# univoque bases and Hausdorff dimensions

Derong Kong  School of Mathematical Science, Yangzhou University, Yangzhou,
JiangSu 225002, People’s Republic of China Mathematical Institute, University
of Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands<EMAIL_ADDRESS>,
Wenxia Li  Department of Mathematics, Shanghai Key Laboratory of PMMP, East
China Normal University, Shanghai 200062, People’s Republic of China
<EMAIL_ADDRESS>,  Fan Lü  Department of Mathematics, Sichuan Normal
University, Chengdu 610068, People’s Republic of China<EMAIL_ADDRESS>and
Martijn de Vries  Tussen de Grachten 213, 1381 DZ, Weesp, The Netherlands
<EMAIL_ADDRESS>
###### Abstract.

Given a positive integer $M$ and a real number $q>1$, a _$q$ -expansion_ of a
real number $x$ is a sequence $(c_{i})=c_{1}c_{2}\cdots$ with
$(c_{i})\in\\{0,\ldots,M\\}^{\infty}$ such that

$x=\sum_{i=1}^{\infty}c_{i}q^{-i}.$

It is well known that if $q\in(1,M+1]$, then each $x\in
I_{q}:=\left[0,M/(q-1)\right]$ has a $q$-expansion. Let
$\mathcal{U}=\mathcal{U}(M)$ be the set of _univoque bases_ $q>1$ for which
$1$ has a unique $q$-expansion. The main object of this paper is to provide
new characterizations of $\mathcal{U}$ and to show that the Hausdorff
dimension of the set of numbers $x\in I_{q}$ with a unique $q$-expansion
changes the most if $q$ “crosses” a univoque base.

Denote by $\mathcal{B}_{2}=\mathcal{B}_{2}(M)$ the set of $q\in(1,M+1]$ such
that there exist numbers having precisely two distinct $q$-expansions. As a
by-product of our results, we obtain an answer to a question of Sidorov (2009)
and prove that

$\dim_{H}(\mathcal{B}_{2}\cap(q^{\prime},q^{\prime}+\delta))>0\quad\textrm{for
any}\quad\delta>0,$

where $q^{\prime}=q^{\prime}(M)$ is the Komornik-Loreti constant.

###### Key words and phrases:

univoque bases and univoque sets and Hausdorff dimensions and generalized
Thue-Morse sequences

###### 1991 Mathematics Subject Classification:

11A63 and 37B10 and 28A78

##  1\. Introduction

Non-integer base expansions have received much attention since the pioneering
works of Rényi [26] and Parry [25]. Given a positive integer $M$ and a real
number $q\in(1,M+1]$, a sequence $(d_{i})=d_{1}d_{2}\cdots$ with _digits_
$d_{i}\in\left\\{0,1,\ldots,M\right\\}$ is called a _$q$ -expansion of $x$_ or
an _expansion of $x$ in base $q$_ if

$x=\pi_{q}((d_{i})):=\sum_{i=1}^{\infty}\frac{d_{i}}{q^{i}}.$

It is well known that each $x\in I_{q}:=[0,M/(q-1)]$ has a $q$-expansion. One
such expansion - the _greedy $q$-expansion_ \- can be obtained by performing
the so called _greedy algorithm_ of Rényi which is defined recursively as
follows: if $d_{1},\ldots,d_{n-1}$ is already defined (no condition if $n=1$),
then $d_{n}$ is the largest element of $\left\\{0,\ldots,M\right\\}$
satisfying $\sum_{i=1}^{n}d_{i}q^{-i}\leq x$. Equivalently, $(d_{i})$ is the
greedy $q$-expansion of $\sum_{i=1}^{\infty}d_{i}q^{-i}$ if and only if
$\sum_{i=n+1}^{\infty}d_{i}q^{-i+n}<1$ whenever $d_{n}<M$, $n=1,2,\ldots$.
Hence if $1<q<r\leq M+1$, then the greedy $q$-expansion of a number $x\in
I_{q}$ is also the greedy expansion in base $r$ of a number in $I_{r}$.

Let ${{\mathcal{U}}}_{q}$ be the _univoque set_ consisting of numbers $x\in
I_{q}$ such that $x$ has a unique $q$-expansion, and let
${{\mathcal{U}}}_{q}^{\prime}$ be the set of corresponding expansions. Note
that a sequence $(c_{i})$ belongs to ${{\mathcal{U}}}_{q}^{\prime}$ if and
only if both the sequences $(c_{i})$ and $(M-c_{i}):=(M-c_{1})(M-c_{2})\cdots$
are greedy $q$-expansions, hence
${{\mathcal{U}}}_{q}^{\prime}\subseteq{{\mathcal{U}}}_{r}^{\prime}$ whenever
$1<q<r\leq M+1$. Many works are devoted to the univoque sets
${{\mathcal{U}}}_{q}$ (see, e.g., [11, 13, 15]). Recently, de Vries and
Komornik investigated their topological properties in [9]. Komornik et al.
considered their Hausdorff dimension in [17], and showed that the dimension
function $D:q\mapsto\dim_{H}{{\mathcal{U}}}_{q}$ behaves like a Devil’s
staircase on $(1,M+1]$. For more information on the univoque set
${{\mathcal{U}}}_{q}$ we refer to the survey paper [16] and the references
therein.

There is an intimate connection between the set ${{\mathcal{U}}}_{q}$ and the
set of _univoque bases_ ${{\mathcal{U}}}={{\mathcal{U}}}(M)$ consisting of
numbers $q>1$ such that $1$ has a unique $q$-expansion over the alphabet
$\left\\{0,1,\ldots,M\right\\}$. For instance, it was shown in [9] that
${{\mathcal{U}}}_{q}$ is closed if and only if $q$ does not belong to the set
$\overline{{{\mathcal{U}}}}$. It is well-known that ${{\mathcal{U}}}$ is a
Lebesgue null set of full Hausdorff dimension (cf. [7, 12, 17]). Moreover, the
smallest element of ${{\mathcal{U}}}$ is the _Komornik-Loreti constant_ (cf.
[18, 19])

$q^{\prime}=q^{\prime}(M),$

while the largest element of ${{\mathcal{U}}}$ is (of course) $M+1$. Recently,
Komornik and Loreti showed in [20] that its closure
$\overline{{{\mathcal{U}}}}$ is a _Cantor set_ (see also, [10]), i.e., a
nonempty closed set having neither isolated nor interior points. Writing the
open set
$(1,M+1]\setminus\overline{{{\mathcal{U}}}}=(1,M+1)\setminus\overline{{{\mathcal{U}}}}$
as the disjoint union of its connected components, i.e.,

(1.1)
$(1,M+1]\setminus\overline{{{\mathcal{U}}}}=(1,q^{\prime})\cup\bigcup(q_{0},q_{0}^{*}),$

the left endpoints $q_{0}$ in (1.1) run over the whole set
$\overline{{{\mathcal{U}}}}\setminus{{\mathcal{U}}}$, and the right endpoints
$q_{0}^{*}$ run through a subset of ${{\mathcal{U}}}$ (cf. [9]). Furthermore,
each left endpoint $q_{0}$ is algebraic, while each right endpoint
$q_{0}^{*}\in{{\mathcal{U}}}$ is transcendental (cf. [21]).

De Vries showed in [8], roughly speaking, that the sets
${{\mathcal{U}}}_{q}^{\prime}$ change the most if we cross a univoque base.
More precisely, it was shown that $q\in{{\mathcal{U}}}$ if and only if
${{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime}$ is
uncountable for each $r\in(q,M+1]$ and $r\in\overline{{\mathcal{U}}}$ if and
only if ${{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime}$ is
uncountable for each $q\in(1,r)$.

The main object of this paper is to provide similar characterizations of
${{\mathcal{U}}}$ and $\overline{{{\mathcal{U}}}}$ in terms of the Hausdorff
dimension of the sets
${{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime}$ after a
natural projection. Furthermore, we characterize the sets ${{\mathcal{U}}}$
and $\overline{{{\mathcal{U}}}}$ by looking at the Hausdorff dimensions of
${{\mathcal{U}}}$ and $\overline{{{\mathcal{U}}}}$ locally.

######  Theorem 1.1.

Let $q\in(1,M+1]$. The following statements are equivalent.

  1. (i)

$q\in{{\mathcal{U}}}$.

  2. (ii)

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})>0$
for any $r\in(q,M+1]$.

  3. (iii)

$\dim_{H}{{\mathcal{U}}}\cap(q,r)>0$ for any $r\in(q,M+1]$.

######  Theorem 1.2.

Let $q\in(1,M+1]$. The following statements are equivalent.

  1. (i)

$q\in\overline{{{\mathcal{U}}}}\setminus(\bigcup\left\\{q_{0}^{*}\right\\}\cup\left\\{q^{\prime}\right\\})$.

  2. (ii)

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q}^{\prime}\setminus{{\mathcal{U}}}_{p}^{\prime})>0$
for any $p\in(1,q)$.

  3. (iii)

$\dim_{H}{{\mathcal{U}}}\cap(p,q)>0$ for any $p\in(1,q)$.

It follows at once from Theorems 1.1 and 1.2 that ${{\mathcal{U}}}$ (or,
equivalently, $\overline{{{\mathcal{U}}}}$) does not contain isolated points.

We remark that the projection map $\pi_{M+1}$ in Theorem 1.1 (ii) can be
replaced by $\pi_{\rho}$ for any $r\leq\rho\leq M+1$. Similarly, the
projection map $\pi_{M+1}$ in Theorem 1.2 (ii) can also be replaced by
$\pi_{\rho}$ with $q\leq\rho\leq M+1$. We also point out that Theorems 1.1 and
1.2 strengthen the main result of [8] where the cardinality of the sets
${{\mathcal{U}}}_{q}^{\prime}\setminus{{\mathcal{U}}}_{p}^{\prime}$ with
$1<p<q\leq M+1$ was determined.

Let $\mathcal{B}_{2}$ be the set of bases $q\in\left(1,M+1\right]$ for which
there exists a number $x\in[0,M/(q-1)]$ having exactly two $q$-expansions. It
was asked by Sidorov [27] whether
$\dim_{H}\mathcal{B}_{2}\cap(q^{\prime},q^{\prime}+\delta)>0$ for any
$\delta>0$, where $q^{\prime}$ is the Komornik-Loreti constant. Since
${{\mathcal{U}}}\subseteq\mathcal{B}_{2}$ (see [27, Lemma 3.1]111This also
follows directly from the observation that $q^{-1}$ has exactly two
$q$-expansions whenever $q\in{{\mathcal{U}}}$.), Theorem 1.1 answers this
question in the affirmative.

######  Corollary 1.3.

$\dim_{H}\mathcal{B}_{2}\cap(q^{\prime},q^{\prime}+\delta)>0$ for any
$\delta>0.$

The rest of the paper is arranged as follows. In Section 2 we recall some
properties of unique $q$-expansions. The proof of Theorems 1.1 and 1.2 will be
given in Section 3.

##  2\. Preliminaries

In this section we recall some properties of the univoque set
${{\mathcal{U}}}_{q}$. Throughout this paper, a _sequence_
$(d_{i})=d_{1}d_{2}\cdots$ is an element of
$\left\\{0,\ldots,M\right\\}^{\infty}$ with each digit $d_{i}$ belonging to
the _alphabet_ $\left\\{0,\ldots,M\right\\}$. Moreover, for a _word_
${\mathbf{c}}=c_{1}\cdots c_{n}$ we mean a finite string of digits with each
digit $c_{i}$ from $\left\\{0,\ldots,M\right\\}$. For two words
${\mathbf{c}}=c_{1}\cdots c_{n}$ and $\mathbf{d}=d_{1}\cdots d_{m}$ we denote
by ${\mathbf{c}}\mathbf{d}=c_{1}\cdots c_{n}d_{1}\cdots d_{m}$ the
concatenation of the two words. For an integer $k\geq 1$ we denote by
${\mathbf{c}}^{k}$ the $k$-times concatenation of ${\mathbf{c}}$ with itself,
and by ${\mathbf{c}}^{\infty}$ the infinite repetition of ${\mathbf{c}}$.

For a sequence $(d_{i})$ we denote its _reflection_ by
$\overline{(d_{i})}:=(M-d_{1})(M-d_{2})\cdots$. Accordingly, for a word
${\mathbf{c}}=c_{1}\cdots c_{n}$ we denote its reflection by
$\overline{{\mathbf{c}}}:=(M-c_{1})\cdots(M-c_{n})$. If $c_{n}<M$ we denote by
${\mathbf{c}}^{+}:=c_{1}\cdots c_{n-1}(c_{n}+1).$ If $c_{n}>0$ we write
${\mathbf{c}}^{-}:=c_{1}\cdots c_{n-1}(c_{n}-1)$.

We will use systematically the lexicographic ordering $<,\leq,>$ and $\geq$
between sequences and between words. For two sequences
$(c_{i}),(d_{i})\in\left\\{0,1,\ldots,M\right\\}^{\infty}$ we say that
$(c_{i})<(d_{i})$ if there exists an integer $n\geq 1$ such that $c_{1}\ldots
c_{n-1}=d_{1}\ldots d_{n-1}$ and $c_{n}<d_{n}$. Furthermore, we write
$(c_{i})\leq(d_{i})$ if $(c_{i})<(d_{i})$ or $(c_{i})=(d_{i})$. Similarly, we
say $(c_{i})>(d_{i})$ if $(d_{i})<(c_{i})$, and $(c_{i})\geq(d_{i})$ if
$(d_{i})\leq(c_{i})$. We extend this definition to words in the obvious way.
For example, for two words ${\mathbf{c}}$ and $\mathbf{d}$ we write
${\mathbf{c}}<\mathbf{d}$ if ${\mathbf{c}}0^{\infty}<\mathbf{d}0^{\infty}$.

A sequence is called _finite_ if it has a last nonzero element. Otherwise it
is called _infinite_. So $0^{\infty}:=00\cdots$ is considered to be infinite.
For $q\in(1,M+1]$ we denote by

$\alpha(q)=(\alpha_{i}(q))$

the _quasi-greedy_ $q$-expansion of $1$ (cf. [6]), i.e., the lexicographically
largest _infinite_ $q$-expansion of $1$. Let $\beta(q)=(\beta_{i}(q))$ be the
_greedy_ $q$-expansion of $1$ (cf. [25]), i.e., the lexicographically largest
$q$-expansion of $1$. For convenience, we set $\alpha(1)=0^{\infty}$ and
$\beta(1)=10^{\infty}$, even though $\alpha(1)$ is not a $1$-expansion of $1$.

Moreover, we endow the set $\left\\{0,\ldots,M\right\\}$ with the discrete
topology and the set of all possible sequences
$\left\\{0,1,\ldots,M\right\\}^{\infty}$ with the Tychonoff product topology.

The following properties of $\alpha(q)$ and $\beta(q)$ were established in
[25], see also [4].

######  Lemma 2.1.

  * (i)

The map $q\mapsto\alpha(q)$ is an increasing bijection from $[1,M+1]$ onto the
set of all infinite sequences $(\alpha_{i})$ satisfying

$\alpha_{n+1}\alpha_{n+2}\cdots\leq\alpha_{1}\alpha_{2}\cdots\quad\textrm{whenever}\quad\alpha_{n}<M.$

  * (ii)

The map $q\mapsto\beta(q)$ is an increasing bijection from $[1,M+1]$ onto the
set of all sequences $(\beta_{i})$ satisfying

$\beta_{n+1}\beta_{n+2}\cdots<\beta_{1}\beta_{2}\cdots\quad\textrm{whenever}\quad\beta_{n}<M.$

######  Lemma 2.2.

  * (i)

$\beta(q)$ is infinite if and only if $\beta(q)=\alpha(q)$.

  * (ii)

If $\beta(q)=\beta_{1}\cdots\beta_{m}0^{\infty}$ with $\beta_{m}>0$, then
$\alpha(q)=(\beta_{1}\cdots\beta_{m}^{-})^{\infty}.$

  * (iii)

The map $q\mapsto\alpha(q)$ is left-continuous, while the map
$q\mapsto\beta(q)$ is right-continuous.

In order to investigate the unique expansions we need the following
lexicographic characterization of ${{\mathcal{U}}}_{q}^{\prime}$ (cf. [4]).

######  Lemma 2.3.

Let $q\in(1,M+1]$. Then $(d_{i})\in{{\mathcal{U}}}_{q}^{\prime}$ if and only
if

$\left\\{\begin{array}[]{lll}d_{n+1}d_{n+2}\cdots<\alpha_{1}(q)\alpha_{2}(q)\cdots&\textrm{
whenever }&d_{n}<M,\\\
d_{n+1}d_{n+2}\cdots>\overline{\alpha_{1}(q)\alpha_{2}(q)\cdots}&\textrm{
whenever}&d_{n}>0.\end{array}\right.$

Note that $q\in{{\mathcal{U}}}$ if and only if $\alpha(q)$ is the unique
$q$-expansion of $1$. Then Lemma 2.3 yields a characterization of
${{\mathcal{U}}}$ (see also, [11] and [19]).

######  Lemma 2.4.

Let $q\in(1,M+1)$. Then $q\in{{\mathcal{U}}}$ if and only if
$\alpha(q)=(\alpha_{i}(q))$ satisfies

$\overline{\alpha(q)}<\alpha_{n+1}(q)\alpha_{n+2}(q)\cdots<\alpha(q)\quad\textrm{for
all}\quad n\geq 1.$

Consider a connected component $(q_{0},q_{0}^{*})$ of
$(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$ as in (1.1). Then there
exists a (unique) word ${\mathbf{t}}=t_{1}\cdots t_{p}$ such that (cf. [9,
21])

$\alpha(q_{0})={\mathbf{t}}^{\infty}\quad\textrm{and}\quad\alpha(q_{0}^{*})=\lim_{n\rightarrow\infty}g^{n}({\mathbf{t}}),$

where $g^{n}=\underbrace{g\circ\cdots\circ g}_{n}$ denotes the $n$-fold
composition of $g$ with itself, and

(2.1)
$g({\mathbf{c}}):={\mathbf{c}}^{+}\overline{{\mathbf{c}}^{+}}\quad\textrm{for
any word}\quad{\mathbf{c}}=c_{1}\ldots c_{k}\textrm{ with }c_{k}<M.$

We point out that the word ${\mathbf{t}}=t_{1}\ldots t_{p}$ in the definitions
of $\alpha(q_{0})$ and $\alpha(q_{0}^{*})$ is called an _admissible block_ in
[21, Definition 2.1] which satisfies the following lexicographical
inequalities: $t_{p}<M$ and for any $1\leq i\leq p$ we have

$\overline{t_{1}\cdots t_{p}}\leq t_{i}\cdots t_{p}t_{1}\cdots
t_{i-1}\quad\textrm{and}\quad t_{i}\cdots t_{p}\;\overline{t_{1}\ldots
t_{i-1}}\leq t_{1}\ldots t_{p}^{+}.$

We also mention that the limit $\lim_{n\rightarrow\infty}g^{n}({\mathbf{t}})$
stands for the infinite sequence beginning with
${\mathbf{t}}^{+}\overline{{\mathbf{t}}}\,\overline{{\mathbf{t}}^{+}}{\mathbf{t}}^{+}\;\overline{{\mathbf{t}}^{+}}{\mathbf{t}}\,{\mathbf{t}}^{+}\overline{{\mathbf{t}}}\cdots,$
and the existence of this limit was shown by Allouche [1].

In this case $(q_{0},q_{0}^{*})$ is called the _connected component generated
by ${\mathbf{t}}$_. The closed interval $[q_{0},q_{0}^{*}]$ is the so called
_admissible interval generated by ${\mathbf{t}}$_ (see [21, Definition 2.4]).
Furthermore, the sequence

$\alpha(q_{0}^{*})=\lim_{n\rightarrow\infty}g^{n}({\mathbf{t}})={\mathbf{t}}^{+}\,\overline{{\mathbf{t}}}\;\overline{{\mathbf{t}}^{+}}\,{\mathbf{t}}^{+}~{}\overline{{\mathbf{t}}^{+}}\,{\mathbf{t}}\,{\mathbf{t}}^{+}\,\overline{{\mathbf{t}}}\cdots$

is a generalized Thue-Morse sequence (cf. [21, Definition 2.2], see also [3]).

The following lemma for the generalized Thue-Morse sequence
$\alpha(q_{0}^{*})$ was established in [21, Lemma 4.2].

######  Lemma 2.5.

Let
$(q_{0},q_{0}^{*})\subset(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$
be a connected component generated by $t_{1}\cdots t_{p}$. Then the sequence
$(\theta_{i})=\alpha(q_{0}^{*})$ satisfies

$\overline{\theta_{1}\cdots\theta_{2^{n}p-i}}<\theta_{i+1}\cdots\theta_{2^{n}p}\leq\theta_{1}\cdots\theta_{2^{n}p-i}$

for any $n\geq 0$ and any $0\leq i<2^{n}p$.

Finally, we recall some topological properties of ${{\mathcal{U}}}$ and
$\overline{{{\mathcal{U}}}}$ which were essentially established in [9, 20]
(see also, [10]).

######  Lemma 2.6.

  * (i)

If $q\in{{\mathcal{U}}}$, then there exists a decreasing sequence $(r_{n})$ of
elements in $\bigcup\left\\{q_{0}^{*}\right\\}$ that converges to $q$ as
$n\rightarrow\infty$;

  * (ii)

If
$q\in\overline{{{\mathcal{U}}}}\setminus(\bigcup\left\\{q_{0}^{*}\right\\}\cup\left\\{q^{\prime}\right\\})$,
then there exists an increasing sequence $(p_{n})$ of elements in
$\bigcup\left\\{q_{0}^{*}\right\\}$ that converges to $q$ as
$n\rightarrow\infty$.

We remark here that the bases $q_{0}^{*}$ are called _de Vries-Komornik
numbers_ which were shown to be transcendental in [21]. By Lemma 2.6 it
follows that the set of de Vries-Komornik numbers is dense in
$\overline{{{\mathcal{U}}}}$.

##  3\. Proofs of Theorems 1.1 and 1.2

###  3.1.  Proof of Theorem 1.1 for (i) $\Leftrightarrow$ (ii).

For each connected component $(q_{0},q_{0}^{*})$ of
$(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$ we construct a sequence
of bases $(r_{n})$ in ${{\mathcal{U}}}$ strictly decreasing to $q_{0}^{*}$.

######  Lemma 3.1.

Let
$(q_{0},q_{0}^{*})\subset(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$
be a connected component generated by $t_{1}\cdots t_{p}$, and let
$(\theta_{i})=\alpha(q_{0}^{*})$. Then for each $n\geq 1$, the number
$r_{n}\in{{\mathcal{U}}}$ determined by

$\alpha(r_{n})=\beta(r_{n})=\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty},$

belongs to ${{\mathcal{U}}}$. Furthermore, $(r_{n})$ is a strictly decreasing
sequence that converges to $q_{0}^{*}$.

###### Proof.

Using (2.1) one may verify that the sequence $(\theta_{i})$ satisfies

$\theta_{2^{n}p+k}=\overline{\theta_{k}}\quad\textrm{for all}\quad 1\leq
k<2^{n}p;\quad\theta_{2^{n+1}p}=\overline{\theta_{2^{n}p}}\,^{+}$

for all $n\geq 0$. Now fix $n\geq 1$. We claim that

(3.1)
$\sigma^{i}\left(\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty}\right)<\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty}$

for all $i\geq 1$, where $\sigma$ is the left shift on
$\left\\{0,\ldots,M\right\\}^{\infty}$ defined by $\sigma((c_{i}))=(c_{i+1})$.
By periodicity it suffices to prove (3.1) for $0<i<2^{n+1}p$. We distinguish
between the following three cases: (I) $0<i<2^{n}p$; (II) $i=2^{n}p$; (III)
$2^{n}p<i<2^{n+1}p$.

Case (I). $0<i<2^{n}p$. Then by Lemma 2.5 it follows that

$\theta_{i+1}\cdots\theta_{2^{n}p}\leq\theta_{1}\cdots\theta_{2^{n}p-i}$

and

$\theta_{2^{n}p+1}\cdots\theta_{2^{n}p+i}=\overline{\theta_{1}\cdots\theta_{i}}<\theta_{2^{n}p-i+1}\cdots\theta_{2^{n}p}.$

This implies (3.1) for $0<i<2^{n}p$.

Case (II). $i=2^{n}p$. Note by [19] that $\alpha_{1}(q^{\prime})=[M/2]+1$ (see
also, [5]), where $[y]$ denotes the integer part of a real number $y$. Then by
using $q_{0}^{*}>q^{\prime}$ in Lemma 2.1 we have

$\theta_{1}=\alpha_{1}(q_{0}^{*})\geq\alpha_{1}(q^{\prime})>\overline{\alpha_{1}(q^{\prime})}\geq\overline{\theta_{1}}.$

This, together with $n\geq 1$, implies

$\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p}=\overline{\theta_{1}\cdots\theta_{2^{n}p}}\,^{+}<\theta_{1}\cdots\theta_{2^{n}p}.$

So, (3.1) holds true for $i=2^{n}p$.

Case (III). $2^{n}p<i<2^{n+1}p$. Write $j=i-2^{n}p$. Then $0<j<2^{n}p$. Once
again, we infer from Lemma 2.5 that

$\theta_{i+1}\cdots\theta_{2^{n+1}p}=\overline{\theta_{j+1}\cdots\theta_{2^{n}p}}\,^{+}\leq\theta_{1}\cdots\theta_{2^{n}p-j}$

and

$\theta_{2^{n}p+1}\cdots\theta_{2^{n}p+j}=\overline{\theta_{1}\cdots\theta_{j}}<\theta_{2^{n}p-j+1}\cdots\theta_{2^{n}p}.$

This yields (3.1) for $2^{n}p<i<2^{n+1}p$.

Note by Lemma 2.5 that

$\sigma^{i}\left(\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty}\right)>\overline{\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty}}$

for any $i\geq 0$. Then by (3.1) and Lemma 2.4 it follows that there exists
$r_{n}\in{{\mathcal{U}}}$ such that

$\alpha(r_{n})=\beta(r_{n})=\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty}.$

In the following we prove $r_{n}\searrow q_{0}^{*}$ as $n\rightarrow\infty$.
For $n\geq 1$ we observe that

$\displaystyle\beta(r_{n+1})=\theta_{1}\cdots\theta_{2^{n+1}p}(\theta_{2^{n+1}p+1}\cdots\theta_{2^{n+2}p})^{\infty}$
$\displaystyle=\theta_{1}\cdots\theta_{2^{n}p}\overline{\theta_{1}\cdots\theta_{2^{n}p}}\,^{+}\overline{\theta_{1}\cdots\theta_{2^{n}p}}\cdots$
$\displaystyle<\theta_{1}\cdots\theta_{2^{n}p}\,(\overline{\theta_{1}\cdots\theta_{2^{n}p}}^{+})^{\infty}=\beta(r_{n}).$

Then by Lemma 2.1 (ii) we have $r_{n+1}<r_{n}$. Note that
$\beta(q_{0}^{*})=\alpha(q_{0}^{*})=(\theta_{i})$, and

$\beta(r_{n})\rightarrow(\theta_{i})=\beta(q_{0}^{*})\quad\textrm{as}\quad
n\rightarrow\infty.$

Hence, we conclude from Lemma 2.2 (iii) that $r_{n}\searrow q_{0}^{*}$ as
$n\rightarrow\infty$. ∎

######  Lemma 3.2.

Let
$(q_{0},q_{0}^{*})\subset(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$
be a connected component generated by $t_{1}\cdots t_{p}$, and let
$(\theta_{i})=\alpha(q_{0}^{*})$. Then for any $n\geq 1$ and any $0\leq
i<2^{n}p$ we have

(3.2)
$\begin{split}&\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\xi_{n}\overline{\xi_{n}})<\theta_{1}\cdots\theta_{2^{n+1}p-i},\\\
&\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\xi_{n}\overline{\xi_{n}^{-}})\leq\theta_{1}\cdots\theta_{2^{n+1}p-i},\\\
&\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\xi_{n}^{-}\xi_{n})<\theta_{1}\cdots\theta_{2^{n+1}p-i},\end{split}$

and thus (by symmetry),

$\displaystyle\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\overline{\xi_{n}}\xi_{n})<\theta_{1}\cdots\theta_{2^{n+1}p-i},$
$\displaystyle\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}\leq\sigma^{i}(\overline{\xi_{n}}{\xi_{n}^{-}})<\theta_{1}\cdots\theta_{2^{n+1}p-i},$
$\displaystyle\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\overline{\xi_{n}^{-}}\,\overline{\xi_{n}})<\theta_{1}\cdots\theta_{2^{n+1}p-i},$

where $\xi_{n}:=\theta_{1}\cdots\theta_{2^{n}p}$.

###### Proof.

By symmetry it suffices to prove (3.2).

Note that $\xi_{n}\overline{\xi_{n}}=\theta_{1}\cdots\theta_{2^{n+1}p}^{-}$
and $\xi_{n}\overline{\xi_{n}^{-}}=\theta_{1}\cdots\theta_{2^{n+1}p}.$ Then by
Lemma 2.5 it follows that

$\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\xi_{n}\overline{\xi_{n}})<\theta_{1}\cdots\theta_{2^{n+1}p-i}$

and

$\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\xi_{n}\overline{\xi_{n}^{-}})\leq\theta_{1}\cdots\theta_{2^{n+1}p-i}$

for any $0\leq i<2^{n}p$.

So, it suffices to prove the inequalities

(3.3)
$\overline{\theta_{1}\cdots\theta_{2^{n+1}p-i}}<\sigma^{i}(\theta_{1}\cdots\theta_{2^{n}p}^{-}\theta_{1}\cdots\theta_{2^{n}p})<\theta_{1}\cdots\theta_{2^{n+1}p-i}$

for any $0\leq i<2^{n}p$. By Lemma 2.5 it follows that for any $0\leq
i<2^{n}p$ we have

$\overline{\theta_{1}\cdots\theta_{2^{n}p-i}}\leq\theta_{i+1}\cdots\theta_{2^{n}p}^{-}<\theta_{1}\cdots\theta_{2^{n}p-i}$

and

$\theta_{1}\cdots\theta_{i}>\overline{\theta_{2^{n}p-i+1}\cdots\theta_{2^{n}p}}.$

This proves (3.3). ∎

######  Lemma 3.3.

Let
$(q_{0},q_{0}^{*})\subset(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$
be a connected component generated by $t_{1}\cdots t_{p}$. Then
$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}^{*}}^{\prime})>0$
for any $r\in(q_{0}^{*},M+1]$.

###### Proof.

Take $r\in(q_{0}^{*},M+1]$. By Lemma 3.1 there exists $n\geq 1$ such that

$r_{n}\in(q_{0}^{*},{r})\cap{{\mathcal{U}}}.$

Write $(\theta_{i})=\alpha(q_{0}^{*})$ and let
$\xi_{n}=\theta_{1}\cdots\theta_{2^{n}p}$. Denote by $X_{A}^{(n)}$ the
subshift of finite type over the states
$\left\\{\xi_{n},\xi_{n}^{-},\overline{\xi_{n}},\overline{\xi_{n}^{-}}\right\\}$
with adjacency matrix

$A=\left(\begin{array}[]{cccc}0&0&1&1\\\ 1&0&0&0\\\ 1&1&0&0\\\
0&0&1&0\end{array}\right).$

Note that
$\alpha(r_{n})=\theta_{1}\cdots\theta_{2^{n}p}(\theta_{2^{n}p+1}\cdots\theta_{2^{n+1}p})^{\infty}$.
Then by Lemmas 3.2 and 2.3 it follows that

(3.4)
$X_{A}^{(n)}\subseteq{{\mathcal{U}}}_{r_{n}}^{\prime}\subseteq{{\mathcal{U}}}_{r}^{\prime}.$

Furthermore, note that

$\displaystyle\xi_{n}\overline{\xi_{n}^{-}}(\overline{\xi_{n}}\xi_{n})^{3}$
$\displaystyle=\theta_{1}\cdots\theta_{2^{n+1}p}(\overline{\theta_{1}\cdots\theta_{2^{n+1}p}}\,^{+})^{3}$
$\displaystyle=\theta_{1}\cdots\theta_{2^{n+2}p}(\overline{\theta_{1}\cdots\theta_{2^{n+1}p}}\,^{+})^{2}$
$\displaystyle>\theta_{1}\cdots\theta_{2^{n+2}p}\overline{\theta_{1}\cdots\theta_{2^{n+1}p}\theta_{2^{n+1}p+1}\cdots\theta_{2^{n+2}p}}\,^{+}$
$\displaystyle=\theta_{1}\cdots\theta_{2^{n+2}p}\theta_{2^{n+2}p+1}\cdots\theta_{2^{n+3}p}.$

Then by Lemmas 2.3 and 3.1 it follows that any sequence starting at

${\mathbf{c}}:=\xi_{n}^{-}\xi_{n}\overline{\xi_{n}^{-}}(\overline{\xi_{n}}\xi_{n})^{3}$

can not belong to ${{\mathcal{U}}}_{r_{n+2}}^{\prime}$. Therefore, by (3.4) we
obtain

(3.5) $\begin{split}X_{A}^{(n)}({\mathbf{c}})&:=\left\\{(d_{i})\in
X_{A}^{(n)}:d_{1}\cdots d_{(2^{n+3}+2^{n})p}={\mathbf{c}}\right\\}\subseteq
X_{A}^{(n)}\setminus{{\mathcal{U}}}_{r_{n+2}}^{\prime}\subset{{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}^{*}}^{\prime}.\end{split}$

Note that the subshift of finite type $X_{A}^{(n)}$ is irreducible (cf. [23]),
and the image $\pi_{M+1}(X_{A}^{(n)})$ is a graph-directed set satisfying the
open set condition (cf. [24]). Then by (3.5) it follows that

$\displaystyle\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}^{*}}^{\prime})$
$\displaystyle\geq\dim_{H}\pi_{M+1}(X_{A}^{(n)}({\mathbf{c}}))$
$\displaystyle=\dim_{H}\pi_{M+1}(X_{A}^{(n)})=\frac{\log\left((1+\sqrt{5})/2\right)}{2^{n}p\log(M+1)}>0.$

∎

The following lemma can be shown in a way which resembles closely the analysis
in [22, Page 2829–2830]. For the sake of completeness we include a sketch of
its proof.

######  Lemma 3.4.

Let
$(q_{0},q_{0}^{*})\subset(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$
be a connected component. Then
$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q_{0}^{*}}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}}^{\prime})=0$.

###### Sketch of the proof.

Suppose that $(q_{0},q_{0}^{*})$ is a connected component generated by
${\mathbf{t}}=t_{1}\cdots t_{p}$. Then

(3.6)
$\alpha(q_{0})={\mathbf{t}}^{\infty}\quad\textrm{and}\quad\alpha(q_{0}^{*})=\lim_{n\rightarrow\infty}g^{n}({\mathbf{t}})={\mathbf{t}}^{+}\,\overline{{\mathbf{t}}}\;\overline{{\mathbf{t}}^{+}}\,{\mathbf{t}}^{+}\cdots,$

where $g(\cdot)$ is defined in (2.1).

For $n\geq 0$ let $\omega_{n}:=g^{n}({\mathbf{t}})^{+}$. Take
$(d_{i})\in{{\mathcal{U}}}_{q_{0}^{*}}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}}^{\prime}$.
Then by using (3.6) and Lemma 2.3 it follows that there exists $m\geq 1$ such
that

(3.7) ${\mathbf{t}}^{\infty}=\alpha(q_{0})\leq
d_{m+1}d_{m+2}\cdots<\alpha(q_{0}^{*})={\mathbf{t}}^{+}\overline{{\mathbf{t}}}\cdots,$

or symmetrically,

(3.8)
${\mathbf{t}}^{\infty}=\alpha(q_{0})\leq\overline{d_{m+1}d_{m+2}\cdots}<\alpha(q_{0}^{*})={\mathbf{t}}^{+}\overline{{\mathbf{t}}}\cdots.$

Suppose $(d_{m+i})\neq{\mathbf{t}}^{\infty}$ and
$(d_{m+i})\neq\overline{{\mathbf{t}}^{\infty}}$. Then there exists $u\geq m$
such that

$d_{u+1}\cdots d_{u+p}={\mathbf{t}}^{+}=\omega_{0}\quad\textrm{or}\quad
d_{u+1}\cdots d_{u+p}=\overline{{\mathbf{t}}^{+}}=\overline{\omega_{0}}.$

  * •

If $d_{u+1}\cdots d_{u+p}=\omega_{0}={\mathbf{t}}^{+}$, then by (3.7) and
Lemma 2.3 it follows that

$d_{u+p+1}\cdots d_{u+2p}=\overline{{\mathbf{t}}^{+}}\quad\textrm{or}\quad
d_{u+p+1}\cdots d_{u+2p}=\overline{{\mathbf{t}}}.$

This implies $d_{u+1}\cdots
d_{u+2p}={\mathbf{t}}^{+}\overline{{\mathbf{t}}^{+}}=\omega_{0}\,\overline{\omega_{0}}$
or $d_{u+1}\cdots
d_{u+2p}={\mathbf{t}}^{+}\overline{{\mathbf{t}}}=\omega_{1}.$

  * •

If $d_{u+1}\cdots d_{u+p}=\overline{\omega_{0}}=\overline{{\mathbf{t}}^{+}}$,
then by (3.8) and Lemma 2.3 it follows that

$d_{u+p+1}\cdots d_{u+2p}={{\mathbf{t}}^{+}}\quad\textrm{or}\quad
d_{u+p+1}\cdots d_{u+2p}={\mathbf{t}}.$

This yields that $d_{u+1}\cdots d_{u+2p}=\overline{\omega_{0}}\,\omega_{0}$ or
$d_{u+1}\cdots d_{u+2p}=\overline{\omega_{1}}$.

Note that for each $n\geq 0$ the word
$g^{n}({\mathbf{t}})^{+}\,\overline{g^{n}({\mathbf{t}})}$ is a prefix of
$\alpha(q_{0}^{*})$. By iteration of the above arguments, one can show that if
$d_{v+1}\cdots d_{v+2^{n}p}=\omega_{n}$, then $d_{v+1}\cdots
d_{v+2^{n+1}p}=\omega_{n}\overline{\omega_{n}}$ or $\omega_{n+1}$.
Symmetrically, if $d_{v+1}\cdots d_{v+2^{n}p}=\overline{\omega_{n}}$, then
$d_{v+1}\cdots d_{v+2^{n+1}p}=\overline{\omega_{n}}\omega_{n}$ or
$\overline{\omega_{n+1}}$.

Hence, we conclude that $(d_{i})$ must end with

${\mathbf{t}}^{*}(\omega_{i_{0}}\overline{\omega_{i_{0}}})^{*}(\omega_{i_{0}}\overline{\omega_{j_{0}}})^{s_{0}}(\omega_{i_{1}}\overline{\omega_{i_{1}}})^{*}(\omega_{i_{1}}\overline{\omega_{j_{1}}})^{s_{1}}\cdots(\omega_{i_{n}}\overline{\omega_{i_{n}}})^{*}(\omega_{i_{n}}\overline{\omega_{j_{n}}})^{s_{n}}\cdots$

or its reflections, where $s_{n}\in\left\\{0,1\right\\}$ and

$0=i_{0}<j_{0}\leq i_{1}<j_{1}\leq i_{2}<\cdots\leq i_{n}<j_{n}\leq
i_{n+1}<\cdots.$

Here $*$ is an element of the set
$\left\\{0,1,2,\ldots\right\\}\cup\left\\{\infty\right\\}$.

Since the length of $\omega_{n}=g^{n}({\mathbf{t}})^{+}$ grows exponentially
fast as $n\rightarrow\infty$, we conclude that
$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q_{0}^{*}}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}}^{\prime})=0$.
∎

###### Proof of Theorem 1.1 for (i) $\Leftrightarrow$ (ii).

First we prove (i) $\Rightarrow$ (ii). If $q=q_{0}^{*}$ is the right endpoint
of a connected component of
$(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$, then by Lemma 3.3 we
have

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})>0\quad\textrm{for
any }\quad r\in(q,M+1].$

Clearly, it is trivial when $q=M+1$. Now we take
$q\in({{\mathcal{U}}}\setminus\left\\{M+1\right\\})\setminus\bigcup\left\\{q_{0}^{*}\right\\}$
and take $r\in(q,M+1]$. By Lemma 2.6 (i) one can find $q_{0}^{*}\in(q,r)$, and
therefore by Lemma 3.3 we obtain

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})\geq\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}^{*}}^{\prime})>0.$

Now we prove (ii) $\Rightarrow$ (i). Take
$q\in(1,M+1]\setminus{{\mathcal{U}}}$. We will show that
$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})=0$
for some $r\in(q,M+1]$. Note that
$\bigcup\left\\{q_{0}\right\\}=\overline{{{\mathcal{U}}}}\setminus{{\mathcal{U}}}$.
Then by (1.1) it follows that

$q\in(1,q^{\prime})\cup\bigcup[q_{0},q_{0}^{*}).$

Therefore, it suffices to prove
$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})=0$
for some $r\in(q,M+1]$. We distinct the following two cases.

Case (I). $q\in(1,q^{\prime})$. Then for any $r\in(q,q^{\prime})$ we have

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})\leq\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime})=0,$

where the last equality follows by [22, Theorem 4.6] (see also, [5, 15]).

Case (II). $q\in[q_{0},q_{0}^{*})$. Then for any $r\in(q,q_{0}^{*})$ we have
by Lemma 3.4 that

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{r}^{\prime}\setminus{{\mathcal{U}}}_{q}^{\prime})\leq\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q_{0}^{*}}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}}^{\prime})=0.$

∎

###  3.2.  Proof of Theorem 1.1 for (i) $\Leftrightarrow$ (iii).

The following property for the Hausdorff dimension is well-known (cf. [14,
Proposition 2.3]).

######  Lemma 3.5.

Let $f:(X,d_{1})\rightarrow(Y,d_{2})$ be a map between two metric spaces . If
there exist constants $C>0$ and $\lambda>0$ such that

$d_{2}(f(x),f(y))\leq Cd_{1}(x,y)^{\lambda}$

for any $x,y\in X$, then $\dim_{H}X\geq\lambda\dim_{H}f(X)$.

######  Lemma 3.6.

Let $q\in{{{\mathcal{U}}}}\setminus\left\\{M+1\right\\}$. Then for any
$r\in(q,M+1)$ we have

$\dim_{H}{{\mathcal{U}}}\cap(q,r)\geq\dim_{H}\pi_{M+1}\left(\left\\{\alpha(p):p\in{{\mathcal{U}}}\cap(q,r)\right\\}\right).$

###### Proof.

Fix $q\in{{{\mathcal{U}}}}\setminus\left\\{M+1\right\\}$ and $r\in(q,M+1)$.
Then Lemma 2.6 yields that ${{\mathcal{U}}}\cap(q,r)$ contains infinitely many
elements. Take $p_{1},p_{2}\in{{\mathcal{U}}}\cap(q,r)$ with $p_{1}<p_{2}$.
Then by Lemma 2.1 we have $\alpha(p_{1})<\alpha(p_{2})$. So, there exists
$n\geq 1$ such that

(3.9)
$\alpha_{1}(p_{1})\cdots\alpha_{n-1}(p_{1})=\alpha_{1}(p_{2})\cdots\alpha_{n-1}(p_{2})\quad\textrm{and}\quad\alpha_{n}(p_{1})<\alpha_{n}(p_{2}).$

This implies

(3.10)
$\begin{split}\pi_{M+1}(\alpha(p_{2}))-\pi_{M+1}(\alpha(p_{1}))&=\sum_{i=1}^{\infty}\frac{\alpha_{i}(p_{2})-\alpha_{i}(p_{1})}{(M+1)^{i}}\\\
&\leq\sum_{i=n}^{\infty}\frac{M}{(M+1)^{i}}=(M+1)^{1-n}.\end{split}$

Note that $r<M+1$. By Lemma 2.1 we have $\alpha(r)<\alpha(M+1)=M^{\infty}$.
Then there exists $N\geq 1$ such that

$\alpha_{1}(r)\cdots\alpha_{N}(r)<\underbrace{M\cdots M}_{N}.$

Therefore, by (3.9) and Lemma 2.3 we obtain

$\sum_{i=1}^{n}\frac{\alpha_{i}(p_{2})}{p_{1}^{i}}\geq\sum_{i=1}^{\infty}\frac{\alpha_{i}(p_{1})}{p_{1}^{i}}=1=\sum_{i=1}^{\infty}\frac{\alpha_{i}(p_{2})}{p_{2}^{i}}>\sum_{i=1}^{n}\frac{\alpha_{i}(p_{2})}{p_{2}^{i}}+\frac{1}{p_{2}^{n+N}}.$

Note that $p_{1},p_{2}$ are elements of ${{\mathcal{U}}}$. Then
$p_{2}>p_{1}\geq q^{\prime}$. This implies

$\displaystyle\frac{1}{(M+1)^{n+N}}$
$\displaystyle<\frac{1}{p_{2}^{n+N}}<\sum_{i=1}^{n}\left(\frac{\alpha_{i}(p_{2})}{p_{1}^{i}}-\frac{\alpha_{i}(p_{2})}{p_{2}^{i}}\right)$
$\displaystyle\leq\sum_{i=1}^{\infty}\left(\frac{M}{p_{1}^{i}}-\frac{M}{p_{2}^{i}}\right)=\frac{M(p_{2}-p_{1})}{(p_{1}-1)(p_{2}-1)}\leq\frac{M(p_{2}-p_{1})}{(q^{\prime}-1)^{2}}.$

Therefore, by (3.10) it follows that

$\pi_{M+1}(\alpha(p_{2}))-\pi_{M+1}(\alpha(p_{1}))\leq(M+1)^{1-n}\leq\frac{(M+1)^{2+N}}{(q^{\prime}-1)^{2}}(p_{2}-p_{1}).$

Furthermore, by Lemma 2.1 it follows that
$\pi_{M+1}(\alpha(p_{2}))-\pi_{M+1}(\alpha(p_{1}))\geq 0$. Hence, by using

$f=\pi_{M+1}\circ\alpha:\quad{{\mathcal{U}}}\cap(q,r)\rightarrow\pi_{M+1}(\left\\{\alpha(p):p\in{{\mathcal{U}}}\cap(q,r)\right\\})$

in Lemma 3.5 we establish the lemma. ∎

######  Lemma 3.7.

Let $(q_{0},q_{0}^{*})$ be a connected component of
$(q^{\prime},M+1)\setminus\overline{{{\mathcal{U}}}}$. Then
$\dim_{H}{{\mathcal{U}}}\cap(q_{0}^{*},r)>0$ for any $r\in(q_{0}^{*},M+1]$.

###### Proof.

Suppose that $(q_{0},q_{0}^{*})$ is a connected component generated by
$t_{1}\cdots t_{p}$. Let $(\theta_{i})=\alpha(q_{0}^{*})$. For $n\geq 2$ we
write $\xi_{n}=\theta_{1}\cdots\theta_{2^{n}p}$, and denote by

$\Gamma_{n}^{\prime}:=\left\\{(d_{i}):d_{1}\cdots
d_{2^{n+1}p}=\xi_{n-1}(\overline{\xi_{n-1}}\,^{+})^{3},\quad(d_{2^{n+1}p+i})\in
X_{A}^{(n)}(\overline{\xi_{n}})\right\\}.$

Here $X_{A}^{(n)}(\overline{\xi_{n}})$ is the follower set of
$\overline{\xi_{n}}$ in the subshift of finite type $X_{A}^{(n)}$ defined in
(3.5). Now we claim that any sequence $(d_{i})\in\Gamma_{n}^{\prime}$
satisfies

(3.11) $\overline{(d_{i})}<\sigma^{j}((d_{i}))<(d_{i})\quad\textrm{for
all}\quad j\geq 1.$

Take $(d_{i})\in\Gamma_{n}^{\prime}$. Then we deduce by the definition of
$\Gamma_{n}^{\prime}$ that

(3.12) $d_{1}\cdots
d_{2^{n+1}p+2^{n-1}p}=\theta_{1}\cdots\theta_{2^{n-1}p}(\overline{\theta_{1}\cdots\theta_{2^{n-1}p}}\,^{+})^{3}\,\overline{\theta_{1}\cdots\theta_{2^{n}p}}.$

We will split the proof of (3.11) into the following five cases.

  1. (a)

$1\leq j<2^{n-1}p$. By (3.12) and Lemma 2.5 it follows that

$\overline{\theta_{1}\cdots\theta_{2^{n-1}p-j}}<d_{j+1}\cdots
d_{2^{n-1}p}=\theta_{j+1}\cdots\theta_{2^{n-1}p}\leq\theta_{1}\cdots\theta_{2^{n-1}p-j},$

and

$d_{2^{n-1}p+1}\cdots
d_{2^{n-1}p+j}=\overline{\theta_{1}\cdots\theta_{j}}<\theta_{2^{n-1}p-j+1}\cdots\theta_{2^{n-1}p}.$

This implies that (3.11) holds for all $1\leq j<2^{n-1}p$.

  2. (b)

$2^{n-1}p\leq j<2^{n}p$. Let $k=j-2^{n-1}p$. Then $0\leq k<2^{n-1}p$. Clearly,
if $k=0$, then by using $\theta_{1}>\overline{\theta_{1}}$ and $n\geq 2$ it
yields that

$\overline{\theta_{1}\cdots\theta_{2^{n-1}p}}<d_{j+1}\cdots
d_{2^{n}p}=\overline{\theta_{1}\cdots\theta_{2^{n-1}p}}\,^{+}<\theta_{1}\cdots\theta_{2^{n-1}p}.$

Now we assume $1\leq k<2^{n-1}p$. Then by (3.12) and Lemma 2.5 it follows that

$\overline{\theta_{1}\cdots\theta_{2^{n-1}p-k}}<d_{j+1}\cdots
d_{2^{n}p}=\overline{\theta_{k+1}\cdots\theta_{2^{n-1}p}}\,^{+}\leq\theta_{1}\cdots\theta_{2^{n-1}p-k},$

and

$d_{2^{n}p+1}\cdots
d_{2^{n}p+k}=\overline{\theta_{1}\cdots\theta_{k}}<\theta_{2^{n-1}p-k+1}\cdots\theta_{2^{n-1}p}.$

Therefore, (3.11) holds for all $2^{n-1}p\leq j<2^{n}p$.

  3. (c)

$2^{n}p\leq j<2^{n}p+2^{n-1}p$. Let $k=j-2^{n}p$. Then in a similar way as in
Case (b) one can prove (3.11).

  4. (d)

$2^{n}p+2^{n-1}p\leq j<2^{n+1}p$. Let $k=j-2^{n}p-2^{n-1}p$. Again by the same
arguments as in Case (b) we obtain (3.11).

  5. (e)

$j\geq 2^{n+1}p$. Note that

$d_{1}\cdots
d_{2^{n+1}p}=\theta_{1}\cdots\theta_{2^{n-1}p}(\overline{\theta_{1}\cdots\theta_{2^{n-1}p}}\,^{+})^{3}>\theta_{1}\cdots\theta_{2^{n+1}p}.$

Then (3.11) follows by Lemma 3.2.

Therefore, by (3.11) and Lemma 2.4 it follows that any sequence in
$\Gamma_{n}^{\prime}$ corresponds to a unique base $q\in{{\mathcal{U}}}$.
Furthermore, by (3.12) and Lemma 3.1 each sequence
$(d_{i})\in\Gamma_{n}^{\prime}$ satisfies

$\alpha(q_{0}^{*})=(\theta_{i})<(d_{i})<\theta_{1}\cdots\theta_{2^{n-1}p}(\overline{\theta_{1}\cdots\theta_{2^{n-1}p}}\,^{+})^{\infty}=\alpha(r_{n-1}).$

Then by Lemma 2.1 it follows that

$\alpha(q)\in\Gamma_{n}^{\prime}\quad\Longrightarrow\quad
q\in{{\mathcal{U}}}\cap(q_{0}^{*},r_{n-1}).$

Fix $r>q_{0}^{*}$. So by Lemma 3.1 there exists a sufficiently large integer
$n\geq 2$ such that

(3.13)
$\Gamma_{n}^{\prime}\subset\left\\{\alpha(q):q\in{{\mathcal{U}}}\cap(q_{0}^{*},r)\right\\}.$

Note by the proof of Lemma 3.3 that $X_{A}^{(n)}$ is an irreducible subshift
of finite type over the states
$\left\\{\xi_{n},\xi_{n}^{-},\overline{\xi_{n}},\overline{\xi_{n}^{-}}\right\\}$.
Hence, by (3.13) and Lemma 3.6 it follows that

$\displaystyle\dim_{H}{{\mathcal{U}}}\cap(q_{0}^{*},r)\geq\dim_{H}\pi_{M+1}(\Gamma_{n}^{\prime})$
$\displaystyle=\dim_{H}\pi_{M+1}(X_{A}^{(n)})$
$\displaystyle=\frac{\log\left((1+\sqrt{5})/2\right)}{2^{n}p\log(M+1)}\;>0.$

∎

###### Proof of Theorem 1.1 for (i) $\Leftrightarrow$ (iii).

First we prove (i) $\Rightarrow$ (iii). Excluding the trivial case $q=M+1$ we
take $q\in{{\mathcal{U}}}\setminus\left\\{M+1\right\\}$. Suppose that
$r\in(q,M+1]$. If $q=q_{0}^{*}$, then by Lemma 3.7 we have
$\dim_{H}{{\mathcal{U}}}\cap(q,r)>0.$

If
$q\in({{\mathcal{U}}}\setminus\left\\{M+1\right\\})\setminus\bigcup\left\\{q_{0}^{*}\right\\}$,
then by Lemma 2.6 (i) there exists $q_{0}^{*}\in(q,r)$. So, by Lemma 3.7 we
have

$\dim_{H}{{\mathcal{U}}}\cap(q,r)\geq\dim_{H}{{\mathcal{U}}}\cap(q_{0}^{*},r)>0.$

Now we prove (iii) $\Rightarrow$ (i). Suppose on the contrary that
$q\in(1,M+1]\setminus{{\mathcal{U}}}$. We will show that
${{\mathcal{U}}}\cap(q,r)=\emptyset$ for some $r\in(q,M+1]$. Take
$q\in(1,M+1]\setminus{{\mathcal{U}}}$. By (1.1) it follows that

$q\in(1,q^{\prime})\cap\bigcup[q_{0},q_{0}^{*}).$

This implies that ${{\mathcal{U}}}\cap(q,r)=\emptyset$ for $r\in(q,M+1]$
sufficiently close to $q$. ∎

###  3.3.  Proof of Theorem 1.2.

###### Proof of Theorem 1.2.

(i) $\Rightarrow$ (ii). Take
$q\in\overline{{{\mathcal{U}}}}\setminus(\bigcup\left\\{q_{0}^{*}\right\\}\cup\left\\{q^{\prime}\right\\})$
and $p\in(1,q)$. By Lemma 2.6 (ii) there exists $q_{0}^{*}\in(p,q)$. Hence, by
Lemma 3.3 it follows that

$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q}^{\prime}\setminus{{\mathcal{U}}}_{p}^{\prime})\geq\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q}^{\prime}\setminus{{\mathcal{U}}}_{q_{0}^{*}}^{\prime})>0.$

(ii) $\Rightarrow$ (i). Suppose on the contrary that
$q\notin\overline{{{\mathcal{U}}}}\setminus(\bigcup\left\\{q_{0}^{*}\right\\}\cup\left\\{q^{\prime}\right\\})$.
Then by (1.1) we have

$q\in(1,q^{\prime}]\cup\bigcup(q_{0},q_{0}^{*}].$

By using Lemma 3.4 it follows that for $p\in(1,q)$ sufficiently close to $q$
we have
$\dim_{H}\pi_{M+1}({{\mathcal{U}}}_{q}^{\prime}\setminus{{\mathcal{U}}}_{p}^{\prime})=0$.

(i) $\Rightarrow$ (iii). Take
$q\in\overline{{{\mathcal{U}}}}\setminus(\bigcup\left\\{q_{0}^{*}\right\\}\cup\left\\{q^{\prime}\right\\})$
and $p\in(1,q)$. By Lemma 2.6 (ii) there exists $q_{0}^{*}\in(p,q)$. Hence, by
Lemma 3.7 it follows that

$\dim_{H}{{\mathcal{U}}}\cap(p,q)\geq\dim_{H}{{\mathcal{U}}}\cap(q_{0}^{*},q)>0.$

(iii) $\Rightarrow$ (i). Suppose
$q\notin\overline{{{\mathcal{U}}}}\setminus(\bigcup\left\\{q_{0}^{*}\right\\}\cup\left\\{q^{\prime}\right\\})$.
Then by (1.1) we have $q\in(1,q^{\prime}]\cup\bigcup(q_{0},q_{0}^{*}].$ So,
for $p\in(1,q)$ sufficiently close to $q$ we have
${{\mathcal{U}}}\cap(p,q)=\emptyset$. ∎

## Acknowledgements

The authors thank the anonymous referees for many useful comments. The first
author was supported by NSFC No. 11401516 and Jiangsu Province Natural Science
Foundation for the Youth No. BK20130433. The second author was supported by
NSFC No. 11271137, 11571144, 11671147 and Science and Technology Commission of
Shanghai Municipality (STCSM) No. 13dz2260400. The third author was supported
by NSFC No. 11601358.

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