††thanks: Contributed equally††thanks: Contributed equally††thanks:
Contributed equally

# Low frequency resistance fluctuations in an ionic liquid gated channel
probed by cross-correlation noise spectroscopy

Bikash C. Barik  Department of Physics, Indian Institute of Technology Bombay,
Mumbai, 400076, India  Himadri Chakraborti  Department of Physics, Indian
Institute of Technology Bombay, Mumbai, 400076, India  SPEC, CEA, CNRS,
Universite Paris-Saclay, CEA Saclay, 91191 Gif sur Yvette, Cedex France
Aditya K. Jain  Department of Physics, Indian Institute of Technology Bombay,
Mumbai, 400076, India  Department of Physics, Royal Holloway, University of
London, Egham, Surrey, TW20 0EX, UK  Buddhadeb Pal  Department of Physics,
Indian Institute of Technology Bombay, Mumbai, 400076, India  Indian
Association for the Cultivation of Science, Kolkata, 700032, India  H.E. Beere
Semiconductor Physics Group, Cavendish Laboratory, University of Cambridge,
Cambridge CB3 0HE, U.K.  D.A. Ritchie  Semiconductor Physics Group, Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, U.K.  K. Das Gupta
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Bombay, Mumbai, 400076, India

###### Abstract

A system in equilibrium keeps “exploring” nearby states in the phase space and
consequently, fluctuations can contain information, that the mean value does
not. However, such measurements involve a fairly complex interplay of effects
arising in the device and measurement electronics, that are non-trivial to
disentangle. In this paper, we briefly analyse some of these issues and show
the relevance of a two-amplifier cross-correlation technique for
semiconductors and thin films commonly encountered. We show that by using
home-built amplifiers costing less than 10 USD/piece one can measure spectral
densities as low as $\sim 10^{-18}-10^{-19}~{}{\rm{V^{2}}{Hz^{-1}}}$. We apply
this method to an ionic liquid gated Ga:ZnO channel and show that the glass
transition of the ionic liquid brings about a change in the exponent of the
low frequency resistance fluctuations. Our analysis suggests that a log-normal
distribution of the Debye relaxation times of the fluctuations and an
increased weight of the long timescale relaxations can give a semi-
quantitative explanation of the observed change in the exponent.

A key problem encountered in all measurements of “noise” is that the recorded
data always contains the noise originating from the sample/DUT (Device under
test) and the measurement circuit, in particular the first stage amplifier.
There are two generic approaches to this aspect.

Figure 1: Circuit diagram of a five probe technique to measure resistance
fluctuations through voltage fluctuations. Equivalent circuit for low noise
amplifiers (LNA) used in cross-correlation measurements. The LNAs are powered
by separate power sources. The dimension of the Hall bar is $\approx
1150\times 100{\rm\mu{m}}$. The typical value of $R_{V}\sim 250\rm k\Omega$.
The input voltage and current noise of the LNAs are shown as $e_{1,n}$,
$e_{2,n}$, $i_{1,n}$ and $i_{2,n}$. The gain of each amplifier is $G$.

  1. 1.

One ensures that the first stage amplifier’s noise is assuredly less than the
DUT. In literature [1] it is often addressed by choosing a passive small-
signal transformer (PST) as the first stage. These transformers with very high
core susceptibility [2, 3] often have noise contribution as low as $0.1{\rm
nV/\sqrt{Hz}}$ and certain more specialised designs can even be operated at
cryogenic temperatures. While these give excellent Common-mode-rejection
(CMRR), the limitation is that the input impedances these can handle are
limited (typically $\ll 1{\rm k\Omega}$). It is important to point out that
the impedance seen by the amplifier is always the “2-terminal” resistance
which includes the typical lead-resistance of a cryostat (or similar
equipment) and ohmic-contact resistance of the DUT itself. Even in cases where
the 4-terminal resistance of the DUT is very small the resistance seen by the
transformer is not necessarily small due to these inevitable contributions
appearing in series. Another aspect is that since no transformer can be truly
d.c. coupled, the very low frequency resistance fluctuations can only be
measured by modulating them with a relatively higher frequency current,
effectively translating the resulting voltage fluctuations to the most
sensitive area of the transformer’s own noise figure.

  2. 2.

Another approach is to use reasonably low noise but d.c. coupled and
relatively inexpensive (in our experiments, the chips used are all priced
below 10 USD) BJT/FET input operational amplifiers, that can handle
significantly higher input impedances but will have an intrinsic input noise
typically a factor of $\sim 10-20$ higher compared to the PST. The resistances
encountered in measurements on semiconductors [4, 5, 6, 7, 8], carbon nanotube
[9, 10], Indium arsenide nanowires [11, 12] and 2D materials [13, 14, 15, 16,
17, 18, 19, 20, 21, 22] are often in the range of $1$-$100$ ${\rm k\Omega}$.
In this method, the voltage signal is fed in parallel to two nearly identical
first-stage amplifiers powered from independent and uncorrelated sources.
Their outputs are then simultaneously digitised (in our experiments with $24$
bit resolution in $\pm 300~{}{\rm mV}$ range and at a rate of up to
$200~{}{\rm kS/sec}$), discrete Fourier transformed (DFT) after passing
through an anti-aliasing filter and then cross-correlated in the frequency
domain. In certain niche applications, like detecting shot noise [23, 24] or
thermal noise [25, 26] in mesoscopic conductors for probing quantum statistics
[24, 27, 28], the granularity of charge [29] etc., the signals are handled
through similar techniques of auto or cross-correlation. This is achieved by
employing an initial-stage cryogenic amplifier (at $\sim 4$ K) with a typical
gain of $5-7$ [30] combined with a room-temperature one. They are often
designed to operate at a few MHz. This approach ensures the primary
amplification of the signal in close proximity to the sample, preventing the
accumulation of undesired signals before being seen by the room temperature
electronics. As described [31, 32] the cross-correlation process largely (but
not entirely) eliminates the uncorrelated noise from the amplifiers
themselves. The resulting output is the spectral density of the signal that
was “common” to both the amplifiers. While this suffices for a large class of
experiments with Gaussian fluctuations, it should be borne in mind that the
original time series cannot be reconstructed.

Figure 2: (a) Shows schematic for noise measurement using a single channel
with a pre-amplification stage, contains excess noise introduced by the
preamplifier and other electronic components, (b) shows schematic for noise
measurements using cross-correlation of two channels with two independently
powered amplifier, cancelling out the excess noise from the electronic
components. (c) shows PSD recorded for various DUTs from $200\ohm$ to $100\rm
k\ohm$ (solid lines) and corresponding average noise floor (dashed line). (d)
The magnitude of the noise floor recorded for different DUTs from a single
channel and two-channel cross-correlation are shown for two different LNAs.
The black dashed line shows the expected thermal noise $4k_{B}TR$.

Our implementation of the cross correlation technique is shown in Fig. 1. The
sample itself forms two arms, assumed to have resistances $r_{1}$ and $r_{2}$,
of a Wheatstone bridge with much larger resistances $R_{v}$ and
$R^{\prime}_{v}$ forming two other arms. One of the large resistances is
adjustable and is used to null the bridge output. We have
$r_{1}\approx{r_{2}}$ and $R_{v},R_{v}^{{}^{\prime}}\gg r_{1},r_{2}$. The
output is then largely immune to small fluctuations of the biasing source
($E_{0}$) and the ambient temperature, as fluctuations arising out of these
will tend to affect $r_{1}$ and $r_{2}$ almost identically. However, a small
resistance fluctuation in one of the arms (due to internal processes like
trapping-de-trapping, thermal generation etc.) will be picked up. This signal
is then fed to both the amplifiers (each of gain $G$), as shown and their
outputs cross-correlated. The expected (cross-correlated) power spectral
density of the voltage fluctuations arising from these resistance
fluctuations, when current $I_{0}$ flows through the DUT is given by Eq.
(1)[1],

$S_{v}(f,I_{0})={G^{2}}\left[S_{v}^{0}(f)+{{I_{0}}^{2}}S^{-}_{r}(f)\right]$
(1)

where $S_{v}^{0}(f)$ is the power spectrum of the background originating from
electronic and thermal noise, $S^{-}_{r}(f)$ is the spectrum of
${\delta}r_{-}(t)={\delta}r_{1}(t)-{\delta}r_{2}(t)$ with long time average
$\left\langle{\delta}r_{1}(t){\delta}r_{2}(t)\right\rangle=0$. The choice of
the first-stage amplifier is guided by a balance between the inevitable
intrinsic input current and the current noise of the operational amplifier
itself. Fig. 1 shows the equivalent circuit with the noise equivalent sources
included as part of the input stages. It is not only the DUT voltage but the
combination (see Fig. 1) of three voltages that would get amplified. They are
related as follows :

$v_{i,\rm{rms}}^{2}=v_{\rm DUT}^{2}+e_{\rm
i,n}^{2}+\left(i_{1,\rm{n}}+i_{2,\rm{n}}\right)^{2}{R_{\rm DUT}}^{2}\ldots
i=1,2$ (2)

Ultimately this determines the practicality of the measurement. We find that a
current noise not exceeding few $fA/\sqrt{\rm Hz}$ and a voltage noise not
exceeding $\sim 5{\rm nV/\sqrt{Hz}}$ are essential for most measurements.
Choosing an amplifier with lower voltage noise but higher current noise is
detrimental once the resistance of the DUT becomes slightly higher, in which
case the noise current flowing through the DUT manifests itself as a much
higher voltage noise over-ruling the benefit of a lower voltage noise.
Although, for very low source resistances, this would be acceptable. However,
our purpose is to design an inexpensive and generic setup covering the typical
range of resistances encountered in semiconductors and similar samples whose
resistances may well go up to several tens of $\rm{k\Omega}$. This is
illustrated with two commonly available operational amplifiers in Fig. 2(d).
If no cross-correlation is used, the spectrum obtained corresponds to the two
upper curves (Uncorrelated). However, once the two data are cross-correlated
the contribution from the amplifiers is suppressed to a large extent and the
correct thermal noise (with no adjustable parameters) is obtained over more
than four orders of magnitude, spanning $10\Omega$ \- $100{\rm k\Omega}$.

Figure 3: (a) Spectrum recorded at lower frequencies for GaAs/AlGaAs
heterostructure for different current values at room temperature, (b)
Resistive noise obtained from the spectra after scaling. The scaling is done
by subtracting the zero current data from each trace and dividing by $I^{2}$
following Eq. (1). Hall data at room temperature is shown in the inset. The
slope, $\rm 3.045\ k\ohm T^{-1}$, corresponds to a two-dimensional carrier
density of $2.07\times{10^{11}}~{}{\rm cm^{-2}}$.

The typical behaviour of $S_{v}(f)$ has a well defined nearly flat high
frequency ($f\geq 100{\rm Hz}$) part and a low frequency ($0.01\leq f\leq
10{\rm Hz}$) $\propto\dfrac{1}{f^{\gamma}}$, with $\gamma\approx{1}$ (Fig.
2(c)). The correctness of the absolute values produced by the cross-
correlation technique can be judged by two observations.

  1. 1.

The $f\gtrsim 100~{}{\rm Hz}$ part of our measurements agrees very closely
with $S_{v}(f)=4{k_{B}}TR$, where $R$ is the 2-probe resistance “seen” by the
amplifier (Fig. 2(c) & (d)) is the well known Johnson-Nyquist noise [33, 34]

  2. 2.

The correctness of the low-frequency part can be established by ascertaining
the magnitude of the Hooge parameter [35, 36] $\alpha_{H}$ given by

$S_{r}(f)=\alpha_{H}\frac{r^{2}}{Nf^{\gamma}}$ (3)

Notice that this requires knowing the total number of carriers ($N$)
participating in the conduction process. We used a well-characterised 2D
electron gas (2DEG) in a GaAs-AlGaAs High electron mobility transistor (HEMT)
$100{\rm nm}$ below the surface and hence mostly free from the effect of
scattering from the charged surface states and oxide monolayers. Hall
measurements on the HEMT yielded a carrier density of
$2.07\times{10^{11}}~{}{\rm cm^{-2}}$ and a room temperature mobility of
$7650~{}{\rm cm^{2}.V^{-1}s^{-1}}$ The dimensions of the etched mesa was
$1250\times 140~{}{\rm\mu{m}}$ and the resistance between the pair of contacts
used was $r~{}=~{}20.5{\rm k\Omega}$ at $T\approx 300~{}{\rm K}$. Using the
data of Fig. 3(b) and the above-mentioned numbers we get
$\alpha_{H}\approx{1.4}\times{10^{-3}}$, which is very close to the values
reported by Hooge [36, 35] when the origin of the noise is primarily in the
bulk.

Figure 4:  (a) At T=300K, the transfer characteristics exhibit a nearly
hysteresis-free nature of the IL-gated device. (The average of multiple curves
with standard deviation is shown as a grey band). A schematic of the device is
shown in the inset. (b) The motion of mobile ions shown, which are frozen
below the glass transition temperature ($T<T_{G}$). (c)-(h) The observed
normalized low-frequency noise at temperatures of 300K, 220K, 195K, 150K,
110K, and 80K respectively. Currents applied are labelled. In all the above
cases (c-h), the gate is always grounded. (i) Normalized noise spectra at 220K
and 80K for applied $V_{G}$= -0.5V. $V_{G}$=0V at 80K is shown for reference.
(j) Normalized noise spectra for ungated ZnO thin films show $1/f$ noise at
$\rm 95K$. (k) Summarizes the behaviour of the exponent and the integrated
noise power over the entire temperature range.

The presence of a parallel layer of charge at some distance from a 2D electron
or hole gas (2DEG/2DHG) is a common feature in heterostructures. Either the
modulation doped layer, a delta doped layer or the charged surface states (in
the case of a shallow 2DEG) create an impurity potential in the plane of the
2D system. This potential, screened by the 2DEG and suitably averaged over its
form factor is one of the major sources of scattering that limits the
transport [37, 38, 39, 40] as well as the quantum lifetime [41, 42]. While for
relatively lower carrier density systems like MOSFETs, GaAs-AlGAs HEMT and
HHMTs (High hole mobility transistors) a dielectric-metal gate is used to
modulate the carrier densities, the technique becomes untenable when the
carrier densities are very high. In several systems of recent interest [43,
44, 45, 46, 47, 48] modulation of carrier densities is possible only via ionic
liquid (IL) gates capable of forming self-organized nano gap capacitors upon
application of gate voltage, resulting in capacitance in the range of $\sim
10-50\mu Fcm^{-2}$ [49, 50, 43, 45, 51, 46, 52]. For the IL gated system, the
electric double layer is effectively a very high density surface charge.
However, systems gated with IL are also inherently very high carrier density
systems and the impurity Coulomb potential would be screened. Thomas-Fermi
screening gives $V(q)\sim\dfrac{1}{q^{2}+k_{TF}^{2}}$ where the Thomas-Fermi
wave vector is given by
$k_{TF}=\left(\dfrac{{e^{2}}D(E_{F})}{{\epsilon_{r}}{\epsilon_{0}}}\right)^{1/2}$.
The density of states at the Fermi level $D(E_{F})$ makes $k_{TF}$
sufficiently large to suppress large angle scattering which has a maximal
effect on transport lifetime (i.e. mobility ). However, it is known that as
the IL nears the glass transition temperature $T_{G}$ from the high
temperature side, the short time fluctuations are gradually frozen out
irrespective of the gate voltage. It is seen experimentally that most of the
time the resistivity of the conducting channel itself does not have any sharp
feature as the IL crosses $T_{G}$ (for DEME-TFSI used in our experiments
$T_{G}\approx 180-200{\rm K}$). However, our investigations show that the
exponent $\gamma$ of the low frequency resistance noise ( $S_{V}(f)\propto
1/f^{\gamma}$, using the method described earlier) in an IL gated ${\rm ZnO}$
has a pronounced change. The results are valid over three to four decades of
frequency. While transport and photo-response using an ionic liquid ($\rm
PEO+LiClO_{4}$) gate has been reported n-ZnO samples [53, 54], the temperature
dependence of low frequency noise in these IL-FET has not been previously
explored.

Our sample (Fig. 4(a)) is a Ga-doped ZnO thin film deposited on c-sapphire by
RF magnetron reactive co-sputtering of Zn and $\rm GaAs$ at 375°C with 3% of
$\rm GaAs$ coverage on the Zn target, in $\rm Ar-O_{2}$ sputtering atmosphere.
Subsequently, these are lithographically patterned into Hall bars with $W=$100
$\mu m$ and $L$=1150 $\mu m$ and $\rm Ti/Au$ contact pads. Hall measurement
(Supplementary data) gives a carrier density of $\rm 9.97\times 10^{19}\
{cm}^{-3}$. The four-terminal resistivity is
$\rho=\left(\frac{W}{L}\right)\times R_{xx}=4.22\times 10^{-3}\ \Omega\
\text{cm}$. This yields a Hall mobility $\mu_{H}=\frac{1}{\rho n_{3D}e}=14.85\
\text{cm}^{2}\ \text{V}^{-1}\ \text{s}^{-1}$, at $\rm 300K$. While applying on
the channel, the contacts were covered with silicone gel to protect them [55].
After application, it was cured in a high vacuum at $\rm 110\degree\ C$ for
$\sim$ 48 hrs. By making the gate electrode much larger than the channel area,
the necessity of a reference electrode was eliminated since most of the
potential will drop across the working electrode [52]. At room temperature,
the gate voltage sweep shows a near $10\%$ variation of the resistance for
applied $V_{G}$ of $\pm 1\ {\rm V}$, shown in Fig. 4(a).

Figure 5: The numerically computed $S_{V}(f)$ using Eq. (4), successfully
replicates the variation of the exponent $\gamma$ observed in the experiment.
(inset) The Lorentzian spectrum (dotted line) and the log normal distribution
function (solid lines) are depicted schematically. The shift of the peak of
the log-normal ultimately controls the power law.

The measurement of noise was carried out from $T\approx 300{\rm K}$ to well
below $T_{G}$, as shown in Fig. 4(c)-(j). We find that for $T>T_{G}$ the
exponent $\gamma\approx 1.2$ with no strong dependence on the gate voltage.
For $T\ll T_{G}$, it is clear that $\gamma\approx 1$, as summarised in Fig.
4(k). The magnitude of the noise has a small peak around the transition
region. The $1/f$ noise at low frequencies is thought to arise from
fluctuations with Debye-type relaxation having distributions of the relaxation
time $\tau$ over several decades of frequencies. The distribution of the
relaxation times $P(\tau)$ indeed affects the power law [35]. We show that by
assuming a physically reasonable log-normal distribution whose mean ($\mu$) is
slightly shifted but the spread ($\sigma$) is essentially kept constant, we
can reproduce in a simple parametric way, variations of the exponent of the
low frequency fluctuations. This can be expressed as,

$\displaystyle S_{v}(f)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{d\tau}P(\tau)\frac{\tau}{1+\left(2\pi{f}\tau\right)^{2}}$
$\displaystyle P(\tau)$ $\displaystyle\propto$
$\displaystyle\frac{1}{\tau}\exp\left(-\frac{(\ln\tau-\mu)^{2}}{2\sigma^{2}}\right)$
(4)

where, $P(\tau)$ is the log normal distribution function. Fig. 5 shows the
results of the numerically computed integral from Eq. (4) with the mean of the
distribution shifted slightly towards the higher values of $\tau$ (as shown in
the inset). The physical interpretation of this is that the weightage of the
faster fluctuations, i.e. smaller values of $\tau$ are slightly reduced. This
is consistent with the expectation that as the temperature is reduced below
the glass transition temperature $T_{G}$ some of the relatively faster
fluctuations are essentially frozen, affecting the distribution over which the
Lorentzian relaxations must be summed over.

In conclusion, we have presented a relatively inexpensive method of measuring
low frequency noise that largely eliminates the noise originating in the
measurement electronics. The method is also suited for a very wide range of
input impedances that are encountered in many systems of current interest. We
applied this method to an IL-gated degenerate semiconductor and showed that
the exponent of the $1/f$ noise changes as the ionic liquid undergoes the
glass transition. We linked this to a physically reasonable distribution of
the relaxation times. While we have not tried to do a microscopic first
principle derivation of this distribution, we note that it is a reasonable and
simply parametrizable distribution that appears to correctly reproduce the
nature of variation of the exponent of the $1/f$ noise observed in our system
as well as several others. The distribution can be further tuned in two
significant ways that may be of physical importance. First, the spread of the
distribution itself may be related to certain aspects of the interfacial
defects. In addition, if there are specific recombination-generation processes
occurring with a certain time scale [56] an additional weight at that time
scale can be added to the smooth background to reproduce the humps that are
sometimes observed in this type of measurement.

Acknowledgements: BCB and AKJ acknowledge financial support through DST-
INSPIRE and UGC fellowships of Govt. of India. Low temperature measurements
were supported by the cryogenic facilities of IIT Bombay. We thank S.K. Appani
and S.S. Major (Dept of Physics, IIT Bombay) for providing the Ga:ZnO samples.

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See pages - of SM.pdf


# Extreme Value Theory for Hurwitz Complex Continued Fractions

Maxim Kirsebom  Maxim Kirsebom, University of Hamburg, Department of
Mathematics, Bundesstrasse 55, 20146 Hamburg, Germany  maxim.kirsebom@uni-
hamburg.de

###### Abstract.

The Hurwitz complex continued fraction is a generalization of the nearest
integer continued fraction. In this paper we prove various results concerning
extremes of the modulus of Hurwitz complex continued fraction digits. This
includes a Poisson law and an extreme value law. The results are based on cusp
estimates of the invariant measure about which information is still limited.
In the process, we get several results concerning extremes of nearest integer
continued fractions as well.

The author is deeply indebted to Seonhee Lim for the many helpful discussions
and the much help and assistance in connection with this paper, much of which
took place during visits to Seoul National University. Thanks also to Gerardo
González Robert for introducing me to this topic and for helpful discussions.
Finally, thanks to the anonymous referees whose suggestions greatly improved
the paper

##  1\. Introduction

Continued fraction expansions of real numbers have long been known as an
interesting and fruitful venue for applications of extreme value theory (EVT).
Interesting because continued fractions digits are almost surely unbounded and
have infinite expectation which raises natural questions about the behaviour
of their largest digits. Fruitful because continued fraction digits are
closely connected to a dynamical system with nice mixing properties and an
explicit invariant measure.

We briefly recall some facts about regular continued fractions necessary for
further discussion. Every $x\in\mathbbm{R}$ may be written as a fraction of
the form

(1)
$x=a_{0}(x)+\cfrac{1}{a_{1}(x)+\cfrac{1}{a_{2}(x)+\cfrac{1}{\ddots}}}:=[a_{0}(x);a_{1}(x),a_{2}(x),\dots],$

where the (possibly finite) sequence $\left\\{a_{i}(x)\right\\}$ consists of
natural numbers which we refer to as the regular continued fraction (RCF)
digits111Often the terminology _partial quotients_ of $x$ is used for the
numbers $a_{i}$, however, we prefer _digits_. Also, in many instances regular
continued fractions are simply referred to as continued fractions. However,
since we will discuss multiple different continued fraction algorithms in this
paper, we will remain specific throughout. of $x$. For simplicity we will
leave out the dependence on $x$ and write $a_{i}$ unless the context requires
specification. $a_{0}$ is the integer part of $x$ hence for $x\in[0,1)$ we
have $a_{0}=0$ and only the fraction in (1) remains. Furthermore, the sequence
$\left\\{a_{i}(x)\right\\}$ is infinite if and only if $x$ is irrational.
Since only the infinite case is of interest for extreme value behaviour and
since the rational numbers make up a null set anyway, we will focus on numbers
in $X:=(0,1)\backslash\mathbbm{Q}$.

The RCF digits of a number $x\in X$ may also be computed using the Gauss map
$T:X\to X$ which is defined by

$T(x):=\left\\{\frac{1}{x}\right\\}=\frac{1}{x}-\left\lfloor\frac{1}{x}\right\rfloor.$

Here $\left\\{x\right\\}$ and $\left\lfloor x\right\rfloor$ denote the
fractional and integer part of $x$ respectively. The $a_{i}$’s are then given
recursively by

$a_{1}:=a_{1}(x)=\left\lfloor\frac{1}{x}\right\rfloor,\quad
a_{n+1}:=a_{n+1}(x)=a_{1}(T^{n}(x)).$

The Gauss map is invariant under the Gauss measure on $X$ which is given by

$\mu_{G}(E)=\frac{1}{\log 2}\int_{E}\frac{1}{1+x}\,dx$

where $E\subset X$ is Lebesgue measurable. Let $\mathcal{E}$ denote the
$\sigma$-algebra of Lebesgue measurable subsets of $X$. As a triple,
$(X,T,\mu_{G})$ constitutes a dynamical systems with many nice and interesting
properties. The first of these is the explicit form of $\mu_{G}$. For many
dynamical system it is possible to prove the existence of an invariant
measure, however there are few systems for which this measure is known
explicitly. Interestingly, the measure was first stated by Gauss in 1800 [1]
but he gave no explanation for how he discovered it and his rationale remains
a mystery to this day. Equipped with the Gauss measure we may think of $X$ as
a probability space and the $a_{i}:X\to\mathbbm{N}$ as a sequence of random
variables. The second property of $(X,T,\mu_{G})$ which we highlight is a
quantitative mixing property with respect to sets belonging to certain
sub-$\sigma$-algebras of $\mathcal{E}$. We define the property in a general
setting.

######  Definition 1 ($\psi$-mixing).

Let $(\Omega,\mathcal{B},\mu)$ denote a probability space and let
$\xi_{j}:\Omega\to\mathbbm{R}$ denote a stationary sequence of random
variables. For natural numbers $u<v$, let $\mathcal{M}_{u,v}$ denote the
smallest $\sigma$-algebra for which $\xi_{u},\dots,\xi_{v}$ are measurable.
Then $\xi_{j}$ is said to be $\psi$-mixing if for any sets
$A\in\mathcal{M}_{1,l}$ and $B\in\mathcal{M}_{l+n,\infty}$ we have

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

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

In the case of RCF digits, the random variables $a_{j}=a_{1}(T^{j-1}(x))$ form
a stationary sequence due to the invariance of the Gauss measure with respect
to $T$. The $a_{j}$’s are known to be $\psi$-mixing with respect to the Gauss
measure and the function $\psi$ is known to vanish at an exponential rate.
This follows from independent work of Kuzmin and Levy in the late 1920’s, see
for example [15] for the proof and a discussion of its history.

###  1.1. Extreme value theory for RCF digits

Extreme value theory for RCF digits came to life in the 1970’s, first through
Galambos in 1972. He used the Gauss measure and $\psi$-mixing to prove the
following extreme value law.

######  Theorem 2 ([7]).

Let $M_{n}=M_{n}(x):=\max_{1\leq i\leq n}a_{n}(x)$. Then for $r>0$ we have

(2) $\lim_{n\to\infty}\mu_{G}\left(M_{n}\leq\frac{nr}{\log
2}\right)=e^{-\frac{1}{r}}.$

Readers familiar with extreme value theory will recognize this as a Frechet
distribution with extremal index equal to 1. Galambos generalized this result
in [8], showing that $\mu_{G}$ may be replaced with any measure absolutely
continuous with respect to Lebesgue. In 1977, Iosifescu gave the following
more general result, proving a Poisson law for general stationary,
$\psi$-mixing sequences.

######  Theorem 3 ([14]).

Let $\xi_{j}:\Omega\to\mathbbm{R}$ denote a stationary and $\psi$-mixing
sequence of random variables. For $\omega\in\Omega$ and $v\in\mathbbm{R}$, set

$S_{n}(\omega,v):=\\#\left\\{j\in\left\\{1,\dots,n\right\\}:\xi_{j}(\omega)>v\right\\}.$

Assume that there exists a sequence of functions
$u_{n}:\mathbbm{R}\to\mathbbm{R}$ such that

(3) $\lim_{n\to\infty}n\mu(\xi_{1}(\omega)>u_{n}(r))=\tau(r)\quad,\quad
r\in\mathbbm{R}$

for some real-valued function $\tau$. Then, for all $r\in\mathbbm{R}$ and any
$j\in\mathbbm{N}$,

(4)
$\lim_{n\to\infty}\mu\left\\{S_{n}(\omega,u_{n}(r))=j\right\\}=e^{-\tau(r)}\frac{\tau(r)^{j}}{j!}.$

It was strightforward for Iosifescu to apply this theorem to RCF digits ([14,
Theorem 2]) since $\psi$-mixing for RCF digits was known and property (3)
follows easily from the form of the Gauss measure. Similar to Galambos, he
gave an argument for why the Poisson law holds for any measure which is
absolutely continuous with respect to Lebesgue. Note that one obtains
Galambos’ theorem by setting $j=0$ in Iosifescu’s result for RCF digits. It is
interesting to note that Iosifescu’s result for RCF digits appeared as early
as 1940 in a paper by Doeblin [5]. However, as Iosifescu explains in [14],
Doeblin’s proof contains a mistake. Galambos was unaware of Doeblin’s result
when proving his extreme value law, and Iosifescu obtained his proof by
applying ideas from Galambos paper to fix Doeblin’s mistake.

The era offered more results concerning the maximum of RCF digits. Galambos
proved an iterated logarithm type theorem for the maximal RCF digits [9],
which was improved by Philipp [19] to give a complete answer to a conjecture
of Erdős. Also in [19], an upper bound on the rate of convergence in (2) was
provided. Diamond and Vaaler [4] showed that the partial maximum is
responsible for the failure of the law of large numbers for RCF digits.

The topic of extreme value statistics for RCF digits has gained interest again
in recent years. Philipp’s rate of convergence was improved by Ghosh, Kirsebom
and Roy [10], while a refinement of Iosifescu’s theorem was presented by
Zweimüller in [30]. Of a slightly different flavour, Chang and Chen [2]
investigated the Hausdorff dimension of certain sets defined via the largest
RCF digits.

###  1.2. Extreme value theory for other CF algorithms

The regular continued fractions considered up to this point are by far not the
only continued fractions in existence. For RCF digits we have a good
understanding of the statistical behaviour of largest digits. For other CF
algorithms however, results of this nature are scarce and many interesting
questions remain open. To our knowledge, the only works on extreme value
theory for other algorithms are the papers by Chang and Ma [3] which considers
the case of Oppenheim CF, by Shen, Xu and Jing [23] which studies the case of
continued fraction defined over the field of formal Laurent series and by
Nakada and Natsui [18] which investigates fibred systems. In a different but
related direction, González Robert [22] recently proved a Borel-Bernstein
Theorem for Hurwitz complex continued fractions. The methods used in [3] are
somewhat different to the ones described for RCF’s since the invariant measure
associated to Oppenheim CF is infinite. Instead, the metric theory is
developed for the Lebesgue measure with respect to which the associated
dynamical system is not $\psi$-mixing. Also [23] is somewhat different in that
an iterated logarithm type result is proven as opposed to a distributional
result.

To the contrary, [18] takes the same approach as described above and applies
it to fibred systems. Many CF algorithms, including several complex CF
algorithms, satisfy the conditions for being fibred systems, see [24] for some
examples.

Under certain assumptions, fibred systems were proven by Waterman [29] to
admit an invariant measure absolutely continuous with respect to Lebesgue and
under further assumptions Schweiger [24], [25] showed that the fibred system
is $\psi$-mixing with respect to this invariant measure. Nakada and Natsui use
this fact to formulate general sufficient conditions on the invariant measure
in order to get analogues of Theorem 2 as well as the results in [4] and [19].

They further proved that these various assumptions are satisfied for the
Jacobi-Perron multidimensional CF algorithm, but for many other CF algorithms
this is not known.

The main aim of this paper is to develop extreme value theory for complex
continued fractions, more specifically the variant introduced by Hurwitz [13].
In the process we “pick up” analogue results for nearest integer continued
fractions which we present first before continuing to the complex continued
fractions and our main results.

###  1.3. Extreme value theory for nearest integer continued fractions

This subsection serves two purposes. First, it allows us to state some new
results concerning extreme value theory for nearest integer continued
fractions (NICF). These results are new in the sense that they appear not to
have been stated elsewhere before. However, aside from a small calculation,
the proofs simply combine results proven elsewhere, hence in that sense the
novelty is limited. Second, the complex continued fractions which are the main
focus of this paper are a generalization of NICF for real numbers. Hence this
subsection serves as a stepping stone between the RCF’s and the Hurwitz
complex continued fractions.

NICF’s work similar to RCF’s, the main difference being that we round to the
nearest integer. We consider the fundamental domain
$X=\left[-\frac{1}{2},\frac{1}{2}\right)$ and for an $x\in X$ we take its
inverse $\frac{1}{x}$ and subtract the integer nearest to it. This brings us
back to $X$ where we repeat the process. This enables us to write $x$ as

(5)
$x=\cfrac{1}{\left[\frac{1}{x}\right]_{\textup{N}}+\cfrac{1}{\left[\frac{1}{\frac{1}{x}-\left[\frac{1}{x}\right]_{\textup{N}}}\right]_{\textup{N}}+\cfrac{1}{\ddots}}},$

where $\left[\cdot\right]_{N}$ denotes the nearest integer function. Similar
to the RCF, the NICF digits may be generated through a dynamical system.
However, the layout of the digits will vary slightly from (5), the benefit
being that our transformation gets a simpler expression. We define the
transformation $T_{\text{N}}:X\to X$ by

$T_{\text{N}}(x)=\begin{cases}\frac{\varepsilon}{x}-\left\lfloor\frac{\varepsilon}{x}+\frac{1}{2}\right\rfloor&,\enskip
x\neq 0\\\ 0&,\enskip x=0.\end{cases}$

where $\varepsilon$ denotes the sign of $x$. Using this definition the modulus
of the nearest integers and their signs are recorded in separate digits. The
NICF expansion of $x\in X$ then becomes

(6)
$x=\cfrac{\varepsilon_{1}}{b_{1}(x)+\cfrac{\varepsilon_{2}}{b_{2}(x)+\cfrac{\varepsilon_{3}}{\ddots}}},$

where

$b_{1}(x)=\left\lfloor\left|\frac{1}{x}+\frac{1}{2}\right|\right\rfloor,\quad
b_{n}(x)=b_{1}(T_{\text{N}}^{n-1}(x))=\left\lfloor\left|\frac{1}{T_{\text{N}}^{n-1}(x)}+\frac{1}{2}\right|\right\rfloor$

and

$\varepsilon_{n}=\textup{sgn}\left(T_{\text{N}}^{n-1}(x)\right).$

In this notation the $\varepsilon_{n}$’s record whether the nearest integer
digits from (5) change sign. Indeed the NICF digits in (5) may be written as

$\varepsilon_{1}b_{1}(x),(\varepsilon_{1}\varepsilon_{2})b_{2}(x),\dots,(\varepsilon_{1}\dots\varepsilon_{n})b_{n}(x),\dots$

The map $T_{\text{N}}$ admits an invariant measure $\mu_{N}$ on
$\left[-\frac{1}{2},\frac{1}{2}\right)$ whose density is given by

$\rho(x)=\begin{cases}\frac{1}{\log(G)}\frac{1}{G+x},&\quad\text{for
}x\in\left[0,\frac{1}{2}\right)\\\
\frac{1}{\log(G)}\frac{1}{G+1+x},&\quad\text{for
}x\in\left[-\frac{1}{2},0\right)\end{cases}$

where $G=\frac{\sqrt{5}+1}{2}$ is the golden ratio. It is also known to be
$\psi$-mixing with respect to this measure. Indeed, the function $\psi(n)$ is
known to decay faster than $O\left(\theta^{n}\right)$ where
$\theta=\frac{3}{4}$. Both of these results were proven by Rieger ([20, 21]).

Consider now $b_{n}:X\to\mathbbm{N}$ as a sequence or random variables. By
$\mu$-invariance $b_{n}$ forms a stationary sequence. Define

$S_{n}(x,r)=\\#\left\\{j\in\left\\{1,\dots,n\right\\}:b_{j}(x)>r\right\\}.$

Now, the only property missing in order to apply Theorem 3 to $S_{n}(x,r)$ is
the property (3). However, this follows from the ensuing calculation. We
shorten notation by writing $\left\\{b_{n}>j\right\\}$ instead of
$\left\\{x\in X:b_{n}(x)>j\right\\}$. Also, we use the notation $\sim$ to
indicate that the related quantities have the same limit. We obtain

$\displaystyle\mu_{N}\left\\{b_{n}>j\right\\}$
$\displaystyle=\mu_{N}\left\\{b_{1}>j\right\\}$
$\displaystyle=\mu_{N}\left\\{\left\lfloor\left|\frac{1}{x}+\frac{1}{2}\right|\right\rfloor>j\right\\}$
$\displaystyle\sim\mu_{N}\left\\{\left|\frac{1}{x}\right|>j\right\\}$
$\displaystyle=\frac{1}{\log
G}\left(\int_{-\frac{1}{j}}^{0}\frac{1}{G+1+x}\,dx+\int_{0}^{\frac{1}{j}}\frac{1}{G+x}\,dx\right)$
$\displaystyle=\frac{1}{\log
G}\left(\log\left(1+\frac{1}{Gj}\right)-\log\left(1+\frac{1}{(G+1)j}\right)\right)$
$\displaystyle=\frac{1}{\log
G}\left(\frac{1}{G}+\frac{1}{G+1}\right)\frac{1}{j}+O\left(\frac{1}{j^{2}}\right)$
$\displaystyle=\frac{1}{\log G}\frac{1}{j}+O\left(\frac{1}{j^{2}}\right).$

In the above, the fact that we did not change the limit by ignoring the floor
function and the added $\frac{1}{2}$ requires a small calculation but is
intuitively clear. Futhermore, we made use of the power series expansion of
$\log(1+x)$. From this we see that

(7) $\lim_{n\to\infty}n\mu_{N}\left\\{b_{n}>\frac{nr}{\log
G}\right\\}=\frac{1}{r}$

which verifies (3) for $u_{n}(r)=\frac{nr}{\log G}$. Hence we have proven the
Poisson law of exceedances for the NICF digits.

######  Theorem 4.

Let $u_{n}(r)=\frac{nr}{\log G}$. For all $r>0$ and any $j\in\mathbbm{N}$

(8) $\lim_{n\to\infty}\mu_{N}\left\\{x\in
X:S_{n}(x,u_{n}(r))=j\right\\}=e^{-\left(\frac{1}{r}\right)}\frac{r^{-j}}{j!}.$

This leads to the immediate corollaries

######  Corollary 5.

Let $M_{n}(x):=\max_{1\leq i\leq n}\left|b_{i}(x)\right|$ and let
$M_{n}^{(k)}(x)$ denote the $k$’th largest element among
$\left\\{b_{1}(x),\dots,b_{n}(x)\right\\}$.

For all $r>0$

$\lim_{n\to\infty}\mu_{N}\left\\{x\in X:M_{n}^{(k)}\leq\frac{nr}{\log
G}\right\\}=e^{-\frac{1}{r}}\sum_{j=0}^{k-1}\frac{r^{-j}}{j!}.$

In particular,

$\lim_{n\to\infty}\mu_{N}\left\\{x\in X:M_{n}(x)\leq\frac{nr}{\log
G}\right\\}=e^{-\frac{1}{r}}.$

The last result is a direct analogue of Galambos’ Theorem for RCF, the
difference appearing only in the constant used to normalize the maximum.

######  Remark 6.

When comparing the RCF with the NICF of certain real numbers, it appears
plausible that the behaviour of their largest digits should be similar. Taking
$\pi$ as an example we have the two different expansions222See
https://oeis.org/A001203 and https://oeis.org/A133593 for more digits of
either expansion.

$\displaystyle\pi_{RCF}$
$\displaystyle=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,\dots]$
$\displaystyle\pi_{NICF}$
$\displaystyle=[3;7,16,294,3,4,5,15,3,2,2,2,2,3,85,3,2,15,3,14,5,2,6,6,\dots].$

One observes a certain similarity in the larger digits, say 292 and 294 as
well as 84 and 85. A big difference is the lack of 1’s in the NICF expansion
leading to shorter intervals between the large digits. Heuristically this
indicates that the partial maximum for the NICF digits should grow faster than
the partial maximum for RCF digits. This is indeed reflected in the extreme
value laws for the two expansions. In the NICF case the normalizing sequence
$nr$ must be multiplied with the larger constant $(\log G)^{-1}>(\log 2)^{-1}$
in order to obtain the same distribution as for the RCF.

We also obtain a rate of convergence for the limits in Theorem 4 and Corollary
5. Let $l_{n}$ be a sequence which satisfies

$l_{n}\theta^{l_{n}}=n.$

where again $\theta=\frac{3}{4}$. Note that in particular $l_{n}=o(\log n)$.

######  Theorem 7.

For all $k\in\mathbbm{N}$ and all $r>0$, the rate of convergence in (12), (13)
and (14) is bounded by

$O\left(\frac{1}{\min\left(r,r^{2}\right)}\frac{l_{n}}{n}\right).$

###### Proof.

The proof follows exactly the proof of Theorem 1.1 of [10]. Only in a few
places, small adjustments have to be made due to the different density of the
invariant measure. The adjustments are done using (7) and the derivations
leading up to it, in particular the power series expansion of $\log(1+x)$. Due
to the simplicity of the adaptations we leave the details to the reader. ∎

##  2\. Complex continued fractions and main results

While there are many CF algorithms for real numbers, the RCF algorithm has a
strong sense of being the most “natural”. This is due to its simplicity,
strong properties and its many connections to other fields like analytic
number theory, dynamical systems and hyperbolic geometry. In the complex realm
no CF algorithm reigns supreme in the same way. Part of the explanation is
that the RCF algorithm for real numbers does not generalize in a meaningful
way to the complex numbers333The argument that follows was kindly presented to
me by Gerardo González Robert.. A naive approach to generalizing the RCF would
see us using complex inversion on the square

$S:=\left\\{z=x+iy\,:\,0<x,y\leq 1\right\\}$

and applying the floor function in both dimensions. More precisely, define the
transformation $T:S\to S$ by

$T(z)=\frac{1}{z}-\left\lfloor\frac{1}{z}\right\rfloor$

where $\left\lfloor\cdot\right\rfloor$ denotes the complex floor function
given by $\left\lfloor z\right\rfloor=\left\lfloor
x+iy\right\rfloor=\left\lfloor x\right\rfloor+i\left\lfloor y\right\rfloor$.
Set $a_{1}(z):=\left\lfloor\frac{1}{z}\right\rfloor$ and
$a_{n+1}(z)=a_{1}(T^{n}(z))$. Using these definitions, uncountably many
different elements of $S$ would be assigned the same sequence of digits
$\left\\{a_{n}\right\\}$. A specific example is the region bounded by the
circles
$C_{1}:=\left\\{z\in\mathbbm{C}:\left|z-\frac{1}{2}\right|=\frac{1}{2}\right\\}$
and
$C_{2}:=\left\\{z\in\mathbbm{C}:\left|z-(\frac{1}{2}+i)\right|=\frac{1}{2}\right\\}$
as well as the line segment connecting $1$ and $1+i$. This region gets mapped
to itself under $T$ and in the process every element $z$ in this region gets
assigned $a_{n}(z)=-i$ for all $n\in\mathbbm{N}$. Clearly this does not lead
to a useful continued fractions representation of numbers in $S$.

Many alternative approaches to complex continued fractions exist, see [16] for
an overview of some of them and their properties and references. The approach
which we study in this paper is a generalization of the NICF algorithm for
real numbers.

###  2.1. Hurwitz complex continued fractions

Denote by $\mathbbm{Z}[i]=\left\\{x+iy:x,y\in\mathbbm{Z}\right\\}$ the
Gaussian integers and denote by $[z]_{i}$ the Gaussian integer nearest to $z$.
We apply the convention that ties are broken by rounding down in both real and
complex part, for example $[z]_{i}=a+bi$ if $z=(a+0.5)+(b+0.5)i$ for
$a,b\in\mathbbm{Z}$. This ensures that $[\cdot]_{i}$ is well-defined, however,
the choice of rounding will play no role for our results since the convention
relates only to a set of measure zero.

Let $B=\left\\{z=x+iy\in\mathbbm{C}:-\frac{1}{2}\leq
x,y<\frac{1}{2}\right\\}$. Analogous to the Gauss map, we define the Hurwitz
map $T:B\to B$ by

(9) $\displaystyle Tz=\frac{1}{z}-\left[\frac{1}{z}\right]_{i}$

For a given $z\in\mathbbm{C}$, consider the sequence $z_{k}$ given by
$z_{0}=z-[z]_{i}$ and

(10) $\displaystyle
z_{k}=Tz_{k-1}=\frac{1}{z_{k-1}}-\left[\frac{1}{z_{k-1}}\right]_{i},\;\;\mathrm{i.e.}\;\;\;z_{k-1}=\frac{1}{\left[\frac{1}{z_{k-1}}\right]_{i}+z_{k}},$

for $k\geq 1$. By setting $a_{0}=a_{0}(z):=[z]_{i}$ and
$a_{k}=a_{k}(z):=\left[\frac{1}{T^{k-1}z_{0}}\right]_{i}$ for $k\geq 1$, we
get the Hurwitz complex continued fractions (HCCF) expansion of $z$ written as

(11) $\displaystyle
z=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots}}}:=[a_{0};a_{1},a_{2},\dots].$

It is well known that this expansion converges and provides a meaningful
representation of complex numbers, see [11] for an introduction to the HCCF
expansion. Let $\mathcal{B}$ denote the $\sigma$-algebra of Borel subsets of
$B$ and let $\lambda$ be the Lebesgue measure on $B$. It is known that there
exists a unique measure on $B$ which is $T$-invariant and absolutely
continuous with respect to Lebesgue (see [11], Section 5.7). We denote this
measure by $\mu$. Thus we have a dynamical system $(T,B,\mu)$ where $T:B\to B$
is a $\mu$-preserving map. Consider the sequence of real-valued random
variables $\left|a_{i}\right|:B\to\mathbbm{R}_{+}$. As previously indicated we
are particularly interested in the occurrence of large values of
$\left|a_{i}(z)\right|$.

We are finally ready to state the main results of this article. The first is a
Poisson law for HCCF, analogue to Iosifescus result applied to RCF’s.

######  Theorem 8.

For $z\in B$ and $v\in\mathbbm{R}_{+}$, let

$S_{n}(z,v):=\\#\left\\{i\in\left\\{1,\dots,n\right\\}:\left|a_{i}(z)\right|>v\right\\}.$

There exist $C>0$ such that for all $r>0$ and any $j\in\mathbbm{N}$

(12) $\lim_{n\to\infty}\mu\left\\{z\in
B:S_{n}(z,Cr\sqrt{n})=j\right\\}=e^{-\left(\frac{1}{r^{2}}\right)}\frac{r^{-2j}}{j!}.$

This leads to the immediate corollaries

######  Corollary 9.

Let $M_{n}(z):=\max_{1\leq i\leq n}\left|a_{i}(z)\right|$ and let
$M_{n}^{(k)}(z)$ denote the $k$’th largest element among
$\left\\{\left|a_{1}(z)\right|,\dots,\left|a_{n}(z)\right|\right\\}$. There
exists $C>0$ such that for all $r>0$

(13) $\lim_{n\to\infty}\mu\left\\{z\in B:M_{n}^{(k)}\leq
Cr\sqrt{n}\right\\}=e^{-\frac{1}{r^{2}}}\sum_{j=0}^{k-1}\frac{r^{-2j}}{j!}.$

In particular,

(14) $\lim_{n\to\infty}\mu\left\\{z\in B:M_{n}(z)\leq
Cr\sqrt{n}\right\\}=e^{-\frac{1}{r^{2}}}.$

Note that (14) provides an analogue of Galambos’s extreme value law for
HCCF’s. The corollary is obtained simply by setting $j=0$ in Theorem 8.

##  3\. The invariant measure with respect to the Hurwitz map

The strategy of proof of Theorem 8 is clear. If we can satisfy the conditions
of Theorem 3 the result follows. Fortunately, the $\psi$-mixing property of
$T$ is known and it is even known to be mixing at an exponetial rate, i.e.
$\psi(n)=O(\theta^{n})$ for some $\theta<1$. This exponential $\psi$-mixing
result was first stated by Nakada in [17, Corollary 2], but the proof relied
on work by Schweiger [28] which turned out to contain a serious gap. As
explained in [6], Schweiger later was able to recover the result with a sub-
exponential rate in [26] and finally the exponential rate in [27], meaning
that Nakada’s result is indeed correct. Note also that the sequence
$\left|a_{i}\right|$ forms a stationary sequence, this is an easy consequence
of $\mu$ being $T$-invariant.

The greater challenge is to satisfy (3) in Theorem 3. For this we need more
specific information about the unique $T$-invariant measure absolutely
continuous with respect to Lebesgue. We first cite the following theorem by
Hensley which is central to our proof.

######  Lemma 10 ([11], Theorem 5.5).

The density function $\rho$ of the measure $\mu$ is continuous except possibly
along the intersections of the circles $|z\pm 1|=1,|z\pm i|=1$ and $|z\pm 1\pm
i|=1$ with $B$. It is real analytic on each of the 12 open regions into which
the interior of $B$ is dissected by these circles. Moreover, the measure $\mu$
is symmetric under complex conjugation and multiplication by $i$.

The 8 arcs and 12 regions in $B$ are denoted by $\gamma_{i}$ and $A_{i}$
respectively. See Figure 1.

Figure 1. Dissection of B.

An important further fact about the density $\rho$ is that the limits

$C_{i}:=\lim_{A_{i}\ni z\to 0}\rho(z),\quad\text{for}\quad i=1,\dots,8$

exist and are strictly positive, i.e $C_{i}>0$. This can be seen with the help
of various facts from recent works of Hiary and Vandehey [12] as well as Ei,
et al [6]. As explained in [12], the density $\rho$ may be expressed as

(15)
$\rho(z)=\int_{V_{i}}\frac{1}{\left|zw+1\right|^{4}}\,dw,\quad\quad\text{if
}z\in A_{i}$

where $V_{i}$ is a certain set related to $A_{i}$ whose exact definition we
will not get into here. The sets $V_{i}$ are not completely understood at this
point (if they were, we would understand the measure $\mu$ much better), but
they were thoroughly studied in [6] where it was shown, among other things,
that they are measurable and of positive Lebesgue measure. Hence we see that
from (15) that the limit for $z\to 0$ inside either of the regions $A_{i}$,
$i=1,\dots,8$, exists and equals the Lebesgue measure of the corresponding set
$V_{i}$. Due to the symmetries mentioned in Theorem 10 the limits inside
$A_{1}$, $A_{2}$, $A_{3}$ and $A_{4}$ must be identical and the same applies
to the limits inside $A_{5}$, $A_{6}$, $A_{7}$ and $A_{9}$. Hence we may set:

(16) $\displaystyle\begin{split}\tilde{C}:=C_{i},\quad\text{for }i=1,2,3,4.\\\
C^{\prime}:=C_{i},\quad\text{for }i=5,6,7,8.\end{split}$

Nakada had much earlier showed [17, Theorem 2] that the measure $\mu$ is
equivalent to the Lebesgue measure, i.e. there exists a constant $Q>0$ such
that for all $E\in\mathcal{B}$

$\frac{1}{Q}\lambda(E)\leq\mu(E)\leq Q\lambda(E).$

### Proof of main results

The main technical lemma of this paper is the following.

######  Lemma 11.

For some constant $H>0$ we have

$\mu\left\\{z\in
B:|a_{1}(z)|>j\right\\}=\frac{H}{j^{2}}+O\left(\frac{1}{j^{3}}\right),\quad\text{
as }j\to\infty.$

###### Proof.

For the purpose of this proof we occasionally shorten notation by writing
$\left\\{\left|a_{1}\right|>j\right\\}$ instead of $\left\\{z\in
B:\left|a_{1}(z)\right|>j\right\\}$.

Set

$A:=\left\\{\left|a_{1}\right|>j\right\\}=\left\\{\left|\left[\frac{1}{z}\right]_{i}\right|>j\right\\},$

hence our goal is to determine the asymptotics of $\mu(A)$. Let
$\varphi:\mathbbm{C}\backslash\left\\{0\right\\}\to\mathbbm{C}\backslash\left\\{0\right\\}$
be given by $\varphi(z)=\frac{1}{z}$ and notice that $\varphi$ is a bijection
which maps disks of radius $r>0$ to complements of disks of radius
$\frac{1}{r}$. Then for
$A^{\prime}=\left\\{z\in\mathbbm{C}:\left|\left[z\right]_{i}\right|\geq
j\right\\}$,

$A=\varphi(A^{\prime}).$

We define two sets $\underline{A^{\prime}}$ and $\overline{A^{\prime}}$ by

$\underline{A^{\prime}}=\left\\{\left|z\right|\geq
j+\frac{1}{\sqrt{2}}\right\\}\enskip\text{and}\enskip\overline{A^{\prime}}=\left\\{\left|z\right|\geq
j-\frac{1}{\sqrt{2}}\right\\}.$

A simple geometric argument (see Figure 2) shows that

(17) $\underline{A^{\prime}}\subset A^{\prime}\subset\overline{A^{\prime}}$

Figure 2 shows $A^{\prime}$, $\underline{A^{\prime}}$ and
$\overline{A^{\prime}}$ for some unspecified $j>2$ as well as the set $B$.
$A^{\prime}$ is the (unbounded) grey region, $\underline{A^{\prime}}$ is the
region _outside_ the green circle while $\overline{A^{\prime}}$ is the region
_outside_ the red circle. The blue dots are the Gaussian integers. The set
$A$, not drawn in the figure, is some neighborhood of $0$ inside $B$.

Figure 2. The sets $B$, $A^{\prime}$, $\underline{A^{\prime}}$ and
$\overline{A^{\prime}}$

Applying $\varphi$ to (17) we get,

$\left\\{\left|z\right|\leq\frac{1}{j+\frac{1}{\sqrt{2}}}\right\\}=\varphi(\underline{A^{\prime}})\subset
A\subset\varphi(\overline{A^{\prime}})=\left\\{\left|z\right|\leq\frac{1}{j-\frac{1}{\sqrt{2}}}\right\\}.$

Taking the $\mu$-measure in the above inclusion followed by a small
computation which makes use of the fact that $\mu$ is equivalent with respect
to Lebesgue, we deduce that

(18)
$\mu(A)=\mu\left\\{\left|z\right|\leq\frac{1}{j}\right\\}+O\left(\frac{1}{j^{3}}\right).$

We now proceed to estimate
$\mu\left\\{\left|z\right|\leq\frac{1}{j}\right\\}$, i.e. the $\mu$-measure of
the ball centered at 0 with radius $\frac{1}{j}$. Set
$D(j)=\left\\{\left|z\right|\leq\frac{1}{j}\right\\}$ and

$D_{i}=D_{i}(j):=D(j)\cap A_{i}\enskip\text{for }i=1,\dots 8.$

Note that the intersection with $A_{9}$, $A_{10}$, $A_{11}$, and $A_{12}$ is
empty for $j$ sufficiently large, hence we may ignore these regions for our
purposes. We have

(19)
$\mu(D(j))=\mu\left(\bigcup_{i=1}^{8}D_{i}\right)=\sum_{i=1}^{8}\mu(D_{i})=4\mu(D_{1})+4\mu(D_{5}).$

The last equality follows from the symmetries noted in Theorem 10. We can now
use the fact that $\rho$ is continuous on each $D_{i}$ and that the limits

$\tilde{C}=\lim_{D_{1}\ni z\to 0}\rho(z)\quad\text{ and }\quad
C^{\prime}=\lim_{D_{5}\ni z\to 0}\rho(z).$

exist and are strictly positive to estimate $\mu(D_{1})$ and $\mu(D_{5})$. Let
$\varepsilon>0$ be given. For $j$ sufficiently large we have

$\displaystyle\left|\rho(z)-\tilde{C}\right|$
$\displaystyle<\varepsilon\enskip\text{if }z\in D_{1}(j),$
$\displaystyle\left|\rho(z)-C^{\prime}\right|$
$\displaystyle<\varepsilon\enskip\text{if }z\in D_{5}(j)$

hence

$\displaystyle(\tilde{C}-\varepsilon)\lambda(D_{1})\leq\mu(D_{1})\leq(\tilde{C}+\varepsilon)\lambda(D_{1}),$
$\displaystyle(C^{\prime}-\varepsilon)\lambda(D_{5})\leq\mu(D_{5})\leq(C^{\prime}+\varepsilon)\lambda(D_{5}).$

We may now estimate $\mu(D)$ as follows

(20) $4(\tilde{C}-\varepsilon)\lambda(D_{1})\leq\mu(D(j))\leq
4(\tilde{C}+\varepsilon)\lambda(D_{1})+4(C^{\prime}+\varepsilon)\lambda(D_{5}).$

The lower bound is motivated by Figure 3 which suggests that $\lambda(D_{5})$
becomes insignificant compared to $\lambda(D_{1})$ when $j$ becomes large. We
make this precise by computing an upper bound for $\lambda(D_{5})$.

Figure 3.

We see from Figure 3 that $D_{5}$ is contained in a rectangle of side length
$\frac{1}{j}$ and $\frac{1}{j^{2}}$. Here the vertical side length is found as
the imaginary value of the intersection point between $\gamma_{1}$ and the
boundary of $D$, i.e. solving

$x^{2}+(y-1)^{2}=1\enskip\text{and}\enskip
x^{2}+y^{2}=\left(\frac{1}{j^{2}}\right),$

leading to $y=\frac{1}{2j^{2}}$ and side length $\frac{1}{j^{2}}$. Hence we
get the bound

$\lambda(D_{5})\leq\frac{1}{j}\frac{1}{j^{2}}=\frac{1}{j^{3}}.$

We also use this bound to estimate $\lambda(D_{1})$. Namely, we have
$\lambda(D(j))=\pi\frac{1}{j^{2}}$ and

$\frac{\pi}{4}\frac{1}{j^{2}}=\frac{1}{4}\lambda(D(j))\geq\lambda(D_{1})=\frac{1}{4}\lambda(D(j))-\lambda(D_{5})\geq\frac{\pi}{4}\frac{1}{j^{2}}-\frac{1}{j^{3}}.$

Inserting these estimates in (20) we get

$(\tilde{C}-\varepsilon)\pi\frac{1}{j^{2}}-4(\tilde{C}-\varepsilon)\frac{1}{j^{3}}\leq\mu(D(j))\leq(\tilde{C}+\varepsilon)\pi\frac{1}{j^{2}}+4(C^{\prime}+\varepsilon)\frac{1}{j^{3}}.$

Since we are interested in the asymptotic behaviour as $j\to\infty$ we can
pick $\varepsilon$ arbitrarily small. Together with (18) we get the conclusion
that for $H:=\pi\tilde{C}$ we have

$\mu\left\\{z\in
B:|a_{1}(z)|>j\right\\}=\frac{H}{j^{2}}+O\left(\frac{1}{j^{3}}\right),\quad\text{
as }j\to\infty.$

This completes the proof of the lemma. ∎

######  Remark 12.

We repeat the point here that followed (15), namely that the constant $H$
could potentially be made explicit if one could find a way of computing the
Lebegue measure of the sets $V_{i}$ which are defined and studied in [6] and
[12].

###### Proof of Theorem 8.

As stated earlier, to finish the proof of Theorem 8 we need to show that
condition (3) in Theorem 3 is satisfied. However, at this stage we only need
to observe that by choosing the sequence $u_{n}(r):=Cr\sqrt{n}$ with
$C:=\sqrt{H}$ we get

$\displaystyle\lim_{n\to\infty}n\mu\left\\{\left|a_{1}\right|>Cr\sqrt{n}\right\\}$
$\displaystyle=\lim_{n\to\infty}n\left(\frac{H}{\left(Cr\sqrt{n}\right)^{2}}+O\left(\frac{1}{\left(Cr\sqrt{n}\right)^{3}}\right)\right)$
$\displaystyle=\frac{1}{r^{2}}+\lim_{n\to\infty}O\left(\frac{1}{\sqrt{n}r^{3}}\right)$
$\displaystyle=\frac{1}{r^{2}},$

and the proof is complete. ∎

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and the first order equation (E11) (used to derive the dictionary) becomes

$\displaystyle\left(\accentset{(1)}{R}_{\mu\nu}-\mathcal{L}_{X}\left(\accentset{(0)}{\bar{R}}_{\mu\nu}\right)\right)\hat{h}^{\mu\alpha}\hat{h}^{\nu\beta}=\left(\frac{\kappa}{2}\rho+\Lambda\right)\hat{h}^{\alpha\beta}.$
(F22)

Because $\accentset{(0)}{\bar{R}}_{\mu\nu}$ has the same properties as
${\bar{R}}_{\mu\nu}$, the resolution of equation (F21) is equivalent as
solving equation (57) in Section 4.3.2. In particular,
$\accentset{(0)}{G}^{\mu}$ corresponds to a Galilean vector, and in Euclidean,
spherical or hyperbolic topologies without anisotropic expansion we have

$\displaystyle\accentset{(0)}{\bar{R}}_{\mu\nu}=2K\tensor[^{\boldsymbol{\tilde{G}}}]{b}{}_{\mu\nu}.$
(F23)

Then, writing $X^{\mu}\eqqcolon
M\tilde{G}^{\mu}+\hat{D}^{\mu}\mathcal{X}+\mathcal{X}^{\mu}$ with
$\mathcal{X}^{\mu}\tau_{\mu}\coloneqq
0\eqqcolon\hat{D}_{\mu}\mathcal{X}^{\mu}$, the additional term in equation
(F22), compared to equation (E11), takes the form

$\displaystyle\mathcal{L}_{X}\left(\accentset{(0)}{\bar{R}}_{\mu\nu}\right)\hat{h}^{\mu\alpha}\hat{h}^{\nu\beta}=4K\left(\hat{D}^{\mu}\hat{D}^{\nu}\mathcal{X}+2\hat{D}^{(\mu}\mathcal{X}^{\nu)}\right).$
(F24)

A consequence is that reperforming the calculation of Section E, we obtain

$\displaystyle\left(\frac{\kappa}{2}\rho+\Lambda\right)\hat{h}^{ij}$
$\displaystyle=\hat{h}^{ij}(\dot{H}+3H^{2})+\hat{h}^{ij}\left[\hat{\Delta}\psi+\hat{\Delta}(H\sigma)\right]$
$\displaystyle\qquad+\hat{D}^{i}\hat{D}^{j}\left[-\Phi+H\sigma+\psi+4K(E+\mathcal{X})\right]$
(F25) $\displaystyle\qquad+4K\hat{D}^{(i}\left(F^{j)}+\mathcal{X}^{j)}\right)$
$\displaystyle\qquad+\left(6K-\hat{\Delta}\right)f^{ij}.$

Therefore, $E$ and $F^{i}$ are not anymore imposed to be zero, instead the
gauge-invariant variables $\mathcal{C}\coloneqq E+\mathcal{X}$ and
$\mathcal{C}^{i}\coloneqq F^{i}+\mathcal{X}^{i}$ are zero.

In conclusion, from either Einstein’s equation (Euclidean case) or the bi-
metric theory (non-Euclidean case), the spacetime metric
$\accentset{\lambda}{g}^{\mu\nu}$ (respectively
$\accentset{\lambda}{g}_{\mu\nu}$) takes the form (70) (respectively (75)) as
function of the gravitational potential, with $\sigma$, $E$ and $F^{i}$ being
free gauge fields in both cases.

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# The Hierarchy Problem and the Top Yukawa

Andreas Bally  Yi Chung 111Speaker and Florian Goertz  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany

###### Abstract

In this talk, an alternative to top partner solutions and its consequences on
phenomenology are discussed. The hierarchy problem from the top loop
contribution is solved by mitigating the top Yukawa coupling at high scales.
In this scenario, the new degrees of freedom appearing at the cut-off scale of
the top loop should then be some new top-philic particles instead of
traditional top partners. The idea can be directly tested through measurements
in top physics, including $t\bar{t}h$, $t\bar{t}$ differential cross section,
and $t\bar{t}t\bar{t}$ cross section.

##  1 Introduction

The hierarchy problem is one of the long-standing problems in the Standard
Model (SM) of particle physics. The problem is growing severe after the
discovery of a light Higgs boson without any other new degrees of freedom. The
Higgs boson’s mass term receives quadratically divergent contributions from
loops of all the particles it interacts with. Among all the loops, the top
quark loop gives the largest contribution as

$\Delta m_{H}^{2}|_{\rm
top}\sim-i\,2N_{c}\,y_{t}^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{2}+m_{t}^{2}}{(k^{2}-m_{t}^{2})^{2}}=-\frac{3}{8\pi^{2}}y_{t}^{2}\left[\Lambda_{\rm
NP}^{2}-3\,m_{t}^{2}\,{\rm ln}\left(\frac{\Lambda^{2}_{\rm
NP}}{m_{t}^{2}}\right)+\cdots\right]\leavevmode\nobreak\ ,$ (1)

where we only kept the $\Lambda_{\rm NP}$-dependent terms.

Traditionally, top partners are introduced to solve the hierarchy problem. The
new contribution $\Delta m_{H}^{2}|_{\rm top\;partner}$ from the top partner
loop cancels the quadratically divergent term $\Lambda_{\rm NP}^{2}$, which is
guaranteed by some new symmetry, such as supersymmetry. However, the symmetry
can not be exact and thus the Higgs mass term is still generated due to the
difference between the top quark mass and the top partner mass as

$\Delta m_{H}^{2}|_{\rm top}+\Delta m_{H}^{2}|_{\rm
top\;partner}\sim-\frac{3}{8\pi^{2}}y_{t}^{2}M_{T}^{2}\,{\rm
ln}\left(\frac{\Lambda_{\rm NP}^{2}}{M_{T}^{2}}\right)\leavevmode\nobreak\ ,$
(2)

where $M_{T}$ is the mass of the top partner. To get the light Higgs mass, the
Naturalness principle suggests top partners with $M_{T}\sim 500$ GeV.

However, after years of searches by the LHC, the bounds on the mass of colored
top partners have reached around $1500$ GeV for both scalar top partners [1,
2] and fermionic top partners [3, 4, 5]. The absence of top partners starts
challenging the naturalness of these traditional models due to the required
fine-tuning. To reduce fine-tuning, one alternative is to introduce uncolored
top partners. Since they are harder to be produced, the bound on their mass is
rather weak. It would be even better if the top partner is a SM singlet, which
is known as Neutral Naturalness, like in Twin Higgs models [6]. However, this
alternative is still based on the idea of symmetry and the cancellation
between $\Delta m_{H}^{2}|_{\rm top}$ and $\Delta m_{H}^{2}|_{\rm
top\;partner}$. In this work [7], we explore another direction that does not
require top partners. The idea is to have a strong dependence of the top
Yukawa coupling on the energy scale $y_{t}=y_{t}(k^{2})$, which can make the
top loop contribution converge. For example, taking

$y_{t}(k^{2})=y_{t}\left(\frac{\Lambda_{T}^{2}}{-k^{2}+\Lambda_{T}^{2}}\right),\
\ \Delta m_{H}^{2}|_{\rm
top}\sim-i\,2N_{c}\int\frac{d^{4}k}{(2\pi)^{4}}\,y_{t}^{2}(k^{2})\frac{k^{2}+m_{t}^{2}}{(k^{2}-m_{t}^{2})^{2}}\leavevmode\nobreak\
\sim-\frac{3}{8\pi^{2}}y_{t}^{2}\Lambda_{T}^{2}\leavevmode\nobreak\ ,$ (3)

where the top loop contribution is controlled by $\Lambda_{T}$, which is the
mass scale of new degrees of freedom responsible for the nontrivial behavior
of top Yukawa coupling.

##  2 Zoom in the Top Yukawa vertex

To realize the idea, the top Yukawa coupling must come from a more intricate
origin, which implies new physics in the top Yukawa vertex at high scales.
There are several possibilities as shown in Fig. 1 and each case (from left to
right) is described as follows (for more details, see our work [7]).

Figure 1: Three possible UV completions for the nontrivial behavior of the top
Yukawa coupling.

The first possibility corresponds to “large $y_{t}$ running”, where an
additional top-philic interaction introduces large running to reduce the top
Yukawa coupling at high scales. Negative contributions to the $y_{t}$ running
already exist in the SM but are not large enough. Therefore, we need a
stronger interaction to bring down the top Yukawa coupling at the desired
scale. The wavy propagator can be either a vector boson or a scalar boson. The
only requirement is that the coupling needs to be strong enough to have an
impact on the top Yukawa coupling.

Second, “radiative $y_{t}$ generation”, where the top Yukawa coupling is not
present at the tree level but is only generated at the one-loop level. In this
scenario, new top-philic bosons (wavy line) and vector-like fermions (double
line) are introduced to connect the Higgs boson and the top quarks. The top
Yukawa is only obtained after integrating out these new degrees of freedom,
which also means, above the mass scale of these intermediate states, the top
Yukawa will drop as desired. The interaction among these new particles should
also be strong enough to reproduce the correct value of the top Yukawa
coupling.

The last one “$y_{t}$ from four-fermion interactions”, different from the
first two scenarios, is only valid for composite Higgs models. In the first
two scenarios, an elementary Higgs and top quark are assumed so the
modification can only happen at the one-loop level, which thus requires strong
couplings. However, if the Higgs is composite, the top Yukawa coupling can
arise from four-fermion interactions. The idea can be traced back to Extended
Technicolor models [8, 9] but now evolved into a modern version with a light
composite Higgs, usually called Extended Hypercolor [10, 11]. In this
scenario, the top Yukawa originates from a dimension-six operator, which will
also decrease when passing the mass scale of the extended-hypercolor bosons.

##  3 Phenomenology

When talking about testing new physics, we always think about searches of new
resonances. However, it is not straightforward in this case as the main idea
is to modify the top Yukawa at high scales. Direct searches require us to
first specify the new degrees of freedom responsible for the modification,
which includes many possibilities as shown in the last section. Moreover, for
each scenario, there are different difficulties. For the first two, the
modification happens at the one-loop level, which means the quantum number of
the new states is not uniquely determined. Also, the requirement of strong
couplings implies some broad resonances, which makes them hard to look for.
For the extended-hypercolor bosons, again, the quantum number is diverse and
depends on the hypercolor group 222The only exception is the top-philic
$Z^{\prime}$ boson which will be studied in our next work.. Overall, it is
hard to perform direct searches due to the requirements of the UV theory and
strong couplings. However, the direct test of the idea can be realized in
other measurements in top physics.

###  3.1 Running top Yukawa and $t\bar{t}h$ differential cross section

Figure 2: The $y_{t}=y_{t}(k^{2})$ can be tested through $t\bar{t}h$
differential cross section.

Starting with the direct probe of the top Yukawa coupling at high scales, the
desired process is $t\bar{t}h$ final state as shown in Fig. 2. The nontrivial
behavior of the top Yukawa coupling at high scales will reveal itself in
differential momentum distributions of $t\bar{t}h$ production [12, 13].
However, based on the $t\bar{t}h$ cross section, it is hard to achieve the
desired sensitivity in the LHC era.

###  3.2 Running top mass and $t\bar{t}$ differential cross section

Figure 3: The $m_{t}=m_{t}(k^{2})$ can be tested through $t\bar{t}$
differential cross section.

Besides probing the top Yukawa coupling at high scales directly, we can also
measure the top quark mass at high scales instead, which is like simply
replacing the Higgs with its VEV. Without a Higgs, the measurement can then be
done from $t\bar{t}$ final states, which has a much larger total cross
section. Such measurement has already been done by the CMS collaboration using
run 2 data with an integrated luminosity of 35.9 fb-1 and the result presents
the running top mass up to $1$ TeV [14], which constraints the idea we
present. However, the result has also been reinterpreted in another theory
paper [15] but considering the energy scale only half of the value in the
original CMS paper. The result is then only sensitive up to an energy scale of
$500$ GeV. In this case, the bound on the top mass running becomes weaker and
light new physics with less fine-tuning is still possible.

###  3.3 Top-philic interaction and $t\bar{t}t\bar{t}$ cross section

Figure 4: New top-philic interactions can be tested through $t\bar{t}t\bar{t}$
cross section.

Modifying top Yukawa coupling always requires some new top-philic interaction,
which will unavoidably introduce additional contributions to the four top
cross section. Generic estimation predicts four top operators with
coefficients $\sim C/\Lambda_{T}^{2}$. To determine the Lorentz structure of
the operators and their coefficients, the UV completion is again required.
However, a sizable contribution is expected and it turns out the four top
quarks cross section might be the most promising channels to look for. In the
SM, the prediction was updated a few months ago with next-to-leading
logarithmic (NLL’) accuracy [16]. From the experimental side, both ATLAS and
CMS just announced the observation [17, 18] in Moriond. The latest result has
started constraining the ”large $y_{t}$ running” scenario with strongly
coupled top-philic bosons. The other scenarios are expected to be confirmed or
ruled out within the LHC era.

##  4 Conclusion

The top quark plays the most important role in the hierarchy problem.
Traditionally, top partners are introduced to cancel the top-loop
contribution. In this talk, we discuss an alternative where the top-loop
contribution is reduced by modifying the running of the top Yukawa coupling.
The new degrees of freedom expected to appear at the Naturalness scale
$\Lambda_{T}\sim 500$ GeV will be new top-philic particles instead of top
partners. Direct tests of the idea can be realized in top physics, including
$t\bar{t}h$, $t\bar{t}$, and $t\bar{t}t\bar{t}$ production, where Naturalness
might still be hidden !

## References

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