# A refinement of ternary Boolean algebras

J. P. Fatelo and N. Martins-Ferreira  School of Technology and Management,
Centre for Rapid and Sustainable Product Development - CDRSP, Polytechnic
Institute of Leiria, P-2411-901 Leiria, Portugal<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

An algebraic structure with two constants and one ternary operation, which is
not completely commutative, is put forward to accommodate ternary Boolean
algebras. When the ternary operation is interpreted as Church’s conditioned
disjunction, Boolean algebras are characterized as a subvariety. Different
interpretations for the ternary operation lead to distinct subvarieties. Rings
and near-rings of characteristic 2 are used to illustrate the procedure.

###### Key words and phrases:

Boolean algebra, ternary Boolean algebra, non-symmetric ternary operation,
rings of characteristic two, near rings.

###### 2020 Mathematics Subject Classification:

06E05, 06D30, 03G25, 16Y30

This work is supported by Fundação para a Ciência e a Tecnologia (FCTUID-
Multi-04044-2019), Centro2020 (PAMI – ROTEIRO/0328/2013- 022158) and
Polytechnic of Leiria through the projects CENTRO-01-0247-FEDER: 069665,
069603, 039958, 039969, 039863, 024533 and also ESTG and CDRSP

Ternary Boolean algebras [15] were introduced by Grau in 1947 to axiomatize
Boolean algebras by means of the ternary operation $xy\circ yz\circ zx$ (see
also [26]). In the same year this operation was used independently by Birkhoff
and Kiss [3] to characterize distributive lattices. Both approaches are
particular cases of median algebras [1, 17, 27]. A different characterization
by Whiteman [29] uses ternary rejection
$\bar{x}\bar{y}\circ\bar{y}\bar{z}\circ\bar{z}\bar{x}$ instead. Although the
set of axioms is distinct in each case, complete commutativity is a common
feature. In 1948, Church [7] shows that it is possible to axiomatize Boolean
algebras in terms of the conditioned disjunction $\bar{y}x\circ yz$ which is
not completely commutative (see also [16]). As observed in 1949 by Jordan [18]
_the theory of groups would retain relatively little of its charms and
richness if it were restricted to the consideration of Abelian groups. It is
therefore reasonable to assume that in the theory of lattices as well,
dispensing with the commutative law could lead to a considerable broadening of
the horizon_. A systematic study of non-commutative Boolean algebras and skew
lattices has been reinforced by Leech [20, 21]. Although skew lattices are not
necessarily commutative, there are good reasons why they should be idempotent
[18] (see also [19, 22]). However, this is too restrictive if we wish to
generalise Boolean algebras from Boolean rings (see [5] for more recent
generalisations).

In this note, we propose a system $(A,p,0,1)$ with two constants $0$ and $1$
and a ternary operation $p$ satisfying conditions (T1)–(T4), presented in
Lemma 1, from which an involution $\bar{x}=p(1,x,0)$ and associative binary
operations $x\cdot y=p(0,x,y)$, $x\circ y=p(x,y,1)$ and $x+y=p(x,y,\bar{x})$
are derived (see also [16]). Every formula specifying $p$ in terms of its
derived operations is equivalent to adding a new axiom and hence it gives rise
to a subvariety. For example, Boolean algebras form a subvariety if and only
if $p(x,y,z)$ is interpreted as Church’s conditioned disjunction
$\bar{y}x\circ yz$ (Theorem 1). Moreover, if specifying $p(x,y,z)$ as
$\bar{y}x+yz$ or $(1+y)x+yz$ then the subvariety of unitary rings with
characteristic 2 is obtained (Theorem 2). While if $p(x,y,z)$ is $x+y(x+z)$
then the result is the subvariety of unitary Abelian (right) near-rings with
characteristic $2$ (Theorem 3). Note that the previous formulas for $p$ are
all equal in the context of Boolean algebras or Boolean rings and are related
to the affine formula $x+y(z-x)$ used in Proposition 1.

Lemma 1 presents the algebraic structure $(A,p,0,1)$ satisfying axioms
(T1)–(T4) and gives the properties needed for the upcoming theorems. Axiom
(T4) alone defines a Church algebra [8, 28] while a Menger algebra of rank 2
(see e.g. [9]) uses axiom (T3). The axioms (T1), (T3) and (T4) have been used
to define proposition algebras [2] while a generalisation of axioms (T2)–(T4)
has been used to study spaces with geodesic paths [11, 13, 14].

######  Lemma 1.

Let $(A,p,0,1)$ be a system consisting of a set $A$, together with a ternary
operation $p$ and two constants $0,1\in A$ satisfying:

  1. (T1)

$p(0,a,1)=a$

  2. (T2)

$p(a,b,a)=a$

  3. (T3)

$p(a,p(b_{1},b_{2},b_{3}),c)=p(p(a,b_{1},c),b_{2},p(a,b_{3},c))$

  4. (T4)

$p(a,0,b)=a=p(b,1,a)$.

For $\bar{a}=p(1,a,0)$, $a\cdot b=p(0,a,b)$, $a\circ b=p(a,b,1)$ and
$a+b=p(a,b,\bar{a})$, the following properties hold:

  1. (L1)

$\bar{1}=0,\quad\bar{0}=1$ and $\overline{\overline{a}}=a$

  2. (L2)

$\overline{p(a,b,c)}=p(\overline{a},b,\overline{c}),\quad
p(a,b,c)=p(c,\bar{b},a)$

  3. (L3)

$\overline{a\cdot b}=\overline{b}\circ\overline{a},\quad\overline{a\circ
b}=\overline{b}\cdot\overline{a},\quad\overline{a+b}=\bar{a}+b=a+\bar{b}$

  4. (L4)

$(A,\cdot,1)$, $(A,\circ,0)$ and $(A,+,0)$ are monoids

  5. (L5)

$a\cdot 0=0=0\cdot a,\quad a\circ 1=1=1\circ a$

  6. (L6)

$a+1=\bar{a}=1+a,\quad 1+1=0$

Furthermore:

  1. (L7)

if, for all $a,b\in A$, $p(0,a,b)=p(a,a,b)$ then the systems
$(A,\cdot,\bar{()},0)$ and $(A,\circ,\bar{()},1)$ are commutative magmas with
$a\cdot\bar{a}=0$ and $a\circ\bar{a}=1$;

  2. (L8)

if $\bar{a}\cdot a=0$ then $a\cdot a=a$; if $\bar{a}\circ a=1$ then $a\circ
a=a$;

  3. (L9)

if either $(A,\cdot,1)$ or $(A,\circ,0)$ is a commutative and idempotent
monoid then $(A,\circ,\cdot,0,1)$ is a bounded distributive lattice;

  4. (L10)

if $a+a=0$ then $+$ is commutative and $(a+b)\cdot c=a\cdot c+b\cdot c$.

###### Proof.

The results are readily obtained from the axioms. Explicit proofs can be found
in [12]. ∎

The following theorem is a refinement of Grau’s ternary Boolean algebra in the
sense that it uses Church’s operation and a systematisation of Hoare’s axioms
considered in [16].

######  Theorem 1.

Suppose that $(A,p,0,1)$ satisfies axioms (T1) to (T4). For
$\bar{a}=p(1,a,0)$, $a\cdot b=p(0,a,b)$, $a\circ b=p(a,b,1)$ and
$a+b=p(a,b,\bar{a})$, the following conditions are equivalent:

  1. (i)

$(A,\circ,\cdot,\bar{()},0,1)$ is a Boolean algebra

  2. (ii)

$(A,+,\cdot,0,1)$ is a Boolean ring

  3. (iii)

$p(a,b,c)=(\bar{b}\cdot a)\circ(b\cdot c)$

  4. (iv)

$p(a,a,b)=a\cdot b$

  5. (v)

$p(a,b,b)=a\circ b$.

###### Proof.

Conditions (iv) and (v) are equivalent by duality (L3). Condition (iv) implies
the commutativity of $\cdot$ and $\circ$ as well as the fact that $\bar{()}$
is a Boolean complement (L7), which in turn implies idempotency (L8) and
distributivity (L9). Associativity is ensured by (L4) and hence (iv) implies
(i). Condition (iii) follows from (i) as the unique solution $p=p(a,b,c)$ to
the system of equations

$\left\\{\begin{array}[]{rcl}p\cdot c&=&((\bar{b}\cdot a)\circ(b\cdot c))\cdot
c\\\ c\circ p&=&c\circ((\bar{b}\cdot a)\circ(b\cdot c))\end{array}\right..$

Condition (iv) follows from (iii). Indeed, (iii) implies
$1=p(1,a,1)=\bar{a}\circ a$ and (by duality) also $\bar{a}\cdot a=0$.
Moreover, $p(a,a,b)=(\bar{a}\cdot a)\circ(a\cdot b)=a\cdot b$. It is readily
checked that (i) and (ii) are equivalent [12]. ∎

Note that the ternary Mal’tsev operation [24] in a group $(ab^{-1}c)$ admits a
similar characterisation (see Theorem 4 in [6]). However, as pointed out by
Birkhoff and Kiss [3], the two operations are quite different: in a group we
have $p(a,b,b)=a=p(b,b,a)$ whereas in a Boolean algebra we have $p(a,b,a)=a$.
Nevertheless, as shown in [25] there are some touching points between (weakly)
Mal’tsev categories and distributive lattices.

Each new interpretation of the ternary operation $p$ satisfying (T1)–(T4) in
terms of its derived operations is equivalent to adding a new axiom and gives
rise to a new subvariety, as Theorem 2 and Theorem 3 illustrate. An example of
a ternary operation obtained from a unitary Abelian near-ring [23], which is
not necessarily determined by its derived operations, is presented in the next
proposition. The algebraic model of the unit interval considered in [10, 11]
is another example.

######  Proposition 1.

If $(A,+,\cdot,0,1)$ is a unitary Abelian (right) near-ring, in which $a\cdot
0=0$, then $(A,p,0,1)$ with $p(a,b,c)=a+b(c-a)$ satisfies the axioms (T1) to
(T4).

###### Proof.

The proof is straightforward.∎

In Theorem 2 below, the formula for $p$ is obtained by writing $a+b(c-a)$ as
$(1-b)a+bc$ and replacing $1-b$ by $1+b$. Recall that a ring (or a near-ring)
of characteristic 2 is such that $b+b=0$ for all $b$, so that $1-b=1+b$.
Moreover, rewriting $(1+b)a+bc$ as $a+b(a+c)$ gives the formula used in
Theorem 3.

######  Theorem 2.

Suppose that $(A,p,0,1)$ satisfies axioms (T1) to (T4). For
$\bar{a}=p(1,a,0)$, $a\cdot b=p(0,a,b)$ and $a+b=p(a,b,\bar{a})$, the
following conditions are equivalent:

  1. (i)

$(A,+,\cdot,0,1)$ is a unitary ring of characteristic $2$

  2. (ii)

$p(a,b,c)=(\bar{b}\cdot a)+(b\cdot c)$

  3. (iii)

$a\cdot(b+c)=(a\cdot b)+(a\cdot c)$.

###### Proof.

It is clear that (i) implies (iii). By (L6), (iii) implies $a+a=0$ and hence,
considering (L4) and (L10), (iii) implies (i). Moreover, when $a+a=0$:

$a+p(a,b,c)=p(a,p(a,b,c),\bar{a})\ =\
p(p(a,a,\bar{a}),b,p(a,c,\bar{a}))=b\cdot(a+c).$ (1)

Consequently, (iii) implies (ii) as $p(a,b,c)=a+ba+bc=(1+b)a+bc.$ It remains
to prove that (ii) implies (iii). By (T2), (L4) and (L6), (ii) implies
$1=p(1,a,1)=\bar{a}+a=1+a+a$ i.e. $a+a=0$. Then (1) and (L10) imply left
distributivity: $a\cdot(b+c)=b+p(b,a,c)=b+(1+a)b+ac=ab+ac.$ ∎

The unique non-commutative ring of order $8$, say consisting of all upper
triangular binary $2$-by-$2$ matrices, illustrates Theorem 2. Note that
addition is the Boolean symmetric difference as in a Boolean ring.

######  Theorem 3.

Suppose that $(A,p,0,1)$ satisfies axioms (T1) to (T4). For
$\bar{a}=p(1,a,0)$, $a\cdot b=p(0,a,b)$ and $a+b=p(a,b,\bar{a})$, the
following conditions are equivalent:

  1. (i)

$(A,+,\cdot,0,1)$ is a unitary (right) near ring of characteristic $2$

  2. (ii)

$p(a,b,c)=a+(b\cdot(a+c))$

  3. (iii)

$a+a=0$

  4. (iv)

$(a+b)\cdot c=(a\cdot c)+(b\cdot c)$.

###### Proof.

It is clear that (i) implies (iii) and, considering (L4) and (L10), (iii)
implies (i). Using (1), condition (iii) implies (ii):

$a+p(a,b,c)=b\cdot(a+c)=a+(a+b\cdot(a+c)).$

Using (L4) and (T4), condition (ii) implies (iii): $0=p(a,1,0)=a+a.$
Properties (L6) and (L10) show that (iii) and (iv) are equivalent. ∎

Several examples of unitary (right) near-rings of characteristic 2 with four
elements can be found. The following example illustrates Theorem 3.
Multiplication is neither commutative nor idempotent and, once again, addition
is the same as Boolean symmetric difference. Note that if the formula
$(1+y)x+yz$ is used as the ternary operation $p$ instead of $x+y(x+z)$ then
(T3) is not satisfied.

$\begin{array}[]{c|cccc}\cdot&0&u&v&1\\\ \hline\cr 0&0&0&0&0\\\ u&0&0&0&u\\\
v&0&u&v&v\\\ 1&0&u&v&1\\\
\end{array}\qquad\quad\begin{array}[]{c|cccc}+&0&u&v&1\\\ \hline\cr
0&0&u&v&1\\\ u&u&0&1&v\\\ v&v&1&0&u\\\ 1&1&v&u&0\end{array}$

The results presented in this note suggest that unitary Abelian near-rings of
characteristic $2$, considered as generalised Boolean rings, provide a way to
explore skew lattices that are not necessarily idempotent. This line of
research goes in the direction advocated by Birkoff and Von Neumann [4] as a
possibility for the logic of quantum mechanics.

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On the Low-SNR Asymptotic Capacity of Optical Wireless Channels with Average-Intensity Constraints

Longguang Li

This paper investigates the capacity of an optical wireless channel with an average-intensity constraint in the low signal-to-noise ratio (SNR) regime. When the average intensity of the input is no larger than a small $\mathcal{E}$, the capacity scales as $\mathcal{E}\sqrt{ \frac{\log \frac{1}{\mathcal{E}}}{2} }$.

§ INTRODUCTION

Intensity-modulation and direct-detection (IM-DD) refers to a transmission scheme where the transmitter uses  the modulated intensity of the optical source as the carrier for information transmission, and the receiver measures incoming intensity to detect the information <cit.>. The IM-DD scheme is widely adopted in current optical wireless communication (OWC) systems due to its low cost and easy implementation. Although the exact characterization of the capacity of an IM-DD based OWC channel is an open problem up to date <cit.>, decent progress has been made from two aspects. One aspect is on deriving capacity upper and lower bounds or characterizing asymptotic capacity in the high or low signal-to-noise ratio (SNR) regime. Capacity bounds and asymptotic results are derived in  <cit.>, and similar results are extended to general multiple-input single- and multiple-output channels <cit.>. The other aspect is on characterizing properties of the capacity-achieving input distribution, e.g., the discreteness and finiteness of its support <cit.>. Recently, bounds on the cardinality of its support were shown in <cit.>.

This paper focuses on exactly characterizing the low-SNR asymptotic capacity of an optical wireless channels under an average-intensity constraint. In the related existing work <cit.>,  when the peak or both the peak and average intensity of the input are limited, the authors exactly characterized the low-SNR asymptotic capacity, which can be expressed in terms of the maximum variance among all admissible input distributions. While in the case where the average intensity of the input is limited, they derived the scaling order of the low-SNR asymptotic capacity. The difficulty in exact characterization of the low-SNR capacity here lies in the fact that the low-SNR capacity is no longer captured by the maximum variance of the input distributions, which can be infinitely large in this case.

Specifically, when the average intensity of the input is limited to be no larger than $\mathcal{E}$, the low-SNR capacity shown in <cit.> scales as $a\mathcal{E}\sqrt{ {\log \frac{1}{\mathcal{E}}} }$, where $\frac{1}{\sqrt{2}}\leq a \leq 2$. In this paper, we prove that the constant $a$ turns out to be $\frac{1}{\sqrt{2}}$, thus the low-SNR asymptotic capacity scaling exactly as $\mathcal{E}\sqrt{ \frac{\log \frac{1}{\mathcal{E}}}{2} }$. The proof leverages tools from the duality capacity expression by carefully choosing the auxiliary distribution, and the data processing inequality.

§ CHANNEL MODEL

Consider an optical intensity channel, and the channel output is given by
Y = X+Z,

where $X$ denotes the channel input, and  $Z$ denotes the Gaussian noise with variance $1$, i.e.,

Since $X$ is proportional to the optical intensity, its support must satisfy
𝗌𝗎𝗉𝗉 X ⊂ℝ^+.

Considering the requirement of illumination quality or energy consumption, the input is subject to

an average-intensity constraint
𝔼 [X]  ≤ℰ.

The single-letter capacity expression of the channel (<ref>) is given by

𝖢 (ℰ) = sup_p_X  satisfying   (<ref>)  and  (<ref>) (X;Y).

To simplify the notation, we use $f(x)\overset{\cdot}{=} g(x)$ to indicate functions $f(x)$ and $g(x)$ satisfy $\lim_{x\rightarrow 0} \frac{f(x)}{g(x)}=1$, and $f(x)\overset{\cdot}{\leq} g(x)$ and  $f(x)\overset{\cdot}{\geq} g(x)$ are defined similarly.

§ MAIN RESULT

The capacity of the channel (<ref>) satisfies
𝖢(ℰ) ·= ℰ√( log1/ℰ/2 ).

It is equivalent to prove
lim sup_ℰ→0 𝖢(ℰ) ·≤ ℰ√( log1/ℰ/2 ),

lim inf_ℰ→0 𝖢(ℰ) ·≥ ℰ√( log1/ℰ/2 ).

We first porve (<ref>). The duality capacity upper bound is given by

(ℰ)    ≤    sup_X D(W(·|X)R(·))

=    sup_X  -∫_-∞^∞ W(y|X)logR(y) y  -1/2log2πe,

where $\const{D}(\cdot\|\cdot)$ denotes the
Kullback-Leibler divergence, $W(\cdot|X)$ denotes the conditional output distribution given the input $X$, and $R(\cdot)$ denotes the auxiliary distribution on the output space. More details on the duality capacity bound can be found in <cit.>.

When $\mathcal{E}$ is sufficiently small, let $t=a\sqrt{\log \frac{1}{\mathcal{E}}}$ with $a>\sqrt{2}$, and $\beta=e^{-\frac{t^2}{2}}$. It is direct to see $\beta \in (0,1)$. We choose the auxiliary distribution $R(\cdot)$ as
R(y) =

1-β/√(2π)𝒬(-t)e^-y^2/2     if  y ≤t,

βe^-(y-t)     otherwise.

The expectation term at the RHS of (<ref>) can be expanded as
-∫_-∞^∞ W(y|X)logR(y) y

=     -∫_-∞^t W(y|X)logR(y) y +  -∫_t^∞ W(y|X)logR(y)  y .

Notice that
-∫_-∞^t W(y|X)logR(y) y

=      -∫_-∞^t
1/√(2π) ^-(y-X)^2/2
(log1-β/√(2π)𝒬(-t) - y^2/2) y

=     𝖤[-log1-β/√(2π)𝒬(-t) (X-t)
+ 1/2(X-t)

+ X^2/2 (X-t) - X+t/2ϕ(X-t) ) ]

≤    -log1-β/√(2π)𝒬(-t) (X-t)
+ 1/2(X-t) + X^2/2 (X-t)

=     -log1-β/√(2πe) (X-t) -log1/𝒬(-t)(X-t)
+ X^2/2 (X-t)

≤      -log1-β/√(2πe)
+  1/2X^2𝒬(X-t)

·=    1/2log2πe + e^-t^2/2+  1/2X^2𝒬(X-t),

where $ \phi(x) \overset{\text{def}}{=} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $\mathcal{Q}(x) \overset{\text{def}}{=} \int_{x}^{\infty}\phi(t)dt$.

-∫_t^∞ W(y|X)logR(y)  y

=      -∫_t^∞ 1/√(2π)^-(y-X)^2/2
-(y-t) ) y

=      -logβ ·(t-X) + (X-t)𝒬(t-X)+ϕ(t-X)

≤     t^2/2
(t-X) + X+ϕ(t-X)

≤     t^2/2
(t-X) + X+ϕ(t)+2X/t

≤    ϕ(t)+ℰ+2ℰ/t  + 1/2 t^2

where (<ref>) follows from the fact $\phi (t-x) \leq \phi(t)+\frac{2x}{t}, \,\, \forall t>0,\,\, x\geq 0$.

Combining (<ref>) and (<ref>), we have
-∫_-∞^∞ W(y|X)logR(y) y

·≤     1/2log2πe + e^-t^2/2+ϕ(t)+ℰ+2ℰ/t+1/2(X^2𝒬(X-t)+t^2𝒬(t-X) ).

Now, we bound the expectation term at the RHS of (<ref>). By the law of total expectation,
=    X^2𝒬(X-t)+t^2𝒬(t-X)X≤tPr(X≤t)

+ X^2𝒬(X-t)+t^2𝒬(t-X)X > tPr(X > t).

The conditional expectation terms at the RHS of (<ref>) can be bounded as
X^2𝒬(X-t)+t^2𝒬(t-X)X≤t    =    X^2+(t^2-X^2)𝒬(t-X)X≤t

≤    Xt+(t+X)(t-X)𝒬(t-X)X≤t

≤    Xt+(t+X)ϕ(t-X)X≤t

≤    tXX≤t +  (t+X)ϕ(t-X)X≤t,

where (<ref>) follows by $x\mathcal{Q}(x)\leq \phi (x), \,\, \forall x \geq 0$. Similarly,
X^2𝒬(X-t)+t^2𝒬(t-X)X > t    =    t^2+(X^2-t^2)𝒬(X-t)X > t

≤    Xt+(X+t)(X-t)𝒬(X-t)X > t

≤    Xt+(X+t)ϕ(X-t)X > t

≤    tXX>t +  (t+X)ϕ(X-t)X>t.

Substituting (<ref>) and (<ref>) into (<ref>), we obtain
≤    tX+ (t+X)ϕ(t-X)

≤    tX+ t(ϕ(t)+2X/t)+X/√(2π)

≤    ℰt+ tϕ(t)+ 3ℰ.

Further substituting (<ref>) into (<ref>), we have
-∫_-∞^∞ W(y|X)logR(y) y
·≤  1/2log2πe + e^-t^2/2+(t/2+1)ϕ(t)+5ℰ/2+2ℰ/t+ ℰt/2.

Recall $t=a\sqrt{\log\frac{1}{\mathcal{E}}}$ and substitute it into (<ref>), we have
e^-t^2/2       ·= ℰ^a^2/2,

(t/2+1)ϕ(t)       ·= a/2ℰ^a^2/2√(log1/ℰ),

2ℰ/t       ·= 2/aℰ/√(log1/ℰ),

ℰ t/2       ·= a/2ℰ√(log1/ℰ).

Since $a > \sqrt{2}$, the last term (<ref>) dominates. Hence
-∫_-∞^∞ W(y|X)logR(y) y
·≤     1/2log2πe +  a/2ℰ√(log1/ℰ).

By (<ref>) we have
𝖢(ℰ)  ·≤ a/2ℰ√(log1/ℰ).

Since $a>\sqrt{2}$ is chosen arbitrarily,
𝖢(ℰ) ·≤ inf_a>√(2) a/2ℰ√(log1/ℰ)     ·=     1/√(2) ℰ√(log1/ℰ).

Eq. (<ref>) is proved.

Eq. (<ref>) was already proved in <cit.>, in which the proof involves quite complicated evaluations on several integral items. Here, we give another proof by using the data processing inequality. Consider a binary input $X_B$ with the distribution
1-ℰ/x_0      if   X_B=0,

ℰ/x_0      if  X_B=x_0,

where $x_0=a \sqrt{\log \frac{1}{\mathcal{E}}}$ with $a>\sqrt{2}$. Given $Y$, denote $\hat{X}_B$ as the estimate of $X_B$ by the maximum a posteriori probability (MAP) decision rule, i.e.,
X̂_B=_X_B Pr(X_B|Y).

Then the error probability $\text{P}_e$ by the MAP rule can be directly calculated as
P_e    =    (1-ℰ/x_0)(x_0/2+log(x_0/ℰ-1)/x_0) +  ℰ/x_0(x_0/2-log(x_0/ℰ-1)/x_0).

For the first term at the RHS of (<ref>), recall $x_0=a \sqrt{\log \frac{1}{\mathcal{E}}}$, then
(1-ℰ/x_0)(x_0/2+log(x_0/ℰ-1)/x_0)    ·=
ϕ(a √(log1/ℰ)/2+log(a √(log1/ℰ)/ℰ-1)/a √(log1/ℰ))/a √(log1/ℰ)/2+log(a √(log1/ℰ)/ℰ-1)/a √(log1/ℰ)

·=    ℰ^1/2(a/2+1/a)^2/√(2π)(a/2+1/a)√(log1/ℰ),

where (<ref>) follows from $\mathcal{Q}\left(\frac{1}{x}\right)\overset{\cdot}{=}x{\phi\left(\frac{1}{x}\right)},\,\,x>0$. Similarly, for the second term,
ℰ/x_0(x_0/2-log(x_0/ℰ-1)/x_0)    ·=
ℰ/a√(1/ℰ)ϕ(a √(log1/ℰ)/2-log(a √(log1/ℰ)/ℰ-1)/a √(log1/ℰ))/a √(log1/ℰ)/2-log(a √(log1/ℰ)/ℰ-1)/a √(log1/ℰ)

·=     ℰ^1+1/2(a/2-1/a)^2/√(2π)(a/2-1/a) √(log1/ℰ)

·=     ℰ^1/2(a/2+1/a)^2/√(2π)(a/2-1/a) √(log1/ℰ) ,

where (<ref>) follows from the fact $a>\sqrt{2}$. Substituting (<ref>) and (<ref>) into (<ref>) yields
·=    ℰ^1/2(a/2+1/a)^2/√(2π)(a/2+1/a)√(log1/ℰ) + ℰ^1/2(a/2+1/a)^2/√(2π)(a/2-1/a)√(log1/ℰ)

·≤    √(2/π)ℰ^1/2(a/2+1/a)^2/(a/2-1/a)√(log1/ℰ).

Since $X_B-Y-\hat{X}_B$ forms a Markov chain, by the data processing inequality, $\mathsf{I}(X_B;Y)$ can be lower bounded by
≥    𝖨(X_B;X̂_B)

=    𝖧(X_B)-𝖧(X_B|X̂_B)

≥    𝖧(X_B)-𝖧_b(P_e),

where $\mathsf{H}_b(\text{P}_e) \overset{\text{def}}{=}  -\text{P}_e\log \text{P}_e -(1-\text{P}_e)\log (1-\text{P}_e)$, and (<ref>) follows by the Fano's inequality.

Recall again $x_0=a \sqrt{\log \frac{1}{\mathcal{E}}}$, and it is direct to see
𝖧(X_B)    =    -ℰ/x_0logℰ/x_0-(1-ℰ/x_0)log(1-ℰ/x_0)

·=    -ℰ/x_0logℰ/x_0

·=    1/a ℰ√(log1/ℰ).

We bound the term $\mathsf{H}_b(\text{P}_e)$ by
𝖧_b(P_e)    =     -P_elogP_e -(1-P_e)log(1-P_e)

·=    -P_elogP_e + P_e -P^2_e

·=    -P_elogP_e

·≤    √(2/π) (a/2+1/a)^2ℰ^1/2(a/2+1/a)^2√(log1/ℰ)/a/2-1/a,

where (<ref>) follows by (<ref>). Substituting (<ref>) and (<ref>) into (<ref>), we obtain
·≥    1/a ℰ√(log1/ℰ) - √(2/π) (a/2+1/a)^2ℰ^1/2(a/2+1/a)^2√(log1/ℰ)/a/2-1/a

·=    1/a ℰ√(log1/ℰ).

·≥  1/a ℰ√(log1/ℰ).

Since $a>\sqrt{2}$ is chosen arbitrarily,
·≥ sup_a>√(2) 1/a ℰ√(log1/ℰ)
·=    1/√(2)ℰ√(log1/ℰ).

Eq. (<ref>) is proved.

§ CONCLUSION

This paper exactly characterized the low-SNR asymptotic capacity of an optical wireless channel when the input is subject to an average-intensity constraint. The techniques used in this paper may be applied to analyze the low-SNR asymptotic capacity of the Poisson channel <cit.>.


# The dimension theory of
semi-regular continued fractions

Yuto Nakajima and Hiroki Takahasi  Department of Mathematics, Keio University,
Yokohama, 223-8522, JAPAN<EMAIL_ADDRESS>Department of
Mathematics, Keio University, Yokohama, 223-8522, JAPAN
<EMAIL_ADDRESS>
###### Abstract.

We consider sets of irrational numbers in $(0,1)$ whose partial quotients
$a_{\sigma,n}$ in the semi-regular continued fraction expansion obey certain
restrictions and growth conditions. We prove that, for any sequence
$\sigma\in\\{-1,1\\}^{\mathbb{N}}$ in the expansion, any infinite subset $B$
of $\mathbb{N}$ and for any function $f$ on $\mathbb{N}$ with values in $[\min
B,\infty)$ and tending to infinity, the set of irrationals such that

$a_{\sigma,n}\in B,\ a_{\sigma,n}\leq f(n)\text{ for all $n\in\mathbb{N}$ and
}a_{\sigma,n}\to\infty\text{ as }n\to\infty$

is of Hausdorff dimension $\tau(B)/2,$ where $\tau(B)$ is the exponent of
convergence of $B$. This considerably extends the result of the second author
[Proc. Amer. Math. Soc. 151, 3645–3653 (2023)] to the semi-regular continued
fraction. We also prove that for any $\sigma\in\\{-1,1\\}^{\mathbb{N}}$ and
any $B\subset\mathbb{N}$, the set of irrationals such that

$a_{\sigma,n}\in B\text{ for all $n\in\mathbb{N}$ and
}a_{\sigma,n}\to\infty\text{ as }n\to\infty$

is also of Hausdorff dimension $\tau(B)/2$. This considerably extends the
solution of Hirst’s conjecture by Wang and Wu [Bull. London Math. Soc. 40,
18–22 (2008)] to the semi-regular continued fraction. To prove these two
results, we exploit the dimension theory developed by Rempe-Gillen and
Urbański: construct non-autonomous iterated function systems well-adapted to
the given restrictions and growth conditions on partial quotients, estimate
the pressure function, and then apply Bowen’s formula.

###### 2020 Mathematics Subject Classification:

Primary 11A55; Secondary 37C45, 37C60

Keywords: continued fractions; Hausdorff dimension; iterated function system
(IFS)

##  1\. Introduction

Each irrational number $x$ in the set $\mathbb{I}=(0,1)\setminus\mathbb{Q}$
has the unique regular continued fraction (RCF) expansion
$x=1/(a_{1}+1/(a_{2}+\cdots))$, where $a_{n}$, $n\geq 1$ belongs to the set
$\mathbb{N}$ of positive integers. For a Lebesgue typical number in
$\mathbb{I}$, Khinchin [14] proved that the frequency with which each integer
$k\in\mathbb{N}$ appears as its RCF partial quotients is $\frac{1}{\log
2}\log\frac{(k+1)^{2}}{k(k+2)}$. Although Khinchin did not use ergodic theory
in his original proof, his result is a consequence of Birkhoff’s theorem
applied to the Gauss map $x\in(0,1]\mapsto 1/x-\lfloor 1/x\rfloor\in[0,1)$
leaving the Gauss measure $\frac{1}{\log 2}\frac{dx}{1+x}$ invariant and
ergodic.

###  1.1. The fractal dimension results of Jarník and Good

Numbers in $\mathbb{I}$ whose RCF partial quotients behave very differently
from the Lebesgue typical ones are not negligible in the sense of the
Hausdorff dimension. This fact lies at the basis of the fractal dimension
theory of continued fractions. A pioneering result in the creation of this
theory is due to Jarník [12], who proved that the set

$\left\\{x\in\mathbb{I}\colon\\{a_{n}(x)\\}\text{ is bounded}\right\\},$

i.e., the set of badly approximable numbers [14, Theorem 23], is of Hausdorff
dimension $1$. Good [7] proved that the set

$\left\\{x\in\mathbb{I}\colon\lim_{n\to\infty}a_{n}(x)=\infty\right\\}$

is of Hausdorff dimension $1/2$. Various extensions and refinements of this
Good’s theorem have been made (see [2, 5, 6, 8, 9, 11, 13, 16, 17, 19, 21, 23]
for example) which determine fractal dimensions of sets of numbers in
$\mathbb{I}$ whose RCF partial quotients obey certain restrictions or growth
conditions.

###  1.2. The conjecture of Hirst

Hirst [9] proved results analogous to that of Good [7] in the case where
$a_{n}$ is restricted to belong to some subsets of $\mathbb{N}$. For an
infinite subset $B$ of $\mathbb{N}$, define the exponent of convergence by

$\tau(B)=\inf\left\\{s\geq 0\colon\sum_{k\in B}k^{-s}<\infty\right\\}.$

In [9], Hirst considered the set

$E(B)=\left\\{x\in\mathbb{I}\colon a_{n}(x)\in B\text{ for all
}n\in\mathbb{N}\text{ and }\lim_{n\to\infty}a_{n}(x)=\infty\right\\},$

and conjectured that

(1) $\dim_{\rm H}E(B)=\frac{\tau(B)}{2}\ \text{ for any }B,$

where $\dim_{\rm H}$ denotes the Hausdorff dimension. He treated the special
case $B=\\{k^{b}\\}_{k\in\mathbb{N}}$, $b$ a positive integer to support his
conjecture (see [9, Theorem 3]). Cusick [2, Theorem 1] proved that the
equality holds in the case $B$ is not too sparse and satisfies what he called
the density assumption. 35 years later than the appearance of Hirst’s paper
[9], his conjecture was confirmed by Wang and Wu [23, Theorem 1.1]. This
solution was extended in [1] to certain infinite iterated function systems
satisfying the so-called open set condition [4].

###  1.3. Sets with restricted, slowly growing partial quotients

In this context, subsets of $E(B)$ of the form

$F(B,f)=\\{x\in E(B)\colon a_{n}(x)\geq f(n)\text{ for all
}n\in\mathbb{N}\\},$

and

$G(B,f)=\left\\{x\in E(B)\colon a_{n}(x)\leq f(n)\text{ for all
}n\in\mathbb{N}\right\\},$

where $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is a function satisfying
$\lim_{n\to\infty}f(n)=\infty$ are also relevant to consider. In [9, p.227],
Hirst conjectured the equality $\dim_{\rm H}F(B,f)=\tau(B)/2$ no matter how
rapidly $f$ grows, and Cusick later showed the case $B=\mathbb{N}$,
$f(n)=2^{2^{2^{n}}}$ as a counterexample (see [2, Lemma 3]). Today it is known
that $\dim_{\rm H}F(B,f)$ can drop from $\tau(B)/2$ and even become $0$ when
$f$ grows very rapidly [1, 2, 6, 13, 16, 17, 24], whereas ${\rm dim}_{\rm
H}G(B,f)$ does not drop at all no matter how slowly $f$ grows to infinity
[21]: Interestingly we have

(2) $\dim_{\rm H}G(B,f)=\frac{\tau(B)}{2}\ \text{ for any $B$ and any $f$ with
$f(\mathbb{N})\subset[\min B,\infty)$.}$

###  1.4. The semi-regular continued fraction

The aim of this paper is to extend the dimension results (1), (2) on the RCF
to the semi-regular continued fraction (SRCF). For each
$\sigma=\\{\sigma_{n}\\}_{n=1}^{\infty}\in\\{-1,1\\}^{\mathbb{N}}$, any
$x\in\mathbb{I}$ has the unique SRCF expansion of the form

(3)
$x=\cfrac{1}{a_{\sigma,1}(x)+\cfrac{\sigma_{1}}{a_{\sigma,2}(x)+\cfrac{\sigma_{2}}{a_{\sigma,3}(x)+\ddots}}}=\frac{\displaystyle{\hfill{1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{\sigma,1}(x)}\hfill}}+\frac{\displaystyle{\hfill{\sigma_{1}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{\sigma,2}(x)}\hfill}}+\frac{\displaystyle{\hfill{\sigma_{2}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{\sigma,3}(x)}\hfill}}+\cdots,$

where $a_{\sigma,n}=a_{\sigma,n}(x)$ are positive integers such that
$\sigma_{n}+a_{\sigma,n}\geq 1$ for all $n\geq 1$. This means that the finite
truncation

$\cfrac{1}{a_{\sigma,1}+\cfrac{\sigma_{1}}{a_{\sigma,2}+\cdots+\cfrac{\ddots}{\displaystyle{\frac{\sigma_{n-1}}{a_{\sigma,n}}}}}}=\frac{\displaystyle{\hfill{1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{\sigma,1}}\hfill}}+\frac{\displaystyle{\hfill{\sigma_{1}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{\sigma,2}}\hfill}}+\cdots+\frac{\displaystyle{\hfill{\sigma_{n-1}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{\sigma,n}}\hfill}}$

converges to $x$ as $n\to\infty$. The SRCF includes several continued
fractions as particular cases. If $\sigma_{n}=1$ for all $n\geq 1$ we obtain
the RCF, and if $\sigma_{n}=-1$ for all $n\geq 1$ we obtain the backward
continued fraction. For other examples, see [3, 10, 15, 18]. Any SRCF can be
transformed into a regular one, as shown in [18]. The converse is also true
[3].

The RCF is generated by iterations of the Gauss map, while there is no single
interval transformation that generates the SRCF. Therefore, developing a
dimension theory on fractal sets determined by the SRCF requires novel
approaches.

###  1.5. Statement of the result

To obtain results analogous to (1) and (2), for each
$\sigma=\\{\sigma_{n}\\}_{n=1}^{\infty}\in\\{-1,1\\}^{\mathbb{N}}$ we consider
sets

$E_{\sigma}(B)=\left\\{x\in\mathbb{I}\colon a_{\sigma,n}(x)\in B\text{ for all
}n\in\mathbb{N}\text{ and }\lim_{n\to\infty}a_{\sigma,n}(x)=\infty\right\\},$

and

$G_{\sigma}(B,f)=\left\\{x\in E_{\sigma}(B)\colon a_{\sigma,n}(x)\leq f(n)\
\mbox{for all }n\in\mathbb{N}\right\\}.$

Clearly we have $\dim_{\rm H}E_{\sigma}(B)\geq\dim_{\rm H}G_{\sigma}(B,f)$.

######  Main Theorem.

For any $\sigma\in\\{-1,1\\}^{\mathbb{N}},$ any infinite subset $B$ of
$\mathbb{N}$ and any function $f:\mathbb{N}\rightarrow[\min B,\infty)$ such
that $\lim_{n\to\infty}f(n)=\infty,$ we have

$\dim_{\rm H}E_{\sigma}(B)=\dim_{\rm H}G_{\sigma}(B,f)=\frac{\tau(B)}{2}.$

This theorem shows that the Hausdorff dimension of the sets obtained in [21,
23] do not depend on $\sigma$. The constant $2$ in the equalities, as in (1),
(2) means that the decay order of the lengths of the unit fundamental
intervals is quadratic.

We would like to position the Main Theorem as a starting point toward the
creation of a dimension theory of the semi-regular continued fractions. Taking
$B=\mathbb{N}$ in the Main Theorem yields an extension of Good’s theorem to
the SRCF.

######  Corollary 1.1.

For any $\sigma\in\\{-1,1\\}^{\mathbb{N}}$, we have

$\dim_{\rm
H}\left\\{x\in\mathbb{I}\colon\lim_{n\to\infty}a_{\sigma,n}(x)=\infty\right\\}=\frac{1}{2}.$

In the same spirit, it would be nice to extend Jarník’s theorem to the SRCF.

###  1.6. The method of proof

The equality (2) was proved in [21] by combining the general upper bound
$\dim_{\rm H}E(B)\leq\tau(B)/2$ in [9, Corollary 1] with a new lower bound
$\dim_{\rm H}G(B,f)\geq\tau(B)/2$, obtained in [21] by constructing a fractal
subset of $G(B,f)$ and estimating its Hausdorff dimension from below. The
argument in [21] remains valid for those $\sigma\in\\{-1,1\\}^{\mathbb{N}}$
which contain only finitely many $-1$, or only finitely many $1$. In order to
treat all other $\sigma$, new ingredients are necessary as we develop below.

To obtain an upper bound $\dim_{\rm H}E_{\sigma}(B)\leq\tau(B)/2$, we slightly
modify the upper bound in [9, Corollary 1] which relies on the well-known
formula for fundamental intervals of the RCF in terms of its convergents [14].
Since this type of formula is not available for the SRCF, we use a bounded
distortion argument.

Establishing the lower bound $\dim_{\rm H}G_{\sigma}(B,f)\geq\tau(B)/2$ is the
most creative part of this paper, and hence deserves a special attention with
a close comparison to [21]. The construction of the fractal set in [21] is
rather rigid, which relies on the ergodic theory of the Gauss map. Key
components are a sequence of finite subsystems with increasing digits, and an
associated sequence of ergodic measures, together with a finite collection of
intervals which approximate the ergodic measure in a particular sense. The
intervals in the subsystems are glued together to form a fractal subset of
$G(B,f)$. The Hausdorff dimension of this set is estimated from below in terms
of measure-theoretic entropies and Lyapunov exponents of the ergodic measures.
To obtain stationary sequences with ergodic arguments (Birkhoff’s and Shannon-
McMillan-Breiman’s theorems), a sufficiently high iteration of each subsystem
is necessary. The number of this high iteration is not controllable, and as a
result, [21] cannot be adapted to constructing a fractal subset of
$G_{\sigma}(B,f)$ in cases $\sigma$ contains infinitely many $-1$ and $1$.

Our strategy for establishing the lower bound is to dispense with the ergodic
arguments in [21] altogether, and introduce a much more flexible construction
exploiting the theory of non-autonomous conformal iterated function systems
(IFSs) developed by Rempe-Gillen and Urbański [20]. In Section 2 we provide
preliminary results, and in Section 3 prove the Main Theorem.

##  2\. Preliminaries

In this section we provide preliminary results on the SRCF and non-autonomous
conformal IFSs. In Section 2.1 we explain a relationship between the SRCF in
(3) and an infinite IFS. In Section 2.2 we introduce non-autonomous conformal
IFSs and recall the result of Rempe-Gillen and Urbański [20]. In Section 2.3
we establish extendibility and distortion bounds for IFSs generating the SRCF.

Figure 1. The graph of the Gauss map $x\in(0,1]\mapsto 1/x-\lfloor
1/x\rfloor\in[0,1)$ (left): that of the Rényi-like map
$x\in(0,1]\mapsto\lfloor 1/x\rfloor-1/x+1\in(0,1]$ (right)

###  2.1. The SRCF as an infinite IFS

Consider an infinite IFS
$\\{\varphi_{1,i}\\}_{i=1}^{\infty}\cup\\{\varphi_{-1,i}\\}_{i=2}^{\infty}$ on
$[0,1]$ defined by

$\varphi_{1,i}(x)=\frac{1}{i+x},\ x\in[0,1]\ \text{ and }\
\varphi_{-1,i}(x)=\frac{1}{i-x},\ x\in[0,1].$

These maps are obtained from inverse branches of the Gauss map
$x\in(0,1]\mapsto 1/x-\lfloor 1/x\rfloor\in[0,1)$ and that of the Rényi-like
map $x\in(0,1]\mapsto\lfloor 1/x\rfloor-1/x+1\in(0,1]$ respectively (see
Figure 1). The SRCF in (3) is generated by this infinite IFS. We have

(4) $\varphi_{1,i}([0,1])=\left[\frac{1}{1+i}\ ,\frac{1}{i}\right]\ \text{ and
}\ \varphi_{-1,i}([0,1])=\left[\frac{1}{i}\ ,\frac{1}{i-1}\right].$

For $\sigma=\\{\sigma_{n}\\}_{n=1}^{\infty}\in\\{-1,1\\}^{\mathbb{N}}$, $n\geq
1$ and $a_{1},\ldots,a_{n}\in\mathbb{N}$ such that $\sigma_{k}+a_{k}\geq 1$
for all $1\leq k\leq n$, we define a fundamental interval

(5)
$[a_{1},\ldots,a_{n}]_{\sigma}=\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{n},a_{n}}([0,1]).$

This is the closure of the interval which consists of points having a (finite
or infinite) SRCF expansion beginning by $a_{1},a_{2},\ldots,a_{n}$. Let $|J|$
denote the Euclidean length of a bounded interval $J\subset\mathbb{R}$.

The next lemma states convergence properties of the SRCF [15, 18, 22] and a
uniqueness property of the expansion. It follows that for each fixed $\sigma$,
the SRCF in (3) is generated by an infinite non-autonomous IFS whose limit set
contains $\mathbb{I}$.

######  Lemma 2.1.

The following hold:

  * (a)

Let $\sigma=\sigma_{1}\sigma_{2}\cdots\in\\{-1,1\\}^{\mathbb{N}}$ and
$a_{1}a_{2}\cdots\in\mathbb{N}^{\mathbb{N}}$ be such that
$\sigma_{n}+a_{n}\geq 1$ for all $n\geq 1$. The set

$\bigcap_{n=1}^{\infty}[a_{1},\ldots,a_{n}]_{\sigma}$

is a singleton, and its unique element equals the semi-regular continued
fraction

$\frac{\displaystyle{\hfill{1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{1}}\hfill}}+\frac{\displaystyle{\hfill{\sigma_{1}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{2}}\hfill}}+\frac{\displaystyle{\hfill{\sigma_{2}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{3}}\hfill}}+\cdots.$

This continued fraction is irrational if and only if $\sigma_{n}+a_{n}\geq 2$
for infinitely many $n\geq 1$.

  * (b)

For any $x\in\mathbb{I}$ and any $\sigma\in\\{-1,1\\}^{\mathbb{N}}$, there
exists a unique $a_{1}a_{2}\cdots\in\mathbb{N}^{\mathbb{N}}$ such that
$\sigma_{n}+a_{n}\geq 1$ for all $n\geq 1$ and

$x\in\bigcap_{n=1}^{\infty}[a_{1},\ldots,a_{n}]_{\sigma}.$

###### Proof.

Let $\sigma=\sigma_{1}\sigma_{2}\cdots\in\\{-1,1\\}^{\mathbb{N}}$ and
$a_{1}a_{2}\cdots\in\mathbb{N}^{\mathbb{N}}$ satisfy $\sigma_{n}+a_{n}\geq 1$
for all $n\geq 1$. By induction based on (4) one can check that for all $n\geq
2$ and all $x\in[0,1]$,

(6)
$\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{n},a_{n}}(x)=\frac{\displaystyle{\hfill{1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{1}}\hfill}}+\frac{\displaystyle{\hfill{\sigma_{1}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{2}}\hfill}}+\cdots+\frac{\displaystyle{\hfill{\sigma_{n-1}}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{a_{n}+\sigma_{n}x}\hfill}}.$

If $\sigma_{n}+a_{n}\geq 2$ for infinitely many $n\geq 1$, then the number in
the right-hand side of (6) converges and the limit is irrational [15, 22].
Otherwise, it converges to a rational number since the continued fraction

$\frac{\displaystyle{\hfill{1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{2}\hfill}}+\frac{\displaystyle{\hfill{-1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{2}\hfill}}+\frac{\displaystyle{\hfill{-1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{2}\hfill}}+\cdots+\frac{\displaystyle{\hfill{-1}\hfill\;\vrule}}{\displaystyle{\vrule\;\hfill{2}\hfill}}+\cdots$

is equal to $1$. The proof of (a) is complete.

Let $x\in\mathbb{I}$ and let $\sigma\in\\{-1,1\\}^{\mathbb{N}}$. We can define
inductively a sequence $a_{1},a_{2},\ldots$ in $\mathbb{N}$ such that
$\sigma_{n}+a_{n}\geq 1$ and $x\in[a_{1},\ldots,a_{n}]_{\sigma}$ for all
$n\geq 1$. If $b_{1},b_{2},\ldots$ is an integer sequence such that
$[a_{1},\ldots,a_{n}]_{\sigma}=[b_{1},\ldots,b_{n}]_{\sigma}$ for all $n\geq
1$, then (4) and the irrationality of $x$ yield $a_{n}=b_{n}$ for all $n\geq
1$. The proof of (b) is complete. $\Box$

###  2.2. Non-autonomous conformal IFSs

Below we summarize the dimension theory in [20] as far as we need them, on
non-autonomous conformal IFS on the Euclidean space $\mathbb{R}^{d}$ of
arbitrary dimension. We note that $d=1$ throughout our application of their
theory in Section 3.2.

Let $W\subset\mathbb{R}^{d}$ be an open set and let $\phi\colon
W\rightarrow\phi(W)$ be a diffeomorphism. We say $\phi$ is conformal if for
any $x\in W$ the differential
$D\phi(x)\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a similarity linear
map: $D\phi(x)=c_{x}\cdot M_{x}$ where $c_{x}>0$ is a scaling factor at $x$
and $M_{x}$ is a $d\times d$ orthogonal matrix. For a conformal map
$\phi\colon W\to\phi(W)$ and a set $A\subset W$, we set

$\|D\phi\|_{A}=\sup\\{|D\phi(x)|\colon x\in A\\},$

where $|D\phi(x)|$ denotes the scaling factor of $\phi$ at $x$.

For each $n\in\mathbb{N}$ let $I^{(n)}$ be a finite set. We introduce index
sets

(7) $I^{\infty}=\prod_{j=1}^{\infty}I^{(j)},\text{ and
}I^{k}_{n}=\prod_{j=n}^{k}I^{(j)}\text{ for }k\geq n.$

For each $n\in\mathbb{N}$ let $\\{\phi_{i}^{(n)}\\}_{i\in I^{(n)}}$ be a
finite collection of self maps of a connected compact set
$X\subset\mathbb{R}^{d}$ such that the closure of its interior $X^{\circ}$
coincides $X$. We assume $X$ is a convex set, or its boundary $\partial X$ is
smooth. For $\omega=\omega_{1}\omega_{2}\cdots\in I^{\infty}$ and
$n,k\in\mathbb{N}$ with $n\leq k$, we set

(8) $\omega|_{n}^{k}=\omega_{n}\cdots\omega_{k}\in I_{n}^{k}\ \text{ and }\
\phi_{\omega|_{n}^{k}}=\phi_{\omega_{n}}^{(n)}\circ\cdots\circ\phi_{\omega_{k}}^{(k)}.$

A non-autonomous conformal IFS on $X$ is a sequence
$\Phi=(\Phi^{(n)})_{n=1}^{\infty}$, $\Phi^{(n)}=\\{\phi_{i}^{(n)}\\}_{i\in
I^{(n)}}$ of collections of conformal self maps of $X$ which satisfies the
following conditions:

  * (A1)

(Open set condition) For all $n\in\mathbb{N}$ and all distinct indices $i,j\in
I^{(n)},$

$\phi_{i}^{(n)}(X^{\circ})\cap\phi_{j}^{(n)}(X^{\circ})=\emptyset.$

  * (A2)

(Conformality) There exists a connected open set $\tilde{X}$ of
$\mathbb{R}^{d}$ containing $X$ such that each $\phi_{i}^{(n)}$ extends to a
$C^{1}$ conformal diffeomorphism
$\tilde{\phi}_{i}^{(n)}\colon\tilde{X}\to\tilde{\phi}_{i}^{(n)}(\tilde{X})\subset\tilde{X}$.

  * (A3)

(Bounded distortion) There exists $C\geq 1$ such that for all $\omega\in
I^{\infty}$ and all $n$, $k\in\mathbb{N}$ with $n\leq k$,

$|D\tilde{\phi}_{\omega|_{n}^{k}}(x_{1})|\leq
C|D\tilde{\phi}_{\omega|_{n}^{k}}(x_{2})|\ \text{ for all
}x_{1},x_{2}\in\tilde{X}.$

  * (A4)

(Uniform contraction) There are constants $0<\gamma<1$ and $L\geq 1$ such that
for all $\omega\in I^{\infty}$ and all $n,k\in\mathbb{N}$ with $k-n\geq L$,

$\|D\phi_{\omega|_{n}^{k}}\|_{X}\leq\gamma^{k-n+1}.$

If $\Phi=(\Phi^{(n)})_{n=1}^{\infty}$ is a non-autonomous conformal IFS, then
(A4) ensures that the set $\bigcap_{n=1}^{\infty}\phi_{\omega|_{1}^{n}}(X)$ is
a singleton for each $\omega\in I^{\infty}$. We define an address map
$\Pi\colon I^{\infty}\to X$ by

$\Pi(\omega)\in\bigcap_{n=1}^{\infty}\phi_{\omega|_{1}^{n}}(X),$

and the limit set of $\Phi$ by

$\Lambda(\Phi)=\Pi(I^{\infty}).$

In order to determine the Hausdorff dimension of the limit set, for $s\geq 0$
we introduce a partition function

$Z_{n}^{\Phi}(s)=\sum_{\omega\in I_{1}^{n}}(\|D\phi_{\omega}\|_{X})^{s},$

and a lower pressure function
$\underline{P}^{\Phi}\colon[0,\infty)\to[-\infty,\infty]$ of $\Phi$ by

$\underline{P}^{\Phi}(s)=\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\log
Z_{n}^{\Phi}(s).$

The lower pressure function has the following monotonicity [20, Lemma 2.6]: if
$0\leq s_{1}<s_{2}$ then
$\underline{P}^{\Phi}(s_{1})=\underline{P}^{\Phi}(s_{2})=\infty$ or
$\underline{P}^{\Phi}(s_{1})=\underline{P}^{\Phi}(s_{2})=-\infty$ or
$\underline{P}^{\Phi}(s_{1})>\underline{P}^{\Phi}(s_{2})$. So, one can define
a critical value

$s(\Phi)={\rm sup}\\{s\geq 0\colon\underline{P}^{\Phi}(s)>0\\}={\rm
inf}\\{s\geq 0\colon\underline{P}^{\Phi}(s)<0\\},$

called the Bowen dimension. Note that (A1) and (A3) together imply
$s(\Phi)\leq d$. We say the non-autonomous conformal IFS $\Phi$ is
subexponentially bounded if

$\lim_{n\to\infty}\frac{1}{n}\log\\#I^{(n)}=0.$

We will use the following result in Section 3.2.

######  Theorem 2.2 (Bowen’s formula, [20, Theorem 1.1]).

Let $\Phi$ be a non-autonomous conformal IFS that is subexponentially bounded.
Then

$\dim_{\rm H}\Lambda(\Phi)=s(\Phi).$

###  2.3. An extendibility and bounded distortion

Below we show that each map in the infinite IFS
$\\{\varphi_{1,i}\\}_{i=1}^{\infty}\cup\\{\varphi_{-1,i}\\}_{i=2}^{\infty}$
admits a $C^{2}$ conformal extension to the interval

$\tilde{X}=\left(-\frac{1}{5},\frac{5}{4}\right)\supset[0,1].$

######  Lemma 2.3.

The following hold:

  * (a)

For each $i\geq 1$, $\varphi_{1,i}$ and $\varphi_{1,i+1}$ extend to $C^{2}$
diffeomorphisms
$\tilde{\varphi}_{1,i}\colon\tilde{X}\to\tilde{\varphi}_{1,i}(\tilde{X})\subset\tilde{X}$
and
$\tilde{\varphi}_{-1,i+1}\colon\tilde{X}\to\tilde{\varphi}_{-1,i+1}(\tilde{X})\subset\tilde{X}$
respectively.

  * (b)

For all $i\geq 3$ we have

$\|D\tilde{\varphi}_{1,i}\|_{\tilde{X}}\leq\frac{1}{2}\ \text{ and }\
\|D\tilde{\varphi}_{-1,i}\|_{\tilde{X}}\leq\frac{1}{2}.$

###### Proof.

We set $\tilde{\varphi}_{1,i}(x)=1/(i+x),\ x\in\tilde{X}$ for $i\geq 1$ and
$\tilde{\varphi}_{-1,i}(x)=1/(i-x),\ x\in\tilde{X}$ for $i\geq 3$. Then
$\tilde{\varphi}_{1,i}$ and $\tilde{\varphi}_{1,-i}$ are $C^{2}$
diffeomorphisms onto their images and satisfy

$\tilde{\varphi}_{1,i}(\tilde{X})=\left(\frac{1}{i+5/4},\frac{1}{i-1/{5}}\right)\subset\tilde{X}\text{
and }$
$\tilde{\varphi}_{-1,i}(\tilde{X})=\left(\frac{1}{i+1/{5}},\frac{1}{i-5/4}\right)\subset\tilde{X}.$

The remaining map $\varphi_{-1,2}$ needs a different treatment since it has
$1$ as a neutral fixed point. Let
$\tilde{\varphi}_{-1,2}\colon\tilde{X}\to\mathbb{R}$ be a $C^{2}$
diffeomorphism onto its image such that $\tilde{\varphi}_{-1,2}(x)=1/(2-x)$
for $x\in(-1/{5},1]$ and $\tilde{\varphi}_{-1,2}(5/4)<5/4$. Then
$\tilde{\varphi}_{-1,2}(\tilde{X})\subset\tilde{X}$ holds. We have verified
(a). Item (b) follows from direct calculations. $\Box$

For the rest of this paper, we denote the $C^{2}$ conformal extensions
$\tilde{\varphi}_{1,i}$, $\tilde{\varphi}_{-1,i}$ in Lemma 2.3 by
$\varphi_{1,i}$, $\varphi_{-1,i}$ for ease of notation. We now establish a
bounded distortion property for these extensions. For $L>1$ put
$\mathbb{N}_{\geq L}=\\{n\in\mathbb{N}\colon n\geq L\\}$.

######  Lemma 2.4.

There exists $C\geq 1$ such that for all $\sigma\in\\{-1,1\\}^{\mathbb{N}}$,
all $a_{1}a_{2}\cdots\in({\mathbb{N}_{\geq 3}})^{\mathbb{N}},$ all
$k\in\mathbb{N}$ and for all $x,y\in\tilde{X}$ we have

$|D(\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{k},a_{k}})(x)|\leq
C|D(\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{k},a_{k}})(y)|.$

###### Proof.

By Lemma 2.3(b), for $\sigma\in\\{-1,1\\}^{\mathbb{N}}$,
$a_{1}a_{2}\cdots\in(\mathbb{N}_{\geq 3})^{\mathbb{N}}$, $k\in\mathbb{N}$,
$1\leq j\leq k$ and for $x,y\in\tilde{X}$ we have

$|\varphi_{\sigma_{j},a_{j}}\circ\cdots\circ\varphi_{\sigma_{k},a_{k}}(x)-\varphi_{\sigma_{j},a_{j}}\circ\cdots\circ\varphi_{\sigma_{k},a_{k}}(y)|\leq\left(\frac{1}{2}\right)^{k-j}|\tilde{X}|.$

A direct calculation shows that there is a constant $C_{0}>0$ such that
$\|D\log|D\varphi_{1,i}|\|\leq C_{0}$ and $\|D\log|D\varphi_{-1,i}|\|\leq
C_{0}$ for all $i\in{\mathbb{N}_{\geq 3}}$. Therefore we obtain

$\begin{split}\log\frac{|D(\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{k},a_{k}})(x)|}{|D(\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{k},a_{k}})(y)|}&\leq\sum_{j=1}^{k}C_{0}\left(\frac{1}{2}\right)^{k-j}|\tilde{X}|\leq
2C_{0}|\tilde{X}|.\end{split}$

The desired inequality holds if we set $C=\exp(2C_{0}|\tilde{X}|)$. $\Box$

##  3\. The Hausdorff dimension

Using the preliminary results in Section 2, we prove main upper and lower
bounds of Hausdorff dimension in Sections 3.1 and 3.2 respectively. In Section
3.3 we combine both bounds and complete the proofs of the Main Theorem.

###  3.1. The upper bound

We slightly modify the argument by Hirst [9, Corollary 1] and prove the
following upper bound of Hausdorff dimension.

######  Proposition 3.1.

Let $\sigma\in\\{-1,1\\}^{\mathbb{N}}$ and let $B$ be an infinite subset of
$\mathbb{N}$. We have

$\dim_{\rm H}E_{\sigma}(B)\leq\frac{\tau(B)}{2}.$

###### Proof.

Let $\varepsilon>0$. Let $L=L(\varepsilon)\geq 3$ be a sufficiently large
integer such that

(9) $(2C)^{\frac{1}{2}(\tau(B)+\varepsilon)}\sum_{k\in B\cap\mathbb{N}_{\geq
L}}\frac{1}{k^{\tau(B)+\varepsilon}}\leq 1,$

where $C>1$ is the distortion constant in Lemma 2.4. We construct a family of
coverings of the set

$E_{\sigma,L}(B)=\\{x\in E_{\sigma}(B)\colon a_{\sigma,n}(x)\geq L\text{ for
all }n\in\mathbb{N}\\},$

by fundamental intervals $[a_{1},\ldots,a_{n}]_{\sigma}$ satisfying
$a_{1}\cdots a_{n}\in(B\cap\mathbb{N}_{\geq L})^{n}$. Lemma 2.4 gives

$\begin{split}\frac{|[a_{1},\ldots,a_{n},a_{n+1}]_{\sigma}|}{|[a_{1},\ldots,a_{n}]_{\sigma}|}&\leq
C\frac{|(\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{n},a_{n}})^{-1}([a_{1},\ldots,a_{n},a_{n+1}]_{\sigma})|}{|(\varphi_{\sigma_{1},a_{1}}\circ\cdots\circ\varphi_{\sigma_{n},a_{n}})^{-1}([a_{1},\ldots,a_{n}]_{\sigma})|}\\\
&=C|\varphi_{\sigma_{n+1},a_{n+1}}([0,1])|\leq\max\\{1,1-\sigma_{n+1}\\}\frac{C}{a_{n+1}^{2}}\leq\frac{2C}{a_{n+1}^{2}}.\end{split}$

Then we have

$\sum_{a_{n+1}\in B\cap\mathbb{N}_{\geq
L}}\frac{|[a_{1},\ldots,a_{n},a_{n+1}]_{\sigma}|^{\frac{1}{2}(\tau(B)+\varepsilon)}}{|[a_{1},\ldots,a_{n}]_{\sigma}|^{\frac{1}{2}(\tau(B)+\varepsilon)}}\leq(2C)^{\frac{1}{2}(\tau(B)+\varepsilon)}\sum_{a_{n+1}\in
B\cap\mathbb{N}_{\geq L}}\frac{1}{a_{n+1}^{\tau(B)+\varepsilon}},$

which does not exceed $1$ by (9). It follows that

$\frac{\sum_{a_{1}\cdots a_{n+1}\in(B\cap\mathbb{N}_{\geq
L})^{n+1}}|[a_{1},\ldots,a_{n+1}]_{\sigma}|^{\frac{1}{2}(\tau(B)+\varepsilon)}}{\sum_{a_{1}\cdots
a_{n}\in(B\cap\mathbb{N}_{\geq
L})^{n}}|[a_{1},\ldots,a_{n}]_{\sigma}|^{\frac{1}{2}(\tau(B)+\varepsilon)}}\leq
1,$

and therefore

$\sum_{a_{1}\cdots a_{n+1}\in(B\cap\mathbb{N}_{\geq
L})^{n+1}}|[a_{1},\ldots,a_{n+1}]_{\sigma}|^{\frac{1}{2}(\tau(B)+\varepsilon)}\leq
1.$

Since $\sup\\{|[a_{1},\ldots,a_{n}]_{\sigma}|\colon a_{1}\cdots
a_{n}\in(B\cap\mathbb{N}_{\geq L})^{n}\\}\to 0$ as $n\to\infty$, we obtain
$\dim_{\rm H}E_{\sigma,L}(B)\leq(\tau(B)+\varepsilon)/2.$

The set $E_{\sigma}(B)$ is covered by the images of sets of the form
$E_{\sigma^{\prime},L}(B)$,
$\sigma^{\prime}=\sigma_{j}\sigma_{j+1}\cdots\in\\{-1,1\\}^{\mathbb{N}}$,
$j\geq 1$ under the infinite IFS
$\\{\varphi_{1,i}\\}_{i=1}^{\infty}\cup\\{\varphi_{-1,i}\\}_{i=2}^{\infty}$.
Since $\varphi_{1,i}$, $\varphi_{-1,i}$ are $C^{1}$ diffeomorphisms and
Hausdorff dimensions do not change under the action of bi-Lipschitz
homeomorphisms, we have $\dim_{\rm
H}E_{\sigma}(B)\leq(\tau(B)+\varepsilon)/2$. Since $\varepsilon>0$ is
arbitrary we obtain the desired inequality. $\Box$

###  3.2. The lower bound

Using the theory of non-autonomous conformal IFSs summarized in Section 2.2,
we prove the following lower bound of Hausdorff dimension.

######  Proposition 3.2.

Let $\sigma\in\\{-1,1\\}^{\mathbb{N}},$ let $B$ be an infinite subset of
$\mathbb{N}$ and let $f\colon\mathbb{N}\rightarrow[\min B,\infty)$ satisfy
$\lim_{n\to\infty}f(n)=\infty.$ Then

$\dim_{\rm H}G_{\sigma}(B,f)\geq\frac{\tau(B)}{2}.$

###### Proof.

We may assume $\tau(B)>0$, for otherwise there is nothing to prove. Let
$\varepsilon\in(0,\tau(B))$. As in [21], we define a strictly increasing
sequence $\\{b_{m}\\}_{m\in\mathbb{N}}$ in $B$ inductively as follows:
$b_{1}=\min B,$ and $b_{m+1}>b_{m}$ is the minimal in $B$ such that

(10) $\sum_{k\in B\cap[b_{m},b_{m+1})}k^{-\tau(B)+\varepsilon}\geq 1.$

Set

(11) $B_{m}=\begin{cases}\\{\min B\\}&\text{for }m=1,\\\
B\cap[b_{m},b_{m+1})&\text{for }m\geq 2.\end{cases}$

Note that $B=\bigcup_{m=1}^{\infty}B_{m}.$ Let $\\{t_{m}\\}_{m\in\mathbb{N}}$
be a sequence of positive integers such that for every $m\geq 2$ we have

(12) $b_{m+1}\leq\inf\left\\{f(n)\colon\sum_{j=1}^{m-1}t_{j}+1\leq
n\leq\sum_{j=1}^{m}t_{j}\right\\},$

and

(13)
$\lim_{m\to\infty}\frac{\log\\#B_{m}}{\sum_{j=1}^{m}t_{j}}=\lim_{m\to\infty}\frac{\log\\#B_{m}}{\sum_{j=1}^{m-1}t_{j}+1}=0.$

Since $\lim_{n\to\infty}f(n)=\infty$, one can choose such a sequence by
induction on $m$.

Put $X=[0,1]$. We now construct a non-autonomous conformal IFS on $X$. For
each integer $1\leq n\leq t_{1}$, we set

$I^{(n)}=B_{1}.$

For each integer $n\geq t_{1}+1$ we pick $m\geq 2$ such that
$\sum_{j=1}^{m-1}t_{j}+1\leq n\leq\sum_{j=1}^{m}t_{j}$, and set

$I^{(n)}=B_{m}.$

Put $I^{\infty}=\prod_{j=1}^{\infty}I^{(j)}$ and
$I^{k}_{n}=\prod_{j=n}^{k}I^{(j)}$ as in (7). For $n\geq t_{1}+1$ we have

$I_{1}^{n}=(B_{1})^{t_{1}}\times\cdots\times(B_{m-1})^{t_{m-1}}\times(B_{m})^{l_{n}},$

where $l_{n}=n-\sum_{j=1}^{m-1}t_{j}$ and

$I^{\infty}=(B_{1})^{t_{1}}\times(B_{2})^{t_{2}}\times\cdots\times(B_{m})^{t_{m}}\times\cdots.$

Let $n\in\mathbb{N}$. For each $i\in I^{(n)}$ we put
$\phi_{i}^{(n)}=\varphi_{\sigma_{n},i},$ and set
$\Phi^{(n)}=\\{\phi^{(n)}_{i}\\}_{i\in I^{(n)}}.$ For
$\omega=\omega_{1}\omega_{2}\cdots\in I^{\infty}$ and $n,k\in\mathbb{N}$ with
$n\leq k$, we set $\omega|_{n}^{k}=\omega_{n}\cdots\omega_{k}\in I_{n}^{k}$
and
$\phi_{\omega|_{n}^{k}}=\phi_{\omega_{n}}^{(n)}\circ\cdots\circ\phi_{\omega_{k}}^{(k)}$
as in (8). Finally we set $\Phi=\\{\Phi^{(n)}\\}_{n=1}^{\infty}.$ Then (A1)
follows from (4), (A2) follows from Lemma 2.3, and (A3) follows from Lemma
2.4. Since $\|D\phi^{(n)}_{i}\|_{X}=\|D\varphi_{\sigma_{n},i}\|_{X}\leq 1/4$
for all $n\geq t_{1}+t_{2}+1$ and all $i\in I^{(n)},$ (A4) holds. Therefore
$\Phi$ is a non-autonomous conformal IFS on $X$.

######  Lemma 3.3.

The limit set $\Lambda(\Phi)$ of $\Phi$ is contained in $G_{\sigma}(B,f)$.

###### Proof.

Let $\Pi$ denote the address map of the non-autonomous conformal IFS $\Phi$.
Let $\omega\in I^{\infty}$. Then $a_{\sigma,n}(\Pi(\omega))\in B$ for all
$n\geq 1$. The first alternative of (11) gives

$a_{\sigma,n}(\Pi(\omega))=\min B\leq f(n)\ \mbox{ if }1\leq n\leq t_{1}.$

For every $m\geq 2$, the second alternative of (11) and (12) yield

$b_{m}\leq a_{\sigma,n}(\Pi(\omega))<b_{m+1}\leq f(n)\ \mbox{ if
}\sum_{j=1}^{m-1}t_{j}+1\leq n\leq\sum_{j=1}^{m}t_{j}.$

As $n\to\infty$ we have $m\to\infty$, $b_{m}\to\infty$ and so
$a_{\sigma,n}(\Pi(\omega))\to\infty$. Hence $\Pi(\omega)\in G_{\sigma}(B,f)$
holds. Since $\omega\in I^{\infty}$ is arbitrary we obtain
$\Lambda(\Phi)\subset G_{\sigma}(B,f)$. $\Box$

######  Lemma 3.4.

The non-autonomous conformal IFS $\Phi$ is subexponentially bounded.

###### Proof.

Recall that for each integer $n\geq t_{1}+1$ we have
$\sum_{j=1}^{m-1}t_{j}+1\leq n$. Then

$0\leq\frac{1}{n}\log\\#I^{(n)}\leq\frac{\log\\#B_{m}}{\sum_{j=1}^{m-1}t_{j}+1}.$

From this and (13) we obtain $\lim_{n\to\infty}(1/n)\log\\#I^{(n)}=0$. $\Box$

Recall that the lower pressure function
$\underline{P}^{\Phi}\colon[0,\infty)\to[-\infty,\infty]$ is given by

$\underline{P}^{\Phi}(s)=\liminf\limits_{n\rightarrow\infty}\frac{1}{n}\log
Z_{n}^{\Phi}(s),\text{ where }Z_{n}^{\Phi}(s)=\sum_{\omega\in
I_{1}^{n}}(\|D\phi_{\omega}\|_{X})^{s}.$

By Lemmas 3.3, 3.4 and Theorem 2.2, the Bowen dimension $s(\Phi)$ satisfies

(14) $\dim_{\rm H}G_{\sigma}(B,f)\geq\dim_{\rm H}\Lambda(\Phi)=s(\Phi).$

In order to estimate the Bowen dimension from below, we estimate the lower
pressure function from below. From Lemma 2.4, there exists a constant $C>1$
such that for all $n\geq 1$, all $i\in I^{(n)}$ and for all $x\in X$ we have

(15) $|D\phi_{i}^{(n)}(x)|\geq
C^{-1}|D\phi_{i}^{(n)}(0)|=C^{-1}\frac{1}{k^{2}},$

where $k\geq 1$ is the integer such that
$\phi_{i}^{(n)}\in\\{\varphi_{1,k},\varphi_{-1,k}\\}$. Since $b_{m}\to\infty$
as $m\to\infty$, for any $\delta>0$ there exists $N\geq 1$ such that for all
$i\geq b_{N+1}$ we have

(16) $C^{-1}i^{-2}\geq i^{-2-\delta}.$

Using the chain rule, (15) and (16) we have

$\begin{split}Z_{n}^{\Phi}(s)=&\sum_{\omega\in
I_{1}^{n}}(\|D(\varphi_{\sigma_{1},\omega_{1}}\circ\varphi_{\sigma_{2},\omega_{2}}\circ\cdots\circ\varphi_{\sigma_{n},\omega_{n}})\|_{X})^{s}\\\
\geq&C^{-Ns}\sum_{\omega\in
I_{1}^{n}}{\omega_{1}}^{-2s}\cdots{\omega_{t_{1}+\cdots+t_{N}}}^{-2s}{\omega_{t_{1}+\cdots+t_{N}+1}}^{-(2+\delta)s}\cdots{\omega_{n}}^{-(2+\delta)s}\\\
=&C^{-Ns}(\min B)^{-2st_{1}}\left(\sum_{k\in
B_{2}}k^{-2s}\right)^{t_{2}}\cdots\left(\sum_{k\in
B_{N}}k^{-2s}\right)^{t_{N}}\\\ &\times\left(\sum_{k\in
B_{N+1}}k^{-(2+\delta)s}\right)^{t_{N+1}}\cdots\left(\sum_{k\in
B_{m}}k^{-(2+\delta)s}\right)^{l_{n}}.\end{split}$

Set $s_{\varepsilon,\delta}=(\tau(B)-\varepsilon)/(2+\delta)$. By (10) we have

$\left(\sum_{k\in
B_{N+1}}k^{-(2+\delta)s_{\varepsilon,\delta}}\right)^{t_{N+1}}\cdots\left(\sum_{k\in
B_{m}}k^{-(2+\delta)s_{\varepsilon,\delta}}\right)^{l_{n}}\geq 1.$

Substituting $s=s_{\varepsilon,\delta}$ and combining the above two
inequalities yield

$Z_{n}^{\Phi}(s_{\varepsilon,\delta})\geq C^{-Ns_{\varepsilon,\delta}}(\min
B)^{-2s_{\varepsilon,\delta}t_{1}}\left(\sum_{k\in
B_{2}}k^{-2s_{\varepsilon,\delta}}\right)^{t_{2}}\cdots\left(\sum_{k\in
B_{N}}k^{-2s_{\varepsilon,\delta}}\right)^{t_{N}}.$

It follows that
$\underline{P}^{\Phi}(s_{\varepsilon,\delta})=\liminf_{n\to\infty}(1/n)\log
Z_{n}^{\Phi}(s_{\varepsilon,\delta})\geq 0$, and hence

(17) $s(\Phi)\geq\frac{\tau(B)-\varepsilon}{2+\delta}.$

Combining (14) and (17), and letting $\delta\to 0$ and then $\varepsilon\to 0$
yields the desired inequality in Proposition 3.2. $\Box$

###  3.3. Proof of the Main Theorem

Let $B$ be an infinite subset of $\mathbb{N}$ and let
$f\colon\mathbb{N}\to[\min B,\infty)$ satisfy $\lim_{n\to\infty}f(n)=\infty$.
Since $E_{\sigma}(B)\supset G_{\sigma}(B,f)$ holds for all
$\sigma\in\\{-1,1\\}^{\mathbb{N}}$, Propositions 3.1 and 3.2 together yield
$\dim_{\rm H}E_{\sigma}(B)=\dim_{\rm H}G_{\sigma}(B,f)=\tau(B)/2$. $\Box$

### Acknowledgments

We are grateful to Mumtaz Hussain for fruitful discussions. This research was
partially supported by the JSPS KAKENHI 19K21835, 20H01811.

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