#  joint normality of representations of numbers: an ergodic approach

Vitaly Bergelson  Department of Mathematics
Ohio State University
Columbus, OH 43210, USA<EMAIL_ADDRESS>and  Younghwan Son
Department of Mathematics
POSTECH
Pohang, 37673, Korea<EMAIL_ADDRESS>
###### Abstract.

We introduce an ergodic approach to the study of joint normality of
representations of numbers. For example, we show that for any integer $b\geq
2$ almost every number $x\in[0,1)$ is jointly normal with respect to the
$b$-expansion and continued fraction expansion. This fact is a corollary of
the following result which deals with pointwise joint ergodicity:

Let $T_{b}:[0,1]\rightarrow[0,1]$ be the times $b$ map defined by
$T_{b}x=bx\,\bmod\,1$ and let $T_{G}:[0,1]\rightarrow[0,1]$ be the Gauss map
defined by $T_{G}(x)=\\{\frac{1}{x}\\}$ for $x\neq 0$ and $T_{G}(0)=0.$ (Here
$\\{\cdot\\}$ denotes the fractional part.) For any $f,g\in
L^{\infty}(\lambda)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T_{b}^{n}x)\,g(T_{G}^{n}x)=\int
f\,d\lambda\cdot\int g\,d\mu_{G}\quad\text{for almost every }x\in[0,1],$

where $\lambda$ is the Lebesgue measure on $[0,1]$ and $\mu_{G}$ is the Gauss
measure on $[0,1]$ given by $\mu_{G}(A)=\frac{1}{\log
2}\int_{A}\frac{1}{1+x}dx$ for any measurable set $A\subset[0,1]$.

We show that the phenomenon of the pointwise joint ergodicity takes place for
a wide variety of number-theoretical maps of the interval and derive the
corresponding corollaries pertaining to joint normality.

We also establish the equivalence of various forms of normality and joint
normality for representations of numbers, hereby providing a general framework
for classical normality results.

††footnotetext: _Mathematics Subject Classification (2020)_ 11K16, 37A44

##  1\. Introduction

For $x\in[0,1)$ and integer $b\geq 2$, consider the $b$-expansion of $x$:

$x=\sum_{n=1}^{\infty}\frac{a_{n}}{b^{n}},$ (1.1)

where $a_{n}\in\\{0,1,\dots,b-1\\}$.111The expansion (1.1) is not unique for
rational numbers of the form $\frac{a}{b^{n}}$. To ensure uniqueness, one can
assume that $a_{n}<b-1$ for infinitely many $n$. This point is actually mute
since we deal in this paper with results which apply to “typical” $x\in[0,1).$
The number $x$ is called simply normal in base $b$ if for any
$v\in\\{0,1,\dots,b-1\\}$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:a_{n}=v\\}}{N}=\frac{1}{b}.$

The number $x$ is called normal in base $b$ if for any $k\in\mathbb{N}$ and
for any finite string $S=(v_{1},\dots,v_{k})\in\\{0,1,\dots,b-1\\}^{k}$, we
have

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:a_{n}=v_{1},\dots,a_{n+k-1}=v_{k}\\}}{N}=\frac{1}{b^{k}}.$ (1.2)

The classical Borel’s normal number theorem ([9]) states that almost every $x$
(with respect to the Lebesgue measure $\lambda$) is normal in base $b$ (or
$b$-normal).

Nowadays this result can be easily obtained by means of the ergodic
theory.222Borel derived his theorem from the law of large numbers. One can
also derive Borel’s theorem by proving first that a number $x\in[0,1)$ is
normal in base $b$ if and only if the sequence $(b^{n}x),n\in\mathbb{N}$, is
uniformly distributed $\bmod\,1$ and invoking a theorem due to Weyl (see [27],
Theorem 4.1 on p.32), which states that for any sequence $(a_{n})$ of distinct
integers, the sequence $(a_{n}x)$ is uniformly distributed $\bmod\,1$ for
almost every $x$. Various equivalent forms of normality are summarized in
Theorem 1.12 below. For historical comments on the notion of normal number and
the controversies that Borel’s paper generated, see Appendix in [6]. Indeed,
consider the map $T_{b}:[0,1]\rightarrow[0,1]$ defined by $T_{b}x=bx\,\bmod
1$. Transformation $T_{b}$ is Lebesgue measure preserving and ergodic, and so
by the pointwise ergodic theorem, we have for any $f\in L^{1}(\lambda),$

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T_{b}^{n}(x))=\int
f\,d\lambda\quad\text{for a.e. }x.$ (1.3)

Let $C_{b}(S)$ be the set of numbers in $[0,1)$ whose first $k$ terms in
$b$-expansion are $(v_{1},\dots,v_{k})$. Applying (1.3) to the characteristic
function $1_{C_{b}(S)}$, we get (1.2).

Ergodic theory also provides a convenient way of dealing with normality of
continued fraction expansions. Let $T_{G}:[0,1]\rightarrow[0,1]$ be the Gauss
map defined by $T_{G}(x)=\\{\frac{1}{x}\\}$ for $x\neq 0$ and $T_{G}(0)=0.$
(Here $\\{\cdot\\}$ denotes the fractional part.) Let $\mu_{G}$ denote the
Gauss measure defined by $\mu_{G}(A)=\frac{1}{\log 2}\int_{A}\frac{1}{1+x}dx$
for any measurable set $A\subset[0,1]$. It is well-known that $T_{G}$ is
$\mu_{G}$-preserving and ergodic. (See [26] and [47].) By invoking the
pointwise ergodic theorem, we have that for any $f\in L^{1}(\mu_{G}),$

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T_{G}^{n}(x))=\int
f\,d\mu_{G}\quad\text{for a.e. }x.$ (1.4)

Consider now a regular continued fraction of a number $x\in[0,1)$:

$x=\cfrac{1}{c_{1}+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}+\cdots}}}=[c_{1},c_{2},c_{3},...],\quad
c_{n}\in\mathbb{N}.$ (1.5)

Given $k\in\mathbb{N}$ and a finite string
$S=(v_{1},\dots,v_{k})\in\mathbb{N}^{k}$, let $C_{G}(S)$ be the set of numbers
in $[0,1)$ whose first $k$ terms in the continued fraction representation in
(1.5) are $(v_{1},\dots,v_{k})$. A number $x\in[0,1)$ is called normal with
respect to the continued fraction expansion (or CF-normal) if for any
$k\in\mathbb{N}$ and for any string $S=(v_{1},\dots,v_{k})\in\mathbb{N}^{k},$

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:c_{n}=v_{1},\dots,c_{n+k-1}=v_{k}\\}}{N}=\mu_{G}(C_{G}(S)).$

It is not hard to show that (1.4) implies that almost every $x\in[0,1)$ is CF-
normal.

It follows that almost every $x\in[0,1)$ is simultaneously $b$-normal and CF-
normal. In this paper we are addressing the following more subtle question: Is
it true that almost every $x$ is jointly $b$-normal and CF-normal? By this we
mean that for any $k,l\in\mathbb{N}$ and for any
$S_{1}=(v_{1},\dots,v_{k})\in\\{0,1,\dots,b-1\\}^{k}$ and
$S_{2}=(w_{1},\dots,w_{l})\in\mathbb{N}^{l}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\left|\\{1\leq n\leq
N:a_{n+i-1}=v_{i}\,(1\leq i\leq k)\text{ and }c_{n+j-1}=w_{j}\,(1\leq j\leq
l)\\}\right|=\lambda(C_{b}(S_{1}))\cdot\mu_{G}(C_{G}(S_{2})).$ (1.6)

Note that if $x$ is jointly $b$-normal and CF-normal, then it is $b$-normal
and CF-normal and moreover

$\displaystyle\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:a_{n+i-1}=v_{i}\,(1\leq i\leq k)\text{ and }c_{n+j-1}=w_{j}\,(1\leq j\leq
l)\\}}{N}$ (1.7) $\displaystyle=\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq
n\leq N:a_{n+i-1}=v_{i}\,(1\leq i\leq
k)\\}}{N}\times\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:c_{n+j-1}=w_{j}\,(1\leq j\leq l)\\}}{N}.$ (1.8)

The answer to this question is YES and is a consequence of the following
theorem.

######  Theorem 1.1.

For any integer $b\geq 2$, for any $f,g\in L^{\infty}(\lambda)$,444Note that
since Lebesgue measure $\lambda$ and Gauss measure $\mu_{G}$ are mutually
absolutely continuous, we have $L^{\infty}(\lambda)=L^{\infty}(\mu_{G})$.

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T_{b}^{n}x)\,g(T_{G}^{n}x)=\int
f\,d\lambda\cdot\int g\,d\mu_{G}$ (1.9)

for almost every $x\in[0,1]$.

Indeed, by Theorem 1.1, formula (1.9) is valid for $f=1_{C_{b}(S_{1})}$ and
$g=1_{C_{G}(S_{2})}$ for any choice of sets $C_{b}(S_{1})$ and $C_{G}(S_{2})$.
Since there are only countably many sets of the form $C_{b}(S_{1})$ and
$C_{G}(S_{2})$, this gives (1.6).

There are numerous examples of concrete constructions of base $b$ normal
numbers and CF-normal numbers. (See, for instance, [10], [11], [12], [39],
[4]). It would be of interest to have concrete examples of numbers $x\in[0,1)$
which are jointly $b$-normal (for some fixed $b$) and CF-normal.

Theorem 1.1 is a special case of a much more general result, which involves
piecewise monotone maps. To give a flavor of this more general result we will
formulate now its special case which involves not only Gauss transformation
$T_{G}$ and times $b$ transformations $T_{b}$, but also
$\beta$-transformations $T_{\beta}$. Recall that for $\beta>1$,
$\beta\notin\mathbb{N}$, $\beta$-transformation $T_{\beta}$ is given by
$T_{\beta}x=\beta x\bmod\,1$. These transformations were introduced by Rényi
([41]) who in particular showed that there exists a unique $T_{\beta}$-ergodic
probability measure $\mu_{\beta}$ which satisfies
$1-\frac{1}{\beta}\leq\frac{d\mu_{\beta}}{d\lambda}\leq\frac{1}{1-1/\beta}$.

######  Theorem 1.2 (Theorem 4.3).

Let $s,t\in\mathbb{N}$. Let $b_{1},\dots,b_{s}$ be distinct integers with
$b_{i}\geq 2$ for all $i=1,2,\dots,s$ and let $\beta_{1},\dots,\beta_{t}$ be
distinct non-integers with $\beta_{i}>1$ for all $i=1,2,\dots,t$.

  1. (1)

For any $f_{1},\dots,f_{s},g_{1},\dots,g_{t}\in L^{\infty}(\lambda)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{s}f_{i}({T_{b_{i}}^{n}x})\prod_{j=1}^{t}g_{j}({T_{\beta_{j}}^{n}x})=\prod_{i=1}^{s}\int
f_{i}\,d\lambda\,\prod_{j=1}^{t}\int g_{j}\,d\mu_{\beta_{j}}$ (1.10)

for almost every $x\in[0,1]$.

  2. (2)

Under the assumption $\log\beta_{j}\neq\frac{\pi^{2}}{6\log 2}$ for $1\leq
j\leq t$, for any $f_{0},f_{1},\dots,f_{s},g_{1},\dots,g_{t}\in
L^{\infty}(\lambda)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{0}(T_{G}^{n}x)\prod_{i=1}^{s}f_{i}({T_{b_{i}}^{n}x})\prod_{j=1}^{t}g_{j}({T_{\beta_{j}}^{n}x})=\int
f_{0}\,d\mu_{G}\,\prod_{i=1}^{s}\int f_{i}\,d\lambda\,\prod_{j=1}^{t}\int
g_{j}\,d\mu_{\beta_{j}}$ (1.11)

for almost every $x\in[0,1]$.

######  Remark 1.3.

The number $\frac{\pi^{2}}{6\log 2}$ is the entropy of $T_{G}$. Also $\log b$
and $\log\beta$ are the entropies of $T_{b}$ and $T_{\beta}$ respectively. So,
entropies of transformations in the above theorem are all distinct; this fact
plays an important role in the proof of Theorem 1.2. When
$\log\beta=\frac{\pi^{2}}{6\log 2}$, we do not know whether the limit of the
averages

$\frac{1}{N}\sum_{n=0}^{N-1}f(T_{G}^{n}x)\,g(T_{\beta}^{n}x),\,N\in\mathbb{N}$

exists almost everywhere or in $L^{2}$.

######  Remark 1.4.

Putting in formula (1.10) $g_{j}=1,(j=1,2,\dots,t)$ gives us the following
statement

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{s}f_{i}({b_{i}^{n}x})=\prod_{i=1}^{s}\int
f_{i}\,d\lambda\quad\text{for a.e. }x.$ (1.12)

This result (which deals with commuting transformations having the same
invariant measure) can be also derived from the work of Berend (see Theorem
5.1 in [5]). A related result due to Maxfield [32] states, in its ergodic
form, that for almost every $s$-tuple $(x_{1},\dots,x_{s})\in[0,1]^{s}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{s}f_{i}({b_{i}^{n}x_{i}})=\prod_{i=1}^{s}\int
f_{i}\,d\lambda.$ (1.13)

Maxfield’s result follows immediately - via the pointwise ergodic theorem -
from the ergodicity of the measure preserving transformation
$T_{b_{1}}\times\cdots\times T_{b_{s}}$ acting on $[0,1]^{s}$. On the other
hand, formula (1.10) (and even its special case (1.12)) is quite a bit less
trivial and manifests a subtle “diagonal phenomenon” which is also behind
Furstenberg’s approach to Szemerédi’s theorem (see [16]).

For a non-integer $\beta>1$, any number $x\in[0,1)$ can be written as

$x=\sum_{n=1}^{\infty}\frac{d_{n}}{\beta^{n}},\,\,d_{n}\in\\{0,1,\dots,[\beta]\\}.$
(1.14)

It is known that almost every $x$ in the interval
$I_{\beta}=[0,\frac{1}{\beta-1}]$ has a continuum of different representation
of the form (1.14) ([13] and [48]). We will be working with a special form of
the representation (1.14) that was introduced in [41] under the name
$\beta$-expansion of $x$ and is defined by the condition $d_{n}=[\beta
T_{\beta}^{n-1}(x)]$ for $n\geq 1$.

Following [23], we say that $x$ is $\beta$-normal if for any finite string
$S=(v_{1},\dots,v_{k})\in\\{0,1,\dots,[\beta]\\}^{k},k\in\mathbb{N}$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:d_{n}=v_{1},\dots,d_{n+k-1}=v_{k}\\}}{N}=\mu_{\beta}(C_{\beta}(S)),$ (1.15)

where $C_{\beta}(S)$ consists of numbers in $[0,1)$, whose first $k$ terms in
the $\beta$-expansion are $(v_{1},\dots,v_{k})$. As in the case of
$b$-expansions, the pointwise ergodic theorem implies that almost every
$x\in[0,1)$ is $\beta$-normal.

######  Remark 1.5.

If $b$ is an integer with $b\geq 2$, for any finite string $S$ in
$\\{0,1,\dots,b-1\\}^{k},k\in\mathbb{N}$, there is a number $x$ such that $S$
appears in the $b$-expansion of $x$, so $C_{b}(S)\neq\emptyset$. However, the
situation for $\beta$-expansions is somewhat different. For instance, if
$\beta=\frac{3}{2}$, then it is not hard to check that
$C_{\beta}(S)=\emptyset$ for $S=(1,1)$.

We will introduce now some definitions which will allow us to formulate a
corollary of Theorem 1.2 in terms of joint normality.

Let $s,t\in\mathbb{N}$. Let $b_{1},b_{2},\dots,b_{s}$ be distinct integers
with $b_{i}\geq 2$ for all $i=1,2,\dots,s$ and let
$\beta_{1},\beta_{2},\dots,\beta_{t}$ be distinct non-integers with
$\beta_{j}>1$ for all $j=1,2,\dots,t$. Consider the following representations
of $x\in[0,1)$:

  1. (1)

$b_{i}$-expansion ($1\leq i\leq s$),

$x=\sum_{n=1}^{\infty}\frac{a_{n}^{(i)}}{b_{i}^{n}},\quad
a_{n}^{(i)}\in\\{0,1,\dots,b_{i}-1\\},$

  2. (2)

$\beta_{j}$-expansion ($1\leq j\leq t$),

$x=\sum_{n=1}^{\infty}\frac{d_{n}^{(j)}}{\beta_{j}^{n}},\quad
d_{n}^{(j)}\in\\{0,1,\dots,[\beta_{j}]\\},$

  3. (3)

continued fraction expansion

$x=\cfrac{1}{c_{1}+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}+\cdots}}}=[c_{1},c_{2},c_{3},...],\quad
c_{n}\in\mathbb{N}.$

For $1\leq i\leq s$, let $V_{i}=(v_{1}^{(i)},\cdots,v_{k_{i}}^{(i)})$ be a
finite string in $\\{0,1,\dots,b_{i}-1\\}^{k_{i}}$, for $1\leq j\leq t$, let
$W_{j}=(w_{1}^{(j)},\cdots,w_{l_{j}}^{(j)})$ be a finite string in
$\\{0,1,\dots,[\beta_{j}]\\}^{l_{j}}$ and let $U=(u_{1},\dots,u_{m})$ be a
finite string in $\mathbb{N}^{m}$. For $x\in[0,1)$, let

$\begin{split}F_{N}^{1}(x,V_{1},\cdots,V_{s},W_{1},\cdots,W_{t})=\\#\\{1\leq
n\leq N:\,&a_{n+p-1}^{(i)}=v_{p}^{(i)},\,1\leq p\leq k_{i},1\leq i\leq s,\\\
&d_{n+q-1}^{(j)}=w_{q}^{(j)},\,1\leq q\leq l_{j},1\leq j\leq t\\}.\end{split}$

Similarly, for $x\in[0,1)$, let

$\begin{split}F_{N}^{2}(x,V_{1},\cdots,V_{s},W_{1},\cdots,W_{t},U)=\\#\\{1\leq
n\leq N:\,&a_{n+p-1}^{(i)}=v_{p}^{(i)},\,1\leq p\leq k_{i},1\leq i\leq s,\\\
&d_{n+q-1}^{(j)}=w_{q}^{(j)},\,1\leq q\leq l_{j},1\leq j\leq
t,c_{n+r-1}=u_{r},\,1\leq r\leq m\\}.\end{split}$

We will say that $x\in[0,1)$ is jointly
$(b_{1},\dots,b_{s},\beta_{1},\cdots,\beta_{t})$-normal if for any strings
$V_{1},\dots,V_{s},W_{1},\dots,W_{t}$,

$\lim_{N\rightarrow\infty}\frac{F_{N}^{1}(x,V_{1},\cdots,V_{s},W_{1},\cdots,W_{t})}{N}=\prod_{i=1}^{s}\lambda(C_{b_{i}}(V_{i}))\,\prod_{j=1}^{t}\mu_{\beta_{j}}(C_{\beta_{j}}(W_{j})),$

and is jointly $(b_{1},\dots,b_{s},\beta_{1},\cdots,\beta_{t},CF)$-normal if
for any strings $V_{1},\dots,V_{s},W_{1},\dots,W_{t},U$,

$\lim_{N\rightarrow\infty}\frac{F_{N}^{2}(x,V_{1},\cdots,V_{s},W_{1},\cdots,W_{t},U)}{N}=\prod_{i=1}^{s}\lambda(C_{b_{i}}(V_{i}))\,\prod_{j=1}^{t}\mu_{\beta_{j}}(C_{\beta_{j}}(W_{j}))\,\mu_{G}(C_{G}(U)),$

where $C_{b_{i}}(V_{i}),C_{\beta_{j}}(W_{j})$ and $C_{G}(U)$ are the so called
cylinder sets, which were defined above and which correspond, respectively, to
strings $V_{i}$ in $b_{i}$-expansion, $W_{j}$ in $\beta_{j}$-expansion, and
$U$ in continued fraction expansion.

We are now in a position to formulate a corollary of Theorem 1.2, which
pertains to joint normality with respect to $b$-expansions, $\beta$-expansions
and continued fraction expansion.

######  Corollary 1.6 (Corollary 4.4).

Let $s,t\in\mathbb{N}$. Let $b_{1},\dots,b_{s}$ be distinct integers with
$b_{i}\geq 2$ for all $i=1,2,\dots,s$ and let $\beta_{1},\dots,\beta_{t}$ be
distinct non-integers with $\beta_{j}>1$ for all $j=1,2,\dots,t$. Then almost
every $x\in[0,1)$ is jointly
$(b_{1},\dots,b_{s},\beta_{1},\dots,\beta_{t})$-normal and also, if
$\log\beta_{j}\neq\frac{\pi^{2}}{6\log 2}$ for any $j=1,2\dots,t$, almost
every $x\in[0,1)$ is jointly
$(b_{1},\dots,b_{s},\beta_{1},\dots,\beta_{t},CF)$-normal.

In this paper we will be mostly working with the class $\mathcal{T}$ which is
comprised of ergodic measure preserving systems $(X,\mathcal{B},\mu,T)$, where
$\mu$ is a probability measure on Borel $\sigma$-algebra $\mathcal{B}$ of
$X=[0,1]$ such that for some $c\geq 1$, $\frac{1}{c}\mu(A)\leq\lambda(A)\leq
c\mu(A)\,\text{for all }A\in\mathcal{B}$ and $T:[0,1]\rightarrow[0,1]$ is a
$\mu$-ergodic555Note that since distinct normalized ergodic $T$-invariant
measures are mutually singular, in light of our assumptions such $T$-invariant
measure $\mu$ is unique., piecewise monotone map, which is equipped with a
(finite or countable) partition $\mathcal{A}$ of $[0,1]$ into (open, closed,
or half open on either end) intervals with nonempty and disjoint interiors
such that for any interval $I\in\mathcal{A}$, the map $T|_{\text{int}(I)}$ is
continuous and strictly monotone. In addition, we will assume that the
following conditions are satisfied:

  1. (1)

$\mathcal{B}=\bigvee_{j=0}^{\infty}T^{-j}\sigma(\mathcal{A})\mod\lambda$,
where $\sigma(\mathcal{A})$ denotes the sub-$\sigma$-algebra generated by
$\mathcal{A}$. (i.e. partition $\mathcal{A}$ generates the $\sigma$-algebra
$\mathcal{B}$.)

  2. (2)

The entropy of the partition $\mathcal{A}$ is finite:
$H(\mathcal{A}):=-\sum_{I\in\mathcal{A}}\mu(I)\log\mu(I)<\infty$.

######  Remark 1.7.

Kolmogorov-Sinai entropy $h(T)$ is given by

$h(T)=\sup\\{h(T,\mathcal{C}):\mathcal{C}\text{ is a measurable partition with
}H(\mathcal{C})<\infty\\},$

where
$h(T,\mathcal{C})=\lim\limits_{n\rightarrow\infty}\frac{1}{n}H\left(\bigvee_{j=0}^{n-1}T^{-j}\mathcal{C}\right)$.
It is well-known (cf. [36] or [51]) that if $H(\mathcal{C})<\infty$, then
$h(T,\mathcal{C})\leq H(\mathcal{C})$ and if $\mathcal{A}$ generates
$\sigma$-algebra $\mathcal{B}$, then $h(T)=h(T,\mathcal{A})$. So it follows
that any system belonging to $\mathcal{T}$ has finite entropy.

For convenience, we will be denoting members of $\mathcal{T}$ by
$(T,\mu,\mathcal{A})$. Note that for any system which appears in Theorem 1.2,
there exists a natural partition $\mathcal{A}$ for which
$(T,\mu,\mathcal{A})\in\mathcal{T}$. Here is the list of these partitions.

  1. (1)

$T_{b}x=bx\,\bmod 1$ ($b\in\mathbb{N}$, $b\geq 2$):

$\mathcal{A}_{b}=\left\\{\left[0,\frac{1}{b}\right),\cdots,\left[\frac{b-1}{b},1\right)\right\\}.$
(1.16)

  2. (2)

$T_{\beta}x=\beta x\,\bmod\,1$ ($\beta>1$, $\beta\notin\mathbb{N}$):

$\mathcal{A}_{\beta}=\left\\{\left[0,\frac{1}{\beta}\right),\cdots,\left[\frac{[\beta]-1}{\beta},\frac{[\beta]}{\beta}\right),\left[\frac{[\beta]}{\beta},1\right)\right\\}.$
(1.17)

  3. (3)

$T_{G}x=\frac{1}{x}\,\bmod\,1$:

$\mathcal{A}_{G}=\left\\{\left(\frac{1}{n+1},\frac{1}{n}\right]:n\in\mathbb{N}\right\\}.$
(1.18)

We are going now to formulate a general result, which contains Theorem 1.2 as
a special case. Before doing so, we have to introduce two technical conditions
regarding the sequence of partitions
$\mathcal{A}^{n}:=\bigvee\limits_{0}^{n-1}T^{-j}\mathcal{A}$,
$n\in\mathbb{N}$.

The first condition deals with entropy equipartition. The classical Shanon-
McMillan-Breiman theorem (it will be discussed in detail in Subsection 3.1)
asserts that for a “typical” $A\in\mathcal{A}^{n}$, $\mu(A)$ is close to
$e^{-nh(T)}$, where $h(T)$ is the entropy of $T$. More precisely, for any
$\epsilon>0$, there is $N=N(\epsilon)$ such that if $n\geq N$, then

$\mu\left(\\{x\in[0,1):x\in A\text{ for some }A\in\mathcal{A}^{n}\text{ with
}e^{-n(h(T)+\epsilon)}\leq\mu(A)\leq e^{-n(h(T)-\epsilon)}\\}\right)\geq
1-\epsilon.$ (1.19)

For our goals, we will need to postulate somewhat sharper condition:

######  Definition 1.8.

$(T,\mu,\mathcal{A})$ has property E if for any $\epsilon>0$, there is a
positive real number $c_{0}=c_{0}(\epsilon)$ such that, for any
$n\in\mathbb{N}$,

$\mu\left(\\{x\in[0,1):x\in A\text{ for some }A\in\mathcal{A}^{n}\text{ with
}e^{-n(h(T)+\epsilon)}\leq\mu(A)\leq e^{-n(h(T)-\epsilon)}\\}\right)\geq
1-\frac{c_{0}}{n}.$ (1.20)

We observe that partitions $\mathcal{A}_{b}$, $\mathcal{A}_{\beta}$,
$\mathcal{A}_{G}$ (see (1.16), (1.17), and (1.18)) satisfy condition (1.20).
Indeed, since for any integer $b\geq 2$, $h(T_{b})=\log b$, condition (1.20)
is trivially satisfied for times $b$ maps $T_{b}$. As for the Gauss
transformation $T_{G}$ and $\beta$-transformations $T_{\beta}$, the details
are provided in Subsection 3.3.

Property M, which is introduced in the following definition, can be
interpreted as a variant of a strong mixing condition, which intertwines
$T$-invariant measure $\mu$ with the iterations $\lambda\circ T^{-n}$ of
Lebesgue measure $\lambda$ as $n\rightarrow\infty$.

######  Definition 1.9.

$(T,\mu,\mathcal{A})$ has property M if there is a sequence
$(b_{n})_{n=1}^{\infty}$ of non-negative real numbers with
$\sum\limits_{n=1}^{\infty}b_{n}<\infty$ such that for any natural number $l$,
for any non-negative integer $n$, if $A\in\sigma(\mathcal{A}^{l})$ and
$B\in\mathcal{B}$,

$|\lambda(A\cap T^{-(n+l)}B)-\lambda(A)\mu(B)|\leq b_{n}.$ (1.21)

We will show in Subsection 3.3 below that the systems appearing in Theorem 1.2
have property M (in addition to having property E).

We are now in a position to formulate the main result of this paper. We will
call the phenomenon encompassed in the following theorem pointwise joint
ergodicity. It will be treated in detail in Section 2.

######  Theorem 1.10 (Theorem 4.1).

Let $T_{i}:[0,1]\rightarrow[0,1],\,1\leq i\leq k$, be measurable
transformations such that for each $i=1,2\dots,k$, there is a partition
$\mathcal{A}_{i}$ and a $T_{i}$-invariant ergodic measures $\mu_{i}$ with
$(T_{i},\mu_{i},\mathcal{A}_{i})\in\mathcal{T}$. Suppose that for each
$i=1,2\dots,k$, $(T_{i},\mu_{i},\mathcal{A}_{i})$ satisfies properties E and
M. If entropy values $h(T_{1}),\cdots,h(T_{k})$ are distinct, then for any
$f_{1},\dots,f_{k}\in L^{\infty}(\lambda)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}({T_{1}^{n}x})f_{2}({T_{2}^{n}x})\cdots
f_{k}({T_{k}^{n}x})=\prod_{i=1}^{k}\int f_{i}\,d\mu_{i}$ (1.22)

for almost every $x\in[0,1]$.

######  Remark 1.11.

While the motivation for establishing Theorem 1.10 comes from the question
related to joint normality, this theorem can also be viewed as a refinement of
some results obtained in [7], which pertain to $L^{2}$-joint ergodicity. It is
worth noting that the $L^{2}$-results obtained in [7] require less restrictive
conditions on $T_{i}$ and, moreover, involve uniform Cesàro averages, i.e.,
have the form

$\lim\limits_{N-M\rightarrow\infty}\frac{1}{N-M}\sum_{n=M+1}^{N}f_{1}({T_{1}^{n}x})f_{2}({T_{2}^{n}x})\cdots
f_{k}({T_{k}^{n}x})=\prod_{i=1}^{k}\int f_{i}\,d\mu_{i}.$

As it was already observed in the work of Rényi ([41]) and Parry ([35]),
ergodic theory provides a natural unifying framework for dealing with various
representations of numbers. As we will see in Section 6, ergodic approach
allows also to prove a rather general theorem which deals with various
equivalent forms of normality and has as a quite special corollary the
following theorem which summarizes various classical results pertaining to
base $b$ normality.

######  Theorem 1.12.

Let $b\geq 2$ be an integer. Let $x\in[0,1)$. The following statements are
equivalent:

  1. (i)

$x$ is normal in base $b$.

  2. (ii)

$(b^{n}x)_{n\in\mathbb{N}}$ is uniformly distributed $\bmod\,1$.

  3. (iii)

For any fixed $m\geq 2$, $x$ is normal in base $b^{m}$.

  4. (iv)

$b^{k}x$ is simply normal in base $b^{m}$ for all $k\geq 0$ and $m\geq 1$.

  5. (v)

$x$ is simply normal in all of the bases $b,b^{2},b^{3},\cdots.$

  6. (vi)

There exists a sequence $(m_{l})_{l\in\mathbb{N}}$ with $m_{1}<m_{2}<\cdots$
such that $x$ is simply normal in all of the bases $b^{m_{l}}$.

######  Remark 1.13.

Borel’s original characterization of base $b$ normality is (iv) (see [9]). The
equivalence of (i) and (iv) was shown by Niven and Zuckerman ([33]). The
characterizations (v) and (vi) were established, respectively, by Pillai
([38]) and Long ([30]). For more details, see Chapter 1, Section 8 in [27] and
Appendix in [6].

A general ergodic form of Theorem 1.12 involves systems belonging to a rather
large family of measure preserving systems which we will denote by
$\tilde{\mathcal{T}}$ and which is defined by relaxing some of the conditions
appearing in the definition of class $\mathcal{T}$. Class
$\tilde{\mathcal{T}}$ is comprised of ergodic measure preserving systems
$(X,\mathcal{B},\mu,T)$, where $X=[0,1]$, $\mathcal{B}$ is the Borel
$\sigma$-algebra, $\mu$ is a probability measure, which is equivalent to
Lebesgue measure $\lambda$, and $T:[0,1]\rightarrow[0,1]$ is a
$\mu$-ergodic777Note that in light of our assumptions such $T$-invariant
measure $\mu$ is unique. map, which is equipped with a (finite or countable)
partition $\mathcal{A}$ of $[0,1]$ into (open, closed, or half open on either
end) intervals with nonempty and disjoint interiors such that for any interval
$I\in\mathcal{A}$, the map $T|_{\text{int}(I)}$ is continuous and strictly
monotone. In addition, we will assume that
$\mathcal{B}=\bigvee_{j=0}^{\infty}T^{-j}\sigma(\mathcal{A})\mod\lambda$,
where $\sigma(\mathcal{A})$ denotes the sub-$\sigma$-algebra generated by
$\mathcal{A}$ (i.e. partition $\mathcal{A}$ generates the $\sigma$-algebra
$\mathcal{B}$.)

As before, we will find it convenient to denote members of
$\tilde{\mathcal{T}}$ by $(T,\mu,\mathcal{A})$.

To formulate a general ergodic form of Theorem 1.12 we utilize the notion of
F-representation which was introduced by E. A. Robinson, Jr. in [42].

Let $(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$, where
$\mathcal{A}=\\{I_{j}:j\in D\\}$, $D$ being a finite or countably infinite
index set. For $x\in X=[0,1)$, introduce a symbolic representation
$(r_{n}(x))_{n=1}^{\infty}$ by putting $r_{n}=r_{n}(x)=j$ if $T^{n-1}x\in
I_{j}$. In [42], this representation is called F-representations and it is
shown that if $T$ is topologically transitive (meaning that for some $x\in X$
the forward orbit $\\{T^{n}x:n\geq 0\\}$ is dense in X) then the
F-representation $(r_{n}(x))_{n=1}^{\infty}$ is unique for almost every $x$.
Note that if $(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$, then $T$ is
transitive, since $T$ is ergodic with respect to $\mu$ (and $\mu$ is
equivalent to $\lambda$).

Write $\mathbf{T}=(T,\mu,\mathcal{A})$. For a string $S=(v_{1},\dots,v_{k})\in
D^{k}$, define the cylinder set
$C_{\mathbf{T}}(S)(=[v_{1},\dots,v_{k}]_{\mathbf{T}}):=\\{y\in[0,1):r_{1}(y)=v_{1},\dots,r_{n}(y)=v_{n}\\}$.
Note that $\\{C_{\mathbf{T}}(S):S\in
D^{k}\\}=\mathcal{A}^{k}\,(\bmod\,\lambda)$ for any $k\in\mathbb{N}$. We say
that $x$ is

  * •

simply ${\mathbf{T}}$-normal if for any $v$ in $D$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{n}=v\\}}{N}=\mu([v]_{\mathbf{T}}).$

  * •

${\mathbf{T}}$-normal if for any $k\in\mathbb{N}$ and any finite string
$S=(v_{1},\dots,v_{k})$ in $D^{k}$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{n}=v_{1},\dots,r_{n+k-1}=v_{k}\\}}{N}=\mu(C_{\mathbf{T}}(S)).$

For $m\in\mathbb{N}$, let
${\mathbf{T}_{m}}=(T^{m},\mu,\mathcal{A}^{m})\in\mathcal{T}$. Note that $x$ is
simply ${\mathbf{T}_{m}}$-normal if $S=(v_{1},\dots,v_{m})$ in $D^{m}$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{m(n-1)+1}=v_{1},\dots,r_{mn}=v_{m}\\}}{N}=\mu(C_{\mathbf{T}}(S)).$

Here is now a general ergodic form of Theorem 1.12.

######  Theorem 1.14 (Theorem 6.1).

Let ${\mathbf{T}}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$ and assume that
$T$ is totally ergodic.888 A measure preserving transformation $T$ is called
totally ergodic if $T^{k}$ is ergodic for all $k\in\mathbb{N}$. Let
$x\in[0,1]$. The following statements are equivalent.

  1. (i)

$x$ is ${\mathbf{T}}$-normal.

  2. (ii)

$(T^{n}x)$ is equidistributed in $[0,1]$ with respect to $\mu$, that is, for
any continuous function $f:[0,1]\rightarrow\mathbb{R}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)=\int f\,d\mu.$

  3. (iii)

For any $m\geq 2$, $x$ is ${\mathbf{T}}_{m}$-normal.

  4. (iv)

$T^{k}x$ is simply ${\mathbf{T}}_{m}$-normal for all $k\geq 0$ and $m\geq 1$.

  5. (v)

$x$ is simply ${\mathbf{T}}_{m}$-normal for all $m\geq 1$.

  6. (vi)

There exists a sequence $(m_{l})_{l\in\mathbb{N}}$ with $m_{1}<m_{2}<\cdots$
such that $x$ is simply ${\mathbf{T}}_{m_{l}}$-normal for any
$l\in\mathbb{N}$.

One also has an analogue of Theorem 1.14 for the notion of joint normality.
Before formulating this result, we have to introduce some definitions.

Let
${\mathbf{T}}^{(i)}=(T_{i},\mu_{i},\mathcal{A}_{i})\in\tilde{\mathcal{T}},\,i=1,2\dots,k$,
where $\mathcal{A}_{i}=\\{I_{j}^{(i)}:j\in D_{i}\\}$ and $D_{i}$ is an index
set, which may be finite or countably infinite. Let
$(r_{n}^{(i)}(x))_{n=1}^{\infty}$ be F-representation of $x$ corresponding to
$\mathbf{T}^{(i)}$, that is, $r_{n}^{(i)}=r_{n}^{(i)}(x)=j$ if
$T_{i}^{n-1}x\in I_{j}^{(i)}$. We say that $x$ is

  * •

jointly simply $({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normal if for
any $v_{1},\dots,v_{k}$, where $v_{i}\in D_{i}$ $(1\leq i\leq k)$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{n}^{(1)}=v_{1},\dots,r_{n}^{(k)}=v_{k}\\}}{N}=\prod_{i=1}^{k}\mu_{i}([v_{i}]_{{\mathbf{T}}_{i}}).$

  * •

jointly $({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normal if for any
finite strings $S_{i}=(v_{1}^{(i)},\dots,v_{m_{i}}^{(i)})$ $(1\leq i\leq k)$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{n}^{(i)}=v_{1}^{(i)},\dots,r_{n+m_{i}-1}^{(i)}=v_{m_{i}}^{(i)}\text{ for
}1\leq i\leq k\\}}{N}=\prod_{i=1}^{k}\mu_{i}(C_{{\mathbf{T}}_{i}}(S_{i})).$

As before, we will use the notation
${\mathbf{T}}_{m}^{(i)}=(T_{i}^{m},\mu_{i},\mathcal{A}_{i}^{m})$. Note that
$x$ is jointly simply
$({\mathbf{T}}_{m}^{(1)},\dots,{\mathbf{T}}_{m}^{(k)})$-normal if for any
finite strings $S_{i}=(v_{1}^{(i)},\dots,v_{m}^{(i)})\in D_{i}^{m}$ $(1\leq
i\leq k)$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{m(n-1)+1}^{(i)}=v_{1}^{(i)},\dots,r_{mn}^{(i)}=v_{m}^{(i)}\text{ for
}1\leq i\leq k\\}}{N}=\prod_{i=1}^{k}\mu_{i}(C_{{\mathbf{T}}_{i}}(S_{i})).$

######  Theorem 1.15 (Theorem 6.6).

Let
${\mathbf{T}}^{(i)}=(T_{i},\mu_{i},\mathcal{A}_{i})\in\tilde{\mathcal{T}},\,i=1,2\dots,k$
and assume that $T_{1}\times\cdots\times T_{k}$ is totally ergodic with
respect to $\mu_{1}\times\cdots\times\mu_{k}$. Let $x\in[0,1]$. The following
statements are equivalent.

  1. (i)

$x$ is jointly $({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normal.

  2. (ii)

$(T_{1}^{n}x,\dots,T_{k}^{n}x)$ is equidistributed in $[0,1]^{k}$ with respect
to $\mu_{1}\times\cdots\times\mu_{k}$: for any continuous function $f$ on
$[0,1]^{k}\rightarrow\mathbb{R}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T_{1}^{n}x,\dots,T_{k}^{n}x)=\int
f\,d(\mu_{1}\times\cdots\times\mu_{k}).$

  3. (iii)

For any $m\geq 2$, $x$ is jointly
$({\mathbf{T}}_{m}^{(1)},\dots,{\mathbf{T}}_{m}^{(k)})$-normal.

  4. (iv)

$x$ is jointly simply
$({\mathbf{T}}_{m}^{(1)},\dots,{\mathbf{T}}_{m}^{(k)})$-normal for all
$m\in\mathbb{N}$.

  5. (v)

There exists a sequence $(m_{l})_{l\in\mathbb{N}}$ with $m_{1}<m_{2}<\cdots$
such that $x$ is jointly simply
$({\mathbf{T}}_{m_{l}}^{(1)},\dots,{\mathbf{T}}_{m_{l}}^{(k)})$-normal for any
$l\in\mathbb{N}$.

######  Remark 1.16.

It is worth noting an important difference between Theorems 1.14 and 1.15 when
$k>1$. Namely, while in light of the pointwise ergodic theorem, for any
$\mathbf{T}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$, almost every $x$ in
$X$ is $\mathbf{T}$-normal, the phenomenon of joint
$({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normality is much more
delicate. For example, the transformation $T_{1}\times\cdots\times T_{k}$ can
be totally ergodic even if for some $i\neq j$, $T_{i}=T_{j}$. However, in this
case there are no jointly
$({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normal points. On the other
hand, Theorem 1.10, which deals with the phenomenon of pointwise joint
ergodicity, implies that if each
$\mathbf{T}^{(i)}=(T_{i},\mu_{i},\mathcal{A}_{i})\in\mathcal{T},(1\leq i\leq
k),$ has properties E and M and $h(T_{1}),\dots,h(T_{k})$ are distinct, then
almost every $x$ is jointly
$({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normal. Similar results hold
for mutually disjoint systems (see Section 5 for the details).

The structure of this paper is as follows. In Section 2, we discuss the notion
of pointwise joint ergodicity and establish a characterization of pointwise
joint ergodicity in terms of equidistribution. The results obtained in this
section will be instrumental for the proof of the main theorem of this paper
(Theorem 1.10 above). In Section 3, we discuss properties E and M and describe
numerous examples of systems which possess properties E and M and belong to
the family $\mathcal{T}$. These examples are used in the subsequent sections
to illustrate our results pertaining to joint normality. Section 4 is devoted
to the proof of Theorem 1.10. In Section 5, we establish additional results on
pointwise joint ergodicity which utilize the notion of disjointness. In
Section 6, we deal with various equivalent forms of normality and joint
normality, which were briefly discussed in the Introduction. In particular, we
provide the proof of Theorem 1.14. We also discuss a variant of Theorem 1.14
for joint normality.

## Acknowledgement

Younghwan Son is supported by the NRF of Korea (NRF-2020R1A2C1A01005446).

##  2\. pointwise joint ergodicity

In this section, we introduce the notion of pointwise joint ergodicity and
present a useful characterization of pointwise joint ergodicity of
transformations $T_{1},\dots,T_{k}$ acting on measurable space
$(X,\mathcal{B})$ in terms of the equidistribution in $X^{k}$ of the orbits
$(T_{1}^{n}x,\dots,T_{k}^{n}x)$, $n=0,1,,2\dots.$

######  Definition 2.1.

Let $T_{1},\dots,T_{k}:X\rightarrow X$ be measurable transformations on a
measurable space $(X,\mathcal{B})$. Assume that for each
$i\in\\{1,2,\dots,k\\}$ there is a $T_{i}$-invariant probability measure
$\mu_{i}$ on $\mathcal{B}$ so that for any $i,j\in\\{1,2,\dots,k\\}$, the
measures $\mu_{i}$ and $\mu_{j}$ are equivalent meaning that they are mutually
absolutely continuous. We say that $T_{1},\dots,T_{k}$ are pointwise jointly
ergodic with respect to $(\mu_{1},\dots,\mu_{k})$ if for any
$f_{1},\dots,f_{k}\in L^{\infty}$,

$\lim\limits_{N\rightarrow\infty}\frac{1}{N}\sum\limits_{n=0}^{N-1}f_{1}(T_{1}^{n}x)f_{2}(T_{2}^{n}x)\cdots
f_{k}(T_{k}^{n}x)=\prod_{i=1}^{k}\int f_{i}\,d\mu_{i}\quad\text{a.e. }x.$

If measures $\mu_{1},\dots,\mu_{k}$ are understood, we just say
$T_{1},\dots,T_{k}$ are pointwise jointly ergodic.

The following useful characterization of pointwise joint ergodicity is
motivated by Lemma 2.1 in [5]. It will be utilized in the proof of our main
theorem (Theorem 1.10) in Section 4.

######  Theorem 2.2.

Let $X$ be a compact metric space and let $\mathcal{B}$ be the Borel
$\sigma$-algebra of $X$. Suppose that $T_{1},\dots,T_{k}:X\rightarrow X$ are
measurable transformations on $(X,\mathcal{B})$ such that for each
$i\in\\{1,2,\dots,k\\}$ there is a $T_{i}$-invariant probability measure
$\mu_{i}$ and for any $i,j\in\\{1,2,\dots,k\\}$, the measures $\mu_{i}$ and
$\mu_{j}$ are mutually absolutely continuous. Then the following conditions
are equivalent.

  1. (i)

$T_{1},\dots,T_{k}$ are pointwise jointly ergodic with respect to
$(\mu_{1},\dots,\mu_{k})$.

  2. (ii)

For a.e. $x\in X$, the sequence of $k$-tuples
$(T_{1}^{n}x,\dots,T_{k}^{n}x),\,n=0,1,2,\dots,$ is equidistributed in $X^{k}$
with respect to $\mu_{1}\times\cdots\times\mu_{k}$:

$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{T_{1}^{n}x}\times\cdots\times\delta_{T_{k}^{n}x}\rightarrow\mu_{1}\times\cdots\times\mu_{k}\text{
in weak* topology}.$

###### Proof.

$(i)\Rightarrow(ii)$: Let $\mu=\mu_{1}\times\cdots\times\mu_{k}$. Denote
$C(Y)$ by the space of continuous functions on a compact space $Y$. We will
use the classical fact that the functions of the form
$f_{1}\otimes\cdots\otimes
f_{k}(x_{1},\dots,x_{k})=\prod_{i=1}^{k}f_{i}(x_{i})$, where $f_{i}\in C(X)$,
are linearly dense in the space $C(X^{k})$. Moreover, there is a countable set
of functions $g_{i}\in C(X)$ such that their tensor product is linearly dense
in $C(X^{k})$. Thus one can find a countable dense subset
$\\{F_{m}:m\in\mathbb{N}\\}$ of $C(X^{k})$, where each $F_{m}$ is a finite sum
of functions of the form $f_{1}\otimes\cdots\otimes f_{k}$.

Condition (i) implies that for each $m\in\mathbb{N}$, there exists a full
measure set $A_{m}$ such that if $x\in A_{m}$, then

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}F_{m}(T_{1}^{n}x,\cdots,T_{k}^{n}x)=\int
F_{m}\,d\mu.$ (2.1)

Let $A=\bigcap\limits_{m=1}^{\infty}A_{m}$. It suffices to show that if $F$ is
a continuous function on $X^{k}$, then for any $x\in A$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}F(T_{1}^{n}x,\cdots,T_{k}^{n}x)=\int
F\,d\mu.$

Let $\epsilon>0$ and let $F_{m}$ satisfy $\|F_{m}-F\|_{\sup}<\epsilon$. Then,
clearly,

$\int|F_{m}-F|\,d\mu<\epsilon$

and

$\frac{1}{N}\sum_{n=0}^{N-1}\left|F(T_{1}^{n}x,\cdots,T_{k}^{n}x)-F_{m}(T_{1}^{n}x,\cdots,T_{k}^{n}x)\right|<\epsilon.$

So we have

$\displaystyle\left|\frac{1}{N}\sum_{n=0}^{N-1}F(T_{1}^{n}x,\cdots,T_{k}^{n}x)-\int
F\,d\mu\right|$
$\displaystyle\leq\frac{1}{N}\sum_{n=0}^{N-1}\left|F(T_{1}^{n}x,\cdots,T_{k}^{n}x)-F_{m}(T_{1}^{n}x,\cdots,T_{k}^{n}x)\right|$
$\displaystyle\quad+\left|\frac{1}{N}\sum_{n=0}^{N-1}F_{m}(T_{1}^{n}x,\cdots,T_{k}^{n}x)-\int
F_{m}\,d\mu\right|+\int|F_{m}-F|\,d\mu$
$\displaystyle\leq\left|\frac{1}{N}\sum_{n=0}^{N-1}F_{m}(T_{1}^{n}x,\cdots,T_{k}^{n}x)-\int
F_{m}\,d\mu\right|+2\epsilon.$

Letting $N\rightarrow\infty$ (and using (2.1)) we get

$\limsup_{N\rightarrow\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}F(T_{1}^{n}x,\cdots,T_{k}^{n}x)-\int
F\,d\mu\right|\leq 2\epsilon.$

Since $\epsilon$ is arbitrary, the desired result follows.

$(ii)\Rightarrow(i)$: Note first that condition (ii) implies that each $T_{i}$
is $\mu_{i}$-ergodic. Indeed, it follows from condition (ii) that there exists
a set $X_{0}$ of full measure such that for any $x\in X_{0}$, for any $f\in
C(X)$, and for any $i=1,2\dots,k$,

$\frac{1}{N}\sum_{n=1}^{N}f(T_{i}^{n}x)\rightarrow\int f\,d\mu_{i}.$

So, by dominated convergence theorem,

$\frac{1}{N}\sum_{n=1}^{N}f(T_{i}^{n}x)\rightarrow\int f\,d\mu_{i}\text{ in
}L^{2}(\mu_{i}).$ (2.2)

It follows by the standard approximation argument that (2.2) holds for any
$f\in L^{2}(\mu_{i})$, which implies that each $T_{i}$ is $\mu_{i}$-ergodic.

We need to prove that for any bounded measurable functions
$f_{1},\dots,f_{k}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)=\prod_{i=1}^{k}\int
f_{i}\,d\mu_{i}\quad\text{for a.e. }x.$

Without loss of generality, we assume that $|f_{i}(x)|\leq 1$ for all $x\in X$
and for $i=1,2,\dots,k$. For any given $m\in\mathbb{N}$, there are continuous
functions $g_{1}^{(m)},\dots,g_{k}^{(m)}$ such that $|g_{i}^{(m)}(x)|\leq 1$
for all $x\in X$ and $\int|f_{i}-g_{i}^{(m)}|d\mu_{i}<\frac{1}{m}$.

By condition (ii), there is a set of full measure $A_{m}^{1}$ such that if
$x\in A_{m}^{1}$, then

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)=\prod_{i=1}^{k}\int
g_{i}^{(m)}\,d\mu_{i},$ (2.3)

and by pointwise ergodic theorem, there is a set of full measure $A_{m}^{2}$
such that if $x\in A_{m}^{2}$, then for any $i=1,2,\dots,k$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}|f_{i}(T_{i}^{n}x)-g_{i}^{(m)}(T_{i}^{n}x)|=\int|f_{i}-g_{i}^{(m)}|\,d\mu_{i}.$
(2.4)

Let $A_{m}=A_{m}^{1}\cap A_{m}^{2}$. We will show that if $x\in A_{m}$, then

$\limsup_{N\rightarrow\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)-\prod_{i=1}^{k}\int
f_{i}\,d\mu_{i}\right|\leq\frac{2k}{m}.$ (2.5)

By triangle inequality,

$\displaystyle\left|\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)-\prod_{i=1}^{k}\int
f_{i}\,d\mu_{i}\right|$
$\displaystyle\leq\frac{1}{N}\sum_{n=0}^{N-1}\left|\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)-\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)\right|$
(2.6)
$\displaystyle+\left|\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)-\prod_{i=1}^{k}\int
g_{i}^{(m)}\,d\mu_{i}\right|$ $\displaystyle+\left|\prod_{i=1}^{k}\int
g_{i}^{(m)}\,d\mu_{i}-\prod_{i=1}^{k}\int f_{i}\,d\mu_{i}\right|$

Let us estimate each of the three terms on the right hand side of (2.6). To
estimate the first term on the right hand side of (2.6), we use the following
identity

$\prod_{i=1}^{k}a_{i}-\prod_{i=1}^{k}b_{i}=(a_{1}-b_{1})b_{2}\cdots
b_{k}+a_{1}(a_{2}-b_{2})b_{3}\cdots b_{k}+\cdots+a_{1}\cdots
a_{k-1}(a_{k}-b_{k}).$ (2.7)

We have

$\left|\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)-\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)\right|\leq\sum_{i=1}^{k}|f_{i}(T_{i}^{n}x)-g_{i}^{(m)}(T_{i}^{n}x)|,$

so

$\frac{1}{N}\sum_{n=0}^{N-1}\left|\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)-\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)\right|\leq\sum_{i=1}^{k}\frac{1}{N}\sum_{n=0}^{N-1}|f_{i}(T_{i}^{n}x)-g_{i}^{(m)}(T_{i}^{n}x)|.$

Thus, if $x\in A_{m}$, by (2.4),

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\left|\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)-\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)\right|\leq\sum_{i=1}^{k}\int|f_{i}-g_{i}^{(m)}|\,d\mu_{i}\leq\frac{k}{m}.$

As for the second term on the right hand side of (2.6), we have by (2.3) that
if $x\in A_{m}$,

$\lim_{N\rightarrow\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}g_{i}^{(m)}(T_{i}^{n}x)-\prod_{i=1}^{k}\int
g_{i}^{(m)}\,d\mu_{i}\right|=0.$

Finally applying (2.7) to the last term on the right hand side of (2.6), we
have that

$\left|\prod_{i=1}^{k}\int g_{i}^{(m)}\,d\mu_{i}-\prod_{i=1}^{k}\int
f_{i}\,d\mu_{i}\right|\leq\sum_{i=1}^{k}\int|g_{i}^{(m)}-f_{i}|\,d\mu_{i}\leq\frac{k}{m}.$

Putting all the things together, we get (2.5).

Let $B=\bigcap_{m=1}^{\infty}A_{m}$. Clearly $B$ is of full measure (with
respect to each $\mu_{i}$). It follows from (2.5) that for any $x\in B$, we
have

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)=\prod_{i=1}^{k}\int
f_{i}\,d\mu_{i},$

which concludes the proof. ∎

##  3\. Number-theoretic transformations possessing properties E and M

In this section we discuss in detail properties E and M, which appear in the
formulation our main theorem (Theorem 4.1 in Section 4). We also describe
numerous examples of systems which possess properties E and M and belong to
the family $\mathcal{T}$, which was introduced in Section 1. These examples
will be used in the subsequent sections to illustrate our results pertaining
to pointwise joint ergodicity and joint normality.

###  3.1. Entropy and property E

In this short section we formulate, for reader’s convenience, the classical
Shannon-McMillan-Breiman theorem as well as a corollary of it which serve as a
motivation for introducing property E.

######  Theorem 3.1 (Shannon-McMillan-Breiman, see for example Section 6.2 in
[36]).

Let $(X,\mathcal{B},\mu,T)$ be a probability measure preserving system and let
$h(T)$ denote the Kolmogorov-Sinai entropy of $T$. Let $\mathcal{A}$ be a
countable measurable partition of $X$ with
$-\sum_{A\in\mathcal{A}}\mu(A)\log\mu(A)<\infty$ and
$\bigvee_{n=0}^{\infty}T^{-n}\sigma(\mathcal{A})=\mathcal{B}\mod\mu$. For
$x\in X$, let $\mathcal{A}(x)$ be the element of $\mathcal{A}$ which contains
$x$. Let $\mathcal{A}^{n}:=\bigvee\limits_{0}^{n-1}T^{-j}\mathcal{A}$. Then

$-\frac{1}{n}\log\mu(\mathcal{A}^{n}(x))\rightarrow h(T)\,\,\text{a.e. and in
}L^{1}.$

The following corollary of Shannon-McMillan-Breiman theorem, which is often
referred to as the entropy equipartition property, states roughly that for
most sets $A\in\mathcal{A}^{n}$, $\mu(A)$ is approximately $e^{-nh(T)}$.

######  Corollary 3.2.

Given $\epsilon>0$, there is $n_{0}$ such that for $n\geq n_{0}$ the elements
of $\mathcal{A}^{n}$ can be divided into two classes
$\mathfrak{g}^{n}=\mathfrak{g}^{n}(\epsilon)$ and
$\mathfrak{b}^{n}=\mathfrak{b}^{n}(\epsilon)$ (“good” and “bad” atoms) such
that

  1. (1)

$\mu\left(\bigcup_{B\in\mathfrak{b}^{n}}B\right)<\epsilon$

  2. (2)

For $A\in\mathfrak{g}^{n}$,

$e^{-n(h(T)+\epsilon)}<\mu(A)<e^{-n(h(T)-\epsilon)}.$ (3.1)

Thus we have that for $n\geq n_{0}$

$\mu\left(\\{x\in[0,1):x\in A\text{ for some }A\in\mathcal{A}^{n}\text{ with
}e^{-n(h(T)+\epsilon)}\leq\mu(A)\leq e^{-n(h(T)-\epsilon)}\\}\right)\geq
1-\epsilon.$ (3.2)

Note that, given $(T,\mu,\mathcal{A})\in\mathcal{T}$ and $n\in\mathbb{N}$,
$\mathcal{A}^{n}=\bigvee\limits_{0}^{n-1}T^{-j}\mathcal{A}$ is also a finite
or countable partition of $[0,1]$ into intervals; we will call these intervals
cylinders of rank $n$. Since $\frac{1}{c}\mu\leq\lambda\leq c\mu$ for some
$c\geq 1$, Corollary 3.2 says that if a cylinder $A$ of rank $n$ belongs to
the class $\mathfrak{g}^{n}$ of good atoms, then

$\frac{1}{c}e^{-n(h(T)+\epsilon)}\leq\lambda(A)\leq ce^{-n(h(T)-\epsilon)}.$

The property E which we will recall here for the convenience of the reader can
be viewed as a formalization of the requirement that most cylinders of rank
$n$ belong to the class $\mathfrak{g}^{n}$ of good atoms.

######  Definition 3.3.

$(T,\mu,\mathcal{A})\in\mathcal{T}$ has property E if for any $\epsilon>0$,
there is a positive real number $c_{0}=c_{0}(\epsilon)$ such that, for any
$n\in\mathbb{N}$,

$\mu\left(\\{x\in[0,1):x\in A\text{ for some }A\in\mathcal{A}^{n}\text{ with
}e^{-n(h(T)+\epsilon)}\leq\mu(A)\leq e^{-n(h(T)-\epsilon)}\\}\right)\geq
1-\frac{c_{0}}{n}.$ (3.3)

###  3.2. Property M and RU maps

In this subsection, we introduce the notion of a RU map and show that any
weakly mixing RU map has property M. We also show that if
$(T,\mu,\mathcal{A})\in\mathcal{T}$ has property M, then $T$ is exact.

We start with recalling the definition of property M.

######  Definition 3.4.

$(T,\mu,\mathcal{A})\in\mathcal{T}$ has property M if there is a sequence
$(b_{n})_{n=1}^{\infty}$ of non-negative real numbers with
$\sum\limits_{n=1}^{\infty}b_{n}<\infty$ such that, for any natural number $l$
and for any non-negative integer $n$, if $A\in\sigma(\mathcal{A}^{l})$ and
$B\in\mathcal{B}$, then

$|\lambda(A\cap T^{-(n+l)}B)-\lambda(A)\mu(B)|\leq b_{n}.$ (3.4)

Define variation of $f$ on $I=[0,1]$ by

$\text{Var}(f):=\sup\sum_{i}|f(x_{i})-f(x_{i-1})|,$

where the supremum ranges over all $0=x_{0}<x_{1}<\cdots<x_{n}=1$.

For every $f\in L^{1}(\lambda)$, set

$\|f\|_{BV}:=\|f\|_{\infty}+V(f),\,\text{where}\,V(f)=\inf\\{\text{Var}(f^{*}):f^{*}=f\,\,\,\text{a.e.
with respect to }\lambda\\}.$

Let $BV=:\\{f\in L^{1}(\lambda):\|f\|_{BV}<\infty\\}$. The space $BV$ is a
Banach space with the norm $\|\cdot\|_{BV}$.

######  Definition 3.5 (cf. [1], p.19 and [2], p.3).

A map $T:[0,1]\rightarrow[0,1]$ is said to be a RU map if there exists a
partition $\mathcal{A}$ of $[0,1]$ into (open, closed, or half open on either
end) intervals with nonempty and disjoint interiors such that

  1. (1)

$T$ is uniformly expanding:

$\inf\\{|T^{\prime}(x)|:x\in\text{int}(A),A\in\mathcal{A}\\}>1$

  2. (2)

$T$ satisfies Rychlik’s condition:

$\sum_{A\in\mathcal{A}}\|1_{TA}v_{A}^{\prime}\|_{BV}<\infty,$

where $v_{A}$, $A\in\mathcal{A}$, denotes the inverse of $T_{|A}:A\rightarrow
TA$.

######  Remark 3.6.

Notice that RU maps are presumed to be equipped with a partition $\mathcal{A}$
of $[0,1]$ (so that the properties (1) and (2) in Definition 3.5 are
satisfied). We will find it convenient to denote RU maps belonging to the
class $\mathcal{T}$ by triples $(T,\mu,\mathcal{A})$ with the understanding
that the partition $\mathcal{A}$ in the triple $(T,\mu,\mathcal{A})$
corresponds to the partition which is stipulated in the Definition 3.5.

The following result is motivated by Theorem 1(a) in the paper by Aaronson and
Nakada ([2]).

######  Theorem 3.7.

If $(T,\mu,\mathcal{A})\in\mathcal{T}$ is a weakly mixing RU map, then it has
property M. More precisely, there is a real number $r$ with $0<r<1$ such that
for any $A\in\sigma(\mathcal{A}^{l})$ and $B\in\mathcal{B}$,

$|\lambda(A\cap T^{-n-l}B)-\lambda(A)\mu(B)|\leq\mu(B)\,O(r^{n}).$

The proof of Theorem 3.7 is the almost same as that of Theorem 1(a) in the
cited above paper by Aaronson and Nakada. We provide the proof for reader’s
convenience. Before embarking on the proof, we introduce some auxiliary facts.
For $(T,\mu,\mathcal{A})\in\mathcal{T}$, let
$\hat{T}:L^{1}(\lambda)\rightarrow L^{1}(\lambda)$ be the corresponding
Frobenius-Perron operator. Then

$\hat{T}^{n}f=\sum_{A\in\mathcal{A}^{n}}1_{T^{n}A}\,v_{A}^{\prime}\,f\circ
v_{A},$

where $v_{A}:T^{n}A\rightarrow A$ is the inverse of the restriction of $T^{n}$
to $A$.

We will need the following two results, which were obtained by Rychlik ([46])
(see also formulas (1) and (2) on p.4 in [2]):

  1. (1)

If $(T,\mu,\mathcal{A})\in\mathcal{T}$ is a RU map, then there is $c_{0}>0$
such that for any $f\in BV$,

$\sum_{A\in\mathcal{A}^{l}}\|1_{T^{l}A}\,v_{A}^{\prime}\,f\circ
v_{A}\|_{BV}\leq c_{0}\|f\|_{BV},\quad l\geq 1.$ (3.5)

  2. (2)

If $T$ is a weakly mixing RU map, then there exist $C>0$ and $r\in(0,1)$ so
that for any $f\in BV$

$\|\hat{T}^{n}f-h\int_{X}f\,d\lambda\|_{BV}\leq Cr^{n}\|f\|_{BV},\quad n\geq
1,$ (3.6)

where $h=\frac{d\mu}{d\lambda}.$

###### Proof of Theorem 3.7.

Let $A\in\sigma(\mathcal{A}^{l})$. Write $A=\bigcup_{i}A_{i}$, where
$A_{i}\in\mathcal{A}^{l}$. Note that

$\hat{T}^{l}1_{A}=\sum_{I\in\mathcal{A}^{l}}1_{T^{l}I}\,v_{I}^{\prime}\,1_{A}\circ
v_{I}=\sum_{A_{i}}1_{T^{l}A_{i}}\,v_{A_{i}}^{\prime}\,1\circ v_{A_{i}},$

where $v_{A_{i}}:T^{l}A_{i}\rightarrow A_{i}$ is given by
$v_{A_{i}}:=({T^{l}}_{|A_{i}})^{-1}$. Then, by (3.5) we get

$\displaystyle\|\hat{T}^{l}1_{A}\|_{BV}$
$\displaystyle\leq\sum_{A_{i}}\|1_{T^{l}A_{i}}\,v_{A_{i}}^{\prime}\,1\circ
v_{A_{i}}\|_{BV}$
$\displaystyle\leq\sum_{I\in\mathcal{A}^{l}}\|1_{T^{l}I}\,v_{I}^{\prime}\,1\circ
v_{I}\|_{BV}\leq c_{0}\|1\|_{BV}=c_{0}$

for some $c_{0}>0$. Notice that

$\lambda(A\cap
T^{-(n+l)}B)=\int(\hat{T}^{n}(\hat{T}^{l}1_{A}))\,1_{B}\,d\lambda\quad\text{and}\quad\lambda(A)\mu(B)=\int\left(h\int\hat{T}^{l}1_{A}\,d\lambda\right)1_{B}\,d\lambda.$

Using (3.6) with $f=\hat{T}^{l}1_{A}$, we obtain

$\|\hat{T}^{n}(\hat{T}^{l}1_{A})-h\int\hat{T}^{l}1_{A}\,d\lambda\|_{\infty}\leq\|\hat{T}^{n}(\hat{T}^{l}1_{A})-h\int\hat{T}^{l}1_{A}\,d\lambda\|_{BV}\leq
Cr^{n}\|\hat{T}^{l}1_{A}\|_{BV}\leq c_{0}Cr^{n}.$

Then,

$\displaystyle|\lambda(A\cap
T^{-(n+l)}B)-\lambda(A)\mu(B)|=\left|\int\left(\hat{T}^{n}(\hat{T}^{l}1_{A})-h\int\hat{T}^{l}1_{A}\,d\lambda\right)1_{B}\,d\lambda\right|$
$\displaystyle\quad\leq\|\hat{T}^{n}(\hat{T}^{l}1_{A})-h\int\hat{T}^{l}1_{A}\,d\lambda\|_{\infty}\lambda(B)\leq
c_{0}Cr^{n}\lambda(B)=\mu(B)O(r^{n}).$

∎

We conclude this section by proving that if
$(T,\mu,\mathcal{A})\in\mathcal{T}$ has property M, then it is exact.

A (non-invertible) measure preserving transformation $T$ on a probability
space $(X,\mathcal{B},\mu)$ is called exact endomorphism if

$\bigcap_{n=0}^{\infty}T^{-n}\mathcal{B}=\\{\emptyset,X\\}\,\,(\bmod\,\mu).$

Rokhlin ([44]) showed that if a measure preserving transformation $T$ on a
probability space $(X,\mathcal{B},\mu)$ is exact, then $T$ has completely
positive entropy (meaning that each of its non-trivial factors has positive
entropy).

######  Theorem 3.8.

If $(T,\mu,\mathcal{A})\in\mathcal{T}$ has property M, then $T$ is exact.

###### Proof.

Assume that $T$ is not exact. Then there is a measurable set
$B\in\bigcap_{n=0}^{\infty}T^{-n}\mathcal{B}$ with $0<\mu(B)<1$ (and so
$0<\lambda(B)<1$). One can find a sequence of measurable sets
$B_{n}\in\bigvee_{i=0}^{n-1}\sigma(T^{-i}\mathcal{A})$ such that
$\lambda(B_{n}\triangle B)\rightarrow 0$ and $\lambda(B_{n})\geq\epsilon_{0}$
for some $\epsilon_{0}>0$ if $n$ is sufficiently large. Since
$B\in\bigcap_{n=0}^{\infty}T^{-n}\mathcal{B}$, for each $n\in\mathbb{N}$ there
exists a measurable set $C_{n}\in\mathcal{B}$ such that $T^{-2n}C_{n}=B^{c}$.
Note that

$\lambda(B_{n}\cap T^{-2n}C_{n})=\lambda(B_{n}\cap
B^{c})\leq\lambda(B_{n}\triangle B)\rightarrow 0.$

By (3.4), we have that

$-b_{n}+\lambda(B_{n}\cap T^{-2n}C_{n})\leq\lambda(B_{n})\mu(C_{n})\leq
b_{n}+\lambda(B_{n}\cap T^{-2n}C_{n}).$

Since $\lambda(B_{n})\geq\epsilon_{0}$ for sufficiently large $n$,
$\mu(C_{n})\rightarrow 0$. Then we have

$0=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{n}\mu(C_{n})=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{n}\mu(T^{-2n}C_{n})=\mu(B^{c}),$

so $\mu(B)=1$, which contradicts the assumption that $\mu(B)<1$ ∎

###  3.3. Examples

In this subsection, we discuss some examples of number-theoretic maps which,
along with an appropriate partition, belong to the class $\mathcal{T}$ and
possess properties E and M.

####  3.3.1. Times $b$ maps: $T_{b}x=bx\mod 1$, $b\geq 2$ an integer

Note that the map $T_{b}x=bx\,\bmod\,1$ is $\lambda$-preserving and ergodic.
It is well-known that the entropy $h(T_{b})$ is $\log b$.
Let

$\mathcal{A}_{b}=\left\\{\left[0,\frac{1}{b}\right),\cdots,\left[\frac{b-1}{b},1\right)\right\\}.$

Then clearly $(T_{b},\lambda,\mathcal{A}_{b})\in\mathcal{T}$.
It is also not hard to check that

  * •

for any $n\in\mathbb{N}$, if $I$ is a cylinder of rank $n$, then
$\lambda(I)=\frac{1}{b^{n}}$, so the set $\mathfrak{b}^{n}$ of bad atoms is
empty, which clearly implies that $(T_{b},\lambda,\mathcal{A}_{b})$ has
property E.

  * •

if $A\in\sigma(\mathcal{A}_{b}^{l})$, $l\in\mathbb{N},$ and $B\in\mathcal{B}$,
then

$\lambda(A\cap T_{b}^{-(n+l)}B)=\lambda(A)\lambda(B)\,\,\text{for any }n\geq
0,$

so $(T_{b},\lambda,\mathcal{A}_{b})$ has property M.

####  3.3.2. Gauss map: $T_{G}x=\frac{1}{x}\,\bmod 1$

It is well-known that Gauss map $T_{G}$ is ergodic with respect to Gauss
measure $d\mu_{G}=\frac{1}{\log 2}\frac{dx}{1+x}.$ Let

$\mathcal{A}_{G}=\left\\{\left(\frac{1}{n+1},\frac{1}{n}\right]:n\in\mathbb{N}\right\\}.$
(3.7)

It is not hard to see that $(T_{G},\mu_{G},\mathcal{A}_{G})\in\mathcal{T}$.

To show that $(T_{G},\mu_{G},\mathcal{A}_{G})$ has property E, recall first
some basic facts about continued fractions. Let $x\in[0,1)$ be an irrational
number. Let $x=[x_{1},x_{2},x_{3},\cdots]$ be the continued fraction expansion
of $x$. Let $p_{n}=p_{n}(x),q_{n}=q_{n}(x)$ denote the numerator and
denominator of the $n$th convergent:

$\frac{p_{n}}{q_{n}}=[x_{1},x_{2},\cdots,x_{n}].$

Let $I_{n}(x)$ denote the cylinder set of rank $n$ containing $x$. The length
of $I_{n}(x)$ is given by

$\lambda(I_{n}(x))=\left|\frac{p_{n}}{q_{n}}-\frac{p_{n}+p_{n+1}}{q_{n}+q_{n+1}}\right|=\frac{1}{q_{n}(q_{n}+q_{n+1})}.$

It is known (see, for instance, Section 14 in [25] or formula (4.18) of
Chapter 1 in [8]) that for a.e. $x$,

$\lim_{n\rightarrow\infty}\frac{1}{n}\log q_{n}(x)=\frac{\pi^{2}}{12\log 2}.$
(3.8)

Then Shannon-McMillan-Breiman Theorem (Theorem 3.1) implies that for a.e. $x$,

$h(T_{G})=\lim_{n\rightarrow\infty}-\frac{\log\mu_{G}(I_{n}(x))}{n}=\lim_{n\rightarrow\infty}-\frac{\log\lambda(I_{n}(x))}{n}=\lim_{n\rightarrow\infty}\frac{1}{n}\log
q_{n}(x)(q_{n}(x)+q_{n-1}(x))=\frac{\pi^{2}}{6\log 2}.$

The fact that $(T_{G},\mu_{G},\mathcal{A}_{G})$ has property E is a
consequence of the following theorem.

######  Theorem 3.9 (cf. Theorem 2 in [14] or Theorem B in [49]).

For any $\delta>0$, there exist $N>0,B>0,\alpha>0$ such that for all $n\geq
N$,

$\lambda\left\\{x:\left|\frac{1}{n}\log q_{n}(x)-\frac{\pi^{2}}{12\log
2}\right|\geq\delta\right\\}\leq B\exp(-\alpha n).$ (3.9)

The following result implies that $T_{G}$ has property M.

######  Theorem 3.10 (cf. [25], [28], [29]).

There is a constant $\gamma$ such that for $A\in\sigma(\mathcal{A}_{G}^{l})$,
$B\in\mathcal{B}$ and for $n\in\mathbb{N}$

$\left|\lambda(A\cap
T_{G}^{-n-l}B)-\lambda(A)\mu_{G}(B)\right|=\lambda(A)\,\mu_{G}(B)\,O(e^{-\gamma
n}).$ (3.10)

####  3.3.3. $\beta$-transformations: $T_{\beta}x=\beta x\,\bmod 1$, $\beta>1$
a non-integer

Rényi showed that there is a $T_{\beta}$-preserving and ergodic measure
$\mu_{\beta}$ such that
$1-\frac{1}{\beta}\leq\frac{d\mu_{\beta}}{d\lambda}\leq\frac{1}{1-1/\beta}$.
(See [41] and also see [35] and [18].)
Let

$\mathcal{A}_{\beta}=\left\\{\left[0,\frac{1}{\beta}\right),\cdots,\left[\frac{[\beta]-1}{\beta},\frac{[\beta]}{\beta}\right),\left[\frac{[\beta]}{\beta},1\right)\right\\}.$

Clearly $(T_{\beta},\mu_{\beta},\mathcal{A}_{\beta})\in\mathcal{T}$.

It is known ([34]) that $h(T_{\beta})=\log\beta$. For $x\in[0,1)$, let
$I_{n}(x)$ denote the cylinder of rank $n$ containing $x$. Then Shanon-
McMillan-Breiman theorem implies that for a.e. $x$,

$\lim_{n\rightarrow\infty}-\frac{\log\lambda(I_{n}(x))}{n}=\lim_{n\rightarrow\infty}-\frac{\log\mu_{\beta}(I_{n}(x))}{n}=h(T_{\beta})=\log\beta.$

The fact that $(T_{\beta},\mu_{\beta},\mathcal{A}_{\beta})$ has property E is
the consequence of the following theorem.

######  Theorem 3.11.

For $\epsilon>0$,

$\lambda\left\\{x:\left|-\frac{\log\lambda(I_{n}(x))}{n}-\log\beta\right|>\epsilon\right\\}\leq\frac{\beta}{\beta-1}e^{-\epsilon
n}.$

###### Proof.

It is obvious that if $A$ is a cylinder of rank $n$, then
$\lambda(A)\leq\frac{1}{\beta^{n}}$. In [41], Rényi showed that number of
cylinders of rank $n$ is less than or equal to
$\frac{\beta}{\beta-1}\beta^{n}$. Then,

$\lambda\\{x:x\in A\text{ for some cylinder }A\text{ of rank }n\text{ such
that }\lambda(A)<\beta^{-n}e^{-\epsilon
n}\\}\leq\frac{\beta}{\beta-1}e^{-\epsilon n}.$

∎

It is not hard to check that $(T_{\beta},\mu_{\beta},\mathcal{A}_{\beta})$ is
a RU map. Also, it is known that $T_{\beta}$ is strongly mixing (see Section
4.5 in [43] or Theorem 1C in [37]), hence weakly mixing, so
$(T_{\beta},\mu_{\beta},\mathcal{A}_{\beta})$ has property M by Theorem 3.7.

####  3.3.4. Linear $\bmod\,1$ transformations.

Times $b$ maps $T_{b}$ and $\beta$-transformations $T_{\beta}$ belong to a
general family of maps $T_{\beta,\gamma}$:

$T_{\beta,\gamma}x=\beta x+\gamma\bmod 1,\,\,(\beta>1,0\leq\gamma<1).$

It is known that if either $\beta\geq 2,0\leq\gamma<1$ or
$1<\beta<2,\gamma=0$, then there is a $T_{\beta,\gamma}$-invariant and ergodic
measure $\mu_{\beta,\gamma}$ such that
$1/c\leq\frac{d\mu_{\beta,\gamma}}{d\lambda}\leq c$ for some $c\geq 1$. (See
[20] and [52].) Also, it is known ([34]) that $h(T_{\beta,\gamma})=\log\beta.$
On the other hand, Parry showed that if $1<\beta<2$ and $\gamma\neq 0$, then
$T_{\beta,\gamma}$ may not have an invariant measure, which is equivalent to
the Lebesgue measure. (See [35]). In what follows we will assume that either
$\beta\geq 2,0\leq\gamma<1$, or $1<\beta<2,\gamma=0$.

Let

$\mathcal{A}_{\beta,\gamma}=\left\\{\left[0,\frac{1-\gamma}{\beta}\right),\left[\frac{1-\gamma}{\beta},\frac{2-\gamma}{\beta}\right)\cdots,\left[\frac{[\beta+\gamma]-1-\gamma}{\beta},\frac{[\beta+\gamma]-\gamma}{\beta}\right),\left[\frac{[\beta+\gamma]-\gamma}{\beta},1\right)\right\\}.$

One readily checks that
$(T_{\beta,\gamma},\mu_{\beta,\gamma},\mathcal{A}_{\beta,\gamma})\in\mathcal{T}$.

Now let us show that
$(T_{\beta,\gamma},\mu_{\beta,\gamma},\mathcal{A}_{\beta,\gamma})$ has
properties E and M. (Note that we only need to consider the case that
$\beta\geq 2,0\leq\gamma<1$.)

It is not hard to check that
$(T_{\beta,\gamma},\mu_{\beta,\gamma},\mathcal{A}_{\beta,\gamma})$ is a RU
map. Also it is known that $T_{\beta,\gamma}$ is strong mixing (indeed, it is
exact ([52])), hence weakly mixing, so
$(T_{\beta,\gamma},\mu_{\beta,\gamma},\mathcal{A}_{\beta,\gamma})$ has
property M by Theorem 3.7.

One also can check that if $\beta\geq 2$, the number of cylinders of rank $n$
for $(T_{\beta,\gamma},\mu_{\beta,\gamma},\mathcal{A}_{\beta,\gamma})$ is
$O(\beta^{n})$; indeed, for $\beta=2$, the number of cylinders of rank $n$ is
less than or equal to $\sum_{i=1}^{n}(2^{i}+1)=2^{n+1}-1$ and for other cases
($\beta>2$), this follows from Lemma 2.1 in [52]. Then by arguing as in the
proof of Theorem 3.11 in Subsection 3.3.3, one concludes that
$(T_{\beta,\gamma},\mu_{\beta,\gamma},\mathcal{A}_{\beta,\gamma})$ has
property E.

##  4\. The proof of main theorem

In this section we prove the main theorem of this article, which we formulate
again for reader’s convenience. Note that the properties E and M, which appear
in the formulation, were introduced in Section 1. (See also Section 3.)

######  Theorem 4.1 (Theorem 1.10).

Let $T_{i}:[0,1]\rightarrow[0,1],1\leq i\leq k$, be measurable transformations
such that for each $i=1,2\dots,k$, there is a partition $\mathcal{A}_{i}$ and
a $T_{i}$-invariant ergodic measures $\mu_{i}$ with
$(T_{i},\mu_{i},\mathcal{A}_{i})\in\mathcal{T}$. Suppose that for each
$i=1,2\dots,k$, $(T_{i},\mu_{i},\mathcal{A}_{i})$ satisfies properties E and
M. If entropy values $h(T_{1}),\cdots,h(T_{k})$ are distinct, then
$T_{1},\dots,T_{k}$ are pointwise jointly ergodic with respect to
$(\mu_{1},\dots,\mu_{k})$.

The following classical lemma will play an instrumental role in the proof of
Theorem 4.1.

######  Lemma 4.2 (See [21], Lemma 1.5 and a historical remark preceding its
formulation.).

Let $(X,\mathcal{B},\mu)$ be a probability space. Let $f_{n}(x)$ be a sequence
of non-negative measurable functions on $X$. Let $u_{n},v_{n}$ be sequences of
real numbers such that

$0\leq u_{n}\leq v_{n}.$

Write $V(N):=\sum\limits_{n=1}^{N}v_{n}$ and assume that
$V(N)\rightarrow\infty$ as $N\rightarrow\infty$. Suppose that there exists a
constant $K$ such that for arbitrary large integers $M_{1},M_{2}\,(1\leq
M_{1}<M_{2})$,

$\int_{X}\left(\sum_{M_{1}\leq n<M_{2}}(f_{n}(x)-u_{n})\right)^{2}\,d\mu\leq
K\sum_{M_{1}\leq n<M_{2}}v_{n}.$ (4.1)

Then for any $\epsilon>0$ and for a.e. $x\in X$,

$\sum_{1\leq n\leq N}f_{n}(x)=\sum_{1\leq n\leq
N}u_{n}+O\left(V(N)^{1/2}(\log(V(N)+2))^{2/3+\epsilon}+\max_{1\leq n\leq
N}u_{n}\right).$ (4.2)

###### Proof of Theorem 4.1.

Recall that $\mathcal{A}_{i}^{m}$ is a finite or countable partition of the
unit interval $[0,1]$ and the elements of $\mathcal{A}_{i}^{m}$ are intervals,
which are called cylinders of rank $m$ (corresponding to
$(T_{i},\mu_{i},\mathcal{A}_{i})$). It is enough to show that for any
$l\in\mathbb{N}$ and for any $A_{i}\in\mathcal{A}_{i}^{l}$, $i=1,2,\dots,k$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x)=\prod_{i=1}^{k}\mu_{i}(A_{i})\quad\text{
for almost every }x.$ (4.3)

To see that the validity for (4.3) implies the result in question observe
first that since the family of cylinders is countably infinite, (4.3) implies
that for almost every $x$, for any $A_{i}\in\mathcal{A}_{i}^{l}$,
$i=1,2,\dots,k$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x)=\prod_{i=1}^{k}\mu_{i}(A_{i}).$
(4.4)

Now, since $\mathcal{B}=\bigvee_{m=1}^{\infty}\sigma(\mathcal{A}_{i}^{m})$ for
all $i=1,2,\dots,k$, any interval $J\subset[0,1]$ can be approximated by
cylinders. Hence, by the standard approximation argument, the fact that (4.4)
holds for any $A_{i}\in\mathcal{A}_{i}^{l},i=1,2,\dots,k,l\in\mathbb{N}$
implies that it holds for any intervals $J_{1},\dots,J_{k}$:

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}1_{J_{i}}(T_{i}^{n}x)=\prod_{i=1}^{k}\mu_{i}(J_{i}).$

This, in turn, implies that for almost every $x$,
$(T_{1}^{n}x,\dots,T_{k}^{n}x)$ is equidistributed with respect to
$\mu_{1}\times\cdots\times\mu_{k}$. The pointwise joint ergodicity of
$T_{1},\dots,T_{k}$ follows then from Theorem 2.2.

We will derive (4.3) by applying Lemma 4.2 to

$f_{n}(x)=\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x),\quad
u_{n}=\prod_{i=1}^{k}\mu_{i}(A_{i}),\quad v_{n}=\max(1,\log n).$

(Note that $f_{n}(x)$ and $v_{n}$ depend on $n$, but $u_{n}$ is a constant
sequence, so we can ignore the last term $\max\limits_{1\leq n\leq N}u_{n}$ in
(4.2).)

Write $h_{i}=h(T_{i})$ for simplicity. Without loss of generality we assume
that $h_{1}<h_{2}<\cdots<h_{k}$. Choose $\epsilon,\delta,\gamma>0$ small
enough so that for $i=2,\dots,k$,

$h_{i}-(1+\delta)h_{i-1}-(2+\delta)\epsilon\geq\gamma.$ (4.5)

Since $(T_{i},\mu_{i},\mathcal{A}_{i})\in\mathcal{T}$ for $i=1,2,\dots,k$,
there exists $c\geq 1$ such that $\frac{1}{c}\mu_{i}\leq\lambda\leq c\mu_{i}$
for all $i=1,2,\dots,k$. For $M_{1},M_{2}\in\mathbb{N}$ with
$M_{1}<M_{2}\leq(1+\delta)M_{1}$, let

$I_{M_{1},M_{2}}=\\{(m,n):M_{1}+l\leq m<n\leq M_{2}-l,n-m\geq
l\\}\subset\mathbb{N}^{2}.$

We will show that (4.3) is the consequence of the following two claims:

Claim 1. For $(m,n)\in I_{M_{1},M_{2}}$,

$\lambda\left(\bigcap_{i=1}^{k}\left(T_{i}^{-m}A_{i}\bigcap
T_{i}^{-n}A_{i}\right)\right)=\prod_{i=1}^{k}\mu_{i}(A_{i})^{2}+c_{m,n},$
(4.6)

where $\sum\limits_{(m,n)\in I_{M_{1},M_{2}}}|c_{m,n}|=O(M_{2}-M_{1})$

Claim 2. For $M_{1}+l\leq m\leq M_{2}-l$,

$\lambda\left(\bigcap_{i=1}^{k}T_{i}^{-m}A_{i}\right)=\prod_{i=1}^{k}\mu_{i}(A_{i})+d_{m},$
(4.7)

where $\sum\limits_{m=M_{1}+l}^{M_{2}-l}|d_{m}|=O(1)$.

Before proving the above claims, let us show that (4.3) follows from (4.6) and
(4.7). First, notice that (4.6) and (4.7) imply that

$\displaystyle\int_{0}^{1}\left(\sum_{n=M_{1}}^{M_{2}}\left(\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x)-\prod_{i=1}^{k}\mu_{i}(A_{i})\right)\right)^{2}\,d\lambda$
$\displaystyle=2\int_{0}^{1}\sum_{(m,n)\in
I_{M_{1},M_{2}}}\left(\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{m}x)-\prod_{i=1}^{k}\mu_{i}(A_{i})\right)\left(\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x)-\prod_{i=1}^{k}\mu_{i}(A_{i})\right)\,d\lambda+O(M_{2}-M_{1})$
$\displaystyle=O\left(\sum_{(m,n)\in
I_{M_{1},M_{2}}}|c_{m,n}|\right)+O\left(\sum_{m=M_{1}+l}^{M_{2}-l}|d_{m}|\right)(M_{2}-M_{1})+O(M_{2}-M_{1})$
$\displaystyle=O(M_{2}-M_{1}).$

Given any $M<N$, partition the interval $[M,N]$ into $O(\log(N-M))$ intervals
of the form $[M_{1},M_{2}]$ with $M_{2}\leq(1+\delta)M_{1}$. Then we have

$\int_{0}^{1}\left(\sum_{n=M+1}^{N}\left(\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x)-\prod_{i=1}^{k}\mu_{i}(A_{i})\right)\right)^{2}\,d\lambda=O((N-M)\log(N-M)).$
(4.8)

Note also that

$\displaystyle(N-M)\log(N-M)$ $\displaystyle\leq(N-M)\log N=N\log N-M\log N$
$\displaystyle\leq N\log N-M\log M=O\left(\sum\limits_{n=M}^{N}\log
n\right)=O\left(\sum\limits_{n=M}^{N}v_{n}\right).$

So Lemma 4.2 implies

$\sum_{1\leq n\leq N}\prod_{i=1}^{k}1_{A_{i}}(T_{i}^{n}x)=\sum_{1\leq n\leq
N}\prod_{i=1}^{k}\mu_{i}(A_{i})+O\left((N\log N)^{1/2}(\log(N\log
N+2))^{2/3+\epsilon}\right),$

which, in turn, gives (4.3).

Now let us prove Claim 1 (the proof of Claim 2 is analogous and is omitted).
We start with two preliminary observations. In what follows, we tacitly assume
that $\epsilon,\delta,\gamma$ satisfy (4.5).

  1. (1)

By Corollary 3.2, for any $i=1,2,\dots,k$ and for all $n\in\mathbb{N}$, we
have a partition

$\mathcal{A}_{i}^{n}=\mathfrak{g}_{i}^{n}\cup\mathfrak{b}_{i}^{n}$

such that

    1. (a)

$\mathfrak{g}_{i}^{n}=\\{I\in\mathcal{A}_{i}^{n}:\frac{1}{c}e^{-n(h_{i}+\epsilon)}\leq\lambda(I)\leq
ce^{-n(h_{i}-\epsilon)}\\}$

    2. (b)

$\mathfrak{b}_{i}^{n}=\mathcal{A}_{i}^{n}\setminus\mathfrak{g}_{i}^{n}$.

Since $(T_{i},\mu_{i},\mathcal{A}_{i})$ has property E, there is a real number
$a=a(\epsilon)>0$ so that for $i=1,2,\dots,k$,

$\lambda\left(\bigcup_{B\in\mathfrak{b}_{i}^{n}}B\right)\leq\frac{a}{n}.$
(4.9)

  2. (2)

Since $(T_{i},\mu_{i},\mathcal{A}_{i})$ has property M, there exists a
sequence $(b_{n})_{n=1}^{\infty}$ of non-negative real numbers with
$\sum\limits_{n=1}^{\infty}b_{n}<\infty$, such that for $i=1,2,\dots,k$, if
$A\in\sigma(\mathcal{A}_{i}^{l})$ and $B\in\mathcal{B}$,

$|\lambda(A\cap T_{i}^{-(n+l)}B)-\lambda(A)\mu_{i}(B)|\leq b_{n}\text{ for any
}n\in\mathbb{N}.$

To show (4.6), we will need the following result:

Claim 3: For $(m,n)\in I_{M_{1},M_{2}}$, there exist sets $I_{1}(k)$ and
$I_{2}(k)$ such that

$\bigcap_{i=1}^{k}(T_{i}^{-m}A_{i}\cap T_{i}^{-n}A_{i})=I_{1}(k)\cup I_{2}(k)$

where

  1. (1)

$I_{1}(k)$ is union of some cylinders in $\mathfrak{g}_{k}^{n+l}$ and
satisfies

$\lambda(I_{1}(k))=\prod\limits_{i=1}^{k}\mu_{i}(A_{i})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}+e^{-\gamma
M_{1}}\right),$

  2. (2)

$I_{2}(k)$ is a set, which is disjoint from $I_{1}(k)$, and satisfies
$\lambda(I_{2}(k))=O\left(\frac{1}{M_{1}}+e^{-\gamma M_{1}}\right)$.

Note that if Claim 3 holds, then we have

$|c_{m,n}|=O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}+e^{-\gamma
M_{1}}\right)$

and so

$\sum_{(m,n)\in I_{M_{1},M_{2}}}|c_{m,n}|=O(M_{2}-M_{1}),$

which implies Claim 1. We prove Claim 3 by induction. Let us first show that
there exist sets $I_{1}(1)$ and $I_{2}(1)$ such that

$T_{1}^{-m}A_{1}\cap T_{1}^{-n}A_{1}=I_{1}(1)\cup I_{2}(1),$

where

  * •

$I_{1}(1)$ is union of some intervals in $\mathfrak{g}_{1}^{n+l}$ and
satisfies

$\lambda(I_{1}(1))=\mu_{1}(A_{1})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}\right)$

  * •

$I_{2}(1)$ is a set, which is disjoint from $I_{1}(1)$, and satisfies
$\lambda(I_{2})=O\left(\frac{1}{M_{1}}\right)$.

Indeed, note that $T_{1}^{-m}A_{1}\cap
T_{1}^{-n}A_{1}\in\sigma(\mathcal{A}_{1}^{n+l})$, so

  * •

$I_{1}(1)$ is union of some intervals in $\mathfrak{g}_{1}^{n+l}$, more
precisely, $I_{1}(1)=\bigcup_{i=1}^{s(1)}J_{i}(1)$ such that
$J_{i}(1)\in\mathfrak{g}_{1}^{n+l}$ for $1\leq i\leq s(1)$,

  * •

$I_{2}(1)$ is union of some intervals in $\mathfrak{b}_{1}^{n+l}$, so
$\lambda(I_{2}(1))\leq\frac{a}{n+l}=O\left(\frac{1}{M_{1}}\right)$.

Moreover, by property M of $(T_{1},\mu_{1},\mathcal{A}_{1})$, we have that

$\displaystyle\lambda(I_{1}(1))$ $\displaystyle=\lambda(T_{1}^{-m}A_{1}\cap
T_{1}^{-n}A_{1})+O\left(\lambda(I_{2}(1))\right)$
$\displaystyle=\lambda(T_{1}^{-m}A_{1}\cap
T_{1}^{-((n-m-l)+(m+l))}A_{1})+O\left(\frac{1}{M_{1}}\right)$
$\displaystyle=\mu_{1}(T_{1}^{-m}A_{1})\cdot\mu_{1}(A_{1})+O\left(b_{n-m-l}+\frac{1}{M_{1}}\right)\quad\quad(\text{by
property M})$ $\displaystyle=\mu_{1}(X\cap
T_{1}^{-(m-M_{1}-l)+(M_{1}+l)}A_{1})\cdot\mu_{1}(A_{1})+O\left(b_{n-m-l}+\frac{1}{M_{1}}\right)$
$\displaystyle=\mu_{1}(A_{1})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}\right)\quad\quad(\text{by
property M})$

To continue the induction, suppose that for $1\leq q<k$, there are sets
$I_{1}(q)$ and $I_{2}(q)$ such that

$\bigcap_{i=1}^{q}(T_{i}^{-m}A_{i}\cap T_{i}^{-n}A_{i})=I_{1}(q)\cup I_{2}(q)$

where for some $s(q)$

  * •

$I_{1}(q)=\bigcup\limits_{i=1}^{s(q)}J_{i}(q)$, where
$J_{i}(q)\in\mathfrak{g}_{q}^{n+l}$ for $1\leq i\leq s(q)$, and

$\lambda(I_{1}(q))=\prod\limits_{i=1}^{q}\mu_{i}(A_{i})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}+e^{-\gamma
M_{1}}\right),$

  * •

$I_{2}(q)$ is a set, which is disjoint from $I_{1}(q)$, and satisfies
$\lambda(I_{2}(q))=O\left(\frac{1}{M_{1}}+e^{-\gamma M_{1}}\right)$.

We have to show that there are sets $I_{1}(q+1)$ and $I_{2}(q+1)$ such that

$\bigcap_{i=1}^{q+1}(T_{i}^{-m}A_{i}\cap T_{i}^{-n}A_{i})=I_{1}(q+1)\cup
I_{2}(q+1)$

where

  * •

$I_{1}(q+1)=\bigcup\limits_{i=1}^{s(q+1)}J_{i}(q+1)$, where
$J_{i}(q+1)\in\mathfrak{g}_{q+1}^{n+l}$ for $1\leq i\leq s(q+1)$, and

$\lambda(I_{1}(q+1))=\prod\limits_{i=1}^{q+1}\mu_{i}(A_{i})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}+e^{-\gamma
M_{1}}\right),$

  * •

$I_{2}(q+1)$ is a set, which is disjoint from $I_{1}(q+1)$, and satisfies
$\lambda(I_{2}(q+1))=O\left(\frac{1}{M_{1}}+e^{-\gamma M_{1}}\right)$,

To conclude the proof, we need first to establish the following statement.

Claim 4. Given $I_{1}(q)=\bigcup\limits_{i=1}^{s(q)}J_{i}(q)$ as above, one
can write

$I_{1}(q)=H_{1}(q)\cup H_{2}(q),$

where

  * •

$H_{1}(q)$ is union of some cylinders in $\mathfrak{g}_{q+1}^{M_{1}}$,

  * •

$\lambda(H_{2}(q))=O(\frac{1}{M_{1}}+e^{-\gamma M_{1}})$.

Proof of Claim 4. Notice that $J_{i}(q)\in\mathfrak{g}_{q}^{n+l}$, so

$\lambda(J_{i}(q))\geq\frac{1}{c}\mu(J_{i}(q))\geq\frac{1}{c}e^{-(n+l)(h_{q}+\epsilon)}\geq\frac{1}{c}e^{-M_{2}(h_{q}+\epsilon)}.$

Since
$\sum\limits_{i=1}^{s(q)}\lambda(J_{i}(q))=\lambda\left(\bigcup\limits_{i=1}^{s(q)}(J_{i}(q))\right)\leq
1$, we have $s(q)\frac{1}{c}e^{-M_{2}(h_{q}+\epsilon)}\leq 1$, which implies

$s(q)\leq ce^{M_{2}(h_{q}+\epsilon)}.$ (4.10)

Assume that $\mathfrak{g}_{q+1}^{M_{1}}$ is comprised of intervals
$U_{1},\dots,U_{s}$, where $U_{i}$ are subintervals of the interval $[0,1]$
having disjoint interiors. If $\mathfrak{b}_{q+1}^{M_{1}}=\emptyset$, then
$U_{1},\dots,U_{s}$ form a partition of the unit interval $[0,1]$. Otherwise,
there exist intervals $V_{1},\dots,V_{t}$ such that
$U_{1},\dots,U_{s},V_{1},\dots V_{t}$ form a partition $\mathcal{P}$ of
$[0,1]$. For each interval $J_{i}(q)$, there exist a finite family of
intervals $I_{i,j}$ $(1\leq j\leq t_{i})$ and two intervals $K_{i,1}$ and
$K_{i,2}$ in $\mathcal{P}$ such that

$\bigcup_{j=1}^{t_{i}}I_{i,j}\subset\overline{J_{i}(q)}\subset\left(\bigcup_{j=1}^{t_{i}}\overline{I_{i,j}}\right)\bigcup\left(\overline{K_{i,1}}\cup\overline{K_{i,2}}\right).$

Let $H_{1}(q)$ be the union of intervals $I_{i,j},1\leq i\leq s(q),1\leq j\leq
t_{i}$ which belong to $\mathfrak{g}_{q+1}^{M_{1}}$. Let
$H_{2}(q)=I_{1}(q)\setminus H_{1}(q)$. Let us show that

$\lambda(H_{2}(q))=O\left(\frac{1}{M_{1}}+e^{-\gamma M_{1}}\right).$

Note that $H_{2}(q)$ is covered by at most $2s(q)$ intervals from
$\mathfrak{g}_{q+1}^{M_{1}}$ and - possibly - by some intervals from
$\mathfrak{b}_{q+1}^{M_{1}}$. Since $\lambda(J)\leq
ce^{-M_{1}(h_{q+1}-\epsilon)}$ for any cylinder
$J\in\mathfrak{g}_{q+1}^{M_{1}}$,

$\lambda(H_{2}(q))\leq\lambda\left(\bigcup_{B\in\mathfrak{b}_{q+1}^{M_{1}}}B\right)+2s(q)ce^{-M_{1}(h_{q+1}-\epsilon)}.$
(4.11)

By (4.5),

$-M_{1}(h_{q+1}-\epsilon)+M_{2}(h_{q}+\epsilon)\leq-
M_{1}(h_{q+1}-(1+\delta)h_{q}-(2+\delta)\epsilon)\leq-\gamma M_{1},$

and so by (4.10)

$2s(q)ce^{-M_{1}(h_{q+1}-\epsilon)}\leq
2c^{2}e^{-M_{1}(h_{q+1}-\epsilon)+M_{2}(h_{q}+\epsilon)}\leq e^{-\gamma
M_{1}}$ (4.12)

By property E,

$\lambda\left(\bigcup_{B\in\mathfrak{b}_{q+1}^{M_{1}}}B\right)\leq\frac{a}{M_{1}}.$
(4.13)

Putting (4.11), (4.12), and (4.13) together, we get

$\lambda(H_{2}(q))\leq\frac{a}{M_{1}}+2c^{2}e^{-M_{1}(h_{q+1}-\epsilon)+M_{2}(h_{q}+\epsilon)}=O\left(\frac{1}{M_{1}}+e^{-\gamma
M_{1}}\right).$

This completes the proof of Claim 4.

By Claim 4,

$\bigcap_{i=1}^{q}(T_{i}^{-m}A_{i}\cap T_{i}^{-n}A_{i})=H_{1}(q)\cup
H_{2}(q)\cup I_{2}(q),$

where $H_{1}(q)\in\sigma(\mathcal{A}_{q+1}^{M_{1}})$ and $\lambda(H_{2}(q)\cup
I_{2}(q))=O(\frac{1}{M_{1}}+e^{-\gamma M_{1}})$.

Note that $H_{1}(q)\cap T_{q+1}^{-m}A_{q+1}\cap
T_{q+1}^{-n}A_{q+1}\in\sigma(\mathcal{A}_{q+1}^{n+l})$, thus one can write

$H_{1}(q)\cap T_{q+1}^{-m}A_{q+1}\cap T_{q+1}^{-n}A_{q+1}=I_{1}(q+1)\cup R,$
(4.14)

where $I_{1}(q+1)=\bigcup_{i=1}^{s(q+1)}J_{i}(q+1)$ with
$J_{i}(q+1)\in\mathfrak{g}_{q+1}^{n+l}$ for $1\leq i\leq s(q+1)$ and $R$ is
union of some intervals in $\mathfrak{b}_{q+1}^{n+l}$, so

$\lambda(R)\leq\frac{a}{n+l}=O\left(\frac{1}{M_{1}}\right).$ (4.15)

Let

$I_{2}(q+1)=\bigcap_{i=1}^{q+1}(T_{i}^{-m}A_{i}\cap T_{i}^{-n}A_{i})\setminus
I_{1}(q+1),$

so $I_{1}(q+1)\bigcup
I_{2}(q+1)=\bigcap\limits_{i=1}^{q+1}(T_{i}^{-m}A_{i}\cap T_{i}^{-n}A_{i}).$
Note that

$I_{2}(q+1)\subset R\cup H_{2}(q)\cup I_{2}(q),$

so $\lambda(I_{2}(q+1))=O\left(\frac{1}{M_{1}}+e^{-\gamma M_{1}}\right)$.

Now, utilizing the fact that $(T_{q+1},\mu_{q+1},\mathcal{A}_{q+1})$ has
property M, we obtain

$\displaystyle\lambda\left(H_{1}(q)\cap T_{q+1}^{-m}A_{q+1}\cap
T_{q+1}^{-n}A_{q+1}\right)$ $\displaystyle=\lambda\left(H_{1}(q)\cap
T_{q+1}^{-m}A_{q+1}\right)\mu_{q+1}(A_{q+1})+O(b_{n-m-l})$
$\displaystyle=\lambda(H_{1}(q))\mu_{q+1}(A_{q+1})^{2}+O(b_{m-M_{1}-l}+b_{n-m-l}).$
(4.16)

(We use that $H_{1}(q)\cap T_{q+1}^{-m}A_{q+1}\in\mathcal{A}_{q+1}^{m+l}$ for
the first equality and
$H_{1}(q)\in\mathcal{A}_{q+1}^{M_{1}}\subset\mathcal{A}_{q+1}^{M_{1}+l}$ for
the second equality.)

Then,

$\displaystyle\lambda(I_{1}(q+1))$ $\displaystyle=\lambda\left(H_{1}(q)\cap
T_{q+1}^{-m}A_{q+1}\cap T_{q+1}^{-n}A_{q+1}\right)+O(\lambda(R))\quad(\text{by
}\eqref{eqn:$I_1(q+1)$})$
$\displaystyle=\lambda(H_{1}(q))\mu_{q+1}^{2}(A_{q+1})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}\right)(\text{by
}\eqref{eqn:R},\eqref{eqn:comp})$
$\displaystyle=\lambda(I_{1}(q))\mu_{q+1}^{2}(A_{q+1})^{2}+O\left(b_{m-M_{1}-l}+b_{n-m-l}+\frac{1}{M_{1}}+e^{-\gamma
M_{1}}\right)(\text{by Claim 4})$

∎

The following result is a special case of Theorem 4.1, which was formulated in
the Introduction.

######  Theorem 4.3 (Theorem 1.2).

Let $s,t\in\mathbb{N}$. Let $b_{1},\dots,b_{s}$ be distinct integers with
$b_{i}\geq 2$ for all $i=1,2,\dots,s$ and let $\beta_{1},\dots,\beta_{t}$ be
distinct non-integers with $\beta_{i}>1$ for all $i=1,2,\dots,t$. Then

  1. (1)

$T_{b_{1}},\dots,T_{b_{s}},T_{\beta_{1}},\dots T_{\beta_{t}}$ are pointwise
jointly ergodic.

  2. (2)

$T_{G},T_{b_{1}},\dots,T_{b_{s}},T_{\beta_{1}},\dots T_{\beta_{t}}$ are
pointwise jointly ergodic if $\log\beta_{i}\neq\frac{\pi^{2}}{6\log 2}$ for
$1\leq i\leq t$.

######  Corollary 4.4 (Corollary 1.6).

Let $s,t\in\mathbb{N}$. Let $b_{1},\dots,b_{s}$ be distinct integers with
$b_{i}\geq 2$ for all $i=1,2,\dots,s$ and let $\beta_{1},\dots,\beta_{t}$ be
distinct non-integers with $\beta_{j}>1$ for all $j=1,2,\dots,t$. Then almost
every $x\in[0,1)$ is jointly
$(b_{1},\dots,b_{s},\beta_{1},\dots,\beta_{t})$-normal and also, if
$\log\beta_{j}\neq\frac{\pi^{2}}{6\log 2}$ for any $j=1,2\dots,t$, almost
every $x\in[0,1)$ is jointly
$(b_{1},\dots,b_{s},\beta_{1},\dots,\beta_{t},CF)$-normal.

##  5\. Disjoint systems and pointwise joint ergodicity

In this section, we deal with an approach to pointwise joint ergodicity which
is based on the notion of disjointness of dynamical systems that was
introduced by Furstenberg in [15]. We remark that while in the previous
sections we were mostly dealing with the maps of the interval, the main
results in this section are quite general. The reader is referred to the
classical literature ([17], [36], [45], [51]) for definitions of mixing
notions which are mentioned in this section.

######  Definition 5.1.

Probability measure preserving systems
$(X_{i},\mathcal{B}_{i},\mu_{i},T_{i}),1\leq i\leq k,$ are mutually disjoint
if the only $T_{1}\times\cdots\times T_{k}$-invariant measure on
$X_{1}\times\cdots\times X_{k}$, which has the property that the projection of
$\mu$ onto $X_{i}$ is $\mu_{i}$ for all $i=1,\dots,k$, is the product measure
$\mu_{1}\times\cdots\times\mu_{k}$.

By slight abuse of language, we will say that $T_{1},\dots,T_{k}$ are mutually
disjoint if it is clear from the context what the systems
$(X_{i},\mathcal{B}_{i},\mu_{i})$ are.

Here are some examples of disjoint measure preserving systems:

  1. (1)

Translations on compact abelian groups and weakly mixing systems are disjoint
(Theorem 6.10 in [45]).

  2. (2)

Zero entropy systems and K-automorphisms are disjoint (Theorem 6.11 in [45]).

  3. (3)

Rigid systems and mildly mixing systems are disjoint (Proposition 4.3 in
[24]).

  4. (4)

If $T_{1}$ is a translation on a compact abelian group, $T_{2}$ is a weakly
mixing rigid transformation, and $T_{3}$ is a mildly mixing transformation,
then it is not hard to show that $T_{1},T_{2},T_{3}$ are mutually disjoint.

  5. (5)

If $\mathsf{X}_{i}=(X,\mathcal{B},\mu_{i},T_{i}),1\leq i\leq k,$ are ergodic,
measurably distal systems and Kronecker factors of $\mathsf{X}_{i}$ are
mutually disjoint, then $T_{1},\dots,T_{k}$ are mutually disjoint (Proposition
8.5 in [7]).

The following simple theorem establishes a relation between disjointness and
pointwise joint ergodicity.

######  Theorem 5.2.

Let $X$ be a compact metric space and let $\mathcal{B}$ be the Borel
$\sigma$-algebra of $X$. Let $\mu_{1},\dots,\mu_{l},\nu_{1},\dots,\nu_{m}$ be
pairwise equivalent probability measures on $(X,\mathcal{B})$. Let
$\mathsf{X}_{i}^{(1)}=(X,\mathcal{B},\mu_{i},S_{i})$, $1\leq i\leq l$, and
$\mathsf{X}_{j}^{(2)}=(X,\mathcal{B},\nu_{j},T_{j})$, $1\leq j\leq m$, be
measure preserving systems such that

  1. (1)

$S_{1},\dots,S_{l}$ are pointwise jointly ergodic with respect to
$(\mu_{1},\dots,\mu_{l})$,

  2. (2)

$T_{1},\dots,T_{m}$ are pointwise jointly ergodic with respect to
$(\nu_{1},\dots,\nu_{m})$,

  3. (3)

$\mathsf{X}^{(1)}=\mathsf{X}_{1}^{(1)}\times\cdots\times\mathsf{X}_{l}^{(1)}$
and
$\mathsf{X}^{(2)}=\mathsf{X}_{1}^{(2)}\times\cdots\times\mathsf{X}_{m}^{(2)}$
are disjoint.

Then, $S_{1},\dots,S_{l},T_{1},\dots,T_{m}$ are pointwise jointly ergodic with
respect to $(\mu_{1},\dots,\mu_{l},\nu_{1},\dots,\nu_{m})$.

###### Proof.

By the assumptions (1) and (2), Theorem 2.2 implies that there exists a full
measure set $A\in\mathcal{B}$ such that if $x\in A$, then

$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{S_{1}^{n}x}\times\cdots\times\delta_{S_{l}^{n}x}\rightarrow\mu_{1}\times\cdots\times\mu_{l}$

and

$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{T_{1}^{n}x}\times\cdots\times\delta_{T_{m}^{n}x}\rightarrow\nu_{1}\times\cdots\times\nu_{l}.$

Now consider the sequence of measures on $X^{l+m}$

$\rho_{N}:=\frac{1}{N}\sum_{n=0}^{N-1}\delta_{S_{1}^{n}x}\times\cdots\times\delta_{S_{l}^{n}x}\times\delta_{T_{1}^{n}x}\times\cdots\times\delta_{T_{m}^{n}x},\quad
N=1,2,\dots.$

Let $\rho$ be a weak* limit point of $(\rho_{N})$ and let us show that
$\rho=\mu_{1}\times\cdots\times\mu_{l}\times\nu_{1}\times\cdots\times\nu_{m}$.
Note that $\rho$ is $(S_{1}\times\cdots\times
S_{l})\times(T_{1}\times\cdots\times T_{m})$-invariant and that the
projections of $\rho$ on $\mathsf{X}^{(1)}$ and $\mathsf{X}^{(2)}$ are,
correspondingly, $\mu_{1}\times\cdots\times\mu_{l}$ and
$\nu_{1}\times\cdots\times\nu_{m}$. So, the desired result follows from the
assumption that $\mathsf{X}^{(1)}$ and $\mathsf{X}^{(2)}$ are disjoint. ∎

The following corollary of Theorem 5.2 provides a convenient criterion of
pointwise joint ergodicity. A special case when $k=2$ and $\mu_{1}=\mu_{2}$
was obtained by Berend (see Theorem 2.2 in [5]; cf. also Theorem I.6 in [15]).

######  Corollary 5.3.

Let $X$ be a compact metric space and let $\mathcal{B}$ be the Borel
$\sigma$-algebra of $X$. Let $\mathsf{X}_{i}=(X,\mathcal{B},\mu_{i},T_{i})$,
$1\leq i\leq k$, be ergodic probability measure preserving systems. Suppose
that $\mu_{1},\dots,\mu_{k}$ are pairwise equivalent and
$\mathsf{X}_{1},\dots,\mathsf{X}_{k}$ are mutually disjoint. Then
$T_{1},\cdots,T_{k}$ are pointwise jointly ergodic with respect to
$(\mu_{1},\dots,\mu_{k})$.

###### Proof.

Take first $k=2$. Since $\mathsf{X}_{1}$ and $\mathsf{X}_{2}$ are disjoint, by
Theorem 5.2, $T_{1},T_{2}$ are pointwise jointly ergodic. Take now $k=3$.
Since $\mathsf{X}_{1},\mathsf{X}_{2}$ and $\mathsf{X}_{3}$ are mutually
disjoint, $\mathsf{X}_{1}\times\mathsf{X}_{2}$ and $\mathsf{X}_{3}$ are
disjoint and so by Theorem 5.2, $T_{1},T_{2},T_{3}$ are pointwise jointly
ergodic. And so on. ∎

######  Theorem 5.4.

Let $X$ be a compact metric space and let $\mathcal{B}$ be the Borel
$\sigma$-algebra of $X$. Let $\mu_{1},\dots,\mu_{l},\nu_{1},\dots,\nu_{m}$ be
pairwise equivalent probability measures on $(X,\mathcal{B})$. Let
$\mathsf{X}_{i}^{(1)}=(X,\mathcal{B},\mu_{i},S_{i})$, $1\leq i\leq l$, and
$\mathsf{X}_{j}^{(2)}=(X,\mathcal{B},\nu_{j},T_{j})$, $1\leq j\leq m$, be
measure preserving systems such that

  1. (1)

each $S_{i}$, $1\leq i\leq l$, has zero entropy,

  2. (2)

each $T_{i}$, $1\leq i\leq m$, is exact.

If $S_{1},\dots,S_{l}$ are pointwise jointly ergodic with respect to
$(\mu_{1},\dots,\mu_{l})$ and $T_{1},\dots,T_{m}$ are pointwise jointly
ergodic with respect to $(\nu_{1},\dots,\nu_{m})$, then
$S_{1},\dots,S_{l},T_{1},\dots,T_{m}$ are pointwise jointly ergodic with
respect to $(\mu_{1},\dots,\mu_{l},\nu_{1},\dots,\nu_{m})$.

###### Proof.

Since a Cartesian product of zero entropy systems has zero entropy,
$S_{1}\times\cdots\times S_{l}$ has zero entropy. It is known ([44]) that
natural extensions of exact transformations are K-automorphisms and a
Cartesian product of K-automorphisms is K-automorphism. Thus,
$T_{1}\times\cdots\times T_{m}$ is a factor of a K-automorphism. It is also
known that K-automorphisms are disjoint from zero entropy systems. Also, if
$\mathsf{X}$ and $\mathsf{Y}$ are disjoint and $\mathsf{Z}$ is a factor of
$\mathsf{X}$, then $\mathsf{Z}$ and $\mathsf{Y}$ are disjoint (Proposition I.1
in [15]). Thus, $S_{1}\times\cdots\times S_{l}$ and $T_{1}\times\cdots\times
T_{m}$ are disjoint, and so Theorem 5.5 follows from 5.2. ∎

Theorems 4.1 and 5.4 lead to the following result.

######  Theorem 5.5.

Let $\mu_{1},\dots,\mu_{k}$ be probability measures on $([0,1],\mathcal{B})$,
where $\mathcal{B}$ is the Borel $\sigma$-algebra of $[0,1]$, such that each
$\mu_{i}$ is equivalent to the Lebesgue measure $\lambda$. For $1\leq i\leq
k$, let $T_{i}:[0,1]\rightarrow[0,1]$ be a $\mu_{i}$-preserving ergodic
transformation. Let $m\leq k$. In addition, assume that

  1. (1)

for $1\leq i\leq m$, $T_{i}$ has zero entropy and $T_{1},\dots,T_{m}$ are
pointwise jointly ergodic,

  2. (2)

for $m+1\leq i\leq k$, $(T_{i},\mu_{i},\mathcal{A}_{i})\in\mathcal{T}$ along
with an appropriate partition $\mathcal{A}_{i}$ such that
$(T_{i},\mu_{i},\mathcal{A}_{i})$ has properties E and M and entropy values
$h(T_{m+1}),\cdots,h(T_{k})$ are distinct.

Then for any bounded measurable functions $f_{1},\dots,f_{k}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{1}({T_{1}^{n}x})f_{2}({T_{2}^{n}x})\cdots
f_{k}({T_{k}^{n}x})=\prod_{i=1}^{k}\int f_{i}\,d\mu_{i}$ (5.1)

for almost every $x\in[0,1]$.

The following extension of Theorem 1.2 (Theorem 4.3) is an immediate corollary
of Theorem 5.5.

######  Corollary 5.6.

Let $\mu_{0}$ be a probability measure on $[0,1]$, which is equivalent to
$\lambda$. Let $S:[0,1]\rightarrow[0,1]$ be a $\mu_{0}$-preserving, ergodic
transformation with $h(S)=0$. Let $s,t\in\mathbb{N}$. Let $b_{1},\dots,b_{s}$
be distinct integers with $b_{i}\geq 2$ for all $i=1,2,\dots,s$ and let
$\beta_{1},\dots,\beta_{t}$ be distinct non-integers with $\beta_{i}>1$ for
all $i=1,2,\dots,t$. Then

  1. (1)

$S,T_{b_{1}},\dots,T_{b_{s}},T_{\beta_{1}},\dots T_{\beta_{t}}$ are pointwise
jointly ergodic:
for any bounded measurable functions
$f_{0},f_{1},\dots,f_{s},g_{1},\dots,g_{t}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{0}(S^{n}x)\prod_{i=1}^{s}f_{i}({b_{i}^{n}x})\prod_{j=1}^{t}g_{j}({T_{\beta_{j}}^{n}x})=\int
f_{0}\,d\mu_{0}\,\prod_{i=1}^{s}\int f_{i}\,d\lambda\,\prod_{j=1}^{t}\int
g_{j}\,d\mu_{\beta_{j}}$

for almost every $x\in[0,1]$.

  2. (2)

$S,T_{G},T_{b_{1}},\dots,T_{b_{s}},T_{\beta_{1}},\dots T_{\beta_{t}}$ are
pointwise jointly ergodic if $\log\beta_{i}\neq\frac{\pi^{2}}{6\log 2}$ for
$1\leq i\leq t$:
for any bounded measurable functions
$f_{0},g_{0},f_{1},\dots,f_{s},g_{1},\dots,g_{t}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f_{0}(S^{n}x)\,g_{0}(T_{G}^{n}x)\,\prod_{i=1}^{s}f_{i}({b_{i}^{n}x})\,\prod_{j=1}^{t}g_{j}({T_{\beta_{j}}^{n}x})=\int
f_{0}\,d\mu_{0}\int g_{0}\,d\mu_{G}\,\prod_{i=1}^{s}\int
f_{i}\,d\lambda\,\prod_{j=1}^{t}\int g_{j}\,d\mu_{\beta_{j}}$

for almost every $x\in[0,1]$.

##  6\. Equivalent forms of normality and joint normality of F-representations

In this section we deal with various equivalent forms of normality and joint
normality, which were briefly discussed in the Introduction. We start with
proving Theorem 1.14 from the Introduction, which we formulate here again for
reader’s convenience.

######  Theorem 6.1 (Theorem 1.14).

Let ${\mathbf{T}}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$ such that $T$ is
totally ergodic. Let $x\in[0,1]$. The following statements are equivalent.

  1. (i)

$x$ is ${\mathbf{T}}$-normal.

  2. (ii)

$(T^{n}x)$ is equidistributed in $[0,1]$ with respect to $\mu$, that is, for
any continuous function $f:[0,1]\rightarrow\mathbb{R}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)=\int f\,d\mu.$
(6.1)

  3. (iii)

For any $m\geq 2$, $x$ is ${\mathbf{T}}_{m}$-normal.

  4. (iv)

$T^{k}x$ is simply ${\mathbf{T}}_{m}$-normal for all $k\geq 0$ and $m\geq 1$.

  5. (v)

$x$ is simply ${\mathbf{T}}_{m}$-normal for all $m\geq 1$.

  6. (vi)

There exists a sequence $(m_{l})_{l\in\mathbb{N}}$ with $m_{1}<m_{2}<\cdots$
such that $x$ is simply ${\mathbf{T}}_{m_{l}}$-normal for any
$l\in\mathbb{N}$.

The proof of Theorem 6.1 rests on two useful lemmas which are of independent
interest. Before formulating and proving these lemmas we need to introduce
some additional terminology.

Let $X$ be a compact metric space and let $M(X)$ denote the space of
probability Borel measures on $X$. For $\mu\in M(X)$, let $\mathcal{R}(X,\mu)$
denote the space of Riemann integrable functions on $X$. Note that
$f\in\mathcal{R}(X,\mu)$ if and only if $\mu(D_{f})=0$, where $D_{f}$ is the
set of points of discontinuity of $f$. Also, note that if the sequence
$\mu_{1},\mu_{2},\dots\in M(X)$ converges to the measure $\mu$ in weak*
topology, then, by the standard approximation argument, for any
$f\in\mathcal{R}(X,\mu)$,

$\int f\,d\mu_{n}\rightarrow\int f\,d\mu.$

We will say that a measure preserving system $(X,\mathcal{B},\mu,T)$ is
admissible if $X$ is a compact metric space, $\mathcal{B}$ is the
$\sigma$-algebra of Borel sets of $X$, $\mu$ is a probability measure on
$\mathcal{B}$, and $T:X\rightarrow X$ is a $\mu$-preserving transformation
with the property that the set $S$ of points of discontinuity of $T$ satisfies
$\mu(S)=0$. Following [19], we will call any such transformation
$\mu$-continuous. Note that if $(X,\mathcal{B},\mu,T)$ is an admissible
system, so is $(X,\mathcal{B},\mu,T^{m})$ for any $m\geq 2$.

Given an admissible system $(X,\mathcal{B},\mu,T)$ and a point $x\in X$ we
will say that $x$ is $(T,\mu)$-generic if for any continuous function $f$ on
$X$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)=\int f\,d\mu,$

or, equivalently,

$\frac{1}{N}\sum_{n=0}^{N-1}\delta_{T^{n}x}\rightarrow\mu\,\,\text{ in weak*
topology.}$

Note that a point $x\in X$ is $(T,\mu)$-generic if and only if for any
$f\in\mathcal{R}(X,\mu)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)=\int f\,d\mu.$

######  Remark 6.2.

Let $\mathbf{T}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$. Then
$([0,1],\mathcal{B},\mu,T)$, where $\mathcal{B}$ is the Borel $\sigma$-algebra
of $[0,1]$, is an admissible system. Moreover, the following statements are
equivalent.

  1. (i)

$x\in[0,1]$ is $(T,\mu)$-generic (i.e. for any $f\in C([0,1])$,
$\lim\limits_{N\rightarrow\infty}\frac{1}{N}\sum\limits_{n=0}^{N-1}f(T^{n}x)=\int
f\,d\mu$).

  2. (ii)

For any interval $I\subset[0,1]$,

$\lim_{N\rightarrow\infty}\frac{\\#\\{0\leq n\leq N-1:T^{n}x\in
I\\}}{N}=\mu(I).$ (6.2)

(One has only to justify (ii) $\Rightarrow$ (i). It is not hard to see that
this implication follows form the fact that linear combinations of
characteristic functions of intervals are dense in $\mathcal{R}([0,1],\mu)$.)

The following lemma is a natural dynamical version of a classical result due
to Pyatetskii-Shapiro. (See Theorem 2 in [40]. See also Theorem 3 in [39] and
Lemma 8.1 in Chapter 1 of [27].)

######  Lemma 6.3.

Let $(X,\mathcal{B},\mu,T)$ be an ergodic admissible system. If $x\in X$ has
the property that there is a constant $C>0$ such that for any non-negative
continuous function $f$ on $X$

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)\leq C\int
f\,d\mu,$ (6.3)

then $x$ is $(T,\mu)$-generic.

###### Proof.

Define a sequence of measures in $M(X)$

$\nu_{N}:=\frac{1}{N}\sum_{n=0}^{N-1}\delta_{T^{n}x},\quad N\in\mathbb{N}.$

It is sufficient to show that $\nu_{N}\rightarrow\mu$ in weak* topology.

Let $\nu$ be a weak* limit point of the sequence $(\nu_{N})$ and let
$(\nu_{N_{k}})$ be a subsequence of $(\nu_{N})$ such that
$\nu_{N_{k}}\rightarrow\nu$. Note that by (6.3), for any nonnegative
continuous function $f$ on $X$,

$\int fd\nu=\lim_{k\rightarrow\infty}\int
f\,d\nu_{N_{k}}=\lim_{k\rightarrow\infty}\frac{1}{N_{k}}\sum_{n=0}^{N_{k}-1}f(T^{n}x)\leq
C\int fd\mu,$

so $\nu\ll\mu$ and hence $\mathcal{R}(X,\mu)\subset\mathcal{R}(X,\nu)$.

Now let us show that $\int f\,d\nu=\int f\circ T\,d\nu$ for any $f\in C(X)$.
Note that since $T$ is $\mu$-continuous, for any $f\in C(X)$, $f\circ
T\in\mathcal{R}(X,\mu)\subset\mathcal{R}(X,\nu)$, and so

$\ \int f\,d\nu_{N_{k}}\rightarrow\int f\,d\nu\quad\text{and}\quad\int f\circ
T\,d\nu_{N_{k}}\rightarrow\int f\circ T\,d\nu.$

Since

$\left|\int f\,d\nu_{N_{k}}-\int f\circ
T\,d\nu_{N_{k}}\right|=\left|\frac{1}{N_{k}}\sum_{n=0}^{N_{k}-1}\left(f(T^{n}x)-f(T^{n+1}x)\right)\right|\leq\frac{2}{N_{k}}\|f\|_{u}\rightarrow
0\quad\text{as }k\rightarrow\infty,$

we get $\int f\,d\nu=\int f\circ T\,d\nu$ for any $f\in C(X)$. This, in turn,
implies that for any $f\in C(X)$

$\int f\circ T^{n}d\nu=\int fd\nu\,\text{ for any }n\in\mathbb{N}.$ (6.4)

Fix a continuous function $f$ on $X$. Since $T$ is ergodic, the pointwise
ergodic theorem implies

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}y)=\int f\,d\mu$
(6.5)

for $\mu-\text{a.e. }y$. Moreover, the fact that $\nu\ll\mu$ implies that
(6.5) also holds for $\nu-\text{a.e. }y$. Note that

$\left|\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}y)\right|\leq\|f\|_{u}\quad\text{for
any }y\in X.$ (6.6)

Using (6.4) and (6.6) and the fact that (6.5) holds for $\nu-\text{a.e. }y$,
we get with the help of the dominated convergence theorem

$\displaystyle\int f\,d\nu$
$\displaystyle=\int\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}y)d\nu(y)=\lim_{N\rightarrow\infty}\int\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}y)d\nu(y)=\int\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}y)d\nu(y)$
$\displaystyle=\int\int f\,d\mu\,d\nu=\int fd\mu.$

Thus $\nu=\mu$, which implies that any limit point of $(\nu_{N})$ is $\mu$,
and so $\nu_{N}\rightarrow\mu$ in weak* topology. ∎

######  Lemma 6.4.

Let $(X,\mathcal{B},\mu,T)$ be a totally ergodic admissible system. Let $m\geq
2$. Then $x\in X$ is $(T,\mu)$-generic if and only if $x$ is
$(T^{m},\mu)$-generic.

###### Proof.

Suppose that $x$ is $(T,\mu)$-generic. Note that for any non-negative
continuous function $f$ on $X$,

$\frac{1}{N}\sum_{n=0}^{N-1}f(T^{mn}x)\leq
m\frac{1}{mN}\sum_{n=0}^{mN-1}f(T^{n}x),$

and hence

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{mn}x)\leq m\int
f\,d\mu.$

By Lemma 6.3, $x$ is $(T^{m},\mu)$-generic.

Now suppose that $x$ is $(T^{m},\mu)$-generic. Let $f$ be a non-negative
continuous function on $X$. Then for any $N\in\mathbb{N}$ we have

$\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)\leq\frac{[N/m]+1}{N}\sum_{r=0}^{m-1}\frac{1}{[N/m]+1}\sum_{n=0}^{[N/m]}f\circ
T^{r}(T^{mn}x).$

Since $T$ is $\mu$-continuous, $f\circ T^{r}$ is Riemann integrable for each
$r=0,1,\dots,m-1$. Therefore

$\lim_{N\rightarrow\infty}\frac{1}{[N/m]+1}\sum_{n=0}^{[N/m]}f\circ
T^{r}(T^{mn}x)=\int f\circ T^{r}\,d\mu=\int f\,d\mu.$

Hence

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}f(T^{n}x)\leq\int
f\,d\mu,$

which, by Lemma 6.3, implies $x$ is $(T,\mu)$-generic. ∎

###### Proof of Theorem 6.1.

For $\mathbf{T}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$, write
$\mathcal{A}=\\{I_{j}:j\in D\\}$, where $D$ is a finite or countably infinite
index set. For a string $S=(a_{1},\dots,a_{k})\in D^{k}$, let
$C_{\mathbf{T}}(S)$ be the cylinder set in $\mathcal{A}^{k}$ corresponding to
$S$. Let $1_{C_{\mathbf{T}}(S)}$ denote the characteristic function of
$C_{\mathbf{T}}(S)$. For $x\in[0,1]$, let $(r_{n}(x))$ be the F-representation
of $x$. Note that for any $k\in\mathbb{N}$ and $S=(a_{1},\dots,a_{k})\in
D^{k}$,

$\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)=\\#\\{1\leq n\leq
N:r_{n}(x)=a_{1},\dots,r_{n+k-1}(x)=a_{k}\\}.$ (6.7)

(i) $\Rightarrow$ (ii): Let $x\in[0,1]$ be $\mathbf{T}$-normal. Then (6.7)
implies that for any $k\in\mathbb{N}$ and $S\in D^{k}$

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)=\int
1_{C_{\mathbf{T}}(S)}\,d\mu.$ (6.8)

By linearity and the monotone convergence theorem, (6.8) implies that for any
$m\in\mathbb{N}$ and $I\in\sigma(\mathcal{A}^{m})$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}1_{I}(T^{n}x)=\int
1_{I}\,d\mu.$ (6.9)

Since the partition $\mathcal{A}$ generates the $\sigma$-algebra
$\mathcal{B}$, the standard approximation argument implies that (6.9) holds
for any interval in $[0,1]$, and so, by Remark 6.2, $x$ is $(T,\mu)$-generic.

(ii) $\Rightarrow$ (i): If $x\in[0,1]$ is $(T,\mu)$-generic, then (6.7) and
(6.2) imply that for any $k\in\mathbb{N}$ and $S=(a_{1},\dots,a_{k})\in D^{k}$

$\lim_{N\rightarrow\infty}\frac{\\#\\{1\leq n\leq
N:r_{n}(x)=a_{1},\dots,r_{n+k-1}(x)=a_{k}\\}}{N}=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)=\mu(C_{\mathbf{T}}(S)),$

so $x$ is $\mathbf{T}$-normal.

(i) $\Leftrightarrow$ (iii) follows from Lemma 6.4.

(i) $\Rightarrow$ (iv): If $x$ is $\mathbf{T}$-normal, then clearly $T^{k}x$
is $\mathbf{T}$-normal for any $k\geq 0$. Then since (i) is equivalent to
(iii), $T^{k}x$ is $\mathbf{T}_{m}$-normal, which gives (iv).

It is clear that (iv) $\Rightarrow$ (v) $\Rightarrow$ (vi).

We will conclude the proof by showing (vi) $\Rightarrow$ (i). Let us assume
that the index set $D$ is countably infinite (the proof for the case when $D$
is finite is analogous and is omitted). By (6.8), it is enough to show that
for any $k\in\mathbb{N}$ and $S=(a_{1},\dots,a_{k})\in D^{k}$

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)=\mu(C_{\mathbf{T}}(S)).$
(6.10)

Let $m_{l}$ be a natural number with $m_{l}>k$. Let $(A_{i})_{i=1}^{\infty}$
be an enumeration of strings in $D^{m_{l}}$. Since $C_{\mathbf{T}}(A_{i})\cap
C_{\mathbf{T}}(A_{j})=\emptyset$ for $i\neq j$ and
$[0,1]=\bigcup\limits_{i=1}^{\infty}C_{\mathbf{T}}(A_{i})\,(\bmod\,\lambda)$,
we have that $\sum\limits_{i=1}^{\infty}\mu(C_{\mathbf{T}}(A_{i}))=1.$ Let
$\epsilon>0$. Take $M\in\mathbb{N}$ such that
$\sum\limits_{i=M+1}^{\infty}\mu(C_{\mathbf{T}}(A_{i}))<\epsilon$.

For $i\in\mathbb{N}$, let $k_{i}\in\mathbb{N}$ denote the number of
occurrences of the string $S$ in the string $A_{i}$. Note that $0\leq
k_{i}\leq m_{l}-k+1$ for all $i=1,2,\dots,$ and

$\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)=\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x)+O\left(\frac{N}{m_{l}}k\right).$
(6.11)

We have

$\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x)=\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=1}^{M}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x)+\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=M+1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x).$
(6.12)

Since $x$ is simply ${\mathbf{T}}^{m_{l}}$-normal,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=1}^{M}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x)=\frac{1}{m_{l}}\sum_{i=1}^{M}k_{i}\mu({C_{\mathbf{T}}(A_{i})}).$
(6.13)

Since for any $i\in\mathbb{N}$, $k_{i}\leq m_{l}$, we get the following
estimate of the second term on the right hand side of (6.12):

$\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=M+1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x)\leq
m_{l}\cdot\\#\\{0\leq
n<\frac{N}{m_{l}}:T^{m_{l}n}x\notin\bigcup_{i=1}^{M}C_{\mathbf{T}}(A_{i})\\}.$

We have

$\displaystyle\lim_{N\rightarrow\infty}\frac{m_{l}}{N}\cdot\\#\\{0\leq
n<\frac{N}{m_{l}}:T^{m_{l}n}x\notin\bigcup_{i=1}^{M}C_{\mathbf{T}}(A_{i})\\}$
$\displaystyle=1-\lim_{N\rightarrow\infty}\frac{m_{l}}{N}\cdot\\#\\{0\leq
n<\frac{N}{m_{l}}:T^{m_{l}n}x\in\bigcup_{i=1}^{M}C_{\mathbf{T}}(A_{i})\\}$
$\displaystyle=1-\sum_{i=1}^{M}\mu(C_{\mathbf{T}}(A_{i}))=\sum_{i=M+1}^{\infty}\mu(C_{\mathbf{T}}(A_{i}))<\epsilon.$

So

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{0\leq
n<\frac{N}{m_{l}}}\sum_{i=M+1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(T^{m_{l}n}x)\leq\epsilon.$
(6.14)

Thus, by (6.11), (6.12), (6.13) and (6.14),

$\limsup_{N\rightarrow\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)-\frac{1}{m_{l}}\sum_{i=1}^{M}k_{i}\mu(C_{\mathbf{T}}(A_{i}))\right|=O\left(\epsilon+\frac{k}{m_{l}}\right).$
(6.15)

Invoking the definition of $k_{i}$, we have for almost every $y\in[0,1)$

$\sum_{i=1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(y)=\sum_{j=0}^{m_{l}-k}1_{C_{\mathbf{T}}(S)}\circ
T^{j}(y),$ (6.16)

and hence

$0\leq\sum_{j=0}^{m_{l}-k}1_{C_{\mathbf{T}}(S)}\circ
T^{j}(y)-\sum_{i=1}^{M}k_{i}1_{C_{\mathbf{T}}(A_{i})}(y)=\sum_{i=M+1}^{\infty}k_{i}1_{C_{\mathbf{T}}(A_{i})}(y).$
(6.17)

Since $\int 1_{C_{\mathbf{T}}(S)}\circ T^{j}\,d\mu=\mu(C_{\mathbf{T}}(S))$ for
$0\leq j\leq m_{l}-k$ and $k_{i}\leq m_{l}$ for all $i\in\mathbb{N}$, from
(6.17) we get

$0\leq(m_{l}-k)\mu(C_{\mathbf{T}}(S))-\sum\limits_{i=1}^{M}k_{i}\mu(C_{\mathbf{T}}(A_{i}))\leq
m_{l}\sum\limits_{i=M+1}^{\infty}\mu(C_{\mathbf{T}}(A_{i}))\leq
m_{l}\epsilon,$

and so

$0\leq\left(1-\frac{k}{m_{l}}\right)\mu(C_{\mathbf{T}}(S))-\frac{1}{m_{l}}\sum\limits_{i=1}^{M}k_{i}\mu(C_{\mathbf{T}}(A_{i}))\leq\epsilon,$

which in turn implies that

$\left|\mu(C_{\mathbf{T}}(S))-\frac{1}{m_{l}}\sum\limits_{i=1}^{M}k_{i}\mu(C_{\mathbf{T}}(A_{i}))\right|=O\left(\epsilon+\frac{k}{m_{l}}\right).$
(6.18)

From (6.15) and (6.18) we get

$\limsup_{N\rightarrow\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)-\mu(C_{\mathbf{T}}(S))\right|=O\left(\epsilon+\frac{k}{m_{l}}\right).$
(6.19)

Since $\epsilon$ is arbitrary, (6.19) implies

$\limsup_{N\rightarrow\infty}\left|\frac{1}{N}\sum_{n=0}^{N-1}1_{C_{\mathbf{T}}(S)}(T^{n}x)-\mu(C_{\mathbf{T}}(S))\right|=O\left(\frac{k}{m_{l}}\right).$
(6.20)

Taking $m_{l}\rightarrow\infty$ in (6.20) gives us (6.10). This concludes the
proof of Theorem 6.1. ∎

######  Remark 6.5.

Let $b\in\mathbb{N},b\geq 2$ and let
$x=\sum\limits_{n=1}^{\infty}\frac{x_{n}}{b^{n}}$ be a $b$-normal number. A
classical theorem due to Wall [50] states that for any $m\in\mathbb{N}$ and
$k\in\mathbb{N}\cup\\{0\\}$, the number
$y=\sum\limits_{n=1}^{\infty}\frac{x_{mn+k}}{b^{n}}$ is also $b$-normal. This
result easily follows form Theorem 1.12 (which is a rather special corollary
of Theorem 6.1). Indeed, Theorem 1.12 implies that if $x$ is normal in base
$b$, then $b^{k}x$ is normal in base $b^{m}$, and so for any $t\in\mathbb{N}$
and any string $S=(v_{1},\dots,v_{t})$ in $\\{0,1,\dots,b-1\\}^{t}$ we have

$\displaystyle\\{0\leq n\leq
N-1:x_{mn+k}=v_{1},x_{m(n+1)+k}=v_{2},\dots,x_{m(n+t-1)+k}=v_{t}\\}$
$\displaystyle\quad=\\{0\leq n\leq N-1:T_{b}^{mn}(T_{b}^{k}x)\in[v_{1}]\cap
T_{b}^{-m}[v_{2}]\cap\cdots\cap T_{b}^{-m(t-1)}[v_{t}]\\}.$

Hence

$\displaystyle\lim_{N\rightarrow\infty}\frac{\\#\\{0\leq n\leq
N-1:x_{mn+k}=v_{1},x_{m(n+1)+k}=v_{2},\dots,x_{m(n+t-1)+k}=v_{t}\\}}{N}$
$\displaystyle\quad\quad=\lambda([v_{1}]\cap T_{b}^{-m}[v_{2}]\cap\cdots\cap
T_{b}^{-m(t-1)}[v_{t}])$
$\displaystyle\quad\quad=\lambda([v_{1},v_{2},\dots,v_{t}])$ (6.21)

Wall’s theorem leads to the question whether a similar phenomenon of
normality-preservation along arithmetic progressions takes place for general
$\mathbf{T}$-normal sequences. This question has, in general, a negative
answer. Indeed, while for any
${\mathbf{T}}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$, we have that
$\mathbf{T}$-normality of $x$ implies the $\mathbf{T}_{m}$-normality of
$T^{k}x$ for any $m\in\mathbb{N}$ and $k\in\mathbb{N}\cup\\{0\\}$, the last
equality in (6.5) may not hold for
${\mathbf{T}}=(T,\mu,\mathcal{A})\in\tilde{\mathcal{T}}$, so arithmetic
progression may not preserve ${\mathbf{T}}$-normality. In fact, it was shown
([22]) that CF-normality is not preserved along arithmetic progression. (See
also [3] for more discussion on the failure of normality preservation.)

We conclude this section by discussing a version of Theorem 6.1 for joint
normality.

######  Theorem 6.6 (Theorem 1.15).

Let
${\mathbf{T}}^{(i)}=(T_{i},\mu_{i},\mathcal{A}_{i})\in\tilde{\mathcal{T}},\,i=1,2\dots,k$
such that $T_{1}\times\cdots\times T_{k}$ is totally ergodic with respect to
$\mu_{1}\times\cdots\times\mu_{k}$. Let $x\in[0,1]$. The following are
equivalent.

  1. (i)

$x$ is jointly $({\mathbf{T}}^{(1)},\dots,{\mathbf{T}}^{(k)})$-normal.

  2. (ii)

$k$-tuple $(x,\dots,x)$ is $(T_{1}\times\cdots\times
T_{k},\mu_{1}\times\cdots\times\mu_{k})$-generic:
$(T_{1}^{n}x,\dots,T_{k}^{n}x)$ is equidistributed in $[0,1]^{k}$ with respect
to $\mu_{1}\times\cdots\times\mu_{k}$.

  3. (iii)

For any $m\geq 2$, $x$ is jointly
$({\mathbf{T}}^{(1)}_{m},\dots,{\mathbf{T}}^{(k)}_{m})$-normal.

  4. (iv)

$x$ is jointly simply
$({\mathbf{T}}^{(1)}_{m},\dots,{\mathbf{T}}^{(k)}_{m})$-normal for all
$m\in\mathbb{N}$.

  5. (v)

There exists a sequence $(m_{l})_{l\in\mathbb{N}}$ with $m_{1}<m_{2}<\cdots$
such that $x$ is jointly simply
$({\mathbf{T}}^{(1)}_{m_{l}},\dots,{\mathbf{T}}^{(k)}_{m_{l}})$-normal.

Similarly to the proof of Theorem 6.1, which is based on Lemma 6.3 and Lemma
6.4, Theorem 6.6 follows from two lemmas, which are natural extensions of
Lemma 6.3 and Lemma 6.4.

Before formulating these lemmas, observe that if
$(X,\mathcal{B},\mu_{i},T_{i})$ is an admissible system for $1\leq i\leq k$,
then so is the product system
$(X^{k},\otimes_{1}^{k}\mathcal{B},\mu_{1}\times\cdots\times\mu_{k},T_{1}\times\cdots\times
T_{k})$. So the following two lemmas are corollaries of, correspondingly,
Lemma 6.3 and Lemma 6.4 applied to the product system
$(X^{k},\otimes_{1}^{k}\mathcal{B},\mu_{1}\times\cdots\times\mu_{k},T_{1}\times\cdots\times
T_{k})$.

######  Lemma 6.7.

Let $(X,\mathcal{B},\mu_{i},T_{i})$ be an ergodic admissible system for $1\leq
i\leq k$. Assume that $T_{1}\times\cdots\times T_{k}$ is ergodic with respect
to $\mu_{1}\times\cdots\times\mu_{k}$.

If $x\in X$ has the property that there is a constant $C>0$ such that for any
non-negative continuous functions $f_{1},\dots,f_{k}$,

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{k}f_{i}(T_{i}^{n}x)\leq
C\prod_{i=1}^{k}\int f_{i}\,d\mu_{i},$

then the $k$-tuple $(x,\dots,x)$ is $(T_{1}\times\cdots\times
T_{k},\mu_{1}\times\cdots\times\mu_{k})$-generic.

######  Lemma 6.8.

Let $(X,\mathcal{B},\mu_{i},T_{i})$ be an ergodic admissible system for $1\leq
i\leq k$ and assume that $T_{1}\times\cdots\times T_{k}$ is totally ergodic
with respect to $\mu_{1}\times\cdots\times\mu_{k}$. Let $x\in X$ and let
$m\geq 2$. Then the $k$-tuple $(x,\dots,x)$ is $(T_{1}\times\cdots\times
T_{k},\mu_{1}\times\cdots\times\mu_{k})$-generic if and only if the $k$-tuple
$(x,\dots,x)$ is $(T_{1}^{m}\times\cdots\times
T_{k}^{m},\mu_{1}\times\cdots\times\mu_{k})$-generic

The proof of Theorem 6.6 follows now from Lemmas 6.7 and 6.8 and is totally
analogous to the proof of Theorem 6.1 above.

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