text:
\part{Results}
\section{Introduction}

One of the prominent limit theorems in classical probability theory is the Poisson Limit Theorem. This is taken advantage of in a variety of probabilistic models describing waiting times until unlikely events occur; due to the Poisson Limit Theorem (PLT) such waiting times are usually well approximated by exponentially distributed variables. It has been a great discovery that, many deterministic systems satisfy the same kind of limit theorems for rare events. \\

Limit distributions of waiting times are most classical for mixing Markov chains, there one considers returns to small cylinders, for example see \cite{pitskel_1991}. As remarked there, this result can be immediately generalized to systems with a Markov partition, the only caveat being that the sets are still cylinders, so geometrically not the most intuitive class. Nonetheless, waiting time limits for returns to small balls can be shown in concrete settings; for example hyperbolic toral automorphisms \cite{DGS}, Gibbs-Markov maps \cite{Poisson_GM}, partially hyperbolic maps \cite{Dhyplim}, some intermittent interval maps \cite{indfixpointCG}, and many more. It is sometimes interesting to also ask for explicit rates of convergence, this can be shown using eg $\alpha$-mixing, see \cite{HSV}, \cite{AS}, \cite{compoundpoisHV}.\\
For systems having a spectral gap for the Perron-Frobenius transfer operator there is a particularly nice method of proving Poisson limits, making use of the relation of hitting and return time limits obtained in \cite{HSV}. The argument has been carried out in \cite{Z3}, \cite{Z2}.\\
Another method is to induce a given system on a small set, such that the induced system has limits for return times, then one can use methods of \cite{HWZ}, \cite{BSTV} to prove the same limits for the original system. This can be used to show return times limits for certain quadratic maps, or expanding maps with an indifferent fixed point.\\
We do not make any claims on completeness of the list of reference given above, for a more complete picture see \cite{ExtremeL}.
\\
Some related topics are extreme value laws \cite{ExtremeFFT}, \cite{ExtremeFFS}, spatiotemporal limits \cite{spatiotempPS}, \cite{Z1} or Borel-Cantelli like Lemmas \cite{borelcantelliHLSW}, \cite{bcHNPV}, \cite{loglargegaps}.
\\
Similar questions can be asked for flows as well, this topic
has not been studied as thoroughly as the question for maps. As shown in \cite{flowPY} for suspension this reduces to the study for maps.\\
Poisson Limit Theorem for flows can be reduced to Poisson Limit Theorem for
time 1 map with the target being the set of points which visit $B(x, r)$ in
the next unit of time.

\section{Preliminaries}
\begin{definition}
Given a probability-preserving ergodic dynamical system $(X,\mathcal{A},\mu,T)$ and a measurable set $A\in \mathcal{A}$, we will define the \textit{first return time to $A$} as
\begin{equation*}
\f{A}(x)=\min(n\geq 1\;|\; T^n(x)\in A).
\end{equation*}
The \textit{first return map} shall be denoted by $T_A(x)=T^{\f{A}(x)}(x)$, and the \textit{sequence of consecutive returns} by
\begin{equation*}
\Phi_A=(\f{A},\f{A}\circ T_A,\f{A}\circ T_A^2,...).
\end{equation*}
\end{definition}

In the following, for some measurable set $A\in \mathcal{A}$ with $\mu(A)>0$, the measure conditional on $A$ shall be given as $\mu_A(B)=\frac{\mu(A\cap B)}{\mu(A)}$.\\
The first important result in the study of $\f{A}$, was Kac's formula, which calculates the expectation as
\begin{equation*}
\int_A \f{A} \d\mu_A = \frac{1}{\mu(A)}.
\end{equation*}
Hence we will study limits of $\mu(A)\f{A}$ as $\mu(A)\rightarrow 0$. More explicitly let $\s{A}{l}$ be a sequence of rare events, that is, each $A_l$ is measurable with $\mu(A_l)\rightarrow 0$, we want to find weak limits of the form
\begin{equation*}
\mu(A_l)\Phi_{A_l}\xRightarrow{\mu} \Phi \;\;\;\as{l},
\end{equation*}
or
\begin{equation*}
\mu(A_l)\Phi_{A_l}\xRightarrow{\mu_{A_l}} \tilde{\Phi} \;\;\;\as{l},
\end{equation*}
where $\Rightarrow$ denotes convergence in distribution. In the above situation we shall call $\Phi$ the \textit{hitting time limit} and $\tilde{\Phi}$ the \textit{return time limit}. An important fact is that the hitting and return time limits are intimately related. In particular an important consequence is the following.

\begin{proposition}\label{hitretrelcor} (\cite[Theorem 2.1]{HSV})
Let $(X,\mathcal{A},\mu,T)$ be an ergodic probability preserving dynamical system and let $\s{A}{l}$ be a sequence of rare events. Then $\Phi\overset{d}{=}\Phi_{Exp}$ is equivalent to $\tilde{\Phi}\overset{d}{=}\Phi_{Exp}$, where $\Phi_{Exp}$ is an iid process of standard exponentially distributed random variables.
\end{proposition}
This suggests that we should expect exponential hitting and return time limits for geometrically sensible sequences of rare events. \\
In the following let $X$ be a $C^r$ Riemannian manifold with $\dim(X)=d$ and assume $\mu\ll m_X$ the volume on $X$, with continuous density. Most of the statements can be reformulated to hold for arbitrary invariant $\mu$, but for the sake of simplicity we shall keep this assumption.
\begin{definition}
\begin{enumerate}[(i)]
\item Let $x^*\in X$ and, for $r>0$, denote by $B_r(x^*)$ the geodesic ball of radius $r$ centred at $x^*$. We will say that \textit{$T$ satisfies the PLT at $x^*$} if
\begin{equation*}
\mu(B_r(x^*))\Phi_{B_r(x^*)}\xRightarrow{\mu} \Phi_{Exp} \;\;\;\textit{ as }r\rightarrow 0.
\end{equation*}
\item Let
\begin{equation*}
PLT:=\{x^*\in X \;|\; \textit{$T$ satisfies the PLT at $x^*$}\}.
\end{equation*}
If $\mu(PLT)=1$ we will say that \textit{$T$ satisfies the PLT almost everywhere}, and if $PLT=X$ we will say that \textit{$T$ satisfies the PLT everywhere}.
\end{enumerate}
\end{definition}
If $T$ is Lipschitz-continuous along the (finite) orbit of a periodic point, then
it \textbf{does not} satisfy the PLT at that point.
To see this note first that the PLT at $x^*$ in particular implies via Theorem \ref{hitretrelcor},
that for each $N\geq 1$,
\begin{equation*}
\mu_{B_r(x^*)}(\f{B_r(x^*)}\leq N)\rightarrow 0 \;\;\;\textit{ as }r\rightarrow 0.
\end{equation*}
Now suppose $x^*$ is a point with period $p$, and $|T^p(x)-T^p(y)|\leq C|x-y|$ near $x^*$ then
\begin{equation*}
\mu_{B_r(x^*)}(\f{B_r(x^*)}\leq p) \geq \mu_{B_r(x^*)}(B_{\frac{r}{C}}(x^*))=\frac{1}{C^d}+o(1).
\end{equation*}

In particular chaotic (in the sense of Devaney) transformations can never satisfy the PLT everywhere, nevertheless we shall construct maps that satisfy the PLT everywhere. The main goal is to prove that a big class of product systems satisfies the PLT (almost) everywhere.

\section{The PLT for (skew-)products}

In this paper we study the PLT for systems that can be written as a product (or skew product of a special type). Therefore let $Y$ be another Riemannian $C^{r'}$-manifold with $\dim(Y)=d'$, and assume $R:Y\rightarrow Y$ preserves a probability measure $\nu\ll m_Y$ with continuous density. Instead of $T:X\rightarrow X$, consider now some  $T:X\times Y\rightarrow X$.
We will prove the PLT for certain systems of the form $S(x,y)=(T(x,y),R(y))$. The case of direct products can be recovered if $T(x,y)=T(x)$ is independent of $y$ (which will be the case for most of our examples). Denote also $T_y(x)=T(x,y)$. We will assume that $T_y$ preserves a probability measure
$\mu$ (independent of $y$). For $n\geq 1$, let $T^n_y(x)=T_{R^{n-1}(y)}(T_{R^{n-2}(y)}(...(T_y(x))))$. For $y\in Y$ we introduce analogously the \textit{fiberwise return times} as
\begin{equation}\label{retdef}
\begin{aligned}
&\f{A,y}^{(1)}(x):=\f{A,y}(x):=\min(j\geq 1\;|\; T_y^{j}(x)\in A)\\
&\f{A,y}^{(n+1)}(x):=\min(j\geq \f{A,y}^{(n)}+1\;|\;T_y^{j}(x)\in A)\\
&\Phi_{A,y}:=(\f{A,y}^{(1)},\f{A,y}^{(2)},\dots).
\end{aligned}
\end{equation}
All of the above can be equivalently expressed as the usual returns times, where we only fix a set in the fiber, e.g
\begin{equation*}
\f{A,y}(x)=\f{A\times Y}(x,y)=\min(j\geq 1\;|\; S^j(x,y)\in A\times Y),
\end{equation*}
but for the sake of the conditions to follow, we think it is best to keep in mind the definitions in \eqref{retdef} as it emphasises that $y$ should be regarded as fixed.

We will list here the main assumptions we make in order to prove the
PLT.

Here and in the following we will use $\varphi$ to denote first return times w.r.t different systems, it will however always be clear which system we mean from the specified set.
\begin{itemize}
\item[(MEM)] For almost all $y\in Y$ there are constants $r>0$, $C>1$ and $\gamma>0$ such that
\begin{equation}\label{ballretasm:mem}
\begin{aligned}
\left|\int_X \right. & \left. \prod_{j=0}^{n-1} f_j \circ T_y^{k_j} \d\mu - \prod_{j=0}^{n-1} \int_X f_j \d\mu\right|\\
&\leq C e^{-\gamma \min_{0\leq j_1 < j_2\leq n-1}|k_{j_1}-k_{j_2}|}\prod_{j=0}^{n-1} ||f_j||_r,
\end{aligned}
\end{equation}
for $n\geq 1$, $f_0,...,f_{n-1}\in C^r$ and $0\leq k_0\leq...\leq k_{n-1}$.
\item[(EE)] There are $r'>0$ and $0<\delta_1<1$ such that
\begin{equation}\label{ballretasm:ee}
\left|\left|\sum_{j=0}^{n-1} f\circ R^j -n\int f \d\nu\right|\right|_{L^2(\nu)} \leq C||f||_{C^{r'}} n^{\delta_1} \;\;\;\forall f\in C^{r'}, \; \forall n\geq 1.
\end{equation}
\item[(LR$(y^*)$)] There is a $c>0$ such that, for $r>0$ and $\nu$-a.e $y\in B_r(y^*)$, we have
\begin{equation}\label{ballretasm:lr}
\f{B_r(y^*)}(y)\geq c |\log(r)|.
\end{equation}
\item[(SLR$(y^*)$)] There is a $\psi:(0,\infty)\rightarrow (0,\infty)$ with $|\log(r)|=o(\psi(r))$ as $r\rightarrow 0$ such that, for $r>0$ and $\nu$-a.e $y\in B_r(y^*)$, we have
\begin{equation}\label{ballretasm:slr}
\f{B_r(y^*)}(y)\geq \psi(r).
\end{equation}
\item[(NSR$(x^*)$)] There is a $\xi:(0,\infty)\rightarrow (0,\infty)$ with $|\log(r)|=o(\xi(r))$ as $r\rightarrow 0$ such that, for $\nu$-a.e $y\in Y$, we have
\begin{equation}\label{noshortreturns:3}
\mu_{B_r(x^*)}(\f{B_r(x^*),y}\leq \xi(r))\rightarrow 0 \;\;\;\textit{ as }r\rightarrow 0.
\end{equation}
\item[(BR$(x^*,y^*)$)] One of the following is satisfied
\begin{itemize}
\item (SLR$(y^*)$),
\item (NSR$(x^*)$) AND (LR$(y^*)$),
\item or (NSR$(x^*)$) AND (LR$(x^*)$)\footnote{With $\f{B_r(x^*),y}$ instead, this is, again, required for almost all $y\in Y$.}.
\end{itemize}
\end{itemize}
~\\

Colloquially we will also refer to (MEM) as exponential mixing of all orders, and to (EE) as the Quantitative Ergodic Theorem or effective ergodicity. Both are standard assumptions
and have been studied for many classes of systems.\\
Conditions (LR), (SLR), and (NSR) all are concerned with the fact that a points in a small ball $B$ cannot return to $B$ too quickly. Sometimes in literature the center $x^*$ or $y^*$ is referred to as a slowly recurrent point. For technical reason we need to distinguish different versions of slow recurrence, (SLR) being the strongest.

\begin{remark}\label{conditionsrem}
\begin{enumerate}[(i)]
\item In the case $T(x,y)=G_{\tau(y)}(x)$, where $G$ satisfies (a continuous version of) (MEM)\footnote{It is in fact enough to assume exponential mixing.} and $\tau$ is bounded from above and below,
the condition (NSR$(x^*)$) at almost every $x^*$.
It was shown in Lemma 4.14 of \cite{loglargegaps}, albeit for maps instead of flows, that $G$ satisfies condition (NSR$(x^*)$) is satisfied\footnote{It is shown that, for every fixed $K>0$,
\begin{equation*}
\mu_{B_r(x^*)}(\f{B_r(x^*),y}\leq K|\log(r)|)\rightarrow 0 \;\;\;\textit{ as }r\rightarrow 0.
\end{equation*}
But then we also have
\begin{equation*}
\mu_{B_r(x^*)}(\f{B_r(x^*),y}\leq K(r)|\log(r)|)\rightarrow 0 \;\;\;\textit{ as }r\rightarrow 0,
\end{equation*}
for some $K(r)\nearrow\infty$ growing slowly enough. This is a routine argument which is left to the reader.
} at almost every $x^*$. Since $\tau$ is bounded, $T$ also has this property.

\item It is shown in \cite[Lemma 5]{BS} that, for a map of positive entropy, condition (LR) is satisfied at almost every point. This remains true for maps of the form $T(x,y)=G_{\tau(y)}(x)$,
for bounded $\tau$, where $G$ has positive entropy.
\item Considering the previous remarks it may seem unnecessary to state condition (SLR).
Note however that none of the conditions can be satisfied at periodic points, and the maps we want to use for $T$ will have plenty periodic points. $SLR(y^*)$ will be useful to show the PLT \textbf{everywhere}, if we can choose $R$ without periodic points.
\item In most of the examples we will have
\begin{equation*}
\left|\left|\sum_{j=0}^{n-1} f\circ R^j -n\int f \d\nu\right|\right|_{L^2(\nu)} \leq C||f||_{CH^{r'}} n^{\delta_1} \;\;\;\forall f\in H^{r'}, \; \forall n\geq 1.
\end{equation*}
Since $C^{r'}\subset H^{r'}$ and
 $||f||_{H^{r'}}\leq ||f||_{C^{r'}}$ for $f\in C^{r'}$, this implies condition $(EE)$.
\end{enumerate}
\end{remark}

\begin{theorem}\label{plt:1}
Assume that $S(x,y)=(T(x,y),R(y))$ satisfies conditions (MEM), (EE), and (BR$(x^*,y^*)$) for some points $x^*\in X, y^*\in Y$. If
\begin{equation}\label{dimdec}
d>\frac{3}{2}\frac{d'(r'+1)}{1-\delta_1}
\end{equation}
then $S$ satisfies the PLT at $(x^*,y^*)$.
\end{theorem}

\begin{corollary}
If $T$ preserves a smooth measure and satisfies (MEM), and $R$ satisfies (EE), then $T\times R$ satisfies the PLT almost everywhere.
\end{corollary}
If the preserved measure is smooth, then, by \cite{emimpliesbernoulli}, $T$ is Bernoulli, in particular, it has positive entropy. (NSR$(x^*)$) and (LR$(x^*)$) are satisfied almost everywhere by Remark \ref{conditionsrem}.

For some applications it will be useful to choose $T(x,y)=G_{\tau(y)}(x)$, where $\int_Y \tau \d\nu=0$. However in this case $T$ will not satisfy condition (MEM). Fortunately we can apply similar techniques if ergodic averages of $\tau$ grow faster than logarithmically. More explicitly denote $\tau_n=\sum_{j=0}^{n-1} \tau\circ R^j$, assume there is a $\zeta:\mathbb{N}\rightarrow(0,\infty)$ with $\log(n)=o(\zeta(n))$ and a $\kappa>0$ such that
\begin{equation*}\tag{BA}
\nu(|\tau_n|<\zeta(n))\leq O(n^{-\kappa}).
\end{equation*}

\begin{theorem}\label{plt:2}
Assume that $S(x,y)=(T(x,y),R(y))$, where $T(x,y)=G_{\tau(y)}(x)$, satisfies conditions (MEM) with $G$ instead of $T$. Suppose that $R$ satisfies (EE), and $\tau$ satisfies (BA).
Let $x^*\in X$, $y^*\in Y$. If there is a $\delta_2>0$ such that for small enough $r>0$ we have
\begin{equation}\label{bigreturnsinplt:2}
\f{B_r(y^*)}\geq r^{-\delta_2} \;\;\;\textit{on }B_r(y^*),
\end{equation}
and
\begin{equation}\label{dimdec:2}
d>\frac{3}{2}\frac{d'(r'+1)}{1-\delta_1}\;\;\;\textit{ and }\;\;\; \kappa>\frac{d'}{\delta_2}.
\end{equation}
then $S$ satisfies the PLT at $(x^*,y^*)$.
\end{theorem}

\begin{remark}\label{pointwiseEE}
If in the situation of Theorem \ref{plt:1} resp. \ref{plt:2}, in the assumption (EE), we assume a pointwise estimate instead of the $L^2$ bound, we can improve the assumption \eqref{dimdec} resp. \eqref{dimdec:2} on the dimension to
\begin{equation*}
d>\frac{d'(r'+1)}{1-\delta_1}.
\end{equation*}
\end{remark}

\section{Examples}
\label{ScEx}
The definitions of the maps in Examples \ref{explethm:1}, \ref{explethm:2} and Lemma \ref{existstau}  are given in section \ref{explesec}.
For most of the examples we present, the choice of $R$ is more interesting then the choice of $T$, mostly because (MEM) implies chaotic behavior and so the PLT in that setting is not surprising.
 We will thus not focus too much on $T$ for this section. We only present some examples here, there are many others one can verify using Theorem \ref{plt:1}.
\begin{example}\label{explethm:1}
Let $T$ be a map satisfying (MEM) on a manifold of sufficiently high\footnote{Sufficient bounds are given in the proofs.} dimension then
\begin{enumerate}[(i)]
\item if $R$ is a diophantine rotation, then $T\times R$ satisfies the PLT everywhere;
\item if $R$ is the time $1$ map of a horocycle flow on $\Gamma\backslash SL_2(\mathbb{R})$ where
$\Gamma$ is a cocompact lattice, then $T\times R$ satisfies the PLT everywhere;
\item if $R$ is a skew-shift, then $T\times R$ satisfies the PLT everywhere.
\end{enumerate}
\end{example}

\begin{example}
Note that we so far only gave examples of maps that satisfy the PLT everywhere. The reason is that, say if $T$ is Anosov, the condition (BR$(x^*,y^*)$) is automatically satisfied for a.e $x^*$ and all $y^*$. Due to
Remark \ref{conditionsrem}(ii) condition (LR$(x^*)$) is satisfied almost everywhere, furthermore, as shown in \cite{DFL}, $T$ itself satisfies the PLT almost everywhere, and so in particular (NSR$(x^*)$) holds almost everywhere. So in order to apply Theorem~\ref{plt:1} on $T\times R$, for some map $R$, we just have to check (EE), i.e there are $L^2$ bounds on equidistribution in $R$. (EE) can be verified for a big class of maps, in particular it holds if $R$ satisfies a summable decay of correlation. For example certain quadratic maps (see \cite{QuadraticYoung}), intermittent interval maps (see \cite{StatPropLM}),\dots
\end{example}

Theorem \ref{plt:2} can be used to construct $T,T^{-1}$ transformations of zero entropy that satisfy the PLT. All that remains to do is to construct a $\tau$ satisfying (BA),
 this can be done with the construction given in \cite{DDKN}.

\begin{lemma}\label{existstau}
Let $R_{\alpha}:\mathbb{T}^{d'}\rightarrow\mathbb{T}^{d'}$ be a diophantine rotation, i.e
\begin{equation*}\tag{(D)}
\left| \langle k, \alpha \rangle -l\right| >C|k|^{-n} \;\;\;\forall k\in \mathbb{Z}^{d'},k\not=0,l\in\mathbb{Z},
\end{equation*}
for some $n\geq 1$. For $\frac{n}{2}<\rho<d'$ there is a $d\geq 1$ and a function $\tau\in C^{\rho}(\mathbb{T}^{d'},\mathbb{R}^d)$ such that $\nu(\tau)=0$, while
\begin{equation*}
\nu(||\tau_n||<\log^2(n))=o(n^{-5}).
\end{equation*}
\end{lemma}

Note that in order to apply Theorem \ref{plt:2} we can always make $d$ as big as we want.

\begin{example}\label{explethm:2}
Let $R=R_{\alpha}$ be a diophantine rotation with $d'=2$ and $n=2$, $\tau$ be the function from Lemma \ref{existstau}, and let $G$ be the Weyl Chamber flow on $SL(d,\mathbb{R}) / \Gamma$, where $\Gamma$ is a uniform lattice. If $d>10$ then $S(x,y)=(G_{\tau(y)}(x),R_{\alpha}(y))$ satisfies the PLT everywhere.
\end{example}

\section{The delayed PLT}

The main tool in the proof will be a PLT, along a subsequence, what we will call a delayed PLT.
This statement in itself is of independent interest, so let us state a version of this here. Let $\alpha=(\alpha^{(n)})_{n\geq 1}$ be a sequence of natural numbers,
we will refer to $\alpha$ as the \textit{delay sequence}, and denote
$\displaystyle \tilde{\alpha}^{(n)}=\sum_{j=1}^n \alpha^{(n)}$. For measurable $A\subset X$ we define the \textit{delayed first return time to $A$ along $\alpha$} as
\begin{equation}\label{retdef:delayed}
\begin{aligned}
&\f{A,\alpha}^{(1)}(x):=\f{A,\alpha}(x):=\min(j\geq 1\;|\; T^{\tilde{\alpha}^{(j)}}(x)\in A)\\
&\f{A,\alpha}^{(n+1)}(x):=\min(j\geq \f{A,\alpha}^{(n)}\;|\;T^{\tilde{\alpha}^{(j)}}(x)\in A)\\
&\Phi_{A,\alpha}:=(\f{A,\alpha}^{(1)},\f{A,\alpha}^{(2)},...).
\end{aligned}
\end{equation}

The main example in this paper will be $\alpha^{(n)}=\varphi_B^{(n)}$, however other choices are possible, for example $\alpha^{(n)}=g(n)$, where $g$ is a polynomial, or $\alpha^{(n)}=p_n$, where $p_n$ is the $n$th prime. \\
Given a rare sequence $\s{A}{l}$, we will distinguish between two cases
\begin{itemize}
\item[1)] the delay sequence is fixed in $l$,
\item[2)] the delay sequence is allowed to vary with $l$.
\end{itemize}

\begin{definition}
\begin{enumerate}[(i)]
\item Let $x^*\in X$ and, for $r>0$, denote by $B_r(x^*)$ the geodesic ball of radius $r$ centred at $x^*$.
Let $(\alpha^{(n)})_{n\geq 1}$ be a sequence of natural numbers. We will say that \textit{$T$ satisfies the delayed PLT along $\alpha$ at $x^*$} if
\begin{equation*}
\mu(B_r(x^*))\Phi_{B_r(x^*),\alpha}\xRightarrow{\mu} \Phi_{Exp} \;\;\;\textit{ as }r\rightarrow 0.
\end{equation*}
\item Let
$PLT(\alpha)=\{x^*\in X \;|\; T$  satisfies the delayed PLT along $\alpha$ at $x^*\}.
$

If $x^*\in PLT(\alpha)$ for all sequences of natural numbers $\alpha$, then we say that \textit{$T$ satisfies the delayed PLT at $x^*$}.
\end{enumerate}
\end{definition}

In case 1) the main result is a straightforward modification of Theorems \ref{plt:1} and \ref{plt:2} and hence will not be proven in detail
\begin{theorem}\label{dplt:1}
Let $\alpha$ be a sequence of positive integers. Assume the conditions of Theorem \ref{plt:1} or \ref{plt:2} hold replacing (EE) in both cases by
\begin{equation*}
\left|\left|\sum_{j=0}^{n-1} f\circ R^{\tilde{\alpha}^{(j)}} -n\int f \d\nu\right|\right|_{L^2(\nu)} \leq C||f||_{C^{r'}} n^{\delta_1} \;\;\;\forall f\in C^{r'}, \; \forall n\geq 1.
\end{equation*}
Then $S$ satisfies the delayed PLT along $\alpha$ at $(x^*,y^*)$.
\end{theorem}

The proof is analogous to the proof of Theorem \ref{plt:1} resp. \ref{plt:2}
(with $\Phi_{B_l}$ replaced by $\Phi_{B_l,\alpha}$ and Lemma \ref{weakapproxlem} replaced by Lemma \ref{robustnesslem}).
Therefore no detailed proof will be given
\footnote{Note however that there is no relation of delayed hitting times to delayed return times as in Proposition \ref{hitretrelcor}.}.

In case 2) the main result is a special case of Proposition \ref{delaykapretprop}(III), however it is worth stating by itself.
\begin{theorem}\label{dplt:2}
Suppose $T$ satisfies conditions (MEM), (SLR$(x^*)$) and (NSR$(x^*)$), then $T$ satisfies the delayed PLT at $x^*$.
\end{theorem}

\part{Proofs}

\section{PLT along varying subsequences}\label{PLTsubssec}
\subsection{Approximation}
In our work we often need to apply the quantitative ergodicity (EE), and mixing condition (MEM), for indicator functions, hence we have to approximate them by functions in $C^r$.
\begin{definition}
\label{DefApprox}
Let $M$ be a $C^r$ manifold with dimension $dim(M)=d$, $\lambda$ be a measure on $M$.

$\bullet$ Let $B\subset M$ be a measurable subset. We say that $B$ is \textit{regularly approximable in $C^r$} if there is a constant $C>0$ such that, for $0<\epsilon<\frac{1}{10}\lambda(B)^{\frac{1}{d}}$, there are $\overline{h},\underline{h}\in C^r$ with $\underline{h}\leq 1_B\leq \overline{h}$ and
for $h\in \{\overline{h},\underline{h}\}$
\begin{equation}\label{regapproxeq}
\displaystyle \lambda(1_B\not= h)\leq\lambda(B)^{\frac{d-1}{d}}\epsilon, \;\textit{ while }\; ||h||_{C^r}\leq C\epsilon^{-r}.
\end{equation}

$\bullet$ Let $\{B_{\alpha}\}$ be measurable subsets of $M$. We say that $\{B_{\alpha}\}$ is \textit{regularly approximable in $C^r$} if there is a constant $C>0$ such that each $B_{\alpha}$ is regularly approximable in $C^r$ with the same constant $C$.
\end{definition}
\begin{lemma}\label{approxlem:2}
Let $M$, $r$, $d$ and $\lambda$ be as in Definition \ref{DefApprox}.
Assume in addition  that $\lambda$ is absolutely continuous w.r.t volume with bounded density. Suppose that $\bigcup_{\alpha} B_{\alpha}$ is relatively compact, and there is an open $U\supset \overline{\bigcup_{\alpha} B_{\alpha}}\supset\bigcup_{\alpha} B_{\alpha}$ and a $C^r$-diffeomorphism $\iota:U\rightarrow V$ for some open set $V\subset \mathbb{R}^d$ such that each $\iota(B_{\alpha})$ is a ball. Then $\{B_{\alpha}\}$ is regularly approximable in $C^r$.
\end{lemma}

\begin{proof}
(i) We may assume that $M=\mathbb{R}^d$ and $\lambda$ is Lebesgue measure, otherwise we pick up another constant, which can be absorbed into $C$. Furthermore we can assume that all $B_{\alpha}$ are balls centered at the origin. In the following fix $\alpha$ and, dropping the $\alpha$ from our notation, let $B=B_t(0)$. We have $\lambda(B)=C_1 t^d$, where $C_1$ is the volume of the $d$-dimensional unit ball.
\\

\noindent(ii) Let $\theta:\mathbb{R}\rightarrow [0,1]$ be a smooth function with $\theta(x)=1$ if $x<0$, and $\theta(x)=0$ if $x>1$. For $t>\epsilon>0$ consider
\begin{equation*}
\hat{\theta}(x)=\theta(\epsilon^{-1}(x-t))
\end{equation*}
then $\hat{\theta}$ is still smooth and $||\hat{\theta}||_{C^r}=\epsilon^{-r}||\theta||_{C^r}$. Consider
$\overline{h}:\mathbb{R}^d\rightarrow[0,1]$ given by
$\displaystyle
\overline{h}(x)=\hat{\theta}(|x|),$
then
\begin{itemize}
\item $\overline{h}$ is smooth, away from the origin because it is the composition of smooth functions, and near the origin it is constant $1$,
\item $\overline{h}(x)=1$ if $|x|<t$ and $\overline{h}(x)=0$ if $|x|>t+\epsilon$,
\item and $||\overline{h}||_{C^r}\leq C_3 \epsilon^{-r}$
where $C_3=rC_2||\theta||_{C^r}$ and
$C_2$ is the $C^r$ norm of the smooth function $x\mapsto |x|$ on $\{t\leq |x|\leq t+\epsilon\}$.
\end{itemize}
Furthermore we have
\begin{align*}
\lambda(\overline{h}\not = 1_B)&=\lambda(t\leq |x|\leq t+\epsilon)=C_1((t+\epsilon)^d-t^d)\\
&\leq C_1 d t^{d-1}\epsilon\leq d C_1^{\frac{1}{d}} \lambda(B)^{\frac{d-1}{d}} \epsilon,
\end{align*}
all the constants can be absorbed in the constant $C$ from the claim, the constant only depends or $r,d$. \\
For $\underline{h}$ repeat the calculations with $\hat{\theta}=\theta(\epsilon^{-1}(x-(t-\epsilon)))$ instead.
\end{proof}

\subsection{Proof of the PLT along varying subsequences}
For our purposes it will not be enough to consider the delayed PLT for a single rare sequence $\s{A}{l}$, rather let $K\geq 1$, and $A_l^{(1)},...,A_l^{(K)}$ be subsets of $X$ such that $\{A_l^{(k)}\}$ is regularly approximable in $C^r$. Assume that there are $\omega^{(1)},...,\omega^{(K)}>0$ and $r_l\rightarrow 0$ such that
\begin{equation}
\label{OmegaK}
\mu(A_l^{(k)})=\omega^{(k)} r_l^d + o(r_l^d).
\end{equation} Given $\kappa_l^{(n)}\in \{1,...,K\}$, for $l,n\geq 1$, define
\begin{align*}
&\phi_{\kappa_l,\alpha_l,y}^{(1)}(x)=\min(n\geq 1\;|\; T_y^{\tilde{\alpha}_l^{(n)})}(x)\in A_l^{(\kappa_l^{(n)})}),\\
&\phi_{\kappa_l,\alpha_l,y}^{(m+1)}(x)=\min(n\geq \phi_{\kappa_l,\alpha_l,y}^{(m)}(x)+1\;|\; T_y^{\tilde{\alpha}_l^{(n)}}(x)\in A_l^{(\kappa_l^{(n)})}),\\
&\Phi_{\kappa_l,\alpha_l,y}=(\phi_{\kappa_l,\alpha_l,y}^{(1)},\phi_{\kappa_l,\alpha_l,y}^{(2)},...).
\end{align*}
Denote
\begin{equation*}
p_{l,t}^{(k)}:=\frac{1}{t}\#\{j=1,...,t_l\;|\;\kappa_l^{(j)} =k\}.
\end{equation*}

For now suppose that there are positive constants $\theta^{(k)}>0$ such that, for all $k=1,...,K$,

\begin{equation}\label{kapprobcon}
p_{l,t_l}^{(k)} \rightarrow \frac{\theta^{(k)}}{\sum_{j=1}^K \theta^{(j)}}=:p^{(k)}\;\;\; \as{l}.
\end{equation}
Later on, when we will prove PLT in product systems, we will choose $\kappa$ and $\theta$ in a specific way\footnote{Say we want to prove a PLT for the system $T\times R$ and sets of the form $B_r(x^*)\times B_r(y^*)$ then we choose $K=1$ and $\theta>0$ such that $\nu(B_r(y^*))=\theta r^{d'} + o(r^{d'})$.}, details shall be given in section \ref{scalePLTsec}, see in particular Lemma \ref{kaplem}.
\\
The main estimate of mixing rates for regularly approximable sets is the following.

\begin{lemma}\label{decorrlem}
Let $m\geq 1$, $A^{(1)},...,A^{(k)}\subset X$ be regularly approximable in $C^r$,  and $1\leq n_1<...<n_k$, then
\begin{equation*}
\left|\mu\left(\bigcap_{i=1}^k T^{-n_i}A^{(i)}\right)-\prod_{i=1}^k \mu (A^{(i)})\right|\leq K
\max_{i=1,...,k} \mu(A^{(i)})^{\frac{d-1}{d} \frac{kr}{kr+1}}
e^{-\frac{\gamma p}{kr+1} },
\end{equation*}
where $\displaystyle p=\min_{i=1,...,k-1} |n_{i+1}-n_i|$ and the constant $K>0$ only depends on $k$ and the constant in the definition of regularly approximable.
\end{lemma}
\begin{proof}

By Lemma \ref{approxlem:2}, for every $\epsilon>0$, there are $h^{(i)}\in C^r$ such that $0\leq h^{(i)}\leq 1_{A^{(i)}}$ and
\begin{equation*}
\mu(1_{A^{(i)}}\neq h^{(i)})\leq \mu({A^{(i)}})^{\frac{d-1}{d}}\epsilon, \; \textit{ while } ||h^{(i)}||_r<C\epsilon^{-r}.
\end{equation*}
We estimate
\begin{align*}
\left|\mu\left( \bigcap_{i=1}^k T^{-n_i}{A^{(i)}}\right)-\prod_{i=1}^k \mu (A^{(i)})\right|&\leq \left|\mu\left( \bigcap_{i=1}^k T^{-n_i}{A^{(i)}}\right)-\int_X \prod_{i=1}^k h^{(i)}\circ T^{n_i}\d\mu \right|\\
&+\left| \int_X \prod_{i=1}^k h^{(i)}\circ T^{-n_i}\d\mu - \prod_{i=1}^k \int_X  h^{(i)}\circ T^{n_i}\d\mu \right|\\
&+ \left|\prod_{i=1}^k \int_X h^{(i)}\circ T^{n_i}\d\mu -\prod_{i=1}^k \mu (A^{(i)})\right|\\
&\leq 4k \max_{i=1,...,k} \mu(A^{(i)})^{\frac{d-1}{d}}\epsilon +\tilde{C}e^{-\gamma p} \epsilon^{-kr}.
\end{align*}
This bound is minimised for
\begin{equation*}
\epsilon^*=\left( \frac{\tilde{C}r}{4k} \max_{i=1,...,k} \mu(A^{(i)})^{-\frac{d-1}{d}} e^{-\gamma p}\right)^{\frac{1}{kr+1}},
\end{equation*}
so
$$
\left|\mu\left( \bigcap_{i=1}^k T^{-n_i}A^{(i)}\right)-\prod_{i=1}^k \mu (A^{(i)})\right|\leq \hat{C}\max_{i=1,...,k} \mu(A^{(i)})^{\frac{d-1}{d} \left(1-\frac{1}{kr+1}\right)} e^{-p\frac{\gamma}{kr+1}}
$$$$
+ \bar{C}\max_{i=1,...,k} \mu(A^{(i)})^{\frac{kr}{kr+1}\frac{d-1}{d}} e^{-\gamma p\left(1-\frac{kr}{kr+1}\right)}$$
$$\leq K \max_{i=1,...,k} \mu(A^{(i)})^{\frac{d-1}{d} \frac{kr}{kr+1}} e^{-\frac{\gamma}{kr+1} p},
$$
where the constant $K>0$ only depends on the constant in the definition of regularly approximable.
\end{proof}

\begin{proposition}\label{delaykapretprop}
Suppose that $T$ satisfies (MEM), $A_l^{(1)},...,A_l^{(K)}\subset X$, for $K\geq 1$
$\kappa_l$ satisfy \eqref{kapprobcon}
for some $p^{(k)}>0$ with $\displaystyle \sum_{k=1}^K p^{(k)}=1$, and let $\alpha_l$ be sequences of natural numbers. Denote $A_l=\bigcup_{k=1}^K A_l^{(k)}$ and suppose that either
\begin{enumerate}[(I)]
\item $\alpha_l$ grows faster than $|\log\mu(A_l)|$ in the sense that
\begin{equation}
\label{GapLogLarge}
|\log\mu(A_l)|=o(\min_{i\geq 2}|\alpha^{(i)}_{l}|),
\end{equation}
\item short returns to $A_l$ are rare in the sense that
\begin{equation}\label{noshortreturns:2}
\mu_{A_l}(\f{A_l}\leq a_l)\rightarrow 0 \;\;\;\as{l},
\end{equation}
for some sequence $\s{a}{l}$ with $|\log(\mu(A_l))|=o(a_l)$, and $\alpha_l$ grows at least as fast $|\log\mu(A_l)|$ in the sense that
\begin{equation}
\label{GapLogLarge:2}
|\log\mu(A_l)|=O(\min_{i\geq 2}|\alpha^{(i)}_{l}|),
\end{equation}
\item or short returns to $A_l$ are rare in the sense that
\begin{equation*}
\mu_{A_l}(\f{A_l}\leq a_l)\rightarrow 0 \;\;\;\as{l},
\end{equation*}
for some sequence $\s{a}{l}$ with $|\log(\mu(A_l))|=o(a_l)$, and
returns are at least logarithmically large, i. e there exists $c>0$ such that
\begin{equation*}
\f{A_l}(x) \geq c \log(\mu(A_l)) \;\;\;\textit{a.e } x\in A_l.
\end{equation*}
\end{enumerate}
Then
\begin{equation*}
\Omega r_l^d \Phi_{\kappa_l,\alpha_l,y}\xRightarrow{\mu} \Phi_{Exp} \;\;\;\as{l},
\end{equation*}
where $\displaystyle \Omega=\sum_{k=1}^K \omega^{(k)} p^{(k)}$ and $\omega^{(k)}$ come from \eqref{OmegaK}.
\end{proposition}
\begin{proof}

(i) Taking a subsequence if necessary, we may assume that there is a $[0,\infty]$-valued process $\Phi=(\phi^{(1)},\phi^{(2)},...)$ such that
\begin{equation*}
\Omega r_l^d \Phi_{\kappa_l,\alpha_l}\xRightarrow{\mu} \Phi \;\;\;\as{l}.
\end{equation*}
Denote $\iota:[0,\infty)^{\mathbb{N}}\rightarrow [0,\infty)^{\mathbb{N}}$ the map
\begin{equation*}
\iota(x_1,x_2,x_3,...)=(x_1,x_1+x_2,x_1+x_2+x_3,\dots).
\end{equation*}
Since $\iota$ is a homeomorphism, showing $\Phi\overset{d}{=}\Phi_{Exp}$ is equivalent to $\iota(\Phi)\overset{d}{=}\iota(\Phi_{Exp})$, where we use the obvious extension of $\iota$ to $[0,\infty]^{\mathbb{N}}$. To show the latter, for $J\geq 1$ and $0<t_1<...<t_J$, we show
\begin{equation}\label{waitingcond}
\mathbb{P}\left(\sum_{j=1}^k \phi^{(j)} \leq t_k, \; k=1, \dots, J\right)=\mathbb{P}\left(\sum_{j=1}^k \phi^{(j)}_{Exp} \leq t_k, \; k=1,\dots ,J\right).
\end{equation}
The trick is is to look at the "dual" object
\begin{equation*}
S_{\kappa_l,\alpha_l}^{(n)}=\sum_{j=1}^n 1_{A_l^{\kappa_l^{(j)}}}\circ T^{\tilde{\alpha}_l^{(j)}}.
\end{equation*}
The important relation here is the following
\begin{equation}
\label{Renewal}
S_{\kappa,\alpha_l}^{(n)}\geq N \; \iff\; \sum_{i=1}^N \f{\kappa,\alpha_l}^{(i)}\leq n.
\end{equation}
Indeed, evaluating both sides at some $x\in X$, the left side says that there are at least $N$ different times $1\leq j_1<...<j_N\leq n$ such that $T^{\tilde{\alpha}_l^{(j)}}(x)\in A_l^{\kappa_l^{(j)}}$. The right hand side expresses that, if $1\leq j_1<...<j_N$ are the first $N$ times that $T^{\tilde{\alpha}_l^{(j)}}(x)\in A_l^{\kappa_l^{(j)}}$, then $\displaystyle j_1+ \sum_{i=2}^N (j_{i}-j_{i-1}) \leq n$.
Let $(P_t)_{t\geq 0}$ be a Poisson process, such that $\Phi_{Exp}$ are the waiting times of $(P_t)$, i.e $(P_t)$ and $\Phi_{Exp}$ are related by
\begin{equation*}
P_t\geq k\iff \sum_{j=1}^k \phi^{(j)}_{Exp} \leq t.
\end{equation*}
The right side of \eqref{waitingcond} is equal to
\begin{equation*}
\mathbb{P}\left(\sum_{j=1}^k \phi^{(j)}_{Exp} \leq t_k, \; k=1,...,J\right)=\mathbb{P}(P_{t_k}\geq k, \; k=1,...,J).
\end{equation*}
Due to \eqref{Renewal} it is enough to show
\begin{equation*}
\left(S_{\kappa_l,\alpha_l}^{\ffloor{t_1}{\Omega r_l^d}},S_{\kappa_l,\alpha_l}^{\ffloor{t_2}{\Omega r_l^d}},...,S_{\kappa_l,\alpha_l}^{\ffloor{t_d}{\Omega r_l^d}}\right)\xRightarrow{\mu} (P_{t_1},...,P_{t_{J}}) \;\;\;\as{l}.
\end{equation*}

(ii)
Taking a further subsequence if necessary, there are $[0,\infty]$-valued $\tilde{P}_{t_1},...,\tilde{P}_{t_d}$ such that
\begin{equation*}
\left(S_{\kappa_l,\alpha_l}^{\ffloor{t_1}{\Omega r_l^d}},S_{\kappa_l,\alpha_l}^{\ffloor{t_2}{\Omega r_l^d}},...,S_{\kappa_l,\alpha_l}^{\ffloor{t_J}{\Omega r_l^d}}\right)\xRightarrow{\mu} (\tilde{P}_{t_1},...,\tilde{P}_{t_J}) \;\;\;\as{l}.
\end{equation*}
We will show that
\begin{enumerate}[(A)]
\item $\tilde{P}_{t_k}-\tilde{P}_{t_{k-1}}$ is Poisson distributed with intensity $t_k-t_{k-1}$ for $k=1,...,d$,
\item and
$
(\tilde{P}_{t_1}-\tilde{P}_{t_0},\tilde{P}_{t_2}-\tilde{P}_{t_1},...,\tilde{P}_{t_J}-\tilde{P}_{t_{J-1}})
$
is an independent vector,
where $t_0=0$.\footnote{This is essentially Watanabe's characterisation of Poisson-processes.}
\end{enumerate}
Clearly $P_0=\tilde{P}_0=0$.\\
For $j=1,\dots,J$
denote $S_{t_j,l}=S_{A_l,\kappa_l}^{\ffloor{t_j}{\Omega r_l^d}}$. Assertions (A) and (B) will follow\footnote{Here we apply the method of moments, see eg \cite{Billprobaandmeasure}.}
once we show that, for all $m_1,...,m_d\geq 1$,
\begin{equation*}
\int_X \prod_{j=1}^{J} {{S_{t_j,l}-S_{t_{j-1},l}} \choose m_j} \d\mu = \prod_{j=1}^{J} \frac{(t_j-t_{j-1})^{m_j}}{m_j!} +o(1) \;\;\;\as{l}.
\end{equation*}
In the rest of the proof fix $J\geq 1$, $0=t_0<t_1<\dots<t_J$ and $m_1,\dots,m_J\geq 1$.\\

(iii)  First, for each $j=1,\dots,J$, rewrite
\begin{equation*}
S_{t_j,l}-S_{t_{j-1},l}=\sum_{i=\ffloor{t_{j-1}}{\Omega r_l^d}+1}^{\ffloor{t_j}{\Omega r_l^d}}
1_{A_l^{(\kappa_l^{(i)})}}\circ T^{\tilde{\alpha}_l^{(i)}}.
\end{equation*}
So
\begin{equation}\label{slbinomprod}
\prod_{j=1}^{J} {{S_{t_j,l}-S_{t_{j-1},l}} \choose m_j}=\prod_{j=1}^d \sum_{\ffloor{t_{j-1}}{\Omega r_l^d}+1\leq k_{1,j} <...< k_{m_j,j} \leq \ffloor{t_j}{\Omega r_l^d}} \prod_{i=1}^{m_j} \xi_{i,j}
\end{equation}
where $\xi_{i,j}=1_{A_l^{(\kappa_l^{(k_{i,j})})}}\circ T^{\tilde{\alpha}_l^{({k_{i,j})}}}$.
To simplify notation we will also denote $m=(m_1+\dots+m_J)$, $\displaystyle \omega=\min_{k=1,...,K} \omega^{(k)}$, $p_l=2m\frac{mr+1}{\gamma}\left|\log( \omega r_l^d)\right|$,
$$ \Delta_l:=\{\textbf{k}=(k_{i,j})_{\substack{j=1,...,J\\i=1,...,m_j}} \;\Big|\; \ffloor{t_{j-1}}{\Omega r_l^d}+1\leq k_{1,j}\!\!<\!\!\dots\!\!<\!\! k_{m_j,j} \leq\!\! \ffloor{t_j}{\Omega r_l^d} \;\textit{ for } j=1,...,d\}$$  and
$$ \Delta'_l:=\{\textbf{k}\in \Delta_l \;|\;\min_{\substack{j=1,...,J, \; i=1,..., m_j\\ j'=1,...,J, \; i'=1,..., m_{j'},\; (j,i)\not=(j',i')}} |\tilde{\alpha}_l^{k_{i,j}}-\tilde{\alpha}_l^{k_{i',j'}}|\leq p_l\} $$
We will split the sum in \eqref{slbinomprod} into two terms
\begin{equation*}
\prod_{j=1}^d {{S_{t_j,l}-S_{t_{j-1},l}} \choose m_j} = M_l+R_l,
\end{equation*}
where
\begin{equation*}
M_l =M_{(t_j),l,m}= \sum_{\textbf{k}\in \Delta_l\setminus \Delta'_l} \prod_{\substack{j=1,...,d \\ i=1,..., m_j}}
\xi_{i,j}
,\quad
R_l =R_{(t_j),l,m} =\sum_{\textbf{k}\in \Delta'_l} \prod_{\substack{j=1,...,d \\ i=1,..., m_j}}
\xi_{i,j}.
\end{equation*}
We will show that
\begin{equation}\label{mlrlcon}
\int_X M_l \d\mu \rightarrow \prod_{j=1}^{J} \frac{(t_j-t_{j-1})^{m_j}}{m_j!} \;\textit{ and }\; \int_X R_l\d\mu \rightarrow 0 \;\;\;\as{l}.
\end{equation}

(iv)
Let us first treat $M_l$. For $l\geq 1$ and $\textbf{k}\in \Delta_l$, by Lemma \ref{decorrlem} we have
$$
\left|\mu\left(\bigcap_{\substack{j=1,...,d \\ i=1,..., m_j}} T^{-\tilde{\alpha}_l^{k_{i,j}}}
A_l^{(\kappa_l^{(k_{i,j})})}\right)-\prod_{j,i} \mu \left(A_l^{(\kappa_l^{(k_{i,j})})}\right)\right|
$$$$
\leq K \max_{j,i} \mu\left(A_l^{(\kappa_l^{(k_{i,j})}}\right)^{\frac{d-1}{d} \frac{mr}{mr+1}}
e^{-\frac{\gamma \min_{i,j} \alpha_l^{k_{i,j}}}{mr+1} }.
$$
For $\textbf{k}\in \Delta_l\setminus \Delta'_l$, this yields
$$
\mu\left(\bigcap_{\substack{j=1,...,d \\ i=1,..., m_j}} T^{-\tilde{\alpha}_l^{k_{i,j}}}
A_l^{(\kappa_l^{(k_{i,j})})}\right)
=r_l^{md}\prod_{\substack{j=1,...,d \\ i=1,..., m_j}} \omega_l^{(\kappa_l^{k_{i,j}})} +o(r_l^{md}),
$$
and the $o$-term does not depend on $\textbf{k}$. Summing over $\textbf{k}\in \Delta_l\setminus \Delta'_l$, and using \eqref{kapprobcon}, yields
\begin{align*}
\int_X M_l \d\mu&=\int_X \sum_{\textbf{k}\in \Delta_l\setminus \Delta'_l} \prod_{\substack{j=1,...,d \\ i=1,..., m_j}} 1_{A_l^{(\kappa_l^{(k_{i,j})})}}\circ T^{\tilde{\alpha}_l^{({k_{i,j})}}} \d\mu\\
&= r_l^{md} \sum_{\textbf{k}\in \Delta_l\setminus \Delta'_l} \prod_{\substack{j=1,...,d \\ i=1,..., m_j}} \omega^{(\kappa_l^{k_{i,j}})} +o(1)\\
&\overset{*}{=}r_l^{md} \prod_{j=1}^J \frac{1}{m_j !} \left(\sum_{k=1}^{K} \ffloor{t_j-t_{j-1}}{\Omega r_l^d} p^{(k)}_{l, \ffloor{t_j-t_{j-1}}{\Omega r_l^d}} \omega^{(k)} \right)^{m_j} +o(1)\\
&=\Omega^{-m} \prod_{j=1}^J \frac{(t_j-t_{j-1})^{m_j}}{m_j !} \left(\sum_{k=1}^K p^{(k)} \omega^{(k)}\right)^{m_j} +o(1)\\
&=\prod_{j=1}^{J} \frac{(t_j-t_{j-1})^{m_j}}{m_j!} +o(1),
\end{align*}
where, in $*$, we consider the unordered sum, i.e the sum over all $(k_{i,j})$ with $\ffloor{t_{j-1}}{\Omega r_l^d}+1\leq k_{i,j} \leq \ffloor{t_j}{\Omega r_l^d}$ and
\footnote{This condition is analogous to $\Delta_l\setminus \Delta'_l$, but without requiring $k_{i,j}<k_{i+1,j}$.}
 \begin{equation*}
\min_{\substack{j=1,...,J, \; i=1,..., m_j\\ j'=1,...,J, \; i'=1,..., m_{j'},\; (j,i)\not=(j',i')}} |\tilde{\alpha}_l^{k_{i,j}}-\tilde{\alpha}_l^{k_{i',j'}}|> p_l.
\end{equation*}
This shows the first assertion of \eqref{mlrlcon}.

(v)
In order to treat $R_l$, first note that under assumption $(I)$ we have $R_l=0$ for big enough $l$.
In the following we focus on assumptions $(II)$ and $(III)$. Note that
\begin{equation*}
\int_X R_l \d\mu=\int_X 1_{R_l\neq 0} R_l \d\mu\leq \mu(\text{supp}(R_l)) \|R_l\|_{L^2}
\end{equation*}
and
\begin{equation*}
\text{supp}(R_l) \subset \bigcup_{s=1}^{\ffloor{t_J}{\Omega r_l^d}} T^{-\tilde{\alpha}_l^{(j)}} \left(A_l  \cap \{\f{A_l}\leq 3p_l\}\right) =:U_l,
\end{equation*}
hence
\begin{equation*}
\mu(\text{supp}(R_l))\leq \mu(U_l) = O(r_l^{-d})\mu(A_l)o(1)=o(1).
\end{equation*}
Therefore,
in order to show $\int_X R_l \d\mu\rightarrow 0$, it is enough to show that $(R_l)_{l\geq 1}$ is bounded in $L^2$. Notice that
\begin{equation*}
R_l^2\leq \sum_{1\leq k_1,...,k_{2m} \leq \ffloor{t_J}{\Omega r_l^d}} \prod_{i=1}^{m} 1_{A_l^{(\kappa_l^{(k_i)})}} \circ T^{\tilde{\alpha}_l^{(i)}}=S_{t_J,l}^{2m}.
\end{equation*}
We may write
\begin{equation*}
S_{t_J,l}^{2m}= \sum_{k=1}^{2m} \stirling{2m}{k} {S_{t_J,l} \choose k} \leq C_m \sum_{k=1}^{2m} {S_{t_J,l} \choose k} \leq C_m \sum_{k=1}^{2m} (M_{t_J,l,k}+R_{t_J,l,k}),
\end{equation*}
where $\stirling{2m}{k}$ are the Stirling numbers of the second kind and
 $\displaystyle C_m=\!\!\max_{k=1,...,2m} \stirling{2m}{k}$. Now the previous parts of the proof show that
\begin{equation*}
\int_X M_{t_J,l,k} \d\mu \;\;\; \textit{is bounded as }l\rightarrow \infty, \textit{ for each fixed }k\geq 1,
\end{equation*}
it remains to show that
\begin{equation*}
\int_X R_{t_J,l,k} \d\mu \;\;\; \textit{is bounded as }l\rightarrow \infty, \textit{ for each fixed }k\geq 1.
\end{equation*}

(vi)
In order to
bound $\int_X R_{t_J,l,m} \d\mu $ for fixed $m$
we first split up $\Delta'_l$ (we adapt all of the notation above for $R_l=R_{t_J,l,m}$, $t=t_J$) into
\begin{equation*}
\Delta_l^{(j)}:=\left\{\textbf{k}=(k_1,...,k_{m})\in \tilde{\Delta}_l\;\left|
\begin{aligned}
&\exists 1\leq i_i < ...< i_{m-j}\leq m \textit{ such that }\\
& |\tilde{\alpha}_l^{k_{i+1}} - \tilde{\alpha}_l^{k_{i}}|\leq \tilde{p}_l \; \forall i\in\{1,...,m\}\setminus\{i_1,...,i_{m-j}\},\\
&\textit{and } |\tilde{\alpha}_l^{k_{i_r+1}}-\tilde{\alpha}_l^{k_{i_r}}|> \tilde{p}_l \; \forall r=1,...,m-j
\end{aligned}
\right. \right\},
\end{equation*}
for $j=1,...,m-1$, so that $\Delta'_l=\bigcup_{j=1}^{m-1} \Delta_l^{(j)}$. Under assumption $(II)$
there is a constant $c>0$ such that for $\rho=2m\frac{mr+1}{c \gamma}$ and big enough $l$ we have
\begin{equation*}
|\Delta_l^{(j)}|\leq {{m}\choose j} {\ffloor{t}{\Omega r_l^d} \choose {{m}-j}} \rho^j\leq m^{m} t^{m}\rho^{m} (\Omega^{-{m}}+1) r_l^{d({m}-j)}.
\end{equation*}
On the other hand, for $\textbf{k}\in \Delta_l^{(j)}$, we can again use Lemma \ref{decorrlem} to estimate
\begin{equation*}
\int_X \prod_{s=1}^{m}  \xi_{s}\d\mu\leq \int_X \prod_{s\in \{1,...,{m}\}\setminus\{i_1,...,i_j\}} \xi_{s} \leq \Omega r_l^{d({m}-j)} +o(r_l^{d({m}-j)}),
\end{equation*}
where
\begin{equation*}
\xi_s=1_{A_l^{\kappa_l^{(k_s)}}}\circ T^{\tilde{\alpha}_l^{(k_s)}}.
\end{equation*}
Summing up we obtain
\begin{equation*}
\int_X R_l \d\mu \leq 2({m})^{{m}+1} t^{m}\rho^{m} (\Omega^{-{m}+1}+\Omega),
\end{equation*}
and we conclude $\int_X R_l \d\mu \rightarrow0$ under assumption $(II)$.

(vii) Finally assume $(III)$. For fixed $j=1,...,m-1$, $l\geq 1$ and $1\leq i_1<...<i_{j}\leq \ffloor{t}{\Omega r_l^d}$ with $\displaystyle \min_{s=1,...,j-1} |\tilde{\alpha}_l^{(i_{s+1})}-\tilde{\alpha}_l^{(i_s)}|\geq p_l$ set
\begin{equation*}
A_{i_1,...,i_j}=\bigcap_{s=1}^j T^{-\tilde{\alpha}_l^{(i_s)}} A_l^{\kappa_l^{(i_s)}}.
\end{equation*}
By Lemma \ref{decorrlem}
\begin{equation*}
\mu(A_{i_1,...,i_j})\leq \omega r_l^{dj} +o(r_l^{dj}),
\end{equation*}
and the $o$ term doesn't depend on $(i_1,...,i_j)$.
For $x\in A_{i_1,...,i_j}$ consider
\begin{equation*}
\mathfrak{K}_{i_1,...,i_j}(x)=\left\{\textbf{k}=(k_1,...,k_m)\in \Delta_l^j\;\left| \begin{aligned}
&\exists r_1,...,r_j\; \textit{ such that } k_{r_s}=i_s \; \forall s=1,...,j, \\
&\textit{ and }\; \prod_{s=1}^{m}  \xi_{s}(x)=1
\end{aligned} \right\} \right..
\end{equation*}
Then, since $\f{A_l}(x) \geq c \log(\mu(A_l))$, we have
\begin{equation*}
|\mathfrak{K}_{i_1,...,i_j}(x)|\leq \left(2\frac{p_l}{dc\log(r_l)}\right)^{m-j}=:\rho^{m-j}.
\end{equation*}
At the same time we have
\begin{equation*}
\text{supp}\left( \prod_{s=1}^{m}  \xi_{s}\right) \subset A_{i_1,...,i_j} \;\;\;\textit{ for } \textbf{k}\in \mathfrak{K}_{i_1,...,i_j}(x) \; \textit{ for some } x.
\end{equation*}
Also every $\textbf{k}\in \tilde{\Delta}_l^{(j)}$ is in some $\mathfrak{K}_{i_1,...,i_j}(x)$, therefore
\begin{align*}
\sum_{\textbf{k}\in \tilde{\Delta}_l^{(j)}} \int_X \prod_{s=1}^{m}  \xi_{s} \d\mu &\leq \sum_{\substack{1\leq i_1<...<i_{j}\leq \ffloor{t}{\Omega r_l^d}\\ \min_{s=1,...,j-1} |\tilde{\alpha}_l^{(i_{s+1})}-\tilde{\alpha}_l^{(i_s)}|\geq p_l}} \int_{A_{i_1,...,i_j}} |\mathfrak{K}_{i_1,...,i_j}(x)| \d\mu(x)\\
&\leq \sum_{\substack{1\leq i_1<...<i_{j}\leq \ffloor{t}{\Omega r_l^d}\\ \min_{s=1,...,j-1}
|\tilde{\alpha}_l^{(i_{s+1})}-\tilde{\alpha}_l^{(i_s)}|\geq \tilde{p}_l}}
\int_{A_{i_1,...,i_j}} \rho^j \d\mu \\
&\leq \rho^j \left(\omega r_l^{dj}+o(r_l^{dj}) \right) \left(\ffloor{t}{\Omega r_l^d}\right)^j\\
&\leq \rho^j t^j \frac{\omega}{\Omega^j} +o(1).
\end{align*}
Summing up over $j$ we get
\begin{equation*}
\int_X R_l \d\mu \leq m \max(1,\rho^{m}) \max(1,t^{m}) \frac{\omega}{\max(1,\Omega^{m})} +o(1).
\end{equation*}
Following the argument in step (v) this shows $\int_X R_l \d\mu \rightarrow0$, hence \eqref{mlrlcon}, in case of assumption (III). This concludes the proof.
\end{proof}

\begin{remark}\label{delaykapretrem}
(i) Note that assumptions (I), (II), or (III) directly correspond to the three possible cases of
(BR$(x^*,y^*)$)--(SLR$(y^*)$), (NSR$(x^*)$) AND (LR$(y^*)$), or (NSR$(x^*)$) AND (LR$(x^*)$) respectively.

(ii)
In all of the examples we give in section \ref{ScEx}, the $\alpha_l$ will satisfy a condition stronger than (I).
In fact, in this set-up, there is a $\delta_2>0$ such that
\begin{equation*}
\min_{i\geq 2}|\alpha^{(i)}_{l}|\geq \mu(A_l)^{-\delta_2},
\end{equation*}
compare also \eqref{bigreturnsinplt:2}\footnote{The constant $\delta_2$ given there is not exactly the same. In the notation there we have to use $\delta_2 \frac{d'}{d}$.}. If this stronger condition is satisfied instead of (I), then we do not need the full strength of exponential mixing in (MEM). Any superpolynomial rate will be enough. Details are given in the proof of
Theorem~\ref{plt:2}, but we shall give a heuristic here.\\
When using mixing of all order with indicators of the form $1_{B_{r_l}(x^*)}$, the error term will contain a term coming from the $C^r$ norm in the definition of regularly approximable.
In this case, using \eqref{regapproxeq}, this term will be of order $C r_l^{-drm}$. To compensate, say the rate of mixing is $\psi$, since the gaps $\alpha_l$ are large we can multiply with $\psi(\min_{i\geq 2}|\alpha^{(i)}_{l}|)$. So we want to show
\begin{equation*}
r_l^{-drm} \psi(r_l^{-d\delta_2})=o(1),
\end{equation*}
for all $m\geq 1$, thus $\psi$ should decay superpolynomially.

\end{remark}

\section{The PLT for rectangles.}
\subsection{The strategy}\label{secThestrategy}

Let us first outline the strategy of proving Theorem \ref{plt:1}. \\
Say we want to show the PLT for the geodesic balls $(Q_l)$ converging to $(x^*,y^*)$, then, for every $\epsilon>0$, we find suitable rectangles\footnote{Using the exponential map, this is a simple exercise in $\mathbb{R}^{d+d'}$. Here we use continuity of the densities of $\mu$ resp. $\nu$!} approximating $Q_l$ in the sense that $\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}\subset Q_l$ and
\begin{equation*}
(\mu\times\nu)_{Q_l}\left( Q_l \setminus \bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)} \right) <\epsilon.
\end{equation*}
Denote $Q_l':=\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}$.
We will use Proposition \ref{delaykapretprop} to show the PLT for $Q'_l$, i.e
\begin{equation*}
(\mu\times\nu)(Q'_l)\Phi_{Q'_l}\rightarrow{(\mu\times \nu)}\Phi_{Exp}\;\;\;\as{l}.
\end{equation*}
Letting $\epsilon\rightarrow0$, the next Lemma will help us conclude that
\begin{equation*}
(\mu\times\nu)(Q_l)\Phi_{Q_l}\rightarrow{(\mu\times \nu)}\Phi_{Exp}\;\;\;\as{l}.
\end{equation*}
\\
The critical approximation here is the following Lemma (see \cite{Z1}).

\begin{lemma}\label{weakapproxlem}
There is a metric
 $D$ defined on the space of probability measures on $[0,\infty]^{\mathbb{N}}$
 and modelling weak convergence of measures\footnote{
In the sense that $\lambda_l\Rightarrow\lambda$ if and only if $D(\lambda_l,\lambda)\rightarrow0$.}
such that
\begin{equation*}
D\left(law_{(\mu\times\nu)_Q}((\mu\times\nu)(Q)\Phi_Q),law_{(\mu\times\nu)_{Q'}}((\mu\times\nu)(Q')\Phi_{Q'}\right)\leq 7(\mu\times \nu)_Q(Q\setminus Q'),
\end{equation*}
for measurable $Q'\subset Q\subset X\times Y$.
\end{lemma}
We shall prove a more general version of Lemma \ref{weakapproxlem}
 and define the metric $D$ in Section \ref{robustnesssec}.

For returns to rectangles, say $A_l\times B_l$, for fixed $y\in Y$, we can ignore all the times $j$ where $R^j(y)\not \in B_l$. Hence
denoting $\alpha_l(y)=\Phi_{B_l}(y)$, we first show that
\begin{equation}\label{delaypoiscon:1}
\mu(A_l)\Phi_{A_l,\alpha_l(y)}\xRightarrow{\mu} \Phi_{Exp} \;\;\;\as{l}.
\end{equation}
To show this, the idea is that, due to assumption (BR), the times $\tilde{\alpha}_l^{(j)}(y)$ are sufficiently far apart to use (MEM), so that $1_{A_l}\circ T_y^{\tilde{\alpha}_l^{(j)}(y)}$ are 'close to independent' and we can use classical arguments. Since \eqref{delaypoiscon:1} is now true for almost every $y\in Y$, we also have
\begin{equation}\label{delaypoiscon:2}
\mu(A_l)\Phi_{A_l,\Phi_{B_l}}\xRightarrow{\mu\times \nu} \Phi_{Exp} \;\;\;\as{l}.
\end{equation}
These are not quite the returns that we wanted to consider. However notice that, in every step, we skip exactly $\f{B_l}$ steps, thus we have the following relation
\begin{equation}\label{rectretsumrel}
\f{A_l\times B_l}(x,y)=\sum_{j=0}^{\f{A_l,\Phi_{B_l}(y)}(x)-1} \f{B_l}\circ R^j_{B_l}(y).
\end{equation}
We now use (EE) to control the ergodic sums of $\varphi_{B_l}$ and show that
\begin{equation*}
\mu(A_l)\nu(B_l)\Phi_{A_l\times B_l}\xRightarrow{\mu\times \nu} \Phi_{Exp} \;\;\;\as{l}.
\end{equation*}
The same idea
works for unions of rectangles $\displaystyle \bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}$.

\subsection{Quantitative Ergodic Theorem}

In section \ref{unifestsec}, it will be convenient to use a pointwise quantitative ergodic Theorem instead of the $L^2$ bound we assume in (EE). We will show first that such pointwise bounds hold for most points.
\\
More explicitly, we will show that for every sequence $(\s{f}{l})$ of functions in $L^2$ and $\epsilon>0$ there are $L_y,M_l\geq 1$ and a constant $K>0$ such that
\begin{equation*}
\left|\sum_{j=1}^n f_l \circ R^j(y) - n\int_Y f_l d\nu \right|\leq K ||f_l||_{C^{r'}} n^{\frac{1+2\delta+\epsilon}{3}} \;\;\;\textit{for a.e }y\in Y, \forall n\geq M_l, l\geq L_y.
\end{equation*}
\\
First let us give a small Lemma.
\begin{lemma}\label{sparselem}
Let $0<\delta<1$, and $k_n =\left\lfloor n^{\delta'} \right\rfloor$, for some $\delta'>1$ with $1>\delta'(1-\delta)$, then for some big enough $M\geq 1$ we have
\begin{equation*}
[M,\infty)\subset \bigcup_{n\geq 1} [k_n-k_n^{\delta},k_n+k_n^{\delta}].
\end{equation*}
\end{lemma}
\begin{proof}
Since $k_n\rightarrow\infty$ as $n\rightarrow\infty$, it is enough to show
\begin{equation*}
k_n+k_n^{\delta}\geq k_{n+1}-k_{n+1}^{\delta},
\end{equation*}
for big enough $n$. Indeed, using the mean value theorem there is $\xi_n\in [n,n+1]$ such that
\begin{align*}
(n+1)^{\delta'}-n^{\delta'}=\delta'\xi_n^{\delta'-1}\leq \delta'(n+1)^{\delta'-1}\leq (n+1)^{\delta\delta'},
\end{align*}
as soon as $n^{\delta\delta'-\delta'+1}>\delta'$.
\end{proof}

\begin{proposition}\label{l2boundprop}
Let $(Y,R,\nu)$ be a probability preserving ergodic dynamical system, $\s{f}{l}$ be a sequence of $C^r$ functions. Suppose there are $0<\delta<1$, $C>0$ and $N_l\geq 1$ such that
\begin{equation*}
\left|\left| \sum_{j=1}^n f_l\circ R^j - n\int_Y f_l\d\nu \right|\right|_{L^2}\leq C||f_l||_r n^{\delta} \;\;\;\forall n\geq N_l,\forall l\geq 1.
\end{equation*}
Then there are $L_y,M_l\geq 1$ and a constant $\tilde{C}>0$ such that
\begin{equation*}
\left|\sum_{j=1}^n f_l \circ R^j(y) - n\int_Y f_l d\nu \right|\leq \tilde{C} ||f_l||_r n^{\delta'} \;\;\;\textit{for a.e }y\in Y, \forall n\geq M_l, l\geq L_y.
\end{equation*}
Furthermore, for any small enough $\epsilon>0$, we can choose
\begin{equation*}
M_l=\max\left(N_l,\left\lceil l^{\frac{-\delta_2}{(\delta-\delta_1)\delta_2+1}+\epsilon}\right\rceil \right)
\end{equation*}
and $\delta'=\frac{1+2\delta+\epsilon}{3}$.
\end{proposition}
\begin{proof}
(1)  W.l.o.g we may assume $\int_Y f_l \d\nu=0$. For $l\geq 1$, $n\geq N_l$, $K>0$ and $\delta_1\in(0,1)$ close to $1$, using the Chebyshev inequality we have
$$
\nu\left( \left|\sum_{j=1}^n f_l\circ R^j\right| > K||f_l||_r n^{\delta_1}\right) \leq \frac{\left|\left| \sum_{j=1}^n f_l\circ R^j\right|\right|_{L^2}^2}{(K||f_l||_r n^{\delta_1})^2}
\leq \frac{C^2}{K^2} n^{2(\delta-\delta_1)}.
$$
Denote $B_n^{l,K}=\left\{\left|\sum_{j=1}^n f_l\circ R^j\right| > K||f_l||_r n^{\delta_1}\right\}$. We will use a Borel-Cantelli argument to show that, for some $K>0$ and for a.e $y$, there are only finitely many $l\geq 1, n\geq M_l$ such that $y\in B_n^{l,K}$.

(2) Note that if
\begin{equation*}
\left|\sum_{j=1}^n f_l\circ R^j(y)\right| > (2K+1)||f_l||_r n^{\delta_1},
\end{equation*}
then
\begin{equation*}
\left|\sum_{j=1}^k f_l\circ R^j(y)\right| > K||f_l||_r k^{\delta_1},
\end{equation*}
for $k\in \mathbb{Z}\cap [n-n^{\delta_1},n+n^{\delta_1}]$. By Lemma \ref{sparselem}, for any $\delta_2>1$ with $1>\delta_2(1-\delta_1)$ we have that there is some $M\geq 1$, only depending on $\delta_1$ and $\delta_2$, such that for each $k\geq M$ there is a $n \geq 1$ such that $B_k^{l,K}\subset B_{k_n}^{l,(2K-1)}$, where $k_n=\left\lfloor n^{\delta_2}\right\rfloor$.
(3) For some small $\tilde{\epsilon}>0$ let
\begin{equation*}
\tilde{M}_l=\left\lceil l^{\frac{-1}{(\delta-\delta_1)\delta_2+1}+\tilde{\epsilon}}\right\rceil.
\end{equation*}
Suppose $2(\delta-\delta_1)\delta_2<-1$ then
\begin{align*}
\sum_{l\geq 1}\sum_{n\geq \tilde{M}_l} \nu(B_{k_n}^{l,(2K+1)})&\leq \sum_{l\geq 1}\sum_{n\geq \tilde{M}_l} \frac{C^2}{(2K+1)^2} n^{2(\delta-\delta_1)\delta_2}\\
&\leq \frac{C^2}{(2K+1)^22(\delta-\delta_1)\delta_2} \sum_{l\geq 1} \tilde{M}_l^{2(\delta-\delta_1)\delta_2+1}
&<\infty.
\end{align*}
By Borel-Cantelli, for a.e $y$ there are only finitely many $l\geq 1$ and $n\geq \tilde{M}_l$ such that $y\in B_{k_n}^{l,(2K+1)}$. It follows that, for large enough $L_y\geq 1$
\begin{equation*}
y\not\in B_n^{l,K}\;\;\; \forall l\geq L_y, n\geq M_l,
\end{equation*}
where
\begin{equation*}
M_l=\max\left(N_l,\left\lceil l^{\frac{-\delta_2}{(\delta-\delta_1)\delta_2+1}+\epsilon}\right\rceil \right),
\end{equation*}
for some small $\epsilon>0$.

(4) Finally for the $\delta_1,\delta_2$ we want
\begin{itemize}
\item $\delta_1>0$,
\item $\delta_2>1$,
\item $(1-\delta_1)\delta_2<1$,
\item and $2(\delta-\delta_1)\delta_2<-1$.
\end{itemize}
For $\delta_1=\frac{1+2\delta+\epsilon}{3}$ and $\delta_2=\frac{3}{2(1-\delta)+2\epsilon}$, these relations are satisfied, making $\epsilon$ smaller if necessary.
\end{proof}

\subsection{Uniform Estimates}\label{unifestsec}
We will show two estimates.

\begin{lemma}\label{indcrlem}
Suppose for every sequence $(\s{f}{l})$ of functions in $L^2$ and $\epsilon>0$ there are $L_y,M_l\geq 1$ and a constant $K>0$ such that
\begin{equation*}
\left|\sum_{j=1}^n f_l \circ R^j(y) - n\int_Y f_l d\nu \right|\leq K ||f_l||_r n^{\delta} \;\;\;\textit{for a.e }y\in Y, \forall n\geq M_l, l\geq L_y.
\end{equation*}
If $B$ is regularly approximable in $C^r$, then
\begin{equation}
\label{IndicatorSum}
\left|\sum_{j=1}^n 1_B\circ R^j(y)-n\nu(B)\right|\leq K_y
n^{1-\frac{1-\delta}{r+1}} \;\;\;\forall n\geq M_y, \textit{ a.e }y\in Y.
\end{equation}
\end{lemma}
\begin{proof}
For each $\epsilon>0$ there are $\underline{h},\overline{h}\in C^r$ with $\underline{h}\leq 1_B\leq \overline{h}$ and
\begin{equation*}
\lambda(1_B\not= h)\leq\nu(B)^{\frac{d-1}{d}}\epsilon, \;\textit{ while }\; ||h||_{C^r}\leq C\epsilon^{-r},
\end{equation*}
for $h\in \{\overline{h},\underline{h}\}$. In particular
\begin{equation*}
\nu(B)-\nu(B)^{\frac{d-1}{d}}\epsilon \leq \int_Y \underline{h} \d\nu \leq \int_Y\overline{h}\d\nu \leq \nu(B)+\nu(B)^{\frac{d-1}{d}}\epsilon.
\end{equation*}
By assumption, for $h\in \{\overline{h},\underline{h}\}$ we have
\begin{equation*}
\left|\sum_{j=1}^n h\circ R^j(y) - n\int_Y h\d\nu\right|\leq C_y||f||_{C^r}n^{\delta} \;\;\;\forall n\geq M_y, \textit{ a.e }y\in Y.
\end{equation*}
Therefore
\begin{align*}
\sum_{j=1}^n 1_B \circ R^j(y) &\geq \sum_{j=1}^n \underline{h}\circ R^j(y)
\geq n\int_Y \underline{h} \d\nu - C_yC\epsilon^{-r}n^{\delta}\\
&\geq n\nu(B)-n\nu(B)^{\frac{d-1}{d}}\epsilon- C_yC\epsilon^{-r}n^{\delta}.
\end{align*}
Likewise
\begin{equation*}
\sum_{j=1}^n 1_B \circ R^j(y)\leq n\nu(B)+n\nu(B)^{\frac{d-1}{d}}\epsilon+ C_yC\epsilon^{-r}n^{\delta}.
\end{equation*}
Hence
\begin{equation*}
\left|\sum_{j=1}^n 1_B \circ R^j(y)-n\nu(B)\right|\leq n\nu(B)^{\frac{d-1}{d}}\epsilon+ C_yC\epsilon^{-r}n^{\delta}.
\end{equation*}
This bound is minimized for
\begin{equation*}
\epsilon^*=\left(\frac{rC_yCn^{\delta}}{n\nu(B)^{\frac{d-1}{d}}}\right)^{\frac{1}{r+1}},
\end{equation*}
so we even have
\begin{align*}
\left|\sum_{j=1}^n 1_B \circ R^j(y)-n\nu(B)\right|&\leq r^{\frac{1}{r+1}}(C_yC)^{\frac{1}{r+1}}n^{1-\frac{1}{r+1}}n^{\frac{\delta}{r+1}} \nu(B)^{\frac{d-1}{d}\left(1-\frac{1}{r+1}\right)}\\
&\;\;\;\;\;+ n^{\frac{r}{r+1}}n^{\delta\left(1-\frac{r}{r+1}\right)}\left(\frac{rC_yC}{\nu(B)^{\frac{d-1}{d}}}\right)^{-\frac{r}{r+1}}\\
&\leq K_y n^{1-\frac{1-\delta}{r+1}},
\end{align*}
where the constant $K_y>0$ only depends on $r$, $C_y$ and $C$.
\end{proof}

\begin{lemma}\label{UClem:1}
Let $\s{B}{l}$ be a sequence of rare events, and assume that
\begin{equation*}
\left|\sum_{j=1}^n 1_{B_l} \circ R^j_{B_l}(y) - n\nu(B_l)\right|\leq C_{l,y} n^{1-\frac{1-\delta}{r+1}} \;\;\;\textit{for a.e }y\in Y, \forall n\geq M_yM_l, l\geq L_y,
\end{equation*}
for some constant $\delta>0$, $M_y\geq 1$ only depending on $y$, $M_l\geq 1$ depending only on $l$, and $C_{l,y}>0$ depending on $l$ and $y$, and $L_y\geq 1$ depending only on $y$. Then
\begin{equation*}
\left|\frac{\nu(B_l)}{sN_l(y)}\sum_{j=0}^{\lceil sN_l(y)\rceil-1} \f{B_l}\circ R^j_{B_l}(y) -1 \right|\leq  3s^{-\frac{1-\delta}{r+1}}+\frac{2}{s}.
\end{equation*}
where
\begin{equation*}
N_l(y)=\max\left(M_l, \fceil{1}{2\nu(B_l)}, \min\left(n\geq 1\;|\; n^{-\frac{1-\delta}{r+1}}\leq \frac{1}{2C_{l,y}}\nu(B_l)\right)\right).
\end{equation*}
\end{lemma}
\begin{proof}
Let $y\in Y$ be as in the statement. To simplify the notation denote $\displaystyle g(n)=n^{1-\frac{1-\delta}{r+1}}$.
For $l\geq L_y$ and $n\geq M_yM_l$ we have
\begin{equation*}
n\nu(B_l)-C_{l,y}g(n) \leq \sum_{j=1}^n 1_{B_l} \circ R^j_{B_l}(y) \leq n\nu(B_l) + C_{l,y}g(n).
\end{equation*}
and thus
$$
\sum_{j=0}^{\lfloor n\nu(B_l)-C_{l,y}g(n) \rfloor-1} \f{B_l}\circ R^j_{B_l}(y) \leq n \leq
\sum_{j=0}^{\lceil n\nu(B_l)+C_{l,y}g(n) \rceil-1} \f{B_l}\circ R^j_{B_l}(y).
$$
Rewrite the summation limits above as
$\displaystyle \left\lfloor n\left(\nu(B_l)-\frac{C_{l,y}g(n)}{n}\right) \right\rfloor-1$
and $\displaystyle
\left\lceil n\left(\nu(B_l)+\frac{C_{l,y} g(n)}{n}\right) \right\rceil-1$ respectively.
For $N_l(y)$ as in the statement and $s\geq M_y$ we have
\begin{align*}
\sum_{j=0}^{\lceil sN_l(y)\rceil-1} \f{B_l}\circ R^j_{B_l}(y) \leq \frac{sN_l(y)}{\nu(B_l)-\frac{g(\lceil sN_l(y)\rceil)}{\lceil sN_l(y)\rceil}}+1.
\end{align*}
Thus
\begin{align*}
&\frac{\nu(B_l)}{sN_l(y)}\sum_{j=0}^{\lceil sN_l(y)\rceil-1} \f{B_l}\circ R^j_{B_l}(y) \leq \frac{\nu(B_l)}{\nu(B_l)-\frac{g(\lceil sN_l(y)\rceil)}{\lceil sN_l(y)\rceil}}+\frac{2}{s}\\
&\leq 1+\frac{\frac{g(\lceil sN_l(y)\rceil)}{\lceil sN_l(y)\rceil}}{\nu(B_l)-\frac{g(\lceil sN_l(y)\rceil)}{\lceil sN_l(y)\rceil}}+\frac{2}{s}
\leq 1+2\frac{g(\lceil s\rceil)}{s+1}+\frac{2}{s}.
\end{align*}

Analogously
$\displaystyle
\frac{\nu(B_l)}{sN_l(y)}\sum_{j=0}^{\lceil sN_l(y)\rceil-1} \f{B_l}\circ R^j_{B_l}(y) \geq  1-2\frac{g(\lceil s\rceil)}{s+1}-\frac{2}{s}.
$
\end{proof}

We shall also use the following.
\begin{proposition}\label{retqergodicprop}
Suppose $R$ satisfies condition (EE), and let $\s{B}{l}$ be a sequence of rare events such that $\{B_l\}$ is regularly approximable. Then for every $\epsilon>0$ and for almost every $y\in Y$
\begin{equation*}
\sup_{l\geq 1}\left|\frac{\nu(B_l)}{sN_l(y)}\sum_{j=1}^{\lceil sN_l(y)\rceil} \f{B_l}^{(j)} (y)-1 \right|\rightarrow 0 \;\;\;\as{s},\; \textit{a.e } y\in Y,
\end{equation*}
where
\begin{equation*}
N_l(y)=\left\lceil \left(\frac{K_y}{\nu(B_l)}\right)^{\frac{3}{2} \frac{r+1}{1-\delta_1}+\epsilon} \right\rceil,
\end{equation*}
for some $K_y>0$ depending only on $y$ and $\epsilon$.
\end{proposition}
\begin{proof}
Take $y\in Y$ as in the conclusion of Proposition \ref{l2boundprop}, then
\begin{equation*}
\left|\sum_{j=1}^n 1_B\circ R^j(y)-n\nu(B)\right|\leq K_yn^{1-\frac{1}{r+1}(1-\delta_1)} \;\;\;\forall n\geq M_y, \textit{ a.e }y\in Y.
\end{equation*}
Now apply Lemma \ref{UClem:1}.
\end{proof}

\subsection{PLT scaled by returns to $\{B_l\}$}\label{scalePLTsec}

For this section let $K\geq 1$, and $A_l^{(1)},...,A_l^{(K)}$ resp. $B_l^{(1)},...,B_l^{(K)}$ be subsets of $X$ resp. $Y$ regularly approximable in $C^r$ resp. $C^{r'}$. Suppose that there are $r_l\searrow 0$, and positive constants $\omega^{(k)},\theta^{(k)}>0$ such that
\begin{equation*}
\mu(A_l^{(k)})=\omega^{(k)} r_l^d+o(r_l^d),\;\textit{ and } \;\nu(B_l^{(k)})=\theta^{(k)}r_l^{d'}+o(r_l^{d'}) \;\;\;\forall l\geq 1, k=1,...,K,
\end{equation*}
and, for each $l$, $B_l^{(1)},...,B_l^{(K)}$ are disjoint. Denote $\displaystyle B_l=\bigcup_{k=1}^K B_l^{(k)}$, which, by disjointness of $B_l^{(1)},...,B_l^{(K)}$, is also regularly approximable in $C^{r'}$, and $\alpha_l=\Phi_{B_l}$.
Consider
\begin{equation*}
\kappa_l^{(n)}(y)=k \;\;\; \textit{ if } \;\; R_{B_l}^n(y)\in B_l^{(k)},
\end{equation*}
which is well-defined as the $B_l^{(k)}$ are disjoint, and
\begin{align*}
&\phi_{\kappa_l,\alpha_l,y}^{(1)}(x)
=\min\left(n\geq 1\;|\; T_y^{\tilde{\alpha}_l^{(n)})}(x)\in A_l^{(\kappa_l^{(n)})}\right),\\
&\phi_{\kappa_l,\alpha_l,y}^{(m+1)}(x)=
\min\left(n\geq \phi_{\kappa_l,\alpha_l,y}^{(m)}(x)+1\;|\; T_y^{\tilde{\alpha}_l^{(n)}}(x)\in A_l^{(\kappa_l^{(n)})}\right),\\
&\Phi_{\kappa_l,\alpha_l,y}=(\phi_{\kappa_l,\alpha_l,y}^{(1)},\phi_{\kappa_l,\alpha_l,y}^{(2)},...).
\end{align*}
Following the steps outlined in \S \ref{secThestrategy} we will first show
\begin{equation*}
\Omega r_l^{d'} \Phi_{\kappa_l,\alpha_l}\xRightarrow{\mu} \Phi_{Exp} \;\;\;\as{l},
\end{equation*}
for $\Omega=\frac{\sum_{k=1}^K \omega^{(k)} \theta^{(k)}}{\sum_{k=1}^K \theta^{(k)}}$, and then use the relation
\begin{equation}\label{unionrectretsumrel:2}
\f{\bigcup_{k=1}^K A_l^{(k)} \times B_l^{(k)}}(x,y)=\sum_{j=0}^{\f{\kappa_l,\Phi_{B_l}(y)}(x)-1} \f{B_l}\circ R^j_{B_l}(y).
\end{equation}
to obtain
\begin{equation}\label{poisconunionrect}
(\mu\times \nu)\left(\bigcup_{k=1}^K A_l^{(k)} \times B_l^{(k)} \right)\Phi_{\bigcup_{k=1}^K A_l^{(k)} \times B_l^{(k)}}\xRightarrow{\mu\times \nu} \Phi_{Exp} \;\;\;\as{l}.
\end{equation}
Denote
\begin{equation*}
p_{l,t}^{(k)}(y):=\frac{1}{t}\#\{j=1,...,t_l\;|\;\kappa_l^{(j)}(y) =k\}=\frac{1}{t}\sum_{j=1}^{t} 1_{B_l^{(k)}}(R_{B_l}^j(y)).
\end{equation*}
We first show that, for each $k$, a.e $y\in Y$, and $t_l=O(r_l^{-d})$, we have
\begin{equation}\label{kapprobcon:2}
p_{l,t_l}^{(k)}(y) \rightarrow \frac{\theta^{(k)}}{\sum_{j=1}^K \theta^{(j)}}=:p^{(k)}\;\;\; \as{l}.
\end{equation}

\begin{lemma}\label{kaplem}
Suppose $R$ satisfies (EE) and
\begin{equation*}
d>\frac{3}{2} \frac{d'(r'+1)}{1-\delta_1}
\end{equation*}
Then $\kappa_l$ satisfies \eqref{kapprobcon:2}.
\end{lemma}
\begin{proof}
Let $\epsilon>0$ be such that $d>\frac{d'(r'+1)}{1-\frac{1+2\delta_1+\epsilon}{3}}$ and denote $\delta'=\frac{1+2\delta_1+\epsilon}{3}$.
(i) Using Lemma \ref{indcrlem}, and disjointness, for each $B_l^{(k)}$, we get a function
$\tilde{g}(n)=O\left(n^{1-\frac{1}{r+1}\left(1-\delta'\right)}\right)$ such that
$$
\left|\sum_{j=1}^n 1_{B_l^{(k)}}\circ R^j(y)-n\nu(B_l^{(k)})\right|\leq \tilde{g}(n), \quad\textit{ and}\quad
\left|\sum_{j=1}^n 1_{B_l}\circ R^j(y)-n\nu(B_l)\right|\leq \tilde{g}(n)
$$
for all $n\geq M_y$.
Lemma \ref{UClem:1} yields
\begin{equation*}
\left|\sum_{i=0}^{n-1} \f{B_l}\circ R_{B_l}^i(y)- \frac{n}{\nu(B_l)}\right|
\leq 3\frac{\tilde{g}(n)}{\nu(B_l)^2}+2,
\end{equation*}
for $n\geq \tilde{N}(y)$, where
\begin{equation*}
\tilde{N}_l(y)=\max\left(M_y,\min\left(n\geq 1\;|\;\frac{\tilde{g}(n')}{n'}\leq \frac{1}{2}\nu(B_l)\;\forall n'\geq n\right)\right).
\end{equation*}
(ii) Rewrite
\begin{equation*}
p_{l,n}^{(k)}=\frac{1}{n}\sum_{i=1}^{\sum_{j=1}^n \f{B_l}\circ R_{B_l}^j(y)} 1_{B_l^{(k)}} \circ R^i.
\end{equation*}
For $\tilde{\alpha}_l^{(n)}=\sum_{j=1}^n \f{B_l}\circ R_{B_l}^j(y)$ we have
\begin{align*}
np_{l,n}^{(k)}(y)&=\sum_{j=1}^{\tilde{\alpha}_l^{(n)}(y)} 1_{B_l^{(k)}}\circ R^j(y)\leq \sum_{j=1}^{\left\lceil\frac{n}{\nu(B_l)}+2\frac{\tilde{g}(n)}{\nu(B_l)^2}+2\right\rceil} 1_{B_l^{(k)}}\circ R^j(y)\\
&\leq \left(\frac{n}{\nu(B_l)}+2\frac{\tilde{g}(n)}{\nu(B_l)^2}+3\right)\nu(B_l^{(k)}) + \tilde{g}\left(\frac{n}{\nu(B_l)}+2\frac{\tilde{g}(n)}{\nu(B_l)^2}+3\right)\\
&\leq n\left(\frac{\nu(B_l^{(k)})}{\nu(B_l)}+2\frac{\tilde{g}(n)\nu(B_l^{(k)})}{n\nu(B_l)^2}\right)+3\nu(B_l^{(k)})+\tilde{g}\left(6\frac{n}{\nu(B_l)}\right)
\end{align*}
for $n\geq \max\left(\tilde{N}_l(y),\left\lceil\nu(B_l)^{-1}\right\rceil\right):=\hat{N}_l(y)$. So we have
\begin{equation*}
p_{l,n}^{(k)}\leq \frac{\nu(B_l^{(k)})}{\nu(B_l)} + \frac{\tilde{g}(n)}{n}r_l^{-d'}O_l(1)+r_l^{d'}O_l(1)+\frac{\tilde{g}(nr_l^{-d'}O_l(1))}{n},
\end{equation*}
where the subscripts in $O_l$ are used to indicate that the error terms only depend on $l$.
The upper bound is similar. So
\begin{align*}
&\left|p_{l,n}^{(k)}-\frac{\theta^{(k)}}{\sum_{j=1}^K \theta^{(j)}}\right|= o_l(1) + \frac{\tilde{g}(n)}{n}r_l^{-d'}O_l(1)+r_l^{d'}O_l(1)+\frac{\tilde{g}(nr_l^{-d'}O_l(1))}{n}\\
&= o_l(1) + O_n\left( n^{-\frac{1}{r'+1}\left(1-\delta_1\right)}\right) r_l^{-d'} O_l(1) + O_n\left( n^{-\frac{1}{r'+1}\left(1-\delta_1\right)}\right)O_l\left(r_l^{d'-\frac{d'}{r+1}\left(1-\delta_1\right)}\right)\\
&=o_l(1) + O_n\left( n^{-\frac{1}{r'+1}\left(1-\delta_1\right)}\right) r_l^{-d'} O_l(1)=o_l(1),
\end{align*}
for $n\geq \overline{N}_l(y)$, where
\begin{equation*}
\overline{N}_l(y)=\max\left(\hat{N}_l(y), \left\lceil r_l^{-\frac{d'(r'+1)}{1-\delta_1}-\epsilon}\right\rceil\right),
\end{equation*}
for some $\epsilon>0$. Choosing $\epsilon>0$ so small that
\begin{equation*}
\frac{d'(r'+1)}{1-\delta_1}+\epsilon<d
\end{equation*}
yields $\overline{N}_l(y)=o\left(r_l^{-d}\right)$, thus proving the Lemma.
\end{proof}

\subsection{Adding in the gaps}

Now all that's left to do is to add back in the gaps. We use \eqref{delaypoiscon:2} to
conclude
\begin{equation*}
(\mu\times \nu)\left(\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}\right) \Phi_{\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}} \xRightarrow{\mu\times\nu} \Phi_{Exp}, \;\;\;\as{l}.
\end{equation*}
To do this we make use of the relation
\begin{equation*}
\f{\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}}=\sum_{j=0}^{\f{\kappa_l,\alpha_l,y}(x)-1} \f{B_l}\circ R^j_{B_l}(y).
\end{equation*}

Let us formulate an abstract result which will allow us to sum up the gaps.
For completeness let us first argue the following probability theory fact.
\begin{lemma}\label{weakzeroconlem}
Let $(\Omega,\mathbb{P})$ be a probability space, and $E_l:\Omega\rightarrow[0,\infty)$ non-negative real random variables, such that there are positive random variables $\mu_l:\Omega\rightarrow (0,1)$ with
\begin{equation*}
\mu_l E_l\xRightarrow{\mathbb{P}}E\;\;\; \as{l},
\end{equation*}
for some non-negative random variable $E$ with $\mathbb{P}(E=0)=0$. Then for any $N_l:\Omega\rightarrow[0,\infty)$ with
\begin{equation*}
\mu_l N_l\rightarrow 0 \;\;\;\as{l} \; \textit{ pointwise a.e}
\end{equation*}
we have
\begin{equation*}
\mathbb{P}(E_l\leq N_l) \rightarrow 0 \;\;\;\as{l}.
\end{equation*}
\end{lemma}
\begin{proof}
Let $\epsilon>0$, there is a $\delta>0$ such that $\mathbb{P}(E\leq \delta)<\epsilon$, and the distribution function of $E$ is continuous at $\delta$. By Jegorow's Theorem, there is a measurable $K\subset\Omega$ with $\mathbb{P}(K^c)<\epsilon$ and
\begin{equation*}
\mu_l N_l \rightarrow0 \;\;\;\as{l} \;\textit{ uniformly on }K.
\end{equation*}
Choose $\tilde{L}\geq 1$ so big that $N_l\leq \delta \mu_l^{-1}$ on $K$ for $l\geq \tilde{L}$. Now choose $L\geq\tilde{L}$ so big that
\begin{equation*}
\mathbb{P}(\mu_l E_l \leq \delta)\leq 2\epsilon\;\;\;\forall l\geq L,
\end{equation*}
it follows that
\begin{equation*}
\mathbb{P}(E_l\leq N_l)\leq \mathbb{P}(\mu_lE_l\leq \delta)+ \mathbb{P}(K^c)\leq 3 \epsilon,
\end{equation*}
for $l\geq L$.
\end{proof}

\begin{proposition}\label{sumweakconprop}
Let $(\Omega,\mathbb{P})$ be a probability space, and $E_l:\Omega \rightarrow \mathbb{N}$ be positive integer valued observables. Assume there are positive real numbers $q_l\searrow0$,
and a $[0,\infty)$-valued random variable $E$ with $\mathbb{P}(E=0)=0$
such that
\begin{equation*}
q_l E_l\xRightarrow{\mathbb{P}} E \;\;\;\as{l},
\end{equation*}

Let $\beta^{(l)}_j:\Omega \rightarrow [0,\infty)$ be non-negative random variables, and assume there are
$N_l:\Omega\rightarrow (0,\infty)$ with $q_l N_l\rightarrow 0$ as $l\rightarrow\infty$ a.e and
positive random variables $b^{(l)}:\Omega \rightarrow (0,\infty)$
such that
\begin{equation*}\tag{UC}
\sup_{l\geq 1}\left|\frac{1}{sN_l b^{(l)}}\sum_{j=1}^{\lceil sN_l\rceil} \beta_j^{(l)} (\omega)-1 \right|\rightarrow 0 \;\;\;\as{s},\; \textit{ a.e. } .
\end{equation*}

Then \hskip2mm
$\displaystyle
\frac{q_l}{b^{(l)}}\sum_{j=1}^{E_l} \beta_j^{(l)} \xRightarrow{\mathbb{P}} E\;\;\;\as{l}
.$
\end{proposition}

\begin{remark}\label{sumweakconrem}
In our context we use $(\Omega,\mathbb{P})=(X\times Y,\mu\times \nu)$, $R_l=\phi_{A_l,\Phi_{B_l}}$, $\beta_j^{(l)}=\f{B_l}\circ R_{B_l}^j$, $q_l=\mu(A_l)$ and $b^{(l)}=\frac{1}{\nu(B_l)}$. The existence of $N_l$ is the content of Proposition \ref{retqergodicprop}.
\end{remark}

\begin{proof}
(i) Let $F$ be the distribution function of $E$ and $C=\{t\;|\; F\textit{ is continuous at }t\}$ the set of its continuities. Let $t\in C$, and $\epsilon>0$ such that $\frac{t}{1+\epsilon},\frac{t}{1-\epsilon}\in C$.
\\
\noindent
(ii) By Jegorow's Theorem, for $l\geq 1$, there is a measurable set $K\subset\Omega$ with $\mathbb{P}(K^c)<\epsilon$
\begin{equation*}
\sup_{l\geq 1}\sup_{\omega\in K} \left|\frac{1}{sN_l b^{(l)}}\sum_{j=1}^{\lceil sN_l\rceil} \beta_j^{(l)} (\omega)-1 \right|\rightarrow 0 \;\;\;\as{s}.
\end{equation*}
\\
(iii)
Choose $S>0$ so big that
\begin{equation*}
(1-\epsilon) b^{(l)}\leq \frac{1}{sN_l}\sum_{j=1}^{\lceil sN_l\rceil} \beta_j^{(l)} \leq (1+\epsilon) b^{(l)}\;\;\;\textit{on }K,\; \forall s\geq S, l\geq 1.
\end{equation*}
Then
\begin{equation*}
(1-\epsilon) q_l E_l\leq \frac{q_l}{b^{(l)}} \sum_{j=1}^{E_l} \beta_j^{(l)} (\omega)\leq (1+\epsilon) q_l E_l\;\;\;\textit{ on }K\cap \{E_l\geq SN_l\},\; \forall l\geq 1.
\end{equation*}
We get
\begin{align*}
&\mathbb{P}\left(\frac{q_l}{b^{(l)}}  \sum_{j=1}^{E_l} \beta_j^{(l)} \leq t\right)  -\mathbb{P}(E_l\leq SN_l)-\mathbb{P}(K^c)\\
&\leq \mathbb{P}\left( K\cap \{E_l\geq SN_l\} \cap \left\{ \frac{q_l}{b^{(l)}}  \sum_{j=1}^{E_l} \beta_j^{(l)} \leq t\right\} \right)
 \leq \mathbb{P}\left( q_l E_l \leq \frac{t}{1-\epsilon} \right).
\end{align*}
Likewise
\begin{equation}\label{LowerB}
\begin{aligned}
\mathbb{P}\left(\frac{q_l}{b^{(l)}}  \sum_{j=1}^{E_l} \beta_j^{(l)} \leq t\right)
& \geq \mathbb{P}\left(K\cap \{E_l\geq SN_l\} \cap \left\{ q_l E_l \leq \frac{t}{1+\epsilon}\right\} \right) \notag \\
&\geq \mathbb{P}\left(q_l E_l \leq \frac{t}{1+\epsilon} \right)-\mathbb{P}(E_l\leq SN_l)-\mathbb{P}(K^c).
\end{aligned}
\end{equation}

Taking $\displaystyle \limsup_{l\rightarrow\infty}$ resp. $\displaystyle \liminf_{l\rightarrow\infty}$ in \eqref{LowerB}
and using Lemma \ref{weakzeroconlem} we obtain
\begin{align*}
&F\left(\frac{t}{1+\epsilon}\right)-\epsilon\leq \liminf_{l\rightarrow \infty} \mathbb{P}\left(\frac{q_l}{b^{(l)}} \sum_{j=1}^{E_l} \beta_j^{(l)} \leq t\right)\\
&\leq \limsup_{l\rightarrow \infty} \mathbb{P}\left(\frac{q_l}{b^{(l)}} \sum_{j=1}^{E_l} \beta_j^{(l)} \leq t\right)
\leq F\left(\frac{t}{1-\epsilon}\right)+\epsilon.
\end{align*}

Since $C\subset (0,\infty)$ is dense, we can let $\epsilon\searrow 0$ while $\frac{t}{1+\epsilon},\frac{t}{1-\epsilon}\in C$, this yields
$\displaystyle
\mathbb{P}\left(\frac{q_l}{b^{(l)}} \sum_{j=1}^{E_l} \beta_j^{(l)} \leq t\right)\rightarrow F(t)\;\;\;\as{l}.
$
\end{proof}

\begin{remark}
\begin{enumerate}[(a)]
\item We can extend this statement to sequences in the following manner: under the assumptions of the proposition, let $E^k_l:\Omega \rightarrow \mathbb{N}$ be such that
\begin{equation*}
\mathbb{P}(E^1_l,E^2_l,...) \xRightarrow{\mu} (E^1,E^2,...) \;\;\;\as{l},
\end{equation*}
for some $E^k:\Omega\rightarrow[0,\infty)$ with $\mathbb{P}(E^k=0)=0$. Then
\begin{equation*}
\frac{q_l}{b^{(l)}}\left(\sum_{j=1}^{E^1_l} \beta_j^{(l)}, \sum_{j=1}^{E^2_l} \beta_j^{(l)},...\right) \xRightarrow{\mathbb{P}} (E^1,E^2,...) \;\;\;\as{l}.
\end{equation*}
The proof of this statement is almost the same as for the proposition, therefore we won't repeat it.
\item The probability measure $\mathbb{P}$ can be replaced by a sequence $\s{\mathbb{P}}{l}$ by also replacing $(UC)$ with
\begin{align*}
\forall \epsilon>0 \textit{ there is are} & \textit{ measurable sets }  K_l\subset \Omega \textit{ with } \limsup_{l\rightarrow\infty} \mathbb{P}_l(K_l^c)<\epsilon \;\textit{ such that}\\
&\sup_{l\geq 1}\sup_{\omega\in K_l} \left|\frac{1}{sN_l b^{(l)}}\sum_{j=1}^{\lceil sN_l\rceil} \beta_j^{(l)} (\omega)-1 \right|\rightarrow 0 \;\;\;\as{s}.
\end{align*}
\end{enumerate}
\end{remark}
\medskip

\begin{proof}[Proof of Theorem $\ref{plt:1}$]
(i) Let $r_l\searrow 0$, and denote by $Q_l=B_{r_l}(x^*,y^*)$ the geodesic ball of radius $r_l$ centred at $(x^*,y*)$, and let $\epsilon>0$. Wlog $r_1$ is small enough that the exponential map at $(x^*,y^*)$ is a diffeomorphism from the ball of radius $2r_1$ in $\mathbb{R}^{d+d'}$ onto $B_{2r_1}(x^*,y^*)$. It is now easy to construct, for some $K\geq 1$, sets $A_l^{(1)},...,A_l^{(K)}$ and $B_l^{(1)},...,B_l^{(K)}$ as in section \ref{PLTsubssec}. Let $\omega^{(k)},\theta^{(k)}>0$ be such that\footnote{Choose $x^*,y^*$ such that the densities of $\mu,\nu$ are positive at the respective points.}
\begin{equation*}
\mu(A_l^{(k)})=\omega^{(k)} r_l^d+o(r_l^d),\;\textit{ and } \;\nu(B_l^{(k)})=\theta^{(k)}r_l^{d'}+o(r_l^{d'}) \;\;\;\forall l\geq 1, k=1,...,K,
\end{equation*}
and set $\Omega=\frac{\sum_{k=1}^K \omega^{(k)} \theta^{(k)}}{\sum_{k=1}^K \theta^{(k)}}$.

Due to Lemma \ref{approxlem:2} all those sets can be chosen to be regularly approximable, such that\footnote{Here we again use the continuity of the density.} $(\mu\times\nu)_{Q_l}\left(Q_l\setminus \bigcup_{k=1}^K A_l^{(k)} \times B_l^{(k)}\right) <\epsilon$ for all $l\geq 1$. Let $\Lambda>0$ be such that $(\mu\times\nu)(Q_l)=\Lambda r_l^{d+d'} +o(r_l^{d+d'})$, then
\begin{equation*}
\left|\Lambda-\Omega\sum_{k=1}^K \theta^{(k)}\right|<\epsilon
\end{equation*}
(ii)
For $y$ as in assumption (MEM), consider $\alpha_l(y)=\Phi_{\bigcup_{k=1}^K B_l^{(k)}}(y)$, and
\begin{equation*}
\kappa_l^{(n)}(y)=k \;\;\; \textit{ if } \;\; R_{\bigcup_{j=1}^K B_l^{(j)}}^n(y)\in B_l^{(k)}.
\end{equation*}
By disjointness $\kappa_l(y)$ is well-defined. By Lemma \ref{kaplem}, $\kappa_l(y)$ satisfies $(\dagger)$, with $a_l=r_l^d$ and $p^{(k)}=\frac{\theta^{(k)}}{\sum_{s=1}^K \theta^{(s)}}$. We can
use Proposition \ref{delaykapretprop}
to obtain
\begin{equation*}
\Omega r_l^d \Phi_{\kappa_l(y),\alpha_l(y)}\xRightarrow{\mu} \Phi_{Exp} \;\;\;\as{l}, \textit{ a.e } y.
\end{equation*}
Since this is true for a.e $y\in Y$, we also have
\begin{equation*}
\Omega r_l^d \Phi_{\kappa_l,\alpha_l}\xRightarrow{\mu\times\nu} \Phi_{Exp} \;\;\;\as{l}.
\end{equation*}
(iii)
Using Lemma \ref{indcrlem} for $B_l^{(k)}$ for each $k=1,...,K$,
and summing over $k$, we get
\begin{equation*}
\left|\sum_{j=1}^n 1_{B_l}\circ R^j(y)-n\nu(B_l)\right|\leq K_y\tilde{g}(n) \;\;\;\forall n\geq N, \textit{ a.e }y\in Y,
\end{equation*}
for some $\tilde{g}(n)=o\left(n^{1-\frac{1}{r'+1}(1-\delta_1)}\right)$ not depending on $l$ and $y$, and $K_y>0$ only depending on $y$. By Lemma \ref{UClem:1}, $\alpha_l$ satisfies $(UC)$ with
$$b^{(l)}=\frac{1}{\nu(B_l)}=\frac{1}{\sum_{k=1}^K \theta^{(k)}}r_l^{-d'}+o(r_l^{-d'})$$ and
\begin{equation*}
N_l=\max\left( \fceil{1}{2\nu(B_l)}, \min\left(n\geq 1\;|\; C\left(n^{\delta_1-1}\right)^{\frac{1}{r'+1}}\leq \frac{1}{2}\nu(B_l)\right)\right),
\end{equation*}
for some constant $C>0$. In particular, $N_l=O\left( r_l^{-d'\frac{r'+1}{1-\delta_1}}\right)=o(a_l)$. Note that, for $Q_l'=\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}$ we have
\begin{align*}
\Phi_{Q_l'}=(\sum_{j=0}^{\f{\kappa_l,\alpha_l}^{(1)}-1} \f{B_l}\circ R^j_{B_l} ,\sum_{j=\f{\kappa_l,\alpha_l}^{(1)}}^{\f{\kappa_l,\alpha_l}^{(2)}-1} \f{B_l}\circ R^j_{B_l},...).
\end{align*}
We can apply Proposition \ref{sumweakconprop}
\begin{equation*}
\Omega \left(\sum_{k=1}^K \theta^{(k)}\right) r_l^{d+d'}\Phi_{Q_l'}\xRightarrow{\mu\times \nu}\Phi_{Exp} \;\;\;\as{l}.
\end{equation*}

(iv) By Proposition \ref{hitretrelcor} also
\begin{equation*}
\Omega \left(\sum_{k=1}^K \theta^{(k)}\right) r_l^{d+d'} (\mu\times\nu)(Q_l')\Phi_{Q_l'}\xRightarrow{{\mu\times \nu}_{Q_l'}}\Phi_{Exp} \;\;\;\as{l}.
\end{equation*}
At the same time, taking a subsequence if necessary, there are $[0,\infty]$-valued processes $\Phi$ and $\tilde{\Phi}$ such that
\begin{align*}
&(\mu\times\nu)(Q_l)\Phi_{Q_l}\xRightarrow{\mu\times \nu}\Phi\;\;\;\as{l},\; \textit{ and}\\
&(\mu\times\nu)(Q_l)\Phi_{Q_l}\xRightarrow{(\mu\times \nu)_{Q_l}}\tilde{\Phi}\;\;\;\as{l}.
\end{align*}

Hence
$
D\left(\Phi_{Exp},\tilde{\Phi}\right)\leq \epsilon,
$
where $D$ is given by Lemma \ref{weakapproxlem}.
Since this is true for every $\epsilon>0$ we have $\tilde{\Phi}\overset{d}{=}\Phi_{Exp}$, and by Corollary \ref{hitretrelcor} also $\Phi\overset{d}{=}\Phi_{Exp}$.
\end{proof}

\section{The skewing time}
Here we will prove Theorem \ref{plt:2}, to do this we will verify that the map $T(x,y)=G_{\tau(y)}(x)$ satisfies superpolynomial mixing of all orders, i.e condition (SPM) of Remark \ref{delaykapretrem}(iii).

\begin{lemma}\label{skewingtimelemma}
In the setting of Theorem \ref{plt:2}, suppose that
\begin{equation*}
\nu(|\tau_n|<\zeta(n))=O(n^{-\kappa}) \;\;\; \as{n},
\end{equation*}
for some $\zeta:\mathbb{N}\rightarrow (0,\infty)$ and $\kappa>\frac{d'}{\delta_2}$, and $\sum_{l\geq 1} r_l^{\frac{1}{2}(d'-\delta_2 \kappa)} <\infty$. Then, for each $t>0$, there is a set $\mathfrak{G}_t$ with $\nu(\mathfrak{G}_t)=1$ such that, for $y\in \mathfrak{G}_t$, there are $L_{y,t}>0$ and sets $\mathcal{B}_{l,y,t}\subset \{1,...,\fceil{t}{\mu(A_l)}\}$ with $|\mathcal{B}_{l,y,t}|=o(\mu(A_l)^{-1})$ such that
$$
|\tau_{\tilde{\alpha}_l^{(n)}}(y)-\tau_{\tilde{\alpha}_l^{(m)}}(y)|\geq
\zeta\left(\tilde{\alpha}_l^{(n)}(y)-\tilde{\alpha}_l^{(m)}(y)\right)
$$$$
\forall l\geq L_{t,y}, 1\leq n< m \leq \fceil{t}{\mu(A_l)}, n\not\in \mathcal{B}_{l,y,t}
$$
where $\alpha_l=\Phi_{B_l}$.
\end{lemma}
\begin{proof}
Fix $t>0$, to keep notation simple we assume $\mu(A_l)=r_l^d+o(r_l^d)$ and $\nu(B_l)=r_l^{d'}+o(r_l^{d'})$, otherwise there is an extra constant in the estimates below.

(i) We call $\displaystyle n\in \left\{1,...,\fceil{t}{\mu(A_l)}\right\}$ a $(l,y)$-bad return if there is a $m>n$ such that
\begin{equation*}
|\tau_{\tilde{\alpha}_l^{(n)}}(y)-\tau_{\tilde{\alpha}_l^{(m)}}(y)|< \zeta(\tilde{\alpha}_l^{(n)}(y)-\tilde{\alpha}_l^{(m)}(y)),
\end{equation*}
denote $\mathcal{B}_{l,y}=\{n\geq 1\;|\; n \textit{ is a } (l,y)-\textit{bad return}\}$. Let $0<\epsilon_1$ We call $y\in Y$ an $l$-bad point if $|\mathcal{B}_{l,y}|<r_l^{-d+\epsilon_1}$.

(ii)  Using Lemma \ref{UClem:1} and Jegorov's theorem, for $\epsilon_2>0$, we can find a measurable $G=G_{\epsilon_2}\subset Y$ with $\nu(G^c)<\epsilon_2$ and an $\tilde{L}\geq 1$ depending on $G$ such that
\begin{equation*}
\tilde{\alpha}_l^{(\ffloor{t}{\mu(A_l})}\leq 2tr_l^{-(d+d')} \;\;\;\forall y\in G, l\geq \tilde{L}.
\end{equation*}

(iii) For $l\geq 1$ denote $G_l=\{\tilde{\alpha}_l^{\fceil{t}{\mu(A_l)}}\leq 2tr_l^{-(d+d')}\}$, we have
$$
\nu( y\in G_l \;|\;  y \textit{ is } l\textit{-bad})\leq \frac{\int_{G_l} |\mathcal{B}_{l,y}| \d\nu}{r_l^{-d+\epsilon}}
\leq r_l^{d-\epsilon} \sum_{n=1}^{\fceil{t}{\mu(A_l)}} \nu(y\;|\; n \textit{ is } (l,y)\textit{-bad})$$
$$\leq r_l^{d-\epsilon} \sum_{j=1}^{\lceil 2tr_l^{-(d+d')}}
\nu \left(T^{-j}\left(\exists i\geq 1\;|\; |\tau_{\tilde{\alpha}_l^{(i)}}|<
\zeta\left(\tilde{\alpha}_l^{(i)}\right)\right)\right) $$
$$\leq r_l^{d-\epsilon}  \sum_{j=1}^{\lceil 2tr_l^{-(d+d')}} \nu(\exists i\geq r_l^{-\delta_2} \;|\; |\tau_i|<\zeta(i))$$
$$\leq 2Kt r_l^{d-\epsilon-d-d'+\delta_2\kappa}\leq 2Kt r_l^{\delta_2\kappa-d'},
$$
for some constant $K>0$, for small enough $\epsilon$ this is summable. An application of the Borel-Cantelli Lemma yields that for almost every $y\in Y$; for big enough $l$, either $y\not \in G_l$ or
$$
|\tau_{\tilde{\alpha}_l^{(n)}}(y)-\tau_{\tilde{\alpha}_l^{(m)}}(y)|\geq \zeta(\tilde{\alpha}_l^{(n)}(y)-\tilde{\alpha}_l^{(m)}(y)) \;\;\;\forall 1\leq n< m \leq \fceil{t}{\mu(A_l)}, n\not\in \mathcal{B}_{l,y,}.
$$
At the same time, by Proposition \ref{retqergodicprop} we have $G_l\nearrow Y$. Thus a.e $y$ is in $G_l$ for big enough $l$, and the conclusion follows.
\end{proof}

\begin{proof}[Proof of Theorem $\ref{plt:2}$]
In order to keep notation simple we will only show the PLT for regularly approximable rectangles, this can be easily extended to geodesic balls, by following the same arguments as in the proof of Theorem~\ref{plt:1}.
\\

(i) For a.e $y$ and $0=t_0<t_1<...<t_J$ choose $L_y=L_{y,t_1+...+t_J}$ and $\mathcal{B}_{l,y}=\mathcal{B}_{l,y,t_1+...+t_J}$ as in Lemma \ref{skewingtimelemma}. For such a $y$ and $l\geq L_y$ (in the following we suppress $y$ from the notation) consider
\begin{equation*}
S_{t_j,l}=\sum_{i=1}^{\fceil{t_j}{\mu(A_l)}} 1_{A_l}\circ T^{\tilde{\alpha}_l^{(i)}} = S'_{t_j,l}+S_{t_j,l}'',
\end{equation*}
where
\begin{equation*}
S'_{t_j,l}=\sum_{\substack{i=1,...,\fceil{t_j}{\mu(A_l)} \\ i\not \in \mathcal{B}_{l}}}  1_{A_l}\circ T^{\tilde{\alpha}_l^{(j)}}.
\end{equation*}
As in Proposition \ref{delaykapretprop}, the first goal is to show
\begin{equation*}
\left( S_{t_1,l}-S_{t_0,l},...,S_{t_J,l}-S_{t_{J-1},l} \right) \xRightarrow{\mu} \left( P_{t_1-t_0},...,P_{t_J-t_{J-1}}\right) \;\;\;\as{l},
\end{equation*}
where $(P_t)$ is a standard Poisson process. Since $||S_{t_j,l}''||_{L^1}\rightarrow 0$ for all $j=1,...,J$, it is equivalent to show
\begin{equation*}
\left( S'_{t_1,l}-S'_{t_0,l},...,S'_{t_J,l}-S'_{t_{J-1},l} \right) \xRightarrow{\mu} \left( P_{t_1-t_0},...,P_{t_J-t_{J-1}}\right) \;\;\;\as{l}.
\end{equation*}
For $m_1,...,m_J\geq 1$ it will be enough to show
\begin{equation}\label{momentsinproofofplt:2}
\int_X \prod_{j=1}^J { {S'_{t_j,l}-S'_{t_{j-1},l}} \choose m_j} \d\mu = \prod_{j=1}^J \frac{(t_j-t_{j-1})^{m_j}}{m_j!}.
\end{equation}
We have
\begin{equation}\label{suminproofofplt:2}
\prod_{j=1}^J { {S'_{t_j,l}-S'_{t_{j-1},l}} \choose m_j} = \sum_{\substack{\fceil{t_{j-1}}{\mu(A_l)}+1\leq k_{j,1}<...<k_{j,m_j} \leq \fceil{t_j}{\mu(A_l)}\\ k_{i,j}\not\in \mathcal{B}_l \textit{ for } j=1,...,J,\; i=1,...,m_j}} \prod_{i,j} 1_{A_l} \circ T^{\tilde{\alpha}_l^{(k_{i,j})}}.
\end{equation}
(ii) Due to assumption (MEM) for $G$, and Lemma \ref{skewingtimelemma}, we have
\begin{equation*}
\left|\int_X \prod_{j=1}^m f_j \circ T^{\tilde{\alpha}_l^{(n_j)}} \d\mu- \prod_{j=1}^m \int_X f_j \d\mu \right|\leq C_y \psi(\min_{j\not =j'}|\tilde{\alpha}_l^{(n_j)}-\tilde{\alpha}_l^{(n_{j'})}|) \prod_{j=1}^m ||f_j||_{C^r},
\end{equation*}
where $\psi(x)=e^{-\gamma \zeta(x)}$ and $\zeta$ is as in assumption (BA), for $f_1,...,f_m\in C^r$ and $1\leq n_1\leq...\leq n_m\leq \fceil{t1+...+t_J}{r_l^d}$ with $n_i\not\in \mathcal{B}_l$. Due to \eqref{bigreturnsinplt:2} and assumption (BA) we have
\begin{equation*}
\psi(\min_{j\not =j'}|\tilde{\alpha}_l^{(n_j)}-\tilde{\alpha}_l^{(n_{j'})}|)=O(r_l^{w_l}),
\end{equation*}
for some $w_l>0$ with $w_l\rightarrow\infty$ as $l\rightarrow\infty$. Approximating $1_{A_l}$ by functions in $C^r$ it is straightforward\footnote{The calculation is analogous to Lemma \ref{decorrlem}.} to show that
\begin{equation*}
\left| \int_X \prod_{i,j} 1_{A_l} \circ T^{\tilde{\alpha}_l^{(k_{i,j})}} \d\mu - \mu(A_l)^{m_1+...+m_J}\right| =o(\mu(A_l)^{m_1+...+m_J}),
\end{equation*}
for $k_{i,j}$ as in \eqref{suminproofofplt:2}. The sum in \eqref{suminproofofplt:2} has
$$\mu(A_l)^{-(m_1+...+m_J)}\prod_{j=1}^J \frac{(t_j-t_{j-1})^m_j}{m_j!}+o(\mu(A_l)^{-(m_1+...+m_J)})$$ many terms, so \eqref{momentsinproofofplt:2} follows.
\end{proof}

\section{Examples}\label{explesec}
Here we verify conditions (EE) and (BR) for the examples listed in Section~\ref{ScEx}.

\subsection{Diophantine rotations}
\label{SSRot}
Let $\alpha\in \left((0,1)\setminus \mathbb{Q}\right)^{d'}$ satisfy a diophantine condition, i.e there are $C>0$ and $n\geq 1$ such that
\begin{equation*}\tag{D}
\left| \langle k, \alpha \rangle -l\right| >C|k|^{-n} \;\;\;\forall k\in \mathbb{Z}^{d'},k\not=0,l\in\mathbb{Z},
\end{equation*}
and $R=R_{\alpha}:x\mapsto x+\alpha \;(mod\;1)$, for $x\in \mathbb{T}^{d'}$, the rotation by $\alpha$. In fact almost all $\alpha$ satisfy $(D)$ for some $n>d'$ (this is a consequence of a higher dimensional version of Khinchin's Theorem, see eg \cite{DiophantineBRV}).\\
Note that $(D)$ implies that there is a   constant $C'>0$. such that
\begin{equation*}
\left|1-e^{2\pi i \langle k,\alpha\rangle}\right| \geq C' |k|^{-n} \;\;\;\forall k\in \mathbb{Z}^{d'}\setminus \{0\}.
\end{equation*}

Property $(BR)$ follows directly from $(D)$. Due to the self-symmetry of $R_{\alpha}$ it is enough to consider returns of $x=0$ to a rectangle $(-r,r)^{d'}$, but
\begin{equation*}
R_{\alpha}^m(0)\in (-r,r)^{d'} \iff |m\alpha - k|<r \;\;\; \textit{for some } k\in \mathbb{Z}^{d'}.
\end{equation*}
Then, by $(D)$, $Cm^{-n}<r$, equivalently $m>(C^{-1}r)^{-\frac{1}{n}}$.
\\
To show effective equidistribution $(EE)$, we solve the homological equation. Let $f\in H^n(\mathbb{T}^{d'})$ with $\int f=0$. Then
\begin{equation*}
f(x)=\sum_{k\in \mathbb{Z}^{d'}} a_k e^{2\pi i \langle k,x\rangle},
\end{equation*}
where $\displaystyle \sum_{k\in \mathbb{Z}^{d'}} |a_k|^2 \sum_{j_1+...+j_{d'}=n} \prod_{i=1}^{d'} |k_i|^{2j_i} < \infty$ and $a_0=0$. To solve $f\!\!=\!\!g-g\circ R_{\alpha}$ we write
\begin{equation*}
g(x)=\sum_{k\in \mathbb{Z}^{d'}} b_k e^{2\pi i \langle k,x\rangle}.
\end{equation*}
By comparing coefficients, this is satisfied for $b_k=\frac{a_k}{1-e^{2\pi i \langle k,\alpha \rangle}}$ for $k\not=0$ and $b_0=0$. We have
$$
\sum_{k\in \mathbb{Z}^{d'}} |b_k|^2  \leq (C')^2 \sum_{k\in \mathbb{Z}^{d'}} |a_k|^2 |k|^{2n}
 \leq (C')^2 \sum_{k\in \mathbb{Z}} |a_k|^2 \sum_{j_1+...+j_{d'}=n} \prod_{i=1}^{d'} |k_i|^{2j_i} <\infty.
$$
In particular $||g||_{L^2}\leq C'||f||_{H^n}$. \\
Thus for every\footnote{If $\int h \not=0$ consider $f=h-\int h$.} $h\in H^n(\mathbb{T}^{d'})$ we have
\begin{equation*}
\left| \left| \sum_{j=1}^J h\circ R_{\alpha}^j - J\int_{\mathbb{T}} h \d\lambda \right| \right|_{L^2} \leq C'' ||h||_{H^n}.
\end{equation*}
Due to Remark \ref{conditionsrem}(iii), condition (EE) is satisfied with $r=n$ and $\delta_1=0$.\\

In this particular case \eqref{dimdec} becomes
$\displaystyle
d>\frac{3}{2}d'(n+1).
$

\subsection{Horocycle flows}
\label{SSHoro}
Consider the classical horocycle flow $h_t$ on (compact) homogeneous space $\Gamma \backslash PSL(2,\mathbb{R})$ generated by \begin{equation*}
h_t=\begin{bmatrix}
1&t\\0&1
\end{bmatrix}.
\end{equation*}
For fixed $t>0$ we will consider the time $t$ map $R=h_t$.\\
\\

Condition (BR) follows from the relation
$h_{e^{2s} t}=g_s\circ h_t\circ g_{-s}$, where $g_s$ is the geodesic flow
\begin{equation*}
g_s=\begin{bmatrix}
e^s&0\\
0&e^{-s}
\end{bmatrix}.
\end{equation*}
Indeed, we will show that there is a constant $c>0$ such that for small enough $r>0$, $0<|s|<cr^{-1}$, and $x,y\in \Gamma / PSL(2,\mathbb{R})$ with $d(x,y)<\frac{c}{2}r^{\frac{1}{2}}$ we have
\begin{equation*}
d(h_s x,y)\geq \frac{c}{2}r^{\frac{1}{2}}.
\end{equation*}
By the triangle inequality it is enough to show $d(h_s x,x)\geq cr^{\frac{1}{2}}$. By compactness choose $c=\inf_x d(h_1 x, x)>0$ (by Hedlund's Theorem there are no periodic orbits). Let $t=\frac{\log(|s|)}{2}$, then
\begin{equation*}
d(h_s x, x)=d(g_t h_{sgn(s)} g_{-t} x, g_tg_{-t}x)\geq e^{-|t|} d( h_{sgn(s)} g_{-t} x, g_{-t}x)\geq cr^{\frac{1}{2}}.
\end{equation*}
For small enough $r$, $g_t$ contracts distances at most by a factor of $e^{-|t|}$. Renaming $r=\frac{c}{2}r^{\frac{1}{2}}$ gives (BR).

In order to show effective ergodicity (EE), we combine
\cite[Corollary 2.8]{FFT} and \cite[Theorem 1.2 and Corollary 1.3]{FF} to conclude that
\begin{equation*}
\left|\left|\sum_{j=0}^{n-1} f\circ R^j -n\int f \d\nu\right|\right|_{L^2(\nu)} \leq C||f||_{C^{12}} \;\;
n^{\alpha+\epsilon} \;\;\;\forall f\in C^{12}, \; \forall n\geq 1,
\end{equation*}
for some\footnote{Denoting $\mu_0>0$ the smallest positive eigenvalue of the Casimir operator, we have $\displaystyle \alpha=\max\left(\frac{1+\sqrt{\max(0,1-4\mu_0)}}{2}, \frac{5}{6}\right) $.}
 $\alpha\in (0, 1)$ and all $\epsilon>0$.
Then Theorem \ref{plt:1} applies with
\begin{equation*}
d>\frac{3}{2}\times \frac{2\times 7}{1-\alpha}=\frac{21}{1-\alpha}.
\end{equation*}

\subsection{Skew shifts}

Let $\alpha\in (0,1)\setminus \mathbb{Q}$ satisfy the diophantine condition $(D)$ for some $n\geq 2$ and $R:\mathbb{T}^2\rightarrow\mathbb{T}^2$ be given by
\begin{equation*}
R(x,y)=(x+\alpha,y+x).
\end{equation*}
Since $R$ has a Diophantine rotation as a factor (BR) is satisfied by \S \ref{SSRot}.

For $k=(k_1,k_2) \in \mathbb{Z}^2$ denote $e_k(x)=e^{2\pi i \langle k,x\rangle}$.
Note that $$\langle e_k,e_{k'}\circ R^j \rangle_{L^2(\mathbb{T}^2)} =\delta^{(k_1,k_2)}_{(k'_1+jk'_2,k'_2)}.$$ For $f\in H^2(\mathbb{T})$ we can write $f=\sum_{k\in \mathbb{Z}^2} a_k e_k$. If $a_{(k_1,0)}\equiv 0$ (in particular $\int f=0$) then
$$
\sum_{j\geq 1} \left| \langle f,f\circ R^j \rangle_{L^2}\right| = \sum_{j\geq 1} \left|
\sum_{k\in \mathbb{Z}^2} a_{(k_1,k_2)} \overline{a_{(k_1+jk_2,k_2)}}\right|\!
\leq \!\left(\sum_{k\in \mathbb{Z}^2} |a_{(k_1,k_2)}| \right)^2 \!\!\!
\leq C||f||_{H^2}^2,
$$
where $C>0$ does not depend on $f$. From this we obtain
\begin{align*}
\left|\left| \sum_{j=1}^J f \circ R^j\right|\right|_{L^2(\mathbb{T}^2)}^2&
= \sum_{j=1}^J \sum_{j'=1}^J \langle f\circ R^{j'},f\circ R^j \rangle_{L^2(\mathbb{T}^2)}
\\&
\leq \sum_{j=0}^{J-1} (J-j) \langle f,f\circ R^j \rangle_{L^2(\mathbb{T}^2)}
\leq CJ||f||_{H^2(\mathbb{T}^2)}^2.
\end{align*}
For general $f\in H^n(\mathbb{T}^2)$ with $\int f=0$, again write $f=\sum_{k\in \mathbb{Z}^2} a_k e_k$ and set
\begin{align*}
f_1=\sum_{k\in \mathbb{Z}^2,k_2\not =0} a_k e_k \;\;\textit{ and }\;\;f_2=\sum_{k\in \mathbb{Z}^2,k_2=0} a_k e_k.
\end{align*}
Applying the above, and the analysis for diophantine rotations, we find
\begin{equation*}
\left|\left| \sum_{j=1}^J f \circ R^j\right|\right|_{L^2(\mathbb{T}^2)}^2\leq (CJ+C')||f||_{H^n(\mathbb{T}^2)}^2.
\end{equation*}
Thus the condition in Proposition \ref{l2boundprop} is satisfied with $\delta=\frac{1}{2}$.
So we can apply Theorem \ref{plt:1} with
$
d>3(n+1).
$

\subsection{Example \ref{explethm:2}}
Recall the definition of the Weyl Chamber flow
on $ \Gamma \backslash SL(d,\mathbb{R})$.
Let $d\geq 3$, and $\Gamma$ be a uniform lattice. Denote by $D_+$ the subgroup of diagonal elements of
$SL(d,\mathbb{R})$ with positive entries. It is easy to see that $D_+$ is isomorphic to $\mathbb{R}^{d-1}$.
$D_+$ acts on $\Gamma \backslash Sl(d,\mathbb{R}) $ by right translation, giving us a $\mathbb{R}^{d-1}$-action. By \cite{MultMixBEG} the action $G$ satisfies (a $\mathbb{R}^{d-1}$ version of) (MEM).\\
\\
The diophantine rotation $R_{\alpha}$ satisfies (EE) and (BR) by \S \ref{SSRot}.
Hence we can apply Theorem \ref{plt:2}.

\section{Robustness of delayed returns}\label{robustnesssec}

Lastly we mention the proof for the delayed PLT. All off the above proofs can be done using $\Phi_{B_l,\alpha}$ instead of $\Phi_{B_l}$, this shows (with the notation from the proof of Theorem \ref{plt:1})
\begin{equation*}
(\mu\times\nu)\left( \bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)}\right) \Phi_{\bigcup_{k=1}^K A_l^{(k)}\times B_l^{(k)},\alpha} \xRightarrow{\mu} \Phi_{Exp}.
\end{equation*}
To conclude, we only need a version of the approximation Lemma \ref{weakapproxlem} for delayed return times.\\

Let $(M,d_M)$ be a compact metric space, let $\s{\tilde{\vartheta}}{n}$ be a sequence of Lipschitz functions on $M$ dense in $C(M)$, and denote $\vartheta_n=\frac{\tilde{\vartheta}_n}{||\tilde{\vartheta}_n||_{Lip}}$. The metric
\begin{equation*}
D_{M}(\lambda,\lambda')=\sum_{n\geq 1} 2^{-n} \left|\int_{[0,\infty]} \vartheta_n \d\lambda- \int_{M} \vartheta_n \lambda\right|,
\end{equation*}
for probability measures $\lambda$ and $\lambda'$, models distributional convergence\footnote{In the sense that $\lambda_n\Rightarrow\lambda$ iff $D_{M}(\lambda,\lambda_n)\rightarrow0$.}.

\begin{lemma}\label{robustnesslem}
Let $(X,\mu,T)$ be a probability-preserving dynamical system, $\alpha$ be a sequence of natural numbers, $\s{A}{l}$ be a sequence of rare events, and $\Phi=(\phi^{(1)},\phi^{(2)},...)$ be a random process in $[0,\infty)$. Assume that, for each $\delta>0$, there is a sequence of rare events $\s{A^{(\delta)}}{l}$ with $A_l^{(\delta)}\subset A_l$ and $\mu_{A_l}(A_l\setminus A_l^{(\delta)})<\delta$ such that
\begin{equation*}
\mu(A_l^{(\delta)})\Phi_{A_l^{(\delta)},\alpha} \xRightarrow{\mu} \Phi \;\;\;;\as{l}, \forall\delta>0.
\end{equation*}
Then
\begin{equation*}
\mu(A_l)\Phi_{A_l,\alpha} \xRightarrow{\mu} \Phi \;\;\;\as{l}.
\end{equation*}
\end{lemma}

\begin{proof}
Taking a subsequence if necessary, we may assume that there is a $[0,\infty]$-valued random process $\Phi'$ with
\begin{equation*}
\mu(A_l)\Phi_{A_l,\alpha} \xRightarrow{\mu} \Phi' \;\;\;\as{l}.
\end{equation*}
For $s,t\in [0,\infty]$ denote $d_{[0,\infty]}(s,t)=|e^{-s}-e^{-t}|$, where by convention $e^{-\infty}=0$, this makes $([0,\infty],d_{[0,\infty]}$ a compact metric space. Then $([0,\infty]^{\mathbb{N}},d_{[0,\infty]^{\mathbb{N}}})$ is a compact metric space with $diam([0,\infty]^{\mathbb{N}})=1$, where
\begin{equation*}
d_{[0,\infty]^{\mathbb{N}}}((s_j),(t_j))=\sum_{j\geq 1} 2^{-j} d_{[0,\infty]}(s_j,t_j).
\end{equation*}
We claim that for every $\epsilon>0$ there exist $\delta_0>0$ and an $L\geq 1$ such that
\begin{equation}\label{robustnesseq:1}
D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_l^{(\delta)})\Phi_{A_l^{(\delta)},\alpha}),law_{\mu}(\mu(A_l)\Phi_{A_l,\alpha})\right)<5\epsilon \;\;\;\forall l\geq L.
\end{equation}
Then taking $l\rightarrow\infty$ shows $D_{[0,\infty]^{\mathbb{N}}}(\Phi,\Phi')<\epsilon$ and the conclusion follows by $\epsilon\rightarrow0$.\\
\\
Let $1>\epsilon>0$. First note that\footnote{For $k\geq 1$ and $s,t\in[0,\infty]$ we have $d_{[0,\infty]}(ks,kt)\geq d_{[0,\infty]}(s,t)\leq |s-t|$.}
\begin{equation*}
D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_l^{(\delta)})\Phi_{A_l^{(\delta)},\alpha}),law_{\mu}(\mu(A_l)\Phi_{A_l^{(\delta)},\alpha})\right)<\delta,
\end{equation*}
so it is enough to show that there exist $\epsilon>\delta_0>0$ and an $L\geq 1$ such that
\begin{equation}\label{robustnesseq:2}
D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_l)\Phi_{A_l^{(\delta)},\alpha}),law_{\mu}(\mu(A_l)\Phi_{A_l,\alpha})\right)<4\epsilon \;\;\;\forall l\geq L.
\end{equation}
Denote $\Phi_{A_l,\alpha}=(\f{A_l,\alpha}^{(1)},\f{A_l,\alpha}^{(2)},...)$ and $\Phi_{A_l^{(\delta)},\alpha}=(\f{A_l^{(\delta)},\alpha}^{(1)},\f{A_l^{(\delta)},\alpha}^{(2)},...)$. Now choose $J\geq 1$ so big that $\sum_{j\geq J} 2^{-j} <\epsilon$, and $T>0$ such that
\begin{equation*}
\mathbb{P}\left(\sum_{j=1}^J \phi^{(j)} >T \right)<\epsilon.
\end{equation*}
For $\delta=\min \left( \frac{\epsilon}{2}, \frac{\epsilon}{2T} \right)$ choose $L\geq 1$ so big that
\begin{equation*}
\mu\left(\sum_{j=1}^J \mu(A_l^{(\delta)}) \f{A_l^{(\delta)},\alpha}^{(j)} >T \right)<2\epsilon \;\;\;\forall l\geq L.
\end{equation*}
Since $\displaystyle \sum_{j=1}^J \f{A_l^{(\delta)},\alpha}^{(j)}>\sum_{j=1}^J \f{A_l,\alpha}^{(j)}$ and $\mu(A_l^{(\delta)})>(1-\delta)\mu(A_l)>\frac{1}{2}\mu(A_l)$, in particular
\begin{equation*}
\mu\left(\sum_{j=1}^J \mu(A_l) \f{A_l,\alpha}^{(j)} > 2T \right)<2\epsilon \;\;\;\forall l\geq L.
\end{equation*}
Now, for $j=1,...,j$, we have
\begin{align*}
\mu & \left(\f{A_l,\alpha}^{(j)}\not= \f{A_l^{(\delta)},\alpha}^{(j))}\right) \leq \mu\left(\f{A_l,\alpha}^{(j)}\not= \f{A_l^{(\delta)},\alpha}^{(J))}\right)\\
&\leq \mu\left(\bigcup_{i=1}^{\ffloor{2T}{\mu(A_l)}} T^{-\tilde{\alpha}^{(i)}} (A_l\setminus A_l^{(\delta)})\right) + \mu\left(\sum_{j=1}^J \mu(A_l) \f{A_l,\alpha}^{(j)} > 2T \right)\\
&\leq \frac{2T}{\mu(A_l)}\mu(A_l)\delta+2\epsilon\leq 3\epsilon.
\end{align*}
Thus
\begin{align*}
&D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_l)\Phi_{A_l^{(\delta)},\alpha}),law_{\mu}(\mu(A_l)\Phi_{A_l,\alpha})\right)\\
&=\sum_{j\geq 1} 2^{-j} \int_X d_{[0,\infty]}(\mu(A_l)\f{A_l,\alpha}^{(j)},\mu(A_l)\f{A_l^{(\delta)},\alpha}^{(j))}) \d\mu
\leq \sum_{j=1}^J 2^{-j} 3\epsilon + \epsilon \leq 4\epsilon
\end{align*}
proving \eqref{robustnesseq:2}.
\end{proof}

\newcommand{\etalchar}[1]{$^{#1}$}

redpajama_set_name: RedPajamaArXiv