# Can we control the amount of useful nonclassicality in a photon added
hypergeometric state?

Priya Malpani¶, Kishore Thapliyal‡,, Anirban Pathak¶,
¶Jaypee Institute of Information Technology, A-10, Sector-62, Noida,
UP-201309, India
‡RCPTM, Joint Laboratory of Optics of Palacky University and Institute of
Physics
of Academy of Science of the Czech Republic, Faculty of Science, Palacky
University,
17\. listopadu 12, 771 46 Olomouc, Czech Republic Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Non-Gaussianity inducing operations are studied in the recent past from
different perspectives. Here, we study the role of photon addition, a non-
Gaussianity inducing operation, in the enhancement of nonclassicality in a
finite dimensional quantum state, namely hypergeometric state with the help of
some quantifiers and measures of nonclassicality. We observed that measures to
characterize the quality of single photon source and anticlassicality lead to
the similar conclusion, i.e., to obtain the desired quantum features one has
to choose all the state parameters such that average photon numbers remains
low. Wigner logarithmic negativity of the photon added hypergeometric state
and concurrence of the two-mode entangled state generated at the output of a
beamsplitter from this state show that nonclassicality can be enhanced by
increasing the state parameter and photon number addition but decreasing the
dimension of the state. In principle, decreasing the dimension of the state is
analogous to holeburning and is thus expected to increase nonclassicality.
Further, the variation of Wigner function not only qualitatively illustrates
the same features as observed quantitatively through concurrence potential and
Wigner logarithimic negativity, but illustrate non-Gaussianity of the quantum
state as well.

##  1 Introduction

On the verge of second quantum revolution [1], nonclassical states are
essential elements to establish quantum dominance. Literature enlightens, a
lot of work has been performed on the study of generation of various
nonclassical states and their properties ([2, 3, 4] and references therein).
To be specific, the quantum states with non-positive Glauber-Sudarshan $P$
function [5, 6], called nonclassical states, are at the core of the second
quantum revolution [1, 7]. The nonclassical states are extremely important for
obtaining quantum advantage in computing [8, 9, 10], communication [11, 12],
cryptography [13], metrology [14, 15], simulation [16, 17], sensing [18, 19],
etc. For instance, many of these applications require squeezed states, having
uncertainty in one of the quadratures less than the corresponding value for
coherent state; entangled states, which cannot be written as a product of
quantum states of individual subsystems; and on-demand single photon sources,
which are necessarily antibunched states.

Generally, a finite dimensional quantum state is referred to as qudit (in a
$d$-dimensional Hilbert space), which can be expressed in the Fock basis as

$\left|\psi_{d}\right\rangle=N_{d}\sum_{n=0}^{d-1}c_{n}\left|n\right\rangle,$
(1)

where $N_{d}$ is the normalization constant, and
$\left|c_{n}\right|^{2}=p_{n}$ is the photon number distribution of the state.
Thus, we know $p_{n}=0\,\forall n\geq d$, which is termed as a hole in the
photon number distribution. Additionally, we know in terms of the Glauber-
Sudarshan $P$ function, we can write the photon number distribution as

$p_{n}=\int P\left(\alpha\right)\left|\left\langle
n|\alpha\right\rangle\right|^{2}d^{2}\alpha,$ (2)

which allows us to conclude that $P\left(\alpha\right)<0$ if $p_{n}=0$ for any
value of $n$. The motivation to study finite dimensional (qudit) states lies
in the fact that some of these states can be reduced mathematically to the
most nonclassical (Fock state) as well as the most classical (coherent state)
states in the limiting cases. Most significant examples of finite dimensional
(qudit) states worth mentioning are a class of intermediate states, such as
hypergeometric [20], binomial [21], negative binomial [22], vacuum filtered
binomial [23], negative hypergeometric [24], photon added binomial [23],
shadowed [25] and shadowed-like [26] negative binomial states, etc. Initially
the investigations on these states were limited to theoretical studies, but
with the advent of advanced experimental techniques, a few groups have
succeeded in generating such states [27, 28, 29] and possibly more to follow.
Due to several applications of qudit states various nonclassical properties
(both lower- and higher-order antibunching, squeezing, sub-Poissonian photon
statistics, etc.) have been studied very rigorously [26, 30, 31, 23]. However,
most of these works were focused on the witnesses of nonclassicality and
relatively less effort has been given on the quantitative aspects of
nonclassicality.

Hypergeometric state is a one-parameter generalization of binomial state and
is also studied in reference to deformed oscillator algebra [20]. Non-
Gaussianity inducing operations, such as photon addition, subtraction, vacuum
filtration, quantum scissors, photon catalysis are used as nonclassicality
inducing and enhancing operations in several quantum state engineering
proposals. Specifically, advantage of photon addition and subtraction in
application of quantum phase in quantum metrology [32], quantum scissors in
teleportation and cryptography [33, 34, 35], entanglement purification ([36]
and references therein), nonclassicality and non-Gaussianity enhancement [37,
32, 23], are reported recently. Further, non-Gaussian states obtained by such
quantum state engineering operations have applications in continuous variable
quantum cryptography and computation ([38, 39, 40] and references therein).
The continuous variable quantum cryptography is considered a better candidate
than corresponding discrete variable counterparts for a metropolitan network
[41] and non-Gaussian operations can further enhance this performance (for
example, see [42, 43]). Moreover, recently engineered qudit states have been
used experimentally in quantum walk [29], which may be a useful resource for
quantum walk based quantum cryptographic schemes [44].

Motivated by above mentioned facts, here, we aim to study the role of photon
addition in the enhancement of nonclassicality in hypergeometric state. The
choice of this state is particularly important as it holds the potential to be
used for realizing all the known applications of finite dimensional states in
quantum technology, and a large set of finite and infinite dimensional quantum
states can be reduced in the limiting cases of the hypergeometric state
(discussed in detail in next section). Specifically, in this paper, we aim to
study nonclassical features of photon added hypergeometric state (PAHS), in
terms of a few quantitative measures of quantumness, namely characterization
of the quality of the single photon source, anticlassicality, concurrence
potential, and Wigner logarithmic negativity. We will further reduce the
corresponding results for a set of other quantum states, composing
hypergeometric, binomial, photon added binomial, coherent, and photon added
coherent states, in the limiting cases.

The rest of the paper is organized as follows. In Section 2, we discuss all
the properties and Wigner function of PAHS and its limiting cases. We
summarize our findings regarding nonclassicality quantifiers to characterize
single photon source, anticlassicality, concurrence potential, and Wigner
logarithmic negativity in the subsequent section. Finally, we conclude our
results in Section 4.

##  2 State to be investigated: Photon added hypergeometric state

Hypergeometric state was introduced by Fu and Sasaki [20] in 1996. It can be
expressed as the superposition of number states in the $M+1$-dimensional space
analogous to Eq. (1) as

$\left|L,M,\eta\right\rangle=\sum_{n=0}^{M}\left[\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\right]^{\frac{1}{2}}\tbinom{L}{M}^{-\frac{1}{2}}\left|n\right\rangle,$
(3)

where probability $\eta\in\left[0,1\right]$ and
$L\geq\max\left\\{M\eta^{-1},M\left(1-\eta\right)^{-1}\right\\}$ are the real
parameters. Also, $\tbinom{N}{l}$ is the binomial coefficient. The quantum
state has photon number distribution $\left|\left\langle n\mid
L,M,\eta\right\rangle\right|^{2}=\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\tbinom{L}{M}^{-1}$
defined as hypergeometric distribution. Note that the hypergeometric
distribution is closely related to the binomial distribution. The former
involves the probability of $n$ successful cases in $M$ draws without
replacement from a total population of $L$ with only $L\eta$ having the
desired feature. While the latter involves the probability of $n$ successful
cases in $M$ draws with replacement. Thus, in the limits of
$L\rightarrow\infty$, the total population size would not change with each
draw and hypergeometric distribution reduces to the binomial distribution.

This quantum state can be transformed to obtain $k$-PAHS by repeatedly
applying a creation operator on the hypergeometric state. The analytical
expression in this case is given by

$\left|L,M,\eta,k\right\rangle=a^{\dagger
k}\left|L,M,\eta\right\rangle=N_{{\rm
PAHS}}\sum_{n=0}^{M}\left[\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\right]^{\frac{1}{2}}\tbinom{L}{M}^{-\frac{1}{2}}\sqrt{\frac{\left(n+k\right)!}{n!}}\left|n+k\right\rangle.$
(4)

where $k$ is the number of photons added. We need to renormalize the state as
photon addition is a non-unitary operation, and the normalization constant for
PAHS can be computed as

$N_{{\rm
PAHS}}=\left\\{\sum_{n=0}^{M}\left[\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\right]\tbinom{L}{M}^{-1}\frac{\left(n+k\right)!}{n!}\right\\}^{-\frac{1}{2}}.$
(5)

Lower- and higher-order antibunching, sub-Poissonian photon statistics and
squeezing in hypergeometric state are reported in [20]. Here, our objective is
to investigate the role of photon addition in enhancement in the nonclassical
features of the state qualitatively using Wigner function [45] as well as with
respect to a set of measures to characterize the quality of single photon
source [46], anticlassicality [47], concurrence potential [48, 49], and Wigner
logarithmic negativity [50].

Before we proceed further it is imperative to mention that the PAHS can be
reduced to a set of finite and infinite dimensional quantum states in limiting
cases (as summarized in Fig. 1). For instance, hypergeometric state can be
obtained from PAHS for $k=0$. As hypergeometric state can be reduced to the
binomial state $\left|\varphi\right\rangle$ in the limits of
$L\rightarrow\infty$, so PAHS gives us photon added binomial state
$\left|\varphi^{\prime}\right\rangle$ in that limit. Further, considering
$M\eta=\alpha$ in the limits of $M\rightarrow\infty$, we can also obtain
coherent state and photon added coherent state with real $\alpha$ parameter
from binomial and photon added binomial states, respectively. Interestingly,
Fock state $\left|M\right\rangle$ and vacuum $\left|0\right\rangle$ can also
be obtained from the binomial state by considering $\eta=1$ and $\eta=0$,
respectively. These limiting cases add value to the present study by clearly
indicating a general nature of the present study. Specifically, as special
cases of the present study, nonclassical and non-Gaussian features of the
limiting states can be visualized and quantified with ease. Some of these
states are already shown to be useful in continuous variable quantum
computation and communication [51, 52], quantum phase estimation [53],
implementation of CNOT gate [28], etc.

Figure 1: (Color online) The complete set of quantum states that can be
reduced from PAHS in the limiting cases. Here, $\left|\varphi\right\rangle$
and $\left|\varphi^{\prime}\right\rangle$ corresponds to binomial and photon
added binomial states, respectively.

Generation of different types of binomial states is proposed theoretically
[54] as well as performed experimentally [27, 28, 29]. Specifically, an
intermediate state can be prepared in a cavity using Jaynes-Cummings
Hamiltonian by passing $M$ atoms through the cavity initially at vacuum ([55]
and references therein). The experimental parameters can be obtained
numerically to generate target state at a target time with the post-selection
of all the atoms in the ground state (which will disentangle it from the
cavity mode) [56]. Therefore, using the same method the hypergeometric state
can be prepared. Further, $k$-photon addition on the cavity mode in
hypergeometric state can be performed by sending $k$ atoms prepared in the
excited state through the cavity and measured subsequently in the ground state
[57]. Independently, photon addition can be performed using spontaneous
parametric down conversion and conditioning on the measurement of one of the
down converted modes [58].

###  2.1 Wigner function of the state

Nonclassicality in phase space is represented by the negative values of
quasiprobability distributions [4, 59], such as Glauber-Sudarshan $P$ function
[5, 6], Wigner function [45]. Here, we discuss the Wigner function to show the
nonclassicality in PAHS qualitatively and subsequently quantify the negative
volume of Wigner function as a measure of nonclassicality [60].

The Wigner function of a pure quantum state in position-momentum space is
defined as [45]

$\begin{array}[]{lcl}W(x,p)&=&\frac{1}{\pi}\intop_{-\infty}^{\infty}\psi^{*}(x+y)\psi(x-y)\exp\left(2ipy\right)dy.\end{array}$
(6)

To compute Wigner function for PAHS we write the state in position space using
the definition of Fock state $|n\rangle$ in the position space [59]

$|n\rangle=\phi_{n}(x)=b_{n}{\rm e}^{-\frac{x^{2}}{2}}H_{n}(x),$ (7)

where $b_{n}=\frac{1}{\sqrt{\pi^{\frac{1}{2}}2^{n}n!}}$ in the units of
$\hbar=1$, and $H_{n}(x)$ is the Hermite polynomial. Consequently, we can
express the finite dimensional qudit state (1) as

$|\psi_{d}(x)\rangle=N_{d}\sum_{n=0}^{d-1}c_{n}|n\rangle=\sum_{n=0}^{d-1}c_{n}\phi_{n}(x).$
(8)

Using Eqs. (6)-(8) we obtain the Wigner function of PAHS as

$\begin{array}[]{lcl}W(x,p)&=&\frac{{\rm
e}^{-\left(x^{2}+p^{2}\right)}}{\pi}\sum\limits_{n,n^{\prime}=0}^{M}N_{{\rm
PAHS}}^{2}\left[\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\tbinom{L\eta}{n^{\prime}}\tbinom{L\left(1-\eta\right)}{M-n^{\prime}}\right]^{\frac{1}{2}}\tbinom{L}{M}^{-1}\sqrt{\frac{\left(n+k\right)!\left(n^{\prime}+k\right)!}{n!n^{\prime}!}}\\\
&\times&(-1)^{n^{\prime}+k}\sqrt{2^{n^{\prime}-n}}\sqrt{\frac{(n+k)!}{(n^{\prime}+k)!}}(ip-x)^{n^{\prime}-n}L_{n+k}^{n^{\prime}-n}\left(2x^{2}+2p^{2}\right),\,\,\,\,\,n+k\leq
n^{\prime}+k,\end{array}$ (9)

where $L_{m}^{a}\left(z\right)$ is generalized Laguerre polynomial.

The Wigner function illustrates nonclassicality in PAHS by the negative values
in Fig. 2. Specifically, we observed that the negative values of the Wigner
function are concentric circles, which are also the signatures of non-Gaussian
behavior of the Wigner function. The number of rings of the negative Wigner
function and non-Gaussian features are decided by the various parameters of
the state, i.e., the number of photon added, dimension, and probability
$\eta$. With an increasing dimension of the state the depth of Wigner minima
decreases while the number of rings increases (cf. Fig. 2 (b) and (c)).
Independently, photon addition increases the number of rings (cf. Fig. 2
(a)-(b)), while Wigner function becomes positive with a decrease in the value
of $\eta$ (cf. Fig. 2 (b) and (d)).

Figure 2:  (Color online) Wigner function of $k$-PAHS for (a) $k=0$ with
$M=10$, $\eta=0.9$, (b) $k=1$ with $M=10$, $\eta=0.9$, (c) $k=1$ with $M=5$,
$\eta=0.9$, and (d) $k=1$ with $M=10$, $\eta=0.75$. All the contour plots are
obtained for the same color scale (given at the bottom) with position $x$ and
momentum $p$ variables shown in the horizontal and vertical axes,
respectively.

##  3 Nonclassicality quantifiers and measures

Here, we quantify the role of the non-Gaussianity inducing operation, i.e.,
photon addition, in the enhancement of nonclassicality in the finite
dimensional intermediate state, i.e., PAHS. In what follows, we report the
quantifiers and measures of nonclassicality with the relevant analytical
expressions. To begin with we will discuss two nonclassicality quntifiers
(e.g., measure to characterize quality of single photon source and
anticlassicality) and subsequently we will discuss nonclassicality measures
(e.g., concurrence potential and Wigner logarithmic negativity).

|
---|---
(a) | (b)
|
(c) | (d)
Figure 3:  (Color online) The dependence of the measure to characterize the
quality of single photon source $\mu$ on (a) $\eta$ for different values of
$k$ in PAHS with $M=5$, (b) $\eta$ for different values of $M$ for single-
PAHS. (c) $\mu$ is compared with that for photon added binomial state and
binomial state as function of $\eta$. In (c), we have also shown $\mu$ for
coherent state and single photon added coherent state with $\alpha=M\eta$ for
$M\rightarrow\infty$. (d) Average photon number $\left\langle
n\right\rangle=\left\langle a^{\dagger}a\right\rangle$ with respect to $\eta$
for different $M$ and $k$ in PAHS.

###  3.1 Measure to characterize quality of single photon source

A quantitative measure which can characterize the quality of the single photon
source [46] can be defined mathematically as

$\mu=\frac{P_{1}}{1-\left(P_{0}+P_{1}\right)},$ (10)

where the probability $P_{i}$ of obtaining exactly $i$ photons in a pulse is
obtained from the corresponding photon number distribution $p_{m}$ as
$P_{i}=p_{m}\left(m=i\right)\forall i\in\left\\{0,1\right\\}$. Specifically,
this quantifies the ratio of single photon pulses with multiphoton pulses in
the concerned state and is expected to be large for a desirable quantum state
of light for single photon generation. The more is the value of $\mu$ for a
state, the better it is as a single photon source to be used in quantum
cryptography. In case of PAHS, the photon number distribution $p_{m}^{{\rm
PAHS}}$ can be obtained as

$\begin{array}[]{lcl}p_{m}^{{\rm PAHS}}&=&\left|\left\langle m\mid
L,M,\eta,k\right\rangle\right|^{2}\\\ &=&\begin{cases}\left|N_{{\rm
PAHS}}\tbinom{L\eta}{m-k}^{\frac{1}{2}}\tbinom{L\left(1-\eta\right)}{M-m+k}^{\frac{1}{2}}\tbinom{L}{M}^{-\frac{1}{2}}\sqrt{\frac{m!}{(m-k)!}}\right|^{2},&m\geq
k\\\ 0,&m<k.\end{cases}\end{array}$ (11)

We have already mentioned that results corresponding to a set of quantum
states can be obtained from Eq. (11) in the limiting cases. Notice that the
probability of detecting a pulse having photon number less than the number of
photon added is zero in Eq. (11). Thus, photon addition ($k>1$) is certainly
not providing any advantage in the context of single photon generation (cf.
Fig. 3 (a)). As far as $k=1$ is concerned, it also reduces the probability of
single photon pulses with respect to the corresponding hypergeometric state
(except at $\eta\rightarrow 0$ and $L\rightarrow\infty$ when the photon added
state reduces to Fock state $\left|n=1\right\rangle$). This is physically
expected and is consistent with our intuition. Additionally, with increase in
the dimension of qudit, average photon number $\left\langle
n\right\rangle=\left\langle a^{\dagger}a\right\rangle$ increases (shown in
Fig. 3 (d)) because the photon number distribution has altered. In other
words, the probability for higher photon number pulses is increasing at the
cost of that for the lower photon number pulses (shown in Fig. 4 (a)), and
therefore, the probability of single photon pulses decreases (cf. Fig. 3 (b)).
We also observed that hypergeometric state performs better than binomial state
as a single photon source for small values of $\eta$ (cf. Fig. 3 (c)). Further
photon addition in all these cases just reduces the quality of these states to
be used as a single photon source. This can also be visualized from the photon
number distribution of hypergeometric state and PAHS in Fig. 4 (a) that due to
a single photon addition probability of zero photon pulses becomes zero. We
have not discussed the case of $k>1$ as the probability of both zero and
single photon pulses becomes zero in that case. For the sake of completeness
we have also shown dependence of the average photon number on all the state
parameters in Fig. 3 (d) which illustrates a monotonous increase in
$\left\langle n\right\rangle$.

It is worth mentioning that the findings for an infinite dimensional state
(coherent and photon added coherent states in this case) cannot be compared
trivially with that of a finite dimensional state. However, in the present
case, photon addition reduces the quality of single photon source from the
coherent state as observed in case of PAHS. In view of the relatively
difficult state preparation, one may not be too optimistic to use the present
state as approximate single photon source. Notice that a single photon added
states reduces to ideal single photon source when the state parameter $\eta$
or $\alpha$ is close to zero. Thus, these photon added states should be a
better candidate for single photon source in comparison to the corresponding
parent state but this quantifier fails to capture this behavior in Fig. 3 (c).
However, the intrinsic beauty of this relatively simple measure will be
further illustrated in the next subsection where we will explore a connection
between $\mu$ and anticlassicality, a well studied concept introduced by
Dodonov et al. in 2003 [47]. The connection may add to the existing physical
meaning of anticlassicality.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 4:  (Color online) (a) The variation of photon number distribution
$p_{m}$ with photon number $m$ for different $\eta$, $M$, and photon addition
$k$. Variation of anticlassicality with (b) $M$ for different photon addition
and $\eta=0.18$ and (c) $\eta$. In (d), we have also shown $A$ for binomial
state and coherent state as well as their single photon added counterparts
with $\alpha=M\eta$ for $M\rightarrow\infty$.

###  3.2 Anticlassicality

Dodonov et al., introduced a measure of nonclassicality as the distance
between the nonclassical state to be studied and Fock state (the most
nonclassical state) [47]. Mathematically, anticlassicality for an arbitrary
quantum state $\rho$ is defined as

$A=\underset{m>0}{{\rm max}}\ p_{m}^{\rho}$ (12)

in terms of its photon number distribution $p_{m}^{\rho}$. In our case, $\rho$
is PAHS and therefore we use $p_{m}^{{\rm PAHS}}$ as defined in Eq. (11) to
obtain anticlassicality using Eq. (12). Incidentally, the variation of
anticlassicality measure shows that increasing $k$, $M$, and $\eta$ have the
same effect: anticlassicality is found to decrease with increase in all these
parameters (cf. Fig. 4 (b) and (c)). This is consistent with the observations
in [47] that anticlassicality decreases with the increasing average photon
number $\left\langle n\right\rangle=\left\langle a^{\dagger}a\right\rangle>1$
while it increases for $0<\left\langle n\right\rangle<1$. However, for the
large values of all these parameters anticlassicality does not change
considerably and thus becomes comparable as observed in Fig. 4 (b) and (c).
Interestingly, for $\eta=0$ PAHS behaves like Fock $\left|n=1\right\rangle$
and vacuum for $k=1$ and 0, respectively. Thus, anticlassicality is unity
(zero) in the former (latter) case. In Ref. [47], it is shown that allowing
$m=0$ as well in the definition (12) the parameter will be unity even in the
latter case. In this scenario, we will observe that modified parameter for
both HS and PAHS to decrease with $\eta$ as well similar to that observed in
variation of $\mu$. Therefore, the observed behavior of anticlassicality
maximized over complete Fock basis is similar to that of the measure of
quality of single photon source. We also observed that anticlassicality of
hypergeometric state is more (less) than that of the binomial state for small
(large) values of $\eta$ (cf. Fig. 4 (d)). Further photon addition in all
these cases increases the average photon number and anticlassicality can be
observed to decrease with $\eta$.

|
---|---
(a) | (b)
Figure 5: (Color online) Concurrence potential for $k$-PAHS as a function of
(a) $\eta$ with $M=5$ and (b) photon addition with $\eta=0.1$.

###  3.3 Concurrence potential

There are several measures of nonclassicality proposed for the purpose of
quantitative analysis of nonclassicality in a quantum state. All these
measures have some inherent limitations (see [61] for discussion). These
measures are often not monotone of each other and therefore, it becomes
important to study either more than one nonclassicality measure or mention the
quantifier used. One such measure, that we are going to use here, attempted to
quantify the amount of nonclassicality present in an arbitrary quantum state
using entanglement measures [49], namely entanglement potential. However,
entanglement measures are also not monotone of each other.

The idea of entanglement potential [49, 62, 63, 64, 65] is based on the fact
that classical states cannot generate entanglement in the output of the
beamsplitter, while mixing of an arbitrary nonclassical state with any
classical state (for example vacuum) on a beamsplitter generates an equivalent
amount of entanglement in the output. Without loss of generality, for the
total input state of the beamsplitter
$\left(|\psi\rangle\otimes|0\rangle\right)=\left(\stackrel{{\scriptstyle[}}{{n}}=0]{M}{\sum}c_{n}|n\rangle\right)\otimes|0\rangle$,
the output of the beamsplitter can be expressed in the Fock basis as

$|\Phi\rangle=U\left(|\psi\rangle\otimes|0\rangle\right)\equiv
U|\psi,0\rangle=\sum_{n=0}^{M}\,\frac{c_{n}}{2^{n/2}}\sum_{j=0}^{n}\sqrt{\tbinom{n}{j}}\,\,|j,\,n-j\rangle.$
(13)

Similarly, mixing PAHS and vacuum state at the beamsplitter, the post
beamsplitter state can be written as

$|\phi\rangle=N_{{\rm
PAHS}}\sum\limits_{n=0}^{M}\left[\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\right]^{\frac{1}{2}}\tbinom{L}{M}^{-\frac{1}{2}}\frac{1}{\sqrt{n!}}\sum\limits_{k_{1}=0}^{n+k}\tbinom{n+k}{k_{1}}\left(\frac{1}{\sqrt{2}}\right)^{k_{1}}\left(\frac{i}{\sqrt{2}}\right)^{n+k-k_{1}}\sqrt{k_{1}!\left(n+k-k_{1}\right)!}|k_{1},\,n+k-k_{1}\rangle,$
(14)

where $i^{2}=-1$. Concurrence is formally defined for a bipartite state
$\rho_{AB}$ as [49]

$C=\sqrt{2\left(1-{\rm Tr}\left(\rho_{B}^{2}\right)\right)},$ (15)

where $\rho_{B}$ is obtained after tracing over subsystem $B$. Mathematical
expression for ${\rm Tr}\left(\rho_{B}^{2}\right)$, in the present case with
$\rho_{AB}=|\phi\rangle\langle\phi|$, is

$\begin{array}[]{lcl}{\rm Tr}\left(\rho_{B}^{2}\right)&=&N_{{\rm
PAHS}}^{4}\sum\limits_{n=0}^{M}\sum\limits_{m=0}^{M}\sum\limits_{r=0}^{M}\left[\tbinom{L\eta}{n}\tbinom{L\left(1-\eta\right)}{M-n}\tbinom{L\eta}{m}\tbinom{L\left(1-\eta\right)}{M-m}\tbinom{L\eta}{r}\tbinom{L\left(1-\eta\right)}{M-r}\tbinom{L\eta}{n-m+r}\tbinom{L\left(1-\eta\right)}{M-n+m-r}\right]^{\frac{1}{2}}\\\
&\times&\left(\frac{1}{2}\right)^{n+2k+r}\tbinom{L}{M}^{-2}\sum\limits_{k_{1}=0}^{n+k}\frac{\left(n+k\right)!\left(m+k\right)!\left(r+k\right)!\left(n-m+r+k\right)!}{\sqrt{n!m!r!\left(n-m+r\right)!}k_{1}!\left(n+k-k_{1}\right)!\left(k+m-k_{1}\right)!\left(r-m+k_{1}\right)!}.\end{array}$
(16)

This can be used to obtain an analytic expression for concurrence potential.
The value of concurrence potential equal to $0$ (nonzero) corresponds to a
separable (entangled) state, i.e., the input state is classical
(nonclassical).

In contrast to anticlassicality and the measure for the characterization of
single photon source, in this case, we can clearly see the advantage of
increasing photon addition and $\eta$ in enhancing the amount of
nonclassicality present in the state. Specifically, nonclassicality can be
observed to increase here with these state parameters by the corresponding
large values of concurrence potential (see Fig. 5). Notice that for higher
values of photon addition nonclassicality decreases with increasing
probability $\eta$ when $\eta\approx 0$ because the quantum state in this case
is very close to Fock state. However, the decrease in the dimension of PAHS
keeping the rest of the parameters unchanged also increases nonclassicality of
the state due to holeburning in the state, which is known as nonclassicality
inducing/enhancing operation. There is a change in the slope of the curve of
concurrence potential with $\eta$ for $\eta=0.5$ as the derivative of $C$ is
not defined because of parameter $L$. This feature is often studied as a
sudden change of entanglement with significance in quantum information
processing [66, 67].

###  3.4 Wigner logarithmic negativity

A measure of nonclassicality was introduced which quantifies the negative
volume of Wigner function [60]. More recently a resource theoretic counterpart
of Wigner volume has been proposed as Wigner logarithmic negativity [50].
Here, we aim to quantify the amount of nonclassicality illustrated by the
Wigner function (9) using Wigner logarithmic negativity [50], which is defined
as

$W\left(\delta\right)=\log\left(\int\int
dxdp\left|W\left(x,p\right)\right|\right),$ (17)

where the integration is performed over both position and momentum quadratures
in the phase space.

All the qualitative observations regarding the dependence of Wigner negativity
on $k$, $M$, and $\eta$ are established quantitatively using Wigner
logarithmic negativity. Specifically, the Wigner logarithmic negativity
increases with photon addition and $\eta$, while increasing dimension reduces
the nonclassicality (see Fig. 6). Specifically, the yellow colored bars (with
dashed lines) in Fig. 6 (a) and (c) is obtained for single-PAHS of dimension
with $M=10$, which can be observed to increase with increase in photon
addition and/or decrease in the dimension of PAHS. Interestingly, the
concurrence potential of the two-mode quantum state generated from PAHS at a
beamspitter (in Fig. 5) and the Wigner logarithmic negativity (in Fig. 6) gave
us the same conclusion.

Thus, both measures of nonclassicality used here show that nonclassicality can
be increased by increasing value of $\eta$ and/or adding single photons in the
state which creates hole in the photon number distribution from lower side
and/or reducing dimension of the state, i.e., burning holes for the large $n$
states.

|  |
---|---|---
(a) | (b) | (c)
Figure 6: (Color online) The variation of Wigner logarithmic negativity as a
function of (a) photon addition for $M=10$ and $\eta=0.9$, (b) $\eta$, and (c)
$M$ for $k=1$ and $\eta=0.9.$ The yellow colored bar (with dashed lines) in
(a) and (c) are obtained for the same set of values of parameters.

##  4 Results and concluding remarks

We proposed PAHS state with an aim to quantify the amount of nonclassicality
in the concerned state and establish the role of quantum state engineering
tools, namely, photon addition and holeburning, in enhancement of
nonclassicality in hypergeometric state. Here, we used a measure to
characterize the quality of single photon sources and anticlassicality, both
of which showed us that these nonclassical features depend on the average
photon numbers. Specifically, with the photon addition, dimension, and
probability $\eta$ the average photon number increases as the photon number
distribution at higher photon numbers increase at the cost of that for lower
photon numbers. Therefore, in view of the present results, we can conclude
that anticlassicality is related to a measure for the characterization of
single photon source. Interestingly, photon subtraction is also known to
increase the average photon number of a quantum state. Therefore, it would be
worth analyzing the role of photon subtraction in changing anticlassicality of
a quantum state, especially when a hole in the photon number distribution at
vacuum is not expected due to single photon subtraction unlike photon
addition.

We have further quantified nonclassicality in PAHS using Wigner logarithmic
negativity and concurrence potential, i.e., the entanglement generated at the
beamspitter. Specifically, we observed that nonclassicality increases by non-
Gaussianity inducing operation photon addition and increasing probability
parameter $\eta$, while nonclassicality reduces by increasing the dimension of
the state. In the attempt of obtaining Wigner logarithmic negativity we also
characterized nonclassicality qualitatively using the Wigner function which
illustrated the non-Gaussian behavior of the quantum state under study.

The present results illustrate the highly nonclassical behavior of the PAHS,
which can be enhanced by photon addition, burning some holes in the phase
space (i.e., reducing the dimension of the state), and optimizing the value of
parameter $\eta$. The choice of the present state allows us to reduce the
corresponding results for a set of quantum states, for instance, binomial
state, phase independent coherent state, and respective photon added states as
well as hypergeometric state. This study can be further extended to the role
of the rest of the quantum state engineering tools in enhancing
nonclassicality and non-Gaussianity of the other finite dimensional quantum
states. We hope the present work will motivate such study and find
applications in some quantum information processing tasks, specially in
quantum cryptography where at one hand single photon sources (which can be
generated using states obtained as limiting cases of PAHS) are required for
discrete variable quantum key distribution and on the other hand continuous
variable quantum cryptography useful for metropolitan quantum network can be
realized using these states.

## Acknowledgment

PM and AP acknowledge the support from DRDO, India project no.
ANURAG/MMG/CARS/2018-19/071. KT acknowledges the financial support from the
Operational Programme Research, Development and Education \- European Regional
Development Fund project no. CZ.02.1.01/0.0/0.0/16_019/0000754 of the Ministry
of Education, Youth and Sports of the Czech Republic.

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# Restricted slowly growing digits for infinite iterated function systems

Gerardo González Robert  Gerardo González Robert, Department of Mathematical
and Physical Sciences, La Trobe University, Bendigo 3552, Australia.
<EMAIL_ADDRESS>,  Mumtaz Hussain  Mumtaz Hussain, Department of
Mathematical and Physical Sciences, La Trobe University, Bendigo 3552,
Australia<EMAIL_ADDRESS>,  Nikita Shulga  Nikita Shulga,
Department of Mathematical and Physical Sciences, La Trobe University, Bendigo
3552, Australia<EMAIL_ADDRESS>and  Hiroki Takahasi  Hiroki
Takahasi, Department of Mathematics, Keio University, Yokohama, 223-8522,
Japan<EMAIL_ADDRESS>
###### Abstract.

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an
attractor $\Lambda(\mathbf{f})$ and for an infinite subset
$D\subseteq\mathbb{N}$, consider the set

$\mathbb{E}(\mathbf{f},D)=\\{x\in\Lambda(\mathbf{f}):a_{n}(x)\in D\text{ for
all }n\in\mathbb{N}\text{ and }\lim_{n\to\infty}a_{n}=\infty\\}.$

For a function $\varphi:\mathbb{N}\to[\min D,\infty)$ such that
$\varphi(n)\to\infty$ as $n\to\infty$, we compute the Hausdorff dimension of
the set

$S(\mathbf{f},D,\varphi)=\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{n}(x)\leq\varphi(n)\text{
for all }n\in\mathbb{N}\right\\}.$

We prove that the Hausdorff dimension stays the same no matter how slowly the
function $\varphi$ grows. One of the consequences of our result is the recent
work of Takahasi (2023), which only dealt with regular continued fraction
expansions. We further extend our result to slowly growing products of (not
necessarily consecutive) digits.

###### 2020 Mathematics Subject Classification:

11K50, 11K55, 11A55, 37C45

Keywords: Iterated Function Systems (IFSs); continued fraction; Hausdorff
dimension.

##  1\. Introduction

The theory of infinite Iterated Function Systems on the unit interval includes
important number systems such as the regular continued fractions, whose
relevance cannot be understated. Recall that each irrational $x\in(0,1)$ has a
unique regular continued fraction expansion $x=1/(a_{1}+1/(a_{2}+\cdots))$,
where each partial quotient $a_{n}=a_{n}(x)$, $n\geq 1$ belongs to the set
$\mathbb{N}$ of positive integers. A pioneering result in the fractal
dimension theory of continued fraction is due to Jarník who proved that the
set of irrationals in $(0,1)$ with bounded partial quotients is of Hausdorff
dimension $1$. Good [6] proved that the set of irrationals in $(0,1)$ whose
partial quotients tend to infinity is of Hausdorff dimension $1/2$. Various
extensions and refinements of Good’s theorem have been made which determine
fractal dimensions of sets of irrationals whose partial quotients obey certain
restrictions or growth conditions, see [2, 3, 5, 7, 14, 17, 18, 23, 24] for
example.

Hirst [8] proved results analogous to that of Good in the case when $a_{n}$’s
belong to some subset of $\mathbb{N}$. For every infinite subset
$D\subseteq\mathbb{N}$, Hirst considered the set

$E(D):=\left\\{x\in(0,1)\setminus\mathbb{Q}\colon a_{n}(x)\in D\text{ for all
}n\in\mathbb{N}\text{ and }\lim_{n\to\infty}a_{n}(x)=\infty\right\\}.$

He introduced the exponent of convergence

$\tau(D):=\inf\left\\{s\geq 0:\sum_{n\in D}n^{-s}<\infty\right\\},$

and proved that the Hausdorff dimension of $E(D)$ does not exceed $\tau(D)/2$.
He also conjectured that $\tau(D)/2$ should be the exact value of the
Hausdorff dimension for this set. About thirty five years after the appearance
of Hirst’s paper, this conjecture was resolved by Wang and Wu in [24, Theorem
1.1].

Naturally, one might wonder that if additional restrictions are introduced on
the growth rate of $a_{n}$, the Hausdorff dimension may drop. In this regard,
there are two natural sets to consider,

$\\{x\in E(D)\colon a_{n}(x)\geq\varphi(n)\text{ for all }n\in\mathbb{N}\\},$
$\left\\{x\in E(D)\colon a_{n}(x)\leq\varphi(n)\text{ for all
}n\in\mathbb{N}\right\\},$

where $\varphi\colon\mathbb{N}\rightarrow\mathbb{N}$ is a function satisfying
$\lim_{n\to\infty}\varphi(n)=\infty$. Interestingly, the dimension of the
first set can drop from $\tau(D)/2$ and even become $0$ when $\varphi$ grows
very rapidly [2, 3, 5, 14, 17, 18, 25] while the dimension of the second set
does not drop no matter how slowly $\varphi$ grows, as precisely stated below.
Let $\dim_{\mathrm{H}}$ denote the Hausdorff dimension.

######  Theorem 1.1 ([23], Theorem 1.1).

Let $D\subseteq\mathbb{N}$ be an infinite subset and
$\varphi\colon\mathbb{N}\to[\min D,\infty)$ such that $\varphi(n)\to\infty$ as
$n\to\infty$. Then,

$\dim_{\mathrm{H}}\left\\{x\in E(D):a_{n}(x)\leq\varphi(n)\text{ for all
}n\in\mathbb{N}\right\\}=\frac{\tau(D)}{2}.$

This ‘no dimension drop’ seems to be a ubiquitous phenomenon, and
generalizations of the above theorem to diverse settings are desired. The
first generalization of Theorem 1.1 was done by Nakajima and the last-named
author in [19] to the semi-regular continued fractions, viewed as a non-
autonomous iterated function system [20]. In this paper, we generalize Theorem
1.1 in two directions while providing a much simpler proof.

First, we treat a question of the slow growth of each digit in an infinite
Iterated Function System (iIFS) (Theorem 1.2). Second, we extend this result
to the weighted products of an arbitrary finite number of digits in the given
iIFS (Theorem 1.3). Below we introduce a definition of an iIFS and state the
main results.

###  1.1. Infinite Iterated Function Systems

Consider a sequence $\mathbf{f}=(f_{n})_{n\geq 1}$ of functions
$f_{n}:[0,1]\to[0,1]$, satisfying

  1. i.

Smoothness Property: $f_{n}\in C^{1}([0,1])$ for each $n\geq 1$;

  2. ii.

Contraction Property: There exists an integer $m$ and a real number
$0<\rho<1$, such that for any $(a_{1},\ldots,a_{m})\in\mathbb{N}^{m}$ and
$x\in[0,1]$,

$0<|(f_{a_{1}}\circ\cdots\circ f_{a_{m}})^{\prime}(x)|\leq\rho<1;$

  3. iii.

Open Set Condition: For any $i\neq j\in\mathbb{N}$, we have $f_{i}((0,1))\cap
f_{j}((0,1))=\emptyset.$

The system $\mathbf{f}=(f_{n})_{n\geq 1}$ is called an infinite iterated
function system (iIFS). By the smoothness and the contraction properties, we
can define a natural projection
$\Pi_{\mathbf{f}}:\mathbb{N}^{\mathbb{N}}\to[0,1]$ as

$\Pi_{\mathbf{f}}(\mathbf{a})=\lim_{n\to\infty}f_{a_{1}}\circ\cdots\circ
f_{a_{n}}(1),$

where $\mathbf{a}=(a_{n})_{n\geq 1}\in\mathbb{N}^{\mathbb{N}}$. Also, we
denote by $\Lambda(\mathbf{f})$ the attractor of the iIFS $(f_{n})_{n\geq 1}$,
that is,

$\Lambda(\mathbf{f})=\Pi_{\mathbf{f}}(\mathbb{N}^{\mathbb{N}}).$

By the open set condition, to all but countably many points $x$ in $\Lambda$
we can associate a unique sequence $(a_{n})_{n\geq 1}$ of integers such that

$x=\lim_{n\to\infty}f_{a_{1}}\circ\cdots\circ f_{a_{n}}(1).$

For a given $x$ we refer to the sequence $(a_{n})_{n\geq 1}=(a_{n}(x))_{n\geq
1}$ as a sequence of digits of $x$.

We say that an iIFS $\mathbf{f}=(f_{n})_{n\geq 1}$ is
$\boldsymbol{\xi}$-regular if in addition it satisfies

  1. iv.

Regularity Property: There exists a sequence of positive numbers
$\boldsymbol{\xi}=(\xi_{n})_{n\geq 1}$ such that for every $\varepsilon>0$
there are positive constants $c_{1},c_{2}$ depending on $\varepsilon$ such
that

$\frac{c_{1}}{\xi_{n}^{1+\varepsilon}}\leq|f^{\prime}_{n}(x)|\leq\frac{c_{2}}{\xi_{n}^{1-\varepsilon}}\quad\text{
for all }n\in\mathbb{N}\text{ and all }x\in[0,1].$

If $\xi_{n}=n^{d}$, the $\boldsymbol{\xi}$-regular iIFS $\mathbf{f}$ is called
$d$-decaying system, as defined by Jordan and Rams in [14]. Finally, we assume
that iIFS satisfies the following condition.

  1. v.

Bounded Distortion Property (BDP): There exists a constant $\kappa\geq 1$ such
that for every $n\in\mathbb{N}$ and $a_{1},\ldots,a_{n}\in\mathbb{N}$ we have

$\left|(f_{a_{1}}\circ\ldots\circ
f_{a_{n}})^{\prime}(x)\right|\leq\kappa\left|(f_{a_{1}}\circ\ldots\circ
f_{a_{n}})^{\prime}(y)\right|\text{ for all }x,y\in[0,1].$

We note that all natural examples of IFS satisfy BDP, see Section 1.3 below.

###  1.2. Statements of results

Let $\mathbf{f}$ be a $\boldsymbol{\xi}$-regular iIFS. For an infinite subset
$D\subseteq\mathbb{N}$ define

$\mathbb{E}(\mathbf{f},D):=\left\\{x\in\Lambda(\mathbf{f}):a_{n}(x)\in D\text{
for all }n\in\mathbb{N},\lim_{n\to\infty}a_{n}(x)=\infty\right\\},$

and

(1) $s_{0}(D;\boldsymbol{\xi}):=\inf\left\\{s\geq 0:\sum_{n\in
D}\frac{1}{\xi_{n}^{s}}<\infty\right\\}.$

For $2$-decaying systems such as the regular continued fractions, we have
$\tau(D)=2s_{0}(D;\boldsymbol{\xi})$. In [2], the solution of Hirst’s
conjecture was generalized to digits in an iIFS in the following way: If
$\mathbf{f}$ satisfies the BDP, then
$\dim_{\mathrm{H}}\mathbb{E}(\mathbf{f},D)=s_{0}(D;\boldsymbol{\xi}).$

For a function $\varphi:\mathbb{N}\to[\min D,\infty)$, define

$S(\mathbf{f},D,\varphi):=\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{n}(x)\leq\varphi(n)\text{
for all }n\in\mathbb{N}\right\\}.$

The first main result of the present paper is the following statement.

######  Theorem 1.2.

Let $\mathbf{f}$ be a $\boldsymbol{\xi}$-regular iIFS with the BDP. For every
infinite subset $D\subseteq\mathbb{N}$ and any function
$\varphi:\mathbb{N}\to[\min D,\infty)$ such that $\varphi(n)\to\infty$ as
$n\to\infty$, we have

$\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)=s_{0}(D;\boldsymbol{\xi}).$

Theorem 1.2 concerns the behavior of a single digit. Kleinbock and Wadleigh
[15] proved that the product of consecutive partial quotients in regular
continued fractions is closely related to improvements of Dirichlet’s Theorem
in Diophantine approximation. Such sets have been further studied in [9, 10,
11] for example. Hence, it is natural to generalise Theorem 1.2 to include
products of digits.

Using Theorem 1.2 we derive a much more general result involving weighted
products of a fixed number of digits. For a fixed $m\in\mathbb{N}$ consider
$\overline{t}=(t_{1},\ldots,t_{m})\in\mathbb{R}_{>0}^{m}$ and a vector
$\overline{g}=(g_{1},\ldots,g_{m})$ of arbitrary functions
$g_{i}\colon\mathbb{N}\ \to\mathbb{N},1\leq i\leq m$. Define a set

$P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g}):=\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{g_{1}(n)}^{t_{1}}(x)\cdots
a_{g_{m}(n)}^{t_{m}}(x)\leq\varphi(n)\text{ for all }n\in\mathbb{N}\right\\}$

and denote $M(D):=\max\\{\min D,(\min D)^{m\max_{i}t_{i}}\\}$. We have the
following theorem.

######  Theorem 1.3.

Let $\mathbf{f}$, $D$ be as in Theorem 1.2 and let
$\varphi:\mathbb{N}\to[M(D),\infty)$ be such that $\varphi(n)\to\infty$ as
$n\to\infty$. For any $\overline{t}\in\mathbb{R}_{>0}^{m}$ and for any
functions $g_{i}\colon\mathbb{N}\to\mathbb{N},1\leq i\leq m$, we have

$\dim_{\mathrm{H}}P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})=\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)=s_{0}(D;\boldsymbol{\xi}).$

Theorem 1.3 means that in fact the Hausdorff dimension of
$P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})$ does not depend on
$m,\overline{t}$ and functions $g_{i}$’s no matter how they behave. In
particular, if we let $g_{i}(n)=n+i$ and $\overline{t}=(1,\ldots,1)$, we get a
set of slowly growing products of consecutive digits

$\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{n+1}(x)\cdots
a_{n+m}(x)\leq\varphi(n)\text{ for all }n\in\mathbb{N}\right\\}.$

If we let $g_{i}(n)=ni$ and $\overline{t}=(1,\ldots,1)$, we get a set of
slowly growing products of digits, located in an arithmetic progression

$\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{n}(x)\cdots
a_{nm}(x)\leq\varphi(n)\text{ for all }n\in\mathbb{N}\right\\}.$

The set of numbers with large products of digits in arithmetic progressions
(that is, numbers $x\in[0,1]$ with $a_{n}(x)\cdots a_{nm}(x)\geq\varphi(n)$
for infinitely many $n$) for the case of the regular continued fractions was
studied in [12].

###  1.3. Examples of infinite iterated function systems

We present several examples of $\boldsymbol{\xi}$-regular iIFS to which our
main results apply.

####  1.3.1. $2$-decaying systems

  * •

Regular Continued fractions. For each $n\in\mathbb{N}$ define
$f_{n}:[0,1]\to[0,1]$ by

$f_{n}(x)=\frac{1}{x+n},\quad x\in[0,1].$

This iIFS is induced by the Gauss map $T:[0,1]\to[0,1]$ defined by $T(0)=0$
and

$T(x)=\frac{1}{x}\mod\,1\quad\text{ for }x\neq 0.$

The contraction constants are $m=2$ and $\rho=\frac{1}{2}$ and the BDP can be
shown as in [19, Lemma 2.4].

  * •

Lüroth expansions. For each $n\in\mathbb{N}$ define $f_{n}:[0,1]\to[0,1]$ by

$f_{n}(x)=\frac{x}{n(n+1)}+\frac{1}{n+1},\quad x\in(0,1],n\in\mathbb{N}.$

This system is induced by the Lüroth map $T:(0,1]\to(0,1]$ defined by

(2) $T(x)=d_{1}(x)(d_{1}(x)-1)\left(x-\frac{1}{d_{1}(x)}\right),\ \
\text{where}\hskip 8.5359ptd_{1}(x)=\left[\frac{1}{x}\right]+1.$

The contraction constants are $m=1$ and $\rho=\frac{1}{2}$ and the BDP holds
for $\kappa=1$. Figure 1 shows the graphs of the Gauss and the Lüroth maps.

Figure 1.  Left: the Gauss map. Right: the Lüroth map.

####  1.3.2. $d$-decaying systems for $d\neq 2$

  * •

Quadratic Gauss map. This is a particular case of $f$-expansion (as defined by
Rényi [21]) which is a $3$-decaying iIFS. Lett $p,q\in\mathbb{R}$ be such that
$p\geq-1$ and $q>0$. Consider the polynomial $P(x)=x^{2}+px+q$ and define
$f:[0,\infty)\to(0,1]$ by $f(x)=\frac{q}{P(x)+x}$. Then the map is defined as

$T(x)=f^{-1}(x)\mod\,1.$

In particular, if we let $p=q=1$, then
$f(x)=\frac{1}{x^{2}+2x+1}=\frac{1}{(x+1)^{2}}$, the map is

$T(x)=\frac{1}{\sqrt{x}}\mod\,1,$

(see Figure 2) and we have

$f_{n}(x)=\frac{1}{(x+n)^{2}},\quad x\in[0,1],n\in\mathbb{N}.$

The parameters of the Contraction Property are $m=2$ and $\rho=\frac{1}{2}$.
Again, we can show the BDP as in [19, Lemma 2.4]. We refer the reader to [22,
Chapters 3 & 13] for more details.

Figure 2. Left: The Quadratic Gauss map. Right: The Generalised Lüroth Series.

####  1.3.3.  $\boldsymbol{\xi}$-regular iIFSs, which are not $d$-decaying for
any $d$.

  * •

Generalised Lüroth series (GLS) was introduced by Barrionuevo et al. in [1].
GLS may have finitely or infinitely many digits. In the latter case, they can
be interpreted as a $\boldsymbol{\xi}$-regular iIFS. Let
$\varepsilon=(\varepsilon_{j})_{j\geq 1}$ be any sequence in $\\{0,1\\}$, and
let $\\{I_{n}=(l_{n},r_{n}]:n\in\mathbb{N}\\}$ be a collection of pairwise
disjoint intervals such that

$\sum_{n\in\mathbb{N}}(r_{n}-l_{n})=1\quad\text{ and}\quad(r_{n}-l_{n})_{n\geq
1}\text{ is non-increasing}.$

Write $I_{\infty}:=[0,1]\setminus\bigcup_{n\in\mathbb{N}}I_{n}$ and define the
maps $T,S,\varepsilon:[0,1]\to[0,1]$ by

$T(x)=\begin{cases}\frac{x-l_{n}}{r_{n}-l_{n}},&\text{ if }x\in I_{n},\\\
0,&\text{ if }x\in I_{\infty},\end{cases}\quad
S(x)=\begin{cases}\frac{r_{n}-x}{r_{n}-l_{n}},&\text{ if }x\in I_{n},\\\
0,&\text{ if }x\in I_{\infty},\end{cases}$
$\varepsilon(x):=\begin{cases}\varepsilon_{n},&\text{ if }x\in I_{n},\\\
0,&\text{ if }x\in I_{\infty}.\end{cases}$

The Generalised Lüroth map $T_{\varepsilon}:[0,1]\to[0,1]$ is given by

$T_{\varepsilon}(x)=\varepsilon(x)S(x)+(1-\varepsilon(x))T(\varepsilon).$

The map $T_{\varepsilon}$ induces the $\xi$-regular iIFS
$\mathbf{f}=(f_{n})_{n\geq 1}$ given by

$f_{n}(x)=\begin{cases}-(r_{n}-l_{n})x+r_{n},&\text{ if }\varepsilon_{n}=1,\\\
(r_{n}-l_{n})x-l_{n},&\text{ if }\varepsilon_{n}=0.\end{cases}$

The parameters of the Contraction Property are $m=1$ and $\rho=r_{1}-l_{1}$.
As for the BDP, we have $\kappa=1$. A non $d$-decaying
$\boldsymbol{\xi}$-regular iIFS is obtained when we consider
$\\{[2^{-n},2^{-n+1}):n\in\mathbb{N}\\}$ and any sequence $\varepsilon$.
Figure 2 shows the map $T_{\varepsilon}$ for $\varepsilon=(1,1,-1,-1,\ldots)$.

The definition of the GLS in [1] considers a proper infinite subset $B$ of
$\mathbb{N}$ rather than $\mathbb{N}$. However, we may assume $B=\mathbb{N}$
after enlisting the elements of $B$ in an increasing manner.

###  1.4. Outline of proofs of the theorems

The rest of this paper consists of two sections, entirely dedicated to proofs
of Theorems 1.2 and 1.3. In the proofs, the upper bounds of the Hausdorff
dimension are rather trivial as they can be obtained from the known results.
The issue is to obtain the lower bounds. A derivation of lower bounds on the
Hausdorff dimension of fractal sets often relies on the mass distribution
principle [4], and the construction of a mass usually becomes intricate
depending on the structure of the fractal set under consideration. In the
proof [23, Theorem 1.1], the construction of masses on fractal sets heavily
relies on the ergodic theory of the Gauss map generating the continued
fraction. Our idea is to dispense with the mass distribution principle
altogether, and instead explicitly construct a well-organized Cantor set whose
Hausdorff dimension can easily be bounded from below. Starting from a
sufficiently large $n$, the compact sets in each layer of our Cantor set
correspond to certain admissible digits. The lower estimate of its Hausdorff
dimension will be a consequence of two properties: the grandchildren of any
compact set appearing in any layer are well separated and the digits defining
the layers tend to infinity at a very slow pace.

To show the lower bound in Theorem 1.3, we construct several subsets of the
set under consideration, eventually showing that for some specific choice of a
growth function, the set from Theorem 1.2 is a subset of a set from Theorem
1.3. The upper bound follows trivially by lifting the restriction on the
growth of products.

##  2\. Lower bound of Hausdorff dimension of certain Cantor subsets

In this section, we prove a lower bound for the Hausdorff dimension of certain
Cantor sets. Similar arguments date back to Jarník in 1928 [13] and they have
been used in Cao-Wang-Wu [2, Theorem 1.1] among others much more recently. Our
setup is inspired by Kleinbock-Weiss [16, Section 2.4].

Denote the diameter of a non-empty set $A\subseteq[0,1]$ by $|A|$. Let
$\\{\mathcal{A}_{n}:n\in\mathbb{N}_{0}\\}$ be a collection of finite families
of pairwise disjoint compact subintervals of $[0,1]$ with a positive diameter
such that $\mathcal{A}_{0}$ has exactly one element and the following
properties hold:

  * •

For all $n\in\mathbb{N}$ and $B\in\mathcal{A}_{n}$, there is some
$A\in\mathcal{A}_{n-1}$ such that $B\subseteq A$.

  * •

For all $n\in\mathbb{N}$ and $A\in\mathcal{A}_{n-1}$, there are at least two
sets $B\in\mathcal{A}_{n}$ such that $B\subseteq A$.

  * •

We have $\delta_{n}:=\max\\{|A|:A\in\mathcal{A}_{n}\\}\to 0$ as $n\to\infty$.

For $n\in\mathbb{N}$ and $A\in\mathcal{A}_{n}$, write
$\operatorname{Kid}_{n}(A):=\\{B\in\mathcal{A}_{n+1}:B\subseteq A\\}$. Under
additional considerations, we provide a lower estimate for the Hausdorff
dimension of the set

$\mathbf{A}_{\infty}:=\bigcap_{n=0}^{\infty}\bigcup_{A\in\mathcal{A}_{n}}A.$

######  Lemma 2.1.

Assume there is some $\varepsilon>0$ such that for all $n\in\mathbb{N}$,
$A\in\mathcal{A}_{n}$, and every $Y,Z\in\operatorname{Kid}_{n}(A)$ with $Y\neq
Z$ we have

(3)
$\min\left\\{|y-z|:y\in\bigcup_{Y^{\prime}\in\operatorname{Kid}_{n+1}(Y)}Y^{\prime},z\in\bigcup_{Z^{\prime}\in\operatorname{Kid}_{n+1}(Z)}Z^{\prime}\right\\}\geq|A|^{1+\varepsilon}.$

If $s>0$ satisfies

(4)
$\sum_{B\in\operatorname{Kid}_{n}(A)}|B|^{s(1+\varepsilon)}\geq|A|^{s(1+\varepsilon)}\quad\text{
for all }\quad A\in\bigcup_{n\in\mathbb{N}}\mathcal{A}_{n},$

then $\dim_{\mathrm{H}}\mathbf{A}_{\infty}\geq s$.

###### Proof.

Take $\delta>0$ and let $\mathcal{G}$ be a $\delta$-covering of
$\mathbf{A}_{\infty}$. Without loss of generality, we may assume $\mathcal{G}$
is finite, consists of open intervals (see, for example, [4, Section 3.4]),
and every element in $\mathcal{G}$ intersects $\mathbf{A}_{\infty}$. For each
$G\in\mathcal{G}$ define

$n(G):=\max\left\\{n\in\mathbb{N}_{0}:\\#\\{A\in\mathcal{A}_{n}:G\cap
A\neq\emptyset\\}=1\right\\}.$

Let us explain why $n(G)$ is well-defined. First, writing
$\mathcal{A}_{0}=\\{K_{0}\\}$, we have $G\cap K_{0}\neq\emptyset$. Second,
take $x\in G\cap\mathbf{A}_{\infty}$ and $r>0$ such that $(x-r,x+r)\subseteq
G$. Since $\delta_{n}\to 0$ as $n\to\infty$, for large $n\in\mathbb{N}$, if
$A_{n}\in\mathcal{A}_{n}$ contains $x$, then $A_{n}\subseteq G$, and thus
$B\subseteq G$ for all $B\in\operatorname{Kid}_{n}(A_{n})$. Therefore, $n(G)$
is the maximum of a non-empty finite set of integers. Let $A(G)$ be the unique
compact interval for which the maximum in the definition of $n(G)$ is
attained. Then there are two different sets
$Y,Z\in\operatorname{Kid}_{n(G)}(A(G))$ intersecting $G$. Pick
$Y^{\prime}\in\operatorname{Kid}_{n(G)+1}(Y)$ and
$Z^{\prime}\in\operatorname{Kid}_{n(G)+1}(Z)$ intersecting $G$. Then, (4)
yields

$|G|\geq\min\\{|y-z|:y\in Y^{\prime},z\in
Z^{\prime}\\}\geq|A(G)|^{1+\varepsilon}.$

As a consequence, we have

$\sum_{G\in\mathcal{G}}|A(G)|^{s(1+\varepsilon)}\leq\sum_{G\in\mathcal{G}}|G|^{s}.$

Clearly, $\mathcal{G}_{1}:=\\{A(G):G\in\mathcal{G}\\}$ also covers
$\mathbf{A}_{\infty}$. Moreover, since every two members in $\mathcal{G}_{1}$
are either disjoint or one is contained in the other, there is a subcover
$\mathcal{G}_{2}\subseteq\mathcal{G}_{1}$ of $\mathbf{A}_{\infty}$ consisting
of pairwise disjoint sets. Let
$N:=\max\\{n\in\mathbb{N}:\mathcal{G}_{2}\cap\mathcal{A}_{n}\neq\emptyset\\}$.
Pick $B\in\mathcal{G}_{2}\cap\mathcal{A}_{N}$ and $A\in\mathcal{A}_{N-1}$ such
that $B\subseteq A$. Since the members of $\mathcal{G}_{2}$ are pairwise
disjoint, $\mathcal{G}_{2}$ covers $\mathbf{A}_{\infty}$, and we cannot go any
level deeper than $N$, we must have
$\operatorname{Kid}_{N-1}(A)\subseteq\mathcal{G}_{2}$. Hence, if
$\mathcal{G}_{3}$ is the cover obtained from $\mathcal{G}_{2}$ by replacing
the sets in $\operatorname{Kid}_{N-1}(A)$ with $A$, we deduce from (4) that

$\sum_{C\in\mathcal{G}_{3}}|C|^{s(1+\varepsilon)}\leq\sum_{C\in\mathcal{G}_{2}}|C|^{s(1+\varepsilon)}.$

After finitely many replacements, we arrive at $\mathcal{A}_{0}=\\{K_{0}\\}$
and thus

$0<|K_{0}|^{s(1+\varepsilon)}\leq\sum_{G\in\mathcal{G}}|G|^{s}.$

Therefore, we obtain $\dim_{\mathrm{H}}\mathbf{A}_{\infty}\geq s$. ∎

##  3\. Proofs of the main results

In this section, we prove Theorem 1.2 and Theorem 1.3.

###  3.1. Proof of Theorem 1.2

We separate the proof into two parts: the upper bound and the lower bound. The
upper bound trivially follows from Theorem 1.1 in [2], because
$S(\mathbf{f},D,\varphi)\subseteq\mathbb{E}(\mathbf{f},D)$, and so

$\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)\leq\dim_{\mathrm{H}}\mathbb{E}(\mathbf{f},D)=s_{0}(D;\boldsymbol{\xi}).$

Now we show the lower bound of $\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)$,
that is

$\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)\geq s_{0}(D;\boldsymbol{\xi}).$

For this purpose, we will construct a suitable Cantor subset
$K(\mathbf{f},D,\mathbf{l},\mathbf{u})\subseteq S(\mathbf{f},D,\varphi)$,
where $\mathbf{l}=(l_{n})_{n\geq 1}$ and $\mathbf{u}=(u_{n})_{n\geq 1}$ are
two slowly increasing sequences of natural numbers to be defined later.

There is nothing to prove if $s_{0}(D;\boldsymbol{\xi})=0$, as we always have
the trivial lower bound $\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)\geq 0.$
Hence, we further assume that $s_{0}(D;\boldsymbol{\xi})>0$. Given
$n\in\mathbb{N}$ and $\overline{a}=(a_{1},\ldots,a_{n})\in\mathbb{N}^{n}$, we
consider

$I_{n}(\overline{a}):=\\{x\in\Lambda:a_{k}(x)=a_{k}\text{ for all
}k\in\\{1,\ldots,n\\}\\}.$

We refer to sets of the form $I_{n}(\overline{a})$ as fundamental intervals.
For readability, let us additionally assume $m=1$ in the Contraction Property.
Then, for all $a\in\mathbb{N}$ and all $x\in[0,1]$ we have
$|f_{a}^{\prime}(x)|\leq\rho$, so

$|I_{n}(\overline{a})|\leq\rho^{n}\quad\text{ for all }n\in\mathbb{N}\text{
and all }\overline{a}\in\mathbb{N}^{n}.$

At the end of the proof, we will explain how to proceed when $m\geq 2$.

Let $s,\varepsilon>0$ satisfy $s(1+\varepsilon)^{2}<s_{0}(D;\boldsymbol{\xi})$
and let $c_{1}=c_{1}(\varepsilon)>0$ and $c_{2}=c_{2}(\varepsilon)>0$ be as in
the definition of the $\boldsymbol{\xi}$-regularity. Let $\kappa\geq 1$ be the
constant from the Bounded Distortion property. Write
$D=\\{d_{1}<d_{2}<d_{3}<\cdots\\}$.

We now construct two sequences $\mathbf{l}$ and $\mathbf{u}$ in $\mathbb{N}$.
First, in the view of $s(1+\varepsilon)<s_{0}(D,\boldsymbol{\xi})$, we may
take a strictly increasing sequence $(r_{k})_{k\geq 1}$ in $\mathbb{N}$ such
that

(5) $r_{k}-k\geq 4\quad\text{ and
}\quad\sum_{n=k}^{r_{k}}\left(\frac{\kappa^{-1}c_{1}}{\xi_{d_{n}}^{1+\varepsilon}}\right)^{s(1+\varepsilon)}\geq
3\quad\text{ for all }k\in\mathbb{N}.$

Let $(N_{j})_{j\geq 1}$ be a strictly increasing sequence of natural numbers
such that for every $j\in\mathbb{N}$ we have

(6) $\displaystyle\varphi(n)$ $\displaystyle\geq d_{r_{j}}+1\quad\text{ for
all }n\in\mathbb{N}_{\geq N_{j}},\;\text{ and }$ (7) $\displaystyle\max$
$\displaystyle\left\\{(\xi_{d_{i}}\xi_{d_{k}})^{1+\varepsilon}:j\leq i\leq
r_{j+1},\ j\leq k\leq
r_{j+2}\right\\}\leq\kappa^{-1}c_{1}^{2}\rho^{-\varepsilon N_{j}}.$

Let $\mathbf{l}=(l_{n})_{n\geq 1}$ and $\mathbf{u}=(u_{n})_{n\geq 1}$ be
defined as follows: for each $n\in\mathbb{N}$, take $j\in\mathbb{N}$ such that

$N_{j}-N_{1}+1\leq n\leq N_{j+1}-N_{1}$

and put

$l_{n}:=j\quad\text{ and }\quad u_{n}:=r_{j}.$

Moreover, for each $n\in\mathbb{N}$ define the set

$D_{n}:=\\{d_{i}\in D\colon l_{n}\leq i<u_{n}\\}.$

We now describe the layers of a Cantor set contained in
$S(\mathbf{f},D,\varphi)$. First, consider

$K_{0}:=\overline{I}_{N_{1}-1}(d_{1},\ldots,d_{1})$

Here and in the following, the bar over $I$ denotes its closure. For all
$n\in\mathbb{N}$ and $(b_{1},\ldots,b_{n})\in\mathbb{N}^{n}$ define

$K_{n}(b_{1},\ldots,b_{n}):=\overline{I}_{N_{1}-1+n}(d_{1},\ldots,d_{1},b_{1},\ldots,b_{n}).$

Define $B_{1}:=D_{1}$. For each $b\in B_{1}$, let $L(b)\in D_{2}$ (resp.
$R(b)\in D_{2}$) be such that $K_{2}(b,L(b))$ (resp. $K_{2}(b,R(b))$) is the
leftmost (resp. rightmost) interval in $\\{K_{2}(b,b_{2}):b_{2}\in D_{2}\\}$
and write

$B_{2}(b):=\left\\{(b,b_{2}):b_{2}\in D_{1}\setminus\\{L(b),R(b)\\}\right\\}.$

Define

$B_{2}:=\bigcup_{b_{1}\in B_{1}}B_{2}(b_{1}).$

In what follows, for any $j,k\in\mathbb{N}$ and any
$\overline{a}=(a_{1},\ldots,a_{j})\in\mathbb{N}^{j}$ and
$\overline{b}=(b_{1},\ldots,b_{k})\in\mathbb{N}^{k}$, we write

$\overline{a}\overline{b}:=(a_{1},\ldots,a_{j},b_{1},\ldots,b_{k}).$

Let us assume that for some $n\in\mathbb{N}$ we have defined a non-empty
finite set $B_{n}\subseteq D_{1}\times\cdots\times D_{n}$. For each
$\overline{b}\in B_{n}$, let $L(\overline{b})\in D_{n+1}$ (resp.
$R(\overline{b})\in D_{n+1}$) be such that
$K_{n+1}(\overline{b}L(\overline{b}))$ (resp.
$K_{n+1}(\overline{b}R(\overline{b}))$) is the leftmost (resp. rightmost)
interval in $\left\\{K_{n+1}(\overline{b}b):b\in D_{n+1}\right\\}$ and write

$B_{n+1}(\overline{b}):=\left\\{\overline{b}b:b\in
D_{n+1}\setminus\\{L(\overline{b}),R(\overline{b})\\}\right\\}.$

Define

$B_{n+1}:=\bigcup_{\overline{b}\in B_{n}}B_{n+1}(\overline{b}).$

Note that each set $B_{n}$ depends on $D_{n}$, which in turn is defined in
terms of $\mathbf{l}$ and $\mathbf{u}$, and hence of $(N_{j})_{j\geq 1}$. We
will apply Lemma 2.1 to estimate the Hausdorff dimension of

$K=K(\mathbf{f},D,\mathbf{l},\mathbf{u}):=\bigcap_{n\in\mathbb{N}}\bigcup_{\overline{b}\in
B_{n}}K_{n}(\overline{b})\subseteq S(\mathbf{f},D,\varphi).$

Let us check that $K$ is indeed a subset of $S(\mathbf{f},D,\varphi)$. Pick
any $x\in K$. It suffices to check the growth and the limit conditions, for
the definition of $K$ implies that every digit of $x$ belongs to $D$. Take
$n\in\mathbb{N}$. If $1\leq n\leq N_{1}-1$, then

$a_{n}(x)=d_{1}\leq\min_{k\in\mathbb{N}}\varphi(k)\leq\varphi(n).$

If $n\geq N_{1}$, let $j\in\mathbb{N}$ be the single number for which
$N_{j}\leq n\leq N_{j+1}-1$. Then we have $N_{j}-N_{1}+1\leq n-N_{1}+1\leq
N_{j+1}-N_{1}$, and so $l_{n-N_{1}+1}=j$, $u_{n-N_{1}+1}=r_{j}$. This yields

$a_{n}(x)\in D_{n-N_{1}+1}=\\{d_{j}<\cdots<d_{r_{j}}\\}.$

As a result, by (6) and the choice of $j$ we obtain

$d_{j}\leq a_{n}(x)\leq d_{r_{j}}\leq\varphi(n).$

Finally, as $n\to\infty$ we have $j\to\infty$, $d_{j}\to\infty$ and so
$a_{n}(x)\to\infty$. We conclude $x\in S(\mathbf{f},D,\varphi)$.

Define $\mathcal{A}_{0}:=\\{K_{0}\\}$ and
$\mathcal{A}_{n}:=\\{K_{n}(\overline{b}):\overline{b}\in B_{n}\\}$ for all
$n\in\mathbb{N}$. Let us verify the assumptions on
$\\{\mathcal{A}_{n}:n\in\mathbb{N}_{0}\\}$ from Section 2. Take
$n\in\mathbb{N}$. Clearly, $\mathcal{A}_{n}$ consists of finitely many
pairwise disjoint compact intervals with positive diameter. The three points
assumed on $\mathcal{A}_{n}$ also hold:

  * •

The first point follows from the construction of $B_{n}\subseteq
D_{1}\times\cdots\times D_{n}$.

  * •

By $r_{n}-n\geq 4$, for any $A\in\mathcal{A}_{n}$ the set
$\operatorname{Kid}_{n}(A)$ has at least two elements.

  * •

Since we have assumed $m=1$ in the Contraction Property, we have
$\delta_{n}\leq\rho^{N_{1}-1}\rho^{n}$.

To check (3) from Lemma 2.1, take $j\in\mathbb{N}$ such that
$N_{j}-N_{1}+1\leq n\leq N_{j+1}-N_{1}$, and $\overline{b}\in B_{n}$. First,
note that if $b_{n+1}\in D_{n+1}$ is such that $\overline{b}b_{n+1}\in
B_{n+1}$ then for all $b_{n+2}\in D_{n+2}$ we have

$\displaystyle|K_{n+2}(\overline{b}b_{n+1}b_{n+2})|$
$\displaystyle\geq|K_{n}(\overline{b})|\frac{\kappa^{-1}c_{1}^{2}}{(\xi_{b_{n+1}}\xi_{b_{n+2}})^{1+\varepsilon}}$
$\displaystyle\geq|K_{n}(\overline{b})|\rho^{\varepsilon N_{j}}$ (by (7))
$\displaystyle\geq|K_{n}(\overline{b})|\rho^{\varepsilon n}$
$\displaystyle\geq|K_{n}(\overline{b})|^{1+\varepsilon}.$

Let $b,c\in D_{n+1}$ and $b^{\prime},c^{\prime}\in D_{n+2}$ be such that
$\overline{b}bb^{\prime},\overline{b}cc^{\prime}\in B_{n+2}$. Then the
construction of $B_{n+2}$ implies that the distance between
$K_{n+2}(\overline{b}bb^{\prime})$ and $K_{n+2}(\overline{b}cc^{\prime})$ is
at least the length of one of the removed intervals:

$\left\\{K_{n+2}(\overline{d}):\overline{d}\in\\{\overline{b}bL(\overline{b}b),\overline{b}bR(\overline{b}b),\overline{b}cL(\overline{b}c),\overline{b}cR(\overline{b}c)\\}\right\\}.$

This length is larger than or equal to
$|K_{n}(\overline{b})|^{1+\varepsilon}$, which shows (3) from Lemma 2.1.

Condition (4) holds because the Bounded Distortion Property and (5) imply

$\begin{split}\sum_{d\in
B_{n+1}(\overline{b})}|K_{n+1}(\overline{b}d)|^{s(1+\varepsilon)}&\geq|K_{n}(\overline{b})|^{s(1+\varepsilon)}\left(\sum_{d\in
D_{n+1}}\left(\frac{\kappa^{-1}c_{1}}{\xi_{d}^{1+\varepsilon}}\right)^{s(1+\varepsilon)}-2\right)\\\
&\geq|K_{n}(\overline{b})|^{s(1+\varepsilon)}.\end{split}$

Therefore, by Lemma 2.1 we obtain
$\dim_{\mathrm{H}}S(\mathbf{f},D,\varphi)\geq\dim_{\mathrm{H}}K\geq s$ as
required.

When $m\geq 2$, we add $m$ digits in each layer of our Cantor set construction
and adjust the parameters accordingly. More precisely, the sequence
$(r_{k})_{k\geq 1}$ remains unchanged. For $(N_{j})_{j\geq 1}$, instead of (6)
we assume that for all $j\in\mathbb{N}$ we have $\varphi(n)\geq d_{r_{j}}+1$
when $n\geq mN_{j}$ and instead of (7) we impose

$\max\left\\{\prod_{k=1}^{2m}\xi_{d_{i_{k}}}^{1+\varepsilon}:j\leq i_{1}\leq
r_{j+1},\ j\leq i_{k}\leq r_{j+2}\text{ for }2\leq k\leq
2m\right\\}\leq\kappa^{-1}c_{1}^{2m}\rho^{-\varepsilon N_{j}}.$

In order to define the sequences $\mathbf{l}=(l_{n})_{n\geq 1}$ and
$\mathbf{u}=(u_{n})_{n\geq 1}$, for each $n\in\mathbb{N}$ take
$j\in\mathbb{N}$ such that

$m(N_{j}-N_{1})+1\leq n\leq m(N_{j+1}-N_{1})$

and put $l_{n}:=j$ and $u_{n}:=r_{j}$. We start with
$K_{0}:=\overline{I}_{mN_{1}-1}(d_{1},\ldots,d_{1})$. Next, for each
$\overline{b}\in B_{1}:=D_{1}\times\cdots\times D_{m}$ we let
$L(\overline{b})$ and $R(\overline{b})$ be the vectors in
$D_{m+1}\times\cdots\times D_{2m}$ such that
$K_{2m}(\overline{a}L(\overline{b}))$ and
$K_{2m}(\overline{b}R(\overline{b}))$ are, respectively, the leftmost and
rightmost intervals in the collection
$\left\\{K_{2m}(\overline{b}\overline{c}):\overline{c}\in
D_{m+1}\times\cdots\times D_{2m}\right\\}$. Afterwards, we write

$B_{2}(\overline{b}):=\left\\{\overline{b}\overline{c}:\overline{c}\in
D_{m+1}\times\cdots\times
D_{2m}\\}\setminus\\{L(\overline{b}),R(\overline{b})\right\\}.$

Finally, we define

$B_{2}:=\bigcup_{\overline{a}\in B_{1}}B_{2}(\overline{b}).$

The rest of the argument now closely follows the case $m=1$. This completes
the proof of Theorem 1.2. ∎

###  3.2.  Proof of Theorem 1.3

The proof is once again split into the upper bound and the lower bound. The
upper bound trivially follows from the definition of the set and [2, Theorem
1.1], as we always have

$P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})\subseteq\mathbb{E}(\mathbf{f},D),$

and so

$\dim_{\mathrm{H}}P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})\leq\dim_{\mathrm{H}}\mathbb{E}(\mathbf{f},D)=s_{0}(D;\boldsymbol{\xi}).$

As for the lower bound, we will step-by-step construct subsets of the original
set $P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})$ and eventually
come to a subset of the form

$P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})\supseteq
S(\mathbf{f},D,\zeta)$

for a particular choice of a function $\zeta:\mathbb{N}\to[\min D,\infty).$ By
the result of Theorem 1.2 this will imply Theorem 1.3.

First, without loss of generality we can assume that $\varphi$ is strictly
increasing, as for any function $\varphi:\mathbb{N}\to[\min D,\infty)$ such
that $\varphi(n)\to\infty$ as $n\to\infty$ we can always find another function
$\phi$ that is strictly increasing and satisfies $\phi(n)<\varphi(n)$ for any
$n\in\mathbb{N}$. This means we can always pass, if needed, to a subset

$P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})\supseteq
P_{m}(\mathbf{f},D,\phi,\overline{t},\overline{g})$

with $\phi$ being a strictly increasing function.

In what follows we assume that $\varphi$ is strictly increasing. It is easy to
see that if one has

$a_{g_{i}(n)}(x)\leq\varphi(n)^{\frac{1}{mt_{i}}}\text{ for each }1\leq i\leq
m,\text{ for all }n\in\mathbb{N},$

then

$a_{g_{1}(n)}^{t_{1}}(x)\cdots a_{g_{m}(n)}^{t_{m}}(x)\leq\varphi(n)\text{ for
all }n\in\mathbb{N}.$

Hence the first subset we consider is

$\begin{split}P_{m}(\mathbf{f},D,\varphi,\overline{t},\overline{g})&\supseteq\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{g_{i}(n)}(x)\leq\varphi(n)^{\frac{1}{mt_{i}}},1\leq
i\leq m,\text{ for all }n\in\mathbb{N}\right\\}\\\ &=:G_{1}.\end{split}$

Now let $I_{i}:=g_{i}(\mathbb{N})$. For each $1\leq i\leq m$ and for each
$k\in I_{i}$, let

$M_{i,k}:=\min\\{r\in\mathbb{N}\colon g_{i}(r)=k\\}.$

Note that $M_{i,k}$ is well-defined as the minimum in the definition is taken
over a non-empty set of natural numbers. The second subset we consider is

$G_{1}\supseteq\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{r_{i}}(x)\leq\varphi(M_{i,r_{i}})^{\frac{1}{mt_{i}}},\text{
for all }r_{i}\in I_{i},1\leq i\leq m\right\\}=:G_{2}.$

For each $1\leq i\leq m$, define a set $C_{i}=\mathbb{N}\backslash I_{i}$ For
any $k\in C_{i}$, let

$\displaystyle R_{i,k}:=\left\\{\begin{split}&\min\\{\\{M_{i,j}\\}_{j\in
I_{i}}\cap\mathbb{N}_{>k}\\},&&\quad\text{ if }\\{M_{i,j}\\}_{j\in
I_{i}}\cap\mathbb{N}_{>k}\neq\emptyset,\\\ &\max\\{\\{M_{i,j}\\}_{j\in
I_{i}}\cap\mathbb{N}_{<k}\\}+k,&&\quad\text{ otherwise.}\end{split}\right.$

Note that $R_{i,k}$ is also well-defined as at least one of the sets
$\\{M_{i,j}\\}_{j\in I_{i}}\cap\mathbb{N}_{>k}$ or $\\{M_{i,j}\\}_{j\in
I_{i}}\cap\mathbb{N}_{<k}$ is non-empty, because $I_{i}$ is always non-empty.

For each $1\leq i\leq m$, define a function
$\zeta_{i}\colon\mathbb{N}\to\mathbb{N}$ by

$\displaystyle\zeta_{i}(n):=\left\\{\begin{split}&\varphi(M_{i,n})^{\frac{1}{mt_{i}}},&&\quad
n\in I_{i},\\\ &\varphi(R_{i,n})^{\frac{1}{mt_{i}}},&&\quad n\in
C_{i}.\end{split}\right.$

It is easy to see that each $\zeta_{i}(n)$ is non-decreasing and tending to
infinity as $n\to\infty.$ Indeed, for each $1\leq i\leq m$ either $I_{i}$ or
$C_{i}$ is infinite. If $I_{i}$ is infinite, then $\\{M_{i,k}\\}_{k\in I_{i}}$
forms a non-decreasing sequence, tending to infinity as $k\to\infty$. If
$C_{i}$ is infinite, then $\\{R_{i,k}\\}_{k\in C_{i}}$ forms a non-decreasing
sequence, tending to infinity as $k\to\infty$. As $\varphi$ is a strictly
increasing function, we get the claim.

Finally, define a function $\zeta\colon\mathbb{N}\to[\min D,\infty)$ by

$\zeta(n):=\min\\{\zeta_{1}(n),\ldots,\zeta_{m}(n)\\}.$

For any $1\leq i\leq m$ and for any $n\in\mathbb{N}$, we have
$\zeta(n)\leq\zeta_{i}(n)$, and so we can pass to the final subset

$G_{2}\supseteq\left\\{x\in\mathbb{E}(\mathbf{f},D):a_{n}(x)\leq\zeta(n),\text{
for all }n\in\mathbb{N}\right\\}=:G_{3}=S(\mathbf{f},D,\zeta).$

As $\zeta(n)$ is non-decreasing and tending to infinity as $n\to\infty$, by
Theorem 1.2 we get the desired conclusion. This completes the proof of Theorem
1.3.∎

### Acknowledgments

GGR, MH, and NS were supported by the Australian Research Council discovery
project grant number 200100994. HT was supported by the JSPS KAKENHI 19K21835,
20H01811.

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