to be an important player in the Bank's work, advancing efforts to boost regional integration within the Economic Community of West African States (ECOWAS) in particular and Africa as a whole.
We will continue to be a strong and committed partner of the African Development Bank, and to lead by example. We have already clearly demonstrated through the partnership that led to the establishment of the Bank's Nigeria Trust Fund, established in 1976. The Trust Fund of more than $500 million has helped to transform the economies and improve the living standards of millions of people in other African countries.
In Nigeria, the AfDB has an active portfolio of close to $6 billion dollars spread over 73 projects across public and private sectors of the Nigerian economy. These projects continue to create thousands of direct and indirect jobs in many parts of our economy.
I am particularly delighted that the Bank is working closely with this administration in addressing the challenges faced by communities and people emerging from the insurgency in the North East and some of the militancy in the Niger Delta Region.
The Bank's Inclusive Basic Service Delivery and Livelihood Empowerment Integrated Programme (IBSDLEIP), aimed at addressing the challenges in the North East in particular, is commendable. The project will support the youth, women and IDPs who have returned to their communities. An estimated population of more than 10 million affected people are expected to directly benefit from this project.
More specifically, I thank the President and Board of Directors of the African Development Bank, for their support to Nigeria at a very difficult time during the 2015/16 economic recession. When we requested support from the Bank, the response was swift. The Bank approved a budget support loan of $1 billion in November 2016. The first tranche of $600m has been fully disbursed and utilized and in my discussions with the President of the Bank, we will be getting the remainder in the form of sector loans, so we are looking forward to the remaining $400m.
The approval by AfDB sent a much-needed positive signals to the markets at a critical time for Nigeria and for this we will ever remain ever grateful.
Your Excellencies, Distinguished Ladies and Gentlemen, as we celebrate the commissioning of this world-class edifice, I wish to reiterate the commitment of the Buhari Administration to the Bank's strategy in Nigeria. The strategy is fully aligned with our own Economic Recovery and Growth Plan 2018 - 2020.
Again, congratulations to the President of the AfDB, Dr Akinwunmi Adesina, the Board of Directors, and the staff of African Development Bank, and all who made this edifice a reality.
I want to thank you for listening and in the next few minutes, we will be commissioning the AfDB new complex.


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HomeTechnologyChatbots
Chatbots in the classroom – the future of education
The teaching of the future must become more individual, more dynamic and more efficient. Most education experts agree on this, but how should a teacher meet these demands? How can we adapt to the individual needs of each student and offer them a tailor-made and unique learning experience? The answer is Chatbot.
That technology cannot be kept out of classrooms forever should be clear even to the most traditional teacher. The younger generations are growing up with tablets, smartphones and computers, and in the near future it will be more than less technology that shapes our everyday lives. It is therefore the task of the educators to prepare the pupils for the world of tomorrow and to make the lessons future-oriented. In addition, technologies can not only make teaching more lively, but also simplify the work of the teacher considerably.
Especially chatbots can complement the daily lessons in a meaningful way and provide the students with individual support, which unfortunately cannot be provided by a single teacher.
Strengthening of commitment and learning success through chatbots
Of course, tablets and smartphones are a distraction in the classroom and some students will try to digitise their attempts to cheat in exams, but therefore completely avoiding the digital change is not a solution either, as it does not live up to its teaching mission of preparing students for everyday life. Chatbots, for example, can significantly increase commitment in the classroom, and content can also be communicated over the long term. While many learners only learn shortly before an exam and are partly overwhelmed with the teaching material, modern chatbot systems already ensure that the material remains stuck in the long run.
The platforms equipped with artificial intelligence, for example, offer continuous feedback sessions to consolidate the material. In addition, they adapt the content to the respective needs of learners and pupils and focus on the people behind the books.
The real challenge for educators is to choose the right chatbot for their students. There are a variety of applications, platforms and styles, each with a different focus on teaching. Whether Gradeproof, Watson or similar, they all follow the purpose to fully engage with the student and his needs and to support, but not replace, the teacher in his mission of education.
No matter how you look at it, chatbots belong in teaching. The future cannot be stopped, but it can be actively shaped. It is up to every teacher to offer their students the best learning experience and to take their role as role model and mentor seriously.
Deutsche Post – a comment on German Angst
The future of mobility
Innovation Alexander Pinker - 30. January 2023
The future of the car is here and it looks incredible! Jetoptera is an innovative flying vehicle that relies on high performance jet propulsion technology. This technology allows the car to take off and land vertically, making it a unique and revolutionary concept. No wings. No unnecessary parts. Simply a vehicle for the skies.
Innovation Alexander Pinker - 1. January 2023
The new year has begun, and in 2023 we can once again expect a lot of new developments and innovations. Here are three trends that are likely to keep us busy in the coming months.


text:
Kuroda Akinobu
黒田明伸
ankuroda@ioc.u-tokyo.ac.jp
The University of Tokyo
Kuroda Akinobu is Professor of East Asian History in the Institute for Advanced Studies on Asia at the University of Tokyo. His research covers comparative studies of monetary history in East Asia, India, Africa, and Europe, as well as specific studies of China's monetary history. The concept of complementarity among monies, which he developed, has had some influence across monetary studies, economic history and anthropology. He has argued that the character of money can be determined according to degrees along two axes: first, to what extent is the relationship between persons making transactions biased in either the direction of cohesiveness or anonymity; and second, to what extent has the conversion between currency for local transactions and that for inter-regional settlements worked according to fixed or flexible conversion rates. He is also the editor of the International Journal of Asian Studies.
His publications include 'What is the Complementarity among Monies?: An Introductory Note' (Financial History Review 15-1, 2008); 'Anonymous Currencies or Named Debts?: Comparison of Currencies, Local Credits and Units of Account between China, Japan and England in the Pre-industrial Era'(Socio Economic Review 11-1, 2013) and Kahei sisutemu no sekaishi: hi-taishosei o yomu (in Japanese)[World history of monetary systems: Interpreting the asymmetric phenomenon] (Iwanami 2003, (new version) 2014, translation in Korean 2005, in Chinese 2007), Chuka teikoku no kozo to sekai keizai (in Japanese)[The Structure of the Chinese Empire and the World Economy] (Nagoya UP 1994).


The State Department has updated its travel alerts for Thailand and Ukraine, responding to an uptick in politically-charged violence affecting both countries.
In Ukraine, demonstrations have flared up after the government opted for closer economic ties with Russia rather than with the EU. Since Thursday, over 100 individuals have died in the Kiev street riots. And anti-government rallies in Bangkok claimed their twelfth casulty—a police officer—on the February 18th.
We've been monitoring the Thai situation for months, and the new travel alert sends the same message: US citizens should avoid protest sites and any large gatherings.
In the Ukraine alert, the State Department urges American travelers to maintain a low profile and to remain indoors at night.
Recent (and ongoing) protests against government deficiencies in Venezuela have not prompted an update to the current travel warning, which restricts Americans from certain areas in Caracas.
Of the three countries, Thailand is by far the largest destination for US travelers. Statistics show that over 800,000 Americans visit the Southeast Asian country annually, compared to the 134,000 that head to Ukraine and the 50,000 to Venezuela.
Visit travel.state.gov for more details, and follow @TravelGov on Twitter for up to date news.