# Ohmic dissipation during the formation of super-Earth

Shi Jia  State Key Laboratory of Lunar and Planetary Sciences, Macau
University of Science and Technology, Macau, People’s Republic of China  CNSA
Macau Center for Space Exploration and Science, Macau, People’s Republic of
China  Wei, Zhong  School of Physics and Astronomy, Sun Yat-Sen University,
Zhuhai, 519082, People’s Republic of China  State Key Laboratory of Lunar and
Planetary Sciences, Macau University of Science and Technology, Macau,
People’s Republic of China  CSST Science Center for the Guangdong-Hong Kong-
Macau Greater Bay Area, Zhuhai, 519082, People’s Republic of China  Cong Yu
School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082,
People’s Republic of China  State Key Laboratory of Lunar and Planetary
Sciences, Macau University of Science and Technology, Macau, People’s Republic
of China  CSST Science Center for the Guangdong-Hong Kong-Macau Greater Bay
Area, Zhuhai, 519082, People’s Republic of China

###### Abstract

Super-Earth population, as one of the representatives of exoplanets, plays an
important role in constraining the planet formation theories. According to the
prediction from core-accretion models, super-Earths should be rare because
their masses are in the range of the critical mass above which they would grow
to be gas giants by runaway gas accretion. In this work, we investigate the
effect of ohmic dissipation on the planetary thermal structure and cooling
contraction as planets accrete gas from their surrounding disks. We find that
the extra heating energy from Ohmic heating deposited into planetary envelopes
can push the planetary radiative-convective boundaries inward and prevent the
planets from cooling, and can even halt accretion. We explore parameter space
to study the dependence of cooling timescale on the input parameters of the
ohmic-dissipation model. Numerical results show that gas accretion can be
halted before runaway gas accretion and the envelope mass is only several
percent of planetary core mass for some parameter sets. Our results suggest
that ohmic dissipation is a potential mechanism to delay the gas accretion and
promote the formation of super-Earths. Future observations may help to
constrain the importance of ohmic dissipation on the super-Earth formation.

Exoplanet evolution(491) — Exoplanet formation(492) — Exoplanet structure(495)
— Magnetic fields(994)—Magnetohydrodynamics(1964)

##  1 Introduction

Super-Earths (or sub-Neptunes), generally defined as the planet mass (or
radius) between that of Earth and Neptune, contribute a significant part of
the current exoplanet population. These low mass planets commonly orbit around
Sun-like stars with orbital periods shorter than 100 days (e.g., Howard et al.
2010; Batalha et al. 2013; Petigura et al. 2013; Zhu et al. 2018). Moreover,
most of super-Earths possess low bulk densities, implying that gas envelopes
should be held outside their solid cores (e.g., Weiss & Marcy 2014; Wu &
Lithwick 2013b; Rogers 2015). Detailed planetary interior models suggest that
the mass ratios of the outer envelope and the inner core are typically from
several up to about ten percent(e.g., Lopez & Fortney 2014; Wolfgang & Lopez
2015).

Currently, the formation of super-Earths is still an open question. A lot of
studies have studied their formation from several conceivable angles. For
example, some concentrated on exploring the conditions for the in situ
formation of super-Earths. Lee et al.(2014) found that super-Earths can be
formed in dusty and gas-poor disks (see also Lee & Chiang 2015 and Lee et al.
2018). Others focused on studying the possible mechanisms that can promote the
formation of super-Earths. For example, giant impact(e.g., Inamdar &
Schlichting 2015; Liu et al. 2015) and tidal heating (Ginzburg & Sari 2017)
could supply extra heating energy to delay or halt the contraction of
protoplanets. Other factors, such as the tidally forced turbulence (Yu 2017)
and planetary rotation (Zhong & Yu 2021), could change the planet’s interior
structure, especially for the location of radiative-convective boundary, and
slow down the envelope contraction. Furthermore, the atmospheric recycling,
which is also referred to entropy advection (Ali-Dib et al. 2020), is also
important to prevent the super-Earths from experiencing runaway accretion and
growing to be gas giants (e.g., Ormel et al. 2015a; Ormel et al. 2015b;
Béthune & Rafikov 2019a; Béthune & Rafikov 2019b).

In this work, we explore the effect of ohmic dissipation on the evolution of
planetary envelope as protoplanets accrete gas from their natal disks. Ohmic
dissipation has been widely used to study the anomalous radius of Hot Jupiters
(e.g., Batygin & Stevenson 2010; Perna et al. 2010; Batygin et al. 2011; Huang
& Cumming 2012; Wu & Lithwick 2013a). Pu & Valencia (2017) also investigate
the radius evolution of mini-Neptunes with Ohmic heating. The ohmic
dissipation of close-in planets can be driven by the thermal dynamics (e.g.,
Showman et al. 2010; Menou 2012). Due to the strong irradiation from host
stars, the surface temperature of close-in planets can be higher than 1000 K,
resulting in the ionization of alkali metals. A thermal wind in the planetary
upper atmosphere is caused by the temperature difference in the atmosphere
between the day side and night side because of the tidal locking of close-in
planet (e.g., Showman et al. 2010; Menou 2012). With the magnetic field from
the close-in planet itself or its host star, induced current in planetary
envelope is then generated and penetrates into planet’s interior. As a result,
heating power from ohmic dissipation is deposited into the planetary envelope.
In addition, the Ohmic dissipation of close-in planets can also be motivated
by the magnetic interaction with their host stars (e.g., Laine & Lin 2008;
Laine & Lin 2012).

It is possible that ohmic dissipation is induced inside the planetary envelope
during planetary gas accretion phase. Although the protoplanets can not obtain
irradiation energy directly from their host stars because they are embedded
into the disks, the close-in protoplanets can still have high surface
temperatures. According to the minimum-mass extrasolar nebula model of Chiang
& Laughlin (2013), the disk temperature at the location with a separation 0.1
AU from host star is on the order of 1000 K, which means that the gas material
with solar-like metallicity at this location can be weakly ionized. The weakly
ionized material is accreted onto the surface of planets and the angular
momentum of accreting gas flow is then transported inward to spin up the
planets (e.g., Dittmann 2021). Atmospheric wind is formed in the planetary
envelope during the accretion process. Therefore, the close-in protoplanet
will undergo the ohmic dissipation while the weakly ionized accreting gas
flows across the planetary intrinsic magnetic fields or the fields from their
host stars.

This work is organized as follows. In Section 2, we describe the methodology
that we used to model the planetary gas accretion with ohmic dissipation and
give the basic parameters and boundary conditions. We then present the
numerical results of planetary thermal structures and cooling contraction in
Section 3. Finally, we discuss the model uncertainties and give a summary in
Section 4.

##  2 Model and method

Coupling the ohmic heating within the planet interior structure to investigate
the gas-accretion process is complex. In the first step, we adopt an
simplified Ohmic heating model from Wu & Lithwick (2013a) and Pu & Valencia
(2017) to construct the planet’s evolution model. A detailed analysis of Ohmic
dissipation is beyond this scope. In this section, we first review the Ohmic
heating mechanism and planet’s interior model. We then describe our method to
model the gas-accretion coupling with the ohmic heating mechanism.

###  2.1 Ohmic dissipation

The intensity of ohmic dissipation in the planetary envelope is determined by
several factors, including the magnetic field acting on the planet, wind speed
and wind zone depth in the envelope, and electrical conductivity of the
material in the envelope. We assume that the planet has a magnetic dipole
field and wind zone with constant wind velocity (Wu & Lithwick 2013a). Ohmic
dissipation in our model is constrained between the outer envelope surface and
the inner core, which is treated as an insulator (e.g., Pu & Valencia 2017).

####  2.1.1 Electrical conductivity

The electrical conductivity of planetary envelope comes from the contribution
of alkali metal ionization in the upper envelope and hydrogen ionization in
the deeper envelope. We adopt the conductivity profile from Pu & Valencia
(2017). The element potassium (K) has the lowest ionization potential for the
alkali metal. The conductivity of alkali metal is dominated by the element K
at low pressures. The conductivity of element K is given by (Pu & Valencia
2017):

$\sigma_{Z}\simeq 1.74\times 10^{3}\left(\frac{T}{1600\
K}\right)^{3/4}\left(\frac{P}{bar}\right)^{(-1/2)}\mathrm{exp\left(\frac{-4.35eV}{2k_{B}T}\right)}$
(1)

Only considering the contribution of element K for the conductivity of alkali
metal is appropriate to our situation. Compared with the full implementation,
this approximation is accurate to within a few percent below the pressures of
$\sim$ 100 bars (Pu & Valencia 2017).

In the deeper part of planetary envelope, hydrogen is ionized by high
pressure. The conductivity of the hydrogen is (e.g., Liu et al. 2008; Huang &
Cumming 2012) expressed as follows:

$\sigma_{X}=\sigma_{0}\
\mathrm{exp\left(\frac{-E_{g}(\rho)}{2k_{B}T}\right)},$ (2)

where $\mathrm{E_{g}=20.3-64.7\rho}$ is in eV, $\rho$ is the density in mol
$cm^{-3}$, and $\sigma_{0}=3.4\times 10^{20}\ s^{-1}\ \mathrm{exp(-44\rho)}$ .

The total conductivity is then given as

$\sigma=\sigma_{X}+\sigma_{Z}$ (3)

####  2.1.2 Ohmic heating model

As the weakly ionized atmospheric flow moves across the intrinsic magnetic
field of the planet or the field from its host star, an induced current is
generated in the planetary envelope. Simultaneously, an induced magnetic field
is produced. In the steady state, the induction equation is expressed as:

$\frac{\partial\textbf{B}}{\partial
t}=\nabla\times(\textbf{v}\times\textbf{B})-\nabla\times\eta(\nabla\times\textbf{B})=0$
(4)

where $\eta$ is the magnetic diffusivity, $\eta=c^{2}/4\pi\sigma$.

The magnetic field acting on the planet is assumed to be a dipolar field (Wu &
Lithwick 2013a):

$\mathrm{\textbf{B}_{dip}=M_{B}\frac{2\cos\theta\textbf{e}_{r}+\sin\theta\textbf{e}_{\theta}}{r^{3}}}$
(5)

where $M_{B}$ is the magnetic dipole moment of the planet,
$M_{B}=\mathrm{BR^{3}}$. $\mathrm{B}$ is the surface dipole field strength.
$\mathrm{R}$ is planetary radius.

Assuming a constant angular velocity $\omega$ in the wind zone, v is given by:

$\textbf{v}=\omega r\ \sin\theta\textbf{e}_{\phi}$ (6)

In the wind zone, the current is given with Ohm’s law:

$\mathrm{\textbf{J}=\sigma\left(-\nabla\Phi+\frac{\textbf{v}\times\textbf{B}_{dip}}{c}\right)}$
(7)

Where $\Phi$ is the electric potential and v is the wind velocity. Below the
wind zone, the current is obtained by:

$\mathrm{\textbf{J}=-\sigma\nabla\Phi}$ (8)

With $\nabla\cdot\textbf{J}=0$, Equations (7) and (8) are solved numerically
(details see Appendix A).

The ohmic power can be obtained with the electric current profile:

$P_{\mathrm{oh}}=\int\frac{J^{2}}{\sigma}dV$ (9)

where $J=\sqrt{J_{r}^{2}+J_{\theta}^{2}}$, $J_{r}$ and $J_{\theta}$ are the
radial current and the meridional current, respectively. The Ohmic power
profile along the radius is expressed as:

$\frac{dP_{\mathrm{oh}}}{dr}=4\pi r^{2}\frac{J^{2}}{\sigma}$ (10)

The boundary conditions for the radial current $J_{r}$ are that $J_{r}=0$ at
the planetary surface $R_{\mathrm{out}}$ and at the core surface $R_{c}$.

###  2.2 Planet model

We adopt the planet interior structure similar to the model from Piso &
Youdin(2014). We employ the realistic equation of state (EOS) for the
hydrogen-helium mixture and dust-free opacity111We adopt the dust free opacity
as our fiducial case since the dust grains could coagulate efficiently (e.g.,
Ormel 2014; Mordasini et al. 2014). For the location of protoplanets we
considered here, a=0.1 AU, the coagulation timescale is several orders of
magnitude smaller than the disk lifetime (e.g., Lee et al. 2014). We also
briefly discuss the envelope accretion with dust opacity in Section 3.3.3.,
details are shown below.

The basic structure equations are as follows:

$\frac{dm}{dr}=4\pi\rho r^{2}$ (11)

$\frac{dP}{dr}=-\rho\frac{Gm}{r^{2}}$ (12)

$\frac{dT}{dr}=\nabla\frac{T}{P}\frac{dP}{dr}$ (13)

where $r$ is radius, $m$ is the mass enclosed within the radius $r$, $P$ is
pressure, $T$ is temperature, $\rho$ is density, $G$ is gravitational
constant. The temperature gradient, $\nabla=d\ lnT/d\ lnP$, is set to the
minimum value of radiative gradient $\nabla_{rad}$ and adiabatic gradient
$\nabla_{ad}$ :

$\nabla=\min(\nabla_{ad},\nabla_{rad}),$ (14) $\nabla_{rad}=\frac{3\kappa
PL}{16\pi\sigma GMT^{4}}$ (15) $\nabla_{ad}=\left(\frac{d\ln T}{d\ln
P}\right)_{ad}$ (16)

where $L$ is the luminosity, $\kappa$ is opacity, and $\sigma$ is the
Stefan–Boltzmann constant.

The ohmic heating model is coupled into the planet structure model. With the
extra energy source from the ohmic heating, the total luminosity is now
expressed as:

$L=L_{cool}+P_{oh}$ (17)

where $L_{cool}$ is the cooling luminosity from the planet convective
interior. $L_{cool}$ is assumed spatially constant inside of planet (e.g.,
Piso & Youdin 2014; Lee et al. 2014). The planetary luminosity profile is
obtained with Equation (10):

$\frac{dL}{dr}=\frac{dP_{\mathrm{oh}}}{dr}$ (18)

The EOS for hydrogen-helium mixture of the envelope used here is from the EOS
table of Chabrier et al.(2019; 2021). The metallicity of EOS that we used is
Z=1-X-Y=0.017 (X=0.708,Y=0.275). The opacity is from the opacity table of
Freedman et al.(2014). The EOS and opacity are solar-like.

The cooling time is given by (Piso & Youdin 2014):

$\Delta t=\frac{-\Delta E+\langle e\rangle\Delta M-\langle P\rangle\Delta
V_{\langle M\rangle}}{\langle L_{\mathrm{cool}}\rangle}$ (19)

where $\Delta$ denotes the difference between the two adjacent snapshots and
the brackets indicate their average. The $e$ in second term of Equation (19)
is the specific energy of the accreting matter: $e=-Gm/r+u$. The third term in
Equation (19) is the work done by the envelope contraction. The total energy
is given as follows:

$E=\int_{M_{c}}^{M}u\ dm-\int_{M_{c}}^{M}\frac{Gm}{r}dm$ (20)

where $u$ is the specific internal energy, which is interpolated from the EOS
table of Chabrier et al.(2019) and Chabrier & Debras (2021). The first term of
Equation (20) is the internal energy and the second term is the gravitational
potential energy. All the terms are calculated at the radiative-convective
boundary.

###  2.3 Boundary conditions

In our fiducial case, the planet core mass $M_{core}$ is set to 5
$\mathrm{M_{\oplus}}$ with constant density $\rho_{core}=7\ g\ {cm}^{-3}$,
same as Lee et al.(2014). The corresponding core radius is 1.6
$\mathrm{R_{\oplus}}$. The outer radius is determined by the minimum value of
the Bondi radius and Hill radius:

$R_{out}=\min(R_{\mathrm{B}},R_{\mathrm{H}})$ (21)

where

$R_{\mathrm{H}}=\left(\frac{M_{p}}{M_{*}}\right)^{1/3}a\approx
40R_{\oplus}\left(\frac{M_{p}}{5M_{\oplus}}\right)^{1/3}a_{0.1}$ (22)
$R_{\mathrm{B}}=\frac{GM_{p}}{c_{s}^{2}}\approx
90{R_{\oplus}}\left(\frac{M_{p}}{5M_{\oplus}}\right)T_{3}^{-1}$ (23)

where $a_{0.1}=a/0.1AU$, $T_{3}=T/1000K$.

The temperature and density of the outer boundary are set to the disk density
and temperature at the planet location. The disk structure is from the
minimum-mass extrasolar nebula model of Chiang & Laughlin (2013):

$T_{\mathrm{d}}=10^{3}\ a_{0.1}^{-3/7}K$ (24)

$\rho_{\mathrm{d}}=6\times 10^{-6}a_{0.1}^{-2.9}g\ {cm}^{-3}$ (25)

We set $a=0.1AU$ as our fiducial case. Correspondingly, the disk temperature
and density are 1000 K and $6\times 10^{-6}\ \mathrm{g\ {cm}^{-3}}$,
respectively.

###  2.4 Calculation procedure

The main goal of our model is to study the planetary interior structure and
cooling contraction considering ohmic heating during the slow gas-accretion
phase. Planetary interior structure and the corresponding ohmic heating
profile are calculated self-consistently for a given set of boundary
conditions and the parameters of the ohmic heating model. If the ohmic heating
profile matches the interior thermal structure, the iteration and integration
of our procedure are completed. The procedure is performed as follows:

(a) Choose a set of initial conditions and boundary conditions. The planetary
thermal structure model is as follows: the planetary core mass
($M_{\mathrm{c}}$) and envelope mass ($M_{\mathrm{envi}}$); the outer boundary
($R_{\mathrm{out}},M_{\mathrm{out}}=M_{\mathrm{c}}+M_{\mathrm{envi}},T_{\mathrm{out}}=T_{\mathrm{d}},\rho_{\mathrm{out}}=\rho_{\mathrm{d}}$);
the inner boundary
($R_{\mathrm{in}}=R_{\mathrm{c}},M_{\mathrm{in}}=M_{\mathrm{c}}$). The ohmic
heating model is as follows: magnetic field $\mathbf{B}$, wind velocity
$\mathbf{v}$ and wind zone depth $d_{\mathrm{w}}$; the outer boundary
($J_{\mathrm{r}}=0$ at $R_{\mathrm{out}}$); the inner boundary
($J_{\mathrm{r}}=0$ at $R_{\mathrm{in}}$).

(b) Assume or guess a total luminosity, $L=L_{\mathrm{cool}}+P_{\mathrm{oh}}$.
Numerically solve the ODE Equations (7), (8), (10), (11)-(13) and (18) from
$R_{\mathrm{out}}$ inward to $R_{\mathrm{in}}$. The Ohmic heating profile [
Equations (7),(8) and (10)] is iterated until it matches the given boundary
conditions of ohmic heating model.

(c) Check the consistency of the envelope mass ($M_{\mathrm{envf}}$) obtained
from step (b) and the initial setting mass ($M_{\mathrm{envi}}$) in step (a).
If the tolerance between $M_{\mathrm{envf}}$ and $M_{\mathrm{envi}}$ is less
than $10^{-5}$, the procedure is finished and the desired planetary thermal
structure is found. If not, return to step (b).

(d) Increase the envelope mass and repeat steps (a)-(c) to obtain a series of
thermal structures. Employ Equation (19) to calculate the cooling time.

##  3 Results

In this section, we present our numerical results for the planetary envelope
thermal structure and cooling contraction taking into account the ohmic
dissipation. For the sake of simplicity, we adopt the Ohmic heating model from
Wu & Lithwick (2013a) whose model assumes a dipolar magnetic field and a
constant wind velocity in the wind zone.

The parameters of ohmic heating model are primarily related to the magnetic
field acting on the planet, wind velocity profile, and wind zone depth. For
the studies of hot Jupiters inflation, the wind velocity is estimated from the
atmosphere’s thermal dynamics because of the tidal locking of close-in planet
(e.g., Showman et al. 2010; Menou 2012), on the order of 1 $\mathrm{km\
s^{-1}}$. The strength of the dipole magnetic field on the planetary surface
is commonly adopted on the order of 1 G. The bottom of the wind zone is
normally set at the location of pressure with about 10 $bar$ (e.g.,Batygin &
Stevenson 2010; Wu & Lithwick 2013a; Pu & Valencia 2017) or estimated with
pressure scale height $H_{\mathrm{P}}$ (e.g., Huang & Cumming 2012; Menou
2012).

Although the exact values of these parameters are unknown for our case, we can
get some references from numerical simulations and observational results. The
atmospheric recycling flow can penetrate to the inner part of the planetary
envelope around 0.3 $R_{out}$, (e.g., Lambrechts & Lega 2017; Kurokawa &
Tanigawa 2018; Zhu et al. 2021). As the planetary mass increases, the
azimuthal velocity of atmospheric flow increases close to the Keplerian
velocity (e.g., Ormel et al.2015a, 2015b; Béthune & Rafikov 2019a). Moreover,
the planetary envelope can be rotational support during the accretion phase
(e.g., Ormel et al.2015a; Béthune & Rafikov 2019a, 2019b). Based on the
results of these studies, we set the wind zone depth (from planetary surface
to the bottom of wind zone) $d_{w}=0.7R_{\mathrm{out}}$ as our fiducial case.
For a 5 $M_{\oplus}$ core locating at 0.1 AU, the Keplerian velocity at the
Hill radius is about $10^{5}\ \mathrm{cm\ s^{-1}}$ and at the surface of
planetary core is about $v_{\mathrm{k}}\simeq 10^{6}\ \mathrm{cm\ s^{-1}}$.
Since we do not know the rotational velocity of the planetary core, we set the
fiducial wind velocity $v_{\mathrm{w}}=0.01\ v_{\mathrm{k}}$, $10^{4}\
\mathrm{cm\ s^{-1}}$.

The magnetic field of Earth-like planets can be generated by the movement of
liquid iron within their cores (e.g., Cuartas-Restrepo 2018). Numerical
simulations show that the surface magnetic field strength of super-Earths is
around 1 G (Driscoll & Olson 2011), depending mainly on the iron-core mass
fractions. Recent ultraviolet observations of HAT-P-11 show that this Neptune
size planet (about 24 $M_{\oplus}$) should pose equatorial magnetic field
strength with 1-5 G (Ben-Jaffel et al. 2022). Moreover, close-in protoplanets
may suffer from the magnetic interaction from their pre-mainsequence hosts
because the observed surface magnetic fields of T Tauri stars can be on the
order of 1000 G (e.g., Johns-Krull 2007; Yang et al. 2008). For our our
fiducial case, the surface magnetic dipole field strength (the magnetic field
strength $B$, hereafter) is set to $B=1G$. Furthermore, we explore the
influence of parameter space of ohmic heating model on the planetary cooling
contraction. In our calculations, we consider wind zone depth $d_{\mathrm{w}}$
from 0.5 $R_{\mathrm{out}}$ (shallow) to 0.9 $R_{\mathrm{out}}$(deep). The
magnetic field strength B spans from 0.1 G up to 10 G. The wind velocity is in
a range from $10^{2}$ to $10^{5}\ \mathrm{cm\ s^{-1}}$.

Examples of planetary envelope thermal structures with the external energy
source from ohmic heating are presented in Section 3.1. In Section 3.2, we
show the variations of planetary luminosity and cooling time with increasing
envelope mass. Parameter space is then explored to study the effect of ohmic
dissipation on the planetary cooling contraction in Section 3.3. We also
explore the influence of ohmic dissipation on the envelope evolution of
protoplanets with dust opacity at the end of Section 3.3.

Figure 1: Thermal structures of a planet with 5 $M_{\oplus}$ core mass during
envelope growth. The planet locates at 0.1 AU. Radial profiles of three
different envelope mass (0.5 $M_{\oplus}$, 1.0 $M_{\oplus}$, 2.0 $M_{\oplus}$)
are shown. From top to bottom, the panels show the pressure profiles,
temperature profiles, density profiles, temperature gradient profiles and
envelope mass profiles, respectively. Thick lines denote the convective zone
and the thin lines indicate the radiative layer. Left-hand panels present the
structures without ohmic dissipation while the right-hand panels display the
structures with ohmic dissipation. The parameters of ohmic dissipation model
are our fiducial case (wind zone depth $d_{w}=0.7R_{\mathrm{out}}$, wind
velocity $v_{\mathrm{w}}=\ 10^{4}\ \mathrm{cm\ s^{-1}}$, magnetic field
strength $B$= 1 G). The radiative-convective boundaries are pushed into the
deeper part of the planet when ohmic dissipation is taken into account, the
details are shown in the panels of temperature profiles. The detailed model
parameters are shown in Table 1.

###  3.1 Planetary thermal structure

Radial profiles of a planet with 5 $M_{\oplus}$ core at 0.1 AU during envelope
growth are shown in Figure 1. The left-hand panels show the interior
structures without Ohmic dissipation while the right-hand panels with ohmic
dissipation. The parameters of ohmic heating model here are our fiducial case
($d_{w}=0.7R_{\mathrm{out}}$, $v_{\mathrm{w}}=\ 10^{4}\ \mathrm{cm\ s^{-1}}$,
B= 1 G). The radiative-convective boundary is an essential factor to describe
the planetary thermal structure and cooling contraction. We can see that the
radiative-convective boundaries of these envelopes move inward when the effect
of ohmic dissipation is considered (see the details of the inner plots of the
temperature profiles and Table 1). Although the ohmic heating power
contributes a large portion of energy ($\sim 65\%$) to planetary luminosity
for the envelope mass with 0.5 $M_{\oplus}$ (Table 1), most of the power
($\sim$ 95%) is directly deposited into the convective zone (see the ohmic
heating power profile in Figure 2). The small fraction of ohmic heating power
contained in the outer envelope has a minor effect on the thermal structure
(see the discussion below). As a result, the radiative-convective boundary is
slight pushed inward (see the parameters at the radiative-convective boundary
in Table 1). As the envelope mass grows, multiple layers appeare in the
envelope (with 1 $M_{\oplus}$), which means that a radiative zone is formed
between two adjacent convective layers, similar to the sandwiched structure
mentioned in Lee et al.(2014). Compared to the low envelope mass (0.5
$M_{\oplus}$) at the initial stage of gas accretion, the radiative-convective
boundaries are obviously pushed inward for envelopes with high mass (1, 2
$M_{\oplus}$).

Figure 2: The radial profiles of temperature (upper panel, solid lines),
conductivity (upper panel, dashed lines) and cumulative ohmic heating power
(bottom panel), corresponding to the thermal structures with ohmic dissipation
in Figure 1. The vertical lines in the bottom panel denote the radiative-
convective boundaries. The vertical-dotted lines in the bottom panel represent
the bottom of the wind zone.

The overall properties of the ohmic power profile is that the most of the
ohmic heating energy is concentrated into the wind zone and the energy
deposited below the wind zone orders of magnitude smaller than that in the
wind zone, as shown in the bottom panel of Figure 2 (e.g., Huang & Cumming
2012; Wu & Lithwick 2013a). From the upper panel of Figure 2, we can see that
the conductivity increases rapidly as temperature increases. The conductivity
several orders of magnitude higher in deep part of planetary envelope than
that in the upper envelope, leading to a sharp reduction of ohmic heating
power (Equation 10). The vertical solid and dotted lines in Figure 2 denote
the radiative-convective boundary and the bottom of the wind zone,
respectively. The effect of ohmic dissipation on the planetary thermal
structure depends on the distribution of Ohmic heating power in the envelope.
For the envelope with 0.5 $M_{\oplus}$, the bottom of wind zone is below the
radiative-convective boundary and most of the ohmic heating energy ($\sim$
95%) is deposited into the convective zone to replace the cooling luminosity.
However, as mentioned earlier, the energy contained in the upper envelope is
not enough to efficiently change the structure of the envelope. The case for
the envelope with 1 $M_{\oplus}$ (2 $M_{\oplus}$) is converse. The bottom of
the wind zone is close to (above) the radiative-convective boundary and about
$90\%$ ($95\%$) of ohmic heating power is concentrated into the outer
envelope. The thermal structure of the envelope is changed obviously (Figure 1
and Table 1). When the envelope mass increases to 2 $M_{\oplus}$, about 75 %
of planetary luminosity is from the Ohmic heating power (Table 1). The effect
of ohmic dissipation becomes important and planetary cooling contraction
starts to slow down significantly, as discussed in next section.

Table 1: Parameters of thermal structures in Figure 1

$M_{\mathrm{env}}$ | Ohmic Dissipation | L | $P_{\mathrm{oh}}$ | $L_{\mathrm{cool}}$ | $P_{\mathrm{oh}}/L$ | $R_{\mathrm{rcb}}$ | $M_{\mathrm{rcb}}$ | $P_{\mathrm{rcb}}$ | $T_{\mathrm{rcb}}$ | $\rho_{\mathrm{rcb}}$
---|---|---|---|---|---|---|---|---|---|---
($M_{\oplus}$) |  | ($\mathrm{erg\ s^{-1}}$) | ($\mathrm{erg\ s^{-1}}$) | ($\mathrm{erg\ s^{-1}}$) |  | ($R_{\oplus}$) | ($M_{\oplus}$) | ($\mathrm{bar}$) | $\mathrm{(K)}$ | $\mathrm{(g\ {cm}^{-3})}$
0.5 | No | $4.011\times 10^{26}$ |  | $4.011\times 10^{26}$ |  | 16.048 | 5.341 | 3.598 | 1662.6 | $6.134\times 10^{-5}$
0.5 | Yes | $4.046\times 10^{26}$ | $2.633\times 10^{26}$ | $1.413\times 10^{26}$ | 0.651 | 15.965 | 5.341 | 3.661 | 1669.8 | $6.214\times 10^{-5}$
1.0 | No | $9.090\times 10^{25}$ |  | $9.090\times 10^{25}$ |  | 14.330 | 5.742 | 9.944 | 1680.0 | $1.677\times 10^{-4}$
1.0 | Yes | $9.486\times 10^{25}$ | $4.571\times 10^{25}$ | $4.915\times 10^{25}$ | 0.482 | 12.355 | 5.704 | 18.339 | 1968.0 | $2.638\times 10^{-4}$
2.0 | No | $3.317\times 10^{25}$ |  | $3.317\times 10^{25}$ |  | 14.556 | 6.570 | 20.463 | 1711.3 | $3.384\times 10^{-4}$
2.0 | Yes | $4.348\times 10^{25}$ | $3.267\times 10^{25}$ | $1.080\times 10^{25}$ | 0.752 | 12.430 | 6.499 | 39.887 | 2009.0 | $5.613\times 10^{-4}$

Note. — $M_{\mathrm{env}}$ is the planetary envelope mass. $L$ is the total
luminosity of the planet. $\mathrm{P_{oh}}$ and $\mathrm{L_{cool}}$ are the
ohmic heating power and planetary cooling luminosity. The subscript
$\mathrm{rcb}$ denotes the radiative-convective boundary. For the thermal
structure with multiple layers (a radiative zone is sandwiched between two
adjacent convective zones, as the structure of the envelope with 1
$M_{\oplus}$ in Figure 1), the parameters ($R_{\mathrm{rcb}}$,
$M_{\mathrm{rcb}}$, $P_{\mathrm{rcb}}$, $T_{\mathrm{rcb}}$,
$\rho_{\mathrm{rcb}}$) are chosen at the inner most boundary.

###  3.2 Cooling contraction with Ohmic dissipation

Figure 3 shows the variations of luminosity and cooling time with the envelope
mass. Without ohmic dissipation (black lines in Figure 3), the planet starts
runaway accretion at about 1 Myr222The time of runaway accretion
($\mathrm{t_{run}}$) is usually defined as the time when the minimum of the
planetary luminosity is reached (e.g., Lee et al. 2014; Yu 2017). (the black-
solid circle in the bottom panel of Figure 3). When the effect of ohmic
dissipation is taken into account, the planetary cooling is modulated by
gravitational contraction and ohmic heating. With the envelope mass
increasing, the luminosity (red solid line), cooling luminosity (red dashed
line) and ohmic heating power (red dotted line) reduces. Until the planetary
luminosity is totally dominated by ohmic heating power
($P_{\mathrm{oh}}/L>90\%$), the contraction of planet is significantly
delayed. We stop the calculation when the ratio of ohmic heating power and
total luminosity is around 0.99 ($P_{\mathrm{oh}}/L\simeq 99\%$) and assume
that gas accretion is halted. The halting mass $M_{\mathrm{halt}}$ is defined
as the envelope mass when the ratio reaches 0.99. For our fiducial parameter
set of the ohmic-heating model, gas accretion is halted at about 2 Myr and
$M_{\mathrm{halt}}=2.1M_{\oplus}$, about 42 % of planetary core mass, as shown
in the bottom panel (red square) of Figure 3. We can see that the planet
avoids runaway gas accretion and stops accreting gas when the effect of ohmic
dissipation is coupled into the envelope accretion.

Figure 3: The luminosity and evolutionary time vary with the envelope mass.
The planetary core mass is 5 $M_{\oplus}$. The black line shows the case
without Ohmic dissipation while the red lines describe the case with ohmic
dissipation. The parameters of ohmic heating model that are used here are the
same as in Figure 1. Upper panel: solid lines represent the total luminosity
of the planet. The dashed line and dotted line denote the cooling luminosity
and the ohmic heating power, respectively. The circle and square symbols in
the bottom panel designate the runaway time ($\mathrm{t_{run}}$) and the
halting mass with omic dissipation ($M_{\mathrm{halt}}$), respectively. The
vertical red-dotted line expresses that the contraction of planet is halted.
Without ohmic dissipation, the planet starts runaway gas accretion at
$\mathrm{t_{run}}\simeq$ 1 Myr. When the effect of ohmic dissipation is
considered, gas accretion is halted at about 2 Myr and
$M_{\mathrm{halt}}=2.1M_{\oplus}$, about 42 % of planetary core mass. For
further details, see Section 3.2.

###  3.3 Parameter study of Ohmic dissipation

In the previous section, only a set of ohmic heating model parameters are
studied, which is not enough to investigate the effect of Ohmic dissipation on
planetary cooling. Moreover, we adopt a simplified ohmic-dissipation model
which assumes that the magnetic filed strength and wind velocity are constant.
This is not realistic during the envelope growth. The properties of magnetic
field and wind zone should change simultaneously during envelope accretion.
Thus, parameter study is needed. In this section, we explore the parameter
space to study the effect of these parameters (i.e., the strength of magnetic,
the wind zone depth, and wind velocity) on the planetary cooling. Envelope
accretion with dust opacity is also briefly discussed.

Figure 4: Evolutionary time varies with envelope mass for different wind
velocities. The magnetic strength is 1 G. The wind zone depth is 0.7
$R_{\mathrm{out}}$. From right-hand to left-hand, the different color lines
shows cases for wind velocity with $v_{\mathrm{w}}=3\times 10^{3}\ \mathrm{cm\
s^{-1}}$ (cyan), $v_{\mathrm{w}}=5\times 10^{3}\ \mathrm{cm\ s^{-1}}$
(blue),$v_{\mathrm{w}}=8\ \times 10^{3}\ \mathrm{cm\ s^{-1}}$ (yellow),
$v_{\mathrm{w}}=1\times 10^{4}\ \mathrm{cm\ s^{-1}}$ (red),
$v_{\mathrm{w}}=1.12\times 10^{4}\ \mathrm{cm\ s^{-1}}$ (magenta),
respectively. The black line denotes the case without ohmic dissipation.
Figure 5: The halting mass varies with the magnetic field strength and wind
velocity. The wind zone depth is 0.7 $R_{\mathrm{out}}$. The planetary core
here is 5 $M_{\oplus}$. The color bar denotes the ratio of halting mass and
the core mass ($\mathrm{M_{halt}/M_{c}}$). The largest values of the ratio
(blue squares) are less than 1 (around 0.9). For the parameters located on the
left-hand side of blue squares, ohmic dissipation has a negligible effect on
the planetary cooling and runaway gas accretion is unavoidable. When the
parameters of ohmic heating model are located on the right side of the red
squares, gas cannot be accreted onto the planet due to the strong ohmic
dissipation.

####  3.3.1 Wind velocity and magnetic field

Figure 4 shows the cooling time with increasing envelope mass for different
wind velocities. The magnetic field strength is 1 G and the wind zone depth
0.7 $R_{\mathrm{out}}$. From right-hand to left-hand, the wind velocity
increases from $3\times 10^{3}$ to $1.12\times 10^{4}\mathrm{cm\ s^{-1}}$. As
wind velocity increases, the gas accretion is significantly delayed and the
halting mass reduces. As the wind velocity reaches $1.12\times
10^{4}\mathrm{cm\ s^{-1}}$, the planet stops contraction and accretion, even
when the envelope mass is just several percent (about 7%) of the core mass
(magenta line in Figure 4). A super-Earth may be formed in this case.

Figure 5 shows the halting mass with different values of magnetic field
strength and wind velocity for the core mass with 5 $M_{\oplus}$ . The wind
zone depth is 0.7 $R_{\mathrm{out}}$. Different colors denote the value of the
ratio of the halting mass and the core mass, as the color bar shown in the
figure. The blue squares in Figure 5 (or Figure 6) represent the largest
values of the ratio can be reached, generally around 0.9 (but less than 1).
The effect of ohmic dissipation is negligible for the parameters located on
the left-hand side of the blue squares. The planet undergoes runaway gas
accretion in this situation. As the wind velocity and magnetic field strength
increase, the halting mass reduces. The halting mass can be several percent of
core mass for some parameter sets, as the red squares shown in Figure 5333When
the parameters are located on the right-hand side of red squares, the ohmic
dissipation is so strong that gas cannot be accreted onto the planet.. If the
planet possess strong magnetic field, a relative small wind velocity can
efficiently halt the gas accretion, and vice versa. The magnetic field acting
on the protoplanets needs to be constrained by future observations.

Figure 5 also implies that the case of small ratio
($\mathrm{M_{halt}/M_{c}}<0.1$) that prefers to form super-Earths can only be
reached in a specific parameter space according to our Ohmic dissipation
model. Although the parameter space can be expanded if different values of
wind zone depth are considered (as discussed in next the section), it possibly
originates from the assumptions that the wind velocity and magnetic field are
constant in our calculations. In real cases, there should be a wind velocity
(magnetic field) profile in the envelope and the velocity (magnetic field)
should evolve with the envelope mass growth. More reliable results may be
obtained if these factors are taken into account. Thus, the importance of
ohmic dissipation on the envelope accretion need to be explored with a more
realistic model.

Figure 6: Same as Figure 5, but the halting mass varies with the wind zone
depth and wind velocity. The magnetic field strength is 1 G.

####  3.3.2 The wind zone depth

Note that most of the ohmic power concentrated into the wind zone (e.g.,
appendix A of Wu & Lithwick 2013a), meaning that the wind zone depth is
essential for the radial distribution of ohmic heating power. The bottom of
the wind zone determines the efficiency of ohmic power deposited into the
inner convective zone. The extent of planetary contraction prevented by ohmic
dissipation depends on how much of the cooling luminosity can be replaced by
ohmic heating power during envelope accretion. With numerical results from
previous studies (e.g., Lambrechts & Lega 2017; Kurokawa & Tanigawa 2018; Zhu
et al. 2021), we set $d_{\mathrm{w}}\ =0.7\ R_{\mathrm{out}}$ as our fiducial
case. Here, we explore the dependence of halting mass on the wind zone depth,
as shown in Figure 6. The strength of magnetic field is 1 G. The wind zone
depth is from 0.5 to 0.9 $R_{\mathrm{out}}$. AS the wind zone depth increases,
the wind velocity required to stop accretion reduces remarkably. We can see
that the planetary gas accretion is efficiently delayed for a deep wind zone
($\mathrm{d_{w}}=0.9\ R_{\mathrm{out}}$) with a wind velocity only
$\mathrm{v_{w}}=\ 10^{2}\ \mathrm{cm\ s^{-1}}$, which is in the order of
$10^{-4}$ Keplerian velocity at the surface of planetary core.

Figure 7: The cooling luminosity and time with the increasing envelope mass
when dust opacity is taken into account. The effect of ohmic dissipation is
not included. $\mathrm{a_{max}}$ is the maximum particle size of dust grains.

####  3.3.3 Dust opacity

Opacity is crucial for the thermal structure of planetary envelope and
dominates envelope contraction. Compared with the dust-free opacity, the
planetary cooling contraction is delayed by employing the dust opacity (e.g.,
Lee et al. 2014; see also Figure 7). However, the opacity of planetary
envelope during gas accretion is still unclear. Due to the efficient
coagulation of dust grains in the envelope, a dust-free opacity may be favored
(e.g., Mordasini et al. 2014; Ormel 2014).

We adopt the combined opacity from Zhu et al. (2021) to study the accretion
process. Figure 7 shows the variations of planetary luminosity and cooling
time with different dust opacities444The effect of ohmic dissipation is not
included here.. For the situation of dust opacity with maximum dust grain size
$a_{\mathrm{max}}=0.001\ \mathrm{cm}$, the runaway accretion time is about 10
Myr, which is similar to the result of the fiducial model from Lee et al.
(2014).

Figure 8: The cooling time for the gas accretion of dusty envelope
($a_{\mathrm{max}}=0.001\ \mathrm{cm}$) with ohmic dissipation. The magnetic
strength is 1 G. The wind zone depth is 0.7 $R_{\mathrm{out}}$. From right-
hand to left-hand, the different color lines show cases for wind velocity with
$v_{\mathrm{w}}=1\times 10^{3}\ \mathrm{cm\ s^{-1}}$ (red),$v_{\mathrm{w}}=3\
\times 10^{3}\ \mathrm{cm\ s^{-1}}$ (blue), $v_{\mathrm{w}}=5\times 10^{3}\
\mathrm{cm\ s^{-1}}$ (magenta), respectively. The black line denotes the case
without ohmic dissipation.

Figure 8 shows the cooling time with ohmic dissipation for the gas accretion
of dusty envelope ($a_{\mathrm{max}}=0.001\ \mathrm{cm}$). The parameters of
ohmic heating model are same as those of Figure 4. We can see that the gas
accretion of dusty envelope can be halted more effectively by ohmic
dissipation relative to that of dust-free envelope. For example, the ratio of
halting mass and planetary core mass is about 0.11 for the dusty envelope when
the wind velocity $v_{\mathrm{w}}=5\times 10^{3}\ \mathrm{cm\ s^{-1}}$, while
the ratio is about 0.74 for the dust-free envelope (Figure 4).

##  4 Summary and Discussion

The mechanism of ohmic dissipation has been widely used to study the radius
inflation of close-in hot planets. For the close-in protoplanets, ohmic
dissipation can also be induced in the planetary envelope as the disk gas is
accreted onto the planetary surface and flows across the planetary intrinsic
magnetic field or the field from its host star. In this work, we employ the
ohmic-dissipation model from Wu & Lithwick (2013a) to numerically investigate
the effect of ohmic dissipation on the interior structure and cooling
contraction of planetary envelopes during the slow accretion phase. We explore
how the planetary cooling changes with different parameter sets of ohmic
dissipation model.

The main findings are as follows:

(1) With the ohmic-dissipation coupling into the planetary thermal structure,
the cooling luminosity is replaced by the extra energy from ohmic heating
deposited into the planetary envelope. As a result, the radiative-convective
boundary of planetary envelope moves inward and envelope contraction slows
down. Planetary envelope accretion is efficiently delayed and even halted when
planetary luminosity is dominated by the ohmic heating power.

(2) For some parameter sets of the ohmic heating model, the planetary envelope
stops contraction and gas accretion is halted before runaway accretion. The
final envelope mass is only several percent of its core mass. Super-Earths may
be formed in situ for these parameter sets of the ohmic heating model.

However, the ohmic-dissipation model and parameters we used have several
limitations. We simply assume that the magnetic field and wind velocity are
constant. Actually, the magnetic field and wind velocity should evolve as the
envelope mass increases. The simple geometries of magnetic field and wind zone
(i.e., wind zone depth and wind velocity profile) that are assumed in our
calculations may not apply in real situations. We use approximate equations to
calculate the conductivity, which may affect the ohmic heating profiles. In
addition, the magnetic field acting on the protoplanets is not well
constrained from observations. We simply assume a magnetic dipole field with
surface strength from 0.1 to 10 G in our calculations. A more accurate ohmic-
dissipation model with different boundary conditions and initial conditions
are required to investigate the gas accretion of protoplanets in future work.
Although our model has these limitations, as a first step, our results show
that ohmic dissipation is a potential mechanism to prevent planetary cooling
contraction and promote the formation of super-Earths. Future observations of
protoplanets may help to constrain the importance of ohmic dissipation on the
super-Earth formation.

We thank the anonymous referee for insightful comments and very useful
suggestions to significantly improve our draft. We thank Dr. Liang Yin for
useful discussions. This work has been supported by the National SKA Program
of China (Grant No. 2022SKA0120101), Macau Science and Technology Development
Fund (File No. 0001/2019/A1, No. 0051/2021/A1 and No. SKL-
LPS(MUST)-2021-2023), the National Key R&D Program of China (No.
2020YFC2201200) and the science research grants from the China Manned Space
Project (No. CMS-CSST- 2021-B09 and CMS-CSST-2021-A10). C.Y. has been
supported by the National Natural Science Foundation of China (grants
11373064, 11521303, 11733010, and 11873103), Yunnan National Science
Foundation (grant Q9 2014HB048), and Yunnan Province (2017HC018).

##  Appendix A Ohmic heating Equation

Equations (7) and (8) can be changed as follows:

${\Phi}_{2}^{\prime\prime}+\left(\frac{d\ln\sigma}{dr}+\frac{2}{r}\right){\Phi}_{2}^{\prime}-\frac{6}{r^{2}}{\Phi}_{2}-\left(\frac{d\ln\sigma}{dr}\frac{2\omega
M_{B}}{3cr^{2}}+\frac{4\omega M_{B}}{cr^{3}}\right)=0,\ \ r_{\mathrm{dep}}\leq
r<R_{\mathrm{out}}$ (A1)
${\Phi}_{2}^{\prime\prime}+\left(\frac{d\ln\sigma}{dr}+\frac{2}{r}\right){\Phi}_{2}^{\prime}-\frac{6}{r^{2}}{\Phi}_{2}=0,\
\ 0<r<r_{\mathrm{dep}}\\\ $ (A2)

where $\Phi$ is expressed with $\Phi=\Phi_{2}(r)P_{2}$ (Wu & Lithwick 2013a),
$P_{2}$ is the Legendre polynomial,
$P_{2}=\frac{1}{2}(3\mathrm{cos}^{2}\theta-1)$. $r_{\mathrm{dep}}$ denotes the
location of the bottom of the wind zone,
$r_{\mathrm{dep}}=R_{\mathrm{out}}-d_{\mathrm{w}}$, where $d_{\mathrm{w}}$ is
the wind zone depth. These two ordinary differential equations can be
numerically solved.

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# (Un)boundedness of directional maximal operators through a notion of “Perron
capacity” and an application

Emma D’Aniello, Anthony Gauvan and Laurent Moonens

###### Abstract

We introduce the notion of Perron capacity of a set of slopes
$\Omega\subset\mathbb{R}$. Precisely, we prove that if the Perron capacity of
$\Omega$ is finite then the directional maximal operator associated
$M_{\Omega}$ is not bounded on $L^{p}(\mathbb{R}^{2})$ for any $1<p<\infty$.
This allows us to prove that the set

$\Omega_{\boldsymbol{e}}=\left\\{\frac{\cos n}{n}:n\in\mathbb{N}^{*}\right\\}$

is not finitely lacunary which answers a question raised by A. Stokolos.

00footnotetext: _2020 Mathematics Subject Classification_ : Primary: 42B25,
26B05; Secondary: 42B35.
_Keywords_ : Lebesgue’s Differentiation Theorem, Directional Maximal
Functions, Lacunary Sets of Finite Order.

##  1 Introduction

It is well known, since the original work of Jensen, Marcinkiewicz and Zygmund
[16], that Lebesgue’s differentiation formula, _i.e._ the equality for a.e.
$x\in\mathbb{R}^{n}$:

$f(x)=\lim_{\begin{subarray}{c}R\ni x\\\ \operatorname{diam}R\to
0\end{subarray}}\frac{1}{|R|}\int_{R}f,$ (1)

holds along $n$-dimensional _rectangles_ $R$ parallel to the coordinate
hyperplanes (called standard rectangles), whenever one has $f\in
L\log^{n-1}L(\mathbb{R}^{n})$ — the latter Orlicz space being, moreover, the
largest for which the above formula holds if one does not assume any extra
hypothesis on the rectangles $R$; the beautiful survey by A. Stokolos [21]
provides a quite exhaustive state of the art on the topic — see also [5] for
sufficient conditions (of geometric nature) on $\mathcal{R}$ ensuring that
$L\log^{n-1}L(\mathbb{R}^{n})$ is the largest space for which (1) holds
whenever $R$ is restricted to belong to the class $\mathcal{R}$.

In dimension $n=2$, the nature of the problem changes when one allows
rectangles $R$ to rotate around one of their vertices, requiring that the
slope of their longest side belongs to a fixed set $\Omega$ (call
$\mathcal{R}_{\Omega}$ the family of such rectangles in the plane). It has
been shown _e.g._ by Cordoba and Fefferman [4] that when
$\Omega=\\{\omega_{k}:k\in\mathbb{N}\\}$ is a lacunary sequence (see below for
a definition), then formula (1) holds for any $f\in L^{p}(\mathbb{R}^{2})$ if
one has $p\in[2,+\infty[$. This was later generalized for arbitrary
$1<p<\infty$ by Nagel, Stein and Wainger [19]. Given a set of slopes $\Omega$,
the question of finding Orlicz spaces (the largest possible) beyond $L^{p}$
spaces, for all elements of which formula (1) holds is often delicate; issues
in these directions have been studied in works involving some of the current
authors, see [7] and [6] for example.

Concerning the possibility of (1) holding for all $f\in
L^{p}(\mathbb{R}^{2})$, $1<p\leqslant\infty$, it has been shown _e.g._ by Katz
[17], Bateman and Katz [2], Hare [14] and more recently by Bateman [1] that
Cantor sets (and even uncountable sets) never give rise to formula (1) for all
$f\in L^{p}(\mathbb{R}^{2})$, $1<p\leqslant\infty$ — Bateman’s work [1] even
gives a complete characterization, even though not always simple to implement
_in practice_ (see below), of sets $\Omega$ for which it does hold in any
$L^{p}(\mathbb{R}^{2})$, $1<p\leqslant\infty$: those are sets one will call
_finitely lacunary_ in the sequel.

Before to state more precisely Bateman’s result, let us mention an important
tool in its proof: the possibility, when a set of slopes $\Omega$ is too
large, to find finite families of rectangles in $\mathcal{R}_{\Omega}$ such
that the union of their (centered) homothetic expansions occupies a much
larger area in the plane than their union does; we call this a _Kakeya blow_.

When no restriction is made on directions of rectangles in play, the
possibility of exhibiting a Kakeya blow is a standard result in measure theory
(see Busemann and Feller’s work [3] for a first construction of this type):

######  Theorem 1.1 (Kakeya Blow).

For any large $A\gg 1$, there exists a finite family of rectangles
$\left\\{R_{i}\right\\}_{i\leqslant N}$ such that we have

$A\left|\bigcup_{i\leqslant N}R_{i}\right|\leqslant\left|\bigcup_{i\leqslant
N}4R_{i}\right|.$

Here, $4R$ stands for the $4$-fold dilation of $R$ by its center.

Kakeya blows have deep implications in harmonic analysis: for example, a
concrete construction proving the above theorem allowed Fefferman to disprove
the Disc multiplier conjecture in [10].

A natural question, given a set of slopes $\Omega$, then becomes the
following: can we still construct Kakeya blows using _only_ rectangles in
$\mathcal{R}_{\Omega}$, namely rectangles the longest side of which makes an
angle $\omega\in\Omega$ with the horizontal axis (see also Stokolos [20] and
Hagelstein and Stokolos [13] concerning the links between directional maximal
operators and multipliers)? In his study of the so-called directional maximal
operator $M_{\Omega}$ defined for a locally integrable
$f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ and $x\in\mathbb{R}^{2}$ as

$M_{\Omega}f(x):=\sup_{x\in
R\in\mathcal{R}_{\Omega}}\frac{1}{\left|R\right|}\int_{R}\left|f\right|,$

Bateman [1] introduced a notion we shall call finitely lacunary set and proved
the following Theorem.

######  Theorem 1.2 (Bateman).

The following conditions are equivalent

  * •

the set $\Omega$ is not finitely lacunary ;

  * •

for any $A\gg 1$, there exists a finite family of rectangles $\\{R_{i}\\}$
such that the longest side of each makes an angle $\omega\in\Omega$ with the
$x$-axis and also satisfying $A\left|\bigcup_{i\in
I}R_{i}\right|\leqslant\left|\bigcup_{i\in I}4R_{i}\right|$ ;

  * •

the operator $M_{\Omega}$ is not bounded on $L^{p}(\mathbb{R}^{2})$ for any
$1<p<\infty$.

Moreover, the following conditions are also equivalent

  * •

the set $\Omega$ is finitely lacunary

  * •

the operator $M_{\Omega}$ is bounded on $L^{p}(\mathbb{R}^{2})$ for any
$1<p<\infty$.

In the latter theorem, it needs to be explained what is meant by “finitely
lacunary”. We recall it here, following the presentation by Kroc and Pramanik
[18]. We start by defining the notion of lacunary sequence and then the notion
of lacunary set of finite order.

######  Definition 1.3 (Lacunary sequence).

We say that a sequence of real numbers $L=(L_{k})$ is a lacunary sequence
converging to $\ell\in\mathbb{R}$ when there holds

$|\ell-\ell_{k+1}|\leqslant\frac{1}{2}|\ell-\ell_{k}|$

for any $k$.

For example the sequences $L_{1}:=\left(\frac{1}{2^{k}}\right)_{k\geqslant 2}$
and $L_{2}:=\left(\frac{1}{k!}\right)_{k\geqslant 4}$ are lacunary.

######  Definition 1.4 (Lacunary set of finite order).

A lacunary set of order $0$ in $\mathbb{R}$ is a set which is either empty or
a singleton. Recursively, for $N\in\mathbb{N}^{*}$, we say that a set $\Omega$
included in $\mathbb{R}$ is a lacunary set of order at most $N+1$ — and write
$\Omega\in\Lambda(N+1)$ — if there exists a lacunary sequence $L$ with the
following properties : for any $a,b\in L$ such that $a<b$ and $(a,b)\cap
L=\emptyset$, the set $\Omega\cap(a,b)$ is a lacunary set of order at most $N$
i.e. $\Omega\cap(a,b)\in\Lambda(N)$.

For example the set

$\Omega:=\left\\{\frac{1}{2^{k}}+\frac{1}{4^{l}}:k,l\in\mathbb{N},l\leqslant
k\right\\}$

is a lacunary set of order $2$. In this case, observe that the set $\Omega$
cannot be re-written as
$\Omega=\left\\{\omega_{k}:k\in\mathbb{N}^{*}\right\\}$ where $(\omega_{k})$
is a monotone sequence, since it has several points of accumulation. We can
finally give a definition of a finitely lacunary set.

Figure 1: A representation of a lacunary sequence and a lacunary set of order
$2$.

######  Definition 1.5 (Finitely lacunary set).

A set $\Omega$ in $\mathbb{R}$ is said to be finitely lacunary if there exists
a finite number of set $\Omega_{1},\dots,\Omega_{M}$ which are lacunary of
finite order such that

$\Omega\subset\bigcup_{k\leqslant M}\Omega_{k}.$

Even if Theorem 1.2 gives a full characterization of the behavior of
directional maximal operators, it is usually hard _in practice_ to decide
whether a set of slopes $\Omega$ is finitely lacunary. For example, it was
known that the set of slopes $\Omega_{0}$ defined by

$\Omega_{0}:=\left\\{\frac{1}{n}:n\in\mathbb{N}^{*}\right\\}$

generates a directional maximal operator $M_{\Omega_{0}}$ which is not bounded
on $L^{p}(\mathbb{R}^{2})$ for any $1<p<\infty$ (see _e.g._ [9, Theorem 2.1]);
using Theorem 1.2 one can hence assert that the set of directions $\Omega_{0}$
is not finitely lacunary. In the case where the set of directions $\Omega$ is
a discrete set which can be ordered in $\mathbb{R}_{+}^{*}$, more precisely in
the case where $\Omega=\\{\frac{1}{u}:u\in U\\}$ and we can write
$U=\left\\{u_{k}:k\in\mathbb{N}^{*}\right\\}\subset\mathbb{R}_{+}^{*}$ for an
increasing sequence $(u_{k})_{k\in\mathbb{N}^{*}}$, Hare and Rönning [15]
provided a quantitative sufficient condition on $U$ which ensures that the
maximal operator associated $M_{U}$ is not bounded on $L^{p}(\mathbb{R}^{2})$.
Precisely, they proved the following Theorem.

######  Theorem 1.6 (Hare-Rönning [15]).

Assume that $U=(u_{k})_{k\in\mathbb{N}^{*}}\subset(0,\pi/2)$ is an increasing
sequence of directions and that one has

$G_{U}:=\sup_{\begin{subarray}{c}k\geqslant 1\\\ l\leqslant
k\end{subarray}}\left(\frac{u_{k+2l}-u_{k+l}}{u_{k+l}-u_{k}}+\frac{u_{k+l}-u_{k}}{u_{k+2l}-u_{k+l}}\right)<\infty.$

Then for any $p\in(1,\infty)$, one has $\|M_{U}\|_{p}=\infty$.

In particular, applying Bateman’s Theorem, it follows that if we have
$G_{U}<\infty$ then $U$ is not finitely lacunary — other uses of Perron trees
have also been made in works involving the current authors, see _e.g._. [8],
[11] and [12]. Hence, usually one does not prove directly that a set of slopes
$\Omega$ is not finitely lacunary (or that it is): in general, it is a
consequence of the (un)-boundedness of the associated maximal operator
$M_{\Omega}$ — obtained by other methods — and Bateman’s Theorem.

Our results are motivated by the following question raised by A. Stokolos :
what can be said (for example) about the set of directions
$\Omega_{\boldsymbol{e}}$ defined as

$\Omega_{\boldsymbol{e}}:=\left\\{\frac{\cos
n}{n}:n\in\mathbb{N}^{*}\right\\}\ ?$

Can one prove that it is finitely lacunary or not? In this text, we prove it
is _not_ finitely lacunary.

##  2 Results

We denote by $\mathcal{R}$ the set containing all rectangles of the plane and
for $R\in\mathcal{R}$, we denote by $\omega_{R}\in[0,\pi)$ the angle that the
longest side of the rectangle $R$ makes with the $x$-axis. For any set
$\Omega\subset\mathbb{R}$ (which should be thought as a set of slopes) we
define the family of rectangle $\mathcal{R}_{\Omega}$ as

$\mathcal{R}_{\Omega}:=\left\\{R\in\mathcal{R}:\tan(\omega_{R})\in\Omega\right\\}.$

We then define the directional maximal operator $M_{\Omega}$ as

$M_{\Omega}f(x):=\sup_{x\in
R\in\mathcal{R}_{\Omega}}\frac{1}{\left|R\right|}\int_{R}|f|.$

We finally define the Perron capacity of a set of slopes $\Omega$ included in
$\mathbb{R}$ as follow.

######  Definition 2.1 (Perron capacity of a set of slopes).

For any set of slopes $\Omega\subset\mathbb{R}$, we define its Perron capacity
as

$P_{\Omega}:=\mathop{\underline{\lim}}_{N\to\infty}\inf_{\begin{subarray}{c}U\subset\Omega\\\
\\#U=2^{N}\end{subarray}}G_{U}\in[2,\infty]$

where $G_{U}=\sup_{\begin{subarray}{c}k,l\geqslant 1\\\ k+2l\leqslant
2^{N}\end{subarray}}\left(\frac{u_{k+2l}-u_{k+l}}{u_{k+l}-u_{k}}+\frac{u_{k+l}-u_{k}}{u_{k+2l}-u_{k+l}}\right)$
if $U=\\{u_{1},\dots,u_{2^{N}}\\}$ with $u_{1}<u_{2}<\cdots<u_{N}$.

Our first result is the following: it gives a sufficient and quantitative
condition on an arbitrary set $\Omega$ which ensures that the associated
maximal operator is unbounded on $L^{p}(\mathbb{R}^{2})$ for any $1<p<\infty$.
In contrast with Hare and Ronning Theorem, we do not assume that our set of
slopes is ordered.

######  Theorem 2.2.

Fix any set of slopes $\Omega\subset\mathbb{R}$ and suppose that we have

$P_{\Omega}<\infty.$

Then for any $p\in(1,\infty)$, one has $\|M_{\Omega}\|_{p}=\infty$.

Loosely speaking, the fact that $P_{\Omega}<\infty$ indicates whether the set
$\Omega$ contains arbitrary large subsets which are (more or less) uniformly
distributed in $\mathbb{R}$. Our second result is an application of Theorem
2.2 and deals with the set of slopes $\Omega_{\boldsymbol{e}}$ defined as

$\Omega_{\boldsymbol{e}}:=\left\\{\frac{\cos
n}{n}:n\in\mathbb{N}^{*}\right\\}.$

Namely, we prove the following Theorem.

######  Theorem 2.3.

The Perron capacity of the set $\Omega_{\boldsymbol{e}}$ is finite i.e. we
have $P(\Omega_{\boldsymbol{e}})<\infty$.

Hence, as a consequence of Theorems 2.3 and 2.2 we know that — for any
$1<p<\infty$ — we have

$\|M_{\Omega_{\boldsymbol{e}}}\|_{p}=\infty.$

As an application of Bateman’s Theorem, we can hence say that the set
$\Omega_{\boldsymbol{e}}$ is not finitely lacunary.

##  3 Proof of Theorem 2.2

Recall that for any set
$U:=\left\\{u_{1},\dots,u_{2^{N}}\right\\}\subset\mathbb{R}_{+}$ with
$u_{1}<u_{1}<\cdots<u_{2^{N}}$ (so that $U$ has $2^{N}$ elements), one define
its Perron factor as

$G_{U}:=\sup_{\begin{subarray}{c}k,l\geqslant 1\\\ k+2l\leqslant
2^{N}\end{subarray}}\left(\frac{u_{k+2l}-u_{k+l}}{u_{k+l}-u_{k}}+\frac{u_{k+l}-u_{k}}{u_{k+2l}-u_{k+l}}\right)\in(0,\infty).$

The following proposition is a careful reading of Hare and Rönning’s work
[15].

######  Proposition 3.1.

There exists $\varepsilon_{0}\in(0,1)$ such that for any
$\alpha\in[1-\varepsilon_{0},1)$ there exists a set $X\subseteq\mathbb{R}^{2}$
for which one has:

$\left|X\right|\leqslant\left[\alpha^{2N}+G_{U}(1-\alpha)^{2}\right]\left|\\{M_{U}\mathbf{1}_{X}>\frac{1}{4}\\}\right|.$

Hence for any $0<\alpha<1$ close enough to $1$, this gives us the following
lower bound for any $p\in(1,\infty)$:

$\|M_{U}\|_{p}^{p}\gtrsim_{p}\frac{1}{\alpha^{2N}+G_{U}(1-\alpha)^{2}},$

where $\gtrsim_{p}$ means that the inequality holds up to a multiplicative
constant that does only depend on $p$.

Indeed, we have

$\|M_{U}\mathbf{1}_{X}\|_{p}^{p}\geqslant\frac{1}{4^{p}}\left|\\{M_{U}\mathbf{1}_{X}>\frac{1}{4}\\}\right|\gtrsim_{p}\frac{\left|X\right|}{\alpha^{2N}+G_{U}(1-\alpha)^{2}}.$

This allows us now to easily conclude the proof of Theorem 2.2. Indeed, fix an
arbitrary set $\Omega\subset\mathbb{R}$ and suppose that its Perron capacity
is finite i.e. assume that one has:

$P_{\Omega}<\infty.$

By definition of $P_{\Omega}$, there thus exists a strictly increasing
sequence of integers $\left\\{N_{k}:k\in\mathbb{N}^{*}\right\\}$ such that for
any $k$ there is a set $U_{k}\subset\Omega$ such that $G(U_{k})<2P_{\Omega}$
and $\\#U_{k}=2^{N_{k}}$. Since there holds $U_{k}\subset\Omega$, we obtain,
for any $0<\alpha<1$ sufficiently close to $1$ and any $k\geqslant 1$:

$\|M_{\Omega}\|_{p}^{p}\geqslant\|M_{U_{k}}\|_{p}^{p}\gtrsim\frac{1}{\alpha^{2N_{k}}+2P_{\Omega}(1-\alpha)^{2}}.$

Since this holds for any $k\geqslant 1$ and any $\alpha$ close to $1$, this
implies that we have $\|M_{\Omega}\|_{p}=\infty$.

##  4 Homogeneous sets

We fix an arbitrary set $U$ in $\mathbb{R}^{*}_{+}$ whose cardinal is $2^{N}$
for some $N\in\mathbb{N}^{*}$. The following proposition shows that it is
equivalent for $U$ to be uniformly distributed and its Perron factor to equal
$2$.

######  Proposition 4.1.

We have $G_{U}=2$ if and only if the elements of $U$ are in arithmetic
progression _i.e._ $U$ is of the form

$U=\\{a+k\delta:1\leqslant k\leqslant 2^{N}\\}$

for some $a\in\mathbb{R}$ and $\delta>0$.

###### Proof.

If $U$ is in arithmetic progression we have easily $G_{U}=2$. On the other
hand if we have $G_{U}=2$ since $x+\frac{1}{x}=2$ if and only $x=1$ we have
for any $k$ (taking $\ell=1$)

$u_{k+2}-u_{k+1}=u_{k+1}-u_{k}$

which concludes the proof. ∎

We will be particularly interested by homogeneous sets that is to say sets $H$
of the form

$H:=H_{a,N}=\\{ka:1\leqslant k\leqslant 2^{N}\\}$

for some integers $a\in\mathbb{N}^{*}$ and $N\in\mathbb{N}$. In particular, we
wish to perturb a little an homogeneous set $H_{a,N}$ into a set $H^{\prime}$
such that the Perron’s constant of $H^{\prime}$ is still bounded bounded. More
precisely, fix any $a,N\in\mathbb{N}^{*}$ and let $\varepsilon$ be an
arbitrary function

$\varepsilon:H_{a,N}\rightarrow(0,1).$

Define then the set $H_{a,N}(\varepsilon)$ as

$H_{a,N}(\varepsilon):=\left\\{(1+\varepsilon(x))x:x\in H_{a,N}\right\\}.$

Figure 2: A representation of a uniformly distributed set and its
perturbation.

######  Proposition 4.2.

Assume that $a\in\mathbb{R}^{*}_{+}$, $N\in\mathbb{N}^{*}$ and
$\varepsilon:H_{a,N}\to(0,1)$ are given. If one has

$2^{N}\|\varepsilon\|_{\infty}\leqslant\frac{1}{2},$

then we have $G_{H_{a,N}}\leqslant 6$.

###### Proof.

To simplify the notations, let for $1\leqslant k\leqslant 2^{N}$,
$h_{k}:=[1+\varepsilon(ka)]ka$. First, observe that for $1\leqslant k\leqslant
2^{N}-1$, one has:

$h_{k+1}-h_{k}=\\{1+\varepsilon[(k+1)a]\\}(k+1)a-[1+\varepsilon(ka)]ka\\\
\geqslant a-ak\varepsilon(ka)=a[1-k\varepsilon(ka)]\geqslant
a[1-(2^{N}-1)\|\varepsilon\|_{\infty}]>0,$

so that there holds $h_{1}<h_{2}<\cdots<h_{2^{N}}$.

Now compute, for $1\leqslant l\leqslant k$ such that $k+2l\leqslant 2^{N}$
(implying in particular that one has $\frac{k}{l}\leqslant 2^{N}-2$):

$\frac{h_{k+2l}-h_{k+l}}{h_{k+l}-h_{k}}\leqslant\frac{la+2^{N}a\|\varepsilon\|_{\infty}}{la-\|\varepsilon\|_{\infty}ka}=\frac{1+\frac{2^{N}}{l}\|\varepsilon\|_{\infty}}{1-\frac{k}{l}\|\varepsilon\|_{\infty}}\leqslant\frac{1+2^{N}\|\varepsilon\|_{\infty}}{1-(2^{N}-2)\|\varepsilon\|_{\infty}}\leqslant
3.$

One obtains a similar inequality for the symmetric ratio. ∎

##  5 Proof of Theorem 2.3

Recall that we have defined

$\Omega_{\boldsymbol{e}}:=\left\\{\frac{n}{\cos
n}:n\in\mathbb{N}^{*}\right\\}$

and that we wish to prove that the Perron capacity of
$\Omega_{\boldsymbol{e}}$ is finite i.e. that we have
$P(\Omega_{\boldsymbol{e}})<\infty$. To do so, we are simply going to prove
that the set $\Omega_{\boldsymbol{e}}$ contains “small perturbations” of
_arbitrarily long homogeneous sets._ Specifically, for any $N\in\mathbb{N}$,
we consider the set

$E(N):=\\{n\in\mathbb{N}^{*}:\exists m\in\mathbb{Z},|n+2\pi m|<2^{-N}\\}.$

To begin with, we claim the following.

######  Claim 1.

For any $N\in\mathbb{N}$ we have $\\#E(N)=\infty$.

###### Proof of the claim.

The claim follows from the fact that $G^{\prime}=\left\\{n+2\pi
m:n\in\mathbb{N},m\in\mathbb{Z}\right\\}$ is dense in $\mathbb{R}$, which can
be shown easily using standard techniques from basic real analysis111See
_e.g._
https://lefevre.perso.math.cnrs.fr/PagesPerso/enseignement/Archives/SSgrpesAdd.pdf
P. Lefèvre’s webpage for a proof..∎

######  Claim 2.

For any $n\in\mathbb{N}$ and any $n\in E(N)$, one has:

$1<\frac{1}{\cos n}\leqslant 1+2^{-2N}.$

###### Proof of the claim.

Choosing $m\in\mathbb{Z}$ such that one has $|n+2\pi m|<2^{-N}$ (which exists
by definition since one has $n\in E(N)$), one obtains, using the inequality
$1-\cos x\leqslant\frac{1}{2}x^{2}$ valid for any $x\in\mathbb{R}$:

$1-\cos n=1-\cos(n+2\pi m)\leqslant\frac{1}{2}(n+2\pi
m)^{2}\leqslant\frac{1}{2}2^{-2N}.$

Note that, in particular, this yields $0<\cos n<1$.

Using the fact that one has $\frac{1}{1-x}\leqslant 1+2x$ for any $0\leqslant
x\leqslant\frac{1}{2}$, one finally computes:

$\frac{1}{\cos n}=\frac{1}{1-(1-\cos n)}\leqslant 1+2(1-\cos n)\leqslant
1+2^{-2N},$

as was announced. ∎

Fix an arbitrary large $N\in\mathbb{N}$ and any integer $a\in
E\left(2N\right)$. We claim the following.

######  Claim 3.

For any $a\in E(2N)$, we have

$H_{a,N}\subset E(N).$

###### Proof.

Fix $a\in E(2N)$ ; by definition we have $m\in\mathbb{Z}$ such that

$\left|a+2\pi m\right|<2^{-2N}.$

Hence for any $1\leqslant k\leqslant 2^{N}$ we have

$\left|ka+2\pi km\right|<k2^{-2N}\leqslant 2^{-N}$

i.e. for any $1\leqslant k\leqslant 2^{N}$ we have $ka\in E(N)$. ∎

Fix now $N\in\mathbb{N}^{*}$ and $a\in E(2N)$ and define
$\varepsilon:H_{a,N}\to(0,1)$ by the formula $1+\varepsilon(n)=\frac{1}{\cos
n}$ for $n\in H_{a,N}$ — we hence see the set:

$\left\\{\frac{n}{\cos n}:n\in
H_{a,N}\right\\}=\left\\{[1+\varepsilon(n)]n:n\in H_{a,N}\right\\},$

as a perturbation of the above type of the set $H_{a,N}$ — . We compute, using
Claim 2:

$2^{N}\|\varepsilon\|_{\infty}\leqslant 2^{-N}\leqslant\frac{1}{2},$

so that Proposition 4.2 yields $G_{H_{a,N}(\varepsilon)}\leqslant 6$.

Finally, observe that we have by construction the inclusion

$H_{a,N}(\varepsilon)\subset\Omega.$

Since this holds for any $N\in\mathbb{N}^{*}$, it follows that we have
$P_{\Omega}\leqslant 6$ as desired.

∎

######  Remark 5.1.

The same conclusion holds if in Theorem 2.3 we replace
$\Omega_{\boldsymbol{e}}:=\left\\{\frac{\cos
n}{n}:n\in\mathbb{N}^{*}\right\\}$ with

$\Omega_{\boldsymbol{s}}:=\left\\{\frac{\sin
n}{n}:n\in\mathbb{N}^{*}\right\\}.$

As an application of Bateman’s Theorem, we can hence say that also the set
$\Omega_{\boldsymbol{s}}$ is not finitely lacunary. The argument is the same
but, in this case, we approximate $\frac{\pi}{2}$ by elements of $G^{\prime}$
and take $1+\varepsilon(x)=\frac{1}{\sin x}$.

### Acknowledgements.

This research has been supported by the _École Normale Supérieure_ , Paris, by
the _“Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro
Applicazioni dell’Istituto Nazionale di Alta Matematica F. Severi”_ and by the
project _Vain-Hopes_ within the program _Valere_ of _Università degli Studi
della Campania “Luigi Vanvitelli”_. It has also been partially accomplished
within the _UMI_ Group _TAA “Approximation Theory and Applications”_.
E. D’Aniello would like to thank the _École Normale Supérieure_ , Paris, for
its warm hospitality in March-April, 2022; L. Moonens would like to thank the
_Dipartimento di Matematica e Fisica_ of the _Universitá degli Studi della
Campania “Luigi Vanvitelli”_ for its warm hospitality in April-May, 2022.

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Emma D’Aniello
Dipartimento di Matematica e Fisica, Università degli Studi della Campania
“Luigi Vanvitelli”, Viale Lincoln n. 5, 81100 Caserta, Italia
<EMAIL_ADDRESS>
Anthony Gauvan
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS UMR8628,
Université Paris-Saclay, Bâtiment 307, F-91405 Orsay Cedex, France
<EMAIL_ADDRESS>
Laurent Moonens
Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, CNRS UMR 8628,
Bâtiment 307, F-91405 Orsay Cedex, France
<EMAIL_ADDRESS>
École Normale Supérieure-PSL University, CNRS UMR 8553, 45, rue d’Ulm, F-75230
Parix Cedex 3, France
<EMAIL_ADDRESS>