# Dynamical Borel–Cantelli Lemma for Recurrence under Lipschitz Twists

Dmitry Kleinbock  Department of Mathematics, Brandeis University, Waltham MA
<EMAIL_ADDRESS>and  Jiajie Zheng  Department of Mathematics, Brandeis
University, Waltham MA<EMAIL_ADDRESS>
(Date: October 19, 2022)

###### Abstract.

In the study of some dynamical systems the limsup set of a sequence of
measurable sets is often of interest. The shrinking targets and recurrence are
two of the most commonly studied problems that concern limsup sets. However,
the zero-one laws for the shrinking targets and recurrence are usually treated
separately and proved differently. In this paper, we introduce a generalized
definition that can specialize into the shrinking targets and recurrence; our
approach gives a unified proof of the zero-one laws for the two problems.

D.K. was supported by NSF grant DMS-1900560. This material is based upon work
supported by a grant from the Institute for Advanced Study School of
Mathematics.

##  1\. Introduction

Throughout the paper, let $(X,d)$ be a separable and compact metric space, and
let $(X,\mu,T)$ be a probability measure preserving system. One of the most
fundamental results in ergodic theory is the Poincaré Recurrence Theorem, see
e.g. [EW, Theorem 2.11], which asserts that almost all points in measurable
dynamical systems return close to themselves under a measure-preserving map;
namely, that

$\mu(R_{T})=1,$ (1.1)

where $R_{T}$ is the set of recurrence for $T$:

$R_{T}:=\\{x\in X:\liminf_{n\to\infty}d(T^{n}x,x)=0\\}.$

One of the first results concerning the speed of recurrence is due to
Boshernitzan in [B]. Namely, assume that the $\alpha$-dimensional Hausdorff
measure of $X$ is zero for some $\alpha>0$. Then

$\liminf_{n\to\infty}n^{1/\alpha}d(T^{n}x,x)=0$

for $\mu$-almost every $x\in X$. In other words, for a function
$\psi:\mathbb{N}\to\mathbb{R}^{+}$ let us define the following set:

$R_{T}(\psi):=\left\\{x\in X:d(T^{n}x,x)<\psi(n)\text{ for infinitely many
}n\in\mathbb{N}\right\\}.$

Then the Poincaré Recurrence Theorem says that the set

$R_{T}=\bigcap_{{\varepsilon}>0}R_{T}({\varepsilon}1_{\mathbb{N}})$

has full measure, and, with the notation

${\psi_{s}(x):=x^{-s},}$ (1.2)

Bosherniztan’s result says that $R_{T}({\varepsilon}\psi_{1/\alpha})$ has full
measure for any ${\varepsilon}>0$ and for any $\alpha$ such that
$\mathcal{H}^{\alpha}(X)=0$.

It is a natural problem to find necessary and sufficient conditions on $\psi$
to guarantee that the set $R_{T}(\psi)$ has measure zero or one. In fact,
under some additional assumptions one expects this condition to be the
convergence/divergence of the sum of measures of the sets

$A_{T}(n,\psi):=\left\\{x\in X:d(T^{n}x,x)<\psi(n)\right\\}.$ (1.3)

And indeed this was proved in several special cases such as [BF, CWW, HLSW];
see also [KKP, DFL, Pe] for similar results.

Note that a topic closely related to recurrence is the so-called shrinking
target problem, which is concerned with determining the speed at which the
orbit of a $\mu$-typical point accumulates near a fixed point $y\in X$. More
precisely, for $y\in X$ one can define the set

$R_{T}^{y}:=\left\\{x\in X:\liminf_{n\to\infty}d(T^{n}x,y)=0\right\\},$

and, more generally, for a function $\psi:\mathbb{N}\to\mathbb{R}^{+}$ define

$R_{T}^{y}(\psi):=\left\\{x\in X:d(T^{n}x,y)<\psi(n)\text{ for infinitely many
}n\in\mathbb{N}\right\\}.$

Equivalently, letting $B(x,r)$ stand for the open ball in $X$ centered in $x$
of radius $r$, we can write $R_{T}^{y}(\psi)=\limsup A^{y}(n,\psi)$, where

$A_{T}^{y}(n,\psi):=\left\\{x\in
X:d(T^{n}x,y)<\psi(n)\right\\}=T^{-n}B\big{(}y,\psi(n)\big{)}.$ (1.4)

Clearly

$\mu(R_{T}^{y})=1\text{ for any $y\in\operatorname{supp}\mu$ \ \ if $T$ is
ergodic};$ (1.5)

furthermore, there have been plenty of results in the literature giving
$0$–$1$ laws for $\mu\big{(}R_{T}^{y}(\psi)\big{)}$. In fact, one can often
use mixing properties of $T$ to conclude that
$\mu\big{(}R_{T}^{y}(\psi)\big{)}$ is equal to zero/one if and only if the
series

$\sum_{n=1}^{\infty}\mu\big{(}A_{T}^{y}(n,\psi)\big{)}=\sum_{n=1}^{\infty}\mu\Big{(}B\big{(}y,\psi(n)\big{)}\Big{)}$

converges/diverges. See [Ph, CK, KM, FMP, HNPV] and many other references.

The goal of the current paper is to study a property unifying these two
settings, and to prove a zero–one law applying to both. Namely, for a Borel
measurable function $f:X\to X$ define $R_{T}^{f}$, the set of $f$-twisted
recurrent points for $T$, by

$R_{T}^{f}:=\left\\{x\in
X:\liminf_{n\to\infty}d\big{(}T^{n}x,f(x)\big{)}=0\right\\}.$

The two previous settings correspond to $f$ being the identity and constant
functions respectively. We will show in the next section that
$\mu(R_{T}^{f})=1$ for any measurable $f$ if $T$ is ergodic and $\mu$ has full
support. Furthermore, one can study the rate of twisted recurrence as follows:
for $\psi:\mathbb{N}\to\mathbb{R}^{+}$ define

$R_{T}^{f}(\psi):=\left\\{x\in X\left|\begin{aligned}
d\big{(}T^{n}x,f(x)\big{)}<\psi(n)\ \ \\\ \text{ for infinitely many
}n\in\mathbb{N}\end{aligned}\right.\right\\},$ (1.6)

so that $R_{T}^{f}=\bigcap_{{\varepsilon}>0}R_{T}^{f}({\varepsilon}1_{X})$. In
general the rate of twisted recurrence can be arbitrary slow, see §2 for
examples. The main goal of the paper is to prove, under assumptions similar to
those of [HLSW], a zero–one law for the sets $R_{T}^{f}(\psi)$ for a large
class of functions $f$.

To state the main result of the paper, we need to adapt and modify the
settings and assumptions from [HLSW]. Throughout the paper we write $a\lesssim
b$ if $a\leq Cb$ for some constant $C>0$, and $a\asymp b$ if $a\lesssim b$ and
$b\lesssim a$.

Our main assumption is that there exist at most countably many pairwise
disjoint open subsets $X_{i}$, $i\in\mathcal{I}$, of $X$ such that
$T|_{X_{i}}$ is continuous and injective for each $i$, and
$\mu(X{\smallsetminus}\cup_{i}X_{i})=0$. Those will be called cylinders of
order $1$. Then for any $m\in{\mathbb{N}}$ one can define

$\mathcal{F}_{m}:=\left\\{X_{i_{1}}\cap T^{-1}X_{i_{2}}\cap\cdots\cap
T^{-(m-1)}X_{i_{m}}:i_{1},\dots,i_{m}\in\mathcal{I}\right\\}$ (1.7)

to be the collection of cylinders of order $m$. Note that for
$J\in\mathcal{F}_{m}$ and $x,y\in J$, the points $T^{n}x$ and $T^{n}y$ are in
the same partition set $X_{i}$ for $0\leq n<m$, and hence $T,\ldots,T^{m}$ are
injective on $J$. Also, since $T$ is continuous, each cylinder in
$\mathcal{F}_{m}$ is open.

Now let us list our assumptions on the measure $\mu$. The first one is Ahlfors
regularity of dimension $\delta>0$; namely, that there exist positive real
numbers $\eta_{1},\eta_{2},r_{0}$ such that

$\eta_{1}r^{\delta}\leq\mu(B(x,r))\leq\eta_{2}r^{\delta}\text{ for any ball
$B(x,r)\subset X$ with }{0<r<r_{0}}.$ (1.8)

As a consequence, since $\mu$ was assumed to be a probability measure, the
space $X$ has finite diameter.

Next, we assume that $(X,\mu,T)$ is uniformly mixing (a property introduced in
[FMP]), that is: there exist a summable sequence of positive real numbers
$(a_{n})_{n\in{\mathbb{N}}}$ such such that

$\displaystyle\left|\mu(E\cap T^{-n}F)-\mu(E)\mu(F)\right|\leq a_{n}\mu(F)$
(1.9) $\displaystyle\text{for any balls $E,F\subset X$ and for all }n\geq 1.$

Note that it was proved in [FMP] that under the aforementioned mixing
assumption, for any $y\in X$ and any $\psi$ the set $R_{T}^{y}(\psi)$ is null
(resp., conull) if the series

$\sum_{n=1}^{\infty}\mu\big{(}B\big{(}y,\psi(n)\big{)}\big{)}\underset{\eqref{ar}}{\asymp}\sum_{n=1}^{\infty}\psi(n)^{\delta}$

converges (resp., diverges). However, in order to similarly treat the sets
$R_{T}^{f}(\psi)$ for more general functions $f$ we will require some more
information on the expanding properties of $T$. For a $m$-cylinder $J$, we
define

$K_{J}:=\inf_{x,y\in J,\,x\neq y}\frac{d(T^{m}x,T^{m}y)}{d(x,y)},$

and impose the following additional assumptions:

  * •

Bounded distortion: There exists a constant $K_{1}>0$ such that

$\displaystyle
K_{1}^{-1}\leq\frac{d(T^{m}x,T^{m}y)/d(x,y)}{d(T^{m}x,T^{m}z)/d(x,z)}\leq
K_{1}$ (1.10) $\displaystyle\text{for all }m\in{\mathbb{N}}$
$\displaystyle\text{ and }x,y,z\in J\in\mathcal{F}_{m}\text{ with $x\neq y$
and }x\neq z.$

  * •

Expanding properties:

$\inf_{J\in\mathcal{F}_{m}}K_{J}\to\infty\text{ as }m\to\infty$ (1.11)

and

$\sup_{m\in{\mathbb{N}}}\sum_{J\in\mathcal{F}_{m}}K_{J}^{-\delta}<\infty.$
(1.12)

  * •

Conformality: There exists a constant $K_{2}\geq 1$ such that

$\displaystyle B(T^{m}x,K_{2}^{-1}K_{J}r)\subset T^{m}B(x,r)\subset
B(T^{m}x,K_{2}K_{J}r)$ (1.13) $\displaystyle\text{ for any
}m\in{\mathbb{N}}\text{ and any ball }B(x,r)\subset J\in\mathcal{F}_{m}.$

######  Remark 1.1.

Notice that the bounded distortion condition (1.10) implies the second
inclusion in (1.13) with $K_{2}$ replaced by $K_{1}$. However the first
inclusion there does not automatically follow from (1.10), hence the need for
an additional condition.

######  Remark 1.2.

We note that conditions (1.8)–(1.13) are essentially equivalent to Conditions
I–V from [HLSW]. Namely:

  * •

(1.8) is a slightly weaker version of [HLSW, Condition I].

  * •

(1.9) replaces [HLSW, Condition II] where the rate of mixing was assumed to be
exponential.

  * •

As for (1.10)–(1.13), in [HLSW] the standing assumption was that the
restriction of $T$ to $X_{i}$ for every $i$ is differentiable and expanding,
namely it was assumed that

${\|D_{x}(T^{-1})\|^{-1}>1\text{ for any }x\in\cup_{i}X_{i}.}$ (1.14)

The role of (1.10) was played there by [HLSW, Condition III] stated as
follows: there exists a constant $K_{1}>0$ such that

$K_{1}^{-1}\leq\frac{d(T^{m}x,T^{m}y)}{d(x,y)\|D_{x}T^{m}\|}\leq K_{1}\
\forall\,m\in{\mathbb{N}}\text{ and }\forall\,x,y\in J\in\mathcal{F}_{m}\text{
with }x\neq y.$

  * •

Similarly, the constant $K_{J}$ for $J\in\mathcal{F}_{m}$ was defined in
[HLSW] by $K_{J}:=\inf_{x\in J}\|D_{x}T^{m}\|$, and the role of (1.11) was
played by $\inf_{J\in\mathcal{F}_{m}}K_{J}>1$ for some $m\in{\mathbb{N}}$,
which, in view of (1.14), is easily seen to be equivalent to
$\inf_{J\in\mathcal{F}_{m}}K_{J}\to\infty\text{ as }m\to\infty$. Conditions IV
and V of [HLSW] are identical to (1.12) and (1.13) respectively.

Examples of dynamical systems satisfying conditions (1.8)–(1.13) include, as
mentioned in [HLSW], $\beta$-transformations

$M_{\beta}:x\mapsto\beta x\mod{1}$

of the unit interval, where $\beta\in{\mathbb{R}}_{>1}$, as well as the Gauss
map. In §7 we add another example to the list: expanding maps defined by
systems of contracting similarities with the open set condition.

Let us now specify the class of functions $f$ which we can treat by our
technique. Say that $f:X\to X$ is Lipschitz if

$\sup_{x,y\in X,\,x\neq y}\frac{d\big{(}f(x),f(y)\big{)}}{d(x,y)}<\infty,$

and that $f$ is piecewise Lipschitz if there exist at most countably many
measurable subsets $Y_{i}$ of $X$ and Lipschitz functions $f_{i}:X\to X$,
$i\in\mathcal{I}$, such that $\mu(X{\smallsetminus}\cup_{i}Y_{i})=0$ and
$f|_{Y_{i}}=f_{i}$ for each $i$. An example: when $X=[0,1]$, the function
$f(x)=\sqrt{x}$ is piecewise Lipschitz but not Lipschitz.

Now we are ready to state our main theorems.

######  Theorem 1.3.

Assume that $(X,\mu,T)$ satisfies conditions (1.8)–(1.13). Then for any
function $\psi:\mathbb{N}\to\mathbb{R}_{+}$ with $\lim_{n\to\infty}\psi(n)=0$
and any piecewise Lipschitz function $f:X\to X$, the set $R_{T}^{f}(\psi)$ is
null if and only if the series

${\sum_{n=1}^{\infty}\psi(n)^{\delta}}$ (1.15)

converges.

It is natural to expect that Theorem 1.3 can be strengthened to the full
measure of $R_{T}^{f}(\psi)$ in the case when the series (1.15) diverges. This
was done in [HLSW] in the case $f=\operatorname{Id}_{X}$. Unfortunately for an
arbitrary Lipschitz function $f$ the full measure conclusion is outside of our
reach. In the following theorems we handle several special cases. First,
employing an argument from [HLSW], we prove

######  Theorem 1.4.

Let $(X,\mu,T)$, $\psi:\mathbb{N}\to\mathbb{R}$ and $f:X\to X$ be as in
Theorem 1.3. Furthermore, assume that

${T\circ f=f\circ T.}$ (1.16)

Then $\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$ whenever the series (1.15) diverges.

Clearly (1.16) holds when $f=\operatorname{const}$ or
$f=\operatorname{Id}_{X}$, but not in general. Next we present an alternative
approach to upgrading Theorem 1.3 to a full measure result, requiring
introducing additional assumptions on $(X,\mu,T)$.

Namely, let $\\{X_{i}\\}_{i\in\mathcal{I}}$ be as defined above. We say the
partition $\\{X_{i}\\}_{i\in\mathcal{I}}$ is pseudo-Markov with respect to $T$
if

  * •

for all $i,j\in\mathcal{I}$, $TX_{i}$ is measurable;

  * •

$TX_{i}\cap X_{j}\neq\varnothing$ implies $X_{j}\subset TX_{i}$ for any
$i,j\in\mathcal{I}$;

  * •

there exists $\tau>0$ such that $\mu(TX_{i})\geq\tau\mu(X)$ for any
$i\in\mathcal{I}$.

######  Theorem 1.5.

Let $(X,\mu,T)$, $\psi:\mathbb{N}\to\mathbb{R}$ and $f:X\to X$ be as in
Theorem 1.3. Furthermore, assume that $\\{X_{i}\\}_{i\in\mathcal{I}}$ is
pseudo-Markov. Then $\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$ whenever the series
(1.15) diverges.

Examples of systems with pseudo-Markov (in fact, truly Markov) partitions
include the Gauss map, the multiplication map $M_{b}$ where $b\geq 2$ is an
integer, and, more generally, conformal expanders described in §7. One can
also show that $\beta$-transformations for some specific $\beta$ admit pseudo-
Markov partitions. This is however not true for arbitrary $\beta$. Yet, the
twisted recurrence set-up was recently considered in [LWW] for $T=M_{\beta}$,
where $\beta>1$ is arbitrary, establishing the conclusion of Theorems 1.4 and
1.5 in that case. Namely they prove

######  Theorem 1.6.

Let $X=[0,1]$, $T=M_{\beta}$, $\mu$ the $M_{\beta}$-invariant probability
measure on $[0,1]$, and let $\psi:\mathbb{N}\to\mathbb{R}$ and $f:X\to X$ be
as in Theorem 1.3. Then $\mu\big{(}R_{T}^{f}(\psi)\big{)}$ is equal to $0$
(resp., $1$) whenever the series (1.15) converges (resp., diverges).

In §8 we show how our methods can be modified to yield an an independent proof
of the above theorem.

We also remark that the paper [DFL] suggests an even more general set-up:
there the authors consider a uniformly Lipschitz function $\Phi:X\times
X\to{\mathbb{R}}$ and under certain assumptions recover zero–one laws for sets
of the form

$\left\\{x\in X\left|\begin{aligned}
\phi_{1}(n)\leq\Phi(x,T^{n}x)\leq\phi_{2}(n)\\\ \text{ for infinitely many
}n\in\mathbb{N}\ \ \end{aligned}\right.\right\\}.$

Our set-up corresponds to $\phi_{1}=0$, $\phi_{2}=\psi$ and
$\Phi(x,y)=d\big{(}f(x),y\big{)}$. It would be interesting to see if the
methods of our paper can be applied to the generalized setting of [DFL].

The structure of the paper is as follows. In §2 we discuss several basic
properties of $f$-twisted recurrence sets and some examples of such sets. In
§3 we prove the convergence part of Theorem 1.3. In §4 we study quasi-
independence properties of the sequence of measurable sets whose limsup set is
given by (1.6). In §§5–6 we consider the divergence case and complete the
proof of Theorems 1.3, 1.4 and 1.5. In §7 we discuss examples of dynamical
systems to which our theorems apply. The final section contains a separate
discussion of $\beta$-transformations and results in proving Theorem 1.6.

### Acknowledgements

The authors are grateful to Dmitry Dolgopyat, Bassam Fayad, Mumtaz Hussain,
Osama Khalil, Bao-Wei Wang and two anonymous referees for helpful discussions.

##  2\. More about $f$-twisted recurrence

We start with several elementary observations concerning sets of $f$-twisted
recurrence.

######  Lemma 2.1.

Let $\psi:\mathbb{N}\to\mathbb{R}^{+}$ be an arbitrary function, and let
$f:X\to X$ be such that there exist at most countably many measurable subsets
$Y_{i}$ of $X$ and functions $f_{i}:X\to X$, $i\in\mathcal{I}$, such that
$\mu(X{\smallsetminus}\cup_{i}Y_{i})=0$,

$f|_{Y_{i}}=f_{i}\text{ and }\mu\big{(}R_{T}^{f_{i}}(\psi)\big{)}=1\text{ for
each }i\in\mathcal{I}.$ (2.1)

Then $\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$.

###### Proof.

Indeed, it follows from (1.6) and (2.1) that

$\mu\big{(}R_{T}^{f}(\psi)\cap Y_{i}\big{)}=\mu\big{(}R_{T}^{f_{i}}(\psi)\cap
Y_{i}\big{)}=\mu(Y_{i})$

for each $i\in\mathcal{I}$. ∎

Let us say that a function is simple if it takes at most countably many
values.

######  Corollary 2.2.

Suppose $T$ is ergodic and $\operatorname{supp}\mu$ is dense in $X$. Then
$\mu(R_{T}^{f})=1$ for any simple function $f:X\to X$.

###### Proof.

Immediate from Lemma 2.1 and (1.5). ∎

######  Lemma 2.3.

Let $(f_{n})_{n\in{\mathbb{N}}}$ be a sequence of functions $X\to X$ such that
$\mu(R_{T}^{f_{n}})=1$ for each $n$. Suppose that $f_{n}\to f$ uniformly on a
set of full measure. Then $\mu(R_{T}^{f})=1$.

###### Proof.

Since $\bigcap_{n}R_{T}^{f_{n}}$ has full measure, for almost every $x\in X$
and each $n\in{\mathbb{N}}$ one has

$\liminf_{k\to\infty}d\big{(}T^{k}x,f_{n}(x)\big{)}=0.$

Fix $\varepsilon>0$; then there exists $N$ so that for all $n>N$,
$d\big{(}f_{n}(x),f(x)\big{)}<\frac{\varepsilon}{2}$ for almost every $x\in
X$; on the other hand, for almost every $x\in X$ such that
$d\big{(}f_{n}(x),f(x)\big{)}<\frac{\varepsilon}{2}$,
$d\big{(}T^{k}x,f_{n}(x)\big{)}<\frac{\varepsilon}{2}$ for infinitely many
$k$. This implies $d\big{(}T^{k}x,f(x)\big{)}<\varepsilon$ for infinitely many
$k$. Since $\varepsilon$ is chosen arbitrarily, we have
$\liminf_{k\to\infty}d\big{(}T^{k}x,f(x)\big{)}=0$. ∎

######  Corollary 2.4.

Suppose that $T$ is ergodic and $\operatorname{supp}\mu$ is dense in $X$. Then
$\mu(R_{T}^{f})=1$ for any Borel-measurable $f:X\to X$.

###### Proof.

Let $\\{x_{n}\\}_{n=1}^{\infty}$ be a dense subset of $X$. Let $\varepsilon>0$
and $f:X\to X$ be a Borel-measurable function. Then
$\\{B(x_{n},\varepsilon)\\}_{n=1}^{\infty}$ covers $X$. Define

$g_{\varepsilon}(x)=x_{n}\text{ where }n=\inf_{m}\\{m:f(x)\in
B(x_{m},\varepsilon)\\}$ (2.2)

Then $g_{\varepsilon}$ is simple and
$||g_{\varepsilon}-f||_{\infty}\leq\varepsilon$. Since $\varepsilon$ is chosen
arbitrarily, $f$ is a uniform limit of simple functions. By Corollary 2.2 and
Lemma 2.3, $x$ is $f$-recurrent for almost every $x\in X$. ∎

Next, let us observe that the properties of sets $R_{T}^{f}(\psi)$ could be
strikingly different from the conclusion of Theorem 1.3 if the assumptions of
that theorem are not imposed. Let us start with the simplest possible non-
trivial111For us ergodic self-maps $T$ of finite sets $X$ will be trivial:
indeed, since those are transitive, it easily follows that $R_{T}^{f}(\psi)=X$
for any $f$ and any positive $\psi$. example of an ergodic dynamical system:
an irrational circle rotation $X={\mathbb{R}}/{\mathbb{Z}}$, $\mu=$ Lebesgue
measure, $T_{\alpha}(x)=x+\alpha\bmod{\mathbb{Z}}$ where
$\alpha\in{\mathbb{R}}{\smallsetminus}{\mathbb{Q}}$. Then the condition
defining the recurrence set

$R_{T_{\alpha}}(\psi)=\\{x:|n\alpha-m|<\psi(n)\text{ for infinitely many
}n\in{\mathbb{N}}\text{ and some }m\in{\mathbb{Z}}\\}$

is independent of $x$; hence $R_{T_{\alpha}}(\psi)$ is either $X$ or
$\varnothing$, and this dichotomy is different for different $\alpha$. More
precisely, Dirichlet’s Theorem implies that $R_{T_{\alpha}}(\psi_{1})=X$ for
any $\alpha$ (see (1.2) for this notation), and the same is true for
$\psi_{1}$ replaced with $\frac{1}{\sqrt{5}}\psi_{1}$, but not with
$c\psi_{1}$ for $c<\frac{1}{\sqrt{5}}$. In particular, $\alpha$ is badly
approximable if and only if $R_{T_{\alpha}}(c\psi_{1})=\varnothing$ for some
$c>0$. On the other hand, the theory of continued fractions shows that for any
positive non-increasing $\psi$ (decaying arbitrarily fast) there exists
$\alpha$ such that $R_{T_{\alpha}}(\psi)$ contains $0$ (and hence coincides
with ${\mathbb{R}}/{\mathbb{Z}}$).

Likewise, studying targets shrinking to $y\in X$ for the above system reduces
to inhomogeneous Diophantine approximation:

$R^{y}_{T_{\alpha}}(\psi)=\\{x:\operatorname{dist}(n\alpha,y-x)<\psi(n)\text{
for infinitely many }n\in{\mathbb{N}}\\}$

According to Minkowski’s theorem [C, Chapter III, Theorem II], for any
irrational $\alpha$ and any $y\in{\mathbb{R}}/{\mathbb{Z}}$, the complement of
$R^{y}_{T_{\alpha}}(\frac{1}{4}\psi_{1})$ is at most countable. A precise
zero-one law for sets $R^{y}_{T_{\alpha}}(\psi)$ again depends on the
Diophantine properties of $\alpha$. For example, it is a theorem of Kurzweil
[K] that $\alpha$ is badly approximable if and only if the following statement
holds: for any non-increasing $\psi$, the set $R^{y}_{T_{\alpha}}(\psi)$ is
null/conull if $\sum_{k=1}^{\infty}\psi(k)$ converges/diverges. However, well
approximable $\alpha$ come with their own convergence/divergence condition on
$\psi$ guaranteeing that $R^{y}_{T_{\alpha}}(\psi)$ is null or conull; see
[FK] for the most general statement.

Clearly the set-up of $f$-twisted recurrence can be similarly and
straightforwardly restated in a Diophantine approximation language:

$R^{f}_{T_{\alpha}}(\psi)=\left\\{x\in
X:\operatorname{dist}\big{(}n\alpha,f(x)-x\big{)}<\psi(n)\text{ for infinitely
many }n\in{\mathbb{N}}\right\\}.$

Thus if $f(x)=x+\beta\bmod{\mathbb{Z}}$ for a fixed $\beta$, then
$R_{T_{\alpha}}(\psi)$ is either $X$ or $\varnothing$; alternatively, if the
pushforward of Lebesgue measure by the map $x\mapsto f(x)-x$ is absolutely
continuous with respect to Lebesgue, then the zero/one law for the sets
$R^{f}_{T_{\alpha}}(\psi)$ depends on the Diophantine properties of $\alpha$
as described in [FK].

The situation is even trickier if one considers irrational rotations of
higher-dimensional tori. Namely, if we let
$X={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ and $\mu=$ Lebesgue measure, then it is
shown in [GP] that for any (arbitrarily slowly decaying) non-increasing
function $\psi$ with $\lim_{t\to\infty}\psi(t)=0$ there exists an ergodic
translation $T_{\alpha}:x\mapsto x+\alpha\bmod{\mathbb{Z}}^{d}$ such that
$\mu\big{(}R^{y}_{T_{\alpha}}(\psi)\big{)}=0$ for any $y\in X$. Moreover, by
suitably reparametrizing the aforementioned example one can construct a smooth
mixing transformation on the three dimensional torus with the same property.
Thus some conditions on the speed of mixing is crucial for a zero-one law as
in Theorem 1.3.

##  3\. The convergence part

In the next two sections we prove Theorem 1.3, thereby assuming that
$(X,\mu,T)$ satisfies conditions (1.8)–(1.13) and fixing
$\psi:\mathbb{N}\to\mathbb{R}^{+}$ with $\lim_{n\to\infty}\psi(n)=0$.
Similarly to (1.3) and (1.4), for an arbitrary $f:X\to X$ let us define

$A_{n}=A_{T}^{f}(n,\psi):=\left\\{x\in
X:d\big{(}T^{n}x,f(x)\big{)}<\psi(n)\right\\}.$ (3.1)

Clearly $R_{T}^{f}(\psi)=\limsup A_{n}$.

Unlike the shrinking target case, corresponding to constant functions $f$, the
sets $A_{n}$ cannot be expressed in the form $T^{-n}B_{n}$ for some balls
$B_{n}$. Our strategy is to consider the intersection of $A_{n}$ with
$f^{-1}B(x_{0},r)$, where $x_{0}\in X$ and $r>0$, and approximate this
intersection by the preimages of some balls under $T$.

######  Lemma 3.1.

For any $x_{0}\in X$, any $r>0$ and any subset $E$ of $f^{-1}B(x_{0},r)$,

$E\cap A_{n}\subset E\cap T^{-n}B\big{(}x_{0},\psi(n)+r\big{)}.$ (3.2)

Furthermore, if $r<\psi(n)$, then

$E\cap T^{-n}B\big{(}x_{0},\psi(n)-r\big{)}\subset E\cap A_{n}.$ (3.3)

###### Proof.

Fix a point $x\in E\cap A_{n}$. Then

$d\big{(}f(x),x_{0}\big{)}<r\text{ and }d\big{(}T^{n}x,f(x)\big{)}<\psi(n),$

which implies that

$d(T^{n}x,x_{0})<d\big{(}T^{n}x,f(x)\big{)}+d\big{(}f(x),x_{0}\big{)}<\psi(n)+r.$

Hence $E\cap A_{n}\subset E\cap T^{-n}B\big{(}x_{0},\psi(n)+r\big{)}$.

On the other hand, fix $x\in E\cap T^{-n}B\big{(}x_{0},\psi(n)-r\big{)}$. Then
$d(f(x),x_{0})<r$ and $d(T^{n}x,x_{0})<\psi(n)-r$. Hence

$d\big{(}T^{n}x,f(x)\big{)}\leq
d(T^{n}x,x_{0})+d\big{(}x_{0},f(x)\big{)}<\psi(n),$

thus $E\cap T^{-n}B\big{(}x_{0},\psi(n)-r\big{)}\subset E\cap A_{n}$. ∎

Choose $n_{0}\in\mathbb{N}$ such that $5\psi(n)<r_{0}$ for all $n>n_{0}$,
where $r_{0}$ is as in (1.8); the next several statements in this section will
be proved for $n>n_{0}$.

######  Lemma 3.2.

Let $B=B\big{(}x_{0},\psi(n)/2\big{)}$ for some $x_{0}\in X$ and $n>n_{0}$.
Then for any open ball $E$ contained in $f^{-1}B$,

$\displaystyle
2^{-\delta}\big{(}\eta_{1}\mu(E)-\eta_{2}a_{n}\big{)}\psi(n)^{\delta}\leq\mu(E\cap
A_{n})\leq\eta_{2}(3/2)^{\delta}\big{(}\mu(E)+a_{n}\big{)}\psi(n)^{\delta},$

with $\delta,\eta_{1},\eta_{2}$ as in (1.8) and $(a_{n})_{n\in{\mathbb{N}}}$
as in (1.9).

###### Proof.

Let $r=\psi(n)/2$, and let $E$ be an open ball contained in
$f^{-1}B(x_{0},r)$. Combining (3.2) with (1.9), we get

$\displaystyle\mu(E\cap A_{n})$ $\displaystyle\geq\mu\Big{(}E\cap
T^{-n}B\big{(}x_{0},\psi(n)-r\big{)}\Big{)}$
$\displaystyle\geq\mu(E)\mu\big{(}T^{-n}B(x_{0},\psi(n)-r)\big{)}-a_{n}\mu\big{(}E\cap
T^{-n}B(x_{0},\psi(n)-r)\big{)}$
$\displaystyle=\mu(E)\mu\Big{(}T^{-n}B\big{(}x_{0},\psi(n)/2\big{)}\Big{)}-a_{n}\mu\Big{(}T^{-n}B\big{(}x_{0},\psi(n)/2\big{)}\Big{)}$
$\displaystyle=\mu(E)\mu\Big{(}B\big{(}x_{0},\psi(n)/2\big{)}\Big{)}-a_{n}\mu\Big{(}B\big{(}x_{0},\psi(n)/2\big{)}\Big{)}$
$\displaystyle\underset{\eqref{ar}}{\geq}\big{(}\eta_{1}\mu(E)-\eta_{2}a_{n}\big{)}2^{-\delta}\psi(n)^{\delta}$

and

$\displaystyle\mu(E\cap A_{n})$ $\displaystyle\leq\mu\Big{(}E\cap
T^{-n}B\big{(}x_{0},\psi(n)+r\big{)}\Big{)}$
$\displaystyle\leq\mu(E)\mu\big{(}T^{-n}B(x_{0},\psi(n)+r)\big{)}+a_{n}\mu\big{(}E\cap
T^{-n}B(x_{0},\psi(n)+r)\big{)}$
$\displaystyle=\mu(E)\mu\Big{(}T^{-n}B\big{(}x_{0},3\psi(n)/2\big{)}\Big{)}+a_{n}\mu\Big{(}T^{-n}B\big{(}x_{0},3\psi(n)/2\big{)}\Big{)}$
$\displaystyle\leq\mu(E)\mu\Big{(}B\big{(}x_{0},3\psi(n)/2\big{)}\Big{)}+a_{n}\mu\Big{(}B\big{(}x_{0},3\psi(n)/2\big{)}\Big{)}$
$\displaystyle\underset{\eqref{ar}}{\leq}\big{(}\mu(E)+a_{n}\big{)}\eta_{2}(3/2)^{\delta}\psi(n)^{\delta},$

establishing the claim. ∎

To prove Theorems 1.3–1.6, in view of Lemma 2.1 it is enough to assume that
$f$ is Lipschitz. Thus for the rest of the paper we let $f:X\to X$ be a
$p$-Lipschitz function for some $p>0$.

The next lemma estimates the measure of the sets $A_{n}$.

######  Lemma 3.3.

For $n>n_{0}$,

$\displaystyle\eta_{2}^{-1}\eta_{1}^{2}10^{-\delta}\psi(n)^{\delta}-{(p/5)^{\delta}}a_{n}$
$\displaystyle\leq\mu(A_{n})\leq{\eta_{1}^{-1}\eta_{2}(3/2)^{\delta}\big{(}\eta_{2}5^{\delta}\psi(n)^{\delta}}+(2p)^{\delta}a_{n}\big{)}.$

###### Proof.

Take $x\in X$, $y\in f^{-1}\\{x\\}$ and $z\in
B\left(y,\frac{\psi(n)}{2p}\right)$. Then by the $p$-Lipshitz condition,
$d\big{(}x,f(z)\big{)}\leq pd(y,z)<\psi(n)/2$. Thus

$B\big{(}y,\psi(n)/p\big{)}\subset f^{-1}B\big{(}x,\psi(n)/2\big{)}.$

We have an open covering

$\left\\{B\big{(}y,\psi(n)/p\big{)}:y\in X\right\\}$

with each $B\left(y,\frac{\psi(n)}{2p}\right)\subset
f^{-1}B\big{(}x,\psi(n)/2\big{)}$ for some $x\in X$.

By Vitali’s covering theorem ($5r$-covering lemma), we can find countably many
disjoint balls
$\left\\{B\left(y_{j},\frac{\psi(n)}{2p}\right)\right\\}_{j\in\mathcal{J}}$
such that

$X\subset\bigcup_{j\in\mathcal{J}}B\left(y_{j},\frac{5\psi(n)}{2p}\right).$
(3.4)

By the disjointness of
$\left\\{B\left(y_{j},\frac{\psi(n)}{2p}\right)\right\\}_{j\in\mathcal{J}}$,
we have

$\sum_{j\in\mathcal{J}}\eta_{1}\left(\frac{\psi(n)}{2p}\right)^{\delta}\leq\sum_{j\in\mathcal{J}}\mu\left(B\Big{(}y_{j},\frac{\psi(n)}{2p}\Big{)}\right)\leq\mu(X)=1.$

Hence
$|\mathcal{J}|\leq\eta_{1}^{-1}\left(\frac{2p}{\psi(n)}\right)^{\delta}$. On
the other hand, by (3.4) we have

$\sum_{j\in\mathcal{J}}\eta_{2}\left(\frac{5\psi(n)}{2p}\right)^{\delta}\geq\sum_{j\in\mathcal{J}}\mu\left(B\Big{(}y_{j},\frac{5\psi(n)}{2p}\Big{)}\right)\geq\mu(X)=1;$

hence
$|\mathcal{J}|\geq\eta_{2}^{-1}\left(\frac{2p}{5\psi(n)}\right)^{\delta}$.

By Lemma 3.2, since for each $j$ we have
$B\left(y_{j},\frac{\psi(n)}{2p}\right)\subset
f^{-1}B\big{(}x_{j},\psi(n)/2\big{)}$ and
$B\left(y_{j},\frac{5\psi(n)}{2p}\right)\subset
f^{-1}B\big{(}x_{j},5\psi(n)/2\big{)}$, it follows that

$\displaystyle\mu(A_{n})\leq$ $\displaystyle\sum_{j\in
J}\mu\left(B\Big{(}y_{j},\frac{5\psi(n)}{2p}\Big{)}\cap A_{n}\right)$
$\displaystyle\leq$ $\displaystyle\sum_{j\in
J}\eta_{2}(3/2)^{\delta}\left[\mu\left(B\Big{(}y_{j},\frac{5\psi(n)}{2p}\Big{)}\right)+a_{n}\right]\psi(n)^{\delta}$
$\displaystyle\leq$ $\displaystyle\
\eta_{1}^{-1}\left(\frac{2p}{\psi(n)}\right)^{\delta}\eta_{2}(3/2)^{\delta}\left[\Big{(}\frac{5\psi(n)}{2p}\Big{)}^{\delta}\eta_{2}+a_{n}\right]\psi(n)^{\delta}$
$\displaystyle{=}$ $\displaystyle\
{\eta_{1}^{-1}\eta_{2}(3/2)^{\delta}\big{(}\eta_{2}5^{\delta}\psi(n)^{\delta}}+(2p)^{\delta}a_{n}\big{)}$

and

$\displaystyle\mu(A_{n})\geq$
$\displaystyle\sum_{j\in\mathcal{J}}\mu\left(B\Big{(}y_{j},\frac{\psi(n)}{2p}\Big{)}\cap
A_{n}\right)$ $\displaystyle\geq$ $\displaystyle\ \sum_{j\in
J}\left[\eta_{1}2^{-\delta}\mu\left(B\Big{(}y_{j},\frac{\psi(n)}{2p}\Big{)}\right)-\eta_{2}2^{-\delta}a_{n}\right]\psi(n)^{\delta}$
$\displaystyle\geq$ $\displaystyle\
\eta_{2}^{-1}\left(\frac{2p}{5\psi(n)}\right)^{\delta}\left[\eta_{1}2^{-\delta}\eta_{1}\Big{(}\frac{\psi(n)}{2p}\Big{)}^{\delta}-\eta_{2}2^{-\delta}a_{n}\right]\psi(n)^{\delta}$
$\displaystyle{=}$ $\displaystyle\
\eta_{2}^{-1}\eta_{1}^{2}10^{-\delta}\psi(n)^{\delta}-\eta_{2}^{-1}{(2p/5)^{\delta}}\eta_{2}2^{-\delta}a_{n},$

finishing the proof of the lemma.∎

######  Proposition 3.4.

$\sum_{n=1}^{\infty}\psi(n)^{\delta}=\infty\iff\sum_{n=1}^{\infty}\mu(A_{n})=\infty$
(3.5)

###### Proof.

By Lemma 3.3 we know that

$\displaystyle\eta_{2}^{-1}\eta_{1}^{2}10^{-\delta}\sum_{n>n_{0}}\psi(n)^{\delta}-{(p/5)^{\delta}}\sum_{n>n_{0}}a_{n}$
$\displaystyle\leq$ $\displaystyle\sum_{n>n_{0}}\mu(A_{n})$
$\displaystyle\leq$ $\displaystyle\
\eta_{1}^{-1}\eta_{2}(3/2)^{\delta}\left(\eta_{2}5^{\delta}\sum_{n>n_{0}}\psi(n)^{\delta}+(2p)^{\delta}\sum_{n>n_{0}}a_{n}\right).$

Since $\\{a_{n}\\}$ is summable, (3.5) holds. ∎

######  Remark 3.5.

Note that Proposition 3.4 and the Borel–Cantelli Lemma immediately imply the
convergence case of Theorem 1.3: if $\sum_{n=1}^{\infty}\psi(n)<\infty$, then
$\mu\big{(}R^{f}_{T}(\psi)\big{)}=\mu(\limsup_{n}A_{n})=0$. Note also that or
this conclusion one only needs the first two conditions of Theorem 1.3, that
is, (1.8) and (1.9); the remaining conditions (1.10)–(1.13) will be used in
the proof of the divergence case.

##  4\. A quasi-independence estimate

Now let us make use of assumptions (1.10)–(1.13). The following lemma was
stated and used in [HLSW]; we prove it here since our set-up is slightly
different.

######  Lemma 4.1.

For $m\in{\mathbb{N}}$, $J$ a cylinder in $\mathcal{F}_{m}$ and for any open
set $U$ contained in $J$, $\mu(T^{m}U)\asymp K_{J}^{\delta}\mu(U)$.

###### Proof.

By (1.8) and (1.13), we know that for all open balls $B\subset J$ with radius
smaller than $r_{0}$, it holds that $\mu(T^{m}B)\asymp K_{J}^{\delta}\mu(B)$.
Let $U\subset J$ be an open subset. Consider the cover

$\mathcal{S}=\big{\\{}B(x,r):x\in U,\ B(x,5r)\subset U,\ r<r_{0}\big{\\}}$

of $U$. By Vitali’s covering theorem, $\mathcal{S}$ has a countable sub-
collection $\mathcal{B}$ of disjoint balls so that

$\bigcup_{B(x,r)\in\mathcal{B}}B(x,r)\subset
U\subset\bigcup_{B(x,r)\in\mathcal{B}}B(x,5r).$

Since $T^{m}$ is injective on $J$,

$\bigcup_{B(x,r)\in\mathcal{B}}T^{m}B(x,r)\subset
T^{m}U\subset\bigcup_{B(x,r)\in\mathcal{B}}T^{m}B(x,5r).$

Hence

$K_{J}^{\delta}\sum_{B(x,r)\in\mathcal{B}}\mu\big{(}B(x,r)\big{)}\asymp\mu(T^{m}U).$

On the other hand,
$\mu(U)\asymp\sum_{B(x,r)\in\mathcal{B}}\mu\big{(}B(x,r)\big{)}$, and the
lemma is proved. ∎

Now recall that we were working with the sets $A_{n}$ defined in (3.1). The
next lemma shows that the intersection of a cylinder of high enough level with
$A_{n}$ is contained in a small ball. Namely, let $m_{0}\geq n_{0}$ be such
that $K_{J}>\max\left\\{\frac{K_{1}\operatorname{diam}(X)}{r_{0}},2p\right\\}$
for all $m>m_{0}$ and $J\in\mathcal{F}_{m}$ (which is possible in view of
(1.11)).

######  Lemma 4.2.

For $m>m_{0}$, for every cylinder $J\in\mathcal{F}_{m}$ and any $z\in J\cap
A_{m}$ there exists a ball of radius

$r=\frac{2\psi(m)}{K_{J}-p},$ (4.1)

say $B(z,r)$, such that

$J\cap A_{m}\subset B(z,r)\cap J.$ (4.2)

###### Proof.

Choose any $x,z\in J\cap A_{m}$. Since $J\in\mathcal{F}_{m}$, in view of
(1.10) we have

$d(T^{m}x,T^{m}z)K_{J}^{-1}\geq d(x,z).$

On the other hand,

$\displaystyle d(T^{m}x,T^{m}z)$ $\displaystyle\leq
d\big{(}T^{m}x,f(x)\big{)}+d\big{(}f(x),f(z)\big{)}+d\big{(}T^{m}z,f(z)\big{)}$
$\displaystyle\leq 2\psi(m)+pd(x,z).$

Then $K_{J}d(x,z)<2\psi(m)+pd(x,z)$, i.e. $d(x,z)<\frac{2\psi(m)}{K_{J}-p}$. ∎

Now let us prove a quasi-independence property of the sequence
$\\{A_{n}\\}_{n}$. For any $m\in{\mathbb{N}}$ and $J\in\mathcal{F}_{m}$ define

$\displaystyle J^{*}:=B(z,r)\cap J,$ (4.3)

where $r$ and $z$ are defined in (4.1) and (4.2).

######  Lemma 4.3.

For all $n>m>m_{0}$ and for each $J\in\mathcal{F}_{m}$,

$\mu(J\cap A_{m}\cap A_{n})\lesssim
K_{J}^{-\delta}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right].$

###### Proof.

Let $J\in\mathcal{F}_{m}$. By Lemma 4.2,

$J\cap A_{m}\cap A_{n}\subset J^{*}\cap A_{n}$

where $J^{*}$ is defined in (4.3). Now let us estimate $\mu(J^{*}\cap A_{n})$.

Case (i): $pr=\frac{2p\psi(m)}{K_{J}-p}\leq\psi(n)$.

Note that for all $x\in B(z,r)$ we have $d\big{(}f(x),f(z)\big{)}<pd(x,z)<pr$,
therefore $B(z,r)\subset f^{-1}B\big{(}f(z),pr\big{)}$. Thus $J^{*}$ is a
subset of $f^{-1}B\big{(}f(z),pr\big{)}$, and we can apply Lemma 3.1 to
$J^{*}$ and obtain

$J^{*}\cap A_{m}\subset J^{*}\cap T^{-n}B\big{(}f(z),\psi(n)+pr\big{)}\subset
J^{*}\cap T^{-n}B\big{(}f(z),2\psi(n)\big{)}.$ (4.4)

Then apply Lemma 4.1 to $J^{*}\cap T^{-n}B\big{(}f(z),2\psi(n)\big{)}$,
getting

$\displaystyle\mu\Big{(}J^{*}\cap T^{-n}B\big{(}f(z),2\psi(n)\big{)}\Big{)}$
$\displaystyle\lesssim K_{J}^{-\delta}\Big{(}T^{m}J^{*}\cap
T^{-(n-m)}B\big{(}f(z),2\psi(n)\big{)}\Big{)}$ $\displaystyle\leq
K_{J}^{-\delta}\mu\Big{(}T^{m}B(z,r)\cap
T^{-(n-m)}B\big{(}f(z),2\psi(n)\big{)}\Big{)}.$

Since $m>m_{0}$, we have $\inf_{J\in\mathcal{F}_{m}}K_{J}>2p$. Then by the
conformality assumption (1.12), we have

$\displaystyle T^{m}B(z,r)$
$\displaystyle=T^{m}B\left(z,\frac{2\psi(m)}{K_{J}-p}\right)$
$\displaystyle\subset
B\left(T^{m}z,K_{2}K_{J}\frac{2\psi(m)}{K_{J}-p}\right)\subset
B\big{(}T^{m}z,K_{2}4\psi(m)\big{)},$

where $K_{2}$ is defined in (1.12). Thus

$\mu(J^{*}\cap A_{n})\lesssim
K_{J}^{-\delta}\mu\Big{(}B\big{(}T^{m}z,4K_{2}\psi(m)\big{)}\cap
T^{-(n-m)}B\big{(}f(z),2\psi(n)\big{)}\Big{)}.$

By the mixing property (1.8),

$\displaystyle\mu(J^{*}\cap A_{n})\lesssim
K_{J}^{-\delta}\mu\Big{(}T^{m}B\big{(}T^{m}z,4K_{2}\psi(m)\big{)}\cap
T^{-(n-m)}B\big{(}f(z),2\psi(n)\big{)}\Big{)}$ $\displaystyle\lesssim\ $
$\displaystyle
K_{J}^{-\delta}\left[\mu\Big{(}B\big{(}T^{m}z,4K_{2}\psi(m)\big{)}\Big{)}\mu\Big{(}B\big{(}z,2\psi(n)\big{)}\Big{)}+a_{n-m}\mu\Big{(}B\big{(}z,2\psi(n)\big{)}\Big{)}\right]$
$\displaystyle\lesssim\ $ $\displaystyle
K_{J}^{-\delta}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}\right].$

Case (ii): $pr=\frac{2p\psi(m)}{K_{J}-p}>\psi(n)$.

We replace the ball $B\big{(}f(z),pr\big{)}$ by a collection of balls of
radius $\psi(n)$. Choose a maximal $\frac{3}{2}\psi(n)$-separated set in
$B\big{(}f(z),pr\big{)}$, denoted by $\\{z_{i}\\}_{1\leq i\leq N_{m,n}}$. Then

$\displaystyle
B\big{(}f(z),pr\big{)}\subset\bigcup_{i=1}^{N_{m,n}}B\big{(}z_{i},\psi(n)\big{)}\subset
B\big{(}f(z),2pr\big{)}.$

Since $\mu$ is Ahlfors regular and $m>n_{0}$,

$\displaystyle
N_{m,n}\asymp\left(\frac{p\psi(m)}{(K_{J}-p)\psi(n)}\right)^{\delta}.$

Since $B\big{(}z_{i},\psi(n)\big{)}\subset
f^{-1}B\big{(}f(z_{i}),p\psi(n)\big{)}$, we can apply Lemma 3.1 to each ball
$B\big{(}z_{i},\psi(n)\big{)}$ with $1\leq i\leq N_{m,n}$ and obtain

$\displaystyle\mu\Big{(}B\big{(}z_{i},\psi(n)\big{)}\cap
A_{n}\Big{)}\underset{\eqref{eq1}}{\leq}$
$\displaystyle\mu\Big{(}B\big{(}z_{i},\psi(n)\big{)}\cap
T^{-n}B\big{(}z_{i},(1+p)\psi(n)\big{)}\Big{)}\quad\begin{aligned}
\end{aligned}$ $\displaystyle\underset{\eqref{em}}{\leq}$
$\displaystyle\mu\Big{(}B\big{(}z_{i},\psi(n)\big{)}\Big{)}\mu\Big{(}B\big{(}z_{i},(1+p)\psi(n)\big{)}\Big{)}+a_{n}\mu\Big{(}B\big{(}z_{i},(1+p)\psi(n)\big{)}\Big{)}$
$\displaystyle\underset{\eqref{ar}}{\lesssim}$
$\displaystyle\left[\psi(n)^{\delta}+a_{n}\right]\psi(n)^{\delta}.$

Now summing over $1\leq i\leq N_{m,n}$, we have

$\displaystyle\mu(J^{*}\cap A_{n})\leq$
$\displaystyle\sum_{i=1}^{N_{m,n}}\mu\Big{(}B\big{(}z_{i},\psi(n)\big{)}\cap
A_{n}\Big{)}$ $\displaystyle\lesssim$
$\displaystyle\left(\frac{p\psi(m)}{K_{J}-p}\right)^{\delta}\left[\psi(n)^{\delta}+a_{n}\right]$
$\displaystyle\lesssim$ $\displaystyle\
K_{J}^{-\delta}[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}].$

Combining the two cases we obtain the desired conclusion. ∎

######  Proposition 4.4.

For $n>m>m_{0}$,

$\mu(A_{m}\cap
A_{n})\lesssim\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}.$

###### Proof.

This follows directly from the previous lemma and (1.12), since

$\displaystyle\mu(A_{m}\cap A_{n})\lesssim$
$\displaystyle\sum_{J\in\mathcal{F}_{m}}K_{J}^{-\delta}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right]$
$\displaystyle\underset{\eqref{qi}}{\lesssim}$
$\displaystyle\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}.$

∎

##  5\. Proof of Theorems 1.3 and 1.4

To prove the divergence case of Theorem 1.3, let us recall

######  Lemma 5.1 (Chung–Erdös inequality, [CE]).

Let $(X,\mu)$ be a probability space, and let $\\{E_{n}\\}_{n}$ be a sequence
of events such that $\sum_{n=1}^{\infty}\mu(E_{n})=\infty$. Then

$\mu\left(\limsup_{n\to\infty}E_{n}\right)\geq\limsup_{N\to\infty}\frac{\left(\sum_{n=1}^{N}\mu(E_{n})\right)^{2}}{\sum_{n,m=1}^{N}\mu(E_{n}\cap
E_{m})}.$

The next lemma is based on the above inequality.

######  Lemma 5.2.

Let $(X,\mu)$ be a probability space. Let $(E_{k})_{k}$ be a sequence of
measurable subsets of $X$, and let $(a_{n})_{n},(b_{n})_{n}$ be sequences of
positive numbers such that $\sum_{n=1}^{\infty}a_{n}<\infty$ and
$\sum_{n=1}^{\infty}b_{n}=\infty$. Assume that for some $s_{1},s_{2},s_{3}>0$
it holds that

$s_{1}(b_{n}-a_{n})\leq\mu(E_{n})\leq s_{3}(b_{n}+a_{n})\quad\text{for all
}n\in{\mathbb{N}}$

and

$\mu(E_{m}\cap E_{n})\leq
s_{2}(b_{m}b_{n}+a_{n-m}b_{n}+a_{n}b_{m})\quad\text{for all }m<n.$

Then $\mu(\limsup_{n\to\infty}E_{n})\geq\frac{s_{1}^{2}}{2s_{2}}$.

###### Proof.

Let us denote $\sum_{k=1}^{\infty}a_{k}=S$. Choose $\varepsilon>0$. On the one
hand,

$\sum_{n=1}^{N}\mu(E_{n})\geq s_{1}\sum_{n=1}^{N}(b_{n}-a_{n})\geq
s_{1}\left(\sum_{n=1}^{N}b_{n}-S\right)\\\
\geq(s_{1}-\varepsilon)\sum_{n=1}^{N}b_{n}$

when $N$ is sufficiently large, because $\sum_{n=1}^{\infty}b_{n}=\infty$. On
the other hand,

$\displaystyle\sum_{m,n=1}^{N}\mu(E_{m}\cap E_{n})$
$\displaystyle=\sum_{n=1}^{N}\mu(E_{n})+2\sum_{1\leq m<n\leq N}\mu(E_{m}\cap
E_{n})$ $\displaystyle\leq
s_{3}\sum_{n=1}^{N}(b_{n}+a_{n})+2s_{2}\sum_{m=1}^{N}\sum_{n=m+1}^{N}(b_{m}b_{n}+a_{n-m}b_{n}+a_{n}b_{m})$
$\displaystyle\leq
s_{3}\sum_{n=1}^{N}b_{n}+s_{3}S+2s_{2}\sum_{m=1}^{N}\sum_{n=m+1}^{N}b_{m}b_{n}+2s_{2}S\sum_{n=1}^{n}b_{n}+2s_{2}S\sum_{m=1}^{n}b_{m}$
$\displaystyle\leq
2s_{2}\left(\sum_{n=1}^{N}b_{n}\right)^{2}+(s_{3}+4s_{2}S)\sum_{n=1}^{N}b_{n}+s_{3}S\leq(2s_{2}+\varepsilon)\left(\sum_{n=1}^{N}b_{n}\right)^{2}$

when $N$ is sufficiently large, again because
$\sum_{n=1}^{\infty}b_{n}=\infty$. Hence by Lemma 5.1

$\mu\left(\limsup_{n\to\infty}E_{n}\right)\geq\limsup_{N\to\infty}\frac{\left((s_{1}-\varepsilon)\sum_{n=1}^{N}b_{n}\right)^{2}}{(2s_{2}+\varepsilon)\left(\sum_{n=1}^{N}b_{n}\right)^{2}}=\frac{(s_{1}-\varepsilon)^{2}}{2s_{2}+\varepsilon},$

and, since $\varepsilon$ was arbitrary, the conclusion follows.∎

###### Proof of Theorem 1.3, the divergence part.

Take $E_{n}=A_{m_{0}+n}$, where $m_{0}$ is chosen prior to Lemma 4.2. By Lemma
3.3, there exist $s_{1},s_{3}>0$ such that

$s_{1}\big{(}\psi(m_{0}+n)^{\delta}-a_{m_{0}+n}\big{)}\leq\mu(E_{n})\leq
s_{3}\big{(}\psi(m_{0}+n)^{\delta}+a_{m_{0}+n}\big{)}$

for any $n\in{\mathbb{N}}$. Also by Proposition 4.4 there exists $s_{2}>0$
such that

$\mu(E_{m}\cap E_{n})\leq
s_{2}\big{(}\psi^{\delta}(m_{0}+m)\psi^{\delta}(m_{0}+n)+a_{n-m}\psi^{\delta}(m_{0}+n)+a_{m_{0}+n}\psi^{\delta}(m_{0}+m)\big{)}$

for $n>m$. By taking $b_{n}=\psi(m_{0}+n)^{\delta}$, the above lemma implies
that
$\mu\big{(}R_{T}^{f}(\psi)\big{)}=\mu(\limsup_{n\to\infty}E_{n})\geq\frac{s_{1}^{2}}{2s_{2}}>0$.
∎

We conclude the section with the proof of Theorem 1.4, that is, a passage from
positive measure to full measure under the assumption that $f$ and $T$
commute. This proof is adapted from [HLSW].

###### Proof of Theorem 1.4.

Suppose that (1.16) holds. Consider the set

$R^{\prime}(\psi):=\left\\{x\in
X:\liminf_{n\to\infty}\psi(n)^{-1}d\big{(}T^{n}x,f(x)\big{)}<\infty\right\\}$

Take a point $x\in R^{\prime}(\psi)\cap f^{-1}X_{i}$. By definition, there
exist $c(x)>0$ and $\\{n_{k}\\}_{k}\subset\mathbb{N}$ so that

$\psi(n_{k})^{-1}d\big{(}T^{n_{k}}x,f(x)\big{)}<c(x)\quad\forall\,k\geq 1$

Let $s(x)$ be a positive real number such that
$B\big{(}f(x),s(x)\big{)}\subset X_{i}$. Since
$\psi(n)\to 0$, take $N\in\mathbb{N}$ such that for all $k\geq N$,

$c(x)\psi(n_{k})<s(x)$

Then $d\big{(}T^{n_{k}}x,f(x)\big{)}<c(x)\psi(n_{k})<s(x)$, therefore
$T^{n_{k}}x\in X_{i}$ for all $k\geq N$. Hence for all $k\geq N$,

$\displaystyle
d\Big{(}T^{n_{k}}(Tx),\big{(}f(Tx)\big{)}\Big{)}\underset{\eqref{eq:comm}}{=}d\Big{(}T^{n_{k}}(Tx),T\big{(}f(x)\big{)}\Big{)}=d\Big{(}T(T^{n_{k}}x),T\big{(}f(x)\big{)}\Big{)}$
$\displaystyle\leq d\big{(}T^{n_{k}}x,f(x)\big{)}<c(x)\psi(n_{k}).$

This implies $R^{\prime}(\psi)\cap(\bigcup_{i}f^{-1}X_{i})\subset
T^{-1}R^{\prime}(\psi)$, thus $\mu\big{(}R^{\prime}(\psi)\smallsetminus
T^{-1}R^{\prime}(\psi)\big{)}=0$. But $R_{T}^{f}(\psi)\subset
R^{\prime}(\psi)$, hence $\mu\big{(}R^{\prime}(\psi)\big{)}>0$, and by the
ergodicity of $T$, $\mu\big{(}R^{\prime}(\psi)\big{)}=1$.

Now we show that $\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$. Take a sequence of
positive numbers $\\{\ell(n):n\geq 1\\}$ such that

$\sum_{n=1}^{\infty}\frac{\psi(n)}{\ell(n)}=\infty\quad\text{and}\quad\lim_{n\to\infty}\ell(n)=\infty.$

Consider $\widetilde{\psi}(n):=\psi(n)/\ell(n)$; then
$R^{\prime}(\widetilde{\psi})$ has full measure, i.e. for $\mu$-almost every
$x\in X$,

$\liminf_{n\to\infty}\widetilde{\psi}(n)^{-1}d\big{(}T^{n}x,f(x)\big{)}<\infty.$

By Egorov’s theorem, for any $\varepsilon>0$ there exists $M>0$ such that the
set

$R_{M}:=\left\\{x\in
X:\frac{\ell(n)}{\psi(n)}d\big{(}T^{n}x,f(x)\big{)}<M\text{ for infinitely
many }n\in\mathbb{N}\right\\}$

is of measure at least $1-\varepsilon$. Then $R_{M}\subset R_{T}^{f}(\psi)$,
by letting $\ell(n)\to\infty$. Since $\varepsilon$ is arbitrary, it implies
$\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$. ∎

##  6\. Proof of Theorem 1.5

We first prove a local version of Lemma 3.3; i.e., fix a ball $B$ with
sufficiently small radius and estimate $\mu(A_{n}\cap B)$ for $n$ sufficiently
large.

######  Lemma 6.1.

For any $r<r_{0}/2$ there exists $n_{r}\in\mathbb{N}$ such that for any open
ball $B=B(x,r)$ in $X$ and all $n>n_{r}$,

$\displaystyle\frac{\eta_{1}}{\eta_{2}20^{\delta}}\mu(B)\left[\frac{\eta_{1}^{2}}{\eta_{2}}\psi(n)^{\delta}-(2p)^{\delta}a_{n}\right]\leq\mu(A_{n}\cap
B)$
$\displaystyle\leq\frac{\eta_{2}^{2}3^{\delta}}{\eta_{1}^{2}}\mu(B)\left[\eta_{2}5^{\delta}\psi(n)^{\delta}+a_{n}(2p)^{\delta}\right]$

###### Proof.

Let $n_{r}\in\mathbb{N}$ be such that

${\frac{5\psi(n)}{p}<\frac{r}{2}}$ (6.1)

for all $n>n_{r}$. As in the proof of Lemma 3.3, we have an open covering

$\left\\{B\Big{(}y,\frac{\psi(n)}{2p}\Big{)}:y\in B\right\\}$

of $B$, with each $B\Big{(}y,\frac{\psi(n)}{2p}\Big{)}\subset
f^{-1}B\big{(}x,\psi(n)/2\big{)}$ for some $x\in X$. Again by Vitali’s
Covering Theorem, we can find countable sub-collection of disjoint balls
$\left\\{B\big{(}y_{j},\psi(n)/2p\big{)}\right\\}_{j\in\mathcal{J}}$ such that

$B\subset\bigcup_{j\in\mathcal{J}}B\big{(}y_{j},5\psi(n)/2p\big{)}.$ (6.2)

Then we have

$\sum_{\begin{subarray}{c}j\in\mathcal{J}\end{subarray}}\eta_{1}\big{(}\psi(n)/2p\big{)}^{\delta}\leq\sum_{\begin{subarray}{c}j\in\mathcal{J}\end{subarray}}\mu\Big{(}B\big{(}y_{j},\psi(n)/2p\big{)}\Big{)}\underset{\eqref{eq:ballsinside}}{\leq}\mu\big{(}B(x,2r)\big{)}\leq
2^{\delta}\frac{\eta_{2}}{\eta_{1}}\mu(B),$

hence

$|\mathcal{J}|\leq\frac{\eta_{2}(4p)^{\delta}\mu(B)}{\eta_{1}^{2}\psi(n)^{\delta}}.$
(6.3)

On the other hand,

$\displaystyle\sum_{\begin{subarray}{c}j\in\mathcal{J}\\\
B\big{(}y,\frac{5\psi(n)}{2p}\big{)}\cap
B(x,r/2)\neq\varnothing\end{subarray}}\eta_{2}\big{(}5\psi(n)/2p\big{)}^{\delta}\geq\
$ $\displaystyle\sum_{\begin{subarray}{c}j\in\mathcal{J}\\\
B\big{(}y,\frac{5\psi(n)}{2p}\big{)}\cap
B(x,r/2)\neq\varnothing\end{subarray}}\mu\Big{(}B\big{(}y_{j},5\psi(n)/2p\big{)}\Big{)}$
$\displaystyle\underset{\eqref{cover5r}}{\geq}$ $\displaystyle\
\mu\big{(}B(x,{r}/{2})\big{)}\geq\frac{\eta_{1}}{2^{\delta}\eta_{2}}\mu(B),$

therefore

$\displaystyle\left|\left\\{j\in\mathcal{J}:B\Big{(}y_{j},\frac{\psi(n)}{2p}\Big{)}\subset
B\right\\}\right|$ (6.4)
$\displaystyle\underset{\eqref{eq:ballsinside}}{\geq}$
$\displaystyle\left|\left\\{j\in\mathcal{J}:B\Big{(}y_{j},\frac{\psi(n)}{2p}\Big{)}\cap
B(x,r/2)\neq\varnothing\right\\}\right|\geq\
\frac{\eta_{1}p^{\delta}\mu(B)}{\eta_{2}^{2}\big{(}5\psi(n)\big{)}^{\delta}}.$

Since for each $j\in\mathcal{J}$ we have
$B\big{(}y_{j},\psi(n)/2p\big{)}\subset f^{-1}B\big{(}x_{j},\psi(n)/2\big{)}$
and $B\big{(}y_{j},5\psi(n)/2p\big{)}\subset
f^{-1}B\big{(}x_{j},5\psi(n)/2\big{)}$, similarly to the proof of Lemma 3.3 we
can write

$\displaystyle\mu(A_{n}\cap B)\leq$
$\displaystyle\sum_{j\in\mathcal{J}}\mu\big{(}B(y_{j},5\psi(n)/2p)\cap
A_{n}\big{)}$ $\displaystyle\underset{\text{Lemma \ref{lemma3}}}{\leq}$
$\displaystyle\ \sum_{j\in\mathcal{J}}\
{\eta_{2}(3/2)^{\delta}}\left[\mu\Big{(}B\big{(}y_{j},5\psi(n)/2p\big{)}\Big{)}+a_{n}\right]\psi(n)^{\delta}$
$\displaystyle\underset{\eqref{upperbound},\,\eqref{ar}}{\leq}\,$
$\displaystyle\
\frac{\eta_{2}(4p)^{\delta}\mu(B)}{\eta_{1}^{2}\psi(n)^{\delta}}{\eta_{2}(3/2)^{\delta}}\left[\eta_{2}\big{(}5\psi(n)/2p\big{)}^{\delta}\psi(n)^{\delta}+a_{n}\psi(n)^{\delta}\right]$
$\displaystyle{=}\ \ \ \ $ $\displaystyle\
\frac{\eta_{2}^{2}3^{\delta}}{\eta_{1}^{2}}\mu(B)\left[\eta_{2}5^{\delta}\psi(n)^{\delta}+a_{n}(2p)^{\delta}\right]$

and

$\displaystyle\mu(A_{n}\cap B)\geq$
$\displaystyle\sum_{\begin{subarray}{c}j\in\mathcal{J}\\\
B(y_{j},\psi(n)/2p)\subset B\end{subarray}}\mu\big{(}B(y_{j},\psi(n)/2p)\cap
A_{n}\big{)}$ $\displaystyle\underset{\text{Lemma }\ref{lemma3}}{\geq}$
$\displaystyle\sum_{\begin{subarray}{c}j\in\mathcal{J}\\\
B(y_{j},\psi(n)/2p)\subset
B\end{subarray}}2^{-\delta}\left[\eta_{1}\mu\big{(}B(y_{j},\psi(n)/2p)\big{)}-\eta_{2}a_{n}\right]\psi(n)^{\delta}$
$\displaystyle\underset{\eqref{lowerbound},\,\eqref{ar}}{\geq}$
$\displaystyle\
\frac{\eta_{1}p^{\delta}\mu(B)}{\eta_{2}^{2}\big{(}5\psi(n)\big{)}^{\delta}}\
2^{-\delta}\left[\eta_{1}^{2}\big{(}\psi(n)/2p)\big{)}^{\delta}-\eta_{2}a_{n}\right]\psi(n)^{\delta}$
$\displaystyle=\ \ \ \ $
$\displaystyle\frac{\eta_{1}}{\eta_{2}20^{\delta}}\mu(B)\left[\frac{\eta_{1}^{2}}{\eta_{2}}\psi(n)^{\delta}-(2p)^{\delta}a_{n}\right],$

finishing the proof. ∎

######  Lemma 6.2.

Let $m>m_{0}$, and let $J$ be a cylinder in $\mathcal{F}_{m}$. Then
$\operatorname{diam}(J)\leq K_{1}\operatorname{diam}(X)K_{J}^{-1}$.

###### Proof.

Let $x,y\in J$. By (1.10) and the definition of $K_{J}$,

$d(x,y)\leq\frac{K_{1}d(T^{m}x,T^{m}y)}{K_{J}},$

hence by (1.10)

$d(x,y)\leq K_{1}\operatorname{diam}(X)K_{J}^{-1},$

which implies the needed upper bound on $\operatorname{diam}(J)$.∎

For the rest of the section, let us assume that
$\\{X_{i}\\}_{i\in\mathcal{I}}$ is pseudo-Markov. Let $0<\tau<1$ be such that
$\mu(TX_{i})\geq\tau\mu(X)$ for all $i\in\mathcal{I}$.

######  Lemma 6.3.

For all $n\in\mathbb{N}$ and for a nonempty cylinder

$J=X_{i_{0}}\cap T^{-1}X_{i_{1}}\cap\cdots\cap
T^{-n+1}X_{i_{n}}\in\mathcal{F}_{n},$

one has $T^{n}J=TX_{i_{n}}$.

###### Proof.

For each $0\leq j<n$, we have $T^{-j+1}X_{i_{j}}\cap
T^{-j}X_{i_{j+1}}\neq\varnothing$, and then $TX_{i_{j}}\cap
X_{i_{j+1}}\neq\varnothing$, so by the pseudo-Markov condition,
$TX_{i_{j}}\supset X_{i_{j+1}}$. Now let $x\in X_{i_{n}}$, then since
$TX_{i_{n-1}}\supset X_{i_{n}}$, there exists some $x_{n-1}\in X_{i_{n-1}}$ so
that $Tx_{n-1}=x$. Similarly, there exists some $x_{n-2}\in X_{i_{n-2}}$ such
that $Tx_{n-2}=x_{n-1}$. Continue this process until we find such $x_{0}\in
X_{i_{0}}$. Then $T^{j-1}x_{0}\in X_{i_{j}}$ for each $0<j<n$, so in
particular $x_{0}\in J$ and $T^{n-1}x_{0}=x$. Hence $T^{n-1}J\supset
X_{i_{n}}$.

For the reverse containment, since $J\subset T^{-n+1}X_{i_{n}}$, it follows
that$T^{n-1}J\subset X_{i_{n}}$. It remains to apply $T$ to both sides of
$T^{n-1}J=X_{i_{n}}$ to conclude the lemma. ∎

######  Lemma 6.4.

Let $m>m_{0}$, and let $J$ be a cylinder in $\mathcal{F}_{m}$. Then
$\mu(J)\gtrsim K_{J}^{-\delta}$.

###### Proof.

Write $J$ in the form

$J=X_{i_{1}}\cap T^{-1}X_{i_{2}}\cap\cdots T^{-m+1}X_{i_{m}};$

then

$K_{J}^{\delta}\mu(J)\underset{\text{Lemma
}\ref{lemma6}}{\asymp}\mu(T^{m}J)\underset{\text{Lemma
}\ref{equality}}{=}\mu(TX_{i_{m}})\geq\tau\mu(X),$

thus $\mu(J)\gtrsim K_{J}^{-\delta}$. ∎

We now prove a local estimate for the quasi-independence of the intersection
of sets $\\{A_{n}\\}_{n}$ with balls.

######  Corollary 6.5.

For any open ball $B=B(x,r)$ in $X$ with $2r<r_{0}$, there exists
$m_{r}\in\mathbb{N}$ so that for all $n>m>m_{r}$,

$\mu(B\cap A_{m}\cap
A_{n})\lesssim\mu(B)\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right],$

where the implicit constant in the above inequality is independent of $B$.

###### Proof.

Let $q>m_{0}$ and let $I\in\mathcal{F}_{q}$. For all $n>m\geq q$, by Lemma 4.3
we get

$\displaystyle\mu(I\cap A_{m}\cap
A_{n})=\sum_{\begin{subarray}{c}J\in\mathcal{F}_{m}\\\ J\subset
I\end{subarray}}\mu(J\cap A_{m}\cap
A_{n})\lesssim\sum_{\begin{subarray}{c}J\in\mathcal{F}_{m}\\\ J\subset
I\end{subarray}}\mu(J^{*}\cap A_{n})$ $\displaystyle\lesssim$
$\displaystyle\sum_{\begin{subarray}{c}J\in\mathcal{F}_{m}\\\ J\subset
I\end{subarray}}K_{J}^{-\delta}\left[[\psi^{\delta}(m)\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}]+[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}]\right]$
$\displaystyle\underset{\text{Lemma }\ref{lemma4.3}}{\lesssim}$ $\displaystyle
K_{I}^{-\delta}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right].$
(6.5)

Now let $B(x,r)$ be an open ball in $X$ with $r<r_{0}/2$. By the expanding
property (1.11), we can take some $m_{r}>m_{0}$ so that for all $q>m_{r}$,

$\inf_{J\in\mathcal{F}_{q}}K_{J}>r^{-1}K_{1}\operatorname{diam}(X);$

Let $q>m_{r}$. Then by Lemma 6.2 for all $J\in\mathcal{F}_{q}$,
$\operatorname{diam}(J)<r$, so

$B(x,r)\subset\bigsqcup_{\begin{subarray}{c}J\in\mathcal{F}_{q},\\\ J\cap
B(x,r)\neq\varnothing\end{subarray}}J\subset B(x,2r).$

Then

$\mu\big{(}B(x,r)\big{)}\leq\sum_{\begin{subarray}{c}J\in\mathcal{F}_{q},\\\
J\cap B(x,r)\neq\varnothing\end{subarray}}\mu(J)\leq\mu\big{(}B(x,2r)\big{)}.$

Therefore by (6.5),

$\displaystyle\mu\big{(}B(x,r)\cap A_{n}\cap A_{m}\big{)}\leq$
$\displaystyle\sum_{\begin{subarray}{c}J\in\mathcal{F}_{q},\\\ J\cap
B(x,r)\neq\varnothing\end{subarray}}\mu(J\cap A_{n}\cap A_{m})$
$\displaystyle\underset{\eqref{localqiest}}{\lesssim}$
$\displaystyle\sum_{\begin{subarray}{c}J\in\mathcal{F}_{q},\\\ J\cap
B(x,r)\neq\varnothing\end{subarray}}K_{I}^{-\delta}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right]$
$\displaystyle\underset{\text{Lemma }\ref{lemma4.3}}{\lesssim}$
$\displaystyle\sum_{\begin{subarray}{c}J\in\mathcal{F}_{q},\\\ J\cap
B(x,r)\neq\varnothing\end{subarray}}\mu(J)\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right]$
$\displaystyle\leq$ $\displaystyle\
\mu\big{(}B(x,2r)\big{)}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right]$
$\displaystyle\leq$ $\displaystyle\
2^{\delta}\frac{\eta_{2}}{\eta_{1}}\mu\big{(}B(x,r)\big{)}\left[\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right].$

∎

Finally let us apply the following generalization of the Lebesgue Density
Theorem to finish the proof of Theorem 1.5. Recall that a probability measure
$\mu$ on $X$ is called doubling if there exists a constant $D>0$ so that for
any $x\in X$ and $r>0$,

$\mu(B(x,2r))\leq D\mu(B(x,r)).$

It is clear that Ahlfors regular measures are doubling.

######  Theorem 6.6 (Lebesgue Density Theorem).

Let $(X,d)$ be a metric space with a Borel doubling probability measure $\mu$,
and let $E$ be a Borel subset of $X$. Suppose there exist constants $C>0$ and
$r_{0}>0$ so that for all balls $B\subset X$ with radii less than $r_{0}$, we
have

$\mu(E\cap B)\geq C\mu(B).$

Then $\mu(E)=1$.

For a proof, see [BDV, §8, Lemma 7].

###### Proof of Theorem 1.5.

By Lemma 6.1 and Corollary 6.5, there exist positive constants
$s_{1},s_{2},s_{3}$ so that for all $B=B(x,r)$ in $X$ with $r<r_{0}/2$ and for
all $n>\max\\{n_{r},m_{r}\\}$,

$\mu(B)s_{1}\left(\psi(n)^{\delta}-a_{n}\right)\leq\mu(B\cap
A_{n})\leq\mu(B)s_{3}\left(\psi(n)^{\delta}+a_{n}\right).$ $\mu(B\cap
A_{n}\cap
A_{m})\leq\mu(B)s_{2}\left(\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right).$

Now take $b_{n}=\psi(n)^{\delta}$ and

$E_{k}=\begin{cases}A_{k}\cap B&\text{if }k>\max\\{n_{r},m_{r}\\}\\\
\varnothing&\text{otherwise.}\end{cases}$ $\mu(\limsup_{n\to\infty}A_{n}\cap
B)=\mu(\limsup_{k\to\infty}E_{k})\geq\frac{\big{(}\mu(B)s_{1})^{2}}{2s_{2}\mu(B)}=\frac{s_{1}^{2}}{2s_{2}}\mu(B).$

Then by the Lebesgue Density Theorem (Theorem 6.6),

$\mu\big{(}R_{T}^{f}(\psi)\big{)}=\mu(\limsup_{n\to\infty}A_{n})=1.$

∎

##  7\. Examples

Here we list several examples of dynamical systems to which our theorems
apply. The first two come from the paper [HLSW]:

  * •

$X=[0,1]$, $T:x\mapsto\beta x\mod{1}$, where $\beta>1$, and $\mu$ is the
$T$-invariant probability measure absolutely continuous with respect to
Lebesgue measure, namely (see [R])

$\mu(E)=\begin{cases}\operatorname{Leb}(E)&\text{if }\beta\text{ is an
integer,}\\\
\frac{1}{\sum_{k=0}^{\infty}\frac{\\{\beta^{k}\\}}{\beta^{k}}}\sum_{k=0}^{\infty}\frac{\operatorname{Leb}\left(E\cap[0,\\{\beta^{k}\\}]\right)}{\beta^{k}}&\text{if
}\beta\text{ is not an integer,}\end{cases}$

where $\\{x\\}$ denotes the fractional part of $x$;

  * •

$X=[0,1]$, $T:x\mapsto\frac{1}{x}\mod{1}$, and $\mu$ is the Gauss measure
given by ${d\mu}=\frac{dx}{(\log 2)(1+x)}$.

Sections 3.1–3.2 of [HLSW] together with Remark 1.2 show that in both cases
the assumptions of Theorem 1.3 are satisfied. In fact, in both cases uniform
mixing with exponential rate was first exhibited in [Ph], together with a
quantitative shrinking target property of these systems.

The pseudo-Markov property holds for the Gauss map but only for some special
$\beta$-transformations. We will prove the full measure in the divergence case
for arbitrary $\beta$-transformations in §8.

Our last example deals with self-similar sets. Let

$\Theta:=\\{\theta_{i}(x):\mathbb{R}\to\mathbb{R}\\}_{i=1}^{L}$

be a system of similarities with

$|\theta_{i}(x)-\theta_{i}(y)|=r_{i}|x-y|\text{ for all }x,y\in\mathbb{R},\
i=1,\ldots,L,$

where $0<r_{i}<1$ for all $i$. Then by [H, Theorem 3.1.(3)] there exists a
unique nonempty compact set $X\subset{\mathbb{R}}$, called the attractor of
the system, such that

$X=\bigcup_{i=1}^{L}\theta_{i}(X).$

Furthermore, we assume that $\Theta$ satisfies the open set condition: that
is, there exists a non-empty bounded open set $U\subset\mathbb{R}$ such that

$\bigcup_{i=1}^{L}\theta_{i}(U)\subset U\text{ and
}\theta_{i}(U)\cap\theta_{j}(U)=\varnothing\text{ for all }i\neq j.$

Then it is known that the Hausdorff dimension of $X$ is equal to the unique
solution $\delta\in[0,1]$ of the equation $\sum_{i=1}^{L}r_{i}^{\delta}=1$
(see [F, Theorem 9.3]). Furthermore, the normalized restriction $\mu$ of the
$\delta$-dimensional Hausdorff measure to $X$ is positive, finite and
satisfies

${\mu=\sum_{i=1}^{L}r^{\delta}_{i}\cdot(\theta_{i})_{*}\mu.}$ (7.1)

(For a proof, see [H, Theorem 4.4.(1)].)

To define the corresponding expanding map and construct the cylinders, we
consider the following lemma from [Sc, Gr]:

######  Lemma 7.1 ([Sc], Theorem 2.2; [Gr], Lemma 3.3).

Let $\\{\theta_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}\\}_{i=1}^{L}$ be a system
of similarities satisfying the open set condition, $X$ its attractor, and
$\mu$ the self-similar measure given by (7.1). Then there exists a nonempty
compact set $A$ with

  * (i)

$\theta_{i}(A)\subset A$ for all $i=1,\ldots,L$;

  * (ii)

$\theta_{i}\big{(}\mathsf{int}(A)\big{)}\cap\theta_{j}\big{(}\mathsf{int}(A)\big{)}=\varnothing$
for each $i\neq j$;

  * (iii)

$\mu\big{(}\mathsf{int}(A)\cap X\big{)}=1$.

We remark that parts (i) and (ii) are stated in [Sc, Theorem 2.2], and part
(iii) follows from the proof of [Gr, Lemma 3.3], where it is shown that
$\mu\big{(}(A\smallsetminus\mathsf{int}(A)\big{)}=0$ and
$X=\operatorname{supp}\mu\subset A$.

Now define

$X_{i}:=\theta_{i}\big{(}\mathsf{int}(A)\cap X\big{)}.$ (7.2)

Each $X_{i}$ is open in $X$ because $\theta_{i}$ is an open map. The
disjointness of $X_{i}$ and $X_{j}$ for $i\neq j$ follows from Lemma 7.1(ii).
Finally, one can write

$\displaystyle\mu\left(\bigcup_{i=1}^{L}X_{i}\right)$
$\displaystyle=\sum_{i=1}^{L}\mu\big{(}\theta_{i}(\mathsf{int}(A)\cap
X)\big{)}=\sum_{i=1}^{L}r_{i}^{\delta}\mu(\mathsf{int}(A)\cap X)$ (by Lemma
7.1(iii)) $\displaystyle=\sum_{i=1}^{L}r_{i}^{\delta}=1=\mu(X).$

Hence one can define the map $T:X\to X$ $\mu$-almost everywhere by setting
$T|_{X_{i}}=\theta_{i}^{-1}|_{X_{i}}$. It follows from (7.1) that $(X,\mu,T)$
is a measure-preserving system. Clearly $T|_{X_{i}}$ is continuous and
injective for every $i$. Therefore the collection $\\{X_{i}\\}_{i=1}^{L}$
satisfies our assumption for being cylinders of order $1$.

For $\mathbf{i}=(i_{1},\dots,i_{m})\in\\{1,\ldots,L\\}^{m}$ let us define

$\theta_{\mathbf{i}}:=\theta_{i_{1}}\circ\cdots\circ\theta_{i_{m}}{\quad\text{and}\quad
r_{\mathbf{i}}:=\prod_{k=1}^{m}r_{i_{k}}}.$

Using the definition (1.7) of cylinders of order $m$ together with (7.2), it
is easy to see that the set $\mathcal{F}_{m}$ of cylinders of order $m$ is
precisely

$\big{\\{}X_{\mathbf{i}}:=\theta_{\mathbf{i}}\big{(}\mathsf{int}(A)\cap
X\big{)}:\mathbf{i}\in\\{1,\ldots,L\\}^{m}\big{\\}},$

and the restriction of $T^{m}$ onto $X_{\mathbf{i}}\in\mathcal{F}_{m}$
coincides with $\theta_{\mathbf{i}}^{-1}$. This, in particular, implies that

${\mu(X_{\mathbf{i}})=r_{\mathbf{i}}^{\delta}}$ (7.3)

and

${K_{J}=\frac{|T^{m}x-T^{m}y|}{|x-y|}=r_{\mathbf{i}}^{-1}\quad\text{for all
}x,y\in J=X_{\mathbf{i}}\in\mathcal{F}_{m}.}$ (7.4)

Denote

$r_{\max}:=\max_{i=1,\dots,L}r_{i}.$

Let us now verify assumptions (1.8)–(1.13) of Theorem 1.3.

  1. (1.8):

By [H, Theorem 5.3(1)(i)],

$\gamma_{1}\leq\liminf_{r\to
0}\frac{\mu\big{(}B(x,r)\big{)}}{r^{\delta}}\leq\limsup_{r\to
0}\frac{\mu\big{(}B(x,r)\big{)}}{r^{\delta}}\leq\gamma_{2}$

for some $0<\gamma_{1}<\gamma_{2}<\infty$ and all $x\in X$. Clearly it implies
that $\mu$ is Ahlfors regular of dimension $\delta$.

  2. (1.9):

($T$ is uniformly mixing) Let $E$ be a non-empty open ball in $X$, let $F$ be
a measurable set in $X$, and let $m\in\mathbb{N}$. Note that for all cylinders
$J=X_{\mathbf{i}}\in\mathcal{F}_{m}$, where
$\mathbf{i}\in\\{1,\ldots,L\\}^{m}$ one can write

$\mu(J\cap T^{-m}F)=r_{\mathbf{i}}^{\delta}\mu(T^{m}J\cap
F)=r_{\mathbf{i}}^{\delta}\mu\big{(}\mathsf{int}(A)\cap
F\big{)}\underset{\eqref{eq:measure}}{=}\mu(J)\mu(F).$

Note that since $E$ is an interval, we can (up to a set of measure zero) write
$E$ as a disjoint union of cylinders of order $m$ and at most two balls
contained in cylinders of order $m$; i.e.,

$E=\Big{(}\bigcup_{J\in\mathcal{F}_{m},\,J\subset E}J\Big{)}\cup E_{1}\cup
E_{2}$

where the unions above are disjoint, and $E_{1},E_{2}$ are contained in some
cylinders $J_{1},J_{2}\in\mathcal{F}_{m}$ respectively, hence have measure not
greater than $r_{\max}^{m\delta}$. Then

$\displaystyle\left|\mu(E\cap T^{-{m}}F)-\mu(E)\mu(F)\right|$ $\displaystyle=\
$ $\displaystyle\Big{|}\sum_{\begin{subarray}{c}J\in\mathcal{F}_{m},\\\
J\subset E\end{subarray}}(J\cap
T^{-{m}}F)-\sum_{\begin{subarray}{c}J\in\mathcal{F}_{m},\\\ J\subset
E\end{subarray}}\mu(J)\mu(F)+\mu(E_{1}\cap T^{-{m}}F)-\mu(E_{1})\mu(F)$
$\displaystyle+\ $ $\displaystyle\mu(E_{2}\cap
T^{-{m}}F)-\mu(E_{2})\mu(F)\Big{|}$ $\displaystyle=\ $
$\displaystyle\Big{|}\mu(E_{1}\cap T^{-{m}}F)-\mu(E_{1})\mu(F)+\mu(E_{2}\cap
T^{-{m}}F)-\mu(E_{2})\mu(F)\Big{|}$ $\displaystyle\leq\ $
$\displaystyle\mu(J_{1}\cap T^{-{m}}F)+\mu(J_{1})\mu(F)+\mu(J_{2}\cap
T^{-{m}}F)+\mu(J_{2})\mu(F)$ $\displaystyle=\ $
$\displaystyle{4}r_{\max}^{m\delta}\mu(F),$

and (1.9) follows with $a_{n}={4}r_{\max}^{n\delta}$.

  3. (1.10):

Follows from (7.4) with $K_{1}=1$.

  4. (1.11):

Again from (7.4), for all $m\in\mathbb{N}$ we have

$\inf_{J\in\mathcal{F}_{m}}K_{J}=\inf_{\mathbf{i}\in\\{1,\ldots,L\\}^{m}}r_{\mathbf{i}}^{-1}=r_{\max}^{-m},$

which goes to $\infty$ as $m\to\infty$.

  5. (1.12):

In view of (7.4), for all $m\in{\mathbb{N}}$ one can write

$\sum_{J\in\mathcal{F}_{m}}K_{J}^{-\delta}=\sum_{\mathbf{i}\in\\{1,\ldots,L\\}^{m}}r_{\mathbf{i}}^{\delta}=(r_{1}^{\delta}+\cdots+r_{m}^{\delta})^{m}=1.$

  6. (1.13):

Also follows from (7.4) with $K_{2}=1$.

It is clear that $\\{X_{i}\\}_{i\in\mathcal{I}}$ is pseudo-Markov. Thus, by
Theorem 1.5, for any function $\psi:\mathbb{N}\to\mathbb{R}^{+}$ and any
Lipshitz function $f:X\to X$, the $f$-twisted recurrence set $R_{T}^{f}(\psi)$
satisfies

  * •

$\mu\big{(}R_{T}^{f}(\psi)\big{)}=0$ if
$\sum_{n=1}^{\infty}\psi(n)^{\delta}<\infty$;

  * •

$\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$ if
$\sum_{n=1}^{\infty}\psi(n)^{\delta}=\infty$.

##  8\. Proof of Theorem 1.6

Let $\beta>1$ be a real number and suppose
$\sum_{n=1}^{\infty}\psi(n)=\infty$. In this section, we will consider the
system

$X=[0,1],\quad T=M_{\beta},\quad\mu\text{ the $M_{\beta}$-invariant measure,}$

and the partition

$\left\\{X_{i}=\left(\frac{i}{\beta},\frac{i+1}{\beta}\right):i=0,1,\ldots,\lfloor\beta\rfloor-1\right\\}\cup\left\\{X_{\lfloor\beta\rfloor}=\left(\frac{\lfloor\beta\rfloor}{\beta},1\right)\right\\}.$
(8.1)

If $\beta$ is an integer, then (8.1) is pseudo-Markov. Furthermore, if $\beta$
satisfies

$\beta^{2}-k\beta-\ell=0,\quad\text{for some }k\geq\ell\in\mathbb{N}^{*}$

then

$TX_{i}=X\ \ \forall i<\lfloor\beta\rfloor\quad\text{and}\quad
TX_{\lfloor\beta\rfloor}=\left(0,\frac{\ell}{\beta}\right),$

and hence (8.1) is also pseudo-Markov. In both cases, we have
$\mu\left(R_{T}^{f}(\psi)\right)=1$. by Theorem 1.5. We now prove the general
case, which is also proved in [LWW].

We will apply Lemmas 6.1 and 4.3. We remark that analogous results are also
proved in [LWW]; however our lemmas are proved in a more general abstract
setting, do not depend on the actual arithmetic and symbolic coding of the
systems, and have a much weaker assumption on the regularity of the cylinders.

We first state some facts about $(X,\mu,T)$.

######  Lemma 8.1 ([R]).

For $(X,\mu,T)$ defined above, $\mu$ is equivalent to $\operatorname{Leb}$ and
$\mu(E)=\frac{1}{\sum_{k=0}^{\infty}\frac{\\{\beta^{k}\\}}{\beta^{k}}}\sum_{k=0}^{\infty}\frac{\operatorname{Leb}\left(E\cap[0,\\{\beta^{k}\\}]\right)}{\beta^{k}}$
for all Lebesgue-measurable set $E$.

For a proof, see [R, Theorem 1].

To state the next lemma, we define the lexicographical order $\prec$. For two
words $\alpha=(\alpha_{1},\alpha_{2},\ldots)$ and
$\beta=(\beta_{1},\beta_{2},\ldots)$, we write $\alpha\prec\beta$ if there
exists $k\in\mathbb{N}$ so that $\alpha_{i}=\beta_{i}$ for all $i\leq k$ and
$\alpha_{k}<\beta_{k}$. We write $\alpha\preceq\beta$ if $\alpha\prec\beta$ or
$\alpha=\beta$. Denote the $\beta$-expansion of $1$ by

$1=\sum_{n=1}^{\infty}\frac{\xi_{n}}{\beta^{n}}.$

If $\\{\xi_{n}\\}_{n}$ is not eventually zero, then we define
$\xi_{n}^{*}=\xi_{n}$ for all $n\in\mathbb{N}$. If $\\{\xi_{n}\\}_{n}$ is
eventually zero, then denote $M_{\xi}:=\max\\{n\in\mathbb{N}:\xi_{n}\neq 0\\}$
and define

$\xi_{n}^{*}=\begin{cases}\xi_{k}&\text{if }n\equiv k\mod{M_{\xi}},\,k>0\\\
\xi_{M_{\xi}}-1&\text{if }n\equiv 0\mod{M_{\xi}}.\end{cases}$

A classical result says that the right-most cylinder has the maximal coding in
lexicographical order, in the following sense:

######  Lemma 8.2 ([P]).

Let $(X,\mu,T)$ and $\\{X_{i}\\}_{i\in\mathcal{I}}$ be defined above, and let

$J=X_{i_{0}}\cap T^{-1}X_{i_{1}}\cap\cdots\cap T^{-n+1}X_{i_{n-1}}.$

Then $J$ is nonempty if and only if

$(i_{j},i_{j+1},\ldots,i_{n-1})\prec(\xi_{1}^{*},\xi_{2}^{*},\ldots),\quad\text{for
each }j=0,\ldots,n-1.$

For a proof, see [P, Theorem 3].

For a cylinder $J\in\mathcal{F}_{n}$ we always have
$\operatorname{Leb}(J)\leq\beta^{-n}$. Let us call $J$ full if the upper bound
is reached; i.e., $\operatorname{Leb}(J)=\beta^{-n}$.

######  Lemma 8.3 ([FW]).

Let $J$ be as in Lemma 8.2. If $J$ is nonempty, then

$X_{i_{0}}\cap T^{-1}X_{i_{1}}\cap\cdots\cap T^{-n+1}X_{j_{n-1}}$

is full for all $j_{n-1}<i_{n-1}$.

For a proof, see [FW, Lemma 3.2(1)].

######  Proposition 8.4.

For all $n\in\mathbb{N}$ and for all $J\in\mathcal{F}_{n}$ with
$J\neq\varnothing$, there exists some full cylinder $I\in\mathcal{F}_{m}$ with

$I\subset
J\quad\text{and}\quad\operatorname{Leb}(I)\geq\frac{\operatorname{Leb}(J)}{\beta}.$

A similar inequality is proved in [LWW].

###### Proof.

Let $J\in\mathcal{F}_{n}$ with $J\neq\varnothing$. By Lemma 8.2, the set

$\mathcal{N}_{J}:=\left\\{k\in\mathbb{N}:J\cap\bigcap_{j=0}^{k}T^{-n-j+1}X_{0}\cap
T^{-n-k}X_{1}\neq\varnothing\right\\}$

is nonempty. Let $k_{J}=\inf\mathcal{N}_{J}$. If $k_{J}=0$, then trivially we
have $J\cap\bigcap_{j=0}^{k_{J}}T^{-n-j+1}X_{0}=J$. Now suppose $k_{J}>0$.
Assume

$J\supsetneq J\cap\bigcap_{j=0}^{k_{J}}T^{-n-j+1}X_{0}.$

Then there exists some $1\leq i<n-1$ such that

$J\cap\bigcap_{j=0}^{k_{J}-1}T^{-n-j+1}X_{0}\cap
T^{-n-k_{J}+1}X_{i}\neq\varnothing.$ (8.2)

But by Lemma 8.3, $i=1$ must satisfy (8.2), which contradicts the minimality
of $k_{J}$ in $\mathcal{N}_{J}$, so

$J\cap\bigcap_{j=0}^{k_{J}}T^{-n-j+1}X_{0}=J.$

By Lemma 8.3

$\operatorname{Leb}\left(J\cap\bigcap_{j=0}^{k_{J}+1}T^{-n-j+1}X_{0}\right)=\beta^{-n-k-1}.$

On the other hand,

$\operatorname{Leb}\left(J\cap\bigcap_{j=0}^{k_{J}}T^{-n-j+1}X_{0}\right)\leq\beta^{-n-k},$

hence

$\operatorname{Leb}(J)\leq\beta\operatorname{Leb}\left(J\cap\bigcap_{j=0}^{k_{J}+1}T^{-n-j+1}X_{0}\right).$

Taking $I=J\cap\bigcap_{j=0}^{k_{J}+1}T^{-n-j+1}X_{0}$, we finish the proof. ∎

Note that $K_{J}=\beta^{n}$ for all $J\in\mathcal{F}_{n}$. Now we can prove
Theorem 1.6.

###### Proof of Theorem 1.6.

Sections 3.1–3.2 of [HLSW] together with Remark 1.2 show that the system
satisfies (1.8)–(1.13). By Lemma 6.1, there exists positive constants
$s_{1},s_{3}$ so that for all $B=(x,r)$ with $r<\frac{1}{2}$ and for all
$n>n_{0}$,

$s_{1}\mu(B)\left(\psi(n)^{\delta}-a_{n}\right)\leq\mu(B\cap A_{n})\leq
s_{3}\mu(B)\left(\psi(n)^{\delta}+a_{n}\right).$ (8.3)

By Lemma 8.1, there exist a constant $\alpha>0$ such that

$\frac{1}{\alpha}\operatorname{Leb}(E)\leq\mu(E)\leq\alpha\operatorname{Leb}(E),\quad\text{for
all Lebesgue-measurable }E\subset X.$

Suppose $I\in\mathcal{F}_{q}$ and $\operatorname{Leb}(I)=\beta^{-q}$. Then by
Lemma 4.3, for all $n>m>q>m_{0}$,

$\displaystyle\mu(I\cap A_{m}\cap A_{n})\lesssim$
$\displaystyle\sum_{\begin{subarray}{c}J\in\mathcal{F}_{m}\\\ J\subset
I\end{subarray}}\beta^{-m}\left(\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right)$
$\displaystyle\leq$ $\displaystyle\
\operatorname{Leb}(I)\left(\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right)$
$\displaystyle\leq$ $\displaystyle\
\alpha\mu(I)\left(\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right),$

so there exists a constant $s_{2}$ such that

$\mu(I\cap A_{m}\cap A_{n})\leq
s_{2}\mu(I)\left(\psi(m)^{\delta}\psi(n)^{\delta}+a_{n-m}\psi(n)^{\delta}+a_{n}\psi(m)^{\delta}\right).$
(8.4)

Note that in this system all cylinders are intervals, so by (8.3) and (8.4)
together with Lemma 5.2, there exists a constant $\gamma>0$ so that for all
$q>m_{0}$ and for all cylinders $I\in\mathcal{F}_{q}$ with
$\operatorname{Leb}(I)=\beta^{-q}$, we have

$\mu\left(I\cap R_{f}^{T}(\psi)\right)\geq\gamma\mu(I).$

By Proposition 8.4, for all cylinders $J\in\mathcal{F}_{n}$, there exists a
cylinder $I\subset J$ with $I\in\mathcal{F}_{q}$,
$\operatorname{Leb}(I)=\beta^{-q}$, and
$\operatorname{Leb}(I)\geq\frac{1}{\beta}\operatorname{Leb}(J)$. If $n>m_{0}$,
then

$\mu\left(J\cap R_{f}^{T}(\psi)\right)\geq\mu\left(I\cap
R_{f}^{T}(\psi)\right)\geq\gamma\mu(I)\geq\frac{\gamma}{\alpha^{2}\beta}\mu(J).$

Now take $B(x,r)\subset X$ with $r<1$. Take $n\in\mathbb{N}$ with
$n>-\log_{\beta}\frac{r}{4}$. Then for all $J\in\mathcal{F}_{n}$,
$\operatorname{diam}(J)\leq 2\beta^{-n}<{r}/{2}$, and

$B\left(x,{r}/{2}\right)\subset\bigsqcup_{\begin{subarray}{c}J\in\mathcal{F}_{n}\\\
J\cap B\left(x,{r}/{2}\right)\neq\varnothing\end{subarray}}J\subset B(x,r).$

Hence

$\displaystyle\mu\left(B(x,r)\cap R_{f}^{T}(\psi)\right)\geq$ $\displaystyle\
\mu\left(\bigsqcup_{\begin{subarray}{c}J\in\mathcal{F}_{n}\\\ J\cap
B\left(x,{r}/{2}\right)\neq\varnothing\end{subarray}}J\cap
R_{f}^{T}(\psi)\right)\geq\sum_{\begin{subarray}{c}J\in\mathcal{F}_{n}\\\
J\cap
B\left(x,{r}/{2}\right)\neq\varnothing\end{subarray}}\frac{\gamma}{\alpha^{2}\beta}\mu(J)$
$\displaystyle\geq$ $\displaystyle\
\frac{\gamma}{\alpha^{2}\beta}\mu\big{(}B(x,{r}/{2})\big{)}\geq\frac{\gamma}{2\alpha^{4}\beta}\mu\big{(}B(x,r)\big{)}.$

Thus it follows from the Lebesgue Density Theorem (Theorem 6.6) that
$\mu\big{(}R_{T}^{f}(\psi)\big{)}=1$. ∎

## References

  * [B] M. Boshernitzan, Quantitative recurrence results, Invent. Math. 113 (1993), no. 3, 617–631.
  * [BDV] V. Beresnevich, D. Dickinson and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846, x+91 pp.
  * [BF] S. Baker and M. Farmer, Quantitative recurrence properties for self-conformal sets, Proc. Amer. Math. Soc. 149 (2021), no. 3, 1127–1138.
  * [C] J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts in Mathematics and Physics, Cambridge University Press, London, 195
  * [CE] K. L. Chung and P. Erdös, On the application of the Borel–Cantelli lemma , Trans. Amer. Math. Soc. 72 (1952), no. 1, 179–186.
  * [CK] N. Chernov and D. Kleinbock, Dynamical Borel-Cantelli lemmas for Gibbs measures, Israel J. Math. 122 (2001), 1–27.
  * [CWW] Y. Chang, M. Wu and W. Wu, Quantitative recurrence properties and homogeneous self-similar sets, Proc. Amer. Math. Soc. 147 (2019), 1453–1465.
  * [DFL] D. Dolgopyat, B. Fayad and S. Liu, Multiple Borel Cantelli Lemma in dynamics and MultiLog law for recurrence, J. Mod. Dyn. 18 (2022), 209–289.
  * [EW] M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, 259, Springer–Verlag London, Ltd., London, 2011, xviii+481 pp.
  * [F] K. Falconer, Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, Ltd., Chichester, 2014. xxx+368 pp.
  * [FK] M. Fuchs and D. H. Kim, On Kurzweil’s 0-1 law in inhomogeneous Diophantine approximation, Acta Arith. 173 (1frm-e016), 41–57.
  * [FMP] L. Fernández, M. V. Melían, and D. Pestana, Quantitative mixing results and inner functions, Math. Ann. 337 (2007), no. 1, 233–251.
  * [FW] A. Fan and B. Wang, On the lengths of basic intervals in beta expansions, Nonlinearity, 25 (2012), no. 5, 1329–1343.
  * [GP] S. Galatolo and P. Peterlongo, Long hitting time, slow decay of correlations and arithmetical properties, Discrete Contin. Dyn. Syst. 27 (2010), no. 1, 185–204.
  * [Gr] S. Graf, On Bandt’s tangential distribution for self-similar measures, Monatsh. Math. 120 (1995), no. 3-4, 223–246.
  * [H] J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747.
  * [HLSW] M. Hussain, B. Li, D. Simmons and B.-W. Wang, Dynamical Borel-Cantelli lemma for recurrence theory, Ergodic Theory Dynam. Systems 42 (2022), no. 6, 1994–2008.
  * [HNPV] N. Haydn, M. Nicol, T. Persson and S. Vaienti, A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems, Ergodic Theory Dynam. Systems 33 (2013), no. 2, 475–498.
  * [K] J. Kurzweil, On the metric theory of inhomogeneous Diophantine approximations, Studia Math. 15 (1955), 84–112.
  * [KKP] M. Kirsebom, P. Kunde and T. Persson, On shrinking targets and self-returning points, Preprint (2020), arXiv:2003.013613.
  * [KM] D. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), no. 3, 451–494.
  * [LWW] F. Lü, B-W. Wang and J. Wu, Diophantine analysis of the expansions of a fixed point under continuum many bases , Preprint (2021), arXiv:arXiv:2103.00546.
  * [P] W. Parry, On the $\beta$-expansion of real numbers, Acta Math. Acad. Sci. Hungar, 11 (1960), 401–416.
  * [Pa] L. Pawelec, Iterated logarithm speed of return times, Bull. Aust. Math. Soc. 96 (2017), 468–478.
  * [Pe] T. Persson, A strong Borel–Cantelli lemma for recurrence, Preprint (2022), arXiv:2202.07344.
  * [Ph] W. Philipp, Some metrical theorems in number theory, Paci. Jour. Math. 20 (1967), no. 1, 109–127.
  * [R] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.
  * [S] B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 259–267.
  * [Sc] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), no. 1, 111–115.