# Superconductivity in metallic hydrogen

D. van der Marel<EMAIL_ADDRESS>Department of Quantum Matter
Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4,
Switzerland  C. Berthod  Department of Quantum Matter Physics, University of
Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland

(August 28, 2024)

###### Abstract

Superconductivity, the amazing phenomenon of lossless transmission of electric
current through metallic wires, requires cooling of the wire to low
temperatures. Metallic hydrogen is considered as the most likely candidate for
superconductivity at very high temperatures, possibly even room temperature.
However, as a result of various approximations used, conflicting theoretical
predictions exist for the range of temperatures where superconductivity is
expected to occur. Here we avoid those approximations and confirm that
metallic hydrogen is indeed a superconductor, but this is limited to
temperatures far below previous estimates. We exploit the “jellium” model
proposed in 1966 by De Gennes, where superconductivity is caused by the
combination of Coulomb repulsion between the electrons and Coulomb attraction
between the protons and the electrons. We find that the superconducting order
develops over an energy range far exceeding the characteristic phonon energy,
and that the phase of the order parameter flips 180 degrees at the
characteristic phonon energy above and below the Fermi energy.

## Introduction

According to the theory of Bardeen Cooper and Schrieffer Bardeen  _et al._
(1957), superconductivity is caused by electron-phonon coupling, the coupling
of the electrons to vibrations of the lattice. A pedagogical description of
this mechanism was introduced by De Gennes using the “jellium model”,
demonstrating that electron-phonon coupling can result in an effective
interaction between electrons which overcomes the Coulomb repulsion between
the electrons De Gennes (1966). De Gennes indicated a number of restrictions
for the range of applicability of the jellium model, notably that the model
assumes the absence of core electrons. This limits the part of the periodic
table where the model can be applied to hydrogen and helium, of which hydrogen
is by far the most promising candidate. In 1968, Ashcroft suggested that the
bulk of Jupiter is composed of hydrogen in the metallic state and that part of
the bulk (with a temperature 100–200 K) may be in a superconducting state
Ashcroft (1968). Various refinements were made in later papers using
Eliashberg strong coupling theory Richardson and Ashcroft (1997); Barbee  _et
al._ (1989); Ashcroft (2004), predicting superconductivity above 200 K. In
1981, Jaffe and Ashcroft pointed out the possibility that metallic hydrogen
may be a superconducting liquid in the density range $3.9\times
10^{23}\leavevmode\nobreak\ \mathrm{cm}^{-3}<n<7.3\times
10^{23}\leavevmode\nobreak\ \mathrm{cm}^{-3}$, with a maximum $T_{c}\approx
141$ K for $n=6.4\times 10^{23}\leavevmode\nobreak\ \mathrm{cm}^{-3}$. The
possibility that pulsars are cold magnetic white dwarfs was discussed, also in
1968, by Ginzburg and Khirzhnits Ginzburg and Kirzhnits (1968). They pointed
out that the superconductivity of a certain layer of the star may be an
essential factor and derived a simple expression for $T_{c}$ for light
elements. Their expression for hydrogen, based on De Gennes’ jellium model, is
reproduced in Eq. (10). In a subsequent paper, Khirzhnits Kirzhnits (1969)
published a slightly different version of this expression, which is reproduced
in Eq. (12). In recent years, superconductivity in hydrogen-rich compounds has
seen a revival, where numerical predictions and experimental confirmation of
superconductivity near and above room temperature have entered the stage Boeri
_et al._ (2022)—and to some extent left it again Garisto (2024). The
calculation of $T_{c}$ for a multi-element material is a complex task, and a
certain degree of coarse-graining of the momentum dependence and energy
dependence of the electron-phonon coupling and the screened Coulomb
interaction is unavoidable.

The values of $T_{c}$ that follow from Ginzburg and Kirzhnits’ expressions are
displayed in Fig. 2. The fact that the results are so different may have to do
with approximations beyond those of the jellium model, that were necessary in
view of limitations of the computational resources available at that time. For
this reason, and without pretence of addressing the pairing mechanism of the
aforementioned more complex materials, we return to the model interaction
proposed by De Gennes and explore its properties within the BCS formalism
Bardeen  _et al._ (1957) without additional approximations.

Figure 1:  Superconducting transition temperature of metallic hydrogen.
Comparison of three calculations of $T_{c}$ in hydrogen at different
densities, each of them using the jellium model De Gennes (1966). Green curve:
Ginzburg and Kirzhnits expression Ginzburg and Kirzhnits (1968) reproduced in
Eq. (10). Orange curve: Kirzhnits’ expression Kirzhnits (1969) reproduced in
Eq. (12). Blue curve: present result. The different phases indicated along the
bottom axis refer to the zero-temperature limit. The labels “solid I” (solid,
phase I) and “s III” (solid, phase III) and IV’ are adapted from Fig. 3 of
Ref. Howie  _et al._ (2015), where IV’ is possibly a liquid phase Eremets and
Troyan (2011). To relate the pressure to the density, the data in Fig. 4 of
Ref. Mao and Hemley (1994) were used. In the region “no stable phase” the
density is below that of solid molecular hydrogen in the limit of zero
pressure Dewar (1899).

Figure 2:  Relevant energy scales of metallic hydrogen as a function of
density. The lower (upper) horizontal axis indicates the density (Wigner-Seitz
radius).

## Superconductivity in the jellium model

Physically, a “jellium” would be realized if the material could be made dense
enough, that not only the electrons but also the nuclei are in a liquid state.
The formal requirement is that the Wigner-Seitz radius (radius of a sphere
whose volume is equal to the volume per particle) of the electrons and that of
the nuclei are both sufficiently small. For the case of hydrogen, this
possibility was proposed—and its properties were explored—in a series of
papers by Ashcroft and collaborators Jaffe and Ashcroft (1981, 1983);
Moulopoulos and Ashcroft (1991); Babaev  _et al._ (2004). A recent review of
the phases of highly compressed hydrogen up to 500 GPa (Carlo Pierleoni in
Ref. Boeri  _et al._ (2022)) does not address this proposal, but points out
that the lack of experimental information about the crystalline structure
together with the large nuclear quantum effects for hydrogen poses a challenge
to ab initio calculations. The jellium model is an interesting alternative
approach since (i) crystallographic information is not required, (ii) the
mathematics is relatively simple, (iii) it is optimally suited for hydrogen
because the model assumes the absence of core electrons, and (iv) the model
rests on only two parameters: the Wigner-Seitz radius of the electrons and the
mass of the nuclei.

Figure 3:  Energy dependence of the interaction and the gap function. a s-wave
coupling function, $\varepsilon_{\mathrm{F}}\upsilon^{s}_{x,0}$, as a function
of energy $\varepsilon_{k}=\varepsilon_{\mathrm{F}}x$ in the case of hydrogen
($m_{N}=m_{p}$, blue) and deuterium ($m_{N}=2m_{p}$, orange) with
$r_{s}=1.44$. b Energy dependence of the order parameter for $r_{s}=1.44$ in
the case of hydrogen ($m_{N}=m_{p}$), and for four temperatures below $T_{c}$
($\sim 17$ K). For $T=0$ the results computed with two different numerical
codes are compared. Open symbols : Numerical code where the integrals are
calculated with the trapezoidal rule (Appendix C). Continuous curve: Numerical
code where the integrals are calculated with a piecewise-continuous function
of $x$ (Appendix D). The difference of these two curves is given in Extended
Data Fig. 1.

At this point, it is useful to introduce the relevant energy scales of the
system. In Fig. 2, the plasmon energy of the electrons $\omega_{p}$, the Fermi
energy of the electrons $\varepsilon_{\mathrm{F}}$, the plasmon energy of the
nuclei $\omega_{0}$, and the gap energy at the Fermi surface
$\Delta_{\mathrm{F}}$ (anticipating for the latter the discussion of the
present manuscript) are displayed as a function of the electron density $n$
and—on the opposite axis— the Wigner-Seitz radius $r_{s}=a_{0}^{-1}(4\pi
n/3)^{-1/3}$, where $a_{0}=0.529177$ Å is the Bohr radius. For the density
range relevant for metallic hydrogen ($r_{s}\sim 1.5$) the plasma energy and
the Fermi energy are of the same order of magnitude, and for all densities
$\omega_{p},\varepsilon_{\mathrm{F}}\gg\omega_{0}\gg\Delta_{\mathrm{F}}$. The
Wigner-Seitz radius plays the role of the coupling constant Kirzhnits (1969),
i.e., $\lambda\approx 0.16\,r_{s}$ in $T_{c}\propto e^{-1/\lambda}$ [see Eq.
(11)], so that for $r_{s}\sim 1.5$ the coupling constant $\sim 0.25$, which is
in the weak coupling regime. Furthermore, since
$\omega_{0}\ll\varepsilon_{\mathrm{F}}$, the system is in the regime where the
motion of the nuclear particles is slow compared to that of the electrons. All
in all, this implies that for the density range of interest the BCS model
Bardeen  _et al._ (1957) can be used. The coupling should be considered strong
for $r_{s}>6$ and predictions with the BCS model become less reliable.

Following De Gennes De Gennes (1966), we adopt as a starting point the Coulomb
interaction screened by electrons and nuclear particles

$V^{\mathrm{eff}}(q,\omega)=\frac{4\pi
e^{2}}{q^{2}}\frac{1}{\epsilon(q,\omega)},$ (1)

where $\epsilon(q,\omega)$ is the dielectric function and
$\bm{q}=\bm{k}-\bm{p}$ represents the transferred momentum. We furthermore
adopt the procedure of De Gennes De Gennes (1966) and Kirzhnits Kirzhnits
(1969), and substitute $\hbar\omega=\varepsilon_{k}-\varepsilon_{p}$. The
argument, provided on pp. 100–102 of Ref. De Gennes (1966), is that with this
substitution the effective interaction has exactly the form of the matrix
element $V_{k,p}$ between an initial state where two electrons are in the
plane wave states $\left|k,-k\right>$ and a final state where the electrons
are in $\left|p,-p\right>$.

For a high-density electron gas, the dielectric function in the jellium
approximation is described by

$\epsilon(q,\omega)=1+\frac{k_{0}^{2}}{q^{2}}-\frac{\omega_{0}^{2}}{\omega^{2}},$
(2)

where $k_{0}$ is the Thomas-Fermi screening wavevector and $\omega_{0}$ is the
plasma frequency of the positively charged nuclei. Expressing the electron
energies and the gap energy in units of the Fermi energy, i.e.,
$\varepsilon_{k}=x\varepsilon_{\mathrm{F}}$,
$\Delta_{k}=\psi_{x}\varepsilon_{\mathrm{F}}$, the gap equation obtains the
following compact form (Appendix B)

$\psi_{x}=-\int_{0}^{\infty}dy\,\sqrt{y}\upsilon^{s}_{x,y}\frac{\psi_{y}}{2\sqrt{(y-1)^{2}+\psi_{y}^{2}}}\tanh\left(\frac{\sqrt{(y-1)^{2}+\psi_{y}^{2}}}{2\tau}\right),$
(3)

where

$\upsilon^{s}_{x,y}=\frac{(x-y)^{2}}{2\sqrt{xy}[4{z}_{0}^{2}-(x-y)^{2}\gamma^{-2}]}\ln\left|\frac{{4(x-y)^{2}-\left(\sqrt{x}-\sqrt{y}\right)^{2}[4{z}_{0}^{2}-(x-y)^{2}\gamma^{-2}]}}{{4(x-y)^{2}-\left(\sqrt{x}+\sqrt{y}\right)^{2}[4{z}_{0}^{2}-(x-y)^{2}\gamma^{-2}]}}\right|$
(4)

is the interaction in the $s$-wave pairing channel, and is displayed in the
top panel of Fig. 3. The two constants in this expression,

$z_{0}^{2}=\frac{4m_{e}}{3m_{N}},\qquad\gamma^{2}=\left(\frac{4}{9\pi^{4}}\right)^{1/3}r_{s},$
(5)

depend uniquely on the nuclear mass $m_{N}$ of the isotope (hydrogen or
deuterium in the present context) and the Wigner-Seitz radius $r_{s}$.
Remarkably, for zero energy shift ($x=y$) the interaction is exactly zero. The
interaction is furthermore negative for small energy shift, passes through a
minimum, changes sign, passes through a positive logarithmic singularity and
asymptotically approaches the Thomas-Fermi screened Coulomb repulsion at high
energies. In the most common implementation of the theory of Bardeen Cooper
and Schrieffer Bardeen  _et al._ (1957) the interaction is assumed to be
constant and attractive in the interval $\pm\omega_{0}$ around the Fermi
energy, corresponding to an interaction at the Fermi level that is both non-
zero and attractive. Given that in the present model the interaction at the
Fermi energy is zero, it is not overwhelmingly obvious that such an
interaction would induce the Cooper instability Cooper (1956). One of the
motivations for the present study was to investigate whether the BCS
variational wave function has a non-trivial solution with this interaction. To
address this question, we solved the gap equation numerically (Appendix C) and
calulated the gap function, the condensation energy (Eq. 36), and the specific
heat (Eq. 25). Instead of substituting
$\hbar\omega=\varepsilon_{k}-\varepsilon_{p}$ in the interaction and solving
the BCS gap equations, it is in principle possible to solve the Eliashberg
equations and calculate $T_{c}$ Eliashberg (1960). While the relevant set of
equations for the present case could be identified without a problem, due to
the simultaneous energy- and momentum dependence of the interaction, the
numerical algorithm poses a computational challenge which we haven’t been able
to overcome yet.

## Results

The resulting gap function is shown in the lower panel of Fig. 3 for hydrogen
assuming a Wigner-Seitz radius $r_{s}=1.44$ at four temperatures below
$T_{c}$. We see that the gap function changes sign near the energy where
$\upsilon^{s}_{x,0}$ changes sign, and shows a negative peak (which is however
not a singularity) near the energy where $\upsilon^{s}_{x,0}$ has a log-
singularity. One of the remarkable features is that the region of finite order
parameter—and temperature evolution thereof—is not limited in any way to a
narrow range around the Fermi energy. Although this doesn’t contribute
significantly to the free energy of the electrons, experimental probes such as
photo-emission, optics and tunneling spectroscopy can in principle pick up the
temperature dependency of the order parameter at energies far beyond the
characteristic energy of the phonons that mediate the pairing interaction.
Another feature—not visible on this scale—is that for large positive energy
$\Delta_{k}=\varepsilon_{\mathrm{F}}\psi_{x}$ is proportional to $1/x$.

Figure 4:  Specific heat divided by temperature for eight different values of
$r_{s}$ in the case of hydrogen ($m_{N}=m_{p}$). Figure 5:  Superconducting
transition temperature as a function of density. Square symbols: $T_{c}$ for
$m_{N}=m_{p}$ (hydrogen) determined from the specific heat jump (Fig. 4). The
vertical dashed line indicates the density of metallic hydrogen Mazin and
Cohen (1995). Open symbols: Effective $T_{c}$ determined from the condensation
energy [Eq. (6) with $A=0.925$] for hydrogen (blue) and deuterium (orange,
$m_{N}=2m_{p}$).

The temperature-dependent specific heat is shown in Fig. 4. The drop in the
specific heat was used to determine $T_{c}$, shown in Fig. 5. The maximum is
reached for $r_{s}\sim 3$, where in the case of hydrogen $T_{c}\sim 30$ K. To
obtain an accurate estimate of $T_{c}$, the gap equation has to be solved for
many temperatures for each $r_{s}$ value. Since this is a time-consuming
calculation, we also tested a less cumbersome method, namely to use the
following proxy for $T_{c}$:

$T_{c}^{*}=A\sqrt{\varepsilon_{\mathrm{F}}E_{c}}/k_{\mathrm{B}}.$ (6)

In the most commonly used implementation of BCS theory, the interaction is
assumed constant between $\pm\omega_{0}$ and zero elsewhere. In this case the
condensation energy _per particle_ is

$E_{c}=\frac{3\Delta_{0}^{2}}{8\epsilon_{F}}$ (7)

and the gap over $T_{c}$ ratio

$\frac{2\Delta_{0}}{k_{\mathrm{B}}T_{c}}=3.53.$ (8)

Combining these two expressions one obtains $A=0.925$. The results for $T_{c}$
and $T_{c}^{*}$ are displayed in Fig. 5 for hydrogen (blue symbols) and
deuterium (orange symbols). The density dependence of $T_{c}^{*}$, shown in
Fig. 5, is described by a dome. We also see that the mass-dependence of
$T_{c}^{*}$ follows closely the conventional isotope effect, i.e., the ratio
$T_{c}^{*}(\mathrm{H})/T_{c}^{*}(\mathrm{D})\sim\sqrt{2}$. The result for
hydrogen is also displayed as the dark blue curve in Fig. 2.

Ashcroft estimated $T_{c}\sim 200$ K for dense hydrogen Ashcroft (1968);
Richardson and Ashcroft (1997) assuming a coupling constant $\lambda\approx
1.5$. The $T_{c}$ values obtained with the jellium model are an order of
magnitude below these estimates as a result of the relatively low coupling
constant which for $r_{s}\sim 1.5$, using Kirzhnits’ expression [see Eq.
(11)], hovers around $\lambda\sim 0.25$. A natural question is what the value
of $T_{c}$ would become if, instead of using the jellium approximation, the
full potential of the positively charged nuclear particles would be taken into
account. Since no structural information is available for densities
corresponding to the metallic phase of hydrogen, it is not possible at the
present stage to predict the superconducting properties for the actual atomic
configuration of solid or liquid metallic hydrogen beyond the jellium
approximation. However, based on what is known for the alkali metals, it is
reasonable to postulate that the Fermi surface would be entirely contained
within the first Brillouin zone and that the energy-momentum dispersion of the
electrons is to a good approximation described by the effective mass
formalism. While such a modification of the band structure would change the
$r_{s}$ value for any given density, the upper limit for $T_{c}(r_{s})$ would
remain 30 K.

Hirsch Hirsch (2020) has argued on the basis of Alfven’s theorem Alfvén (1942)
that for the Meissner effect to occur it is essential that the charge carriers
have negative effective mass; otherwise the transition would not be
reversible. On the one hand we have demonstrated in the present study that,
with De Gennes’ effective electron-electron interaction, the BCS variational
wave-function of free electrons—which have a positive mass—has a non-trivial
ground state with a finite order parameter. On the other hand, according to
Hirschs’ argument there can be no Meissner effect in this model in view of the
positive mass. If this argument is correct, this would imply that interacting
electrons can condense into a state with a BCS-type order parameter but, in
view of the positive mass of the electrons this state of matter differs in a
fundamental way from a superconductor and does not have a Meissner effect. We
nonetheless continue to use the terminology of superconductivity for the
remainder of the discussion and return to the questions about
superconductivity in white dwarf stars and in the interior of Jupiter
mentioned in the introduction. To begin with the latter, the black-body
temperature provides a lower limit of the temperature of a planet, which in
the case of Jupiter is 109.9 K Williams (2021). For white dwarfs the situation
is more complicated. For the coldest one that has been observed an upper limit
of 3000 K was estimated Kaplan  _et al._ (2014). This does not exclude that
the temperature is lower than 30 K, however, given the current age of the
known universe it is unlikely that sufficient time has elapsed for a white
dwarf to cool down to such a low temperature. All in all it seems unlikely
both for the case of Jupiter or for white dwarfs that the temperature is low
enough (lower than 30 K) to contain metallic hydrogen in the superconducting
state.

## Conclusions

The superconducting pairing in metallic hydrogen involves an intricate
interplay of the Coulomb repulsion between the electrons and an attractive
phonon-mediated pairing mechanism. One of the consequences is that the
superconducting order parameter has a 180∘ phase shift at the characteristic
energy of the phonons. The outcome for the superconducting critical
temperature ($T_{c}<30$ K) is much lower than the $T_{c}$ predicted in Refs.
Ashcroft (1968); Richardson and Ashcroft (1997); Barbee  _et al._ (1989);
Ashcroft (2004) and previous results using the same model Ginzburg and
Kirzhnits (1968); Kirzhnits (1969). For energies far above the phonon energy
range, the amplitude of the order parameter is of the same order of magnitude
as that at the Fermi surface.

## Acknowledgements

We gratefully acknowledge valuable discussions with Peter Hirschfeld and Louk
Rademaker.

##  Appendix A Previous expressions for $T_{c}$ using the jellium model

Ginzburg and Kirzhnits Ginzburg and Kirzhnits (1968) used the following
expressions

$T_{c}=\hbar\omega_{0}\exp{\left(-\frac{1}{NV}\right)}\quad\text{with}\quad
NV=\frac{e^{2}}{\hbar
v_{\mathrm{F}}}=\left(\frac{4}{9\pi}\right)^{1/3}r_{s}\quad\text{and}\quad\omega_{0}^{2}=\frac{4\pi
ne^{2}}{m_{p}}.$ (9)

The combination of these expressions gives

$k_{\mathrm{B}}T_{c}=\mbox{Hr}\sqrt{\frac{3m_{e}}{m_{N}}}\frac{1}{r_{s}^{3/2}}\exp{\left[-\left(\frac{9\pi}{4}\right)^{1/3}\frac{1}{r_{s}}\right]}.$
(10)

where Hr $\approx 27$ eV is the Hartree energy. Kirzhnits Kirzhnits (1969)
used

$\Delta=\alpha\omega_{0}\exp{\left(-\frac{8}{\pi^{2}\alpha}\right)}\quad\text{with}\quad\alpha=\frac{e^{2}}{\pi
v_{\mathrm{F}}}=\left(\frac{4}{9\pi^{4}}\right)^{1/3}r_{s}$ (11)

and the same definition of $\omega_{0}$, stating that the critical temperature
$T_{c}$ is connected with this quantity in the usual manner. Combining these
expressions gives

$k_{\mathrm{B}}T_{c}=\mbox{Hr}\frac{e^{\gamma}}{\pi}\left(\frac{2}{3\pi^{2}}\right)^{2/3}\sqrt{\frac{3m_{e}}{m_{N}}}\frac{1}{r_{s}^{1/2}}\exp{\left[-2\left(\frac{12}{\pi}\right)^{2/3}\frac{1}{r_{s}}\right]}$
(12)

where $\gamma\approx 0.577$ is the Euler constant.

##  Appendix B The gap equation

If the BCS variational wave function is applied to a system of band electrons
with energy-momentum dispersion $\epsilon_{k}$ and an electron-electron
interaction $V^{\mathrm{eff}}_{\bm{k},\bm{p}}$, one obtains the following
expression for the grand potential Rickayzen (1965)

$\displaystyle\Omega$
$\displaystyle=\sum_{\bm{k}}\left[-2k_{\mathrm{B}}T\ln\left(1+e^{-E_{k}/k_{\mathrm{B}}T}\right)+\xi_{k}-E_{k}+\frac{|\Delta_{k}|^{2}}{E_{k}}\tanh\left(\frac{E_{k}}{2k_{\mathrm{B}}T}\right)\right]+\sum_{\bm{k},\bm{p}}\frac{\Delta_{k}\Delta_{p}^{*}V^{\mathrm{eff}}_{\bm{k},\bm{p}}}{4E_{k}E_{p}\nu}\tanh\left(\frac{E_{p}}{2k_{\mathrm{B}}T}\right)$
(13)

where $\xi_{k}=\varepsilon_{k}-\mu$, $\mu$ is the chemical potential, and
$E_{k}=\sqrt{\xi_{k}^{2}+|\Delta_{k}|^{2}}$. In equilibrium, where
$\Delta_{k}$ corresponds to the set of values that minimizes $\Omega$, the
number of electrons follows from the relation

$N=-\frac{\partial\Omega}{\partial\mu}=\sum_{\bm{k}}\left[1-\frac{\xi_{k}}{E_{k}}\tanh\left(\frac{E_{k}}{2k_{\mathrm{B}}T}\right)\right]$
(14)

We reserve the symbols $N_{0}$ for the number of electrons, $n_{0}$ for the
electron density and $\varepsilon_{\mathrm{F}}$ for the chemical potential at
$T=0$ and in the absence of superconducting order. $V^{\mathrm{eff}}_{k,p}$
represents the effective interaction between the electrons. Here we consider
the screened Coulomb interaction De Gennes (1966)

$V^{\mathrm{eff}}(q,\omega)=\frac{4\pi
e^{2}}{q^{2}}\frac{1}{\epsilon(q,\omega)}$ (15)

For $\epsilon(q,\omega)$ we adopt the the jellium model:

$\epsilon(q,\omega)=1+\frac{k_{0}^{2}}{q^{2}}-\frac{\omega_{0}^{2}}{\omega^{2}}$
(16)

where

$k_{0}^{2}={\frac{4e^{2}m_{e}k_{\mathrm{F}}}{\pi\hbar^{2}}},\hskip
28.45274pt\omega_{0}^{2}={\frac{4\pi ne^{2}}{m_{N}}}$ (17)

so that

$V^{\mathrm{eff}}(q,\omega)=\frac{4\pi
e^{2}\omega^{2}}{k_{0}^{2}\omega^{2}+(\omega^{2}-\omega_{0}^{2})q^{2}}$ (18)

To obtain a more compact formulation we multiply both sides of Eq. (13) with
$1/(N_{0}\varepsilon_{\mathrm{F}})=3\pi^{2}/(\nu
k_{\mathrm{F}}^{3}\varepsilon_{\mathrm{F}})$, divide the electron energies by
the Fermi energy and adopt Ashcroft’s notation Ashcroft (2004) $\gamma$ for
the Thomas-Fermi factor. Concretely we define the following set of
dimensionless quantities

$\displaystyle w$
$\displaystyle=\frac{\Omega}{N_{0}\varepsilon_{\mathrm{F}}},$ $\displaystyle
x$ $\displaystyle=\frac{\varepsilon_{k}}{\varepsilon_{\mathrm{F}}},$
$\displaystyle y$
$\displaystyle=\frac{\varepsilon_{p}}{\varepsilon_{\mathrm{F}}},$
$\displaystyle z$
$\displaystyle=x-y=\frac{\hbar\omega}{\varepsilon_{\mathrm{F}}}$ (19)
$\displaystyle\psi_{x}$
$\displaystyle=\frac{\Delta_{k}}{\varepsilon_{\mathrm{F}}},$
$\displaystyle\tau$
$\displaystyle=\frac{k_{\mathrm{B}}T}{\varepsilon_{\mathrm{F}}},$
$\displaystyle z_{0}$ $\displaystyle=\sqrt{\frac{4m_{e}}{3m_{N}}},$
$\displaystyle\gamma$
$\displaystyle=\frac{k_{0}}{2k_{\mathrm{F}}}=\left(\frac{2}{3\pi^{2}}\right)^{1/3}\sqrt{r_{s}}.$

Since $\psi_{k}$ is isotropic for the s-wave pairing assumed here,
$V^{\mathrm{eff}}_{\bm{k}\bm{p}}$ is the only term in the free-energy
expression that depends on the angular coordinates of $\bm{k}$ and $\bm{p}$,
and only on the relative angle of these two. This way we obtain

$\displaystyle w$
$\displaystyle=\frac{3}{4}\int_{0}^{\infty}dx\,\sqrt{x}\left[-2\tau\ln\left(1+e^{-E_{x}/\tau}\right)+\xi_{x}-E_{x}+\frac{|\psi_{x}|^{2}}{E_{x}}\tanh\left(\frac{E_{x}}{2\tau}\right)\right]+$
$\displaystyle\quad+\frac{3}{4}\int_{0}^{\infty}dx\,\sqrt{x}\int_{0}^{\infty}dy\,\sqrt{y}\frac{\psi_{x}\psi_{y}^{*}\upsilon^{s}_{x,y}}{4E_{x}E_{y}}\tanh\left(\frac{E_{x}}{2\tau}\right)\tanh\left(\frac{E_{y}}{2\tau}\right),$
(20)

where $\xi_{x}=x-\mu/\varepsilon_{\mathrm{F}}$,
$E_{x}=\sqrt{\xi_{x}^{2}+|\psi_{x}|^{2}}$ and

$\upsilon^{s}_{x,y}=\frac{z^{2}}{2\sqrt{xy}[4{z}_{0}^{2}-z^{2}\gamma^{-2}]}\ln\left|\frac{{4z^{2}-\left(\sqrt{x}-\sqrt{y}\right)^{2}[4{z}_{0}^{2}-z^{2}\gamma^{-2}]}}{{4z^{2}-\left(\sqrt{x}+\sqrt{y}\right)^{2}[4{z}_{0}^{2}-z^{2}\gamma^{-2}]}}\right|$
(21)

is the interaction in the $s$-wave pairing channel. The equilibrium condition

$\forall x:\qquad\frac{\partial w}{\partial{\psi_{x}}}=0$ (22)

results in the gap equation

$\psi_{x}=-\int_{0}^{\infty}dy\,\sqrt{y}\frac{\psi_{y}\upsilon^{s}_{x,y}}{2E_{y}}\tanh\left(\frac{E_{y}}{2\tau}\right).$
(23)

In a bulk 3-dimensional sample the macroscopic charge has the effect of
maintaining the number of electrons inside the sample constant. Consequently
the chemical potential is temperature dependent, which can be observed
experimentally as a temperature dependence of the workfunction Rietveld  _et
al._ (1992). To account for this effect, the condition (14) must be applied
simultaneously with the solution of the gap equation. Since the shift of
chemical potential is of order $\Delta^{2}/4\varepsilon_{\mathrm{F}}$, the
effect becomes relevant in superconductors where $\Delta$ is of comparable
size as $\varepsilon_{\mathrm{F}}$ which, given the hierarchy of energy scales
shown in Fig. 2, clearly doesn’t apply to the system considered here. For this
reason it makes no significant difference if the gap equation is solved in the
canonical ensemble or in the grand canonical ensemble. Here we follow the
latter approach and fix the chemical potential at
$\mu=\varepsilon_{\mathrm{F}}$.

The entropy is

$S(T)=-\frac{dw}{d\tau}k_{\mathrm{B}}=\frac{3}{2}k_{\mathrm{B}}\int_{0}^{\infty}dx\sqrt{x}\left[\frac{\ln({1+e^{E_{x}/\tau}})}{1+e^{E_{x}/\tau}}+\frac{\ln({1+e^{-E_{x}/\tau}})}{1+e^{-E_{x}/\tau}}\right]$
(24)

The specific heat is obtained by (numerical) differentiation of $S(T)$

$C(T)=T\frac{d}{dT}S(T)$ (25)

In Fig. 6 entropy, specific heat and order parameter at the Fermi surface are
displayed as a function of temperature for the case $r_{s}=2.08$.

Figure 6:  Temperature dependence of the order paramater at the Fermi surface,
the entropy and the specific heat for $r_{s}=2.08$ in the case of hydrogen
($m_{N}=m_{p}$) calculated with the trapezoidal rule (“Numerical
implementation” in Methods). The noise of $C(T)/T$ in the lower panel is
stronger than that of the entropy, because it was calculated taking the
numerical derivative of $S(T)$.

##  Appendix C Numerical implementation

Energy interval | No. of equidistant energy values
---|---
$-1.0<\xi_{j}<0.1$ | $n$
$-0.1<\xi_{j}<-4\sqrt{\psi_{F}^{2}+\tau^{2}}$ | $4n$
$-4\sqrt{\psi_{F}^{2}+\tau^{2}}<\xi_{j}<4\sqrt{\psi_{F}^{2}+\tau^{2}}$ | $n$
$4\sqrt{\psi_{F}^{2}+\tau^{2}}<\xi_{j}<0.1$ | $4n$
$0.1<\xi_{j}<1$ | $n$
$1<\xi_{j}<100$ | $n$
$100<\xi_{j}<10^{4}$ | $n$
$10^{4}<\xi_{j}<\infty$ | analytical continuation
Table 1:  Discretization of the energy axis used in numerical integrals.

It is easy to demonstrate that at $T_{c}$ the gap function is real, apart from
an arbitrary phase that doesn’t depend on $k$ and that we can set to zero. It
is in principle possible that at an additional transition takes place to a
state with a $k$-dependent phase. Since this requires a finite amplitude of
the order parameter, it can only happen at a temperature lower than $T_{c}$ so
that it wouldn’t influence $T_{c}$. Since our purpose is to determine $T_{c}$,
we can limit the present discussion to a real-valued order parameter. In this
case the expression of the grand potential formulated for numerical coding is
:

$\displaystyle w$
$\displaystyle=\frac{3}{4}\sum_{h}\left[-2\tau\ln\left(1+e^{-X_{h}/\tau}\right)+\xi_{h}-X_{h}+\frac{\psi_{h}(2\psi_{h}-\phi_{h})}{2X_{h}}\tanh\left({X_{h}}/{2\tau}\right)\right]D_{h}$
(26) where $\displaystyle\xi_{h}$ $\displaystyle=x_{h}-1$ (27) $\displaystyle
X_{h}$ $\displaystyle=\sqrt{\xi_{h}^{2}+\psi_{h}^{2}}$ (28) $\displaystyle
D_{h}$ $\displaystyle=\sqrt{x_{h}}dx_{h}$ (29) $\displaystyle\phi_{h}(n)$
$\displaystyle=-\sum_{j=1}^{N}\left[\frac{\psi_{j}(n)\tanh\left({X_{j}/2\tau}\right)\upsilon^{s}(x_{h},x_{j})}{2X_{j}}\right]D_{j}-\left[\frac{\psi_{N}(n)\tanh\left({X_{N}/2\tau}\right)\upsilon^{s}(x_{h},x_{N})}{2X_{N}}\right]\times\frac{2}{3}x_{N}^{3/2}$
(30)

The second term on the right hand side is the analytical integral in the
interval $\\{x_{N};\infty\\}$, taking into account that for $x_{j}\gg 1$ both
$\upsilon^{s}(x_{i},x_{j})\propto x_{j}^{-1}$ and $\psi_{j}(n)\propto
x_{j}^{-1}$. For $x_{j}\gg 1$ we can use $X_{j}\approx x_{j}$. Consequently
the integrand is proportional to $x_{j}^{-5/2}$, from which follows the second
term. The first and second derivatives of the grand potential are

$\displaystyle\frac{dw}{d\psi_{i}}$
$\displaystyle=\frac{3}{4}\left[\frac{\xi_{i}^{2}\tanh\left({X_{i}}/{2\tau}\right)}{X_{i}^{3}}+\frac{\psi_{i}^{2}}{2\tau
X_{i}^{2}\cosh^{2}({X_{i}}/{2\tau})}\right]\left[\psi_{i}-\phi_{i}\right]D_{i}$
(31) $\displaystyle\frac{\partial^{2}w}{\partial\psi_{i}^{2}}$
$\displaystyle=\frac{3}{4}\frac{\psi_{i}}{X_{i}^{3}}\left[\frac{3\xi_{i}^{2}+\psi_{i}^{2}}{2\tau
X_{i}^{2}\cosh^{2}\left({X_{i}}/{2\tau}\right)}-\frac{\psi_{i}^{2}\tanh\left({X_{i}}/{2\tau}\right)}{4\tau^{2}\cosh^{2}\left({X_{i}}/{2\tau}\right)}-\frac{3\xi_{i}^{2}\tanh\left({X_{i}}/{2\tau}\right)}{X_{i}^{2}}\right]\left[\psi_{i}-\phi_{i}\right]\delta_{i,m}D_{i}+$
$\displaystyle\quad+\frac{3}{4}\frac{1}{X_{i}^{2}}\left[\frac{\xi_{i}^{2}\tanh\left({X_{i}}/{2\tau}\right)}{X_{i}}+\frac{\psi_{i}^{2}}{2\tau\cosh^{2}\left({X_{i}}/{2\tau}\right)}\right]D_{i}+$
$\displaystyle\quad+\frac{3}{8}\frac{\upsilon^{s}(x_{i},x_{i})}{X_{i}^{2}X_{i}^{2}}\left[\frac{\xi_{i}^{2}\tanh\left({X_{i}}/{2\tau}\right)}{X_{i}}+\frac{\psi_{i}^{2}}{2\tau\cosh^{2}\left({X_{i}}/{2\tau}\right)}\right]\left[\frac{x_{i}^{2}\tanh\left({X_{i}}/{2\tau}\right)}{X_{i}}+\frac{\psi_{i}^{2}}{2\tau\cosh^{2}\left({X_{i}}/{2\tau}\right)}\right]D_{i}^{2}$
(32)

At the mininum we have $dw/d\psi_{i}=0$ for all $\psi_{i}$. We search for the
minimum using the steepest descent method, i.e., we iterate

$\psi_{i}(n+1)=\psi_{i}(n)-\frac{\eta}{2}\frac{\partial
w(n)/\partial\psi_{i}(n)}{\partial^{2}w(n)/\partial\psi_{i}(n)^{2}}$ (33)

where $\eta<1$ to ensure convergence of the iteration. The first step
consisted of solving the equation for $r_{s}=50$, and this process was
repeated down to $r_{s}=0.5$ using the output for $\psi_{i}$ as the starting
values for each subsequent $r_{s}$ value. For the calculation of the
temperature dependence, for each $r_{s}$ value the temperature was increased
from 0 to $0.85\psi_{F}$ where $\psi_{F}$ is the order parameter at
$\varepsilon_{\mathrm{F}}$, using the output for $\psi_{i}$ as the starting
values for each subsequent temparature. Integrations were done using the
trapezium method. The energy mesh used is specified in the Table 1

In each new iteration $\psi_{F}$ is taken to be the output value of previous
iteration. Consequently, if the order parameter converges to a small value the
set of energy values is adjusted correspondingly to a finer mesh around
$\varepsilon_{\mathrm{F}}$. Numerical verification indicates that $n=500$ is
sufficiently large, giving a grand total of 6500 energy points at which the
order parameter needs to be calculated self-consistently.

Although integrals over a log-divergence are convergent, jitter is unavoidable
in the present iterative procedure due to finite sampling of the energy axis.
To dampen the jitter we replace in Eq. 4

$\ln\left|\frac{4z^{2}-\left(\sqrt{x}-\sqrt{y}\right)^{2}[4z_{0}^{2}-z^{2}\gamma^{-2}]}{4z^{2}-\left(\sqrt{x}+\sqrt{y}\right)^{2}[4z_{0}^{2}-z^{2}\gamma^{-2}]}\right|\hskip
8.53581pt\mbox{with}\hskip
8.53581pt\ln\left|\frac{4(z^{2}+i\delta^{2})-\left(\sqrt{x}-\sqrt{y}\right)^{2}[4z_{0}^{2}-z^{2}\gamma^{-2}]}{4(z^{2}+i\delta^{2})-\left(\sqrt{x}+\sqrt{y}\right)^{2}[4z_{0}^{2}-z^{2}\gamma^{-2}]}\right|$
(34)

where $\delta=2/n$. At equilibrium the grand potential becomes

$w=\frac{3}{4}\int_{0}^{\infty}dx\sqrt{x}\left[-2\tau\ln\left(1+e^{-E_{x}/\tau}\right)+\xi_{x}-E_{x}+\frac{\psi_{x}^{2}}{2E_{x}}\tanh\left(\frac{E_{x}}{2\tau}\right)\right]$
(35)

where $\psi_{x}$ is order parameter at the free energy minimum. For the
purpose of displaying the energy dependence we choose the sign of $\psi_{x}$
to be positive at the Fermi surface. The condensation energy is the energy
saving at $T=0$ of the superconducting state compared to the trivial
($\bm{\psi}=0)$ solution of the gap-equation:

$e_{c}=w_{0}^{(n)}-w_{0}^{(sc)}$ (36)

Figure 7:  Difference of the gap-function at $T=0$ calculated with two
different numerical codes shown in Fig. 3.
$\Delta(\epsilon)=\Delta_{cont}(\epsilon)-\Delta_{discr}(\epsilon)$, where the
integrals in the numerical code for $\Delta_{discr}(\epsilon)$ are calculated
with the trapezoidal rule (Appendix C), and those for
$\Delta_{cont}(\epsilon)$ with a piecewise-continuous function of $x$
(Appendix D).

##  Appendix D Solution without discretization

The interaction $v^{s}_{x,y}$ entering the gap equation (23) presents
logarithmic singularities. This may lead to inaccuracies when the integral is
evaluated using a trapezoidal rule. We have investigated this issue by means
of an alternative approach, where the gap function $\psi_{x}$ is represented
as a piecewise-continuous function of $x$. In each piece, beside polynomials
of arbitrary order, we use a variety of functions to capture the precise
behavior of $\psi_{x}$, including logarithms and power laws. $\psi_{y}$ being
a continuous function, the integral can be split at the singularities of
$v^{s}_{x,y}$ and evaluated to the desired accuracy using Gaussian
quadratures, without discretization error. At each iteration towards self-
consistency, we fit our set of functions to $\psi_{x}$ in sub-domains that are
recursively split until a satisfactory fit is found. This method is much
slower, but produces results that are hardly distinguishable from those
obtained by discretizing the integral (see Fig. 3). In Fig. 7 the difference
of the two results is displayed. The differences are small, and most
pronounced near the singularities. We conclude that our results are not
significantly affected by discretization errors.

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# Some extreme value theory for $\theta$-expansions

Gabriela Ileana Sebe<EMAIL_ADDRESS>Faculty of Applied Sciences,
Politehnica University of Bucharest,
Splaiul Independentei 313, 060042, Bucharest, Romania  Gheorghe Mihoc-Caius
Iacob Institute of Mathematical Statistics and Applied Mathematics of the
Romanian Academy, Calea 13 Sept. 13, 050711 Bucharest, Romania  Dan Lascu
<EMAIL_ADDRESS>Mircea cel Batran Naval Academy, 1 Fulgerului, 900218
Constanta, Romania

###### Abstract

The main aim of this paper is to develop extreme value theory for
$\theta$-expansions. We get the limit distribution of the largest value of
$\theta$-continued fraction mixing stationary stochastic process and some
related results. These are analogous to J.Galambos and W.Philipp theorems for
the regular continued fractions. We also have to note that a Borel-Bernstein
type theorem plays an important role.

###### keywords:

$\theta$-expansions, Borel-Bernstein type theorem, extreme value theory,
Fréchet law, $\psi$-mixing.

##  1 Introduction

The investigation initiated by Bhattacharya and Goswami [2] in random number
generation, led to concept of continued fraction expansion of a number in
terms of an irrational $\theta\in(0,1)$. This new expansion of positive reals,
different from the regular continued fraction (RCF)- expansion is called
$\theta$-expansion. We mention that the case $\theta=1$ refers to RCF-
expansions.

For a fixed $\theta\in(0,1)$, Chakraborty and Rao [4] have considered a
generalization of the Gauss map $T_{\theta}:[0,\theta]\to[0,\theta]$,

$T_{\theta}(x):=\left\\{\begin{array}[]{ll}{\displaystyle\frac{1}{x}-\theta\left\lfloor\frac{1}{x\theta}\right\rfloor}&{\displaystyle\hbox{if
}x\in(0,\theta],}\\\ \\\ 0&\hbox{if }x=0.\end{array}\right.$ (1.1)

Here $\left\lfloor\cdot\right\rfloor$ stands for integer part. Then every
$x\in\left(0,\theta\right)$ can be expanded into a finite or infinite
$\theta$-expansion

$x=\frac{1}{\displaystyle a_{1}\theta+\frac{1}{\displaystyle
a_{2}\theta+\frac{1}{\displaystyle
a_{3}\theta+\ddots}}}=:[a_{1}\theta,a_{2}\theta,a_{3}\theta,\ldots],$ (1.2)

where

$a_{1}=a_{1}(x):=\left\\{\begin{array}[]{ll}\left\lfloor\displaystyle\frac{1}{x\theta}\right\rfloor&\hbox{if
}x\neq 0,\\\ \\\ \infty&\hbox{if }x=0\end{array}\right.$

and

$a_{n}=a_{n}(x):=a_{1}\left(T_{\theta}^{n-1}(x)\right),\quad
n\in\mathbb{N}_{+}:=\left\\{1,2,3,\ldots\right\\},$

with $T_{\theta}^{0}(x)=x$. The positive integers $a_{n}\in\mathbb{N}_{+}$ are
called the digits or partial quotients  of $x$.

Let $\mathcal{B}_{[0,\theta]}$ denote the $\sigma$-algebra of all Borel
subsets of $[0,\theta]$. It is obvious that the digits $a_{n}$,
$n\in\mathbb{N}_{+}$, are random variables which are defined almost surely on
$\left([0,\theta],\mathcal{B}_{[0,\theta]}\right)$ with respect to any
probability measure on $\mathcal{B}_{[0,\theta]}$ that assigns probability $0$
to the set of rationals in $[0,\theta]$. An example of such a measure is
Lebesgue measure $\lambda_{\theta}$ on $[0,\theta]$.

It was shown in [4, 11] that this expansion has many of the usual properties
of RCFs. A natural question is whether the dynamical system given by the
transformation $T_{\theta}$ admits an absolutely continuous invariant
probability like the Gauss measure in the case $\theta=1$. Chakraborty and Rao
[4] have identified that for certain values of $\theta$ (for example, if
$\theta^{2}=\frac{1}{m}$, $m\in\mathbb{N}_{+}$) the invariant measure for the
transformation $T_{\theta}$ as

$\mathrm{d}\gamma_{\theta}:=\frac{1}{\log\left(1+\theta^{2}\right)}\frac{\theta\,\mathrm{d}x}{1+\theta
x}.$ (1.3)

Moreover, if $\theta^{2}=\frac{1}{m}$, $m\in\mathbb{N}_{+}$,
$[a_{1}\theta,a_{2}\theta,a_{3}\theta,\ldots]$ is the $\theta$-expansion of
any $x\in(0,\theta)$ if and only if the following conditions hold:

  1. (i)

$a_{n}\geq m$ for any $n\in\mathbb{N}_{+}$

  2. (ii)

in the case when $x$ has a finite expansion, i.e.,
$x=[a_{1}\theta,a_{2}\theta,\ldots,a_{n}\theta]$, then $a_{n}\geq m+1$.

It was proved in [4] that the dynamical system $([0,\theta],T_{\theta})$ is
ergodic and the measure $\gamma_{\theta}$ is invariant under $T_{\theta}$,
that is, $\gamma_{\theta}(A)=\gamma_{\theta}(T^{-1}_{\theta}(A))$ for any
$A\in{\mathcal{B}}_{[0,\theta]}$. Therefore, $(a_{n})_{n\in\mathbb{N}_{+}}$ is
a strictly stationary sequence on
$([0,\theta],{\mathcal{B}}_{[0,\theta]},\gamma_{\theta})$.

For more results about $\theta$-expansions see [5, 10, 11, 12] and references
therein.

Every irrational $x\in(0,\theta)\setminus\mathbb{Q}=:\Omega$ has an infinite
$\theta$-expansion. Note that for all $n\in\mathbb{N}_{+}$, $a_{n}(x)\geq m$
and
$T^{n}_{\theta}([a_{1}\theta,a_{2}\theta,\ldots])=[a_{n+1}\theta,a_{n+2}\theta,\ldots]$.

For all $n\in\mathbb{N}_{+}$, we call the finite truncation of (1.2)

$\frac{p_{n}(x)}{q_{n}(x)}=[a_{1}(x)\theta,a_{2}(x)\theta,\ldots,a_{n}(x)\theta]$

the $n$-th convergent of the $\theta$-expansion of $x$.

For every infinite $\theta$-expansion $[a_{1}\theta,a_{2}\theta,\ldots]$ the
sequences $\\{p_{n}\\}_{n\geq-1}$ and $\\{q_{n}\\}_{n\geq-1}$ can be obtained
by the following recursive relations

$\displaystyle p_{n}(x)$ $\displaystyle=$ $\displaystyle a_{n}(x)\theta
p_{n-1}(x)+p_{n-2}(x),\quad$ (1.4) $\displaystyle q_{n}(x)$ $\displaystyle=$
$\displaystyle a_{n}(x)\theta q_{n-1}(x)+q_{n-2}(x),\quad$ (1.5)

with $p_{-1}(x):=1$, $p_{0}(x):=0$, $q_{-1}(x):=0$ and $q_{0}(x):=1$. Then by
induction, we have

$p_{n-1}(x)q_{n}(x)-p_{n}(x)q_{n-1}(x)=(-1)^{n},\quad n\in\mathbb{N}.$ (1.6)

By using (1.4) and (1.5), we can verify that

$x=\frac{p_{n}(x)+T^{n}_{\theta}(x)p_{n-1}(x)}{q_{n}(x)+T^{n}_{\theta}(x)q_{n-1}(x)},\quad
n\geq 1.$ (1.7)

Using (1.6) and (1.7) we obtain

$\left|x-\frac{p_{n}(x)}{q_{n}(x)}\right|=\frac{1}{q_{n}(x)\left(\left(T^{n}_{\theta}(x)\right)^{-1}q_{n}(x)+q_{n-1}(x)\right)},\quad
n\geq 1.$ (1.8)

Since
$a_{n+1}(x)\theta\leq\left(T^{n}_{\theta}(x)\right)^{-1}\leq(a_{n+1}(x)+1)\theta$,
using (1.4) and (1.5) in (1.8) we get

$\frac{1}{q_{n}(x)(q_{n+1}(x)+\theta
q_{n}(x))}\leq\left|x-\frac{p_{n}(x)}{q_{n}(x)}\right|\leq\frac{1}{q_{n}(x)q_{n+1}(x)},\quad
n\geq 1.$ (1.9)

From (1.5), we have that $q_{n}(x)\geq\theta$, $n\in\mathbb{N}_{+}$. Further,
also from (1.5) and by induction we have that

$q_{n}(x)\geq\left\lfloor\frac{n}{2}\right\rfloor\theta^{2}.$ (1.10)

Finally, from (1.9) and (1.10) it follows that
$[a_{1}(x)\theta,a_{2}(x)\theta,\ldots,a_{n}(x)\theta]\to x$ as $n\to\infty$.
Relation (1.9) means that the degree of accuracy of this approximation depends
on the growth rate of partial quotients.

In the case of RCFs, Borel [3] and Bernstein [1] gave a result called Borel-
Bernstein theorem or “$0-1$” law on describing the growth rate of partial
quotients in the sense of Lebesgue measure. Our first result, Theorem 3.1 is
an analogy of Borel-Bernstein theorem for $\theta$-expansions. We also show in
Section 5 that the Borel-Bernstein type theorem plays an important role in the
case of $\theta$-expansions.

In Sections 4 and 5 we state some results concerning extreme value theory for
$\theta$-expansions. These results are new in the sense that they appear not
to have been stated elsewhere before.

Extreme value theory for RCF digits first emerged in the 1970’s. The results
of Galambos [6, 7] concerning the maximal RCF digit have been improved by
Philipp [9] to give a complete answer to a conjecture of Erdös.

In Section 4 we derive a Fréchet law concerning the partial maxima of the
growth rate of the digit sequence. Theorems 4.5 and 4.6 extend previous work
of Galambos [6] and Philipp [9] on the asymptotic behavior of the largest
digit of RCF-expansions. To get these results we mention that we need the
$\theta$-continued fraction mixing property, and we also need a condition on
the speed of the convergence of mixing.

In Section 5 we give some iterated logarithm results (Theorem 5.2 and
Corollary 5.4) for the largest digit of $\theta$-expansions.

##  2 Preliminaries

Let us fix $\theta^{2}=1/m$, $m\in\mathbb{N}_{+}$. Putting
$\mathbb{N}_{m}:=\\{m,m+1,\ldots\\}$, $m\in\mathbb{N}_{+}$, the partial
quotients $a_{n}$, $n\in\mathbb{N}_{+}$, take positive integer values in
$\mathbb{N}_{m}$.

We now introduce a partition of the interval $[0,\theta]$ which is natural to
the $\theta$-expansions. Such a partition is generated by the cylinders of
rank $n$. For any $n\in\mathbb{N}_{+}$ and
$i^{(n)}=(i_{1},\ldots,i_{n})\in\mathbb{N}_{m}^{n}$, define the $n$-th
cylinder of $\theta$-expansion by

$C\left(i^{(n)}\right)=\\{x\in\Omega:a_{k}(x)=i_{k}\mbox{ for
}k=1,\ldots,n\\},$

where $C\left(i^{(0)}\right)=[0,\theta]$. For any $i\in\mathbb{N}_{m}$, we
have

$C\left(i\right)=\left\\{x\in\Omega:a_{1}(x)=i\right\\}=\left(\frac{1}{(i+1)\theta},\frac{1}{i\theta}\right).$
(2.1)

If $n\in\mathbb{N}_{+}$ and $i_{n}\in\mathbb{N}_{m}$, we will write

$C(a_{1},\ldots,a_{n})=C\left(i^{(n)}\right).$

Next we recall some known results for later use. From the definition of
$T_{\theta}$ and (1.7) we have for any $n\in\mathbb{N}_{+}$ and
$(a_{1},\ldots,a_{n})\in\mathbb{N}_{m}^{n}$,

$C(a_{1},\ldots,a_{n})=\left\\{\begin{array}[]{lll}\left[\displaystyle\frac{p_{n}}{q_{n}},\frac{p_{n}+\theta
p_{n-1}}{q_{n}+\theta q_{n-1}}\right)&\quad\mbox{if $n$ is even},\\\ \\\
\left(\displaystyle\frac{p_{n}+\theta p_{n-1}}{q_{n}+\theta
q_{n-1}},\frac{p_{n}}{q_{n}}\right]&\quad\mbox{if $n$ is odd}.\\\
\end{array}\right.$ (2.2)

Using (1.6) we get

$\displaystyle\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n}\right)\right)=\frac{1}{\theta}\left|\frac{p_{n}}{q_{n}}-\frac{p_{n}+\theta
p_{n-1}}{q_{n}+\theta q_{n-1}}\right|=\frac{1}{q_{n}(q_{n}+\theta
q_{n-1})}=\frac{1}{q^{2}_{n}(1+\theta s_{n})},$ (2.3)

where $s_{n}=\frac{q_{n-1}}{q_{n}}$, $n\in\mathbb{N}_{+}$ and $s_{0}=0$. Since
$s_{n}\in[0,\theta]$, it follows from (2.3) that

$\frac{1}{2q^{2}_{n}}\leq\frac{1}{(1+\theta^{2})q^{2}_{n}}\leq\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n}\right)\right)\leq\frac{1}{q^{2}_{n}}.$
(2.4)

It is of interest to calculate the approximate proportion of the $n$-th level
cylinder set $C\left(a_{1},\ldots,a_{n}\right)$ that is occupied by each of
the $(n+1)$-th level cylinder sets $C\left(a_{1},\ldots,a_{n},k\right)$.
Notice that the endpoints of the interval $C\left(a_{1},\ldots,a_{n},k\right)$
are $\frac{p_{n+1}}{q_{n+1}}$ and $\frac{p_{n+1}+\theta p_{n}}{q_{n+1}+\theta
q_{n}}$ with $p_{n+1}=k\theta p_{n}+p_{n-1}$ and $q_{n+1}=k\theta
q_{n}+q_{n-1}$. So we obtain

$\frac{p_{n+1}}{q_{n+1}}=\frac{k\theta p_{n}+p_{n-1}}{k\theta
q_{n}+q_{n-1}},\quad\frac{p_{n+1}+\theta p_{n}}{q_{n+1}+\theta
q_{n}}=\frac{(k+1)\theta p_{n}+p_{n-1}}{(k+1)\theta q_{n}+q_{n-1}}.$

Directly computation yields that

$\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n},k\right)\right)=\frac{1}{(k\theta
q_{n}+q_{n-1})((k+1)\theta
q_{n}+q_{n-1})}=\frac{1}{k^{2}q^{2}_{n}\left(\theta+\displaystyle\frac{s_{n}}{k}\right)\left(\left(1+\displaystyle\frac{1}{k}\right)\theta+\displaystyle\frac{s_{n}}{k}\right)}.$

By (2.3) it follows that

$\displaystyle\frac{\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n},k\right)\right)}{\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n}\right)\right)}$
$\displaystyle=$ $\displaystyle\frac{q^{2}_{n}(1+\theta
s_{n})}{k^{2}q^{2}_{n}\left(\theta+\displaystyle\frac{s_{n}}{k}\right)\left(\left(1+\displaystyle\frac{1}{k}\right)\theta+\displaystyle\frac{s_{n}}{k}\right)}$
$\displaystyle=$ $\displaystyle\frac{1+\theta
s_{n}}{k^{2}\left(\theta+\displaystyle\frac{s_{n}}{k}\right)\left(\left(1+\displaystyle\frac{1}{k}\right)\theta+\displaystyle\frac{s_{n}}{k}\right)}.$

Since

$\theta^{2}<\left(\theta+\displaystyle\frac{s_{n}}{k}\right)\left(\left(1+\displaystyle\frac{1}{k}\right)\theta+\displaystyle\frac{s_{n}}{k}\right)<\theta^{2}\left(1+\displaystyle\frac{1}{k}\right)\left(1+\displaystyle\frac{2}{k}\right)<6\theta^{2}<6,$

for $k\geq m$, we find that

$\frac{1}{6k^{2}}<\frac{\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n},k\right)\right)}{\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n}\right)\right)}<\frac{1+\theta^{2}}{k^{2}\theta^{2}}=\frac{m+1}{k^{2}}.$
(2.5)

Next, we give some lemmas for later use.

######  Lemma 2.1.

Let $k\geq m$, then

$\frac{1}{6k^{2}}<\lambda_{\theta}\left(\bigcup_{a_{1},\ldots,a_{n}\geq
m}C\left(a_{1},\ldots,a_{n},k\right)\right)<\frac{m+1}{k^{2}}.$

###### Proof.

Using (2.5), since $\displaystyle\sum_{a_{1},\ldots,a_{n}\geq
m}\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{n}\right)\right)=1$, the proof
is completed. ∎

We recall the following well-known and extremely useful result.

######  Lemma 2.2 (Borel-Cantelli).

Let $(X,{\cal X},\mu)$ be a measurable space. Let $\\{C_{n}\\}_{n\geq 1}$ be a
sequence of ${\cal X}$-measurable sets and define the $lim-sup$ set

$C_{\infty}=\limsup_{n\to\infty}C_{n}=\bigcap_{n\geq 1}\bigcup_{m\geq
n}C_{m}=\\{x\in X:x\in C_{n}\mbox{ for infinitely many
}n\in\mathbb{N}_{+}\\}.$

Then, if $\displaystyle\sum_{n\geq 1}\mu(C_{n})<\infty$, we have that
$\mu(C_{\infty})=0$.

##  3 A Borel-Bernstein-type theorem

Our first main result is the following theorem.

######  Theorem 3.1 (Borel-Bernstein-type theorem).

Let $\varphi:\mathbb{N}_{+}\to(0,+\infty)$ be a function and

$A_{\varphi}=\\{x\in\Omega:a_{n}(x)>\varphi(n)\,\mbox{ for infinitely many
}\,n\in\mathbb{N}_{+}\\}.$

Then we have

$\lambda_{\theta}(A_{\varphi})=\left\\{\begin{array}[]{lll}0&\quad\mbox{if
}\displaystyle\sum_{n\geq 1}\frac{1}{\varphi(n)}<\infty,\\\ \\\
1&\quad\mbox{if }\displaystyle\sum_{n\geq 1}\frac{1}{\varphi(n)}=\infty.\\\
\end{array}\right.$

###### Proof.

Let $A_{n}=\\{x\in\Omega:a_{n}(x)>\varphi(n)\\}$, thus
$A_{\varphi}=\displaystyle\limsup_{n\to\infty}A_{n}=\displaystyle\bigcap_{j\geq
1}\displaystyle\bigcup_{n\geq j}A_{n}$. By Lemma 2.1, one has

$\lambda_{\theta}(A_{n})=\lambda_{\theta}\left(\bigcup_{a_{1},\ldots,a_{n-1}\geq
m}\,\bigcup_{k>\varphi(n)}C\left(a_{1},\ldots,a_{n-1},k\right)\right)<\sum_{k\geq\left\lfloor\varphi(n)\right\rfloor+1}\frac{m+1}{k^{2}}<\frac{m+1}{\left\lfloor\varphi(n)\right\rfloor}<\frac{2(m+1)}{\varphi(n)}.$

Thus the convergence of Borel-Cantelli lemma enables us to conclude that
$\lambda_{\theta}(A_{\varphi})=0$ when $\displaystyle\sum_{n\geq
1}\frac{1}{\varphi(n)}<\infty$.

Suppose now $\displaystyle\sum_{n\geq 1}\frac{1}{\varphi(n)}=\infty$. Notice
that

$\lambda_{\theta}(A_{\varphi})=\lambda_{\theta}\left(\bigcap_{j\geq
1}\,\bigcup_{n\geq
j}A_{n}\right)=1\iff\lambda_{\theta}(A^{c}_{\varphi})=\lambda_{\theta}\left(\bigcup_{j\geq
1}\,\bigcap_{n\geq j}A^{c}_{n}\right)=0.$

Thus we need only to show $\lambda_{\theta}\left(\displaystyle\bigcap_{n\geq
j}A^{c}_{n}\right)=0$, where

$A^{c}_{n}=\\{x\in\Omega:a_{n}(x)\leq\varphi(n)\\}.$

Let

$B_{j,\ell}=\bigcap_{j<n\leq j+\ell}A^{c}_{n}.$

Then

$\lambda_{\theta}\left(\bigcap_{n\geq
j+1}A^{c}_{n}\right)=\lim_{\ell\to\infty}\lambda_{\theta}(B_{j,\ell}).$

By the definition of $B_{j,\ell}$, we have

$B_{j,1}=\bigcup_{a_{1},\ldots,a_{j}\geq m}\,\bigcup_{m\leq
k\leq\varphi(j+1)}C\left(a_{1},\ldots,a_{j},k\right)=\bigcup_{a_{1},\ldots,a_{j}\geq
m}\left(C(a_{1},\ldots,a_{j})\setminus\bigcup_{k>\varphi(j+1)}C\left(a_{1},\ldots,a_{j},k\right)\right).$

By Lemma 2.1, we obtain that

$\sum_{k\geq\left\lfloor\varphi(j+1)\right\rfloor+1}\frac{\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{j},k\right)\right)}{\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{j}\right)\right)}>\sum_{k\geq\left\lfloor\varphi(j+1)\right\rfloor+1}\frac{1}{6k^{2}}>\sum_{k\geq\left\lfloor\varphi(j+1)\right\rfloor+1}\frac{1}{6k(k+1)}>\frac{1}{6(\varphi(j+1)+1)}.$

Hence

$\lambda_{\theta}(B_{j,1})\leq\sum_{a_{1},\ldots,a_{j}\geq
m}\lambda_{\theta}\left(C\left(a_{1},\ldots,a_{j}\right)\right)\cdot\left(1-\frac{1}{6(\varphi(j+1)+1)}\right)=1-\frac{1}{6(\varphi(j+1)+1)}.$

Since

$\displaystyle B_{j,\ell+1}$ $\displaystyle=$
$\displaystyle\\{x\in\Omega:a_{j+1}\leq\varphi(j+1),\ldots,a_{j+\ell+1}\leq\varphi(j+\ell+1)\\}$
$\displaystyle=$ $\displaystyle\\{x\in
B_{j,\ell}:a_{j+\ell+1}\leq\varphi(j+\ell+1)\\},$
$\lambda_{\theta}(B_{j,\ell+1})\leq\left(1-\frac{1}{6(\varphi(j+\ell+1)+1)}\right)\lambda_{\theta}(B_{j,\ell}).$

By induction,

$\lambda_{\theta}(B_{j,\ell})\leq\prod^{\ell}_{i=1}\left(1-\frac{1}{6(\varphi(j+i)+1)}\right)\leq\prod^{\ell}_{i=1}\left(1-\frac{1}{12\varphi(j+i)}\right)\leq\exp\left(-\sum_{i=1}^{\ell}\frac{1}{12\varphi(j+i)}\right).$

Here we use the fact that $1-x\leq\exp(-x)$ if $x\geq 0$. One has
$\displaystyle\lim_{\ell\to\infty}\lambda_{\theta}(B_{j,\ell})=0$ by the fact
that $\displaystyle\sum_{n\geq 1}\frac{1}{\varphi(n)}=\infty$. Therefore,
$\lambda_{\theta}\left(\displaystyle A^{c}_{\varphi}\right)=0$, which complete
the proof. ∎

######  Corollary 3.2.

For $\lambda_{\theta}$\- a.e. $x\in[0,\theta]$, we have that

$a_{n}(x)>n\log n\,\mbox{ for infinitely many }\,n\in\mathbb{N}_{+},$

whereas for every $\varepsilon>0$, we have that

$a_{n}(x)<n(\log n)^{1+\varepsilon}\,\mbox{ for all sufficiently large
}\,n\in\mathbb{N}_{+}.$

###### Proof.

This follows from Theorem 3.1 immediately on the observation that

$\sum_{n\geq 1}\frac{1}{n\log n}=\infty\,\mbox{ and }\sum_{n\geq
1}\frac{1}{n(\log n)^{1+\varepsilon}}<\infty$

for all $\varepsilon>0$. ∎

##  4 The asymptotic behavior of the largest digit in $\theta$-expansions

In the sequel we shall use the fundamental facts of the metric theory of
$\theta$-expansions. One of these facts is that the stochastic process arising
from $\theta$-expansion digits has the $\psi$-mixing property.

######  Definition 4.1 ($\psi$-mixing).

Let $(X,\cal{X},\mu)$ denote a probability space and let
$\xi_{j}:X\to\mathbb{R}$ denote a stationary sequence of random variables. For
any $k\in\mathbb{N}_{+}$ let
$\mathcal{B}^{k}_{1}=\sigma(\xi_{1},\ldots,\xi_{k})$ and
$\mathcal{B}_{k}^{\infty}=\sigma(\xi_{k},\xi_{k+1},\ldots)$ denote the
$\sigma$-algebras generated by the random variables $\xi_{1},\ldots,\xi_{k}$,
respectively $\xi_{k},\xi_{k+1},\ldots$. Then $\\{\xi_{j}\\}$ is said to be
$\psi$-mixing if for any sets $A\in\mathcal{B}^{k}_{1}$ and
$B\in\mathcal{B}_{k+n}^{\infty}$ we have

$\left|\mu(A\cap B)-\mu(A)\mu(B)\right|\leq\psi(n)\mu(A)\mu(B),$

where $\psi:\mathbb{N}_{+}\to\mathbb{R}$ is a function for which $\psi(n)\to
0$ as $n\to\infty$.

The random variables $a_{n}(x)$, $n\in\mathbb{N}_{+}$, form a stationary
sequence due to the invariance of the measure $\gamma_{\theta}$ with respect
to $T_{\theta}$.

######  Lemma 4.2.

For all $n\in\mathbb{N}_{+}$ and $w\in\mathbb{N}_{m}$,

$\gamma_{\theta}(a_{n}(x)\geq
w)=\frac{1}{\log\left(1+\theta^{2}\right)}\log\left(1+\frac{1}{w}\right)=:p_{\theta}(w).$

###### Proof.

Using (2.1) and the fact that $(a_{n})_{n\in\mathbb{N}_{+}}$ is a strictly
stationary sequence as the transformation $T_{\theta}$ is measure-preserving
with respect to $\gamma_{\theta}$, we have

$\displaystyle\gamma_{\theta}(a_{n}(x)\geq w)$ $\displaystyle=$
$\displaystyle\gamma_{\theta}(a_{1}(x)\geq
w)=\displaystyle\frac{1}{\log\left(1+\theta^{2}\right)}\displaystyle\sum_{k\geq
w}\displaystyle\int^{\frac{1}{k\theta}}_{\frac{1}{(k+1)\theta}}\frac{\theta\mathrm{d}x}{1+\theta
x}$ $\displaystyle=$
$\displaystyle\frac{1}{\log\left(1+\theta^{2}\right)}\sum_{k\geq
w}\left(\log\left(1+\frac{1}{k}\right)-\log\left(1+\frac{1}{k+1}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\log\left(1+\theta^{2}\right)}\log\left(1+\frac{1}{w}\right).$

∎

######  Lemma 4.3.

Let $\\{f_{\theta,n}(x)\\}_{n\geq 1}$ be a sequence of functions
$f_{\theta,n}\in C^{2}[0,\theta]$ defined recursively by

$f_{\theta,n+1}(x)=\sum_{i\geq m}\left(f_{\theta,n}\left(\frac{1}{\theta
i}\right)-f_{\theta,n}\left(\frac{1}{x+\theta i}\right)\right),\quad
n\in\mathbb{N}$

with $f_{\theta,0}(0)=0$ and $f_{\theta,0}(\theta)=1$.

Set

$g_{\theta,n}(x)=(\theta x+1)f^{\prime}_{\theta,n}(x),\,\,x\in[0,\theta].$
(4.1)

Then

$\left\|g^{\prime}_{\theta,n}\right\|\leq
q^{n}_{\theta}\cdot\left\|g^{\prime}_{\theta,0}\right\|,\,n\in\mathbb{N}_{+}$
(4.2)

with

$q_{\theta}:=m\left(\sum_{i\geq
m}\left(\frac{m}{i^{3}(i+1)}+\frac{i+1-m}{i(i+1)^{3}}\right)\right)<1.$ (4.3)

Here $\left\|\cdot\right\|$ stands for the supremum norm.

###### Proof.

Since

$g_{\theta,n+1}(x)=\sum_{i\geq
m}P_{\theta,i}(x)g_{\theta,n}\left(u_{\theta,i}(x)\right),$

where

$P_{\theta,i}(x):=\frac{\theta x+1}{(x+\theta
i)(x+\theta(i+1))}=\frac{1}{\theta}\left[\frac{1-\theta^{2}i}{x+\theta
i}-\frac{1-\theta^{2}(i+1)}{x+\theta(i+1)}\right]$

and

$u_{\theta,i}(x):=\frac{1}{x+\theta i}$

we have

$g^{\prime}_{\theta,n+1}(x)=\sum_{i\geq m}\frac{1-\theta^{2}(i+1)}{(x+\theta
i)(x+\theta(i+1))^{3}}f^{\prime}_{\theta,n}(\alpha_{\theta,i})-\sum_{i\geq
m}\frac{P_{\theta,i}(x)}{(x+\theta
i)^{2}}f^{\prime}_{\theta,n}(u_{\theta,i}(x)),$

where $u_{\theta,i+1}(x)<\alpha_{\theta,i}<u_{\theta,i}(x)$. Then

$\left\|g^{\prime}_{\theta,n+1}\right\|\leq\left\|g^{\prime}_{\theta,n}\right\|\cdot\max_{x\in[0,\theta]}\left(\sum_{i\geq
m}\frac{\theta^{2}(i+1)-1}{(x+\theta i)(x+\theta(i+1))^{3}}+\sum_{i\geq
m}\frac{P_{\theta,i}(x)}{(x+\theta i)^{2}}\right).$

Using that

$\frac{\theta^{2}(i+1)-1}{(x+\theta i)(x+\theta(i+1))^{3}}\leq
m^{2}\frac{\theta^{2}(i+1)-1}{i(i+1)^{3}}$

and

$\sum_{i\geq m}\frac{P_{\theta,i}(x)}{(x+\theta i)^{2}}\leq m^{2}\sum_{i\geq
m}\frac{1}{i^{3}(i+1)},$

then we get (4.2) and (4.3). ∎

The random variables $a_{n}(x)$, $n\in\mathbb{N}_{+}$, are not independent.
However, they satisfy a $\psi$-mixing condition. Actually, the sequence
$\\{a_{n}\\}_{n\in\mathbb{N}_{+}}$ is $\psi$-mixing under $\gamma_{\theta}$
and the function $\psi$ vanishes at an exponential rate.

######  Lemma 4.4.

For any sets $A\in\mathcal{B}^{k}_{1}=\sigma(a_{1},\ldots,a_{k})$ and
$B\in\mathcal{B}_{k+n}^{\infty}=\sigma(a_{k+n},a_{k+n+1},\ldots)$ we have

$\left|\gamma_{\theta}(A\cap
B)-\gamma_{\theta}(A)\gamma_{\theta}(B)\right|\leq
K_{\theta}q^{n}_{\theta}\gamma_{\theta}(A)\gamma_{\theta}(B),$ (4.4)

where $0<q_{\theta}<1$ and $K_{\theta}$ is a positive constant.

###### Proof.

Let $C_{k}$ be the $k$-th cylinder with endpoints
$\displaystyle\frac{p_{k}}{q_{k}}$ and $\displaystyle\frac{p_{k}+\theta
p_{k-1}}{q_{k}+\theta q_{k-1}}$. Let

$f_{\theta,n}(x)=\gamma_{\theta}\left(T^{n+k}_{\theta}(\omega)<x\,|\,C_{k}\right):=\frac{\gamma_{\theta}\left(\left(T^{n+k}_{\theta}(\omega)<x\right)\cap
C_{k}\right)}{\gamma_{\theta}(C_{k})}$

be the conditional distribution function of $T^{n+k}_{\theta}(\omega)$ given
$C_{k}$. Obviously $\left(\left(T^{k}_{\theta}(\omega)<x\right)\cap
C_{k}\right)$ is an interval with endpoints $\displaystyle\frac{p_{k}}{q_{k}}$
and $\displaystyle\frac{p_{k}+x\theta p_{k-1}}{q_{k}+x\theta q_{k-1}}$. Thus
we obtain

$f_{\theta,0}(x)=\frac{1}{\gamma_{\theta}(C_{k})}\frac{(-1)^{k}}{\log\left(1+\theta^{2}\right)}\left(\log\left(1+\theta\frac{p_{k}+x\theta
p_{k-1}}{q_{k}+x\theta
q_{k-1}}\right)-\log\left(1+\theta\frac{p_{k}}{q_{k}}\right)\right).$

If $g_{\theta,n}$ is defined as in Lemma 4.3 let us put

$K_{\theta}:=\sup_{x\in[0,\theta]}\left|g^{\prime}_{\theta,0}(x)\right|=\left\|g^{\prime}_{\theta,0}\right\|.$

Hence by (4.1) and (4.2)

$\left|f^{\prime}_{\theta,n}(x)-\frac{\theta}{(\theta
x+1)\log\left(1+\theta^{2}\right)}\right|\leq\frac{\left|g_{\theta,n}(x)-g_{\theta,n}(0)\right|}{\theta
x+1}+\frac{\left|g_{\theta,n}(0)-\frac{\theta}{\log\left(1+\theta^{2}\right)}\right|}{\theta
x+1}$ (4.5)

and

$\left|g_{\theta,n}(x)-g_{\theta,n}(0)\right|=\left|\int_{0}^{x}g^{\prime}_{\theta,n}(t)\mathrm{d}t\right|\leq\left\|g^{\prime}_{\theta,n}\right\|\cdot
x\leq K_{\theta}q^{n}_{\theta}x.$

Also, for some $0<v_{\theta}<1$

$\displaystyle 1$ $\displaystyle=$ $\displaystyle
f_{\theta,n}(\theta)=\int_{0}^{\theta}f^{\prime}_{\theta,n}(t)\mathrm{d}t=\int_{0}^{\theta}\frac{g_{\theta,n}(t)}{\theta
t+1}\mathrm{d}t$ $\displaystyle=$ $\displaystyle
g_{\theta,n}(0)\frac{\log\left(1+\theta^{2}\right)}{\theta}+\int_{0}^{\theta}\frac{g_{\theta,n}(t)-g_{\theta,n}(0)}{\theta
t+1}\mathrm{d}t$ $\displaystyle=$ $\displaystyle
g_{\theta,n}(0)\frac{\log\left(1+\theta^{2}\right)}{\theta}+v_{\theta}K_{\theta}q^{n}_{\theta}\left(1-\frac{\log\left(1+\theta^{2}\right)}{\theta^{2}}\right)$

and so

$g_{\theta,n}(0)=\frac{\theta}{\log\left(1+\theta^{2}\right)}+v_{\theta}\frac{K_{\theta}q^{n}_{\theta}}{\theta}\left(1-\frac{\theta^{2}}{\log\left(1+\theta^{2}\right)}\right).$

Thus, from (4.5)

$\displaystyle\left|f^{\prime}_{\theta,n}(x)-\frac{\theta}{(\theta
x+1)\log\left(1+\theta^{2}\right)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{K_{\theta}q^{n}_{\theta}}{\theta
x+1}+\frac{K_{\theta}q^{n}_{\theta}}{\theta}\left(\frac{\theta^{2}}{\log\left(1+\theta^{2}\right)}-1\right)\frac{1}{\theta
x+1}$ (4.6) $\displaystyle<$
$\displaystyle\frac{K_{\theta}q^{n}_{\theta}}{\theta
x+1}\frac{\theta}{\log\left(1+\theta^{2}\right)}.$ (4.7)

Integrating (4.6) over $F$ we obtain

$\left|\gamma_{\theta}\left(T^{-(n+k)}_{\theta}(F)\,\left|\,C_{k}\,\right.\right)-\gamma_{\theta}(F)\right|\leq
K_{\theta}q^{n}_{\theta}\gamma_{\theta}(F).$

Since each $A\in\mathcal{B}^{k}_{1}$ is a countable union of disjoint $C_{k}$
we obtain (4.4) and thus the proof is complete.

∎

Define

$L_{N}:=\max_{1\leq n\leq N}a_{n}(x),\quad x\in\Omega.$

In the sequel we discuss the asymptotic behavior of the largest digit $L_{N}$.

######  Theorem 4.5.

For any $y>0$, we have

$\lim_{N\to\infty}\gamma_{\theta}\left(x\in\Omega:L_{N}(x)<\frac{Ny}{\log\left(1+\theta^{2}\right)}\right)=\exp\left(-\frac{1}{y}\right).$
(4.8)

###### Proof.

$1$st step. Let

$A_{n}=\\{x\in\Omega:a_{n}(x)\geq w\\},$

which means

$\bigcap^{N}_{n=1}A^{C}_{n}=\\{x\in\Omega:L_{N}(x)<w\\}=:B_{N}.$

Given that $B_{N}$ represents the event where none of the $A_{n}$ occurs, the
Poincaré identity reveals that

$\gamma_{\theta}(B_{N})=\sum_{k=0}^{N}(-1)^{k}S_{k}$ (4.9)

with

$S_{0}=1,\,S_{k}=\sum_{1\leq n_{1}<n_{2}<\ldots<n_{k}\leq
N}\gamma_{\theta}\left(A_{n_{1}}\cap\ldots\cap A_{n_{k}}\right).$

Thus, equation (4.9) provides an expression for the distribution function of
$L_{N}$. By selecting
$w=\left\lfloor\frac{Ny}{\log(1+\theta^{2})}\right\rfloor$, we demonstrate
that the tail $\displaystyle\sum_{k\geq Z}S_{k}$, where $Z$ is a sufficiently
large but fixed value, can be made arbitrarily small.

By repeatedly applying equation (4.4) and referring to Lemma 4.2, we obtain
that

$\gamma_{\theta}\left(A_{n_{1}}\cap\ldots\cap
A_{n_{k}}\right)\leq(1+K_{\theta})^{k-1}\gamma_{\theta}(A_{n_{1}})\gamma_{\theta}(A_{n_{2}})\cdot\ldots\cdot\gamma_{\theta}(A_{n_{k}})<(1+K_{\theta})^{k}p^{k}_{\theta}(w).$
(4.10)

For sufficiently large values of $N$, we obtain

$w=\left\lfloor\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right\rfloor\geq\frac{1}{2}\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)},$
(4.11)

whence

$p_{\theta}(w)\leq\frac{1}{w\log\left(1+\theta^{2}\right)}\leq\frac{2}{Ny}.$

Therefore

$\displaystyle\sum_{k\geq Z}S_{k}$ $\displaystyle<$ $\displaystyle\sum_{k\geq
Z}\frac{N!}{(N-k)!k!}(1+K_{\theta})^{k}p^{k}_{\theta}(w)\leq\sum_{k\geq
Z}\frac{N!}{(N-k)!k!}N^{-k}\left(\frac{2(1+K_{\theta})}{y}\right)^{k}$ (4.12)
$\displaystyle\leq$ $\displaystyle\sum_{k\geq
Z}\frac{1}{k!}\left(\frac{2(1+K_{\theta})}{y}\right)^{k}<\frac{1}{Z!}\left(\frac{4K_{\theta}}{y}\right)^{Z}\exp\left(\frac{4K_{\theta}}{y}\right)\leq\varepsilon$

as the value of $Z$ is increased sufficiently.

$2$nd step. Let’s divide $S_{k}$ into two separate terms when considering
$k<Z$:

$S_{k}=S^{*}_{k}+R_{k}.$ (4.13)

Here, $S^{*}_{k}$ represents the sum over all $n_{1}<n_{2}<\ldots<n_{k}$ with
$n{i+1}-n_{i}\geq t$ ($i\geq 1$), where $t$ is a positive integer determined
as follows. Let $\eta>0$ be an arbitrary real number, and let $t$ be the
smallest integer $n$ such that $K_{\theta}q^{n}_{\theta}<\eta$. Next, we
proceed to estimate $S_{k}^{*}$. Using repeated applications of (4.4) and
another reference to Lemma 4.2, we find that for any term belonging to
$S^{*}_{k}$,

$\gamma_{\theta}\left(A_{n_{1}}\cap\ldots\cap
A_{n_{k}}\right)=p^{k}_{\theta}(w)\left(1+\mathcal{O}_{k}(\eta)\right),\,n_{i}+t\leq
n_{i+1}.$

In the estimation of $S^{*}_{k}$, the constant involved in
$\mathcal{O}_{k}(\eta)$ depends exclusively on $k$. Hence

$S^{*}_{k}=\frac{(N-(t-1)(k-1))!}{(N-(t-1)(k-1)-k)!k!}p^{k}_{\theta}(w)\left(1+\mathcal{O}_{k}(\eta)\right).$
(4.14)

In order to estimate $R_{k}$ in (4.13), it’s important to observe that the
overall estimation (4.10) is applicable to each of its individual terms and
that its number of terms is

$\frac{N!}{(N-k)!k!}-\frac{(N-(t-1)(k-1))!}{(N-(t-1)(k-1)-k)!k!}={o}\left(N^{k}\right).$

We thus have

$R_{k}={o}\left(N^{k}p^{k}_{\theta}(w)\right).$ (4.15)

Considering that

$p_{\theta}(w)=p_{\theta}\left(\left\lfloor\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right\rfloor\right)=\left(1+\mathcal{O}(N^{-1})\right)\frac{1}{Ny}$

by (4.13), (4.14) and (4.15) we can deduce that

$S_{k}=\left(1+\mathcal{O}_{k}(\eta)\right)\frac{y^{-k}}{k!}+o_{N}(1),$ (4.16)

where $k$ is fixed and $o_{N}(1)\to 0$ as $N\to\infty$.

$3$rd step. Finally, by (4.9), (4.12) and (4.16), we establish that for any
given positive integer $Z$

$\gamma_{\theta}\left(L_{N}<\left\lfloor\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right\rfloor\right)=\sum_{k=0}^{Z-1}(-1)^{k}\left(1+\mathcal{O}_{k}(\eta)\right)\frac{y^{-k}}{k!}+o_{N}(1)+o_{Z}(1),$

the last term approaches $0$ as $Z\to\infty$. Letting $N\to\infty$, and then
$\eta\to 0$, we deduce that for any positive integer $Z$

$\lim_{N\to\infty}\gamma_{\theta}\left(L_{N}<\left\lfloor\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right\rfloor\right)=\sum_{k=0}^{Z-1}(-1)^{k}\frac{y^{-k}}{k!}+o_{Z}(1).$

Given that the left-hand side remains independent of $Z$, letting
$Z\to\infty$, we achieve the limit relation (4.8) while taking into account
that the argument $w$ in $\\{L_{N}<w\\}$ is considered to be an integer in
$\mathbb{N}_{m}$. Since

$\gamma_{\theta}\left(L_{N}<\left\lfloor\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right\rfloor\right)\leq\gamma_{\theta}\left(L_{N}<\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right)\leq\gamma_{\theta}\left(L_{N}<\left\lfloor\displaystyle\frac{Ny}{\log\left(1+\theta^{2}\right)}\right\rfloor+1\right)$

the proof is complete. ∎

######  Theorem 4.6.

For any $0<\delta<1$ and $y>0$, we have

$\gamma_{\theta}\left(x\in\Omega:L_{N}(x)<\frac{Ny}{\log\left(1+\theta^{2}\right)}\right)=\exp\left(-\frac{1}{y}\right)+\mathcal{O}\left(\exp\left(-(\log
N)\right)^{\delta}\right)$ (4.17)

where the constant involved in $\mathcal{O}$ depends exclusively on $\delta$.

###### Proof.

We follow the proof of Theorem 4.5 with a particular choice of $Z$ and $t$. We
choose $Z=\displaystyle\left\lfloor\frac{\log N}{\log\log N}\right\rfloor$.
For a specific $0<\delta<1$, we choose $\delta<\delta^{\prime}<1$,
$\varepsilon>0$ and $\zeta>0$ so that $1-\delta^{\prime}>\varepsilon+\zeta$.
We assume that $y\geq(\log N)^{-\delta}$. Applying Stirling’s formula, we
derive

$\frac{1}{Z!}~{}\frac{1}{\sqrt{2\pi}\,Z^{Z+1/2}\exp(-Z)}\asymp\frac{\exp\left(\frac{\log
N}{\log\log N}\right)}{\left(\frac{\log N}{\log\log N}\right)^{\frac{\log
N}{\log\log N}+\frac{1}{2}}}.$

For $N$ sufficiently large,

$e\leq\left(\frac{\log N}{\log\log N}\right)^{\zeta}.$

Thus, we obtain

$\exp\left(\frac{\log N}{\log\log N}\right)\leq\left(\left(\frac{\log
N}{\log\log N}\right)^{\zeta}\right)^{\frac{\log N}{\log\log N}}<N^{\zeta}.$

Furthermore, for $N$ sufficiently large

$\left(\frac{\log N}{\log\log N}\right)^{\displaystyle\frac{\log N}{\log\log
N}}>N^{1-\varepsilon}.$

As a result, we obtain

$\frac{1}{Z!}\ll\frac{1}{N^{1-\varepsilon-\zeta}}.$ (4.18)

Alternatively, it is obvious that

$\left(\frac{4K_{\theta}}{y}\right)^{Z}\exp\left(\frac{4K_{\theta}}{y}\right)\leq\left(4K_{\theta}\left(\log
N\right)^{\delta}\right)^{\displaystyle\frac{\log N}{\log\log
N}}\exp\left(4K_{\theta}\left(\log
N\right)^{\delta}\right)<N^{\delta^{\prime}}$

when $N$ is sufficiently large.. Finally, using (4.12), we obtain

$\sum_{k\geq Z}S_{k}\ll N^{-a}$ (4.19)

for $0<a<1-\varepsilon-\zeta-\delta^{\prime}$.

Setting $t=\left\lfloor(\log N)^{2}\right\rfloor$, we estimate $R_{k}$ for
$k<Z$

$\displaystyle R_{k}$ $\displaystyle\leq$
$\displaystyle\left(\frac{N!}{(N-k)!k!}-\frac{(N-(t-1)(k-1))!}{(N-(t-1)(k-1)-k)!k!}\right)(1+K_{\theta})^{k}p^{k}_{\theta}(w)\ll
t\,Z\,N^{k-1}(2K_{\theta})^{k}p^{k}_{\theta}(w)$ $\displaystyle\ll$
$\displaystyle(\log N)^{3}\frac{1}{\log\log
N}N^{k-1}(2K_{\theta})^{k}p^{k}_{\theta}(w)\ll(\log
N)^{3}N^{k-1}\left(\frac{4K_{\theta}}{Ny}\right)^{k}.$

In cases where $\displaystyle\frac{4K_{\theta}}{y}>1$, we proceed to evaluate,
for $N$ sufficiently large,

$R_{k}\leq\displaystyle\frac{1}{N^{1-\varepsilon}}\left(\displaystyle\frac{4K_{\theta}}{y}\right)^{Z}\leq\frac{1}{N^{1-\varepsilon}}\left(4K_{\theta}(\log
N)^{\delta}\right)^{\displaystyle\frac{\log N}{\log\log N}}<N^{-a}$ (4.20)

for $0<a<1-\varepsilon-\delta$. When $\displaystyle\frac{4K_{\theta}}{y}<1$,
the estimation becomes relatively straightforward. We can select the value of
$a$ to be the same as that in equation (4.19).

As a result, the number of terms in $S^{*}_{k}$, $k<Z$, is given by
$\displaystyle\frac{N!}{(N-k)!k!}+\mathcal{O}\left(N^{k-1}(\log
N)^{3}\right)$. We have

$\displaystyle S^{*}_{k}$ $\displaystyle=$
$\displaystyle\left(\displaystyle\frac{N!}{(N-k)!k!}+\mathcal{O}\left(N^{k-1}(\log
N)^{3}\right)\right)\left(1+\mathcal{O}\left(N^{-1}\right)\right)^{k}\left(Ny\right)^{-k}\left(1+\beta
K_{\theta}q^{(\log N)^{2}}_{\theta}\right)^{k}$ (4.21) $\displaystyle=$
$\displaystyle\displaystyle\frac{y^{-k}}{k!}+\mathcal{O}\left(N^{-a}\right)$

where $|\beta|\leq 1$. Subsequently, using (4.20) and (4.21), we deduce that
$S_{k}=\displaystyle\frac{y^{-k}}{k!}+\mathcal{O}\left(N^{-a}\right)$. In
conclusion, using (4.19), we obtain

$\gamma_{\theta}(B_{N})=\sum_{k=0}^{Z-1}\left((-1)^{k}\frac{y^{-k}}{k!}+\mathcal{O}\left(N^{-a}\right)\right)+\mathcal{O}\left(N^{-a}\right)=\sum_{k=0}^{Z-1}(-1)^{k}\frac{y^{-k}}{k!}+\mathcal{O}\left(N^{-a^{\prime}}\right)=\exp\left(-\frac{1}{y}\right)+\mathcal{O}\left(N^{-a^{\prime}}\right),$

with $0<a^{\prime}<a$. ∎

##  5 Some iterated logarithm results

We begin with the following quantitative Borel-Cantelli lemma.

######  Lemma 5.1.

[8] Let $\\{E_{N}\\}_{n\geq 1}$ be a sequence of measurable sets in a
probability space $(X,\cal{X},\mu)$. Denote by $A(N,x)$ the number of integers
$n\leq N$ such that $x\in E_{n}$, i.e., $A(N,x)=\displaystyle\sum_{n\leq
N}\chi_{E_{n}}(x)$, where $\chi_{E_{n}}$ is the characteristic function of
$E_{n}$. Define

$\varphi(N):=\sum_{n\leq N}\mu(E_{n}).$

Suppose that there exists a convergent series $\displaystyle\sum_{k\geq
1}c_{k}$ with $c_{k}\geq 0$ such that for all integers $n>\ell$ we have

$\mu(E_{n}\cap E_{\ell})\leq\mu(E_{n})\mu(E_{\ell})+\mu(E_{n})c_{n-\ell}.$
(5.1)

Then for any $\varepsilon>0$

$A(N,x)=\varphi(N)+\mathcal{O}\left(\varphi^{1/2}(N)\log^{3/2+\varepsilon}\varphi(N)\right)\quad\mu\mbox{-a.s.}$
(5.2)

######  Theorem 5.2.

For a.e. $x\in[0,\theta]$ we have

$\liminf_{N\to\infty}\frac{L_{N}(x)\log\log
N}{N}=\frac{1}{\log\left(1+\theta^{2}\right)}.$

###### Proof.

Since for all $A\in\mathcal{B}_{[0,\theta]}$

$\frac{\lambda_{\theta}(A)}{\left(1+\theta^{2}\right)\log\left(1+\theta^{2}\right)}\leq\gamma_{\theta}(A)\leq\frac{\lambda_{\theta}(A)}{\log\left(1+\theta^{2}\right)}$

the measures $\gamma_{\theta}$ and $\lambda_{\theta}$ are equivalent. Hence we
make the proof for all $x$ except a set of $\gamma_{\theta}$-measure $0$.
Consider integers $M$ and $N$ with $M,N\geq 0$. Define

$L(M,N,x):=\max_{M<n\leq M+N}a_{n}(x),$ $\varphi(n):=\frac{n}{\log\log
n\log\left(1+\theta^{2}\right)}$

and

$E_{k}:=\left(x\in\Omega:L\left(k^{2k},k^{2(k+1)},x\right)\leq\varphi\left(k^{2(k+1)}\right)\right).$

Due to the $T_{\theta}$-invariance of $\gamma_{\theta}$, we can deduce from
Theorem 4.6 that, for any integer $k\geq k_{0}$,

$\displaystyle\gamma_{\theta}(E_{k})$ $\displaystyle=$
$\displaystyle\gamma_{\theta}\left(x\in\Omega:L\left(k^{2k},k^{2(k+1)},x\right)\leq\varphi\left(k^{2(k+1)}\right)\right)$
(5.3) $\displaystyle=$
$\displaystyle\gamma_{\theta}\left(x\in\Omega:L\left(0,k^{2(k+1)},x\right)\leq\varphi\left(k^{2(k+1)}\right)\right)$
$\displaystyle\geq$ $\displaystyle\frac{1}{2}\exp\left(-\log\log
k^{2(k+1)}\right)\geq\frac{1}{8}(k\log k)^{-1}.$

Obviously $E_{k}$ depends only on $a_{n}(x)$ with $k^{2k}<n\leq
k^{2(k+1)}+k^{2k}$. Consequently, according to Lemma 4.4, we can establish
that for any pair of integers $k<\ell$

$\left|\gamma_{\theta}(E_{k}\cap
E_{\ell})-\gamma_{\theta}(E_{k})\gamma_{\theta}(E_{\ell})\right|\leq
K_{\theta}q^{\ell-k}_{\theta}\gamma_{\theta}(E_{k})\gamma_{\theta}(E_{\ell}),$

since $(k+1)^{2(k+1)}-k^{2(k+1)}-k^{2k}\geq 1$.

As a result, the implication from Lemma 5.1 suggests that $x\in E_{k}$ for
infinitely many $k$ (a.e. $x$) given that $\varphi(N)\gg\log\log N$ according
to (5.3).

On the other hand, by Lemma 4.2

$\displaystyle\gamma_{\theta}(F_{k})$ $\displaystyle:=$
$\displaystyle\gamma_{\theta}\left(x\in\Omega:L\left(0,k^{2k},x\right)\geq\varphi\left(k^{2(k+1)}\right)\right)$
$\displaystyle\leq$ $\displaystyle\sum_{n\leq
k^{2k}}\gamma_{\theta}\left(x\in\Omega:a_{n}(x)\geq\varphi\left(k^{2(k+1)}\right)\right)=k^{2k}p_{\theta}\left(\varphi\left(k^{2(k+1)}\right)\right)$
$\displaystyle\leq$ $\displaystyle k^{2k}\log\log k^{2(k+1)}\cdot
k^{-2(k+1)}\leq k^{-3/2}.$

Therefore, according to Lemma 5.1 $x\in F_{k}$ only for finitely many $k$
(a.e. $x$). Thus

$x\in E_{k}\setminus
F_{k}=\left(x\in\Omega:L\left(0,k^{2k}+k^{2(k+1)},x\right)\leq\varphi\left(k^{2(k+1)}\right)\right)$

for infinitely many $k$ (a.e. $x$), which implies that
$L\left(0,k^{2(k+1)},x\right)\leq\varphi\left(k^{2(k+1)}\right)$ holds for
infinitely many $k$ (a.e. $x$). Hence,

$\liminf_{N\to\infty}\frac{L_{N}(x)\log\log
N}{N}\leq\frac{1}{\log\left(1+\theta^{2}\right)}\,\,\mbox{ a.e.}$ (5.4)

Now, we proceed to prove the converse inequality. Let $b>1$. Again by Theorem
4.6

$\displaystyle\gamma_{\theta}(G_{k})$ $\displaystyle:=$
$\displaystyle\gamma_{\theta}\left(x\in\Omega:L\left(0,\left\lfloor
b^{k}\right\rfloor,x\right)\leq b^{-2}\varphi\left(\left\lfloor
b^{k+1}\right\rfloor\right)\right)$ $\displaystyle\ll$
$\displaystyle\exp\left(-b\log\log b^{k}\right)\ll k^{-b}.$

By Lemma 5.1, since $\sum k^{-b}<\infty$, it follows that $x\in G_{k}$ only
for finitely many $k$ (a.e. $x$), which means that

$L\left(0,\left\lfloor
b^{k}\right\rfloor,x\right)>b^{-2}\varphi\left(\left\lfloor
b^{k+1}\right\rfloor\right)$

holds for all $k\geq k_{0}(x,b)$. For a given value of $N$ such that
$\left\lfloor b^{k}\right\rfloor\leq N<b^{k+1}$ where $k\geq k_{0}(x,b)$ since
$L\left(0,\left\lfloor b^{k}\right\rfloor,x\right)\leq L_{N}(x)$ and
$\varphi(N)\leq\varphi\left(\left\lfloor b^{k+1}\right\rfloor\right)$ we
conclude that

$L_{N}(x)>b^{-2}\varphi(N)\,\mbox{ a.e. }x.$

Since this holds for any $b>1$ we obtain

$\liminf_{N\to\infty}\frac{L_{N}(x)\log\log
N}{N}\geq\frac{1}{\log\left(1+\theta^{2}\right)}\,\,\mbox{ a.e.}$

By (5.4) the proof is completed. ∎

There is no analogous result for Theorem 5.2 with a finite nonzero superior
limit. This follows from the following theorem.

######  Theorem 5.3.

Let $\\{\varphi(n)\\}_{n\geq 1}$ be a positive nondecreasing sequence. Then
for a.e. $x\in[0,\theta]$

$L_{N}(x)>\varphi(N)$ (5.5)

has finitely many or infinitely many solutions in integers $N$ according as
the series

$\sum_{n\geq 1}\frac{1}{\varphi(n)}$ (5.6)

converges or diverges.

###### Proof.

Indeed, if $\sup\varphi(n)<\infty$, then the divergence of (5.5) implies,
according to Theorem 3.1, that $a_{n}(x)>\varphi(n)$ holds for infinitely many
$n$ (a.e. $x$).

On the other hand, when $\varphi(n)\nearrow\infty$, the behavior of (5.5) is
determined by whether the inequality $a_{n}(x)>\varphi(n)$ holds finitely or
infinitely often. This, in turn, leads to the conclusion that, by Theorem 3.1,
this behavior holds for a.e. $x$ based on whether the series (5.6) converges
or diverges. ∎

######  Corollary 5.4.

Let $\\{\varphi(n)\\}_{n\geq 1}$ be as in Theorem 5.3. Then for a.e.
$x\in[0,\theta]$

$\limsup_{N\to\infty}\frac{L_{N}(x)}{\varphi(N)}$ (5.7)

is either $0$ or $\infty$.

###### Proof.

We distinguish the cases where the series (5.6) converges or diverges. If the
series (5.6) converges we choose a monotone sequence $\\{\alpha_{n}\\}_{n\geq
1}$ tending to $\infty$ but so slowly that still $\displaystyle\sum_{n\geq
1}\frac{\alpha_{n}}{\varphi(n)}<\infty$. Therefore in accordance with Theorem
5.3, the inequality $L_{N}(x)>\displaystyle\frac{\varphi(N)}{\alpha_{N}}$
holds only for finitely many $N$ (a.e. $x$). Hence (5.7) vanishes for a.e.
$x$.

If the series (5.6) diverges, we consider a monotone sequence
$\\{\alpha_{n}\\}_{n\geq 1}$ tending to $0$ such that
$\displaystyle\sum_{n\geq 1}\frac{\alpha_{n}}{\varphi(n)}=\infty$. Hence,
$L_{N}(x)>\displaystyle\frac{\varphi(N)}{\alpha_{N}}$ holds for infinitely
many $N$ (a.e. $x$) and thus (5.7) is infinite for a.e. $x$. ∎

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