# Model-Independent Determination of Isotopic Cross Sections per Fission for
Reactor Antineutrinos

Yu-Feng Li<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Zhao Xin
<EMAIL_ADDRESS>(corresponding author) Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China  School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

( August 27, 2024, )

###### Abstract

Model-independent reactor isotopic cross sections per fission are determined
by global fits of the reactor antineutrino data from High-Enriched Uranium
(HEU) reactor rates, Low-Enriched Uranium (LEU) reactor rates, and reactor
fuel evolution data. Taking account of the implicit quasi-linear relationship
between the fission fractions of ${}^{239}\text{Pu}$ and ${}^{241}\text{Pu}$
in the LEU reactor data, the Inverse-Beta-Decay (IBD) yields and their
correlations of the fissionable isotopes ${}^{235}\text{U}$,
${}^{238}\text{U}$, and Pu’s are obtained. The data-driven isotopic IBD yields
provide an anomaly-free model for the reactor isotopic cross sections per
fission, where better than 1% accuracy of the expected reactor IBD yields can
be achieved for future experiments.

##  I Introduction

Reactor antineutrinos originate from the beta decays of the neutron-rich
fission fragments produced by the fission of four main isotopes
${}^{235}\text{U}$, ${}^{238}\text{U}$, ${}^{239}\text{Pu}$, and
${}^{241}\text{Pu}$ [1, 2, 3]. The nuclear reactors are one of the most
important sources of electron antineutrinos, which are applied to the study of
the fundamental properties of neutrinos [4], from the first discovery of the
neutrino [5] to the studies of the neutrino oscillation phenomenon [6, 7, 8,
9] and new physics beyond the Standard Model (SM) [10, 11].

The theoretical fluxes of reactor antineutrinos are crucial inputs for reactor
experiments, which can be obtained with two complementary methods [2, 3]. One
is usually called the conversion method, in which the antineutrino fluxes are
converted from the measured cumulative beta spectra, such as those of
${}^{235}\text{U}$, ${}^{239}\text{Pu}$, and ${}^{241}\text{Pu}$ at Institut
Laue-Langevin (ILL) [12, 13, 14] or at Kurchatov Institute (KI) [15] from the
thermal-neutron induced fission process, and that of ${}^{238}\text{U}$ at FRM
II in Garching [16] from the fission process induced by fast neutrons. The
other method is the ‘ab initio’ summation method, which directly sums the
energy spectra of all beta decay branches from all the fission fragments.
These two methods have motivated us to construct different theoretical models
for reactor antineutrino fluxes. The Huber-Mueller (HM) model is arguably the
most well-known reactor flux model, in which the antineutrino fluxes of
${}^{235}\text{U}$, ${}^{239}\text{Pu}$ and ${}^{241}\text{Pu}$ are obtained
by Huber [17] with the conversion method, and the flux of ${}^{238}\text{U}$
is obtained by Mueller et al. [18] using the summation method. Hayen,
Kostensalo, Severijns, and Suhonen [19] improved the conversion method by
including the forbidden transitions, and proposed a correction to the HM
model, which is abbreviated as the HKSS model. In Ref. [15], a new measurement
of the cumulative beta spectra of ${}^{235}\text{U}$ is considered and a
corrected ${}^{235}\text{U}$ antineutrino flux is obtained. Moreover, in the
KI model, a converted ${}^{238}\text{U}$ antineutrino flux is also presented
based on the measured beta spectra of ${}^{238}\text{U}$ in FRM II. On the
other hand, with the development of the nuclear database and the Pandemonium
free data [20, 21, 22, 23, 24, 25, 26], the updated summation model by
Estienne, Fallot et al. [27] is also presented (EF model). More details on the
properties and comparisons of these models are summarized in Ref. [28]. In
addition, some other studies on the theoretical reactor antineutrino fluxes
[13, 29, 30, 31, 32, 33] are also established to offer precise predictions of
the experimental observation.

However, the experimental measurements of reactor antineutrino rates and
energy spectra have shown anomalous results compared to the theoretical
predictions. One is a $6\%$ deficit between the experimental and theoretical
reactor antineutrino rates [34] (i.e., the rate anomaly). The other is a
spectral excess between the experimental and theoretical reactor antineutrino
spectra [35, 36, 37] (i.e., the shape anomaly). These anomalies have
questioned the foundation of all the aforementioned reactor antineutrino flux
models. In Ref. [28], it is shown that both the EF summation model and the KI
conversion model tend to reduce the ${}^{235}\text{U}$ flux, and provide a
plausible solution to the reactor rate anomaly, but the solution to the
reactor shape anomaly is still elusive. Therefore, it would be important to
obtain a model-independent determination of the reactor antineutrino fluxes,
which is the main focus of this work. By using the quasi-linear relationship
between the fission fractions of ${}^{239}\text{Pu}$ and ${}^{241}\text{Pu}$,
we present a method to directly obtain the isotopic Inverse-Beta-Decay (IBD)
yields from global fits of the reactor antineutrino data from High-Enriched
Uranium (HEU) reactor rates, Low-Enriched Uranium (LEU) reactor rates, and
reactor fuel evolution data. The IBD yields and their correlations of the
fissionable isotopes ${}^{235}\text{U}$, ${}^{238}\text{U}$, and Pu’s are
obtained, and can be applied to future reactor experiments. Taking the
representative values of fission fractions for typical commercial reactors,
the accuracy of the expected IBD yield can be better than 1%.

This work is organized as follows. In Section II, we discuss how we deduce the
data-driven isotopic IBD yields. In Section III, we provide results of the
global fits of the reactor data. We present and discuss different fitting
results of the HEU reactor rates, fuel evolution data and LEU reactor rates
from Section III.1 to Section III.3. Moreover, the application how to apply
the data-driven isotopic IBD yields to the prediction of future reactor
experiments is illustrated in Section III.4. Finally, the concluding remarks
are summarized in Section IV.

##  II Model independent method

The reactor IBD yield $\sigma_{f,a}$ is used to illustrate the measured
reactor antineutrino rates known as cross section per fission as well:

$\displaystyle\sigma_{f,a}=\sum_{i}f_{i}^{a}\sigma_{i}\,,$ (1)

where $a$ represents the sequence of the experiments, $\sigma_{i}$ is the
isotopic IBD yield for a certain fissionable isotope ($i=$235, 238, 239 and
241 for ${}^{235}\text{U}$, ${}^{238}\text{U}$, ${}^{239}\text{Pu}$ and
${}^{241}\text{Pu}$, respectively), and $f_{i}^{a}$ is the fission fraction of
the isotope $i$ in the experiment $a$. For a certain fissionable isotope $i$,
whose isotopic IBD yield is defined as

$\displaystyle\sigma_{i}=\int_{E_{\nu}^{\text{thr}}}^{E_{\nu}^{\text{max}}}\Phi_{i}(E_{\nu})\sigma_{\text{IBD}}(E_{\nu})dE_{\nu},$
(2)

where $E_{\nu}$ is the antineutrino energy, $\Phi(E_{\nu})$ is the
antineutrino flux and $\sigma_{\text{IBD}}(E_{\nu})$ is the IBD cross section
[38, 39]. The theoretical isotopic IBD yield for a certain isotope $i$ from
different flux models [18, 16, 17, 27, 19, 15] has been revisited in [28], and
will be used in the current analysis.

The nuclear reactors of reactor antineutrino experiments can be divided into
two categories: the commercial reactors mainly composited with
${}^{235}\text{U}\,(\sim 50-60\%)$ and ${}^{239}\text{Pu}\,(\sim 25-35\%)$,
which is also known as the LEU nuclear power reactors; and the research
reactors with practically pure ${}^{235}\text{U}$ known as HEU research
reactors. Therefore, the reactor antineutrino data are dominated by
${}^{235}\text{U}$ and ${}^{239}\text{Pu}$ and are provided in Table 1, which
is reordered from Table IV of Ref. [28] by considering different types of
reactors. In Table 1, $a$ is the index of each experiment, $f^{a}_{i}$’s are
the effective fission fractions of the four isotopes,
$\sigma_{f,a}^{\text{exp}}$ is the experimental IBD yield in units of
$10^{-43}\text{cm}^{2}/\text{fission}$, $\delta_{a}^{\text{exp}}$ is the total
relative experimental statistical plus systematic uncertainty,
$\delta_{a}^{\text{cor}}$ is the part of the relative experimental systematic
uncertainty which is correlated in each group of experiments indicated by the
braces; $L_{a}$ is the source-detector distance. In addition, we also consider
the fuel evolution data from Daya Bay [40] and RENO [41], each of which
provided the experimental IBD yields as a function of the fission fractions.

rate | $a$ | Experiment | $f^{a}_{235}$ | $f^{a}_{238}$ | $f^{a}_{239}$ | $f^{a}_{241}$ | $\sigma_{f,a}^{\text{exp}}$ |  $\delta_{a}^{\text{exp}}$ [%] |  $\delta_{a}^{\text{cor}}$ [%] |  $L_{a}$ [m]
---|---|---|---|---|---|---|---|---|---|---
LEU | 1 | Chooz | 0.496 | 0.087 | 0.351 | 0.066 | 6.12 | 3.2 | 0 | $\approx 1000$
2 | Palo Verde | 0.600 | 0.070 | 0.270 | 0.060 | 6.25 | 5.4 | 0 | $\approx 800$
3 | Daya Bay | 0.564 | 0.076 | 0.304 | 0.056 | 5.94 | 1.5 | 0 | $\approx 550$
4 | RENO | 0.571 | 0.073 | 0.300 | 0.056 | 5.85 | 2.1 | 0 | $\approx 411$
5 | Double Chooz | 0.520 | 0.087 | 0.333 | 0.060 | 5.71 | 1.1 | 0 | $\approx 415$
6 | Bugey-4 | 0.538 | 0.078 | 0.328 | 0.056 | 5.75 | 1.4 |  }220pt[1.4] | $15$
7 | Rovno91 | 0.614 | 0.074 | 0.274 | 0.038 | 5.85 | 2.8 |  | $18$
8 | Rovno88-1I | 0.607 | 0.074 | 0.277 | 0.042 | 5.70 | 6.4 |  }220pt[3.1] }520pt[2.2] | $18$
9 | Rovno88-2I | 0.603 | 0.076 | 0.276 | 0.045 | 5.89 | 6.4 |  | $17.96$
10 | Rovno88-1S | 0.606 | 0.074 | 0.277 | 0.043 | 6.04 | 7.3 |  }345pt[3.1] | $18.15$
11 | Rovno88-2S | 0.557 | 0.076 | 0.313 | 0.054 | 5.96 | 7.3 |  | $25.17$
12 | Rovno88-3S | 0.606 | 0.074 | 0.274 | 0.046 | 5.83 | 6.8 |  | $18.18$
13 | Bugey-3-15 | 0.538 | 0.078 | 0.328 | 0.056 | 5.77 | 4.2 |  }320pt[4.0] | $15$
14 | Bugey-3-40 | 0.538 | 0.078 | 0.328 | 0.056 | 5.81 | 4.3 |  | $40$
15 | Bugey-3-95 | 0.538 | 0.078 | 0.328 | 0.056 | 5.35 | 15.2 |  | $95$
16 | Gosgen-38 | 0.619 | 0.067 | 0.272 | 0.042 | 5.99 | 5.4 |  }320pt[2.0] }420pt[3.8] | $37.9$
17 | Gosgen-46 | 0.584 | 0.068 | 0.298 | 0.050 | 6.09 | 5.4 |  | $45.9$
18 | Gosgen-65 | 0.543 | 0.070 | 0.329 | 0.058 | 5.62 | 6.7 |  | $64.7$
HEU | 19 | ILL | 1 | 0 | 0 | 0 | 5.30 | 9.1 |  | $8.76$
20 | Krasnoyarsk87-33 | 1 | 0 | 0 | 0 | 6.20 | 5.2 |  }220pt[4.1] | $32.8$
21 | Krasnoyarsk87-92 | 1 | 0 | 0 | 0 | 6.30 | 20.5 |  | $92.3$
22 | Krasnoyarsk94-57 | 1 | 0 | 0 | 0 | 6.26 | 4.2 | 0 | $57$
23 | Krasnoyarsk99-34 | 1 | 0 | 0 | 0 | 6.39 | 3.0 | 0 | $34$
24 | SRP-18 | 1 | 0 | 0 | 0 | 6.29 | 2.8 | 0 | $18.2$
25 | SRP-24 | 1 | 0 | 0 | 0 | 6.73 | 2.9 | 0 | $23.8$
26 | STEREO | 1 | 0 | 0 | 0 | 6.34 | 2.5 | 0 | $9-11$
Table 1:  List of the experiments which measured the absolute reactor
antineutrino fluxes. For each experiment numbered with the index $a$:
$f^{a}_{235}$, $f^{a}_{238}$, $f^{a}_{239}$, and $f^{a}_{241}$ are the
effective fission fractions of the four isotopes ${}^{235}\text{U}$,
${}^{238}\text{U}$, ${}^{239}\text{Pu}$, and ${}^{241}\text{Pu}$,
respectively; $\sigma_{f,a}^{\text{exp}}$ is the experimental IBD yield in
units of $10^{-43}\text{cm}^{2}/\text{fission}$; $\delta_{a}^{\text{exp}}$ is
the total relative experimental statistical plus systematic uncertainty,
$\delta_{a}^{\text{cor}}$ is the part of the relative experimental systematic
uncertainty which is correlated in each group of experiments indicated by the
braces; $L_{a}$ is the source-detector distance.

Based on the intrinsic reactor fuel burnup relation, it is widely conceived
that there is a quasi-liner relationship [42] between the fission fractions of
${}^{241}\text{Pu}$ and ${}^{239}\text{Pu}$ in LEU reactors:

$\displaystyle f_{241}=k\cdot f_{239}\,,$ (3)

where $f_{239}$ and $f_{241}$ are the fission fractions of ${}^{239}\text{Pu}$
and ${}^{241}\text{Pu}$, respectively, and $k$ is the fitting linear
coefficient from reactor rates or evolution data, which are shown in Table 2
for different groups of data sets. In this work, we always use the value from
the combined data with $k=0.177$. Therefore, the IBD yield for a certain LEU
reactor antineutrino experiment can be expressed as

$\displaystyle\sigma_{f,a}=$ $\displaystyle
f_{235}^{a}\cdot\sigma_{235}+f_{238}^{a}\cdot\sigma_{238}+f_{239}^{a}\cdot\sigma_{239}+f_{241}^{a}\cdot\sigma_{241}$
$\displaystyle=$ $\displaystyle
f_{235}^{a}\cdot\sigma_{235}+f_{238}^{a}\cdot\sigma_{238}+f_{239}^{a}\cdot\sigma_{\text{Pu}}+(f_{241}^{a}-k\cdot
f_{239}^{a})\cdot\sigma_{241}$ $\displaystyle=$ $\displaystyle
f_{235}^{a}\cdot\sigma_{235}+f_{238}^{a}\cdot\sigma_{238}+f_{239}^{a}\cdot\sigma_{\text{Pu}}+\Delta
f^{a}\cdot\sigma_{241},$ (4)

where $a$ represents the index of the experiment, and

$\displaystyle\sigma_{\text{Pu}}=\sigma_{239}+k\cdot\sigma_{241}\,,$ (5)

is the combined isotopic IBD yield of Pu’s, with $k$ describing the linear
relationship between ${}^{239}\text{Pu}$ and ${}^{241}\text{Pu}$. $\Delta
f^{a}=f_{241}^{a}-k\cdot f_{239}^{a}$ is the residual fission fraction of
${}^{241}\text{Pu}$, which are shown in Figure 1(a) for LEU reactor rates
(left panel) and Figure 1(b) (right panel) for fuel evolution data. From the
figure, one can note that most of the residual fission fractions of
${}^{241}\text{Pu}$ are smaller than $1\%$, which indicates that model
dependence on the IBD yield corrections of the residual ${}^{241}\text{Pu}$
should be negligible.

Data sets |  $k$
---|---
LEU reactors rates |  $0.173$
Daya Bay |  $0.183$
RENO |  $0.185$
Evolution data |  $0.184$
Combined data |  $0.177$
Table 2: The fitting coefficients $k$’s obtained from different groups of
reactor data. In our analysis, we always use the value from the combined data
with $k=0.177$.

(a) LEU reactor rates

(b) Fuel evolution data

Figure 1: Residual fission fractions of ${}^{241}\text{Pu}$ for the LEU rate
data evaluated from Table 1 and fuel evolution data from Daya Bay[40] and RENO
[41] . Most of data points are smaller than $1\%$, while only two points
(Rovon 91 and Palo Verde) are slightly larger than $1\%$.

The least-squares function based on Wilks’ theorem [43] adopted in this work
can be written as follows:

$\displaystyle\chi^{2}=\sum_{a,b}\left(\sigma_{f,a}^{\text{exp}}-\sigma_{f,a}^{\text{fit}}\right)(V^{\text{exp}})^{-1}_{ab}\left(\sigma_{f,b}^{\text{exp}}-\sigma_{f,b}^{\text{fit}}\right),$
(6)

where $(a,b)$ are the indexes of the reactor rates or fuel evolution data,
$V^{\text{exp}}$ is the experimental covariance matrix,
$\sigma_{f,a}^{\text{exp}}$ is the experimental IBD yield, and according to
Eq. (4), $\sigma_{f,a}^{\text{fit}}$ is the predicted IBD yield to be fitted
in the analysis and can be expressed as

$\displaystyle\sigma_{f,a}^{\text{fit}}=f_{235}^{a}\cdot\sigma_{235}^{\text{fit}}+f_{238}^{a}\cdot\sigma_{238}^{\text{fit}}+f_{239}^{a}\cdot\sigma_{\text{Pu}}^{\text{fit}}+\Delta
f^{a}\cdot\sigma_{241}^{\text{mod}},$ (7)

where $\sigma_{235}^{\text{fit}}$, $\sigma_{238}^{\text{fit}}$ and
$\sigma_{\text{Pu}}^{\text{fit}}$ are the physical parameters which represent
the best-fit isotopic IBD yields for ${}^{235}\text{U}$, ${}^{238}\text{U}$,
and Pu’s, respectively. Meanwhile, $\Delta f^{a}\cdot\sigma_{241}^{\rm{mod}}$
is the model correction for the residual ${}^{241}\text{Pu}$.

##  III Fits of reactor experimental data

In this section, we shall present the analysis from the fits of reactor data
including reactor rates listed in Table 1 and reactor fuel evolution data from
Daya Bay [40] and RENO [41]. Eq. (6) and Eq. (7) will be used to fit with the
reactor data, and the HM model will be used for the corrections of residual
${}^{241}\text{Pu}$. The fitting results are shown in Table 3, Figure 2 and
Figure 3, and detailed analyses are provided in the following subsections.

In Section III.1, we show the fit of HEU reactor rates, which exclusively
constrain the ${}^{235}\text{U}$ isotopic IBD yield. In Section III.2, the
combined fit of fuel evolution data and HEU reactor rates will be presented,
which clearly illustrates the complementary role of each data set. In Section
III.3, the LEU reactor rate data will be added in the fit, providing a better
constraint on the ${}^{238}\text{U}$ isotopic IBD yield. The fitting results
based on all data sets are also illustrated in Section III.3. Finally, in
Section III.4 we prove the robustness of the data-driven model against the
model variations of the residual ${}^{241}\text{Pu}$, and apply to the
prediction of future reactor experiments.

| HEU rates | Evolution data | Evolution data + HEU rates | LEU rates | Evolution data + rates
---|---|---|---|---|---
$\chi^{2}_{\text{min}}$ |  $8.9$ |  $7.6$ |  $17.5$ |  $8.6$ |  $25.7$
NDF |  $7$ |  $13$ |  $21$ |  $15$ |  $37$
GoF |  $26\%$ |  $87\%$ |  $68\%$ |  $90\%$ |  $92\%$
$\sigma_{235}$ |  $6.36\pm 0.08$ |  $3.60^{+2.7}_{-1.1}$ |  $6.36\pm 0.08$ |  $6.85\pm 0.55$ |  $6.37\pm 0.08$
$\sigma_{238}$ |  $/$ |  $44.74^{+15}_{-37}$ |  $7.25\pm 1.88$ |  $0.68\pm 8.81$ |  $6.63\pm 1.30$
$\sigma_{\rm{Pu}}$ |  $/$ |  $1.52^{+4.0}_{-1.5}$ |  $5.60\pm 0.21$ |  $6.37\pm 2.13$ |  $5.64\pm 0.20$
Table 3: The model-independent IBD yields from the fits of different groups of
data sets. The units are $\times\,10^{-43}\,\rm{cm^{2}/fission}$. Figure 2:
One-dimensional and two-dimensional marginalized distribution with $1\sigma$,
$2\sigma$ and $3\sigma$ contours of $\sigma_{235}$, $\sigma_{238}$ and
$\sigma_{\text{Pu}}$ provided by a fit of evolution data + HEU reactor rates.
The red solid, dashed and dotted contours show the allowed regions of HEU
reactor rates at $1\sigma$, $2\sigma$ and $3\sigma$. The blue solid, dashed
and dotted contours show the allowed regions of evolution data at $1\sigma$,
$2\sigma$ and $3\sigma$. The darkgreen, lawngreen and yellow shaded regions
show the results of the allowed regions of evolution data + HEU reactor rates.
The three black curves show one-dimensional $\Delta\chi^{2}$ profiles for
$\sigma_{235}$, $\sigma_{238}$ and $\sigma_{\rm{Pu}}$ fitted with evolution
data + HEU reactor rates.  Figure 3: One-dimensional and two-dimensional
marginalized distribution with $1\sigma$, $2\sigma$ and $3\sigma$ contours of
$\sigma_{235}$, $\sigma_{238}$ and $\sigma_{\text{Pu}}$ provided by a fit of
evolution data + reactor rates. The blue solid, dashed and dotted contours
show the allowed regions of evolution data + HEU reactor rates at $1\sigma$,
$2\sigma$ and $3\sigma$ The darkgreen, lawngreen and yellow shaded regions
show the results of the allowed regions of evolution data + reactor rates. The
three black curves show one-dimensional $\Delta\chi^{2}$ profiles for
$\sigma_{235}$, $\sigma_{238}$ and $\sigma_{\rm{Pu}}$ fitted with evolution
data + reactor rates.

###  III.1 HEU reactors rates

For HEU reactors (i.e., the research reactors), the antineutrinos are
practically emitted from the beta decay generated from the fission products of
${}^{235}\text{U}$. Thus in Eq. (6), we shall only work with the isotopic IBD
yield of ${}^{235}\text{U}$ and the theoretical prediction for HEU reactor
rates in Eq. (7) can further be expressed as:

$\displaystyle\sigma_{f,a}^{\text{fit}}=\sigma_{235}^{\text{fit}}\,.$ (8)

The fitting results of HEU reactor rates are shown in the first column of
Table 3 and red lines in Figure 2. The red solid, dashed and dotted lines are
illustrated as the allowed regions of the ${}^{235}\text{U}$ isotopic IBD
yield at 1$\sigma$, 2$\sigma$ and 3$\sigma$ confidence levels respectively.
The best-fit value of $\sigma_{235}$ is $(6.36\pm
0.08)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$, as shown in the first column of
Table 3, which is a deficit about $6\%$ compared to the prediction from the HM
model, corresponding to a deviation about $4.7\,\sigma$ ($2.0\,\sigma$) lower
than the HM model, where the first number is the deviation from the central
value of the HM model, and the second number quoted in parentheses is the
deviation from HM model considering the model uncertainty. This result
supports the hypothesis that ${}^{235}\text{U}$ is the main isotope which is
responsible to the reactor rate anomaly [15, 44, 45, 40, 41].

###  III.2 Evolution data + HEU reactor rates

The evolution of the experimental IBD yield during the reactor fuel burning in
multiple burnup cycles has been measured at Daya Bay [40] and RENO [41], which
provides a powerful probe to constrain the reactor flux models and new physics
beyond the SM [44, 45, 46, 47, 48, 49, 50]. By constraining the isotopic IBD
yields of ${}^{241}\text{Pu}$ and ${}^{238}\text{U}$ from the HM model, it is
shown that $\sigma_{235}$ is 8% smaller than the HM model while $\sigma_{239}$
is consistent with the prediction [40, 41]. In order to remove the model
dependence, we analyze the fuel evolution data from Daya Bay and RENO using
three independent isotopic IBD yields $\sigma_{235}$, $\sigma_{238}$ and
$\sigma_{\text{Pu}}$ defined in Eq. (7). The fitting results are shown in the
second column of Table 3 and blue lines of Figure 2, where the blue solid,
dashed and dotted contours are the allowed regions of fuel evolution data at
$1\sigma$, $2\sigma$ and $3\sigma$ confidence levels respectively. One can
observe that the fuel evolution data themselves cannot measure the individual
isotopic IBD yields, but provide stringent constraints on the linear
combinations of the three components. Therefore, in order to break the
parameter degeneracy, it is quite important to combine the fuel evolution data
and HEU reactor rates in the analysis.

In the third column of Table 3 and shaded regions of Figure 2, we illustrate
the combined analysis of the fuel evolution data and HEU reactor rates, where
the darkgreen, lawngreen and yellow regions are shown for the $1\sigma$,
$2\sigma$ and $3\sigma$ confidence levels respectively. As expected, we can
obtain the closed allowed regions for all three isotopic IBD yields. Firstly
$\sigma_{235}=(6.36\pm 0.08)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$ is
obtained, which is identical to that from the HEU rates-only analysis.
Secondly, a meaningful measurement of $\sigma_{\text{Pu}}=(5.60\pm
0.21)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$ and $\sigma_{238}=(7.25\pm
1.88)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$ has been obtained by virtue of
the synergy of two different data sets, in which the IBD yield of Pu’s is 2.2%
larger than the HM prediction, and that of ${}^{238}\text{U}$ is less than the
model by around 30%, corresponding to the significance of $0.4\sigma$ and
$1.4\sigma$ respectively, by taking account of the theoretical uncertainty of
the HM model. Our results on $\sigma_{235}$ and $\sigma_{\text{Pu}}$ are both
larger than those obtained in Ref. [40, 41] by constraining the IBD yields of
${}^{241}\text{Pu}$ and ${}^{238}\text{U}$, but their relative ratio is
consistent with the evolution-only fit, which demonstrates that only the
relative ratios of $\sigma_{235}$, $\sigma_{\text{Pu}}$ and $\sigma_{238}$ are
properly constrained. With the inclusion of HEU reactor rates, the isotopic
IBD yield of ${}^{235}\text{U}$ is fixed and those of Pu’s and
${}^{238}\text{U}$ are also decoupled, resulting in a complete and accurate
determination of all three isotopic IBD yields. Note that the value of
$\sigma_{238}$ deviates from the HM prediction with a significance of
$1.4\sigma$. Whether this value is due to statistical fluctuation or hints at
the problem of the ${}^{238}\text{U}$ flux requires future accurate
measurements, in particular for those from HEU reactor rates.

###  III.3 Evolution data + reactor rates

Model | $\sigma_{235}$ | $\sigma_{238}$ | $\sigma_{239}$ | $\sigma_{241}$ | $\sigma_{\text{Pu}}$
---|---|---|---|---|---
This work | $6.37\pm 0.08$ | $6.63\pm 1.30$ | $/$ | $/$ | $5.64\pm 0.20$
HM | $6.74\pm 0.17$ | $10.19\pm 0.83$ | $4.40\pm 0.13$ | $6.10\pm 0.16$ | $5.48\pm 0.16$
EF | $6.29\pm 0.31$ | $10.16\pm 1.02$ | $4.42\pm 0.22$ | $6.23\pm 0.31$ | $5.52\pm 0.23$
HKSS | $6.82\pm 0.18$ | $10.28\pm 0.84$ | $4.45\pm 0.13$ | $6.17\pm 0.16$ | $5.54\pm 0.16$
KI | $6.41\pm 0.14$ | $9.53\pm 0.48$ | $4.40\pm 0.13$ | $6.10\pm 0.16$ | $5.48\pm 0.16$
Table 4: Comparison of this data-driven isotopic IBD yield model with the
theoretical reactor flux models. The theoretical predictions of the HM, EF,
HKSS, and KI models are taken from Ref. [28]. The IBD yields are given in
units of $\left[10^{-43}\,\rm{cm^{2}/fission}\right]$.

In addition to the HEU reactor rates and fuel evolution data, there are also
reactor rates data from LEU reactors with different values of fission
fractions, as shown in Table 1. In the fourth column of Table 3, we present
the fit of all LEU reactors, in which $\sigma_{235}$, $\sigma_{\text{Pu}}$ are
nicely constrained, but $\sigma_{238}$ turns out to be vanishing and only
upper bound is obtained due to the positivity of all the isotopic IBD yields.
In order to further reveal the role of LEU rates data, the combined fit of
fuel evolution data and HEU reactor rates is compared to the combined fit of
fuel evolution data and all reactor rates in Figure 3, where the blue solid,
dashed and dotted contours show the allowed regions of the first combined fit
at $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels respectively, and the
darkgreen, lawngreen and yellow shaded regions show the allowed regions of the
second combined fit. Note that in the combined fit of evolution data and all
reactor rates, two data points from the Daya Bay and RENO rates in Table 1
have been excluded to avoid double counting. From the figure one can observe
that the inclusion of the LEU rates has negligible effects on the isotopic IBD
yields of $\sigma_{235}$, $\sigma_{\text{Pu}}$, but the improvement on
$\sigma_{238}$ is significant, where the relative uncertainty is reduced from
26% to 19%, and the significance of deviation from the HM model is also
increased from $1.4\sigma$ to $2.3\sigma$, raising a question on the
theoretical IBD yield of ${}^{238}\text{U}$ from both the summation and
conversion models. The best fits and associated uncertainties of
$\sigma_{235}$, $\sigma_{\text{Pu}}$ and $\sigma_{238}$ from all reactor data
are also provided in the last column of Table 3, where the isotopic IBD yields
for ${}^{235}\text{U}$, Pu’s and ${}^{238}\text{U}$ are $\sigma_{235}=(6.37\pm
0.08)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$, $\sigma_{\text{Pu}}=(5.64\pm
0.20)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$ and $\sigma_{238}=(6.63\pm
1.30)\,\times\,10^{-43}\,\rm{cm^{2}/fission}$, respectively. A comparison of
this data-driven model with the theoretical reactor flux models is illustrated
in Table 4, where the theoretical predictions of the HM, EF, HKSS, and KI
models are taken from Ref. [28]. From the table, it is shown that the data-
driven IBD yield for ${}^{235}\text{U}$ is consistent with EF and KI models
within $1\sigma$, but smaller than HM and HKSS models by around 2-3$\sigma$.
Meanwhile, the data-driven IBD yield for ${}^{238}\text{U}$ is also smaller
than all the theoretical models by around 2-3$\sigma$, and
$\sigma_{\text{Pu}}$ for Pu’s is consistent with all models within $1\sigma$.
These conclusions are consistent with those in Ref. [28], but from a rather
different viewpoint by decomposing the possible isotopic contributions.
Finally, from Figure 3 it is evident that possible correlations between
different isotopic contributions should be considered when applying this data-
driven model to the prediction of future reactor experiments.

###  III.4 Discussion and application

|  $\sigma^{\rm HM}_{241}$ ($\sigma^{\rm KI}_{241}$) | $\sigma^{\rm HKSS}_{241}$ | $\sigma^{\rm EF}_{241}$
---|---|---|---
$\chi^{2}_{\text{min}}$ |  $25.7$ |  $25.7$ |  $25.7$
NDF |  $37$ |  $37$ |  $37$
GoF |  $92\%$ |  $92\%$ |  $92\%$
$\sigma_{235}$ |  $6.37\pm 0.08$ |  $6.37\pm 0.08$ |  $6.37\pm 0.08$
$\sigma_{238}$ |  $6.63\pm 1.30$ |  $6.66\pm 1.30$ |  $6.68\pm 1.30$
$\sigma_{\rm{Pu}}$ |  $5.64\pm 0.20$ |  $5.63\pm 0.20$ |  $5.63\pm 0.20$
Table 5:  The data-driven isotopic IBD yields from global fits of all reactor
data based on the residual ${}^{241}\text{Pu}$ corrections from different
theoretical models in Ref. [28]. The IBD yields are given in units of
$\left[10^{-43}\,\rm{cm^{2}/fission}\right]$.

As previously mentioned, in the analysis of the data-driven isotopic IBD
yields, there is still a tiny level of model dependence on the corrections of
the residual ${}^{241}\text{Pu}$. To prove the robustness of the data-driven
model against the model variations, we have performed parallel global fits of
all reactor data by using the residual ${}^{241}\text{Pu}$ corrections from
different theoretical models in Ref. [28]. The results are shown in Table 5,
in which the variations of the data-driven isotopic IBD yields are as small as
$0.2\,\%$ and $0.7\,\%$ for $\sigma_{\rm{Pu}}$ and $\sigma_{238}$
respectively. Meanwhile $\sigma_{235}$ keeps unchanged as expected. In this
case the variations for the expected IBD yields would be at the level of 0.1%
considering the typical fission fractions of commercial reactors. Therefore,
the data-driven isotopic IBD yields are practically model-independent. In
summary, we have demonstrated the robustness of the isotopic IBD yields for
${}^{235}\text{U}$, ${}^{238}\text{U}$ and Pu’s and are going to propose the
following data-driven model for the isotopic IBD yields

$\displaystyle\left\\{\begin{aligned} \sigma_{235}&=(6.37\pm
0.08)\,\times\,10^{-43}\,\rm{cm^{2}/fission}\\\ \sigma_{238}&=(6.63\pm
1.30)\,\times\,10^{-43}\,\rm{cm^{2}/fission}\\\ \sigma_{\text{Pu}}&=(5.64\pm
0.20)\,\times\,10^{-43}\,\rm{cm^{2}/fission}\,,\end{aligned}\right.$ (9)

and the corresponding correlation matrix

$\displaystyle\rho=\left(\begin{array}[]{ccc}1.000&-0.751&0.571\\\
-0.751&1.000&-0.875\\\ 0.571&-0.875&1.000\\\ \end{array}\right)\,,$ (13)

where $\sigma_{i}$’s ($i=235,\,238,\,\text{and}\,\text{Pu}$) are from the fits
of all reactor data (i.e., evolution data + reactor rates) based on the
residual ${}^{241}\text{Pu}$ correction from $\sigma_{241}^{\text{HM}}$.

Figure 4: Predicted IBD yields for a future reactor experiment with both the
model independent IBD yields and theoretical flux models in Ref. [28]. The IBD
yields are given in units of $\left[10^{-43}\,\rm{cm^{2}/fission}\right]$.

Next as an application, we adopt the model-independent isotopic IBD yields to
predict the expected IBD yield of a future reactor antineutrino experiment
”$A$” with typical fission fractions
($f_{235}^{A},\,f_{238}^{A},\,f_{239}^{A},\,f_{241}^{A}$):

$\displaystyle\sigma_{A}=f_{235}^{A}\cdot\sigma_{235}+f_{238}^{A}\cdot\sigma_{238}+f_{239}^{A}\cdot\sigma_{\text{Pu}}+\Delta
f^{A}\cdot\sigma_{241}^{\rm{HM}}\,.$ (14)

By using the isotopic IBD yields from Eq. (9) and the correlation matrix of
Eq. (13), we can estimate the predicted IBD yields of a future experiment with
(${}^{235}\text{U}:{}^{238}\text{U}:{}^{239}\text{Pu}:{}^{241}\text{Pu})=(0.577:0.076:0.295:0.052$)
[51]:

$\displaystyle\sigma_{f}=[5.84\pm(0.04)_{\rm{MI}}\pm(0.0004)_{\rm{HM}}]\times\,10^{-43}\,\rm{cm^{2}/fission}\qquad(0.7\%\;{\rm
precision})\,,$ (15)

where the first and second terms represent the uncertainties originating from
model-independent (MI) isotopic IBD yields and the HM model input of
$\sigma_{241}$, respectively. Note that the latter one are completely
negligible, and here a sub-percent precision (i.e., 0.7% precision) of the
expected IBD yield has been obtained. A comparison of the predicted IBD yields
using both the model-independent method and theoretical flux models in Ref.
[28] is illustrated in Figure 4. From this comparison, we conclude that the
model-independent method has the best precision among all the models, which is
because the data-driven isotopic IBD yields have a specific form of the
correlation matrix as in Eq. (13), resulting in sizable cancellations between
different isotopic contributions. Note that the model-independent method
predicts a smaller IBD yield than the HM and HKSS models, free from the
reactor antineutrino anomaly.

If the considered reactors are different from the pressurized water reactor
(PWR), the linear relationship of Pu’s cannot fully account for all the Pu
contributions. Taking the European reactors with the mixed oxide (MOX)
technology as an example, the typical fission fractions are given as
${}^{235}\text{U}:{}^{238}\text{U}:{}^{239}\text{Pu}:{}^{241}\text{Pu}=0.000:0.080:0.708:0.212$
[52]. In this case, the predicted IBD yield is

$\displaystyle\sigma_{f}=\left[5.05\pm(0.07)_{\rm{MI}}\pm(0.01)_{\rm{HM}}\right]\times\,10^{-43}\,\rm{cm^{2}/fission}\qquad(1.4\%\,\text{precision})\,,$
(16)

which has an uncertainty of 1.4% and the contribution from
$\sigma^{\rm{HM}}_{241}$ is larger than that in Eq. (15), but still much
smaller than the model-independent part.

Finally before finishing this section, we would like to illustrate the
robustness of the analysis framework used in this study, in which the model-
independent isotopic IBD yields are obtained based on the linear relationship
of Pu’s and a fixed average ratio of $k$. Firstly, we consider different
values of $k$ in the linear fit of different data sets (i.e.,
$k=0.173,0.177,0.185$). The model-independent isotopic IBD yields
$\sigma_{235}$, $\sigma_{238}$ and the ratio of
$r_{\text{Pu}}=\sigma_{\rm{Pu}}/(\sigma_{239}^{\rm{HM}}+k\cdot\sigma_{241}^{\rm{HM}})$
will be essentially unchanged, which demonstrates that the predicted IBD yield
for a certain reactor neutrino experiment also keeps the same. Secondly, when
a $6\%$ fractional uncertainty of $k$ is taken into account in the $\chi^{2}$
function with a pull term, the fitting results are practically the same as
Table 3 and Eq. (13). Thirdly, a quadratic relationship of the fission
fractions of Pu’s is also considered, and the resulting IBD yield of a reactor
neutrino experiment is also consistent with the cases of the linear
relationship. All these aforementioned stress tests demonstrate the robustness
and credibility of the fitting framework, and ensure the model-independent
isotopic IBD yields reliable.

##  IV Conclusion

The latest measurements from reactor antineutrino rates and spectra have shown
anomalous results compared with the theoretical reactor flux models. In this
work, we have obtained a model-independent reactor isotopic cross sections per
fission from the global fits of the reactor antineutrino data from HEU reactor
rates, LEU reactor rates, and reactor fuel evolution data. By using the
implicit quasi-linear relationship between the fission fractions of
${}^{239}\text{Pu}$ and ${}^{241}\text{Pu}$ in the LEU reactor data, the IBD
yields of ${}^{235}\text{U}$, ${}^{238}\text{U}$, and Pu’s can be properly
constrained in the global fits, where the HEU reactor rates practically
provide a complete determination of the isotopic IBD yield of
${}^{235}\text{U}$, and the fuel evolution data put stringent constraints on
the linear combinations of $\sigma_{235}$, $\sigma_{\text{Pu}}$ and
$\sigma_{238}$. In addition, the inclusion of LEU reactor rates can further
reduce the uncertainty of the isotopic IBD yield of ${}^{238}\text{U}$. We
have also demonstrated the robustness of the data-driven isotopic IBD yields
and proposed a model-independent method to calculate the expected IBD yield of
future reactor experiments, in which better than 1% accuracy can be achieved
for those using typical commercial reactors. Our method can also be
generalized to the study of reactor antineutrino spectra from different
reactor experiments or experiments with different fission fractions. It would
be important to disentangle the isotopic contributions [53, 54] to the reactor
shape anomaly, which will be reported in a future separated work.

###### Acknowledgements.

Z. Xin is grateful to Dr. Jianrun Hu for the helpful discussion. The work of
Y.F. Li and Z. Xin was supported by National Natural Science Foundation of
China under Grant Nos. 12075255 and 11835013, by Beijing Natural Science
Foundation under Grant No. 1192019, by the Key Research Program of the Chinese
Academy of Sciences under Grant No. XDPB15. Y.F. Li is also grateful for the
support by the CAS Center for Excellence in Particle Physics (CCEPP).

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# A Complex Borel-Bernstein Theorem

Gerardo González Robert  Facultad de Ciencias, Universidad Nacional Autónoma
de México
Ciudad de México, México<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Zero-one laws are a central topic in metric Diophantine approximation. A
classical example of such laws is the Borel-Bernstein theorem. In this note,
we prove a complex analogue of the Borel-Bernstein theorem for complex Hurwitz
continued fractions. As a corollary, we obtain a complex version of Khinchin’s
theorem on Diophantine approximation.

##  1\. Introduction

For over a century, number theory and probability have had a fruitful
relationship. A particularly fecund environment for this alliance is the
theory of regular continued fractions, which we shall denote by

$[a_{0};a_{1},a_{2},\ldots]_{\mathbb{R}}=a_{0}+\frac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots}}}.$

The Borel-Bernstein Theorem is a classical result in this area, its proof can
be found in [3] (Theorem 1.11.).

######  Theorem 1 (Borel-Bernstein).

Let $\mathbf{u}=(u_{n})_{n\geq 1}$ be a sequence of positive real numbers and
define

$E_{\mathbb{R}}(\mathbf{u}):=\left\\{z=[0;a_{1},a_{2},\ldots]_{\mathbb{R}}\in[0,1]:a_{n}\geq
u_{n}\;\text{ for infinitely many }n\in\mathbb{N}\right\\}.$

If $\mathfrak{m}_{1}$ denotes the Lebesgue measure on $[0,1)$, then

$\mathfrak{m}_{1}(E_{\mathbb{R}}(\mathbf{u}))=\begin{cases}0,\text{ if
}\displaystyle\sum_{n\in\mathbb{N}}u_{n}^{-1}<+\infty,\\\ 1,\text{ if
}\displaystyle\sum_{n\in\mathbb{N}}u_{n}^{-1}=+\infty.\end{cases}$

In this note, we show a complex analogue of the Borel-Bernstein theorem using
Hurwitz continued fractions (defined on Section 2). Hurwitz continued
fractions allow us to represent every complex number as a finite or infinite
continued fraction of the form

$[a_{0};a_{1},a_{2},a_{3},\ldots]:=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots}}},$

where each $a_{n}$ is a Gaussian integer. In this context, a unit square
$\mathfrak{F}$ centered at the origin (defined on Section 2) will play the
role the interval $[0,1)$ plays within the regular continued fraction theory.

######  Theorem 2 (Main result).

Let $\mathbf{u}=(u_{n})_{n\geq 1}$ be a sequence of positive real numbers and
define

$E(\mathbf{u})=\left\\{z=[0;a_{1},a_{2},\ldots]\in\mathfrak{F}:|a_{n}|\geq
u_{n}\;\text{ for infinitely many }n\in\mathbb{N}\right\\}.$

If $\mathfrak{m}$ is the Lebesgue measure on $\mathfrak{F}$, then

$\mathfrak{m}(E(\mathbf{u}))=\begin{cases}0,\text{ if
}\displaystyle\sum_{n\in\mathbb{N}}u_{n}^{-2}<+\infty,\\\ 1,\text{ if
}\displaystyle\sum_{n\in\mathbb{N}}u_{n}^{-2}=+\infty.\end{cases}$

We obtain a complex version of Khinchin’s theorem on Diophantine approximation
(Theorem 1.10 in [3]) as a consequence of Theorem 2.

######  Corollary 3.

Given $\psi:\mathbb{R}_{\geq 1}\to\mathbb{R}_{>0}$, define the set

$K(\psi)=\left\\{\xi\in\mathbb{C}:\left|\xi-\frac{p}{q}\right|\leq\psi(|q|)\;\text{
for infinitely many }p,q\in\mathbb{Z}[i],q\neq 0\right\\}.$

If $x\mapsto x^{2}\psi(x)$ is non-increasing, then

$\mathfrak{m}(K(\psi))=\begin{cases}0,\text{ if }\displaystyle\sum_{n\geq
1}n^{3}\psi(n)^{2}<+\infty,\\\ \text{full},\text{ if }\displaystyle\sum_{n\geq
1}n^{3}\psi(n)^{2}=+\infty.\end{cases}$

A. Nogueira studied a Borel-Bernstein theorem for some multidimensional
continued fractions algorithms, such as the Perron-Jacobi and Brun algorithm
[15]. Some results resembling Corollary 3 are well known. In 1952, W. J.
Leveque introduced a complex continued fraction expansion and he used it to
show a Khinchin-type result (Theorem 5 in [13]). In 1982, with the aid of
disjoint spheres and Bianchi groups, D. Sullivan obtained a result stronger
than Corollary 3 (see Theorem 5 in [16]). In 2006, V. Beresnevich, D.
Dickinson, and S. Velani strengthened Sullivan’s result by applying their work
on ubiquitous systems. They proved that Corollary 3 still holds if we replace
“$x\mapsto x^{2}\psi(x)$ is non-increasing” by “$\psi$ is non-increasing”
(Theorem 7 in [1]). Moreover, in the recent paper [7], Y. He and Y. Xiong used
Hurwitz continued fractions to compute the Hausdorff dimension of $K(\psi)$
for non-increasing functions $\psi$ such that $\psi(x)$ tends to $0$ when $x$
tends to $+\infty$. We can combine Theorem 8 in [1] and Theorem 1.2 in [7] to
obtain the next result:

######  Theorem 4.

Let $\psi:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ be a non-increasing function such
that $\psi(x)\to 0$ when $x\to+\infty$. Define

$\lambda:=\lambda(\psi)=\liminf_{x\to+\infty}\frac{-\log\psi(x)}{\log(x)}$

and let $\operatorname{D}(\psi)$ be the Hausdorff dimension of $K(\psi)$. If
$\lambda>2$, then $\operatorname{D}(\psi)=\tfrac{4}{\lambda}$ and
$\operatorname{D}(\psi)=2$ whenever $\lambda\leq 2$. Furthermore, if
$\mathcal{H}^{\operatorname{D}(\psi)}$ denotes the
$\operatorname{D}(\psi)$-Hausdorff measure on $\mathbb{C}$, we have

$\mathcal{H}^{\operatorname{D}(\psi)}(K(\psi))=\begin{cases}0,\text{ if
}\sum_{n\geq 1}\psi(n)^{\operatorname{D}({\psi})}n^{3}<+\infty,\\\ \text{full,
if }\sum_{n\geq 1}\psi(n)^{\operatorname{D}({\psi})}n^{3}=+\infty.\end{cases}$

When we choose $\varepsilon>0$ and $\psi(x)=x^{-(1+\varepsilon)}$, Theorem 4
is precisely the complex Jarník-Besicovitch Theorem (Theorem 6.1 in [5]). A
nice discussion of Khinchin-type results and a simple proof of the complex
Jarník-Besicovitch Theorem can be found in [5]. A related multidimensional
theorem was proven by M. Hussain in [10]. None of these papers use on any
Borel-Bernstein-type result. However, they do rely on the convergence part of
the Borel-Cantelli Lemma, whose proof can be found in almost every measure
theory book.

######  Lemma 5 (Borel-Cantelli lemma, convergence part).

Let $(X,\mathscr{A},\mu)$ be a measure space and $(A_{n})_{n\geq 1}$ a
sequence in $\mathscr{A}$. If $\sum_{n}\mu(A_{n})<+\infty$, then
$\mu(\limsup_{n}A_{n})=0$.

The organization of this note is as follows: in the next section, we define
Hurwitz continued fractions and prove some useful lemmas; afterwards, we prove
Theorem 2 and Corollary 3.

Notation. The natural numbers, $\mathbb{N}$, are the positive integers and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$. If $(x_{n})_{n\geq 1}$ and
$(y_{n})_{n\geq 1}$ are two sequences of non-negative real numbers, $x_{n}\ll
y_{n}$ means that there is some constant $\kappa>0$ such that $x_{n}\leq\kappa
y_{n}$ holds for sufficiently large $n\in\mathbb{N}$. We write $x_{n}\asymp
y_{n}$ if $x_{n}\ll y_{n}$ and $y_{n}\ll x_{n}$ are true. Whenever the
constant implied by $\ll$ depends on some object $\alpha$, we shall write
$\ll_{\alpha}$. For any $A\subseteq\mathbb{C}$, $A^{*}:=A\setminus\\{0\\}$,
$\operatorname{Cl}(A)$ is the closure of $A$, $\operatorname{int}A$ is the
interior of $A$, $zA:=\\{za:a\in A\\}$ for a given $z\in\mathbb{C}$, and, if
$A$ is finite, $\\#A$ is the number of elements contained in $A$. For any
$z\in\mathbb{C}$ and $r>0$, we write
$\mathbb{D}(z;r):=\\{w\in\mathbb{C}:|w-z|<r\\}$,
$\overline{\mathbb{D}}(z;r):=\operatorname{Cl}(\mathbb{D}(z;r))$,
$\mathbb{E}(z):=\mathbb{C}\setminus\overline{\mathbb{D}}(z;1)$, and
$\overline{\mathbb{E}}(z):=\operatorname{Cl}(\mathbb{E}(z))$. We also write
$\|z\|=\max\\{\Re(z)|,|\Im(z)|\\}$ for any $z\in\mathbb{C}$. More notation and
terminology is defined throughout the text.

##  2\. Hurwitz Continued Fractions

Let us define Hurwitz continued fraction. Denote by
$\lfloor\cdot\rfloor:\mathbb{R}\to\mathbb{Z}$ the usual floor function, that
is, $\lfloor x\rfloor$ is the largest integer less than or equal to $x$ for
any real number $x$. Define the function $[\cdot]:\mathbb{C}\to\mathbb{Z}[i]$
by

$\forall
z\in\mathbb{C}\quad[z]=\left\lfloor\Re(z)+\frac{1}{2}\right\rfloor+i\left\lfloor\Im(z)+\frac{1}{2}\right\rfloor,$

and the sets

$\mathfrak{F}:=\\{z\in\mathbb{C}:[z]=0\\},\quad\mathfrak{F}^{*}:=\mathfrak{F}\setminus\\{0\\},\quad\mathfrak{F}^{\prime}:=\mathfrak{F}\setminus\mathbb{Q}(i).$

Let $a_{1}:\mathfrak{F}^{*}\to\mathbb{Z}[i]$ be given by $a_{1}(z)=[z^{-1}]$
and $T:\mathfrak{F}\to\mathfrak{F}$ by

$\forall z\in\mathfrak{F}\quad
T(z)=\begin{cases}\frac{1}{z}-\left[\frac{1}{z}\right],\text{ if }z\neq 0,\\\
0,\text{ if }z=0.\end{cases}$

As usual, $T^{0}:\mathfrak{F}\to\mathfrak{F}$ is the identity map and
$T^{n}=T^{n-1}\circ T$ for $n\in\mathbb{N}$. The function $T$ is a complex
analogue of the usual Gauss map.

Let $z$ be any complex number. Define the possibly finite sequence of Gaussian
integers $(a_{n}(z))_{n\geq 0}$ as follows. First, $a_{0}(z):=[z]$ and, as
long as $T^{n-1}(z-a_{0}(z))\neq 0$, put

$a_{n}(z)=a_{1}\left(T^{n-1}(z-a_{0}(z))\right).$

The Hurwitz continued fraction (HCF) of $z$ is

$[a_{0};a_{1},a_{2},\ldots]:=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots}}}.$

We shall refer to the sequence $(a_{n}(z))_{n\geq 0}$ as the Hurwitz elements
of $z$ and we will omit the dependence on $z$ if there is no risk of
ambiguity.

Several properties of regular continued fractions hold too in the complex
case, although their known proofs might be much more complicated (for example,
the third point of proposition 6 below). Among the properties that remain
true, we have that the HCF of any $z\in\mathbb{Q}(i)$ is finite and equals
$z$. Also, when $z$ is an irrational complex number, that is
$z\in\mathbb{C}^{\prime}:=\mathbb{C}\setminus\mathbb{Q}(i)$, the sequence
$(a_{n})_{n\geq 1}$ is infinite and

$\lim_{n\to\infty}[a_{0};a_{1},\ldots,a_{n}]=z$

(see [9] for a proof).

######  Proposition 6.

Let $z$ belong to $\mathbb{C}^{\prime}$, let $(a_{n})_{n\geq 1}$ be its
Hurwitz elements and define the sequences of Gaussian integers $(p_{n})_{n\geq
0}$, $(q_{n})_{n\geq 0}$ by

$\begin{pmatrix}p_{-1}&p_{-2}\\\
q_{-1}&q_{-2}\end{pmatrix}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\forall n\in\mathbb{N}\quad\begin{pmatrix}p_{n}\\\
q_{n}\end{pmatrix}=\begin{pmatrix}p_{n-1}&p_{n-2}\\\
q_{n-1}&q_{n-2}\end{pmatrix}\begin{pmatrix}a_{n}\\\ 1\end{pmatrix}.$

The following statements hold for all $n\in\mathbb{N}_{0}$:

  1. 1.

$q_{n}p_{n-1}-q_{n-1}p_{n}=(-1)^{n}$,

  2. 2.

If $z_{n}=T^{n-1}(z-a_{0})$, then
$z=\displaystyle\frac{p_{n-2}z_{n}+p_{n-1}}{q_{n-2}z_{n}+q_{n-1}}$,

  3. 3.

$|q_{n}|<|q_{n+1}|$.

  4. 4.

$\tfrac{p_{n}}{q_{n}}=[a_{0};a_{1},\ldots,a_{n}]$.

The proofs of 3 and 4 can be found in [9]. The rest of the theorem is proven
in a broader context as Propositions 3.3 and 3.7 in [4]. Proposition 6 also
holds for finite HCF and any integer $n$ such that the corresponding objects
are defined.

We shall now explore the geometry of $T$. Let
$\iota:\mathbb{C}^{*}\to\mathbb{C}^{*}$ be the complex inversion,
$\iota(z)=z^{-1}$, then

(1)
$\iota[\mathfrak{F}]:=\overline{\mathbb{E}}(-1)\cap\overline{\mathbb{E}}(i)\cap\mathbb{E}(1)\cap{\mathbb{E}(-i)},$

(see the paragraph on notation by then end of the introduction) and thus

(2) $\forall z\in\mathbb{C}^{\prime}\quad\forall
n\in\mathbb{N}\quad|a_{n}|\geq\sqrt{2}.$

Hence, if for any $\mathbf{a}=(a_{1},\ldots,a_{n})\in\mathbb{Z}[i]^{n}$ we
define the cylinder

$\mathcal{C}_{n}(\mathbf{a}):=\\{z\in\mathfrak{F}:a_{1}(z)=a_{1},\ldots,a_{n}(z)=a_{n}\\},$

we obtain from (2) that $\mathcal{C}_{1}(a)=\varnothing$ for
$a\in\\{0,1,-1,i,-i\\}$. Equation (1) along with some direct computations show
that $T[\mathcal{C}_{1}(a)]=\mathfrak{F}$ when $|a|\geq\sqrt{8}$. In general,
given $a\in\mathbb{Z}[i]$ with $|a|\in\\{\sqrt{2},2,\sqrt{5}\\}$, the set
$T[\mathcal{C}_{1}(a)]$ may assume fourteen different forms and, if we
overlook the boundaries, it may assume only four different forms or any of
their rotations by right angles (see Section 2 in [6]). Moreover, if for any
$n\in\mathbb{N}$ and $\mathbf{a}\in\mathbb{Z}[i]^{n}$ we write

$\mathfrak{F}_{n}(\mathbf{a}):=T^{n}[\mathcal{C}_{n}(\mathbf{a})],$

an inductive argument shows that the set
$\\{\mathfrak{F}_{n}(\mathbf{a}):n\in\mathbb{N},\mathbf{a}\in\mathbb{Z}[i]^{n}\\}\setminus\\{\varnothing\\}$
is finite and we may conclude that

$\\{\mathfrak{m}\left(\mathfrak{F}_{n}(\mathbf{a})\right):n\in\mathbb{N},\mathbf{a}\in\mathbb{Z}[i]^{n}\\}\setminus\\{0\\}$

is non-empty and finite (cfr. Section 2 in [6]).

It may happen that $\mathfrak{F}_{n}(\mathbf{a})\neq\varnothing$ but
$\mathfrak{m}(\mathfrak{F}_{n}(\mathbf{a}))=0$. Given $n\in\mathbb{N}$, we say
that $\mathbf{a}\in\mathbb{Z}[i]^{n}$ is regular if
$\mathcal{C}_{n}(\mathbf{a})$ has non-empty interior and irregular otherwise.
An infinite sequence in $\mathbb{Z}[i]$ is regular if all its prefixes are
regular and irregular otherwise. Finally, a complex number is regular if the
sequence of its HCF elements is regular and it is irregular in other case.
Some properties follow directly from the definitions, inductive arguments, and
our previous discussions. For example, if $\mathbf{a}\in\mathbb{Z}^{n}$ is
irregular and $\mathcal{C}_{n}(\mathbf{a})\neq\varnothing$, then
$\mathcal{C}_{n}(\mathbf{a})$ is contained in some arc or in some line
segment. Therefore, the set of irregular complex numbers is null relative to
the Lebesgue measure. Also, for any $n\in\mathbb{N}$ and any regular
$\mathbf{a}\in\mathbb{Z}[i]^{n}$ we have

(3) ${\text{for some }}k\in\\{0,1,2,3\\}\quad
i^{k}\operatorname{int}\mathfrak{F}_{1}({1-i})\subseteq\mathfrak{F}_{n}(\mathbf{a})\subseteq\mathfrak{F}$

(cfr. Lemma 1 in [6]). In what follows, we ignore irregular numbers, for they
form a $\mathfrak{m}$-null set. However, for completeness sake, we exhibit an
irregular sequence. If $n\in\mathbb{N}_{\geq 2}$, then the set

$\iota\left[\mathfrak{F}_{1}(-1+i)\right]\cap\\{z+1-in:z\in\mathfrak{F}\\}\neq\varnothing$

is a segment of the line determined by $\Re(z)=\frac{1}{2}$, so $(-1+i,1-in)$
is irregular.

We will not delve into the structure of the shift space associated to the
dynamical system $(\mathfrak{F},T)$. We content ourselves with pointing out
that it is rather complicated (see section 5.3 in [8]).

For any $z\in\mathbb{C}$, write $\|z\|:=\max\\{|\Re(z)|,|\Im(z)|\\}$. Hence,
for every $m\in\mathbb{N}_{\geq 3}$ we have

$\displaystyle\\#\left(\\{b\in\mathbb{Z}[i]:\|b\|=m\\}\cap\iota\left[\mathfrak{F}\right]\right)$
$\displaystyle=8m,$
$\displaystyle\\#\left(\\{b\in\mathbb{Z}[i]:\|b\|=m\\}\cap\iota\left[\operatorname{int}\mathfrak{F}_{1}({1-i})\right]\right)$
$\displaystyle=2m-1.$

For $m=2$, the first set contains fourteen elements while the second set has
three elements. In view of (3), we can conclude the next estimate:

######  Lemma 7.

Let $n\in\mathbb{N}$ and $\mathbf{a}\in\mathbb{Z}[i]^{n}$ be such that
$\mathcal{C}_{n}(\mathbf{a})$ is regular, then

(4) $\forall m\in\mathbb{N}_{\geq
2}\qquad\\#\;\left\\{b\in\mathbb{Z}[i]:\|b\|=m\right\\}\cap\iota\left[\mathfrak{F}_{n}(\mathbf{a})\right]\asymp
m.$

######  Lemma 8.

Every $z=[0;a_{1},a_{2},\ldots]\in\mathfrak{F}^{\prime}$ satisfies

(5) $\forall
n\in\mathbb{N}_{0}\qquad\left|z-\frac{p_{n}}{q_{n}}\right|\leq\frac{4+2\sqrt{2}}{|a_{n+1}||q_{n}|^{2}}.$

###### Proof.

Take $z=[0;a_{1},a_{2},\ldots]\in\mathfrak{F}^{\prime}$ and $n\in\mathbb{N}$.
Write $z_{j}=[0;a_{j},a_{j+1},a_{j+2},\ldots]$ for every $j\in\mathbb{N}$ and
let $(p_{n})_{n\geq 0}$, $(q_{n})_{n\geq 0}$ be as in Proposition 6, then

$\left|z-\frac{p_{n}}{q_{n}}\right|=\left|\frac{p_{n}z_{n+1}^{-1}+p_{n-1}}{q_{n}z_{n+1}^{-1}+q_{n-1}}-\frac{p_{n}}{q_{n}}\right|=\frac{1}{|q_{n}|^{2}\left|z_{n+1}^{-1}+\frac{q_{n-1}}{q_{n}}\right|}.$

(see Proposition 6). Thus, we can conclude (5) if we obtain an absolute
constant $\kappa_{1}>0$ such that

(6) $\forall
n\in\mathbb{N}\quad\left|z_{n+1}^{-1}+\frac{q_{n-1}}{q_{n}}\right|\geq\kappa_{1}|a_{n+1}|.$

Take $n\in\mathbb{N}$. Assume that $|a_{n+1}|\geq 2$. Put

${\kappa_{1}}=1-\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)=\frac{2-\sqrt{2}}{4}.$

Since $|q_{n-1}|<|q_{n}|$ and $|z_{n+2}|\leq 1/\sqrt{2}$, we have

$\left|{z_{n+1}^{-1}}+\frac{q_{n-1}}{q_{n}}\right|=\left|a_{n+1}+{z_{n+2}}+\frac{q_{n-1}}{q_{n}}\right|\geq|a_{n+1}|-\left(1+\frac{1}{\sqrt{2}}\right)\geq{\kappa_{1}}|a_{n+1}|.$

Now, assume that $|a_{n+1}|=\sqrt{2}$. Since
$\operatorname{Cl}(\mathfrak{F})\subseteq\mathbb{D}(0;1)$, the closed set
$\iota[\operatorname{Cl}(\mathfrak{F})]$ is entirely contained in the
complement of the compact set $\overline{\mathbb{D}}(0;1)$, so

$\kappa_{2}:=\inf\left\\{|w-w^{\prime}|:w\in\iota[\operatorname{Cl}(\mathfrak{F})],\;w^{\prime}\in\overline{\mathbb{D}}(0;1)\right\\}>0,$

and hence

$\left|{z_{n+1}^{-1}}+\frac{q_{n-1}}{q_{n}}\right|\geq\kappa_{2}=\frac{\kappa_{2}}{\sqrt{2}}|a_{n+1}|.$

It is not hard to show that $\tfrac{\kappa_{2}}{\sqrt{2}}>\kappa_{1}$, so (6)
also holds in this case. Because $\kappa_{1}^{-1}=4+2\sqrt{2}$, the lemma is
proven. ∎

Once again, let us introduce some notation. For any $n\in\mathbb{N}$,
$\mathbf{a}=(a_{1},\ldots,a_{n})\in\mathbb{Z}[i]^{n}$, $m\in\mathbb{N}$, and
$\mathbf{b}=(b_{1},\ldots,b_{m})\in\mathbb{Z}[i]^{m}$, write
$\mathbf{a}\mathbf{b}:=(a_{1},\ldots,a_{n},b_{1},\ldots,b_{m})$, and define
the sets $\Omega_{R}(n):=\\{\mathbf{c}\in\mathbb{Z}[i]^{n}:\mathbf{c}\text{ is
regular}\\}$ and

$R(\mathbf{a};m):=\left\\{\mathbf{c}\in\mathbb{Z}[i]^{m}:\mathbf{a}\mathbf{c}\in\Omega_{R}(n+m)\right\\}.$

######  Lemma 9.

If $n\in\mathbb{N}$, $\mathbf{a}\in\Omega_{R}(n)$, and $b\in R(\mathbf{a};1)$,
then

$\frac{\mathfrak{m}\left(\mathcal{C}_{n+1}(\mathbf{a}b)\right)}{\mathfrak{m}\left(\mathcal{C}_{n}(\mathbf{a})\right)}\asymp\frac{1}{|b|^{4}}.$

Lemma 9 is the complex version of a well known estimate for regular continued
fractions (see Equation (57) in [11] for a precise statement and its proof).

###### Proof.

Let $n,\mathbf{a},b$ be as in the statement and take $(p_{n})_{n\geq 0}$ and
$(q_{n})_{n\geq 0}$ as in Proposition 6. The restriction of $T^{n}$ to
$\mathcal{C}_{n}(\mathbf{a})$ is a bijection onto
$\mathfrak{F}_{n}(\mathbf{a})$ with inverse
$t_{\mathbf{a},n}:\mathfrak{F}_{n}(\mathbf{a})\to\mathcal{C}_{n}(\mathbf{a})$

$\forall z\in\mathfrak{F}_{n}(\mathbf{a})\quad
t_{\mathbf{a},n}(z)=\frac{p_{n-1}z+p_{n}}{q_{n-1}z+q_{n}}=\frac{p_{n-1}}{q_{n-1}}+\frac{{(-1)^{n-1}}}{q_{n-1}(q_{n-1}z+q_{n})}.$

Since $|q_{n-1}|<|q_{n}|$ and
$\mathfrak{F}_{n}(\mathbf{a})\subseteq\mathbb{D}\left(0;1/\sqrt{2}\right)$,
$t_{\mathbf{a},n}$ is holomorphic and

$\forall z\in\mathfrak{F}_{n}(\mathbf{a})\quad
t_{\mathbf{a},n}^{\prime}(z)=\frac{{(-1)^{n}}}{(q_{n-1}z+q_{n})^{2}}.$

When we identify $\mathbb{C}$ with $\mathbb{R}^{2}$, the function
$t_{\mathbf{a},n}$ is differentiable on $\mathfrak{F}_{n}(\mathbf{a})$ and, if
$Dt_{\mathbf{a},n}(x,y)$ denotes its real derivative at the point $(x,y)$, we
get for all $z=x+iy$ in $\mathfrak{F}_{n}(\mathbf{a})$

$|\det
Dt_{\mathbf{a},n}(x,y)|={|t_{\mathbf{a},n}^{\prime}(z)|}=\frac{1}{|q_{n-1}z+q_{n}|^{4}}=\frac{1}{|q_{n}|^{4}\left|\frac{q_{n-1}}{q_{n}}z+1\right|^{4}}$

(see p. 33 in [12]) and, since every $z\in\mathfrak{F}$ satisfies

$1-\frac{1}{\sqrt{2}}\leq 1-|z|\leq
1-\left|\frac{q_{n-1}}{q_{n}}\right||z|\leq\left|\frac{q_{n-1}}{q_{n}}z+1\right|\leq\left|\frac{q_{n-1}}{q_{n}}z\right|+1<2,$

we may conclude that $|\det Dt_{\mathbf{a},n}|\asymp|q_{n}|^{-4}$. Therefore,
the Theorem of Change of Variable (Theorem 3.7.1 in [2], Vol. 1., p. 194)
implies that every Borel set $B\subseteq\mathbb{C}$ satisfies

$\mathfrak{m}\left(t_{\mathbf{a},n}[B\cap\mathfrak{F}_{n}(\mathbf{a})]\right)\asymp\frac{\mathfrak{m}\left(B\cap\mathfrak{F}_{n}(\mathbf{a})\right)}{|q_{n}|^{4}}.$

In particular, choosing $B=\mathfrak{F}$ and using (3), we get
$\mathfrak{m}(\mathcal{C}_{n}(\mathbf{a}))\asymp|q_{n}|^{-4}$. If we apply a
similar argument on $\mathcal{C}_{n+1}(\mathbf{a}b)$, we arrive at

$\frac{\mathfrak{m}\left(\mathcal{C}_{n+1}(\mathbf{a}b)\right)}{\mathfrak{m}\left(\mathcal{C}_{n}(\mathbf{a})\right)}\asymp\frac{|q_{n}|^{4}}{|bq_{n}+q_{n-1}|^{4}}=\frac{1}{\left|b+\frac{q_{n-1}}{q_{n}}\right|^{4}}.$

Finally, by $|q_{n-1}|<|q_{n}|$ and $\sqrt{2}\leq|b|$ (see (2)), the constant
${\kappa_{3}}=4(\sqrt{2}-1)^{-4}$ verifies

$\frac{1}{2^{4}|b|^{4}}\leq\frac{1}{(|b|+1)^{4}}\ll\frac{\mathfrak{m}\left(\mathcal{C}_{n+1}(\mathbf{a}b)\right)}{\mathfrak{m}\left(\mathcal{C}_{n}(\mathbf{a})\right)}\ll\frac{1}{(|b|-1)^{4}}\leq\frac{{\kappa_{3}}}{|b|^{4}}.\qed$

##  3\. Proof of the Main Theorem

Let $\mathbf{u}=(u_{n})_{n\geq 1}$ be a sequence in $\mathbb{R}_{>0}$. Instead
of working with $E(\mathbf{u})$, we will use more manageable sets. For any
$\kappa>0$ define

$E^{\infty}(\kappa\mathbf{u}):=\left\\{z=[0;a_{1},a_{2},\ldots]\in\mathfrak{F}:\kappa
u_{n}\leq\|a_{n}\|\quad\text{for infinitely many }n\in\mathbb{N}\right\\}$

and $E^{\infty}(\mathbf{u}):=E^{\infty}(1\mathbf{u})$. Note that

(7) $E^{\infty}(\mathbf{u})\subseteq E(\mathbf{u})\subseteq
E^{\infty}\left(\frac{1}{\sqrt{2}}\mathbf{u}\right),$

because $\|z\|\leq|z|\leq\sqrt{2}\|z\|$ for any $z\in\mathbb{C}$.

######  Lemma 10.

If $\sum_{n}u_{n}^{-2}<+\infty$, then
$\mathfrak{m}(E^{\infty}(\kappa\mathbf{u}))=0$ for every $\kappa>0$.

###### Proof.

Define the sequence of sets $(E_{n}^{\infty})_{n\geq 1}$ by

(8) $\forall n\in\mathbb{N}\qquad
E_{n}^{\infty}:=\left\\{[0;a_{1},a_{2},\ldots]\in\mathfrak{F}^{\prime}:u_{n}\leq\|a_{n}\|\right\\}$

For any $k\in\mathbb{N}$ and $\mathbf{a}\in\Omega_{R}(k)$, Lemmas 7 and 9 give

$\displaystyle\mathfrak{m}\left(E_{k+1}^{\infty}\cap\mathcal{C}_{k}(\mathbf{a})\right)$
$\displaystyle=\sum_{\|b\|\geq
u_{k+1}}\mathfrak{m}\left(\mathcal{C}_{k+1}(\mathbf{a}b)\right)$
$\displaystyle\asymp\sum_{\|b\|\geq
u_{k+1}}\frac{\mathfrak{m}\left(\mathcal{C}_{k}(\mathbf{a})\right)}{|b|^{4}}$
$\displaystyle\ll\mathfrak{m}\left(\mathcal{C}_{k}(\mathbf{a})\right)\sum_{n\geq
u_{k+1}}n^{-3}{\leq\frac{\mathfrak{m}\left(\mathcal{C}_{k}(\mathbf{a})\right)}{(u_{k+1}-1)^{2}},}$

Note that there is no loss of generality in assuming that $u_{k+1}>1$. Letting
$\mathbf{a}$ run along $\mathbb{Z}[i]^{k}$, we arrive at
$\mathfrak{m}(E_{k+1}^{\infty}){\ll(u_{k+1}-1)^{-2}}$. Hence, by the Borel-
Cantelli lemma, $\mathfrak{m}\left(E^{\infty}(\mathbf{u})\right)=0$. Finally,
for any $\kappa>0$, the convergence of $\sum_{k}u_{k}^{-2}$ implies that of
$\sum_{k}(\kappa u_{k})^{-2}$ and the previous argument shows
$\mathfrak{m}\left(E^{\infty}(\kappa\mathbf{u})\right)=0$. ∎

######  Lemma 11.

If $\sum_{n\geq 1}u_{n}^{-2}=+\infty$, then
$\mathfrak{m}\left(E^{\infty}(\kappa\mathbf{u})\right)=1$ for every
$\kappa>0$.

###### Proof.

As in the proof of Lemma 10, it is enough to show the result for $\kappa=1$.
Let $(E_{n}^{\infty})_{n\geq 1}$ be as in (8) and define

$\forall n\in\mathbb{N}\qquad B_{n}:=\bigcap_{j\geq
n}\mathfrak{F}^{\prime}\setminus E_{j}^{\infty}.$

Note that $(B_{n})_{n\geq 1}$ is increasing and

$\mathfrak{F}^{\prime}\setminus
E^{\infty}(\mathbf{u})=\bigcup_{n\in\mathbb{N}}B_{n}.$

Take $k\in\mathbb{N}$, $\mathbf{a}\in\Omega_{R}(k)$ and let $0<c_{1}<1$ be a
constant (independent of $k$ and $\mathbf{a}$) such that for any
$k\in\mathbb{N}$ and any $\mathbf{a}\in\Omega_{R}(k)$ we have

$\forall b\in
R(\mathbf{a};1)\qquad\mathfrak{m}\left(\mathcal{C}_{k+1}(\mathbf{a}b)\right)\geq\frac{c_{1}}{|b|^{4}}\mathfrak{m}\left({\mathcal{C}_{k}}(\mathbf{a})\right)$

(cfr. Lemma 9). Let $0<c_{2}<1$ be such that for every $M>0$

$\sum_{j\geq M}\frac{1}{j^{3}}>c_{2}\frac{1}{(M+1)^{2}}.$

Therefore, after choosing a suitable constant $0<c_{3}<1$ with the aid of
Lemma 7 and writing $c=c_{1}c_{2}c_{3}$, we have

$\sum_{\begin{subarray}{c}\|b\|\geq M\\\ b\in
R(\mathbf{a};1)\end{subarray}}\mathfrak{m}\left({\mathcal{C}_{k+1}}(\mathbf{a}b)\right)>c_{1}{c_{3}}\mathfrak{m}\left(\mathcal{C}_{k}(\mathbf{a})\right)\sum_{j\geq
M}\frac{1}{j^{3}}>c\frac{\mathfrak{m}\left(\mathcal{C}_{k}(\mathbf{a})\right)}{(M+1)^{2}},$

and thus

(9) $\sum_{\|b\|\leq
M}\mathfrak{m}\left(\mathcal{C}_{k+1}(\mathbf{a}b)\right)\leq\left(1-\frac{c}{(M+1)^{2}}\right)\mathfrak{m}\left(\mathcal{C}_{k}(\mathbf{a})\right)$

Define for each $n\in\mathbb{N}$ the set

${F(\mathbf{a},k,n)=\bigcup_{\mathbf{b}}\mathcal{C}_{k+n}(\mathbf{a}\mathbf{b}),}$

where $\mathbf{b}=(b_{1},\ldots,b_{n})$ runs along the $n$-tuples that belong
to $R(\mathbf{a};n)$ and that satisfy $\|b_{j}\|<u_{k+j}$ for all
$j\in\\{1,\ldots,n\\}$. Then, applying (9) recursively,

(10) $\forall n\in\mathbb{N}_{\geq
2}\quad\mathfrak{m}\left(F(\mathbf{a},k,n)\right)\leq\mathfrak{m}\left(F(\mathbf{a},k,1)\right)\prod_{j=2}^{n}\left(1-\frac{c}{(1+u_{k+j})^{2}}\right).$

Because $1+x\leq e^{x}$ holds for $x\in\mathbb{R}$, the equality

$B_{k+1}\cap\mathcal{C}_{k}(\mathbf{a})=\bigcap_{n\in\mathbb{N}}F(\mathbf{a},k,n)$

and (10) imply
$\mathfrak{m}\left(B_{k+1}\cap\mathcal{C}_{k}(\mathbf{a})\right)=0$. Taking
the union along $\mathbf{a}\in\mathbb{Z}[i]^{k}$, we obtain
$\mathfrak{m}(B_{k+1})=0$ and hence

$\mathfrak{m}\left(\mathfrak{F}^{\prime}\setminus
E^{\infty}(\mathbf{u})\right)=\mathfrak{m}\left(\bigcup_{k\in\mathbb{N}}B_{k}\right)=0.\quad\qed$

###### Proof of Theorem 2.

On the one hand, if $\sum_{n}u_{n}^{-2}<+\infty$, then Lemma 10 and (7) give

$\mathfrak{m}\left(E(\mathbf{u})\right)\leq\mathfrak{m}\left(E^{\infty}\left(\frac{1}{\sqrt{2}}\mathbf{u}\right)\right)=0.$

On the other hand, $\sum_{n}u_{n}^{-2}=+\infty$ and Lemma 11 imply
$\mathfrak{m}(\mathfrak{F}\setminus E^{\infty}(\mathbf{u}))=0$ and

$1=\mathfrak{m}\left(E^{\infty}(\mathbf{u})\right)\leq\mathfrak{m}\left(E(\mathbf{u})\right)\leq\mathfrak{m}(\mathfrak{F})=1.\qed$

###  3.1. Proof of Corollary 3

The well known existence of a Borel probability measure on $\mathfrak{F}$
which is equivalent to $\mathfrak{m}$ and $T$-ergodic yields the following
result (see propositions 1 and 2 in [14]):

######  Proposition 12.

There exists a positive constant $B>0$ such that for $\mathfrak{m}$-almost
every $z\in\mathfrak{F}$ the sequence $(q_{n})_{n\geq 0}$ defined as in
Proposition 6 satisfies

$\lim_{n\to\infty}\frac{1}{n}\log|q_{n}|=B.$

Take any function $\psi:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$. As in the regular
continued fraction context, we only need $x\mapsto x^{2}\psi(x)$ to be non-
increasing in the divergence case. Observe that, because $z-[z]$ belongs to
$\mathfrak{F}$ for any $z\in\mathbb{C}$, the set $K(\psi)$ is the union of
Gaussian integral translates of $K(\psi)\cap\mathfrak{F}$. Therefore, it
suffices to show Corollary 3 for $K(\psi)\cap\mathfrak{F}$.

###### Proof.

Assume that $\sum_{n}n^{3}\psi(n)^{2}<+\infty$. Define for each
$n\in\mathbb{N}$ the sets

$\displaystyle Z_{n}=\\{z\in\mathbb{Z}[i]:\|z\|=n\\},$ $\displaystyle\quad
C_{n}=\\{z\in\mathbb{Z}[i]:\|z\|<n\\},$ $\displaystyle K_{n}=$
$\displaystyle\bigcup_{\begin{subarray}{c}p\in C_{n}\\\ q\in
Z_{n}\end{subarray}}\overline{\mathbb{D}}\left(\frac{p}{q},\psi(|q|)\right)\cap\mathfrak{F}.$

Then, we have that

$\mathfrak{m}(K_{n})\ll\\#Z_{n}\,\\#C_{n}\psi(n)^{2}\ll n^{3}\psi(n)^{2},$

so $\sum_{n}\mathfrak{m}(K_{n})<+\infty$ and, by the Borel-Contelli Lemma,
$\mathfrak{m}(K(\psi)\cap\mathfrak{F})=0$. Then, the discussion prior to the
proof implies $\mathfrak{m}(K(\psi))=0$.

Now assume that $\sum_{n}n^{3}\psi(n)^{2}=+\infty$ and that $x\mapsto
x^{2}\psi(x)$ is non-increasing for $x>0$. Then, $y\mapsto y^{3}\psi(y)^{2}$
is non-increasing for $y>1$. Let $D$ be a positive number strictly larger than
$B$ (given as in proposition 12) and define
$\tilde{\Phi}:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ by

$\forall x\in\mathbb{R}_{>0}\qquad\tilde{\Phi}(x)=e^{2Dx}\psi(e^{Dx}).$

Hence, $\tilde{\Phi}$ is non-increasing and for $0<t<T$ we have

$\int_{t}^{T}\tilde{\Phi}^{2}(x)\mathrm{d}x=\int_{t}^{T}e^{4Dx}\psi(e^{Dx})^{2}\mathrm{d}x=\frac{1}{D}\int_{e^{Dt}}^{e^{DT}}y^{3}\psi(y)^{2}\mathrm{d}y$

and we conclude $\sum_{n}\tilde{\Phi}(n)^{2}=+\infty$ by letting
$T\to+\infty$.

Write $\kappa=4+2\sqrt{2}$ and define $\Phi=\kappa^{-1}\tilde{\Phi}$. By the
main theorem, for almost all $z=[0;a_{1},a_{2},\ldots]\in\mathfrak{F}$ the
inequality $|a_{n+1}|\geq\Phi(n)^{-1}$ holds infinitely often. Also, because
of Proposition 12, we have $|q_{n}|<e^{Dn}$ for sufficiently large $n$ almost
everywhere. Therefore, by Lemma 8, for almost every $z\in\mathfrak{F}$ there
are infinitely many $n\in\mathbb{N}$ satisfying

$\left|z-\frac{p_{n}}{q_{n}}\right|\leq\frac{\kappa}{|a_{n+1}||q_{n}|^{2}}\leq\frac{\tilde{\Phi}(n)}{|q_{n}|^{2}}\leq\frac{\tilde{\Phi}\left(D^{-1}\log|q_{n}|\right)}{|q_{n}|^{2}}=\psi(|q_{n}|).\qed$

##  4\. Acknowledgements

I thank Yann Bugeaud, Mauricio González Soto, Mumtaz Hussain, Simon
Kristensen, and the referees for their helpful comments. I especially thank M.
Hussain for suggesting this problem.

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