Conditions for rational weak mixing
Jon. Aaronson
We present conservative
1208.3505.tex
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\begin{document}\title{ Conditions for rational weak mixing  }
\author{Jon. Aaronson}
\address{\ \ School of Math. Sciences, Tel Aviv University,
69978 Tel Aviv, Israel.}
\email{aaro@tau.ac.il}%\date{July 24th 2012}

%\begin{abstract}We present conservative\end{abstract}
\subjclass[2010]{37A40 (37A25, 37A30, 60K15)}
%\keywords{Probability preserving transformation, spectral type, rigidity, IP-convergence, Dirichlet measure,  eigenvalue group.}

\thanks{ The author's research was partially
supported by Israel Science Foundation grant No. 1114/08. The author would like to thank   the University of Surrey, for hospitality provided when this paper was written. }


\begin{abstract}We exhibit  rationally ergodic,  weakly mixing  measure preserving transformations which are not subsequence  rationally weakly mixing and give a condition for smoothness of renewal sequences. \end{abstract}
\maketitle\markboth{Jon. Aaronson \copyright  2012}{weak mixing}
\

Rational weak mixing of a measure preserving transformation was defined and considered in [A1] to which we refer for definitions.
\

We show by  example in \S1 that
weak rational ergodicity and  weak mixing does not imply subsequence rational weak mixing. The main examples are dyadic towers with super-growth sequences (defined below).  We also give zero type examples.  See [DGPS] for related examples.
 %and a condition for smoothness of renewal sequences.
\

A Markov shift is conservative, ergodic iff the associated stochastic matrix is irreducible and recurrent, and in this case is rationally ergodic. It is  weakly mixing iff its associated renewal sequences are aperiodic, and (subsequence) rationally weakly mixing iff the associated renewal sequences are (subsequence) smooth (see [A1]).
\

It is not known whether every aperiodic, recurrent renewal sequence  is subsequence smooth, or whether smoothness implies Orey's strong ratio limit property. See \S8 in [A1].
\

Smoothness of the renewal sequence $u$ would  follow e.g. from the property $\sum_{n\ge 1}|u_n-u_{n+1}|<\infty$. This  is known for positively recurrent, aperiodic,  renewal sequences and is  conjectured in for all aperiodic  renewal sequences (see \cite{K} \S1.6(iv)).
\

In \S2, we give a sufficient condition establishing smoothness e.g. when $\sum_{n\ge 1}u_n$ is $1$-regularly varying. This  condition entails the property $\sum_{n\ge 1}(u_n-u_{n+1})^2<\infty$.

\

\

\section*{\S1 Examples}


\subsection*{1.1 Dyadic towers over the dyadic adding machine}
\


Let $\Om:=\{0,1\}^\Bbb N,$ and let $P\in\mathcal P(\Om)$ be symmetric product measure: $P=\prod(\frac12,\tfrac12)$, and let $\tau:\Om\to\Om$ be the dyadic odometer
defined by
$$\tau(1,\dots,1,0,\om_{\ell+1},\dots)=(0,\dots,0,1,\om_{\ell+1},\dots)$$
where $\ell=\ell(\om):=\min\,\{n\ge 1:\ \om_n=0\}$.
\

An increasing sequence $q\in\incss$ is a {\it growth sequence} as in \cite{A2} if $q_n>\sum_{1\le k<n}q_k$.
\


The {\it dyadic cocycle } $\varphi:\Om\to\Bbb N$ associated to the growth sequence $q\in\incss$ is  defined by
\begin{align*}\tag{\Football}\varphi(\om):=q_{\ell(\om)}-\sum_{k=1}^{\ell(\om)-1}q_k=\sum_{n\ge 1}q_n(\tau(\om)_k-\om_k).\end{align*}
\
The {\it dyadic tower} with the growth sequence $q$ is
the tower over $(\Om,\B(\Om),P,\tau)$ with height function $\vf$, namely
 $( X,\B( X), m, T)$ with
 \begin{align*}&X:=\{(x,n)\in\Om\x\Bbb N:\ 1\le \vf(x)\},\ m(A\x\{n\}:=P(A\cap [\vf\ge n]),\\ & T(x,n)=\begin{cases} & (x,n+1)\ \ \ \ \vf(x)\ge n+1;\\ & (\tau x,1)\ \ \ \ \vf(x)=n.\end{cases}
 \end{align*}

\subsubsection*{Rational ergodicity of $T$}


Recall from [A2] that $(X,\B,m,T)$ is (boundedly) rationally ergodic with return sequence
$$a_n(T)\asymp 2^{c(n)}\ \text{\rm where}\ c(n) =\min\,\{k\ge 1:\ q_k\ge n\}.$$

\subsubsection*{Weak mixing of $T$}
\

By [AN], $T$ is weakly mixing iff $G_2(q)=\{0\}$ where
$$G_2(q):=\{t\in\Bbb T:\ \sum_{n\ge 1}\|q_nt\|^2<\infty\}$$
where $\|x\|:=\min_{N\in\Bbb Z}|x-n|$.
\

By lemma 3 and theorem 2 in \cite{P} (see also \cite{AHL}), if $q_{n+1}=a_nq_n+1$ where $a_n\in\Bbb N,\ \sum_n\tfrac1{a_n^2}=\infty$, then $G_2(q)=\{0\}$ and $T$ is  weakly mixing. See also \cite{AFS}.
\subsubsection*{Negation of subsequence rational weak mixing}
\

Let $q\in\incss$ be a   growth sequence. For $\e\in\mathcal E:=\{\eta\in\{-1,0,1\}^\Bbb N:\ \ \eta_n\to 0\}$, let
$$N_\e:=\sum_{k\ge 1}\e_kq_k.$$
It is easy to see that $\e\mapsto N_\e$ ($\mathcal E\to\Bbb Z$) is injective if
$q$ is a {\it super growth sequence} in the sense that $q_n>2\sum_{1\le k<n}q_k$.
\

For $\e\in\mathcal E$, we have that $N_\e>0$ iff
$\e_{\kappa_{\max}}=1$ where $\kappa_{\max}(\e):=\max\,\{k\ge 1:\e_k\ne 0\}$.
\

Write $\mathcal E^+:=\{\e\in\mathcal E:\ \e_{\kappa_{\max}}=1\}$ and $\|\e\|:=\sum_{n\ge 1}|\e_n|$.

\


We claim that
\Smi {\em A dyadic tower with a super growth sequence cannot be subsequence, rationally weakly mixing.}
\

Using the above, it is easy to construct super growth sequences $q\in\incss$ with $G_2(q)=\{0\}$
and hence with  weakly mixing dyadic towers.
\


The claim (\Smiley) will follow from
\begin{align*}\tag{\dsrailways}m(\Om\cap T^{-n}\Om)=\begin{cases}& \frac1{2^{\|\e\|}}\ \ \ n=N_\e,\ \e\in\mathcal E;\\ & 0\ \ \ \ \ \text{\rm else.}\end{cases}\end{align*}
\demo{Proof of (\dsrailways)\ \ \ {\rm (see [HK])}}
\

 Let $\mathcal N\ge 1$ and $x\in\Om$, then $T^\mathcal N x\in\Om$ iff
 $\exists\ N\ge 1$ so that $\vf_N(x)=\mathcal N$.

By (\Football),
$$\mathcal N=\vf_N(x)=\sum_{k\ge 1}q_k(\tau^N(\om)_k-\om_k)=\sum_{k\ge 1}\e_kq_k=:N_\e$$ for some
$\e\in\mathcal E^+$.
\



We now show that
$m(\Om\cap T^{-N_\e}\Om)=\frac1{2^{|\e|}}.$
\

If $n_1<n_2<\dots<n_k,\ m_1<m_2<\dots m_\ell$ and
$$\e_{n_i}=1,\ \e_{m_j}=-1\ \ \&\ \ \e_n=0\ \ \text{\rm else},$$
then
$$\Om\cap T^{-N_\e}\Om=\{\om\in\Om:\ \om_{n_i}=1\ \forall\ i\ \&\ \e_{m_j}=0\ \ \forall\ j\}$$
and $T^{N_\e}\om=G\om$ where $G:\Om\to\Om$ is defined by
$$G(x)_k=\begin{cases}& 1-x_k\ \ \ \ \ k\in\{n_i\}_i\cup\{m_j\}_j;\\ &x_k\ \ \ \ \text{\rm else}.\end{cases}$$
Thus
$$m(\Om\cap T^{-N_\e}\Om)=m(\{\om\in\Om:\ \om_{n_i}=1\ \forall\ i\ \&\ \e_{m_j}=0\forall\ j\})=\frac1{2^{\|\e\|}}.\ \ \ \text{\text{\Checkedbox}(\dsrailways)}$$
\demo{Proof of (\Smiley)}
\

It suffices to show that
$$\varliminf_{n\to\infty}\frac1{2^{c(n)}}\sum_{k=1}^n|u_k-u_{k+q_1}|>0$$
where $u_n=u_n(\Om):=m(\Om\cap T^{-n}\Om)$.
\


 To see this, we restrict summation to
 $k=N_\e$ where $\e\in\mathcal E^+\  \&\ \e_1=0$; noting  $N_\e\le n$ iff $\kappa_{\max}\le c(n)$.
 \

 Here
$$u_{N_\e}=\frac1{2^{\|\e\|}},\ u_{N_\e+q_1}=\frac1{2^{\|\e\|+1}}$$ and
 $$u_{N_\e}-u_{N_\e+q_1}=\frac12\cdot u_{N_\e}.$$
 \

 Thus
 \begin{align*}\sum_{k=1}^n|u_k-u_{k+q_1}|&\ge
 \sum_{\e\in\mathcal E^+,\ \e_1=0,\ \kappa_{\max}\le c(n)}|u_{N_\e}-u_{N_\e+q_1}|\\ &=\frac12\sum_{\e\in\mathcal E^+,\ \e_1=0,\ \kappa_{\max}\le c(n)}u_{N_\e}\\ &=
 \frac12\sum_{\e\in\mathcal E^+,\ \e_1=0,\ \kappa_{\max}\le c(n)}\frac1{2^{\|\e\|}}.\end{align*}
 Now
\begin{align*}\sum_{\e\in\mathcal E^+,\ \e_1=0,\ \kappa_{\max}\le c(n)}\frac1{2^{\|\e\|}}&=
 \sum_{\emptyset\ne F\subset\Bbb N\cap [2,c(n)]}\ \ \sum_{\e\in\mathcal E^+,\ \text{\tt\tiny supp}\,\e=F}\frac1{2^{\#F}}.
  \end{align*}
  For fixed $F$,
  \begin{align*}\#\{\e\in\mathcal E,\ \text{\tt supp}\,\e=F\}=2^{\#F}.\end{align*} Thus
  \begin{align*}\sum_{k=1}^n|u_k-u_{k+q_1}|&\ge \frac12\sum_{\e\in\mathcal E^+,\ \e_1=0,\ \kappa_{\max}\le c(n)}\frac1{2^{\|\e\|}}\\ &=\frac12\#\{\emptyset\ne F\subset\Bbb N\cap [2,c(n)]\}\\ &\ge
  \frac18\cdot 2^{c(n)}.\ \ \ \text{\text{\Checkedbox}}
  \end{align*}
  \subsection*{1.2 A zero type example}
  \

  Let $(X,\B,\mu,T)$ be a weakly mixing, dyadic tower with a super-growth sequence (as above) and let $(Y,\mathcal C,\nu,S)$ be a conservative, aperiodic  ergodic Markov shift with $A\in M(S),\ u(A)$ Kaluza and with $S\x T$ conservative.
  \

\smallskip\f{\Large\bf\Yinyang\ \ \ } {\em The transformation $(Y\x X,\mathcal C\otimes\B,\nu\x\mu,S\x T)$ is zero-type, rationally ergodic and weakly mixing, but not subsequence, rationally weakly mixing.}
\demo{Proof}
\

  Evidently $S\x T$ is zero type.
  \

  To see that $S\x T$ is weakly mixing, let
  $R$ be an ergodic, probability preserving transformation. It follows (from the weak mixing of $T$) that $R\x T$ is conservative, ergodic. Since $S$ is the natural extension of a mildly mixing transformation, we have that
  $$(S\x T)\x R=S\x (T\x R)$$
  is conservative, ergodic. This shows that $S\x T$ is weakly mixing.
  \


  Next, we claim that $S\x T$ is  rationally ergodic with $A\x\Om\in R(S\x T)$.
 \pf It suffices to consider the one sided Markov shift $(Y_+,\mathcal C_+,\nu,\sigma)$ and show that $\s\x T$ is  rationally ergodic with $A\x\Om\in R(\s\x T)$.
  \

  Write $u_k:=\nu(A\cap \sigma^{-n}A)$, then $v_k:=u_k-u_{k+1}>0$ and on $A\x\Om$, the transfer operator is given by
 \begin{align*}\widehat{\sigma\x T}^n(1_{A\x\Om})(y,\om)=\widehat{\sigma}^n(1_{A})(y)1_{\Om}(T^{-n}\om)=u_n1_{\Om}(T^{-n}\om).\end{align*}
 Thus, for $(y,\om)\in A\x\Om$,
   \begin{align*}\sum_{1\le k\le n}\widehat{\sigma\x T}^k(1_{A\x\Om})(y,\om)&=\sum_{1\le k\le n}u_k1_{\Om}(T^{-k}\om)\\ &=
   \sum_{1\le k\le n}u_k(S_k^{(T^{-1})}(1_{\Om})(\om)-S_{k-1}^{(T^{-1})}(1_{\Om})(\om))\\ &=\sum_{1\le k\le n}v_kS_k^{(T^{-1})}(1_{\Om})(\om)+u_nS_n^{(T^{-1})}(1_{\Om})(\om)\\ &\le M\sum_{1\le k\le n}v_ka_k^{(T)}(\Om)+Mu_na_n^{(T)}(\Om)\\ &= M\sum_{1\le k\le n}u_k\mu(\Om\cap T^{-k}\Om)+2Mu_na_n^{(T)}(\Om)\\ &\le 3Ma_n^{(S\x T)}(A\x\Om).
   \end{align*}
   Rational ergodicity of $S\x T$ and $A\x\Om\in R(S\x T)$ follow from this.
   \

   To see that $S\x T$ is not rationally weakly mixing, consider $E,\  F\in R(S\x T)$ given by
   $$E:=A\x\Om,\ F:=A\x T^{-q_1}\Om,$$ then for $n=N_\e$ with $\e_1=0$, we have
   \begin{align*}\nu\x\mu(E\cap (S\x T)^{-n}E)&-\nu\x\mu(F\cap (S\x T)^{-n}F)\\ &=
   u_n(\mu(\Om\cap T^{-n}\Om)-\mu(\Om\cap T^{-(n+q_1)}\Om))\\ &=
   \frac12u_n\mu(\Om\cap T^{-n}\Om).
   \end{align*}
   Noting that
   \begin{align*}a_n(S\x T) &\sim\sum_{\e\in\mathcal E^+,\ N_\e\le n}u_{N_\e}\frac1{2^{\|\e\|}}\\ &\asymp
   \sum_{\e\in\mathcal E^+,\ \e_1=0,\ N_\e\le n}u_{N_\e}\frac1{2^{\|\e\|}}\end{align*}
   we see that
   \begin{align*}\sum_{1\le k\le n}|\nu\x\mu(E\cap (S\x T)^{-k}E)&-\nu\x\mu(F\cap (S\x T)^{-k}F)|\\ &=
   \sum_{1\le k\le n}u_n|\mu(\Om\cap T^{-n}\Om)-\mu(\Om\cap T^{-(n+q_1)}\Om)|\\ &\ge
   \sum_{\e\in\mathcal E^+,\ \e_1=0,\ N_\e\le n}u_{N_\e}\frac1{2^{\|\e\|}}\\ &\asymp a_n(S\x T).\ \ \text{\text{\Checkedbox}\Yinyang}
   \end{align*}

   \section*{\S2 Smoothness of renewal sequences}

Suppose that  $u=u^{(f)}=(u_0,u_1,\dots)$ is an aperiodic, recurrent, renewal  sequence with lifetime distribution $f\in\mathcal P(\mathbb N)$.
\

The renewal sequence $u$ is called {\it smooth} if
$$\sum_{k=1}^n|u_k-u_{k+1}|=o(a_n)\ \text{as}\ n\to\infty\ \ \text{where}\ a_n=a_n^{(u)}:=\sum_{k=1}^nu_k.$$
It follows from  [GL] that if $u=(u_0,u_1,\dots)$ is an aperiodic, recurrent, renewal  sequence and $a^{(u)}$ is $\a$-regularly varying with $\a\in (0,1)$, then $u$ is smooth (see [A1]). The case $\a=1$ follows from the next proposition (which is related to proposition 8.3 in [A1]).
\

For  $u=u^{(f)}$, let $c_N:=f([N,\infty))$ and $L(N):=\sum_{k=1}^Nc_k$.
\proclaim{Proposition }\ \ \ If $u=(u_0,u_1,\dots)$ is an aperiodic, recurrent, renewal  sequence with \begin{align*}\tag{\Pointinghand}\varlimsup_{N\to\infty}\frac{Nc_N}{L(N)}<\frac1{\sqrt{5}+1},\end{align*}
 then $u$ is smooth.\endproclaim
\f{\bf Remark}\ \ If  $a^{(u)}$ is $t$-regularly varying for some $t>\frac{\sqrt{5}}{\sqrt{5}+1}$ , then by the renewal equation and Karamata theory,
$L(N)\propto\frac{N}{a^{(u)}(N)}$ is $(1-t)$-regularly varying, $Nc_N\sim (1-t)L(N)$ and (\Pointinghand) holds.
\demo{Proof}
We show first that
\begin{align*}\tag{\Football}\sum_{k=1}^\infty(u_k-u_{k+1})^2<\infty.\end{align*}
Let $u=u^{(f)}$ and let $c_N:=f([N,\infty))$,  $M(N):=\sum_{1\le n\le N} nf_n$ and $V(N):=\sum_{1\le n\le N} n^2f_n$.
\

By (\Pointinghand), $\exists\ R<\frac1{\sqrt 5+1}$ so that
%Since $L(N):=\sum_{k=1}^Nc_k$ is slowly varying as $N\to\infty$, we have that
$Nc_N\le RL(N))$ for large $N$. It follows that
$$NC_N\le RL(N)=R(M(N)+NC_N)$$ whence
\begin{align*}\tag{\Bicycle}NC_N\le \frac{R}{1-R}\cdot M(N)\end{align*}
for large $N$ where $\frac{R}{1-R}<\frac1{\sqrt 5}.$
\

In particular
$$M(N)\ \asymp\ L(N)\ \ \ \ \text{as}\  N\to\infty;$$
and
$$\sum_{n=1}^\infty\frac1{n^2M(n)^2}<\infty.$$
We'll use these to prove (\Football).

\begin{comment}
\proclaim{Lemma}\ \ \

\begin{align*}\sum_{n=1}^\infty\frac1{1+V(n)^2}<\infty\ &\Lra\ \sum_{n=1}^\infty(u_n-u_{n+1})^2<\infty\\ & \Lra\
\sum_{n=1}^\infty\frac1{1+(V(n)+nM(n)+n^2c_n)^2}<\infty.\end{align*}
\endproclaim

\demo{Proof}\ \ The first implication is  proposition 8.3(i) of [A]. We prove the second.\end{comment}
\

 By  Parseval's formula, and the renewal equation,
\begin{align*} \int_{-\pi}^\pi\frac{|\th|^2d\th}{|1-f(\th)|^2}<\infty\ \ \iff\ \sum_{n=1}^\infty(u_n-u_{n+1})^2<\infty\end{align*}
where $f(\th):=\sum_{n=1}^\infty f_ne^{in\th}$. By aperiodicity, $\sup_{\e\le|\th|\le\pi}|f(\th)|<1\ \forall\ \e>0$ whence
 (using symmetry)
 \begin{align*}\int_{-\pi}^\pi\frac{|\th|^2d\th}{|1-f(\th)|^2}<\infty\ \ \iff\ \ \int_0^{\e}\frac{\th^2d\th}{|1-f(\th)|^2}<\infty\ \text{\tt\small for some}\ \ \e>0.\end{align*}
For $|\th|<\frac{\pi}2$,
 \begin{align*}1-f(\th)&=
 \sum_{n=1}^\infty f_n(2\sin^2(\frac{n\th}2)-i\sin(n\th))\\ &=
 (\sum_{1\le n\le \frac{2}{\pi|\th|}}+ \sum_{n> \frac{2}{\pi|\th|}})f_n(2\sin^2(\frac{n\th}2)-i\sin(n\th)) \end{align*}
 and
 \begin{align*}|1-f(\th)|\ge |\sum_{1\le n\le \frac{2}{\pi|\th|}}|-| \sum_{n> \frac{2}{\pi|\th|}}|.\end{align*}
 Now
 \begin{align*}|\sum_{1\le n\le \frac{2}{\pi|\th|}}|&=\sqrt{\(\sum_{1\le n\le \frac{2}{\pi|\th|}}f_n2\sin^2(\frac{n\th}2)\)^2+\(\sum_{1\le n\le \frac{2}{\pi|\th|}}f_n\sin(n\th)\)^2} \\ &\ge
 |\sum_{1\le n\le \frac{2}{\pi|\th|}}f_n\sin(n\th)|
  \\ &\ge\frac{2}{\pi}\cdot|\th|
 \sum_{1\le n\le \frac{2}{\pi|\th|}}nf_n\\ &=
 \frac{2}{\pi}\cdot|\th| M(\frac{2}{\pi|\th|});\end{align*}
 and
 \begin{align*}| \sum_{n> \frac{2}{\pi|\th|}}|&=|\sum_{n>\frac{2}{\pi|\th|}}f_n(2\sin^2(\frac{n\th}2)-i\sin(n\th))|\\ &\le \sqrt{\(\sum_{n>\frac{2}{\pi|\th|}}f_n2\sin^2(\frac{n\th}2)\)^2+\(\sum_{n> \frac{2}{\pi|\th|}}f_n\sin(n\th)\)^2} \\ &\le \sqrt{5}c_{\frac{2}{\pi|\th|}}.\end{align*}
 By assumption $\exists\ \D>0$ so that for  $|\th|<\D$,
 $$c_{\frac{2}{\pi|\th|}}\le R'\frac{2|\th|}{\pi}M(\frac{2}{\pi|\th|})$$ where
 $R'=\frac{R}{1-R}=\frac1{\sqrt 5}(1-\eta)$ for some $\eta>0$;
 whence
 \begin{align*}|1-f(\th)| & \ge \frac{2}{\pi}\cdot|\th|M(\frac{2}{\pi|\th|})-\sqrt{5}c_{\frac{2}{\pi|\th|}}\\ &\ge
 \frac{2}{\pi|\th|}M(\frac{2}{\pi|\th|})(1-\sqrt 5 R')\\ &=\frac{2\eta}{\pi|\th|}M(\frac{2}{\pi|\th|}).\end{align*}



Let $N\ge 1,\ \D>\frac{\pi}N$, then
\begin{align*}\int_0^{\frac{\pi}N}\frac{\th^2d\th}{|1-f(\th)|^2} &\ll \int_0^{\frac{\pi}N}\frac{\th^2d\th}{(\th M(\frac{2}{\pi|\th|}))^2}\\ &=\int_0^{\frac{\pi}N}\frac{d\th}{M(\frac{2}{\pi|\th|})^2}\\ &=
\sum_{n=1}^\infty\int_{\frac{\pi}{n+1}}^{\frac{\pi}n}\frac{d\th}{M(\frac{2}{\pi|\th|})^2}\\ &\asymp \sum_{n=1}^\infty\frac1{n^2M(n)^2}<\infty.\ \ \ \text{\Checkedbox(\Football)}\end{align*}
To see smoothness, by assumption
$$\log L(N)\sim \sum_{k=1}^N\frac{c_k}{L(k)}\le \sum_{k=1}^N\frac{1}{\sqrt 5 k}+ O(1)=\frac{1}{\sqrt 5 }\log N+O(1)$$
whence
$$L(N)=O(N^{\frac{1}{\sqrt 5 }}).$$
Using the renewal equation, $a_u(n)\asymp\frac{n}{L(n)}$, whence
$a_u(n)\gg N^{1-\frac{1}{\sqrt 5 }}$ and
$$\frac{\sqrt{n}}{a_n}\ll \frac1{N^{\frac12-\frac{1}{\sqrt 5 }}}\xyr[n\to\infty]{}0.$$
By {\tt CSI}
$$\frac1{a_u(n)}\sum_{k=1}^n|u_k-u_{k+1}|\le \frac{\sqrt{n}}{a_u(n)}\cdot\sqrt{\sum_{k=1}^\infty(u_k-u_{k+1})^2}\xyr[n\to\infty]{}0.\  \ \text{\Checkedbox}$$
\

\begin{thebibliography}{99999}
\bibitem[A1]{A1} Aaronson, J. \textit{ Rational weak mixing in infinite measure spaces.}
arXiv:1105.3541

\bibitem[A2]{A2}   J. Aaronson,   {\em Rational ergodicity, bounded rational ergodicity and some continuous
 measures on the circle.}
A collection of invited papers on ergodic theory.
 Israel J. Math.  33  (1979),  no. 3-4, 181--197 (1980).

\bibitem[AFS]{AFS} T. Adams, N. Friedman, and C. E. Silva. {\em Rank-one weak mixing
for nonsingular transformations.} Israel J. Math., {\bf 102} (1997)  269--281.
 \bibitem[AHL]{AHL} J. Aaronson, M. Hosseini,  M. Lema\'nczyk, {\em  IP-rigidity and eigenvalue groups },arXiv:1203.2257.
 \bibitem[AN]{AN} J. Aaronson, M.G. Nadkarni, {\em $L_\infty$ eigenvalues and $L_2$ spectra of non-singular transformations}, Proc.\ London Math.\ Soc.\ 930 55 9 (1987), 538-570.

\bibitem[DGPS]{DGPS}Dai I,  Garcia X,  P\v{a}durariu T,
Silva  CE,   {\em On rationally ergodic and rationally
weakly mixing rank-one transformations}, Preprint.

\bibitem[GL]{GL} Garsia, A.;  Lamperti, J. \textit{A discrete renewal theorem with infinite mean.}
 Comment. Math. Helv. \textbf{ 37}  (1962/1963), 221--234.

\bibitem[HK]{HK}    Hajian, A. ;  Kakutani, S. {\em Example of an ergodic measure preserving transformation on an infinite
 measure space.}
   Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio
 State Univ., Columbus, Ohio, 1970)
 pp. 45--52 Springer, Berlin , 1970.
 \bibitem[K]{K}\  Kingman, J. F. C.  \textit{Regenerative phenomena}.
Wiley Series in Probability and Mathematical Statistics.
John Wiley\& Sons Ltd., London-New
 York-Sydney,  1972.
 \bibitem[P]{P}  F. Parreau,  {\em Ergodicit\'e et puret\'e des produits de Riesz}, Ann.
Inst. Fourier, {\bf 40} (1990)  391--405.
 \end{thebibliography}
\end{document} 
1208.3505.extracted_bib
\begin{thebibliography}{99999}
\bibitem[A1]{A1} Aaronson, J. \textit{ Rational weak mixing in infinite measure spaces.}
arXiv:1105.3541

\bibitem[A2]{A2}   J. Aaronson,   {\em Rational ergodicity, bounded rational ergodicity and some continuous
 measures on the circle.}
A collection of invited papers on ergodic theory.
 Israel J. Math.  33  (1979),  no. 3-4, 181--197 (1980).

\bibitem[AFS]{AFS} T. Adams, N. Friedman, and C. E. Silva. {\em Rank-one weak mixing
for nonsingular transformations.} Israel J. Math., {\bf 102} (1997)  269--281.
 \bibitem[AHL]{AHL} J. Aaronson, M. Hosseini,  M. Lema\'nczyk, {\em  IP-rigidity and eigenvalue groups },arXiv:1203.2257.
 \bibitem[AN]{AN} J. Aaronson, M.G. Nadkarni, {\em $L_\infty$ eigenvalues and $L_2$ spectra of non-singular transformations}, Proc.\ London Math.\ Soc.\ 930 55 9 (1987), 538-570.

\bibitem[DGPS]{DGPS}Dai I,  Garcia X,  P\v{a}durariu T,
Silva  CE,   {\em On rationally ergodic and rationally
weakly mixing rank-one transformations}, Preprint.

\bibitem[GL]{GL} Garsia, A.;  Lamperti, J. \textit{A discrete renewal theorem with infinite mean.}
 Comment. Math. Helv. \textbf{ 37}  (1962/1963), 221--234.

\bibitem[HK]{HK}    Hajian, A. ;  Kakutani, S. {\em Example of an ergodic measure preserving transformation on an infinite
 measure space.}
   Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio
 State Univ., Columbus, Ohio, 1970)
 pp. 45--52 Springer, Berlin , 1970.
 \bibitem[K]{K}\  Kingman, J. F. C.  \textit{Regenerative phenomena}.
Wiley Series in Probability and Mathematical Statistics.
John Wiley\& Sons Ltd., London-New
 York-Sydney,  1972.
 \bibitem[P]{P}  F. Parreau,  {\em Ergodicit\'e et puret\'e des produits de Riesz}, Ann.
Inst. Fourier, {\bf 40} (1990)  391--405.
 \end{thebibliography}