# Uncovering the nature of transient and metastable non-equilibrium phases in
1$T$-TaS2

Tanusree Saha  Corresponding author<EMAIL_ADDRESS>Laboratory
of Quantum Optics, University of Nova Gorica, 5001 Nova Gorica, Slovenia
Arindam Pramanik  Department of Theoretical Physics, Tata Institute of
Fundamental Research, Mumbai 400005, India  Barbara Ressel  Laboratory of
Quantum Optics, University of Nova Gorica, 5001 Nova Gorica, Slovenia
Alessandra Ciavardini  Laboratory of Quantum Optics, University of Nova
Gorica, 5001 Nova Gorica, Slovenia  Fabio Frassetto  CNR-Institute of
Photonics and Nanotechnologies (CNR-IFN), 35131 Padova, Italy  Federico
Galdenzi  Laboratory of Quantum Optics, University of Nova Gorica, 5001 Nova
Gorica, Slovenia  Luca Poletto  CNR-Institute of Photonics and
Nanotechnologies (CNR-IFN), 35131 Padova, Italy  Arun Ravindran  Laboratory of
Quantum Optics, University of Nova Gorica, 5001 Nova Gorica, Slovenia  Primož
Rebernik Ribič  Elettra Sincrotrone Trieste, Strada Statale 14 km 163.5, 34149
Trieste, Italy  Giovanni De Ninno  Laboratory of Quantum Optics, University of
Nova Gorica, 5001 Nova Gorica, Slovenia  Elettra Sincrotrone Trieste, Strada
Statale 14 km 163.5, 34149 Trieste, Italy

###### Abstract

Complex systems are characterized by strong coupling between different
microscopic degrees of freedom. Photoexcitation of such materials can drive
them into new transient and long-lived hidden phases that may not have any
counterparts in equilibrium. By exploiting femtosecond time- and angle-
resolved photoemission spectroscopy, we probe the photoinduced transient phase
and the recovery dynamics of the ground state in a complex material: the
charge density wave (CDW)- Mott insulator 1$T$-TaS2. We reveal striking
similarities between the band structures of the transient phase, where the
Mott electronic order and CDW lattice order are suppressed, and the
(equilibrium) structurally undistorted metallic phase. Following the transient
phase, we find that the restoration of the Mott and CDW ordering begins around
the same time. This highlights that the Mott transition is tied to the CDW
structural distortion, although earlier studies have shown that the collapse
of Mott and CDW phases are decoupled from each other. Interestingly, as the
suppressed order starts to recover, a long-lived metastable phase emerges
before the material recovers to the ground state. Our results demonstrate that
it is the CDW lattice order that drives the material into this metastable
phase, which is indeed a commensurate CDW-Mott insulating phase but with a
smaller CDW amplitude. Moreover, we find that the long-lived state emerges
only under strong photoexcitation and has no evidence when the photoexcitation
strength is weak.

Materials dominated by strong electron-electron and electron-lattice
interactions can undergo phase transitions to insulating ground states,
exhibiting charge and lattice orderingch6_mott1961 ; ch6_mott1968 ;
ch6_gruner1994 ; ch6_imada1998 ; ch6_knox1964 ; ch6_zandt1974 ;
ch6_gebhard2000 . Under non-equilibrium conditions, such systems display a
collapse of charge and lattice ordering of the ground state, as well as the
occurrence of novel or hidden phases which are thermally inaccessible under
equilibriumch6_ichikawa2011 ; ch6_stojchevska2014 . Ultrafast pump-probe
techniques have paved the way to delve into the non-equilibrium regime of
matterch6_hilton2006 ; ch6_bovensiepen2012 ; ch6_miller2014 . Solid-state
systems exhibiting some intriguing phases, such as Mottch6_V2O3_1983 ;
ch6_boer1937 ; ch6_limelette2003 ; ch6_janod2015 ; ch6_katsufuji1994 ;
ch6_ma1993 , charge density wave (CDW)ch6_wang2003 ; ch6_moncton1975 ;
ch6_boubeche2021 ; ch6_blackburn2013 ; ch6_hashimoto2014  and
excitonicch6_cercellier2007 ; ch6_bucher1991 ; ch6_du2017 ; ch6_lu2017 , are
being extensively studied using ultrafast spectroscopic and diffraction
methods in the femtosecond time domain. The relevant timescales of quenching
dynamics, photoinduced phase transitionsch6_wall2011 ; ch6_rohwer2011 ;
ch6_smallwood2011 ; ch6_hellmann2012 ; ch6_peterson2011 ; ch6_tomeljak2009 ;
ch6_schmitt2011 ; ch6_okomoto2011  and the emergence of metastable
phasesch6_stojchevska2014 ; ch6_vaskivskyi2015 ; ch6_sun2018 ; ch6_shi2019
are the topics of great interest. While the quenching occurs instantaneously
in Mott insulators and the timescale is set by the electronic hopping time
given by the bandwidthch6_wall2011 ; ch6_moritz2010 , the Peierls-CDW
materials exhibit quenching times that are comparable to the timescales of
the, slower, lattice-driven processesch6_tomeljak2009 ; ch6_schmitt2008 . For
excitonic insulators, carrier screening time, given by the plasma frequency,
determines the characteristic timescalech6_rohwer2011 .

The layered CDW-Mott insulator 1$T$-TaS2 is a prominent example of complex
system since both electron-electron and electron-lattice interactions are
simultaneously strong. It exhibits a manifold of electronic and structurally
ordered phasesch6_hellmann2012 ; ch6_smith2006 ; ch6_perfetti2006 ;
ch6_perfetti2008 ; ch6_sohrt2014 ; ch6_wang2020 : at high temperatures ($T>$
550 K), the system has an undistorted hexagonal structure and is metallic
while cooling results in the formation of various CDW phases - incommensurate
$\rightarrow$ nearly-commensurate $\rightarrow$ commensurate. Below the
critical temperature for the commensurate CDW (CCDW) phase, $T_{C}=180$ K, a
periodic lattice distortion gives rise to the formation of “Star-of-
David”-shaped clusters consisting of thirteen Ta atoms. Fig. 1(a) shows a
schematic of the lattice reconstruction in the plane of Ta atoms and its
Brillouin zone in the metallic and CCDW phases of 1$T$-TaS2. The
$\sqrt{13}\times\sqrt{13}$ superlattice splits the Ta $5d$ valence band into
three subband manifolds, such that the narrow half-filled band at the Fermi
level $E_{F}$ becomes favorable for a Mott-Hubbard transitionch6_smith2006 ;
ch6_fazekas1979 . Previous trARPES studies have shown an instantaneous
collapse of the Mott gap at $E_{F}$ on timescales $<50$ fs after
photoexcitationch6_hellmann2012 ; ch6_peterson2011 ; ch6_sohrt2014 ;
ch6_ligges2018 . In addition, the CDW gap between the Ta $5d$ subbands was
found to melt faster than the lattice vibrational timescale, suggesting that
electron correlations might play a vital role in the CDW
orderingch6_peterson2011 ; ch6_sohrt2014 . A prompt collapse of charge
ordering was also shown using ultrafast core-level PESch6_hellmann2010 .
Ultrafast electron diffraction studies have identified a suppression of the
periodic lattice distortion in the nearly-CCDW phase from the optically
induced change in the spatial distribution of the electron
densitych6_eichberger2010 . Lately, single-shot time-resolved techniques were
able to capture the emergence of a persistent “hidden” phase in
1$T$-TaS2ch6_stojchevska2014 ; ch6_vaskivskyi2015 ; ch6_sun2018 ;
ch6_vaskivskyi2016 ; ch6_gerasimenko2019 ; ch6_stahl2020 ; ch6_ravnik2021 ;
ch6_gao2022 . However, different characteristics of such a state can be
manifested by tuning the experimental conditionsch6_avigo2016 .

Figure 1: (a) (Top) In-plane structural distortion in the CCDW phase of
1$T$-TaS2 produces “Star-of-David” clusters having inequivalent “a”, “b”, and
“c” atoms. Red and blue dashed lines indicate the unit cells in the CCDW and
unreconstructed phases, respectively. The arrows indicate the displacement of
the Ta atoms from their initial positions. (Bottom) Brillouin zone in the
unreconstructed (blue) and distorted phases (red) with the high-symmetry
points $\Gamma,M,K$. (b) A schematic of the pump-probe experimental geometry
where the electric field $\vec{E}$ of s- and p-polarized photons are indicated
by blue (along y-axis) and green (in the xz-plane) double-headed arrows,
respectively. In all experimental plots, the pump fluence refers to the
incident peak energy density (in mJ/cm2), determined from the expression:
$2E_{p}/(\pi w^{2})$, where $E_{p}$ is the energy per pulse and $w$ is the
beam waist at the sample position.

Even though this material has been extensively studied, there has been minimal
emphasis on the state of charge and lattice ordering in the non-equilibrium
transient phase. Moreover, the majority of the studies have focussed on the
early stages of the dynamics, i.e., on the collapse rather than the recovery
to the ground state. In the present work, we address the above scenario in
1$T$-TaS2 by studying its electronic band structure in the transient phase, as
well as the recovery dynamics of the electronic and lattice order. We choose
band structure as the spectroscopic parameter since its various features such
as bandwidth, dispersion of the band and binding energy provide information
about the lattice order, which plays a prominent role in the ground state of
1$T$-TaS2. Angle-resolved photoemission spectroscopych6_damascelli2003
(ARPES) in the ultrafast time domain is employed to systematically track the
temporal evolution of Ta $5d$ subbands in the CCDW-Mott phase. Our time-
resolved ARPES (trARPES) study demonstrates that, after optical excitation,
the material enters a transient phase which bears a striking correspondence
with the high-temperature unreconstructed phase. Interestingly, the recovery
of the Mott and CDW dynamics, after traversing the transient phase, is
observed to commence around the same time. It is important to note that
although the suppression of Mott-CDW electronic order and CDW lattice order is
known to occur on two distinct time scales in 1$T$-TaS2ch6_peterson2011 ;
ch6_sohrt2014 , the presence of a single timescale observed for the order re-
establishment emphasizes that Mott physics is indeed coupled to the CCDW
ordering in this material. Moving further, we find that the material recovers
to a long-lived hidden phase that is primarily governed by the lattice order
of the CDW. Moreover, our results predict that the hidden phase is a CCDW-Mott
insulating phase but with a reduced CDW amplitude. Lastly, we also demonstrate
that the emergence of a long-lived metastable state is observed only at high
photoexcitation strengths and has no signatures under weak photoexcitation.

Figure 2: (a) Time evolution of the electronic band structure in 1$T$-TaS2
about $M$-point (along $MK$ direction). The peak positions of the energy
distribution curves (EDCs) have been plotted as a function of $k_{||}$ at each
pump-probe delay $\Delta t$. (b) ARPES snapshots acquired before and after
($\Delta t$ = +300 fs) photoexcitation. (c) Corresponding EDC stacking where
the blue curve represents the EDC at $M$. The black curves are guide to the
eye for the band dispersion. (d) Comparison of the band dispersion before
photoexcitation and in the transient state of the system, where there is an
energy shift towards $E_{F}$ and the band is more dispersive. All the data
correspond to a high pump fluence of 3.6 mJ/cm2 and the dashed lines in (b)
and (c) indicate $E_{F}$. Binding energy is abbreviated to B. E. Figure 3: (a)
Temporal evolution of the EDCs at early pump-probe delays integrated over a
$k_{||}$ range of $\pm 0.1$ Å-1 about $\Gamma$-point (along $\Gamma M$
direction). (b) ARPES snapshots acquired before and after ($\Delta t$ = +300
fs) photoexcitation. (c) Corresponding EDC stacking where the blue curve
denotes the EDC at $\Gamma$. Smooth curves are guide to the eye to emphasize
the change in the band dispersion around $\Gamma$ in the transient phase. (d)
ARPES snapshots about $\Gamma$ taken at different delays using s-polarized
probe pulses. (e) Corresponding EDC stacking. The smooth black line indicates
the flat upper Hubbard band and its dynamics is obtained by changing the probe
polarization from horizontal (p-pol) to vertical (s-pol). The data acquired
using p-polarized and s-polarized probe pulses correspond to pump fluences of
3.2 mJ/cm2 and 3.6 mJ/cm2, respectively, and the dashed lines indicate
$E_{F}$. Binding energy is abbreviated to B. E.

Single crystals of 1$T$-TaS2 were purchased from HQ Graphenech6_hqgraphene .
The trARPES experiments were performed at the CITIUS high-harmonic generation
(HHG) light sourcech6_citius . The system is driven by a mode-locked
Ti:Sapphire laser delivering 800 nm (1.55 eV) pulses, with a duration of 35 fs
at a repetition rate of 5 kHz. The photon energy of the probe was selected by
a monochromator grating with off-plane geometry, which preserved the pulse
durationch6_poletto . A probe energy of $\sim$ 20 eV (harmonic 13 of the
fundamental laser) was used in the experiments due to the higher
photoionization cross-section of the Ta 5$d$ bands and a high photon flux. To
preserve the ultrafast response, the energy resolution of the source was
limited to about 150 meV. This allowed us to achieve a temporal resolution of
around 50 fs. A schematic of the experimental geometry showing the
polarization of pulses is shown in Fig. 1(b). All the measurements were
performed at an initial sample temperature $T=100$ K ($<T_{C}$) and a base
pressure $<$ 1x10-10 mbar. p-polarized pump and probe pulses [green arrows in
Fig. 1(b)] were used for the obtained data, unless specified.

Figure 4: (a) Time evolution of the electronic band dispersion about $M$-point
(along $MK$ direction). For each pump-probe delay $\Delta t$, the peak
position of the EDCs are plotted as a function of $k_{||}$. (b) ARPES
snapshots acquired before and after ($\Delta t$ = +3.5 ps) photoexcitation.
(c) Corresponding EDC stacking where the blue curve represents the EDC at $M$.
The black curves are guide to the eye for the band dispersion. (d) Comparison
of the band dispersion before photoexcitation and in the long-lived state of
the system. The energy shifts around the band minimum and maximum are
indicated by arrows. (e) ARPES snapshots acquired before and after ($\Delta t$
= 3.5 ps) photoexcitation about $\Gamma$-point (along $\Gamma M$ direction).
(f) Temporal evolution of the EDCs at longer delays integrated over a $k_{||}$
range of $\pm 0.1$ Å-1 about $\Gamma$. All the data correspond to a high pump
fluence of 3.6 mJ/cm2 and the dashed lines in (b), (c), (e) denote $E_{F}$.
Binding energy is abbreviated to B. E.

We will refer to the (equilibrium) electronic band structure of 1$T$-TaS2
reported in Ref. [44] while presenting the trARPES results on different Ta
$5d$ subbands. Firstly, we will demonstrate the nature of the photoinduced
transient phase by characterizing the evolved band structure. For a high
photoexcitation strength (3.6 mJ/cm2), the time evolution of the Ta $5d$
subband along high symmetry $MK$ direction (we call it $B_{2}$
band)ch6_sohrt2014  is plotted in Fig. 2(a). We observe that a shift in
binding energy towards $E_{F}$ and an enhancement of the bandwidth
characterize the evolution, which occurs on a 200 fs-timescale. Since the
timescale corresponds to half an oscillation cycle of the CDW amplitude mode
ch6_duffey1976 ; ch6_demsar2002 , the temporal changes indicate the collapse
of CDW lattice order after photoexcitation. Subsequent recovery of the
suppressed order is observed to occur after 300 fs (red and yellow circles).
The characterization of the transient phase at pump-probe delay $\Delta
t=+300$ fs is presented in Figs. 2(b)-(d). A band shift of the minima by 0.16
eV towards $E_{F}$ accompanied by a substantial increase of the bandwidth [see
Fig. 2(d)] are in excellent agreement with the dispersion of the $B_{2}$ band
in the unreconstructed phasech6_sohrt2014 . According to theoretical
calculationsch6_smith2006 ; ch6_sohrt2014 , the dispersion crosses $E_{F}$ at
$k_{||}$ away from $M$, which is however, not evident in our data at 300 fs.
This is because $B_{2}$ might have traversed such a feature within a few 10s
of fs before 300 fs and could not be captured due to a large time interval (50
fs) used in the experiments. This particular characteristic of the dispersion
is reported in Ref. [31].

We will now look at the dynamics of the lower Hubbard band (LHB)ch6_sohrt2014
along high symmetry $\Gamma M$ direction at a similar photoexcitation strength
(3.2 mJ/cm2). The EDCs at various time delays extracted from the
$k$-integrated trARPES spectrum are shown in Fig. 3(a). The early dynamics
show a collapse of the Mott phase as the spectral weight in LHB is suppressed
and transferred to binding energies at and above $E_{F}$, similar to earlier
studiesch6_perfetti2006 ; ch6_peterson2011 ; ch6_hellmann2012 . The recovery
of the spectral weight begins after 300 fs; it is to be noted that this is the
same time at which the CDW lattice order starts to reform in Fig. 2(a). In
spite of the established scenario where the suppression of electronic and
lattice order occurs on different time scalesch6_peterson2011 , we find that
the re-establishment of Mott electronic order and CDW lattice order begins at
the same time. This provides evidence that the CCDW lattice reconstruction is
the mechanism behind the Mott transition in this materialch6_sohrt2014 ;
ch6_smith2006 . Figures 3(b)-(e) display the characteristics of the band
structure in the transient phase at $\Delta t=+300$ fs. We find that the
spectral weight from LHB has shifted to an energy band above $E_{F}$, which is
(i) dispersive about $\Gamma$ unlike the flatness of LHB, (ii) the band minima
lies at $\approx$ -0.1 eV [see Figs 3(b), (c)]. [It is to be noted that the
dispersive feature beyond $\pm 0.15$Å-1 in Figs. 3(b), (c) (left panels) is a
contribution from other Ta $5d$ subbands.] More importantly, the dispersive
band at 300 fs does not correspond to the flat upper Hubbard band (UHB). This
has been verified from the UHB dynamics that could be tracked at 20 eV probe
energy by changing the polarization of the probe pulses [see Fig. 1(b)] from
horizontal (p-pol) to vertical (s-pol). In Figs. 3(d) and 3(e), the UHB lying
at $\approx$ -0.25 eV can be distinctly observed at $\Delta t=+50$ fs, which
eventually shifts towards $E_{F}$ with time. At $\Delta t=+300$ fs, the UHB
lies across $E_{F}$ and cannot be spectrally resolved as shown in Fig. 3(e)
(right panel). All the above features of the dispersive band have a close
resemblance to the band structure of the unreconstructed metallic phase about
$\Gamma$ch6_sohrt2014 . Therefore, our trARPES results at early time delays
show that under the destruction of electronic and lattice ordering in the
CCDW-Mott phase, the material enters a transient phase which has remarkable
similarities with the unreconstructed metallic phase.

Figure 5: (a) ARPES snapshots of the Ta $5d$ subband at 0.5 eV below $E_{F}$
along $\Gamma M$ direction taken before and after photoexcitation: delay
$\Delta t$ = - 1 ps (left), $\Delta t$ = + 30 ps at pump fluence 4.2 mJ/cm2
(middle), $\Delta t$ = + 3 ps at pump fluence 1.2 mJ/cm2 (right). (b)
Corresponding stacked EDCs representing the band dispersion. Smooth black
curves are guide to the eye for the dispersion and dashed lines denote
$E_{F}$. (c) Peak positions of the EDCs plotted as a function of $k_{||}$ at
various time delays for high fluence, 4.2 mJ/cm2. (d) The same for low
fluence, 1.2 mJ/cm2. The data at high fluence shows the presence of a long-
lived state. Binding energy is abbreviated to B. E.

Now, we will move on to the recovery dynamics and identify the nature of the
phase where it settles at longer time delays. Figure 4 captures such dynamics
under strong photoexcitation (3.6 mJ/cm2) for the probed Ta $5d$ subbands
($B_{2}$ and LHB). We observe that, as the relaxed lattice structure of the
transient phase starts to recover after 300 fs, there is only a partial
recovery of the lattice order till $\Delta t=+600$ fs shown in Fig. 4(a). We
call it partial since $B_{2}$ band does not exhibit the dispersion
corresponding to that of before photoexcitation ($\Delta t=-1.2$ ps). Any
further recovery occurs on extremely long time scales which can be clearly
identified from the negligible changes in the band dispersion from 600 fs to
$\Delta t=+3.5$ ps. This signifies the emergence of a long-lived metastable
state in photoexcited 1$T$-TaS2. The ARPES snapshots taken before and after
(3.5 ps) photoexcitation, and their EDCs are shown in Fig. 4(b) and Fig. 4(c),
respectively. In the long-lived hidden phase, $B_{2}$ exhibits a weaker band
dispersion in comparison to the transient phase [compare red and yellow curves
in Fig. 4(a)]. However, the band minima is still shifted by $\approx 0.08$ eV
towards $E_{F}$ and $B_{2}$ has a larger bandwidth with respect to the ground
state dispersion [see Fig. 4(d)]. On the other hand, the dynamics of the LHB
display a complete recovery of the Mott phase. This can be claimed from the
following features of LHB at $\Delta t=+3.5$ ps: (i) the spectral weight
recovery in LHB and no additional weight at the tail of the EDC in Fig. 4(f),
and (ii) the peak of the EDC lying at a similar binding energy as that of
before photoexcitation [see Figs. 4(e) and 4(f)]. However, the recovery of LHB
intensity slows down after 600 fs, with no pronounced changes at longer time
delays. It is not known whether such slow dynamics of LHB can be linked to the
destruction of CDW order and will require fluence-dependent studies in the
future to make any further comments.

Finally, we will look at the features of the metastable state under strong
(4.2 mJ/cm2) and weak (1.2 mJ/cm2) photoexcitation by tracking the dynamics of
the Ta $5d$ subband lying at 0.5 eV below $E_{F}$ (we call it
$B_{1}$)ch6_sohrt2014  in Fig. 5. For a high photoexcitation strength, the
band dispersion at long time delays is stronger and shifted towards $E_{F}$
while this is not the case at a low photoexcitation strength [see Figs. 5(c),
(d)]. We show the data at $\Delta t=+30$ ps for pump fluence 4.2 mJ/cm2 in
Figs. 5(a), (b) to emphasize that the dispersion (CDW lattice order) has not
recovered even at longer times. The quantitative changes in the band structure
at $\Delta t=+2$ ps are persistent till $\Delta t=+30$ ps and longer under
strong photoexcitation in Fig. 5(c). This, once again, provides evidence that
the system is driven to a long-lived metastable state prior to the complete
recovery of the CDW lattice order. On the contrary, we do not find any
signatures of the metastable state under weak photoexcitation since the small
bandshifts are completely recovered within $\Delta t=+3$ ps [compare black and
green curves in Fig. 5(d)]. At low photoexcitation strengths ($\sim 1.3$
mJ/cm2 in this study), the LHB dynamics show complete recovery of the Mott
phase. Hence, the long time dynamics of the Ta $5d$ subbands (LHB, $B_{1}$,
$B_{2}$) provide insights into the metastable phase in 1$T$-TaS2, which is a
hidden phase having no counterparts in equilibrium.

The correspondence between the (photoinduced) transient and (equilibrium)
structurally undistorted phases imply that the ordering in the CCDW-Mott phase
is completely destroyed as the lattice order relaxes to the undistorted
metallic phase. Although the recovery of both the CDW and Mott phases begin at
the same time, the CDW phase undergoes only a partial recovery while the Mott
phase fully recovers within one ps. The metastable state attained by the
system after its partial recovery does not correspond to any of the thermally
accessible equilibrium phases. The signatures of the metastable phase are
exhibited only by $B_{1}$ and $B_{2}$ bands, while LHB shows no evidence of
such a long-lived state. Since the LHB is derived from electron-electron
interactions and $B_{1}$ and $B_{2}$ have dominant contribution from electron-
lattice interactionsch6_smith2006 , it can be inferred that it is primarily
the interaction of the electrons with the lattice that pushes the material
towards a long-lived state. Such a state could be mediated by mode-selective
electron-phonon coupling due to the destruction of CDW order, as has been
shown for a similar compound 1$T$-TaSe2ch6_shi2019 .

It is the electronic and lattice configuration in the low-temperature CCDW
phase, which makes it susceptible to a Mott-Hubbard transition. Even though
the CDW phase is not observed to reform completely, the ordering of the
electronic and lattice degrees of freedom is such that the intracluster
Coulomb repulsion ($U$) is larger than the electronic hopping strength ($W$),
i.e., $U/W\gtrsim 1.2$ch6_perfetti2008 . This tends to localize the electrons
at the atomic sites, leading to the recovery of the Mott phase. Therefore, it
can be deduced that the metastable state is indeed a Mott insulating phase but
with a reduced CDW amplitude as compared to the CCDW phase in equilibrium. A
clear and direct investigation of the structural configuration in the
metastable non-equilibrium phase can be obtained from time-resolved electron
diffraction, which will be used in future studies to probe the long-lived
hidden phases in this compound. It is also important to identify the critical
fluence above which such a long-lived hidden phase emerges. Further time-
resolved studies in this direction would involve a deeper investigation of how
the microscopic interactions evolve as the material changes its state under
non-equilibrium.

In summary, we demonstrated the characteristics of the non-equilibrium phases
in photoexcited 1$T$-TaS2 using time-resolved ARPES. In the transient phase,
the Mott-CDW order is suppressed and the band structure has excellent
resemblance with that of the unreconstructed metallic phase. The Mott and CDW
orders begin recovering around the same time, but only to settle in a long-
lived metastable phase. In this “hidden” phase, 1$T$-TaS2 is a CCDW-Mott
insulator but with a reduced CDW amplitude and the emergence of this phase is
driven by the lattice order. In addition, the metastable state emerges only
under strong photoexcitation of the system. A distinct characterization of
these phases provides deeper insights into the state of charge and lattice
order under non-equilibrium and the prominent role played by the different
degrees of freedom in governing these phases in a complex system.

We are thankful to E. Nicolini and G. Bortoletto for characterization of the
samples using Laue diffraction. We acknowledge fruitful discussions with M.
Capone, Z. Bacciconi and A. Amaricci. This work was supported by the FLAG-ERA
grant DIMAG, by the Research Foundation – Flanders (FWO), the Agence Nationale
pour la Recherche (ANR), the Deutsche Forschungsgemeinschaft (DFG), the
Slovenian Research Agency (ARRS).

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\[
t<t+\veps<t + 2\veps<s_{n,B}(M)\leq 2
\quad\text{ for all }
n\in\Na_{\geq N_3}.
\]

Define $\kappa_7\colon= 81000$ and let $k_0\in\Na$ be such that every $k\in\Na_{\geq k_0+1}$ has the following properties:

\begin{equation}\label{EQ:COND-ON-nk-01}
m_k\geq N_3,
\end{equation}

\begin{equation}\label{EQ:COND-ON-nk-02}
\frac{n_k}{m_k} \leq 1 + \frac{2\veps}{t+2\veps},
\end{equation}

\begin{equation}\label{EQ:COND-ON-nk-03}
\frac{\log \kappa_7}{2k\veps \log B} + \frac{\log(B^{2N_3}(2M+1)^{4N_3}) + \log(6^8\cdot 350) - \log\kappa_5^2 - \log \kappa(\hat{s})}{2k\veps \log B}
\leq
\frac{1}{k-1} \sum_{j=k_0+1}^{k} n_j
\end{equation}

Condition (<ref>) is actually attained by every large $k$ because (<ref>) implies

\[
\lim_{k\to\infty} \frac{1}{k}\sum_{j=1}^k n_j = \infty.
\]

For notational convenience, for any $k\in\Na$ and any $j\in \{1, \ldots, k\}$, we write $q_{m_j}$ rather than the cumbersome $q_{m_j}(a_{n_{j-1}+1}, \ldots, a_{n_j-1})$. Lastly, for

\[
c_0\colon=  (2M+1)^{4n_{k_0}}B^{4(n_1 + \ldots + n_{k_0})},
\]

Proposition <ref> guarantees

\begin{equation}\label{Eq:CondOn_c0}
\prod_{j=1}^{k_0} \frac{1}{\lfloor B^{n_j}\rfloor} \left( \frac{5\sqrt{2}}{B^{m_j}\|q_{m_j}\|^2}\right)^{s_{m_j,B}(M)}
\leq
\leq
c_0\prod_{j=1}^{k_0} \left( \frac{5\sqrt{2}}{B^{2n_j}\|q_{m_j}\|^2}\right)^{t+\veps}.
\end{equation}

Although all of these parameters depend on $t$, the order we used to define them is relevant. We have the following dependencies: $\veps=\veps(t)$, $\tilde{\veps}=\tilde{\veps}(\veps)$, $N_3=N_3(\veps, \tilde{\veps})$, $k_0=k_0(\veps, N_3)$, and $c_0=c_0(k_0)$.

Take $n\in\Na$ such that $n\geq n_{k_0+1}$ and $\bfa\in D_n$. We now estimate $\mu(A_n(\bfa))$.

§.§.§ Case $n=n_k$

For any $\bfa=(a_1,\ldots, a_{n_k})\in D_{n_k}$, the definition of $\mu$ and Proposition <ref> yield

\begin{align}
\mu(A_{n_k}(\bfa))
\frac{1}{2^k}
\prod_{j=1}^k \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{s_{m_j,B}(M)} \nonumber\\
\frac{1}{2^k}
\prod_{j=1}^{k_0} \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{s_{m_j,B}(M)}\;
\prod_{j=k_0+1}^k \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{s_{m_j,B}(M)} \nonumber\\
&\leq \frac{1}{2^k} c_0
\prod_{j=1}^{k_0} \left( \frac{5\sqrt{2}}{B^{2n_j}\|q_{m_j}\|^2} \right)^{t + \veps} \;
\prod_{j=k_0+1}^k \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{t + 2\veps} && \text{(by \eqref{EQ:COND-ON-nk-01} and \eqref{Eq:CondOn_c0})} \nonumber\\
\frac{(5\sqrt{2})^{2k_0}}{2^k} c_0
\prod_{j=1}^{k_0} \left( \frac{ 1 }{B^{2n_j}\|q_{m_j}\|^2} \right)^{t + \veps} \;
\prod_{j=k_0+1}^k \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{t +2\veps} \nonumber\\
\frac{50^{k_0}}{2^k} c_0
\prod_{j=1}^{k_0}  \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^2)^{t + \veps}}  \;
\prod_{j=k_0+1}^k \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{t +2\veps} \label{Eq:n=n_k01}
\end{align}

Let us focus on the last product:

\begin{align}
\prod_{j=k_0+1}^k \frac{1}{\lfloor B^{n_j}\rfloor^2} \left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{t+2\veps}
&\leq  \prod_{j=k_0+1}^k \frac{2^2}{ B^{2n_j}} \left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{t+2\veps} \nonumber\\
&= \prod_{j=k_0+1}^k 4  \left( \frac{5\sqrt{2}}{\|q_{m_j}\|^2} \right)^{t+2\veps}\; \prod_{j=k_0+1}^k \frac{1}{B^{2n_j + m_j (t+2\veps)}} \nonumber\\
&\leq \prod_{j=k_0+1}^k 4  \left( \frac{5\sqrt{2}}{\|q_{m_j}\|^2} \right)^{t+2\veps} \;\prod_{j=k_0+1}^k \frac{1}{B^{(n_j + m_j)(t+2\veps)}}  \nonumber\\
\prod_{j=k_0+1}^k \frac{200}{\|q_{m_j}\|^{2(t+2\veps)}}  \;
\prod_{j=k_0+1}^k \frac{1}{B^{(n_j + m_j)(t+2\veps)}}  \nonumber\\
\prod_{j=k_0+1}^k \frac{200}{\|q_{m_j}\|^{2(t+2\veps)}} \;
\prod_{j=k_0+1}^k \frac{1}{B^{2n_j( t + \veps)}} && \text{(by \eqref{EQ:COND-ON-nk-02})} \nonumber\\
\prod_{j=k_0+1}^k \frac{1}{\|q_{m_j}\|^{2(t+\veps)}} \;
\prod_{j=k_0+1}^k \frac{1}{B^{2n_j( t + \veps)}} \nonumber\\
\prod_{j=k_0+1}^k \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^{2})^{t+\veps}} \;
\prod_{j=k_0+1}^k \frac{1}{B^{2n_j \veps}} \label{Eq:n=n_k02}.
\end{align}

We substitute (<ref>) in (<ref>) to obtain

\begin{align}
\mu(A_{n_k}(\bfa))
\frac{50^{k_0}}{2^k} c_0
\prod_{j=1}^{k_0}  \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^2)^{t + \veps}}
\prod_{j=k_0+1}^k \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^{2})^{t+\veps}}
\prod_{j=k_0+1}^k \frac{1}{B^{2n_j \veps}} \nonumber\\
100^k \frac{c_0}{4^{k_0}}
\prod_{j=1}^{k_0}  \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^2)^{t + \veps}}
\prod_{j=k_0+1}^k \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^{2})^{t+\veps}}
\prod_{j=k_0+1}^k \frac{1}{B^{2n_j \veps}} \nonumber\\
100^k \frac{c_0}{4^{k_0}}
\prod_{j=1}^{k}  \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^2)^{t + \veps}}
\prod_{j=k_0+1}^k \frac{1}{B^{2n_j \veps}} \nonumber\\
100^k c_0
\prod_{j=1}^{k}  \frac{ 1 }{(B^{2n_j}\|q_{m_j}\|^2)^{t + \veps}}
\left(\prod_{j=k_0+1}^k \frac{1}{B^{2n_j}}\right)^{\veps} \nonumber\\
100^k c_0
\prod_{j=1}^{k}  \frac{ 1 }{(\lfloor B^{n_j} \rfloor^2\|q_{m_j}\|^2)^{t + \veps}}
\left(\prod_{j=k_0+1}^k \frac{1}{B^{2n_j}}\right)^{\veps} \label{Eqn:Aux01}
\end{align}

Recall that for every $j\in\Na$ we have $\|a_{n_j}\|\leq 2\lfloor B^{n_j}\rfloor + 1$, so $\|a_{n_j}\|\leq 3\lfloor B^{n_j}\rfloor$ and

\begin{align*}
\prod_{j=1}^k  \frac{1}{(\lfloor B^{n_j}\rfloor^2\|q_{m_j}\|^2)^{t+\veps}}
\prod_{j=1}^k \left( \frac{3^2}{(3\lfloor B^{n_j}\rfloor)^2 \|q_{m_j}\|^2} \right)^{t+\veps} \nonumber\\
\prod_{j=1}^k \left( \frac{9}{\|a_{n_j}\|^2 \|q_{m_j}\|^2} \right)^{t+\veps} \nonumber\\
9^{k(t+ \veps)} \left( \frac{1}{6^{2k-1} \|q_{n_k}(\bfa)\|}\right)^{2(t+\veps)} && \text{(Proposition \ref{Propo:BndTn})} \\
\left( \frac{9^{t+\veps}}{6^{4t}}\right)^k 6^{2t} \frac{1}{\|q_{n_k}(\bfa)\|^{2t+2\veps}} \\
\frac{1}{\|q_{n_k}(\bfa)\|^{2t+2\veps}}.
\end{align*}

Hence, plugging the previous inequality into (<ref>), we have

\begin{align*}
\mu(A_{n_k}(\bfa))
c_0 8100^k 6^4
\left(\prod_{j=k_0+1}^k \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n_k}(\bfa)\|^{2t+2\veps}} \nonumber\\
c_0 8100^k 6^4
\left(\prod_{j=k_0+1}^k \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\kappa_5^t} |A_{n_k}(\bfa)|^{t} &&\text{(Proposition \ref{Propo:CotaAn})} \\ \nonumber\\
c_0 8100^k 6^4 \frac{1}{\kappa_5^t}
\left(\prod_{j=k_0+1}^k \frac{1}{B^{2n_j}}\right)^{\veps}
|A_{n_k}(\bfa)|^{t} \nonumber\\
c_0 6^4 \frac{\kappa_7^k}{\kappa_5^2}
\left(\prod_{j=k_0+1}^k \frac{1}{B^{2n_j}}\right)^{\veps}
|A_{n_k}(\bfa)|^{t} \nonumber\\
&\leq c_0 |A_{n_k}(\bfa)|^{t}.
\end{align*}

The last inequality follows from (<ref>). This finishes the proof for the case $n=n_k$. We state the next corollary for reference.

If $r\in\Na_{\geq k_0}$ and $\bfa\in D_{n_r}$, then

\begin{align*}
\mu(A_{n_r}(\bfa)) &\leq \frac{1}{2^r}\prod_{j=k_0+1}^r \frac{1}{\lfloor B^{n_j}\rfloor^2} \left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{s_{m_j,B}(M)} \nonumber\\
&\leq c_0 6^4 \kappa_7^r \left(\prod_{j=k_0+1}^r \frac{1}{B^{2n_j}}\right)^{\veps} \frac{1}{\|q_{n_r}(\bfa)\|^{2t+2\veps }}. \nonumber
\end{align*}

§.§.§ Case $n=n_k-1$

Take $k\in\Na_{k_0+1}$, $\bfa\in D_{n_k-1}$ and let $b\in\scD$ be such that $\bfa b\in D_{n_k}$. Then, using Proposition <ref> and Corollary <ref>,

\begin{align*}
\mu(A_{n_k-1}(\bfa)) &
\leq 14
\lfloor B^{n_k}\rfloor^2 \mu\left( A_{n_k}(\bfa b)\right) \nonumber\\
&\leq c_0 \frac{14}{2}
\left(\frac{1}{ 2^{k-1} } \prod_{j=1}^{k-1} \frac{1}{\lfloor B^{n_j}\rfloor^2}
\left( \frac{5\sqrt{2}}{B^{m_j} \|q_{m_j}\|^2} \right)^{s_{m_j,B}(M)}\right)
\left( \frac{5\sqrt{2}}{\|q_{m_k}\|^2 B^{m_k}}\right)^{s_{m_k,B}(M)}\nonumber\\
&\leq 7c_0 \kappa_7^{k-1} 6^4
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n_{k-1}}\|^{2t+2\veps}}
\left( \frac{5\sqrt{2}}{\|q_{m_k}\|^2 B^{m_k}}\right)^{ s_{m_k,B}(M) }\nonumber\\
&\leq 7c_0 \kappa_7^{k-1} 6^4 \left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps} \frac{1}{\|q_{n_{k-1}}\|^{2t+2\veps}} \left( \frac{5\sqrt{2}}{\|q_{m_k}\|^2 B^{m_k}}\right)^{ t+ 2\veps}\nonumber\\
&\leq 7c_0 \kappa_7^{k-1}6^4
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{50}{(\|q_{n_{k-1}}\|\|q_{m_k}\|)^{2(t+\veps)}}
\frac{1}{\|q_{m_k}\|^{2\veps} B^{m_k(t+2\veps)}}\nonumber\\
&\leq 7c_0 \kappa_7^{k-1} 6^4
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{50}{(\|q_{n_{k-1}}\|\|q_{m_k}\|)^{2(t+\veps)}}
\frac{1}{B^{m_k(t+2\veps)}}\nonumber\\
&\leq {7}c_0 \kappa_7^{k-1} 50 \cdot 6^4
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\left(\frac{6}{\|q_{n_{k}-1}\|}\right)^{2(t+\veps)}
\frac{1}{B^{m_k(t+2\veps)}} && \text{(Proposition \ref{PROP:BD-CONTINUANT})}\nonumber\\
&\leq c_0 \kappa_7^{k-1} 350\cdot 6^8
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{ \|q_{n_{k}-1}\| ^{2(t+\veps)}}
\frac{1}{B^{tn_k }}\nonumber\\
&\leq c_0 \kappa_7^{k-1} 350\cdot 6^8
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{ (B^{n_k }\|q_{n_{k}-1}\|^{2})^t} \nonumber\\
&\leq c_0 \kappa_7^{k-1} 350\cdot 6^8
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{ (\lfloor B^{n_k} \rfloor \|q_{n_{k}-1}\|^{2})^t} \nonumber\\
&\leq c_0 \kappa_7^{k-1} 350\cdot 6^8
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\kappa_5^t} |A_{n_k-1}(\bfa)|^{t} &&\text{(Proposition \ref{Propo:CotaAn})} \nonumber\\
&\leq c_0 \kappa_7^{k-1} 350\cdot 6^8 \kappa_5^{-2}
\left(\prod_{j=k_0+1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
|A_{n_k-1}(\bfa)|^{t} \nonumber\\
&\leq c_0
|A_{n_k-1}(\bfa)|^{t} \nonumber
\end{align*}

§.§.§ Case $n_{k-1}<n<n_k$

\[
l'\colon=n_k-n-1 \text{ and } l\colon=n-n_{k-1}.
\]

Letting $\bfb$ run along the set $\{\bfc\in \sfR(l'): \bfa\bfc\in D_{n_k-1}\}$, applying Corollary <ref>, and writing $\bfa[\alpha,\beta]=(a_{\alpha}, a_{\alpha+1}, \ldots, a_{\beta} )$ for natural numbers $\alpha<\beta$, we obtain

\begin{align}
\mu(A_n(\bfa))
\sum_{\bfb} \mu(A_{n_k-1}(\bfa\bfb))\nonumber\\
&=\sum_{\bfb} \mu(A_{n_{k-1}}(a_1,\ldots, a_{n_{k-1}}))
\left( \frac{5\sqrt{2}}{B^{m_k}\|q_{m_k}(\bfa[n_{k-1}+1,n]\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
c_0 \kappa_7^{k-1} 6^4
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n_{k-1}}(\bfa)\|^{2t+2\veps}} \sum_{\bfb}
\left( \frac{5\sqrt{2}}{B^{m_k}\|q_{m_k}(\bfa[n_{k-1}+1,n]\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
&\leq c_0 \kappa_7^{k-1} 6^4
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n_{k-1}}(\bfa)\|^{2t+2\veps}} \sum_{\bfb} \left( \frac{(5\sqrt{2})^2}{B^{m_k}\|q_{l}(\bfa[n_{k-1}+1,n])\|^2\|q_{l'}(\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
&\leq c_0 \kappa_7^{k-1} 6^4
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps} \frac{(5\sqrt{2})^{s_{m_k,B(M)}}}{\|q_{n_{k-1}}(\bfa)\|^{2t+2\veps}\|q_{l}(\bfa[n_{k-1}+1,n])\|^{2s_{m_k,B}(M)}} \sum_{\bfb} \left( \frac{ 5\sqrt{2} }{B^{m_k}\|q_{l'}(\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
c_0 50 \kappa_7^{k-1} 6^4
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\left(\frac{ 1}{\|q_{n_{k-1}}(\bfa)\|\,\|q_{l}(\bfa[n_{k-1}+1,n])\|}\right)^{2t+ 2\veps} \sum_{\bfb} \left( \frac{ 5\sqrt{2} }{B^{m_k}\|q_{l'}(\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
&\leq c_0 50 \kappa_7^{k-1}  6^4
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\left(\frac{6}{\|q_{n}(\bfa)\|}\right)^{2t+ 2\veps} \sum_{\bfb} \left( \frac{ 5\sqrt{2} }{B^{m_k}\|q_{l'}(\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
c_0 50 \kappa_7^{k-1} 6^8
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n}(\bfa)\|^{2t+ 2\veps}}
\sum_{\bfb} \left( \frac{5\sqrt{2}}{B^{m_k}\|q_{l'}(\bfb)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
c_0 50 \kappa_7^{k-1} 6^8
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n}(\bfa)\|^{2t+ 2\veps}}
g_{l'}(s_{m_k,B}(M);\scD(M),B). \label{Eq:CotaMedCyl01}
\end{align}

Then, we must give an upper estimate of $g_l(s_{m_k,B}(M);\scD(M),B)$ in order to bound (<ref>). Since $l+l'=m_k$, in view of (<ref>), we have

\[
\kappa(\hat{s})g_{l'}(s_{m_k,B}(M);\scD(M),B)g_l(s_{m_k,B}(M);\scD(M),B)
\leq
\]

\[
\leq \frac{1}{\kappa(\hat{s})g_{l}(s_{m_k,B}(M);\scD(M),B)},
\]

and we are after a lower bound of $g_{l}(s_{m_j,B}(M);\scD(M),B)$. We will get it taking two cases into account: $l\leq N_3$ and $l> N_3$. If $l\leq N_3$, then

\begin{align*}
g_{l}(s_{m_k,B}(M);\scD(M),B) &= \sum_{\bfc\in\Delta_l(M)} \left( \frac{5\sqrt{2}}{B^l\|q_l(\bfc)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
&\geq \frac{1}{B^{N_3s_{m_k,B}(M)}} \sum_{\bfc\in\Delta_l(M)} \frac{1}{\|q_l(\bfc)\|^{2s_{m_k,B}(M)}} \nonumber\\
&\geq \frac{1}{B^{2N_3} (2M+1)^{4N_3}}\nonumber
\end{align*}

and, by (<ref>),

\begin{align*}
\mu(A_n(\bfa))
\frac{c_0 50 \kappa_7^{k-1} 6^8 }{\kappa(\hat{s})}
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{1}{\|q_{n}(\bfa)\|^{2t+ 2\veps}}
B^{2N_3} (2M+1)^{4N_3} \\
\frac{c_0 50 \kappa_7^{k-1} 6^8 }{\kappa(\hat{s})}
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
B^{2N_3} (2M+1)^{4N_3}
\frac{|\clC_n(\bfa)|^{t+\veps}}{\kappa_5^{t+\veps}} \\
\frac{ c_0 50 \kappa_7^{k-1} 6^8}{\kappa(\hat{s}) }
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
B^{2N_3} (2M+1)^{4N_3}
\frac{|A_n(\bfa)|^{t+\veps}}{\kappa_5^{t+\veps}} \\
\frac{ c_0 50 \kappa_7^{k-1} 6^8 \kappa_5^2}{\kappa(\hat{s}) }
B^{2N_3} (2M+1)^{4N_3}
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
|A_n(\bfa)|^{t+\veps} \\
&\leq c_0 |A_n(\bfa)|^{t}.
\end{align*}

Now assume that $l>N_3$. The definitions of $N_3$ and $\tilde{\veps}>0$ yield

\begin{align*}
g_{l}(s_{m_j,B}(M);\scD(M),B) &= \sum_{\bfc\in\Delta_l(M)} \left( \frac{5\sqrt{2}}{B^l\|q_l(\bfc)\|^2}\right)^{s_{m_k,B}(M)} \nonumber\\
&\geq \sum_{\bfc\in\Delta_l(M)} \left( \frac{5\sqrt{2}}{B^l\|q_l(\bfc)\|^2}\right)^{s_{l,B}(M) + \tilde{\veps}} \nonumber\\
&\geq \sum_{\bfc\in\Delta_l(M)} \frac{1}{(B^l(2M+1)^{2l})^{\tilde{\veps}}} \left( \frac{5\sqrt{2}}{B^l\|q_l(\bfc)\|^2}\right)^{s_{l,B}(M)} \nonumber\\
&= \frac{1}{(B^l(2M+1)^{2l})^{\tilde{\veps}}} g_{l}(s_{l,B}(M);\scD(M),B) \nonumber\\
&= \frac{1}{(B^l(2M+1)^{2l})^{\tilde{\veps}}} \geq \frac{1}{\gamma^{l\veps}} \geq \frac{1}{\gamma^{n\veps}}.
\end{align*}

The previous inequality and (<ref>) imply

\begin{align*}
\mu(A_n(\bfa))
\frac{c_0 50 \kappa_7^{k-1} 6^8}{\hat{\kappa}(s)}
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{\gamma^{n\veps}}{\|q_{n}(\bfa)\|^{2t+ 2\veps}} \\
\frac{c_0 50 \kappa_7^{k-1} 6^8}{\hat{\kappa}(s)}
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{\gamma^{n\veps}}{\|q_{n}(\bfa)\|^{2\veps}}
\frac{1}{\|q_{n}(\bfa)\|^{2t}} \\
\frac{c_0 50 \kappa_7^{k-1} 6^8\psi^{\veps}}{\kappa(\hat{s})\kappa_5^t}
\left(\prod_{j=k_0 + 1}^{k-1} \frac{1}{B^{2n_j}}\right)^{\veps}
\frac{\gamma^{n\veps} }{\psi^{n\veps}}
c_0 |A_n(\bfa)|^t.
\end{align*}

§.§.§ Estimate of $\mu(\Dx(z,r))$

In the context of regular continued fractions or Lüroth series, the measure of the balls is bounded by estimating the distance between two different fundamental sets (see <cit.>). The strategy must be modified for Hurwitz continued fractions. Let us illustrate why with a simple example. When $M=5$ and $n_1\geq 3$, both $a=3+3i$ and $2+3i$ belong to $D_1$ but $A_1(a)\cap A_1(b)\neq\vac$ (see Figure <ref>). We may thus loose any hope of obtaining a non-trivial lower bound of the distance between any two fundamental sets of the same level. This is when our discussion on the geometry of Hurwitz continued fractions pays off.

The sets $\iota[A_1(2+3i)]$ and $\iota[A_1(3+3i)]$.

Let $\tilde{\rho}$ and $\rho$ be as in Theorems <ref> and <ref>, respectively. Set $\rho' \colon= \min\{\tilde{\rho},\rho\}$. For each $\bfa\in \scD^{\Na}$ such that $\prefi(\bfa;n)\in D_n$ for all $n\in\Na$, define

\[
G_{n}(\bfa) \colon= \frac{\rho'}{\kappa_3} |A_n(\bfa)|,
\]

where $\kappa_3$ is as in Corollary <ref>. Let $r_0$ be

\[
\colon=
\min \left\{ G_n(\bfa): n\in \{1,\ldots, n_{k_0}\}, \, \bfa\in D_n\right\}.
\]

There is a constant $\nu=\nu(M,B,t)>0$ such that $\mu(\Dx(z;r)) \leq \nu r^t$ for every $z\in F_M(B)$ and every positive number $r<r_0$.

Take a real number $r$ with $0<r<r_0$, $z\in F_M(B)$, and $\bfa\in D_n$ such that $z\in A_n(a_1, \ldots, n)$ for all $n\in \Na$. Let $n\in\Na$ be such that

\[
G_{n+1}(a_1,\ldots, a_n,a_{n+1})
\leq
G_{n}(a_1,\ldots, a_n).
\]

We examine three different cases for $n$.

§.§.§ Case $n=n_k-1$

First, let us further assume that either $\omfF_n(\bfa)=\omfF$ or $\omfF =  \omfF_1(1+2i)$ or some of their images under any $f\in\Di_8$. Then, the disc $\Dx(z;r)$ intersects exactly one fundamental set of order $n$. We now consider two possibilites:

\[
r\leq  \kappa_3 |A_{n_k}(a_1,\ldots, a_{n_k})|
\text{ or }
\kappa_3 |A_{n_k}(a_1,\ldots, a_{n_k})|<r.
\]

In the first option, $\Dx(z;r)$ intersects at most six cylinders of level $n_k$, which are $\clC_n(\prefi(\bfa;n))$ and five of its neighbors, so it intersects at most six fundamental sets of level $n_k$. This gives

\[
\mu(\Dx(z;r))
\leq
6\kappa_5 c_0|A_{n_k}(\bfa)|^t
6\kappa_5 c_0\frac{\kappa_3}{\tilde{\rho}} r^t.
\]

Now assume that $\kappa_3 |A_{n_k}(a_1,\ldots, a_{n_k})|<r$. We claim that every $d\in \scD$ such that $(a_1,\ldots, a_{n_k-1}, d)\in D_{n_k}$ satisfies

\begin{equation}\label{EQ:Caso_nk-1_b1}
\leb\left( \clC_{n_k}(a_1,\ldots, a_{n_k-1},d)\right)
\geq
\frac{\kappa_1}{6^42^5}\frac{ 1 }{\|q_{n_k}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^2},
\end{equation}

\begin{equation}\label{EQ:Caso_nk-1_b2}
\frac{1}{\nu_1}
\leq
\frac{|A_{n_k}(a_1,\ldots, a_{n_k-1}, a_{n_k})|}{|A_{n_k}(a_1,\ldots, a_{n_k-1}, d)|}
\leq
\nu_1
\end{equation}

for some universal constant $\nu_1>1$. By Proposition (<ref>) and the choice of $n_1$, we have

\begin{align*}
\frac{|A_{n_k}(a_1,\ldots, a_{n_k-1}, a_{n_k})|}{|A_{n_k}(a_1,\ldots, a_{n_k-1}, d)|} &= \frac{|\clC_{n_k}(a_1,\ldots, a_{n_k-1}, a_{n_k})|}{|\clC_{n_k}(a_1,\ldots, a_{n_k-1}, d)|} \\
\frac{\kappa_1}{2} \frac{|q_{n_k}(a_1,\ldots, a_{n_k-1}, d)|^2}{|q_{n_k}(a_1,\ldots, a_{n_k-1}, a_{n_k})|^2} \\
\frac{\kappa_1\cdot 25\cdot 9}{2} \frac{|q_{n_k-1}(a_1,\ldots, a_{n_k-1})|^2|d|^2}{|q_{n_k-1}(a_1,\ldots, a_{n_k-1})|^2|a_{n_k}|^2} \\
\frac{225\kappa_1}{2} \frac{|d|^2}{|a_{n_k}|^2} \\
225 \kappa_1  \frac{\|d\|^2 }{\|a_{n_k}\|^2} \leq  675 \kappa_1.
\end{align*}

The last estimate holds because we have $\tfrac{1}{3}\leq \frac{\|d\|}{\|a_{n_k}\|}\leq 3$ and $\nu_1\colon=675 \kappa_1$ works.

Write $\nu_2\colon= 1 + \frac{\nu_1}{\kappa_3}$. We conclude from (<ref>) that

\[
A_{n_k}(a_1,\ldots, a_{n_k-1},d)\cap \Dx(z;r)\neq \vac
\;\text{ implies }\;
A_{n_k}(a_1,\ldots, a_{n_k-1},d)\subseteq \Dx(z;\nu_2 r).
\]

Therefore, if

\[
\mathsf{N}\colon= \#\left\{\bfd\in D_{n_k}: \Dx(z;r)\cap A_{n_k}(\bfd)\neq\vac\right\},
\]

we have

\[
\mathsf{N}
\leq
6^4 2^5\pi \nu_2^2  \|q_{n_k}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^4 r^2,
\]

because, by Proposition <ref>,

\begin{align*}
\leb\left( \clC_{n_k}(a_1,\ldots, a_{n_k-1},d)\right) &\geq
\frac{\kappa_1 }{2^2\|q_{n_k}(a_1,\ldots, a_{n_k-1},d)\|^4} \\
\frac{\kappa_1 6^{-4} }{2^2\|q_{n_k-1}(a_1,\ldots, a_{n_k-1})\|^4 \|d\|^4} \\
\frac{\kappa_1 6^{-4} }{2^2\|q_{n_k-1}(a_1,\ldots, a_{n_k-1})\|^4 2^4\lfloor B^{n_k}  \rfloor^2 \left( 1+  \frac{1}{2\lfloor B^{n_k}\rfloor} \right)^2 } \\
&= \frac{\kappa_1}{6^42^7}\frac{ 1 }{\|q_{n_k-1}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^2 }.
\end{align*}

We thus conclude

\begin{align}
\mu\left( \Dx(z;r)\right) &\leq \frac{\mu\left( A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right)} {\#\{d\in\scD:\bfa d\in D_{n_k}\}} \mathsf{N} \nonumber\\
\frac{6^4 2^5\pi \nu_2^2}{2\lfloor B^{n_k}\rfloor^2}   \mu\left( A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right) \|q_{n_k}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^4 r^2 \nonumber\\
6^4 2^4\pi \nu_2^2   \mu\left( A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right) \|q_{n_k}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^2 r^2. \label{EQ:Cota-muDx-01}
\end{align}

Moreover, since $A_{n_k-1}(\bfa)$ is the only fundamental set of level $n_k-1$ that $\Dx(z;r)$ intersects, we have

\begin{equation}\label{EQ:Cota-muDx-02}
\mu\left( \Dx(z;r) \right)
\mu\left( \Dx(z;r)\cap A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right)
\leq
\mu\left(A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right).
\end{equation}

Combining inequalities (<ref>) and (<ref>) and recalling that $t<2$, we conclude that for some constant $\nu_3$ we have

\begin{align*}
\mu\left( \Dx(z;r)\right) &\leq \mu\left( A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right) \min\left\{ 1, 6^4 2^4\pi \nu_2^2 \|q_{n_k}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^2 r^2\right\} \\
&\leq \mu\left( A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right) 1^{1-\tfrac{t}{2}} \left(6^4 2^4\pi \nu_2^2 \|q_{n_k-1}(a_1,\ldots, a_{n_k-1})\|^4 \lfloor B^{n_k}  \rfloor^2 r^2\right)^{\frac{t}{2}} \\
&\leq \mu\left( A_{n_k-1}(a_1,\ldots, a_{n_k-1})\right) \|q_{n_k-1}(a_1,\ldots, a_{n_k-1})\|^{2t} \lfloor B^{n_k}  \rfloor^{t} r^{t} \\
&\leq c_0\nu_3 r^{t}.
\end{align*}

We now appeal to the second geometric construction. Assume that $\omfF_n(\bfa)$ is equal to either $\omfF_n(2)$, $ \omfF_1(1+i)$ or any of their images under elements of $\Di_8$. Then, $\Dx(z;r)$ intersects at most two fundamental sets of level $n_k-1$ and they are subsets of neighboring $n_{k-1}$ cylinders, say $A_{n_k-1}(a_1,\ldots, a_{n_k-1})$ and $A_{n_k-1}(\bfb)$. Hence,

\[
\mu\left( \Dx(z;r) \right)
\mu\left( \Dx(z;r)\cap A_n(a_1,\ldots, a_{n_k-1}) \right) + \mu\left( \Dx(z;r)\cap A_n(b_1,\ldots, b_{n_k-1}) \right).
\]

Propping on the above argument we may then say that

\begin{align*}
\mu\left( \Dx(z;r) \right)
\nu_3 c_0|A_{n_k-1}(a_1,\ldots, a_{k_n-1})|^t + \nu_3c_0|A_{n_k-1}(b_1,\ldots, b_{k_n-1})|^t \\
&\leq c_0\nu_3 (1+\kappa_3) r^t.
\end{align*}

§.§.§ Case $n=n_k-2$

Observe that for any $\bfb\in D_{n_k-2}$  and any $c,d\in \scD$ such that $\bfa c, \bfa d\in D_{n_k-1}$, the definition of $\mu$ gives

\begin{align*}
\frac{\mu\left( A_{n_k-1}(\bfb c)\right)}{\mu\left( A_{n_k-1}(\bfb d)\right)}
&= \left( \frac{\|q_{m_k}(\bfb d)\|^2}{ \| q_{m_k}(\bfb c)\|^2} \right)^{s_{m_k,B}(M)} \\
&\leq \left(1800 \frac{\|d\|^2}{\|c\|^2}\right)^{s_{m_k,B}(M)}  \\
&\leq \left(1800 M^2 \right)^{ s_{m_k,B}(M)}   \\
&\leq 1800^2 M^4. && \text{ (Proposition \ref{Le:LemPrel-03})}
\end{align*}

As a consequence, estimating $\#\{x\in\scD:\bfb c\in D_{n_k-1}\}$,

\[
\mu\left( A_{n_k-2}(\bfb) \right)
\leq
8\kappa_5 1800^2 M^4 (2M+1)^2 c_0 \mu\left( A_{n_k-1}(\bfb c) \right)
\]

Finally, since $\Dx(z;r)$ intersects at most six fundamental sets of level $n_k-2$, one of them being $A_{n_k-2}(a_1, \ldots, a_{n_k-2})$ and the others are contained in the neighboring cylinders of $\clC_{n_k-2}(a_1, \ldots, a_{n_k-2})$, we have

\[
\mu\left( \Dx(z;r)\right)
\leq
48\kappa_5 1800^2 M^4 (2M+1)^2 c_0 r^t.
\]

§.§.§ Case $n_{k} \leq n \leq n_{k+1}-3$

For any $\bfb=(b_1,\ldots, b_n)\in D_{n}$ and any $c\in \scD$, we may express $\mu(A_{n+1}(\bfb c))$ as follows (these sums run along the words $\bfc\in \sfR(n_{k+1}- n - 1)$ such that $\bfb \,c\, \bfc\in D_{n_{k+1}-1}):$

\begin{align*}
\mu\left( A_{n+1}(\bfb c)\right) &=  \sum_{ \bfc} \mu\left( A_{n_{k+1}-1}(\bfb c\bfc )\right)\\
&=  \sum_{\bfc} \mu\left( A_{n_{k+1}-1}( \prefi(\bfb;n_{k-1}) )\right) \left( \frac{5\sqrt{2}}{B^{m_{k+1}} \|q_{m_{k+1}}(b_{n_k},\ldots,b_n,c,\bfc)\|^2}\right)^{s_{n_k-n-2,B}(M)} \\
&= \frac{\mu\left( A_{n_{k+1}-1}(\prefi(\bfb;n_{k-1}) )\right) (5\sqrt{2})^{s_{n_k-n-2,B}(M)}}{B^{m_{k+1} s_{n_k-n-2,B}(M)}} \sum_{\bfc}   \frac{ 1 }{ \|q_{m_{k+1}}(b_{n_k},\ldots,b_n,c,\bfc)\|^{2s_{n_k-n-2,B}(M)}}
\end{align*}

On the one hand,

\begin{align*}
\sum_{\bfc}   &\frac{ 1 }{ \|q_{m_{k+1}}(b_{n_k},\ldots,b_n,c,\bfc)\|^{2s_{n_k-n-2,B}(M)}} \leq \\
&\leq \left(\frac{50}{\|q_{n-n_k+1}(b_{n_k},\ldots,b_{n})\| \|c\|} \right)^{2s_{n_k-n-2,B}(M)} \sum_{\bfc}\frac{1}{\|q_{n_{k+1}-1-n}(\bfc)\|^{2s_{n_k-n-2,B}(M)}} \\
&\leq \left(\frac{50}{\kappa_5 \| q_{n-n_k+1}(b_{n_k},\ldots, b_{n})\| \|c\|} \right)^{2s_{n_k-n-2,B}(M)} \sum_{\bfc} |\clC_{n_{k+1}-n-2}(\bfc)|^{s_{n_k-n-2,B}(M)}.
\end{align*}

On the other hand,

\begin{align*}
\sum_{\bfc}   &\frac{ 1 }{ \|q_{m_{k+1}}(b_{n_k},\ldots, b_n,c,\bfc)\|^{2s_{n_k-n-2,B}(M)}} \geq \\
&\geq \left(\frac{1}{36\|q_{n-n_k+1}(b_{n_k},\ldots, b_{n})\| \|c\|} \right)^{2s_{n_k-n-2,B}(M)} \sum_{\bfc}\frac{1}{\|q_{n_{k+1}-1-n}(\bfc)\|^{2s_{n_k-n-2,B}(M)}} \\
&\geq \left(\frac{1}{72 \| q_{n-n_k+1}(b_{n_k}, \ldots b_{n})\| \|c\|} \right)^{2s_{n_k-n-2,B}(M)} \sum_{\bfc} |\clC_{n_{k+1}-n-2}(\bfc)|^{s_{n_k-n-2,B}(M)}.
\end{align*}

Hence, if $d\in \scD$ is such that $\bfb d$ also belongs to $D_{n+1}$ and we take $\nu_5\colon= \nu_5(M,B)\colon=\sup\{s_{n,B}(M):n\in\Na\}$ (see Proposition <ref>), then

\begin{align*}
\frac{\mu\left(A_{n+1}(\bfb c)\right)}{\mu\left(A_{n+1}(\bfb d)\right)} &\leq \left( 3600 \frac{\|d\|}{\|c\|}\right)^{2s_{n_k-n-2,B}(M)} \frac{\sum_{ \bfc } \left|\clC_{n_k-n-2}(\bfc)\right|^{s_{n_k-n-2,B}(M)} }{\sum_{ \bfd } \left|\clC_{n_k-n+1}(\bfd)\right|^{s_{n_k-n-2,B}(M)} } \\
&\leq  3600^{2\nu} M^{2\nu} \frac{\sum_{ \bfc } \left|\clC_{n_k-n-2}(\bfc)\right|^{s_{n_k-n-2,B}(M)} }{\sum_{ \bfd } \left|\clC_{n_k-n+1}(\bfd)\right|^{s_{n_k-n-2,B}(M)} } \\
&\leq  4\cdot 3600^{2\nu_5} M^{2\nu_5}.
\end{align*}

The last inequality is due to the symmetries of cylinders and considering $\omfF_{n+1}(\bfb c)=\omfF$ in the numerator and $\omfF_{n+1}(\bfb c)=\omfF_1(1+i)$ in the denominator. We may now argue as in the case $n=n_k-2$ to conclude the existence of a constant $\nu_6=\nu_6(M,B,t)>0$ such that $\mu\left(\Dx(z;r)\right)
\leq \nu_6 r^t$.

For any $M\in\Na_{\geq 4}$, the Mass Distribution Principle applied to $F_M(B)$ tells us that $s_B(\scD(M))\leq \dimh(F_M(B))$. Therefore, by Lemmas <ref> and Lemma <ref>,

\[
\lim_{M\to\infty} s_B(M)
\leq
\dimh(F(B)).
\]

The proof of Theorem <ref> is flexible enough for us to assume that the terms of the sequence $(n_j)_{j\geq 1}$ lie in a prescribed infinite subset of $\Na$. Let $\scL \subseteq \Na$ be infinite and, for each $B>1$, define

\[
\colon=
\left\{ z\in \mfF: \|a_n\|\geq B^n \text{ for infinitely many }n\in \scL\right\}.
\]

If $\scL\subseteq \Na$ is infinite and $B>1$, then $\dimh\left(E(B;\scL)\right) = s_B$.

§ A COMPLEX ŁUCZAK THEOREM

This section is dedicated to the proof of Theorem <ref>. First, observe that we can replace the complex absolute value $|\,\cdot\,|$ with $\|\,\cdot\,\|$. Indeed, let $b,c>1$ be given. Define the sets

\begin{align*}
\widetilde{\Xi}(b,c) &\colon=  \{\xi\in\mfF: c^{b^n}\leq \|a_n(\xi)\| \text{ for all } n\in\Na\}, \\
\Xi(b,c) &\colon=  \{\xi\in\mfF: c^{b^n}\leq \|a_n(\xi)\| \text{ for infinitely many } n\in\Na\}.
\end{align*}

Since $\|z\|\leq |z|$ for all $z\in \Cx$, we have

\[
\widetilde{\Xi}(b,c)
\subseteq
\{\xi\in\mfF: c^{b^n}\leq |a_n(\xi)| \text{ for all } n\in\Na\}.
\]

Consider $\veps= \frac{c-1}{2}>0$. Then, $c-\veps>1$ and every large $n\in\Na$ satisfies

\[
(c-\veps)^{b^n} \leq \frac{1}{\sqrt{2}} c^{b^n},
\]

\[
\{\xi\in\mfF: c^{b^n}\leq  |a_n(\xi) | \text{ for infinitely many } n\in\Na\}
\subseteq
\Xi(b,c- \veps).
\]

Hence, Theorem <ref> is a consequence of Theorem <ref> below.

If $b,c>1$, then

\[
\dimh\left( \Xi(b,c)\right)
\dimh\left( \widetilde{\Xi}(b,c)\right)
\frac{2}{b+1}.
\]

It is clear that $\widetilde{\Xi}(b,c)$ is contained in $\Xi(b,c)$. Then, in order to prove Theorem <ref>, it suffices to show

\[
\frac{2}{b+1}
\leq
\dimh\left( \widetilde{\Xi}(b,c)\right)
\quad\text{ and }\quad
\dimh\left( \Xi (b,c)\right)
\leq
\frac{2}{b+1}.
\]

§.§ Upper bound of Hausdorff dimension

The overall strategy resembles T. Łuczak's proof. Nevertheless, as should be clear by now, Hurwitz continued fractions present certain difficulties that prevent the arguments for real continued fractions to carry over into the complex plane.

Given $\xi=[0;a_1,a_2,a_3,\ldots]\in\mfF$ and $n\in\Na$, we have

\[
\left| \xi - \frac{p_n}{q_n}\right|
\frac{2}{\|q_n\|\|q_{n+1}\|}.
\]

This follows from

\[
\left| \xi - \frac{p_n}{q_n}\right|
\leq
\left| \xi - \frac{p_{n+1}}{q_{n+1}}\right|
\left| \frac{p_{n+1}}{q_{n+1}} - \frac{p_n}{q_n} \right|
\frac{1}{|q_{n+1}|^2}+ \frac{1}{|q_{n+1} q_n|}
\leq
\frac{2}{|q_{n}||q_{n+1}|}
\leq
\frac{2}{\|q_{n}\|\|q_{n+1}\|} .
\]

Let $b , c$ be real numbers $> 1$. Let $\alpha$ and $d$ be two real numbers such that

\[
\quad
\text{ and }
\quad
\frac{2}{1+b}
\]

In what follows, given a complex number $\xi=[0;a_1,a_2,a_3,\ldots]\in\mfF\setminus\QU(i)$, we write $q_n(\xi)\colon= q_n(a_1,\ldots, a_n)$ for all $n\in\Na$. For each $n\in\Na$ and each $\xi\in \mfF$ define

\[
I_{n,\xi} \colon= \Dx\left(\frac{p_n(\xi)}{q_n(\xi)}; \frac{ 2 }{\|q_n( \xi)\|^{1+d}}\right),
\quad
K_{n,\xi} \colon= \Dx\left( \frac{p_n(\xi)}{q_n(\xi)} ; \frac{2}{\|q_n( \xi)\|^{2(1+d)}}\right),
\]

and, for each $q\in\Na$,

\[
\clI_q\colon= \left\{ I_{n,\xi}: \xi\in\mfF, \, q=\|q_n(\xi)\| \geq c^{\frac{d^n}{3}}\right\},
\quad
\clK_q\colon= \left\{ K_{n,\xi}: \xi\in\mfF, \, \|q_n(\xi)\|=q \right\} .
\]

If $\xi\in \Xi(b,c)$, then there are infinitely many $n\in\Na$ such that

\begin{equation}\label{EQ:LUC:UP:00}
\|q_{n+1}(\xi)\|
\geq
\max\left\{ \|q_n(\xi)\|^d, c^{d^n}\right\}
\end{equation}

Given $m\in\Na$, we can pick $k\in \Na_{>m}$ such that

\begin{equation}\label{EQ:LUC:UP:01}
\|q_m(\xi)\| < c^{b^kd^{m-k}},
\quad\text{ and }\quad
c^{b^k} < \|a_k(\xi)\|.
\end{equation}

Indeed, the first condition is attained for every large $k$ because $b>d>1$. The second condition is true for infinitely many natural numbers  by the definition of $\Xi(b,c)$. Furthermore, the strict growth of the continuants allows us to further impose the condition

\[
\frac{|q_k|}{\sqrt{2}} \left( 1- \frac{\sqrt{2}}{2}\right)>1.
\]

Let $f_k\colon \Na\to \RE$ be the function given by

\[
f_k(N)= c^{b^kd^{N-k}}
\quad\text{ for all } N\in\Na.
\]

Thus, the first inequality in (<ref>) is $\|q_m(\xi)\| < f_k(m)$. Moreover, by Proposition <ref>,

\[
\|q_k(\xi) \|
\geq
|a_k|   \frac{|q_k|}{\sqrt{2}} \left( 1- \frac{\sqrt{2}}{2}\right)
\geq
\|a_k\| > f_k(k).
\]

Let $n\in\{m,\ldots, k-1\}$ be the largest integer less than $k$ such that $\|q_n(\xi)\|< f_k(n)$, then

\[
\|q_{n+1}(\xi)\|
\geq
\left( f_k(n)\right)^{d}
\geq
\|q_n(\xi)\|^d.
\]

Finally, $b>d$ yields $b^kd^{n-k}>d^n$, which implies $ f_k(n)^{d}> c^{d^{n+1}}$ and

\[
\|q_{n+1}(\xi)\|
\geq
\left( f_k(n)\right)^{d}
\geq
\max\left\{ \|q_n(\xi)\|^d, c^{d^{n+1}}\right\}.
\]

We have

\[
\Xi(b,c)
\subseteq
\bigcap_{m\in\Na}
\left(
\bigcup_{q=m}^{\infty} \mathcal{I}_q \cup \bigcup_{q=m}^{\infty} \mathcal{K}_q
\right).
\]

Pick any $\xi\in\Xi(b,c)$ and $m\in\Na$. Take $n\in\Na_{\geq m}$ satisfying (<ref>) and  $q\colon=\|q_n(\xi)\|$. We may assume that $q>n\geq m$, because $(\|q_j(\xi)\|)_{j\geq 0}$ grows exponentially. There are two possibilities:

\[
\|q_n(\xi)\| \geq c^{\frac{d^n}{3}}
\quad \text{ or } \quad
\|q_n(\xi)\| < c^{\frac{d^n}{3}}.
\]

If the first inequality holds, then, by Proposition <ref>,

\[
\left| \xi - \frac{p_n(\xi)}{q_n(\xi)}\right|
\frac{2}{\|q_n(\xi)\|\|q_{n+1}(\xi)\|}
\leq
\frac{2}{\|q_n(\xi)\|^{1+d}}
\]

and $\xi\in I_{n,\xi}$, thus $\xi \in \clI_{q}$. When the second inequality holds, the choice of $n$ yields

\[
\|q_n(\xi)\|^{1+2d}
< c^{\frac{d^n}{3}(1+2d)}
< c^{d^{n+1}}
\leq \|q_{n+1}(\xi)\|.
\]

Then, by Lemma <ref>, we have

\[
\left| \xi - \frac{p_n(\xi)}{q_n(\xi)}\right|
\frac{2}{\| q_n(\xi) \| \| q_{n+1}(\xi) \|}
\leq
\frac{2}{\| q_n(\xi)\|^{2(1+d)}},
\]

which means $\xi\in K_{n,\xi}$, thus $\xi \in \clK_q$.

Set $\beta\colon=5\sqrt{2}$. For each $m,k\in\Na$, define

\[
\Cas(m,k)
\colon=
\left\{\bfa=(a_1,\ldots, a_k) \in \scD^k: \prod_{j=1}^k \|a_j\|\leq m \beta^{k-1}\right\}.
\]

Given $m,n\in\Na$, let $a\in\scD$ be such that $\|a\|\leq m\beta^{k-1}$. For any $\bfb\in\scD^{k-1}$, we have $a\bfb\in \Cas(m,k+1)$ if and only if $\bfb\in\Cas\left( \left\lfloor \frac{m\beta}{\|a\|} \right\rfloor, k\right)$.

Write $\bfb=(b_1,\ldots,b_k)$. Then, $a\bfb\in\Cas(m,k+1)$ if and only if

\[
\prod_{j=1}^k\|b_j\| \leq \frac{\beta^{k-1}\beta m}{\|a\|},
\]

which is equivalent to $\bfb\in \Cas\left(\left\lfloor \frac{\beta m}{\|a\|} \right\rfloor, k\right)$.

If $m,k\in\Na$ and $m\geq 2$, then

\[
\Cas(m,k+1)
\bigcup_{\|a\|\leq \beta^{k} m} \Cas\left(\left\lfloor \frac{ \beta m}{\|a\|} \right\rfloor,k\right).
\]

\[
\#\Cas(m,k+1)
\leq
8\sum_{j=1}^{\lfloor \beta^k m\rfloor} j\, \#\Cas\left(\left\lfloor \frac{ \beta m}{j}\right\rfloor ,k\right).
\]

The first assertion is obvious from the definition of $\Cas(\,\cdot\, ,\,\cdot\,)$ and Proposition <ref>. The second assertion holds, because $\#\left\{ a\in\scD: \|a\|=j\right\} = 8j$ for all $j\in\Na$.

For $m\in\Na_{\geq 2}$ and $n\in\Na$, we have

\[
\#\Cas(m,n)
\leq
m^2 8^n \beta^{2(n-1)} \left( 1+ n \log\beta +\log m \right)^{n-1}.
\]

The proof is by induction on $n$. For all $m\in\Na$ we have $\#\Cas(m,1)\leq 8m^2$. This is obvious when $m=1$. If $m\geq 2$, then

\[
\#\Cas(m,1)
= (2m+1)^2 -5 \leq 4m^2 \left( 1+\frac{1}{2m}\right) < 8m^2.
\]

Assume that the next inequality is true for any $n=N\in\Na$ and any $m\in\Na$:

\[
\#\Cas(m,N)
\leq
m^2 8^N \beta^{2(N-1)} \left( 1+ N \log\beta +\log m \right)^{N-1}.
\]

Then, for $n=N+1$,

\begin{align*}
\#\Cas(m,N+1)
8\sum_{j=1}^{\lfloor \beta^N m\rfloor} j \,
\#\Cas\left(\left\lfloor \frac{ \beta m}{j}\right\rfloor ,N\right) \\
\sum_{j=1}^{\lfloor \beta^N m\rfloor} j
\frac{\beta^2 m^2}{j^2} 8^N \beta^{2(N-1)} \left( 1+ N \log\beta +\log \frac{\beta m}{j} \right)^{N-1} \\
\sum_{j=1}^{\lfloor \beta^N m\rfloor}
\frac{ 1 }{j}  \left( 1+ N \log\beta +\log \frac{\beta m}{j} \right)^{N-1}\\
\left( 1+ (N+1) \log\beta +\log m  \right)^{N-1}
\sum_{j=1}^{\lfloor \beta^N m\rfloor} \frac{1}{j} \\
\left( 1+ (N+1) \log\beta +\log m  \right)^{N-1}
(\log(\beta^N m) +1 ) \\
\left( 1+ (N+1) \log\beta +\log m  \right)^{N-1}
\left( 1+ (N+1) \log\beta +\log m  \right) \\
\left( 1+ (N+1) \log\beta +\log m  \right)^{N}.
\end{align*}

If $m,n \in\Na$ and $m\geq 2$, then

\[
\#\bigcup_{k=1}^n \Cas(m,k)
\leq
8m^2 \left( 8\beta^2 (1+ {(n-1)}\log\beta + \log m)\right)^n.
\]

By Proposition <ref>,

\begin{align*}
\#\bigcup_{k=1}^n \Cas(m,k)
\sum_{k=1}^n 8^k \beta^{2(k-1)}m^2(1+k\log\beta + \log m)^{k-1} \\
8m^2 \sum_{k=1}^n \left(8\beta^2 (1+k\log\beta + \log m) \right)^{k-1} \\
8m^2 \sum_{k=1}^{n} \left(8\beta^2 (1 + n \log\beta + \log m) \right)^{k-1}.
\end{align*}

Then, considering $\tau=8\beta^2 (1+ n \log\beta + \log m)>1$,

\begin{align*}
\#\bigcup_{k=1}^n \Cas(m,k)
8m^2 \sum_{k=1}^{n} \tau^{k-1}\\
8m^2 \left( \frac{\tau^n-1}{\tau-1}\right) \\
8m^2 \tau^n \\
8m^2  \left( 8\beta^2 (1 + n \log\beta + \log m)\right)^n.
\end{align*}

If $m>c$, then $\bigcup_{q=1}^m \clI_q$ contains at most

\[
8m^2 \left( 8\beta^2 (1 + \log_d(3\log_c m ) \log\beta + \log m)\right)^{\log_d(3\log_c m )+1}
\]

Consider $n=\lfloor \log_d(3\log_c m )\rfloor$. For every $k\in\{1,\ldots,n\}$ and $\bfa=(a_1,\ldots,a_k)\in\Omega(k)$, by Proposition <ref> we have

\[
\frac{1}{\beta^{k-1}}\prod_{j=1}^k \|a_j\|\leq \|q_k(\bfa)\|.
\]

Therefore, the set

\begin{equation}\label{EQ:LUC:UP:02:05}
\left\{ \bfa\in\Omega(k): 1\leq k\leq n, \; c^{\frac{d^k}{3}} \leq \|q_k(\bfa)\|\leq m\right\}
\end{equation}

is contained in

\[
\left\{\bfa\in \scD^k: 1 \leq k \leq n, \; \prod_{j=1}^k \|a_j\|\leq \beta^{k-1}m\right\}
\bigcup_{k=1}^n \Cas(m,k).
\]

By the choice of $n$, the inequality $c^{\frac{d^k}{3}} \leq m$ implies $d^k\log c\leq 3\log m$ and, thus, $k\leq n$. Then, since each $I_{k,\xi}\in \bigcup_{q=1}^m \clI_q$ determines a single word $\bfa\in\Omega(k)$ contained in the set in (<ref>), we get

\begin{align*}
\#\bigcup_{q=1}^m \clI_q
&\leq \#\bigcup_{k=1}^n \Cas(m,k) \\
&\leq 8m^2  \left( 8\beta^2 (1 + n \log\beta + \log m)\right)^n\\
&\leq 8m^2 \left( 8\beta^2 (1 + n \log\beta + \log m)\right)^{\log_d(3\log_c m ) } \\
&\leq 8m^2 \left( 8\beta^2 (1 + \log_d(3\log_c m )\log\beta + \log m)\right)^{\log_d(3\log_c m ) }.
\end{align*}

Given a countable collection $\{A_q:q\in\Na\}$ of subsets of $\Cx$, we write

\[
\Lambda_{\alpha}(\{A_q:q\in\Na\})
\colon=
\sum_{q\geq 1} |A_q|^{\alpha}.
\]

We have

\[
\lim_{m\to\infty} \Lambda_{\alpha}\left( \bigcup_{q=m}^{\infty} \clI_q \cup \bigcup_{q=m}^{\infty} \clK_q\right) = 0.
\]

Let us show first that

\begin{equation}\label{EQ:LUC:UP:02}
\lim_{m\to\infty} \Lambda_{\alpha}\left( \bigcup_{q=m}^{\infty} \clI_q \right) = 0.
\end{equation}

Fix $r\in\Na$ such that $r\geq \max\{2,c\}$ and define $c_1 \colon= \log_d\log_c r +  \log 3$ and $c_2=\frac{1}{\log d}$. Pick an arbitrary $j\in\Na$, then

\[
\log_d(3\log_c(r^j))
=c_1 + c_2\log j.
\]

Observe that for all $q\in\{r^{j-1},\ldots, r^j\}$ and each $I_{l,\zeta}\in\clI_q$, we have

\[
\left| I_{l,\zeta}\right|
\frac{4}{\|q_l(\zeta)\|^{1+d}}
\leq
\frac{4}{r^{(j-1)(1+d)} }.
\]

As a consequence, using Proposition <ref> with $m=r^j$, we get

\begin{align*}
\Lambda_{\alpha} \left( \bigcup_{q=r^{j-1}}^{r^j} \clI_q\right)
\frac{ 4^{\alpha}}{r^{(j-1)(d+1)\alpha}} \,
\#\bigcup_{q=1}^{r^j}\clI_q \\
\frac{4^{\alpha}r^{\alpha(d+1)}}{r^{\alpha(d+1)j}}  8r^{2j} \left( 8\beta^2 (1 + \log_d(3\log_c r^j ) \log\beta + \log r^j)\right)^{\log_d(3\log_c r^j ) } \\
\frac{4^{\alpha}r^{\alpha(d+1)} 8}{r^{j(\alpha(d+1)-2)}} \left( 8\beta^2 (1 + \log_d(3\log_c r^j ) \log\beta + j\log r)\right)^{c_2\log j + c_1 }.
\end{align*}

Then, for

\[
c_3\colon=8\beta^2(1 + c_1),\quad
c_4\colon= 8\beta^2c_2,\quad
c_5\colon= 8\beta^2\log r, \quad
\text{ and }\quad
\veps\colon=  \alpha(1+d) - 2 >0,
\]

we have

\begin{align*}
\Lambda_{\alpha} \left( \bigcup_{q=r^{j-1}}^{r^j} \clI_q\right)
\frac{4^{\alpha}r^{\alpha(d+1)} 8}{r^{j(\alpha(d+1)-2)}}
\left( c_5j + c_4\log j + c_3  \right)^{c_2\log j + c_1 } \\
4^{\alpha}r^{\alpha(d+1)} 8 \,
\frac{\left( c_5j + c_4\log j + c_3  \right)^{c_2\log j + c_1 }}{(r^{\veps})^j},
\end{align*}

and, since,

\[
\lim_{j\to\infty} \frac{ \left( c_5j + c_4\log j + c_3  \right)^{c_2\log j + c_1 + 1}}{( r^{\veps}  )^j} = 0,
\]

we conclude

\[
\sum_{j=1}^{\infty} \Lambda_{\alpha} \left( \bigcup_{q=r^{j-1}}^{r^j} \clI_q\right) < \infty
\]

and (<ref>) follows.

We now show that

\begin{equation}\label{EQ:LUC:UP:03}
\lim_{n \to \infty} \, \Lambda_{\alpha}\left( \bigcup_{q\geq m} \clK_q\right)=0.
\end{equation}

For each $q\in\Na$, the collection $\clK_q$ has at most $8q^2$ discs of diameter $4q^{-2(1+d)}$, so

\[
\Lambda_{\alpha}(\clK_q) \leq \frac{ 4^{\alpha}\cdot 8 q^2}{q^{2(1+d)\alpha}}.
\]

Thus, for any $m\in\Na$ and $\veps$ as above,

\[
\Lambda_{\alpha}\left( \bigcup_{q\geq m} \clK_q\right)
\leq
4^{\alpha}\cdot 8 \sum_{q\geq m} \frac{1}{q^{2(1+d)\alpha -2}}
4^{\alpha}\cdot 8 \sum_{q\geq m} \frac{1}{q^{2+4\veps}}
\]

which gives (<ref>).

§.§.§ Proof of Theorem <ref>. Upper bound

By Proposition <ref> and taking $\alpha$ arbitrarily close to $1/ (1+b)$, we arrive at

\[
\dimh\left( \Xi(b, \xi)\right) \leq \frac{2}{1+b}.
\]

§.§ Lower bound of Hausdorff dimension

We obtain the desired bound applying the Mass Distribution Principle. Take $M_0\in\Na$ such that

\[
\geq
\log_b\log_c \left( \frac{8\left(1 + \frac{\sqrt{2}}{2} \right)\left(2 + \frac{\sqrt{2}}{2} \right)}{\kappa_1\left(1 - \frac{\sqrt{2}}{2} \right)} + \frac{\sqrt{2}}{2}\right).
\]

Note that $M_0\geq \log_b\log_c(3)$. For every $n\in\Na$, define

\begin{align*}
\widetilde{D}_n
\left\{d\in\scD: c^{b^{M_0+n+1}} \leq \|a_j\|\leq 2c^{b^{M_0+n+1}} \right\}, \\
\Pre(n)
\prod_{j=1}^n \widetilde{D}_j.
\end{align*}

\[
J_0\colon=\bigcup_{a_1\in\widetilde{D}_1} \oclC_1(a_1)
\]

and, for $n\in\Na$ and $\bfa\in \Pre(n)$, put

\[
\colon=
S_n\left(\bfa; c^{b^{M_0+n}}, 2c^{b^{M_0+n}}\right)
=\bigcup_{b\in \widetilde{D}_{n+1}} \oclC_{n+1}(\bfa b).
\]

Define the set

\[
\colon=
\bigcap_{n\in\Na}\bigcup_{\bfa\in \Pre(n)} J_n(\bfa).
\]

By Proposition <ref>, $E\subseteq \widetilde{\Xi}(b,c)$, so it is enough to show that

\begin{equation}\label{Ec:LoBndLuc}
\dimh(E)\geq \frac{2}{1+b}.
\end{equation}

§.§.§ A measure on $E$

For every $a_1\in\widetilde{D}_1$, define

\[
\tilde{\mu}(J_1(a_1))
\colon=
\frac{1}{\#\widetilde{D}_1}.
\]

Assume that we have already defined $\tilde{\mu}(J(\bfa))$ for all $\bfa\in\Pre(n)$. For $(a_1,\ldots, a_n,a_{n+1})\in\Pre(n+1)$, we put

\[
\tilde{\mu} \left( J_{n+1}(a_1,\ldots, a_n,a_{n+1})\right)
\colon=
\frac{1}{\#\widetilde{D}_{n+1}}
\tilde{\mu}\left(J_{n}(a_1,\ldots, a_n)\right).
\]

We are distributing uniformly the mass on each step. Then, $\tilde{\mu}$ actually defines a measure by the Daniell-Kolmogorov Consistency Theorem.

§.§.§ Estimate of $\mu(J_n(a_1,\ldots,a_n))$

We estimate $\mu(J_n(a_1,\ldots,a_n))$ in terms of $|J_n(a_1,\ldots,a_n)|$.

For $k\in\Na$, we have

\[
\#\widetilde{D}_k
\geq
8c^{2b^{M_0 + k +1}}.
\]

If $k\in\Na$, then

\begin{align*}
\#\widetilde{D}_k
\left( 2\left\lfloor 2c^{2b^{M_0+k+1}} \right\rfloor +1 \right)^2 - \left( 2\left\lceil 2c^{2b^{M_0+k+1}} -1 \right\rceil + 1 \right)^2 \\
\left( 4c^{b^{M_0+k+1} } -1 \right)^2
\left( 2c^{b^{M_0+k+1}} + 1 \right)^2 \\
\left( 2c^{2b^{M_0+k+1}} - 2 \right)  \\
\left( 1 - \frac{1}{c^{b^{M_0+k+1}}} \right)
12\,\frac{2}{3} c^{2b^{M_0+k+1}}
8 c^{2b^{M_0+k+1}}.
\end{align*}

Write $t\colon= \frac{2}{1+b}$.

There exists some $\gamma_1>0$ such that

\begin{equation}\label{EQ:LUC:LB:01}
\tilde{\mu}\left(J_n(\bfa )\right)
\leq
\frac{ \gamma_1 }{c^{tb^{M_0+n+1}}|q_n(\bfa)|^{2t}}
\quad
\text{ for all } n\in\Na \text{ and } \bfa\in \Pre(n).
\end{equation}

Take any $n\in\Na$ and $\bfa\in \Pre(n)$. We obtain (<ref>) by showing that the product

\[
\tilde{\mu}\left(J_n( \bfa )\right) c^{tb^{M_0+n+1}}|q_n(\bfa)|^{2t}
\]

is bounded above by some constant depending on $b$ and $c$. Equivalently, if

\[
\log\left( \tilde{\mu}\left(J_n( \bfa )\right)\right) + \log\left( c^{tb^{M_0+n+1}}|q_n(\bfa)|^{2t}\right)
\]

is bounded. We work each term of the sum separately. On the one hand,

\begin{align*}
\log\left( \tilde{\mu}\left(J_n( \bfa )\right)\right)
\sum_{j=1}^n \log\left( \frac{1}{\#\widetilde{D}_j}\right) \\
\sum_{j=1}^n \log\left( \frac{1}{8} \right) + \log\left( \frac{1}{c^{b^{M_0+j+1}} } \right) \\
-n\log 8 - \sum_{j=1}^n 2b^{M_0 + j + 1}\log c\\
-n\log 8 - 2b^{M_0 + 2} \log c \sum_{j=1}^n b^{j - 1}\\
-n\log 8 - 2b^{M_0 + 2} \left( \frac{b^n-1}{b-1} \right) \log c \\
-n\log 8 - 2b^{M_0 + 2}b^{n-1} \left( \frac{1 - b^{-n}}{1 - b^{-1}} \right) \log c \\
-n\log 8 - 2b^{M_0 + n +1} \left( \frac{1 - b^{-n}}{1 - b^{-1}} \right) \log c \\
-n\log 8 - b^{M_0 + n +1} \frac{2b}{b-1}\left( 1 - \frac{1}{b^n}\right) \log c.
\end{align*}

On the other hand, writing $ \bfa =(a_1,\ldots, a_n)$,

\[
\leq \prod_{j=1}^n |a_j| + 1
\leq \prod_{j=1}^n \sqrt{2}\| a_j \| + 1
\leq 2^{\frac{n}{2}} \prod_{j=1}^n \|a_j\| + \frac{1}{\sqrt{2}}
\leq 8^{\frac{n}{2}} \prod_{j=1}^n \|a_j\|,
\]

\[
\log |q_n(\bfa)|
\leq
\frac{n}{2}\log 8 + \sum_{j=1}^n \log \|a_j\|
\]

and, hence,

\begin{align*}
\log\left( c^{tb^{M_0+n+1}}|q_n(\bfa)|^{2t}\right)
tb^{M_0+n+1} \log c + 2t\log |q_n(\bfa)| \\
tb^{M_0+n+1} \log c + 2t\left( \frac{n}{2}\log 8 + \sum_{j=1}^n \log \|a_j\|  \right)\\
tb^{M_0+n+1} \log c + t n\log 8 + 2t\sum_{j=1}^n \log \|a_j\|  \\
tb^{M_0+n+1} \log c + 3tn\log 2 + 2t \left( \sum_{j=1}^n \log 2 + b^{M_0+j+1}\log c\right) \\
tb^{M_0+n+1} \log c + 3tn\log 2 + 2t n\log 2 +  2t  b^{M_0+2} \left( \sum_{j=1}^n b^{j-1}\right)\log c \\
tb^{M_0+n+1} \log c + 5nt\log 2  +  2t b^{M_0+2} \left(  \frac{b^n-1}{b-1} \right)\log c \\
tb^{M_0+n+1} \log c + 5nt\log 2  +  2t b^{M_0+n+1} \left(  \frac{1 - b^{-n}}{1-b^{-1}} \right)\log c \\
\left( 1 + 2\left( \frac{1 - b^{-n}}{1-b^{-1}}  \right) \right) \log c
+ 5nt\log 2 \\
\left( 1 + \frac{2b}{b-1} \left( 1 - \frac{1}{b^n}\right)  \right) \log c
+ 5nt\log 2.
\end{align*}

As a consequence,

\begin{align}
\log &\left( \tilde{\mu}\left(J_n( \bfa )\right)\right) + \log\left( c^{tb^{M_0+n+1}}|q_n(\bfa)|^{2t}\right)< \nonumber\\
-n\log 8 - b^{M_0 + n +1} \frac{2b}{b-1}\left( 1 - \frac{1}{b^n}\right) \log c
\left( 1 + \frac{2b}{b-1} \left( 1 - \frac{1}{b^n}\right)  \right) \log c
+  5nt\log 2 \nonumber\\
b^{M_0+n+1}\log c  \left(
t + \frac{2tb}{b-1} \left( 1 - \frac{1}{b^n}\right)    -  \frac{2b}{b-1}\left( 1 - \frac{1}{b^n}\right) \right) +  n(5t \log 2 - \log 8)  \nonumber\\
b^{M_0+n+1}\log c  \left(
\frac{2}{b+1} + \frac{4b}{b^2-1} \left( 1 - \frac{1}{b^n}\right)    -  \frac{2b}{b-1}\left( 1 - \frac{1}{b^n}\right) \right) +   n(5t \log 2 - \log 8) . \label{EQ:LUC:LB:02}
\end{align}

Since $b>1$, we have

\[
\frac{2}{b+1} + \frac{4b}{b^2-1}   -  \frac{2b}{b-1}
-2 \left(\frac{b-1}{b+1}\right)<0
\]

so the coefficient of $b^{M_0+n+1}$ in (<ref>) is negative for large $n$, and (<ref>) holds for some $\gamma_1>0$.

There exists some $\gamma_2 >0$ such that

\[
\tilde{\mu}\left( J_n( \bfa )\right) \leq \gamma_2  |J_n( \bfa )|
\quad
\text{ for all large }n\in\Na \text{ and all } \bfa\in\Pre(n).
\]

§.§.§ Estimates of $\tilde{\mu}(\Dx(z;r))$

\[
r_0 \colon= \frac{1}{2} \min\{|J_1(a)|:a\in D_1\}.
\]

There is a constant $\gamma_3=\gamma_3(b,c)>0$ such that $\mu(\Dx(z;r))<\gamma_3r^t$ whenever $z\in E$ and $0<r<r_0$.

Let $z=[0;a_1,a_2,\ldots] \in E$ be arbitrary and take any real number $r$ such that $0<r< r_0$. Let $n\in\Na$ be the natural number determined by

\[
|J_{n+1}(a_1,\ldots,a_n, a_{n+1})|
\leq
|J_{n}(a_1,\ldots,a_n, a_{n})|.
\]

\[
\widetilde{\gamma} \colon= \frac{1}{16\left( 1+ \frac{\sqrt{2}}{2}\right)\left( 2 +  \frac{\sqrt{2}}{2}\right)}.
\]

We consider two possible cases: either

\begin{equation}\label{EQ:LUC:LB:03}
|J_{n+1}(a_1,\ldots,a_n, a_{n+1})|
\leq
\widetilde{\gamma}\,|\clC_{n+1}(a_1,\ldots,a_n, a_{n+1})|
\end{equation}

\begin{equation}\label{EQ:LUC:LB:04}
\widetilde{\gamma}\,|\clC_{n+1}(a_1,\ldots,a_n, a_{n+1})|
\leq
|J_{n}(a_1,\ldots,a_n, a_{n})|.
\end{equation}

Suppose that (<ref>) is true. We claim that $\Dx(z;r)$ intersects exactly one cylinder of level $n+1$ and, hence, only one set of the form $J_{n+1}(b_1,\ldots, b_{n+1})$. The condition imposed on $M_0$ implies

\[
\left| T^{n+1}(z)\right|
\frac{\kappa_1\left( 1 - \frac{\sqrt{2}}{2}\right)}{8\left( 1 + \frac{\sqrt{2}}{2}\right) \left( 2 + \frac{\sqrt{2}}{2}\right) },
\]

\[
\left| \frac{1}{T^{n+1}(z)}\right|
\geq
\| a_{n+2}(z) \| - \left|T^{n+2}(z)\right|
\geq c^{b^{M_0}} - \frac{\sqrt{2}}{2}
\frac{8\left( 1 + \frac{\sqrt{2}}{2}\right)\left( 2 + \frac{\sqrt{2}}{2}\right)}{\kappa_1 \left( 1 - \frac{\sqrt{2}}{2}\right)}.
\]

\begin{align*}
\left| \frac{p_{n+1}(z)}{q_{n+1}(z)} -  z \right|
\left| Q_{n+1}(0;(a_1,\ldots, a_{n+1}) -  Q_{n+1}(T^{n+1}(z);(a_1,\ldots, a_n, a_{n+1}) \right| \\
&\leq \frac{|T^{n+1}(z)|}{|q_{n+1}(a_1,\ldots, a_{n+1})|^2\left| 1 + \frac{q_{n}}{q_{n+1}}\right| } \\
&\leq \frac{1}{8\left( 1 + \frac{\sqrt{2}}{2}\right)\left( 2 + \frac{\sqrt{2}}{2}\right)}|\clC_{n+1}(a_1,\ldots,a_n)|.
\end{align*}

As a consequence, for all $w\in \Dx(z;\widetilde{\gamma}\;|\clC_{n+1}(a_1,\ldots,a_{n+1})|)$ we have

\begin{align*}
\left| \frac{p_{n+1}(z)}{q_{n+1}(z)} -  w \right|
\left| \frac{p_{n+1}(z)}{q_{n+1}(z)} -  z \right| + |z - w| \\
\left( \frac{1}{8\left( 1 + \frac{\sqrt{2}}{2}\right)\left( 2 + \frac{\sqrt{2}}{2}\right)} + \widetilde{\gamma} \right)|\clC_{n+1}(a_1,\ldots,a_n)|\\
\frac{1}{4\left( 1 + \frac{\sqrt{2}}{2}\right)\left( 2 + \frac{\sqrt{2}}{2}\right)}|\clC_{n+1}(a_1,\ldots,a_n)|.
\end{align*}

In other words, we obtain

\begin{equation*}
\Dx(z;  \widetilde{\gamma}|\clC_{n+1}(a_1,\ldots, a_{n+1})|)
\subseteq
\Dx\left( \frac{p_{n+1}(z)}{q_{n+1}(z)}; \frac{|\clC_{n+1}(a_1,\ldots, a_{n+1})|}{4\left( 1+ \frac{\sqrt{2}}{2}\right)\left( 2+ \frac{\sqrt{2}}{2}\right) } \right).
\end{equation*}

By Proposition <ref> and

\[
\Dx(0;2^{-1})\subseteq \mfFc=\mfFc_{n+1}(a_1,\ldots,a_{n+1}),
\]

we have

\[
\Dx\left( \frac{p_{n+1}(z)}{q_{n+1}(z)}; \frac{|\clC_{n+1}(a_1,\ldots, a_{n+1})|}{4\left( 1+ \frac{\sqrt{2}}{2}\right)\left( 2+ \frac{\sqrt{2}}{2}\right) } \right)
\subseteq
\clCc_{n+1}(a_1,\ldots,a_{n+1}).
\]

As a consequence, the next estimate holds

\begin{align*}
\tilde{\mu}\left( \Dx(z;r)\right)
\tilde{\mu}\left( \Dx(z;r) \cap J_{n+1}(a_1,\ldots, a_{n+1}) \right) \\
\tilde{\mu}\left(J_{n+1}(a_1,\ldots, a_{n+1}) \right)
\leq
\gamma_2 |J_{n+1}(a_1,\ldots, a_{n+1})|^t
\leq
\gamma_2 r^t .
\end{align*}

Let us now assume (<ref>). Define

\[
\tilde{\sfN}
\colon=
\#\left\{h\in\widetilde{D}_{n+1}: J_{n+1}(a_1,\ldots,a_n,h)\cap \Dx(z;r)\neq \vac\right\}.
\]

In view of

\[
\bigcup_{a\in \widetilde{D}_{n+1}} \oclC_1(a)
\subseteq
\Dx\left( 0;2^{-1}\right)
\subseteq
\mfFc,
\]

for each $a\in \widetilde{D}_{n+1}$, we apply $Q_{n+1}(\cdot;(a_1,\ldots, a_n,a))$ and obtain

\[
J_{n+1}(a_1,\ldots, a_n,a)
\subseteq
Q_{n+1}\left(\Dx\left( 0;2^{-1}\right);(a_1,\ldots, a_n,a)\right)
\subseteq
\clCc_{n+1}(a_1,\ldots, a_n,a).
\]

Therefore, since $\clCc_{n+1}(a_1,\ldots, a_n,a)$ contains only one set of the form $J_{n+1}(b_1,\ldots,b_{n+1})$,

\[
\tilde{\sfN}
\leq
\#\left\{a\in\widetilde{D}_{n+1}: Q_{n+1}\left(\Dx\left( 0;2^{-1}\right);(a_1,\ldots, a_n,a)\right)\cap \Dx(z;r)\neq \vac\right\}.
\]

Moreover, if $\leb$ is the Lebesgue measure on $\Cx$, there is an absolute constant $\gamma_3>0$ such that

\begin{align*}
\leb\left( Q_{n+1}\left(\Dx\left( 0;2^{-1}\right);(a_1,\ldots, a_n,a)\right)\right) &\geq
\gamma_3 \frac{\pi}{4} \frac{1}{|q_{n+1}(a_1,\ldots, a_n,a)|^4} \\
\gamma_3 \frac{\pi}{36} \frac{1}{|q_{n}(a_1,\ldots, a_n)|^4|a|^4} \\
\gamma_3 \frac{\pi}{36\cdot 4} \frac{1}{|q_{n}(a_1,\ldots, a_n)|^4c^{4b^{M_0+n+2}}},
\end{align*}

(see the proof of <cit.>), so

\[
\tilde{\sfN} \frac{\gamma_3 \pi}{36\cdot 4} \frac{1}{|q_{n}(a_1,\ldots, a_n)|^4c^{4b^{M_0+n+2}}}
\leq
\pi r^2,
\]

which gives

\[
\tilde{\sfN}
\leq
\frac{36\cdot 4}{ \gamma_3}
|q_{n}(a_1,\ldots, a_n)|^4c^{4b^{M_0+n+2}} r^2.
\]

From $0<t<1$ we get $0<\frac{t}{2}<\frac{1}{2}$, hence

\begin{align*}
\tilde{\mu} \left(\Dx(z;r)\right)
\tilde{\mu} \left(J_n(a_1,\ldots, a_n)\right)
\min\left\{ 1, \tilde{\sfN} \frac{1}{\#\widetilde{D}_{n+1}} \right\} \\
\tilde{\mu} (J_n(a_1,\ldots, a_n)
\min\left\{ 1, \frac{36\cdot 4}{ \gamma_4  }c^{4b^{M_0 + n + 2}}|q_n(\bfa)|^4 r^2 \frac{1}{8c^{2b^{M_0+ n + 2}}}  \right\} \\
\tilde{\mu} \left(J_n(a_1,\ldots, a_n)\right)
\min\left\{ 1, 18 c^{2b^{M_0 + n + 2}}|q_n(\bfa)|^4 r^2   \right\} \\
\tilde{\mu}\left(J_n(a_1,\ldots, a_n)\right)
18^{\frac{t}{2}} c^{tb^{M_0 + n + 2}}|q_n(\bfa)|^{2t} r^t \\
\tilde{\mu}\left(J_n(a_1,\ldots, a_n)\right)
18 c^{b^{M_0 + n + 1}}|q_n(\bfa)|^{2t} r^t  \\
18\gamma_2 r^t.
\end{align*}

§.§.§ Proof of Theorem <ref>. Lower bound

The Mass Distribution Principle and Proposition <ref> imply <ref>. Hence, since $E\subseteq \widetilde{\Xi}(b,c)$,

\[
\frac{2}{1+b}
\leq
\dimh\left( \widetilde{\Xi}(b,c)\right).
\]

A slight modification of the proof of the lower bound in Theorem <ref> tells us that for every $b,c>1$ and $\delta>0$ we have

\[
\dimh\left( \widetilde{\Xi}(b,c)\setminus \widetilde{\Xi}(b,c+\delta) \right )
\dimh\left( \Xi(b,c)\setminus \Xi(b,c+\delta) \right )
\frac{2}{b+1}.
\]

§ PROOF OF THEOREM <REF>

* Assume that $B=1$. Let $\veps>0$ be arbitrary and consider the infinite set

\[
\scL_{\veps}
\colon=
\left\{ n\in\Na : \Phi(n) \leq (1+\veps)^n\right\}.
\]

Then, we have

\[
\left\{ z\in \mfF: \|a_n\|\geq (1+\veps)^n \text{ for infinitely many }n\in \scL\right\}
\subseteq  E_{\infty}(\Phi),
\]

so $s_{1+\veps}\leq \dimh(E_{\infty}(\Phi))$, by Corollary <ref>. In view of Proposition <ref>, we conclude

\[
2 = \lim_{\veps \searrow 0} s_{1+\veps}\leq \dimh(E_{\infty}(\Phi)) \leq 2.
\]

* Assume that $1<B<\infty$. For each $\veps>0$, define the infinite set

\[
\scL_{\veps}'
\colon=
\left\{ n\in\Na: \Phi(n) \leq (B+ \veps)^n\right\}.
\]

\[
\left\{z\in \mfF: \|a_n\|\geq (B+\veps)^n \text{ for infinitely many } n\in \scL_{\veps}'\right\}
\subseteq
\]

\[
\subseteq
\left\{z\in \mfF: \|a_n\|\geq (B - \veps)^n \text{ for infinitely many } n\in \Na \right\};
\]

therefore, $s_{B-\veps} \leq \dimh(E_{\infty}(\Phi))\leq s_{B+\veps}$ and, since $B\mapsto s_B$ is continuous (Proposition <ref>), $\dimh(E_{\infty}(\Phi))=s_B$.

* Assume that $B=\infty$. Pick $\veps>0$. The assumption $b=1$ implies that the set

\[
\scL
\colon=
\left\{n\in\Na: \Phi(n) \leq e^{(1+\veps)^n} \right\}
\]

is infinite. Then, we have

\begin{align*}
\left\{ z\in\mfF: \|a_n(z)\|\geq e^{(1+\veps)^n} \text{ for all }n\in\Na\right\}
\left\{ z\in\mfF: \|a_n(z)\|\geq e^{(1+\veps)^n} \text{ for all }n\in\scL \right\} \\
&\subseteq E_{\infty}(\Phi)
\end{align*}

and Theorem <ref> implies

\[
\dimh\left( E_{\infty}(\Phi)\right)
\geq \frac{2}{2+\veps}.
\]

Letting $\veps\to 0$, we conclude $\dimh\left( E_{\infty}(\Phi)\right) \geq 1$. Moreover, $B=\infty$ implies that for any $C>B$ the inequality $\frac{\log \Phi(n)}{n}\geq \log C$ holds for infinitely many $n\in\Na$, hence

\[
\subseteq
\left\{ z\in\mfF: C^n\leq \|a_n(z)\| \text{ for infinitely many} n\in\Na \right\}
\]

and $\dimh\left( E_{\infty}(\Phi) \right) \leq s_{C}$. We let $C\to\infty$ and recourse to Lemma <ref> to conclude $\dimh\left( E_{\infty}(\Phi) \right) \leq 1$, so

\[
\dimh\left( E_{\infty}(\Phi) \right) = 1.
\]

Suppose that $1<b<\infty$. Then, for any $\veps>0$, the inequality $\Phi(n)\leq e^{(b+\veps)^n}$ holds infinitely often and $\Phi(n)\geq e^{(b-\veps)^n}$ is true for every large $n$. As a consequence, since the set

\[
\left\{ z\in \mfF: \|a_n(z)\|\geq e^{(b+\veps)^n} \text{ for all } n\in\Na \right\}
\]

is contained in

\[
\left\{ z\in \mfF: \|a_n(z)\|\geq e^{(b+\veps)^n}\geq \Phi(n) \text{ for infinitely many } n\in\Na \right\}
\subseteq
\]

\[
\subseteq
\left\{ z\in \mfF: \|a_n(z)\|\geq \Phi(n) \geq e^{(b-\veps)^n} \text{ for infinitely many } n\in\Na \right\},
\]

Theorem <ref> tells us that

\[
\frac{2}{b + 1 + \veps}
\leq
\dimh\left( E_{\infty}(\Phi)\right)
\leq
\frac{2}{b+1-\veps}.
\]

Letting $\veps\to 0$, we arrive at $\dimh\left( E_{\infty}(\Phi)\right) = \frac{2}{b+1}$.

To finish the proof, assume that $b= \infty$. Then, $\Phi(n)\geq e^{c^n}$ for all $c>0$ and every sufficiently large $n\in\Na$; that is,

\[
\subseteq
\left\{ z\in\mfF: \| a_n(z)\|\geq e^{c^n} \text{ for infinitely many }n \in\Na \right\}
\]

An application of Theorem <ref> leads us to $\dimh\left(E_{\infty}(\Phi)\right) \leq \frac{2}{1 + c}$ and letting $c\to \infty$, we conclude $\dimh\left(E_{\infty}(\Phi)\right) =0$.

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B. Adamczewski and Y. Bugeaud, On the complexity of algebraic
numbers. II. Continued fractions, Acta Math., 195 (2005), pp. 1–20.

[2]
T. Andreescu, D. Andrica, and Z. Feng, 104 number theory problems,
Birkhäuser Boston, Inc., Boston, MA, 2007.
From the training of the USA IMO team.

[3]
Y. Bugeaud, Sets of exact approximation order by rational numbers,
Math. Ann., 327 (2003), pp. 171–190.

[4]
height 2pt depth -1.6pt width 23pt, Approximation by
algebraic numbers, vol. 160 of Cambridge Tracts in Mathematics, Cambridge
University Press, Cambridge, 2004.

[5]
height 2pt depth -1.6pt width 23pt, Automatic continued
fractions are transcendental or quadratic, Ann. Sci. Éc. Norm.
Supér. (4), 46 (2013), pp. 1005–1022.

[6]
J. W. S. Cassels, An introduction to Diophantine approximation,
Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge
University Press, New York, 1957.

[7]
N. Chevallier, Gauss lattices and complex continued fractions, Pure
Appl. Math. Q., 17 (2021), pp. 1785–1860.

[8]
S. G. Dani and A. Nogueira, On orbits of ${\rm SL}(2,{\Bbb Z})_+$
and values of binary quadratic forms on positive integral pairs, J. Number
Theory, 95 (2002), pp. 313–328.

[9]
height 2pt depth -1.6pt width 23pt, Continued fractions
for complex numbers and values of binary quadratic forms, Trans. Amer. Math.
Soc., 366 (2014), pp. 3553–3583.

[10]
H. Ei, S. Ito, H. Nakada, and R. Natsui, On the construction of the
natural extension of the Hurwitz complex continued fraction map, Monatsh.
Math., 188 (2019), pp. 37–86.

[11]
K. Falconer, Techniques in fractal geometry, John Wiley & Sons,
Ltd., Chichester, 1997.

[12]
A.-H. Fan, L.-M. Liao, B.-W. Wang, and J. Wu, On Khintchine
exponents and Lyapunov exponents of continued fractions, Ergodic Theory
Dynam. Systems, 29 (2009), pp. 73–109.

[13]
G. González Robert, Good's theorem for Hurwitz continued
fractions, Int. J. Number Theory, 16 (2020), pp. 1433–1447.

[14]
height 2pt depth -1.6pt width 23pt, Purely periodic and
transcendental complex continued fractions, Acta Arith., 194 (2020),
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