# Simulations of thermal surface waves in a protoplanetary disk using 1+1D
approximation

Ya. N. Pavlyuchenkov, L. A. Maksimova, V. V. Akimkin
Institute of Astronomy RAS, Moscow, Russia

(Received 30.10.2021; revised 17.01.2022; accepted 24.01.2022
<EMAIL_ADDRESS>
###### Abstract

Heating by the central star is one of the key factors determining the physical
structure of protoplanetary disks. Due to the large optical thickness in the
radial direction, disk midplane regions are heated by the infrared radiation
from the disk surface (atmosphere), which in turn is directly heated by the
star. It was previously shown that interception of the stellar radiation by
inhomogeneities on the disk surface can cause perturbations that propagate
towards the star. In this work, we investigate the occurrence of such waves
within a detailed 1+1D numerical model of the protoplanetary disk. We confirm
the previous findings that in the disk, that is optically thick to its own
radiation, the surface perturbations indeed occur and propagate towards the
star. However, contrary to some analytical predictions, the thermal waves in
sufficiently massive disks affect only the upper layers without significant
fluctuations of temperature in the midplane. Our results indicate the need to
study this instability within more consistent hydrodynamic models.

DOI: 10.31857/S000462992205005X

##  1 Introduction

Solving the problem of the formation of stars and planets is an important
astrophysical and worldview challenge. However, despite the success in
building a general pictures of stars and planets formation, details of the
protoplanetary disks evoltion (precursors of planetary systems) and the role
of various physical processes have not yet been fully clarified. This is
primarily due to the variety of physical processes in protoplanetary disks and
their complex interaction. In protoplanetary disks, the conditions for
occurrence of a wide variety of dynamic instabilities are satisfied. Their
development can affect both the observational manifestations and the general
disk evolution.

Recent observations made with the ALMA telescope array have shown that the
disk surface density distributions do not correspond to smooth power laws
often used by theorists. On the contrary, on the scale of tens to hundreds of
au bright rings and dark gaps are common phenomena (ALMA Partnership et al.
(2015); Andrews (2020); Huang et al. (2018)). For now, these rings and gaps
are most often explained by the influence of invisible planets (Baruteau et
al. (2014); Dong, Zhu & Whitney (2015); Dipierro et al. (2016); Bae, Zhu &
Hartmann (2017); Zhang et al. (2018)). However, many alternative scenarios
have been proposed to explain these features. One of them is the instability
associated with the interception of stellar radiation by the surface
inhomogeneities of the disk.

This instability (known as irradiation instability or thermal wave
instability) can cause the emergence of surface waves traveling towards the
star, and ring structures in circumstellar disks (D’Alessio et al. (1999);
Dullemond (2000); Watanabe & Lin (2008); Siebenmorgen & Heymann (2012)). The
mechanism of this physical instability is as follows. If a small bump forms on
the surface of the disk, then the illuminated side of the bump, facing the
star, receives more starlight and heats up. The warm elements of the bump heat
the lower layers of the disk with their radiation, as a result the formed bump
grows. Since the side of the bump facing the star heats up more strongly than
the opposite side, the perturbation also begins to move towards the star.

Recently, significant progress has been made in the study of this instability,
see Ueda, Flock & Birnstiel (2021) and Wu & Lithwick (2021). In particular, in
Wu & Lithwick (2021) within the framework of an analytical approximation, it
is shown that this instability actually takes place. Moreover, the key role in
the development of instability is the detailed account of dependence of $H/h$
on the distance (where $H$ is the optical height of the disk, and $h$ is the
characteristic hydrostatic scale height). In addition to the analytical model,
the authors of Wu & Lithwick (2021) also presented semi-analytical model where
the evolution of the midplane temperature is derived using the heating
function calculated by the radiation transfer code RADMC-3D.

The purpose of this work is to study the irradiation instability within a more
detailed numerical model of the protoplanetary disk, where the transfer of
stellar radiation is treated in two-dimensional approach while the non-
stationary thermal disk structure is calculated on the entire vertical scale
of the disk.

##  2 Quasi-hydrostatic 1+1D model of a protoplanetary disk

Figure 1: Illustration of the ray tracing on the adopted discrete grid. When
integrating the radiation transfer equation along the ray from the star to the
current cell, all intersections of the ray with the cell boundaries are found
(shown as black dots in the right panel), which is used for accurate
calculation of total optical depth. The final segments of rays belonging to
the cells in which heating is calculated are highlighted in red.  Figure 2:
Diagram explaining the calculation of the heating function in the cell.

To calculate the evolution of the disk, we adopt the model described in
Vorobyov & Pavlyuchenkov (2017); Pavlyuchenkov et al. (2020); Maksimova,
Pavlyuchenkov & Tutukov (2020). This model solves radiative transfer problem,
taking into account the heating by stellar and interstellar radiation, as well
as the diffusion of thermal (infrared) radiation of the disk itself. The
vertical disk structure is derived consistently with the temperature
calculation. Diffusion of thermal radiation in the model is calculated only in
the vertical ($z$) direction. To model it, we solve system of moment transport
equations in the Eddington approximation:

$\displaystyle c_{\rm V}\dfrac{\partial T}{\partial t}=\kappa_{\rm
P}c(E-aT^{4})+S$ (1) $\displaystyle\dfrac{\partial E}{\partial
t}-\dfrac{\partial}{\partial z}\left(\dfrac{c}{3\rho\kappa_{\rm
R}}\dfrac{\partial E}{\partial z}\right)=-\kappa_{\rm P}c(E-aT^{4}),$ (2)

where $T$ is the medium temperature, $E$ is the density of radiation energy,
$z$ is the vertical coordinate, $\rho$ is the gas+dust volume density, $c_{V}$
is the heat capacity of the gas-dust medium, $c$ is the speed of light, $a$ is
the radiative constant, $\kappa_{\rm P}$ [cm2 g-1] is the Planck-averaged
absorption coefficient (per unit mass of the gas-dust mixture), $\kappa_{\rm
R}$ [cm2 g-1] is the Rosseland-averaged extinction coefficient
(absorption+scattering), and $S$ [erg s-1 g-1] is the heating function (per
unit mass of gas-dust medium) by the stellar and interstellar radiation,
$S=S^{*}+S_{\rm bg}$.

The UV intensity required for the calculation of the heating function $S^{*}$
is found for each cell by the direct integration of the radiative transfer
equation from the star to the considered element throughout the entire disk.
This two-dimensional procedure differs from the method described in Vorobyov &
Pavlyuchenkov (2017), where the angle between the direction to the star and
the surface of the disk was assumed to be constant, and thus the problem there
was reduced to one-dimensional one. In the modified method, when integrating
transport equation along the ray all intersections with the cell boundaries
are identified (see Fig. 1), which is used for exact calculation of the
optical depth.

The key aspect in calculating the heating function by stellar radiation within
our model is the consideration of the radial density gradient inside the cell.
In our model, the heating function $s^{*}$ [erg cm-3 s-1] by the stellar
radiation is calculated as follows:

$s^{*}=\rho^{*}\kappa\frac{L\exp{(-\tau)}}{4\pi
R^{2}}\,\left(\frac{1-\exp{(-\Delta\tau})}{\Delta\tau}\right)$ (3)

where $L$ is the luminosity of the star, $\kappa$ [cm2 g-1] is the absorption
coefficient for stellar radiation, $R$ is the radial distance from the star to
the cell center, $\tau$ is the the total optical depth along the ray up to the
point of entry into the cell, $\Delta\tau=\kappa\rho^{*}\Delta l$ is the
optical thickness of the cell itself along the ray, $\Delta l$ is the length
of the ray segment inside the cell,
$\rho^{*}=\dfrac{1}{4}\left(\rho_{L}+2\rho_{i}+\rho_{r}\right)$ is the average
density along the ray. For the deriving of the Equation (3) from the formal
solution of the radiative transfer equation we assumed that the density along
the ray inside the cell changes linearly from $\rho_{L}$ to $\rho_{i}$ and
from $\rho_{i}$ to $\rho_{R}$, see diagram in Fig. 2. The values $\rho_{L}$
and $\rho_{R}$, in turn, are found using linear density interpolation between
the center of the current cell and the centers of adjacent left and right
cells:

$\displaystyle\rho_{L}=\rho_{i}+\frac{x_{i}-x_{L}}{x_{i}-x_{i-1}}\left(\rho_{i-1}-\rho_{i}\right)$
(4)
$\displaystyle\rho_{R}=\rho_{i}+\frac{x_{R}-x_{i}}{x_{i+1}-x_{i}}\left(\rho_{i+1}-\rho_{i}\right).$
(5)

Thus, when calculating the heating function, we use the density averaged over
neighboring radial cells rather than the central value (which is adopted in
the procedure for calculating the vertical disk structure and assumed to be
constant inside the cell). In the absence of such (or more precise) averaging,
the cell with larger $R$ does not affect cells with smaller $R$, and as a
consequence, there will be no mechanism for the emergence of thermal wave
traveling from outside to inside. With accounting for the density
interpolation, the potential mechanism for the appearance of irradiation
instability is the following. Let us consider two adjacent columns, inner and
outer. Let the outer column heats up, so the density $\rho_{i+1}$ in its upper
layers increases, because the characteristic scale of the disk height
increases. Increased density in the outer cells leads to an increase in the
average density $\rho^{*}$ in adjacent inner cells, which leads to an increase
of $s^{*}$. Further development of the process depends on the characteristic
thermal time scale and geometric effects of the radiation interception by the
inner column.

Heating by the interstellar UV radiation is calculated as:

$S_{\rm bg}=\kappa_{\rm uv}W\,\sigma
T^{4}\exp{(-\tau)}\left(\frac{1-\exp{(-\Delta\tau})}{\Delta\tau}\right),$ (6)

where $\kappa_{\rm uv}$ is the absorption coefficient for the interstellar
radiation, $W=10^{-14}$, $T_{\rm bg}=10^{4}$ K are the dilution and
temperature of the interstellar radiation, $\tau$ is the optical depth from
the surface disk to the current volume element along the vertical direction,
$\Delta\tau$ is the optical thickness of the current element, and $\sigma$ is
the Stefan-Boltzmann constant. Heating by the interstellar radiation is
included in the model, since it can be comparable to the heating from the star
in the outer parts of the disk at the moments of eclipses of the star by the
emerging inhomogeneities of disk surface.

The solution of equations (1)–(2) is found using a fully implicit scheme,
which is similar to the numerical scheme for solving the quasilinear equation
of thermal conductivity with variable coefficients, described in the book
Kalitkin (1978). With linearizing equations and using Newton’s method,
iterations can diverge at a sufficiently large time step. This situation is
rare but still occurs. When divergence of the iterative Newton process
appears, we split the time step into substeps. The method allows us to
calculate thermal evolution in all disk locations, including optically thick
regions in which the characteristic times of heating and cooling processes are
comparable with dynamic times.

The thermal structure calculation in this model is closely related to the
restoration of the vertical disk structure under the assumption of the local
hydrostatic equilibrium, which is found from the following equation:

$\frac{k_{B}}{\mu_{\rm g}m_{\rm H}}\dfrac{d(\rho
T)}{\rho\,dz}=-\dfrac{GM_{\ast}}{r^{3}}z,$ (7)

where $k_{B}$ is the Boltzmann constant, $\mu_{\rm g}=2.3$ is the mean
molecular weight, $m_{\rm H}$ is the atomic mass unit, $G$ is the
gravitational constant, and $M_{\ast}$ is the mass of the star. Note that when
restoring the vertical structure, we do not take into account the self-gravity
of the disk, which is justified for the disk parameters used in this work. The
calculation of the vertical structure makes it possible to obtain complete
information about the distribution of the density and temperature in the disk.
For solving the equation of hydrostatic equilibrium an implicit method is also
used.

The important condition for the effectiveness of these methods is the optimal
choice of a spatial grid in the $z$-direction. The spatial grid should resolve
all a priori unknown features of the solution (density and temperature
gradients) taking into account significant limitation on the number of cells
and large variation of gas density (up to 10 orders of magnitude). We have
developed an algorithm for the construction and adaptive modification of such
a grid, based on approximate fast solution of the equation of hydrostatic
equilibrium. This method of restoring the disk vertical structure with the
calculation of radiation transfer has been extensively tested and compared
with other methods. In the stationary mode, the temperature distribution is in
good consistence with the results of disk structure modeling obtained by other
authors. In the non-stationary regime, the characteristic times of arrival to
thermal equilibrium correspond to analytical estimates. More details with
description of this method can be found in the article by Vorobyov &
Pavlyuchenkov (2017).

The model assumes that the only source of opacity is dust, and the
temperatures of gas and dust are equal. Dust to gas mass ratio is assumed to
be constant throughout the disk and equal to 0.01. A feature of the thermal
model is the use of Planck and Rosseland temperature-dependent opacities.
These coefficients are taken from Pavlyuchenkov et al. (2020), where they are
described in detail.

Figure 3: The structure of the protoplanetary disk with $\Sigma_{0}=10^{2}$ g
cm-2 at 10 yr after the start of evolution. Shown are the distribution of the
logarithm of the gas number density (top left), the logarithm of the
temperature (bottom left), the logarithm of the UV heating function (top
right), and the logarithm of the IR energy density.

The main input parameters of the model are the mass and luminosity of the
star, which we assume to be solar, as well as the distribution of the surface
density, which is chosen by us in the form:

$\Sigma(R)=\Sigma_{0}\left(1-e^{-\left(\frac{R}{R_{0}}\right)^{p}}\right)\left(\frac{R}{R_{au}}\right)^{q},$
(8)

where $\Sigma_{0}$ is the surface density near the inner boundary $R_{au}$=1
au, $q=-1$ is the slope of the density distribution, $R_{0}$=3 au, $p=8$ is
the density distribution smoothing parameters near the inner edge of the disk.
The inner and outer boundaries of the disk are equal to 1 and 100 au,
respectively. Note that in the absence of smoothing the surface density
distribution near the inner disk boundary, the inner cells intercept most of
the stellar radiation and strongly influence the structure and evolution of
the disk, making it difficult to analyze the surface instability itself.
Initial disk state is calculated under the assumption of a constant angle of
entry of radiation into the disk within the thermal model Vorobyov &
Pavlyuchenkov (2017). The time step is chosen constant and equal to $0.1$ yr,
which is less than the characteristic thermal time scales at the adopted model
parameters. The calculation is carried out on a grid of 200 radial $\times$
120 vertical cells.

The difference between the presented numerical model and analytical model used
in Wu & Lithwick (2021) is the treatment of the detailed two-dimensional
structure of the disk. In our case, the heating by the stellar UV radiation is
calculated using direct integration of radiation transfer equation, while the
intristic diffusive thermal radiation is simulated over whole vertical extent.

##  3 Results

Modeling the disk evolution within the framework of the described 1+1D
approach shows that perturbations spontaneously arise in the disk, propagating
towards the star. The characteristics of these perturbations strongly depend,
in particular, on the assumed surface density of the disk. Consider simulation
results for a disk with $\Sigma_{0}=10^{2}$ g cm-2 (corresponding to the disk
mass $7\times 10^{-3}$ M⊙). Fig. 3 shows the disk structure (distributions of
the gas number density, temperature, UV heating and energy density of IR
radiation) at the moment 10 yr after the start disk evolution. In the vicinity
of 35 au one can see a bend in the surface geometry, behind which the
characteristic scale height of the disk increases. This perturbation
propagates inward. Behind the bend, the stellar radiation heating function
increases in the disk atmosphere, which is associated with a steeper angle of
stellar radiation entry into the disk. The maximum of the heating function in
the bend vicinity is associated with the temperature maximum. The temperature
is also increased in the disk midplane at some distance behind the bend, which
is due to finite time scale for heating of the inner layers by IR radiation of
the surface layers. In the vicinity of the perturbed layer the energy density
of infrared radiation is enhanced, which also indicates stronger heating of
this layer.

Figure 4:  Left panel: evolution of midplane temperature $T_{\rm m}$ for the
disk model with $\Sigma_{0}=10^{2}$ g cm-2. The distance to the star is shown
on the horizontal axis, time is along the vertical axis. The black line shows
the dependence of the characteristic thermal time scale $t_{\rm therm}$ on the
distance for a time moment of 30 yr. Right panel: evolution of $T_{\rm m}$ for
a model without interpolation of density in the radial direction in the
calculation of the heating rate by the stellar radiation.

The space-time diagram showing the evolution of the midplane temperature is
shown in the left panel of Fig. 4. Inward moving waves stand out well on it,
they correspond to maxima moving from right to left with increasing time.
Waves extend from the outer edge of the disk to the region of $\approx 3$ au,
which is transparent to the UV radiation of the star (due to its low density
because of the smoothing of the $\Sigma(R)$ distribution). This diagram also
includes the distribution of the characteristic thermal time calculated as:

$t_{\rm therm}=\frac{3}{8}\frac{c_{p}\,\Sigma\,\tau_{\rm IR}}{\sigma T_{\rm
m}^{3}},$ (9)

where $\tau_{\rm IR}=\kappa_{P}\Sigma$ is the optical depth in vertical
direction for intrinsic thermal radiation, $T_{\rm m}$ is the midplane
temperature, $c_{p}$ is the specific heat, and $\sigma$ is the
Stefan–Boltzmann constant. Equation (9) is obtained under assumption that the
disk is optically thick to its own thermal radiation, and serves as an
estimate for the characteristic heating (or cooling) time scale of the
midplane layers to the temperature of $T_{\rm m}$. Periodicity of waves of
$\approx 3$ yr is close to the maximum characteristic thermal time scale
$\approx 7$ yr, i.e. the perturbed regions of the disk do warm up to the
midplane.

The right panel of Fig. 4 shows the evolution of the midplane temperature for
a disk model with the same parameters, but without interpolation of the
density over the cell in radial direction during the calculation of the
heating rate by the stellar radiation, i.e. where $\rho^{*}=\rho_{i}$ is
assumed, see Equation (3). In this case, several bumps appear in the disk,
intercepting the radiation of the star and shading areas behind them. The slow
movement of these humps towards the star at the initial stages of the disk
evolution is associated with the establishing the self-consistent solution
over the thermal time scale. Regular surface waves do not arise in such a
model. This simulation shows the importance of using high approximation
schemes when integrating the radiative transfer equation in this problem.

Figure 5:  Disk model for $\Sigma_{0}=10^{3}$ g cm-2 at 10 yr after the start
of evolution.

Figure 6: Evolution of temperature distributions for the disk model with
$\Sigma_{0}=10^{3}$ g cm-2. Left top panel: midplane temperature for 20 yr.
Left bottom panel: disk surface temperature. Top right panel: temperature at
half surface density from the upper boundary of the disk to the equator. Right
bottom panel: long-term evolution of midplane temperature over 1000 yr. The
black curve shows the dependence of the characteristic thermal time $t_{\rm
therm}$ on distance.

Fig. 5 shows the distributions of physical quantities for the model with
enhanced surface density, $\Sigma_{0}=10^{3}$ g cm-2, at the moment of 10 yr
from the beginning of evolution. The structure of this disk is inhomogeneous.
Two perturbations can be distinguished in the energy density distribution of
IR radiation: in the vicinity of 10 and 40 au. The morphology of these
disturbances generally repeats the pattern described for the previous model.
However, unlike the previous model, the heated area in the vicinity of
disturbances does not reach the midplane. This can be clearly seen from the
top left panel of Fig. 6, which shows evolution of the midplane temperature
distribution in the first 20 yr. The waves do not appear in this diagram, with
the exception of weak oscillations in vicinity of 3 au. The inward propagating
waves, however, are clearly visible on surface temperature distributions
(lower left panel of Fig. 6) and temperatures at half surface density up to
equator (upper right panel of Fig. 6). Over the considered time interval, the
disturbances are generated in the vicinity of 20 au and spread inward in
within 3 yr. As they approach the inner boundary of the disk, the propagation
velocity of perturbations decreases. The period of passage of these waves is
much shorter than the thermal time scale $t_{\rm therm}$ calculated for the
entire thickness of the disk. This is evident from the bottom right panel of
Fig. 6, where the distribution of thermal time scale for the entire thickness
of the disk is plotted. The time $t_{\rm therm}$ is about hundreds of years
for the region between 3 and 20 au. On the same panel we show the evolution of
midplane temperature of the disk during 1000 yr, i.e. over time comparable
with the characteristic thermal time. In this diagram, the waves inside 20 au
are also not identified but quasi-periodic perturbations propagating inward
from the outer edge of the disk are visible with a occurrence period
$\approx$100 yr. Obviously, the waves in the outer region of the disk are
identified on midplane temperature distribution due to the fact that the
characteristic thermal times in this region are significantly shorter than in
more internal parts, i.e. disk being perturbed on the surface in these regions
has time to warm up to the equator.

##  4 Discussion

The main purpose of this work is to test the possibility of spontaneous
excitation of surface thermal waves in protoplanetary disks. To achieve this
goal, we updated the previously developed 1+1D disk model with a more accurate
(two-dimensional) calculation of the disk heating by the stellar radiation. In
the modified model, heating is calculated by integrating the radiative
transfer equation along the entire disk taking into account the radial density
gradient inside the cell, for which the heating function is calculated. This
allows us to account for geometric effects that are of key importance for this
task. The simulation results showed that the instability does indeed arise,
leading to the emergence of inward moving perturbations. Thus, we confirm the
previously obtained by other authors conclusions on possible spontaneous
excitation of the surface thermal waves in protoplanetary disks.

In our disk model, we focuse on the occurrence of thermal surface waves, as
far as feasible “in a pure form” in connection with the fact that there are
still questions of their formation. Therefore, many processes essential for
disk evolution are not taken into account in this model. In particular, we do
not take into account internal viscous heating, which can play an important
role in the evolution of the disk, providing an irregular mode of accretion
(we studied this effect earlier in Pavlyuchenkov et al. (2020); Maksimova,
Pavlyuchenkov & Tutukov (2020). Neglecting the viscous heating is equivalent
to assuming low accretion rate through the disk.

Surely, at temperatures of the order of thousands of degrees, the assumption
of our model that the opacity is due only to dust is not fulfilled. At such
temperatures dust evaporation becomes important, and gas begins to dominanate
in opacity. At high temperatures other processes become significant, such as
the dissociation of hydrogen. In this case, one can expect the appearance of
various instabilities in the disk and the complex dynamics associated with
them. This is a separate issue requiring detailed study. We note that with our
model parameters such high temperatures are obtained only in the most inner
($<3$ au) regions of the disk and do not affect more distant regions, where
the thermal waves originate and begin to propagate.

In our model, we also do not take into account dust sedimentation, migration,
growth and destruction, which play a critical role in protoplanetary disks.
Taking this processes into account can significantly modify or even completely
suppress the instability of the disk associated with the effects of self-
shadowing. We also note that the mere presence of dust leads to its own
instabilities, such as streaming instability (Youdin & Goodman, 2005).
Streaming instability now becomes very popular for explaining turbulence and
formation of planetary embryos. The study of these processes is a separate
direction for which complex dynamical models are developed. In this work, we
do not consider all these processes, postponing the study of their interaction
with the irradiation instability for the future.

A significant difference between our results and the conclusions of the
analytical model by Wu & Lithwick (2021) are characteristic timescales for
propagation of perturbations. According to the analytical model presented in
Wu & Lithwick (2021), characteristic propagation time of perturbations
corresponds to the thermal time scale for the entire vertical column of the
disk. This result is natural, since in frame of that analytical model, the
height of the disk is determined exclusively by the midplane temperature. In
our model, the propagation times for optically thick disks turn out to be
significantly smaller than the thermal time scale $t_{\rm term}$. This is
explained by the fact that traveling perturbations do not affect the entire
thickness of the disk, i.e. wave excitation mechanism can work in the sub-
surface layers.

The obtained results indicate the need to study the irradiation instability
using more consistent models. Indeed, the 1+1D approach we used is based on
several key approximations that can significantly distort the real picture.
Such approximations are: 1) the hydrostatic equilibrium in the vertical
direction; 2) the absence of the thermal radiation diffusion in the radial
direction; 3) the lack of dynamical interaction between disk regions in the
radial direction. The influence of these effects on the excitation of surface
thermal waves should be explored within the framework of a two-dimensional or
three-dimensional hydrodynamic models with a detailed calculation of the
radiative transfer. Particular attention in such model should be paid to the
treatment of the heating function by the stellar radiation.

## ACKNOWLEDGMENTS

The authors thank the reviewer for valuable comments and constructive
suggestions for improving this article.

## Funding

The reported study was funded by RFBR according to the research project
20-32-90103. VVA was supported by the Foundation for the Advancement of
Theoretical Physics and Mathematics ”BASIS” (20-1-2-20-1).

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# Commuting probability for approximate subgroups of a finite group

Eloisa Detomi  Dipartimento di Matematica “Tullio Levi-Civita”, Università di
Padova, Via Trieste 63, 35121 Padova, Italy
<EMAIL_ADDRESS>,  Marta Morigi  Dipartimento di Matematica, Università
di Bologna
Piazza di Porta San Donato 5
40126 Bologna
Italy<EMAIL_ADDRESS>and  Pavel Shumyatsky  Department of Mathematics,
University of Brasilia
Brasilia-DF
70910-900 Brazil<EMAIL_ADDRESS>
###### Abstract.

For subsets $X,Y$ of a finite group $G$, we write $\mathrm{Pr}(X,Y)$ for the
probability that two random elements $x\in X$ and $y\in Y$ commute. This paper
addresses the relation between the structure of an approximate subgroup
$A\subseteq G$ and the probabilities $\mathrm{Pr}(A,G)$ and
$\mathrm{Pr}(A,A)$. The following results are obtained.

Theorem 1.1: Let $A$ be a $K$-approximate subgroup of a finite group $G$, and
let $\mathrm{Pr}(A,G)\geq\epsilon>0$. There are two $(\epsilon,K)$-bounded
positive numbers $\gamma$ and $K_{0}$ such that $G$ contains a normal subgroup
$T$ and a $K_{0}$-approximate subgroup $B$ such that $|A\cap
B|\geq\gamma\;{\mathrm{max}}\\{|A|,|B|\\}$ while the index $[G:T]$ and the
order of the commutator subgroup $[T,\langle B\rangle]$ are
$(\epsilon,K)$-bounded.

Theorem 1.2: Let $A$ be a $K$-approximate subgroup of a finite group $G$, and
let $\mathrm{Pr}(A,A)\geq\epsilon>0$. There are two $(\epsilon,K)$-bounded
positive numbers $\gamma$ and $s$, and a subgroup $C\leq G$ such that $|C\cap
A^{2}|>\gamma|A|$ and $|C^{\prime}|\leq s$. In particular, $A$ is contained in
the union of at most $\gamma^{-1}K^{2}$ left cosets of the subgroup $C$.

It is also shown that the above results admit approximate converses.

###### Key words and phrases:

Commuting probability, approximate subgroups, conjugacy classes, centralizers

###### 2020 Mathematics Subject Classification:

20E45; 20P05; 20N99

##  1\. Introduction

If $G$ is a finite group and $X,Y$ are subsets of $G$, we write
$\mathrm{Pr}(X,Y)$ for the probability that two random elements $x\in X$ and
$y\in Y$ commute. Thus,

$\mathrm{Pr}(X,Y)=\frac{|\\{(x,y)\in X\times Y\mid xy=yx\\}|}{|X|\,|Y|}.$

The number $\mathrm{Pr}(G,G)$ is called the commuting probability of $G$. It
is well-known that $\mathrm{Pr}(G,G)\leq 5/8$ for any nonabelian group $G$.
Another important result is the theorem of P. M. Neumann [7] which states that
if $G$ is a finite group and $\epsilon$ is a positive number such that
$\mathrm{Pr}(G,G)\geq\epsilon$, then $G$ has a nilpotent normal subgroup $R$
of nilpotency class at most $2$ such that both the index $[G:R]$ and the order
of the commutator subgroup $[R,R]$ are $\epsilon$-bounded (see also [4]).

Throughout the article we use the expression “$(a,b,\dots)$-bounded” to mean
that a quantity is bounded from above by a number depending only on the
parameters $a,b,\dots$.

There are several recent papers studying $\mathrm{Pr}(H,G)$, where $H$ is a
subgroup of $G$ (see for example [3, 5, 6]). In particular, it was proved in
[3, Proposition 1.2] that if $H$ is a subgroup of a finite group $G$ and
$\mathrm{Pr}(H,G)\geq\epsilon>0$, then there is a normal subgroup $T\leq G$
and a subgroup $B\leq H$ such that the indices $[G:T]$ and $[H:B]$, and the
order of the commutator subgroup $[T,B]$ are $\epsilon$-bounded.

Lately there also has been a considerable interest in studying approximate
subgroups of finite groups.

Let $K$ be a positive real number. A subset $A$ of a finite group $G$ is said
to be a $K$-approximate subgroup of $G$, or simply a $K$-approximate group, if
$A$ contains $1$ and the inverse of each of its elements, and if there exists
$E\subseteq G$ with $|E|\leq K$ such that $A^{2}\subseteq EA$.

Here and throughout, given a positive integer $j$ and a subset $X$ of a group
$G$, we write $X^{j}$ for the set of all products $x_{1}\dots x_{j}$, where
$x_{i}\in X$. The definition of approximate subgroups was introduced by Tao in
[9]. Since then many important results on the subject have been established.
In particular, Breuillard, Green and Tao essentially described the structure
of finite approximate subgroups [2]. The reader is referred to the book [10]
for detailed information on these developments.

In the present paper we examine the relation between the structure of an
approximate subgroup and the commuting probability. Using the well known
analogy between approximate groups and groups, we aim at extending the above
mentioned group-theoretical results to approximate subgroups. On the one side,
we employ the group-theoretical machinery already available, and, on the other
side, ad hoc techniques for approximate subgroups.

Proposition 1.2 in [3] says, roughly speaking, that a subgroup $H$ of a finite
group $G$ has many commuting elements with $G$ if and only if $H$ has a large
subgroup which almost commutes with a large normal subgroup of $G$. Replacing
$H$ with an approximate subgroup $A$ we obtain that $A$ has many commuting
elements with $G$ if and only if it is commensurate with another approximate
subgroup $B$ which generates a subgroup $H$ of $G$ with the same properties as
in the aforementioned proposition. More formally, our result is as follows.

######  Theorem 1.1.

Let $A$ be a $K$-approximate subgroup of a finite group $G$, and let
$\mathrm{Pr}(A,G)\geq\epsilon>0$. There are two $(\epsilon,K)$-bounded
positive numbers $\gamma$ and $K_{0}$ such that $G$ contains a normal subgroup
$T$ and a $K_{0}$-approximate subgroup $B$ such that

  1. (i)

$|A\cap B|\geq\gamma\;{\mathrm{max}}\\{|A|,|B|\\}$, and

  2. (ii)

the index $[G:T]$ and the order of the subgroup $[T,\langle B\rangle]$ are
both $(\epsilon,K)$-bounded.

Throughout, $\langle X\rangle$ denotes the subgroup generated by a subset $X$
of a group.

Next, we study approximate subgroups $A$ such that
$\mathrm{Pr}(A,A)\geq\epsilon>0$. Roughly speaking, Neumann’s theorem says
that if a finite group has many commuting elements then it has a large
subgroup which is almost abelian, in the sense that it has a small commutator
subgroup. It turns out that if $A$ is an approximate subgroup with many
commuting elements then a big part of $A^{2}$ is contained in a subgroup $C$
with small commutator subgroup, and $A$ itself is contained in boundedly many
cosets of $C$.

######  Theorem 1.2.

Let $A$ be a $K$-approximate subgroup of a finite group $G$ and assume that
$\mathrm{Pr}(A,A)\geq\epsilon>0$. There are two $(\epsilon,K)$-bounded
positive numbers $\gamma$ and $s$, and a subgroup $C\leq G$ such that

  1. (i)

$|C\cap A^{2}|>\gamma|A|$, and

  2. (ii)

the commutator subgroup of $C$ is of order at most $s$.

Moreover $A$ is contained in the union of at most $\gamma^{-1}K^{2}$ left
cosets of the group $C$.

Furthermore, we show that each of the above theorems admits an “approximate”
converse.

######  Proposition 1.3.

Let $A,B$ be subsets and $T$ a subgroup of a finite group $G$. Set
$\gamma=|A\cap B|/|A|$, $n=[G:T]$ and $m=|[T,\langle B\rangle]|$. Then
$\mathrm{Pr}(A,G)\geq\frac{\gamma}{nm}$.

######  Proposition 1.4.

Let $A$ be a $K$-approximate subgroup of a finite group $G$, and let $C\leq G$
be a subgroup. Set $\gamma=|C\cap A^{2}|/|A|$ and $s=|C^{\prime}|$. Then

$\mathrm{Pr}(A^{2},A^{2})\geq\frac{\gamma^{2}}{K^{4}s}.$

Note that some properties of the commuting probability of subgroups cannot be
extended verbatim to approximate subgroups but they do hold when replacing an
approximate subgroup $A$ with $A^{2}$ (compare Proposition 2.6 and Example
2.8). Clearly, $A^{2}$ is a $K^{2}$-approximate subgroup, whenever $A$ is a
$K$-approximate subgroup (if $A^{2}\subseteq EA$, then $A^{4}\subseteq
EA^{3}\subseteq E^{2}A^{2}$).

The next section contains some general comments on the commuting
probabilities. In Section 3 we prove Theorems 1.1 and 1.2. The last section is
devoted to Propositions 1.3 and 1.4.

##  2\. General comments on commuting probabilities

Note that if $X,Y$ are subsets of a finite group $G$, we have

$\mathrm{Pr}(X,Y)=\frac{1}{|Y|}\sum_{y\in
Y}\frac{|C_{X}(y)|}{|X|}=\frac{1}{|X|}\sum_{x\in X}\frac{|C_{Y}(x)|}{|Y|}.$

It was proved in [3, Lemma 2.3] that if $K$ and $N$ are subgroups of a finite
group $G$, with $N$ normal in $G$, then

$\mathrm{Pr}(K,G)\leq\mathrm{Pr}\left(KN/N,G/N\right)\,\,\mathrm{Pr}(N\cap
K,N).$

We are interested in finding an “approximate” variant of this result. We start
by looking at symmetric subsets.

######  Proposition 2.1.

Let $G$ be a finite group, $N$ a normal subgroup of $G$, and assume that $A$
is a symmetric subset of $G$. Then

$\mathrm{Pr}(A,G)\leq\frac{|A^{5}|}{|A|}\,\mathrm{Pr}\left(\frac{AN}{N},\frac{G}{N}\right)\,\mathrm{Pr}(A^{4}\cap
N,N).$

###### Proof.

Let $\bar{G}=G/N$ and $\bar{A}=\\{hN\mid h\in A\\}=\\{h_{1}N,\dots,h_{r}N\\}$.
Note that $A$ is contained in the union of the subsets $h_{i}(N\cap A^{2})$,
for $i=1,\dots,r$. Indeed, $A\subseteq\bigcup_{i}h_{i}N$ and if $a\in h_{i}N$,
then $h_{i}^{-1}a=n$ for some $n\in N\cap A^{2}$ and so $a=h_{i}n\in
h_{i}(N\cap A^{2})$. Therefore

$\displaystyle|A|\,|G|\,\mathrm{Pr}(A,G)$ $\displaystyle=$
$\displaystyle\sum_{x\in A}|C_{G}(x)|\leq\sum_{hN\in\bar{A}}\left(\sum_{x\in
h(N\cap A^{2})}\frac{|NC_{G}(x)|}{|N|}|C_{N}(x)|\right)$ $\displaystyle\leq$
$\displaystyle\sum_{hN\in\bar{A}}\left(\sum_{x\in h(N\cap
A^{2})}|C_{\bar{G}}(hN)|\,|C_{N}(x)|\right)$ $\displaystyle=$
$\displaystyle\sum_{hN\in\bar{A}}|C_{\bar{G}}(hN)|\left(\sum_{x\in h(N\cap
A^{2})}|C_{N}(x)|\right)$ $\displaystyle=$
$\displaystyle\sum_{hN\in\bar{A}}|C_{\bar{G}}(hN)|\left(\sum_{y\in
N}|C_{h(N\cap A^{2})}(y)|\right).$

If $C_{h(N\cap A^{2})}(y)\neq\emptyset$ take $y_{0}\in h(N\cap A^{2})\cap
C_{G}(y)$ and observe that $h(N\cap A^{2})\subseteq y_{0}(N\cap A^{4})$.
Indeed, $y_{0}=hu$ with $u\in N\cap A^{2}$, whence $h=y_{0}u^{-1}$ and
$h(N\cap A^{2})\subseteq y_{0}u^{-1}(N\cap A^{2})\subseteq y_{0}(N\cap
A^{4})$. Therefore

$\displaystyle C_{h(N\cap A^{2})}(y)$ $\displaystyle=$ $\displaystyle h(N\cap
A^{2})\cap C_{G}(y)$ $\displaystyle\subseteq$ $\displaystyle y_{0}C_{N\cap
A^{4}}(y),$

whence $|C_{h(N\cap A^{2})}(y)|\leq|C_{N\cap A^{4}}(y)|$. It follows that

$\displaystyle|A|\,|G|\,\mathrm{Pr}(A,G)$ $\displaystyle\leq$
$\displaystyle\left(\sum_{hN\in\bar{A}}|C_{\bar{G}}(hN)|\right)\left(\sum_{y\in
N}|C_{N\cap A^{4}}(y)|\right)$ $\displaystyle=$
$\displaystyle\Big{(}|\bar{A}|\,|\bar{G}|\,\mathrm{Pr}(\bar{A},\bar{G})\Big{)}\Big{(}|N|\,|N\cap
A^{4}|\,\mathrm{Pr}(N\cap A^{4},N)\Big{)}.$

Thus

$\mathrm{Pr}(A,G)\leq\frac{|\bar{A}|\,|N\cap
A^{4}|}{|A|}\,\mathrm{Pr}(\bar{A},\bar{G})\,\mathrm{Pr}(N\cap A^{4},N).$

As $|\bar{A}|\,|N\cap A^{4}|\leq|A|^{5}$ by [10, Lemma 2.6.3], the result
follows. ∎

The previous proposition says, in particular, that in a homomorphic image
$\bar{G}$ of $G$ the commuting probability of $\bar{A}$ in $\bar{G}$ is
controlled in terms of $\mathrm{Pr}(A,G)$. The next lemma deals with the
commuting probability $\mathrm{Pr}(\bar{A},\bar{A})$.

######  Proposition 2.2.

Let $G$ be a finite group, $N$ a normal subgroup of $G$, and assume that $A$
is a symmetric subset of $G$. Then

$\mathrm{Pr}(A,A)\leq\frac{|A^{3}|\,|A^{5}|}{|A|^{2}}\,\mathrm{Pr}\left(\frac{AN}{N},\frac{AN}{N}\right)\,\mathrm{Pr}(A^{4}\cap
N,A^{2}\cap N).$

###### Proof.

The proof is very similar to the proof of Proposition 2.1. Thus we will sketch
it, avoiding repetitions. Let $\bar{G}=G/N$ and $\bar{A}=\\{hN\mid h\in
A\\}=\\{h_{1}N,\dots,h_{r}N\\}$. As shown in Proposition 2.1, $A$ is contained
in the union of the subsets $h_{i}(N\cap A^{2})$, for $i=1,\dots,r$. Moreover,
for $x\in A$,

$C_{A}(x)\subseteq\bigcup_{hN\in C_{A}(x)N/N}hC_{N\cap A^{2}}(x),$

since, for $a,h\in C_{A}(x)$, the equality $aN=hN$ implies $h^{-1}a\in N\cap
A^{2}$. Therefore, as $C_{A}(x)N/N\subseteq C_{\bar{A}}(xN)$, we have

(1) $|C_{A}(x)|\leq|C_{\bar{A}}(xN)|C_{N\cap A^{2}}(x)|.$

It follows that

$\displaystyle|A|\,|A|\,\mathrm{Pr}(A,A)$ $\displaystyle=$
$\displaystyle\sum_{x\in A}|C_{A}(x)|\leq\sum_{hN\in\bar{A}}\left(\sum_{x\in
h(N\cap A^{2})}|C_{A}(x)|\right)$ $\displaystyle\leq$
$\displaystyle\sum_{hN\in\bar{A}}\left(\sum_{x\in h(N\cap
A^{2})}|C_{\bar{A}}(hN)|\,|C_{N\cap A^{2}}(x)|\right)$ $\displaystyle=$
$\displaystyle\sum_{hN\in\bar{A}}|C_{\bar{A}}(hN)|\left(\sum_{x\in h(N\cap
A^{2})}|C_{N\cap A^{2}}(x)|\right)$ $\displaystyle=$
$\displaystyle\sum_{hN\in\bar{A}}|C_{\bar{A}}(hN)|\left(\sum_{y\in{N\cap
A^{2}}}|C_{h(N\cap A^{2})}(y)|\right).$

As in the proof of Proposition 2.1, the inequality $|C_{h(N\cap
A^{2})}(y)|\leq|C_{N\cap A^{4}}(y)|$ holds for every $h\in A$. Therefore

$|A|\,|A|\,\mathrm{Pr}(A,A)\leq\left(\sum_{hN\in\bar{A}}|C_{\bar{A}}(hN)|\right)\left(\sum_{y\in{N\cap
A^{2}}}|C_{N\cap A^{4}}(y)|\right)$
$=\Big{(}|\bar{A}|\,|\bar{A}|\,\mathrm{Pr}(\bar{A},\bar{A})\Big{)}\Big{(}|{N\cap
A^{2}}|\,|N\cap A^{4}|\,\mathrm{Pr}(N\cap A^{4},{N\cap A^{2}})\Big{)}.$

Thus

$\mathrm{Pr}(A,A)\leq\frac{|\bar{A}|^{2}\,|N\cap A^{4}||{N\cap
A^{2}}|}{|A|^{2}}\,\mathrm{Pr}(\bar{A},\bar{A})\,\mathrm{Pr}(N\cap
A^{4},{N\cap A^{2}}).$

As $|\bar{A}|\,|N\cap A^{4}|\leq|A|^{5}$ and $|\bar{A}|\,|N\cap
A^{2}|\leq|A|^{3}$ by [10, Lemma 2.6.3], the result follows. ∎

As $|A^{n}|/|A|\leq K^{n-1}$ for any $K$-approximate subgroup $A$ and for
every integer $n\geq 2$, the following corollary is a straightforward
consequence of the above propositions.

######  Corollary 2.3.

Let $G$ be a finite group, $N$ a normal subgroup of $G$, and assume that
$A\subseteq G$ is a $K$-approximate subgroup. Then

  * •

$\mathrm{Pr}(A,G)\leq K^{4}\,\mathrm{Pr}(AN/N,G/N)\,\mathrm{Pr}(A^{4}\cap
N,N),$

  * •

$\mathrm{Pr}(A,A)\leq K^{6}\,\mathrm{Pr}(AN/N,AN/N)\,\mathrm{Pr}(A^{4}\cap
N,A^{2}\cap N).$

In particular,

  * •

$\mathrm{Pr}(AN/N,G/N)\geq(1/{K^{4}})\mathrm{Pr}(A,G)$,

  * •

$\mathrm{Pr}(AN/N,AN/N)\geq(1/K^{6})\mathrm{Pr}(A,A).$

More generally, the above corollary holds in the case where $A$ is a symmetric
set containing $1$ which has small tripling, i.e. $|A^{3}|\leq T|A|$. To see
this, observe that $|A^{5}|\leq T^{3}|A|$ by [10, Proposition 2.5.3].

If $H_{1}\leq H_{2}$ are subgroups of a finite group $G$, then

(2) $\mathrm{Pr}(H_{1},G)\geq\mathrm{Pr}(H_{2},G)$

(see [5, Theorem 3.7]).

As we will show in Example 2.8, the above inequality does not hold if $H_{1}$,
$H_{2}$ are $K$-approximate subgroups. Indeed, if $H_{1}$ is a $K$-approximate
subgroup contained in $H_{2}$, then $\mathrm{Pr}(H_{1},G)$ might be
arbitrarily small compared to $\mathrm{Pr}(H_{2},G)$ (even if $H_{2}$ is a
subgroup). We will show however that $\mathrm{Pr}(H_{1}^{2},G)$ is bounded
away from zero (see Proposition 2.6). In the particular case where $H_{1}$ is
a subgroup of $G$ and $H_{2}$ is a $K$-approximate subgroup containing
$H_{1}$, we get

$\mathrm{Pr}(H_{1},G)\geq\frac{1}{K}\mathrm{Pr}(H_{2},G)$

(see Corollary 2.7).

To prove these results, we need a preliminary elementary lemma.

######  Lemma 2.4.

Assume that $G$ is a finite group, $g\in G$ and $A$ is a symmetric subset of
$G.$ Then

  * (a)

$|C_{A}(g)|\,|g^{A}|\leq|A^{2}|$.

  * (b)

$|A|\leq|C_{A^{2}}(g)|{|g^{A}|}$.

###### Proof.

Note that if $g^{A}=\\{g^{a_{1}},\dots,g^{a_{r}}\\}$, then the subsets
$C_{A}(g)a_{i}$, for $i=1,\dots,r$, are pairwise disjoint subsets of $A^{2}$,
so

$\left|C_{A}(g)\right|\left|g^{A}\right|=\left|\bigcup_{1\leq i\leq
r}C_{A}(g)a_{i}\right|\leq\left|A^{2}\right|,$

and (a) holds.

To prove (b), observe that whenever $a,b\in A$ are such that $g^{a}=g^{b}$, we
have $ab^{-1}\in C_{G}(g)\cap A^{2}=C_{A^{2}}(g)$. ∎

The following result holds for approximate subgroups.

######  Lemma 2.5.

Assume that $G$ is a finite group, $g\in G$ and $A\subseteq G$ is a
$K$-approximate subgroup. Then $|g^{A^{n}}|\leq K^{n-1}|g^{A}|$ for every
$n\geq 1$.

###### Proof.

As $A^{2}\subseteq EA$ where $|E|\leq K$, we have $A^{n}\subseteq E^{n-1}A$.
Since $A$ is symmetric, every element $h\in A^{n}$ can be written as the
inverse of an element $ea\in A^{n}\subseteq E^{n-1}A$, with $e\in E^{n-1}$ and
$a\in A$. So

$g^{h}=g^{(ea)^{-1}}=(g^{a^{-1}})^{e^{-1}},$

and since there are at most $K^{n-1}$ elements in $E^{n-1}$, we deduce that
$|g^{A^{n}}|\leq K^{n-1}|g^{A}|.$ ∎

######  Proposition 2.6.

Let $G$ be a finite group, and let $A_{1}\subseteq A_{2}$ be symmetric subsets
of $G$ such that $|A_{1}^{2}|\leq K|A_{1}|$ and $|A_{2}^{2}|\leq
K^{\prime}|A_{2}|$. Then for any subset $B\subseteq G$ we have

$\mathrm{Pr}(A_{1}^{2},B)\geq\frac{1}{KK^{\prime}}\mathrm{Pr}(A_{2},B).$

###### Proof.

Let $g\in G$. As $|A_{1}^{2}|\leq K|A_{1}|$, it follows from Lemma 2.4 (b)
that

$\frac{|C_{A_{1}^{2}}(g)|}{|{A_{1}^{2}}|}\geq\frac{1}{K|g^{A_{1}}|}.$

Clearly, as $A_{1}\subseteq A_{2}$, we have $1/{|g^{A_{1}}|}\geq
1/{|g^{A_{2}}|}$. Moreover, as $|A_{2}^{2}|\leq K^{\prime}|A_{2}|$, we deduce
from Lemma 2.4 (a) that

$\frac{1}{|g^{A_{2}}|}\geq\frac{|C_{A_{2}}(g)|}{K^{\prime}|{A_{2}}|}.$

Therefore

$\frac{|C_{A_{1}^{2}}(g)|}{|{A_{1}^{2}}|}\geq\frac{1}{K|g^{A_{1}}|}\geq\frac{1}{K|g^{A_{2}}|}\geq\frac{|C_{A_{2}}(g)|}{KK^{\prime}|{A_{2}}|}.$

We conclude that

$\mathrm{Pr}(A_{1}^{2},B)=\frac{1}{|B|}\sum_{g\in
B}\frac{|C_{A_{1}^{2}}(g)|}{|{A_{1}^{2}}|}\geq\frac{1}{KK^{\prime}|B|}\sum_{g\in
B}\frac{|C_{A_{2}}(g)|}{|{A_{2}}|}=\frac{1}{KK^{\prime}}\mathrm{Pr}(A_{2},B),$

as claimed. ∎

######  Corollary 2.7.

Let $G$ be a finite group, and let $H$ be a subgroup contained in a
$K$-approximate subgroup $A\subseteq G$. Then for any subset $B\subseteq G$ we
have

$\mathrm{Pr}(H,B)\geq\frac{1}{K}\mathrm{Pr}(A,B).$

###### Proof.

The result a straightforward consequence of the previous lemma, taking into
account that $H$ is a $1$-approximate subgroup and $H^{2}=H$. ∎

######  Example 2.8.

Here we show that for any positive integer $K\geq 2$ and $\epsilon>0$ there is
a finite group $G$ with a subgroup $H$ and a $K$-approximate subgroup
$A\subseteq H$ of size at least $K|H|/2^{K}$ such that
$\mathrm{Pr}(A,G)\leq\epsilon\;\mathrm{Pr}(H,G)$. In particular,
${\mathrm{Pr}(A,G)}/{\mathrm{Pr}(H,G)}$ cannot be bounded away from zero in
terms of $|A|/|H|$ and $K$.

Let $V$ be an elementary abelian $2$-group of rank $n\geq K$, and let
$g_{1},\dots,g_{n}$ be a basis of $V$. Denote by $g$ the automorphism of $V$
which cyclically permutes $g_{1},\dots,g_{n}$. Consider the natural semi-
direct product $N=V\langle g\rangle$ and let $G=N\times U$, where $U$ is a
finite abelian group. Then $Z=Z(G)=\langle g_{1}g_{2}\cdots g_{n}\rangle\times
U$. Set $z=|Z|=2\;|U|$ and note that

$|G|=n2^{n}|U|=n2^{n-1}z.$

Let $k=K-1$ and set

$A=\bigcup_{1\leq i\leq k}g_{i}Z\cup\\{1\\}\quad\text{and }\quad H=\langle
A\rangle=\langle g_{1},\dots,g_{k}\rangle Z.$

Note that $A$ is symmetric because the $g_{i}$ have order $2$, moreover
$A^{2}\subseteq BA$, where $B=\\{1,g_{1},\dots,g_{k}\\}$ has size $k+1=K$, so
$A$ is a $K$-approximate subgroup. Moreover, as $k<n$, we have

$|A|=kz+1,\quad|H|=2^{k}z,\quad\frac{|A|}{|H|}=\frac{kz+1}{2^{k}z}>\frac{k}{2^{k}}.$

Since $|a^{G}|=n$ for every $1\neq a\in A$, we have

$\displaystyle\mathrm{Pr}(A,G)$ $\displaystyle=$
$\displaystyle\frac{1}{|A|}\sum_{a\in
A}\frac{|C_{G}(a)|}{|G|}=\frac{1}{|A|}\left(\sum_{1\neq a\in
A}\frac{|C_{G}(a)|}{|G|}+\frac{|C_{G}(1)|}{|G|}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{|A|}\left(\frac{|A|-1}{n}+1\right)=\frac{1}{kz+1}\left(\frac{kz}{n}+1\right)<\frac{kz+n}{kzn}.$

As $H$ is contained in $VU$, which is an abelian subgroup of index $n$ in $G$,
whenever $x\in H$ we have $[G:C_{G}(x)]\leq n$. So we can estimate
$\mathrm{Pr}(H,G)$ as follows:

$\displaystyle\mathrm{Pr}(H,G)$ $\displaystyle=$
$\displaystyle\frac{1}{|H|}\sum_{x\in
H}\frac{|C_{G}(x)|}{|G|}=\frac{1}{|H|}\left(\sum_{x\in H\setminus
Z}\frac{|C_{G}(x)|}{|G|}+\sum_{x\in Z}\frac{|C_{G}(x)|}{|G|}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{|H|}\left(\frac{|H|-z}{n}+z\right)=\frac{1}{2^{k}z}\left(\frac{2^{k}z-z+nz}{n}\right)$
$\displaystyle>$
$\displaystyle\frac{1}{2^{k}z}\left(\frac{nz}{n}\right)=\frac{1}{2^{k}}.$

Therefore

$\frac{\mathrm{Pr}(A,G)}{\mathrm{Pr}(H,G)}<\frac{kz+n}{kzn}\,2^{k}=\left({\frac{1}{n}+\frac{1}{kz}}\right)\,{2^{k}}.$

This can be arbitrarily small if $n$ and $|U|$ are chosen large enough.

######  Example 2.9.

In the previous example $A$ was quite small compared to the subgroup
$H=\langle A\rangle$, as ${|A|}/{|H|}=({kz+1})/({2^{k}z})<({k+1})/{2^{k}}$.
Now we show that, in the notation of the previous example, we can choose $G$,
$A$, and a $K$-approximate subgroup $A_{0}\subseteq G$ containing $A$, such
that $|A|/|A_{0}|>k/(k+1)$ and $\mathrm{Pr}(A,G)\leq\mathrm{Pr}(A_{0},G)$.
Moreover the value of

$\mathrm{Pr}(A,G)/\mathrm{Pr}(A_{0},G)$

cannot be bounded away from zero in terms of $|A|/|A_{0}|$ and $K$.

Let $G,A,Z$ be as in Example 2.8 and let $A_{0}=Z\cup A$. Note that
$A^{2}\subseteq BA$ and $A_{0}^{2}\subseteq BA_{0}$, where
$B=\\{1,g_{1},\dots,g_{k}\\}$, so both $A$ and $A_{0}$ are $K$-approximate
subgroups.

Note that $|A|=kz+1$ and $|A_{0}|=(k+1)z$, where $z=|Z|$, and so
$|A|/|A_{0}|=(kz+1)/(k+1)z>k/(k+1)$. We know from the calculations in Example
2.8 that

$\mathrm{Pr}(A,G)<\frac{kz+n}{kzn}=\frac{1}{n}+\frac{1}{kz}.$

Moreover, as $A_{0}=A\cup Z$ and $|C_{G}(y)|/|G|\geq\frac{1}{n}$ for every
$y\in A$ (see Example 2.8),

$\displaystyle\mathrm{Pr}(A_{0},G)$ $\displaystyle=$
$\displaystyle\frac{1}{|A_{0}|}\left(\sum_{y\in
Z}\frac{|C_{G}(y)|}{|G|}+\sum_{1\neq y\in A}\frac{|C_{G}(y)|}{|G|}\right)$
$\displaystyle\geq$ $\displaystyle\frac{1}{(k+1)z}\left(z+\frac{kz}{n}\right)$
$\displaystyle>$ $\displaystyle\frac{1}{k+1}.$

Therefore

(3)
$\frac{\mathrm{Pr}(A,G)}{\mathrm{Pr}(A_{0},G)}<\left(\frac{1}{n}+\frac{1}{kz}\right)\left(k+1\right).$

If $k$ is fixed and the group $G$ is chosen with $n$ and $|U|=z/2$ arbitrary
large, the righthand side of (3) becomes arbitrary small.

##  3\. Proof of the main results

We start with two preliminary results, the first one being purely group
theoretical.

######  Lemma 3.1.

Let $m\geq 1$, and let $G$ be a finite group containing a subgroup $B$ such
that $[G:C_{G}(x)]\leq m$ for all $x\in B$. Then there is a normal subgroup
$T\leq G$ such that the index $[G:T]$ and the order of the commutator subgroup
$[T,B]$ are $m$-bounded.

###### Proof.

Note that

$\displaystyle\mathrm{Pr}(B,G)\geq\frac{1}{m}.$

By Proposition 1.2 of [3] there exists a normal subgroup $R$ of $m$-bounded
index in $G$ and a subgroup $U$ of $m$-bounded index in $B$ such that $[R,U]$
has $m$-bounded order. By Remark 2.6 in [3], the normal closure $[R,U]^{G}$
has $m$-bounded order.

Passing to the quotient over $[R,U]^{G}$, we can assume that $R\leq C_{G}(U)$.
Take $m$-boundedly many elements $b_{1},\dots,b_{r}$ in $B$ such that
$B=\langle b_{1},\dots,b_{r},U\rangle$. Since $[G:C_{G}(b_{i})]\leq m$ for
every $i=1,\dots,r$ and $C_{G}(U)$ has $m$-bounded index in $G$, the
intersection $C$ of $C_{G}(U)$ and all $r$ subgroups $C_{G}(b_{i})$ has
$m$-bounded index in $G$. So $C$ contains a normal subgroup $T$ of $G$ of
$m$-bounded index with $[T,B]=1$. This concludes the proof.∎

The following lemma tells us that if $A$ is an approximate subgroup and
$g_{1},\dots,g_{s}$ are elements of $G$ having few $A$-conjugates, then any
element of the subgroup $\langle g_{1},\dots,g_{s}\rangle$ has few
$A$-conjugates.

######  Lemma 3.2.

Let $G$ be a finite group, $A$ a $K$-approximate subgroup of $G$, and
$g_{1},\dots,g_{s}\in G$ elements such that $|g_{i}^{A}|\leq m$ for every
$i=1,\dots,s$. Then there exists a $(K,m,s)$-bounded integer $u$ such that
$|g^{A}|\leq u$ for every $g\in\langle g_{1},\dots,g_{s}\rangle.$

###### Proof.

It is sufficient to show that there exists a $(K,m,s)$-bounded integer $u$ and
elements $d_{1},\dots,d_{u}\in G$ such that

(4) $A\subseteq\bigcup_{1\leq i\leq u}C_{A^{2^{s}}}(g_{1},\dots,g_{s})d_{i}.$

Use induction on $s$. Let $s=1$ and
$g_{1}^{A}=\\{g_{1}^{a_{1}},\dots,g_{1}^{a_{m}}\\}$. If $a\in A$, then there
exists $k$ such that $g_{1}^{a}=g_{1}^{a_{k}}$, thus $aa_{k}^{-1}\in
C_{A^{2}}(g_{1})$ and $a=(aa_{k}^{-1})a_{k}$. Hence,

$A\subseteq\bigcup_{1\leq i\leq m}C_{A^{2}}(g_{1})a_{i},$

as desired.

Now assume that the result is true for $s-1$, that is, there are
$(K,m,s)$-boundedly many elements $h_{1},\dots,h_{v}\in G$ such that

$A\subseteq\bigcup_{1\leq i\leq v}C_{A^{2^{s-1}}}(g_{1},\dots,g_{s-1})h_{i}.$

Set $D=C_{A^{2^{s-1}}}(g_{1},\dots,g_{s-1})$. Note that by Lemma 2.5 the size
of the class $g_{s}^{D}$ is $(K,m,s)$-bounded. Write

$g_{s}^{D}=\\{g_{s}^{b_{1}},\dots,g_{s}^{b_{r}}\\}$

for suitable ${b_{1}},\dots,{b_{r}}\in D$.

As above, it follows that $D\subseteq\bigcup_{1\leq i\leq
r}C_{D^{2}}(g_{s})b_{i}$. In particular, we have

$\displaystyle D\subseteq\bigcup_{1\leq i\leq
r}C_{D^{2}}(g_{s})b_{i}\subseteq\bigcup_{1\leq i\leq
r}C_{A^{2^{s}}}(g_{1},\dots,g_{s})b_{i}.$

We know that $A\subseteq\bigcup_{1\leq i\leq v}Dh_{i}$, whence (4) follows
with $u=rv$. This establishes the lemma. ∎

Built on the ideas of P. M. Neumann’s theorem [7], we get the next
proposition, which contains the core of the proofs of Theorem 1.1 and Theorem
1.2.

######  Proposition 3.3.

Let $G$ be a finite group containing a $K_{1}$-approximate subgroup $H$ and a
$K_{2}$-approximate subgroup $U$ such that $\mathrm{Pr}(H,U)\geq\epsilon>0$.
Then there exists a symmetric subset $X$ of $H$, with $1\in X$, and two
positive numbers $K_{0}$ and $m$ depending only on $K_{1},K_{2}$ and
$\epsilon$ such that

  * •

$|X|\geq\frac{\epsilon}{2}|H|$,

  * •

$X^{2}$ is a $K_{0}$-approximate subgroup of $G$,

  * •

$|X^{2}|\leq K_{0}|X|$,

  * •

$|y^{U}|\leq m$ for every $y\in\langle X^{2}\rangle$.

###### Proof.

Set

$X=\\{x\in H\mid|x^{U}|\leq 2K_{2}/\epsilon\\}=\\{x\in H\mid
1/|x^{U}|\geq\epsilon/(2K_{2})\\}.$

As $|U^{2}|\leq K_{2}|U|$, from Lemma 2.4 (a) it follows that, for any $g\in
H\setminus X$,

$|C_{U}(g)|\leq\frac{K_{2}|U|}{|g^{U}|}\leq\frac{\epsilon
K_{2}|U|}{2K_{2}}=\frac{\epsilon|U|}{2},$

whence

$\displaystyle\epsilon|H|\,|U|$ $\displaystyle\leq$ $\displaystyle|\\{(x,y)\in
H\times U\mid xy=yx\\}|=\sum_{x\in H}|C_{U}(x)|$ $\displaystyle\leq$
$\displaystyle\sum_{x\in X}|U|+\sum_{x\in H\setminus X}\frac{\epsilon}{2}|U|$
$\displaystyle\leq$ $\displaystyle|X||U|+(|H|-|X|)\frac{\epsilon}{2}|U|$
$\displaystyle\leq$ $\displaystyle|X||U|+\frac{\epsilon}{2}|H|\,|U|.$

Therefore $({\epsilon}/{2})|H|\leq|X|$, that is,

$|X|\geq\alpha|H|$

for $\alpha={\epsilon}/{2}$. Thus,

$|X^{2}|\leq|H^{2}|\leq K_{1}|H|\leq(K_{1}/\alpha)|X|$

and also

$|X^{3}|\leq|H^{3}|\leq K_{1}^{2}|H|\leq(K_{1}^{2}/\alpha)|X|,$

hence $X$ has tripling $K_{1}^{2}/\alpha$.

It follows that $B=X^{2}$ is a $K_{0}$-approximate subgroup where $K_{0}$
depends on $K_{1}$ and $\alpha$ only (see Proposition 2.5.5 in [10]).

Note that $|y^{U}|\leq(K_{2}/\alpha)^{2}$ for every $y\in B$.

Let $E$ be a minimal subset of $G$ such that $B^{2}\subseteq EB$ and $|E|\leq
K_{0}$. By minimality of $E$, for every element $e\in E$ there are
$b_{1},b_{2},b_{3}\in B$ such that $b_{1}b_{2}=eb_{3}$ and so every element
$e\in E$ can be written as a product of at most $3$ elements of $B$. Therefore
$|e^{U}|\leq(K_{2}/\alpha)^{6}$ for every $e\in E$.

It follows from Lemma 3.2 that there exists a $(K_{0},K_{2},\alpha)$-bounded
integer $n$ such that $|g^{U}|\leq n$ for every $g\in\langle E\rangle.$ Note
that $B^{i}\leq\langle E\rangle B$ for every $i\geq 2$, and so there exists a
$(K_{0},K_{2},\alpha)$-bounded integer $m$ such that $|y^{U}|\leq m$ for every
$y\in\langle B\rangle$. As $K_{0}$ and $\alpha$ depend only on $K_{1}$ and
$\epsilon$, the proof is complete. ∎

Now we are ready to proceed with the proof of our main results. For the
reader’s convenience we restate Theorem 1.1 here.

Theorem 1.1. Let $A$ be a $K$-approximate subgroup of a finite group $G$ such
that $\mathrm{Pr}(A,G)\geq\epsilon>0$. There are two $(\epsilon,K)$-bounded
positive numbers $\gamma$ and $K_{0}$ such that $G$ contains a normal subgroup
$T$ and a $K_{0}$-approximate subgroup $B$ such that

  1. (i)

$|A\cap B|\geq\gamma\;{\mathrm{max}}\\{|A|,|B|\\}$, and

  2. (ii)

the index $[G:T]$ and the order of the subgroup $[T,\langle B\rangle]$ are
both $(\epsilon,K)$-bounded.

###### Proof.

Apply Proposition 3.3 with $H=A$, $U=G$, $K_{1}=K$ and $K_{2}=1$. Deduce that
there exists a subset $X$ of $A$ and two $(K,\epsilon)$-bounded numbers
$K_{0}$ and $m$ such that $B=X^{2}$ is a $K_{0}$-approximate subgroup while
$|B\cap A|\geq|X|\geq\frac{\epsilon}{2}|A|$ and $|y^{G}|\leq m$ for every
$y\in\langle B\rangle$. Note also that

$|B\cap
A|\geq\frac{\epsilon}{2}|A|\geq\frac{\epsilon}{2K}|A^{2}|\geq\frac{\epsilon}{2K}|B|,$

so we can take $\gamma={\epsilon}/({2K})$.

It follows from Lemma 3.1 applied to the subgroup $\langle B\rangle$ that
there exists a normal subgroup $T\leq G$ such that the index $[G:T]$ and the
order of the commutator subgroup $[T,\langle B\rangle]$ are $m$-bounded. Since
$m$ is $(K,\epsilon)$-bounded, the result follows. ∎

We now deal with Theorem 1.2.

Theorem 1.2. If $A$ is a $K$-approximate subgroup of a finite group $G$
satisfying $\mathrm{Pr}(A,A)\geq\epsilon>0$, then there are two
$(\epsilon,K)$-bounded positive numbers $\gamma$ and $s$, and a subgroup
$C\leq G$ such that $|C\cap A^{2}|>\gamma|A|$ and $|C^{\prime}|\leq s$.
Moreover, $A$ is contained in the union of at most $\gamma^{-1}K^{2}$ left
cosets of the group $C$.

###### Proof.

We apply Proposition 3.3 with $H=U=A$ and $K_{1}=K_{2}=K$ and deduce that
there exists a subset $X$ of $A$ and two positive numbers $K_{0}$ and $m$
depending only on $K$ and $\epsilon$ such that $|X|\geq\frac{\epsilon}{2}|A|$,
the set $B=X^{2}$ is a $K_{0}$-approximate subgroup, and $|y^{A}|\leq m$ for
every $y\in\langle B\rangle$.

As $|B^{2}|\leq K_{0}|B|$, it follows from Lemma 2.4 (b) that for every
$y\in\langle B\rangle$ we have

$|C_{B}(y)|\geq\frac{|B|}{K_{0}|y^{X}|}\geq\frac{|B|}{K_{0}|y^{A}|}\geq\eta|B|,$

where $\eta=1/({K_{0}m})$. Since

$|C_{B}(y)|\geq\eta|B|,$

for every $y\in\langle B\rangle$, we have

$\mathrm{Pr}(B,\langle B\rangle)=\frac{1}{|\langle B\rangle|}\sum_{y\in\langle
B\rangle}\frac{|C_{B}(y)|}{|B|}\geq\eta.$

Now we apply Proposition 3.3 with $H=B$ and $U=\langle B\rangle$, where
$K_{1}=K_{0}$ and $K_{2}=1$, to deduce that there exists a symmetric subset
$Y$ of $B$ and two positive numbers $K_{3}$ and $n$ depending only on $K_{0}$
and $\eta$ such that $|Y|\geq\frac{\eta}{2}|B|$ and $Y^{2}$ is a
$K_{3}$-approximate subgroup satisfying $|y^{\langle B\rangle}|\leq n$ for
every $y\in\langle Y^{2}\rangle$.

It follows from Theorem 1.1 in [8] that the commutator subgroup of the
subgroup $\langle(Y^{2})^{\langle B\rangle}\rangle$ has $(K_{3},n)$-bounded
order. As $\langle Y^{2}\rangle=\langle Y\rangle$, we deduce that the order of
the commutator subgroup of $\langle Y\rangle$ is $(K_{3},n)$-bounded.

Taking into account that $Y\subseteq B=X^{2}\subseteq A^{2}$, write

$|A|\leq\frac{2}{\epsilon}|X|\leq\frac{2}{\epsilon}|X^{2}|\leq\frac{2}{\epsilon}\left(\frac{2}{\eta}|Y|\right),$

and

(5) $|\langle Y\rangle\cap A^{2}|\geq|Y\cap
A^{2}|=|Y|\geq\frac{\epsilon\eta}{4}|A|=\gamma|A|,$

for $\gamma=\epsilon\eta/4$.

Finally, as

$|AY|\leq|A^{3}|\leq K^{2}|A|\leq K^{2}\frac{4}{\epsilon\eta}|Y|,$

it follows from [10, Lemma 2.4.4] (Ruzsa’s covering lemma) that there exists a
subset $F\subseteq A$ with $|F|\leq 4K^{2}/(\epsilon\eta)$ such that
$A\subseteq FY^{2}$. Therefore $A$ is contained in the union of at most
$4K^{2}/(\epsilon\eta)$ left cosets of $\langle Y\rangle$ (note that this
follows also from (5) and Lemma 7.1.5 in [10]). ∎

##  4\. The converse statements

In this section we prove Proposition 1.3 and Proposition 1.4, which are
roughly converse to Theorem 1.1 and Theorem 1.2, respectively.

We will also often use without mention the fact that if $S,T$ are subsets of a
finite group $G$ and $g\in S$ then $|g^{T}|\leq|[S,T]|$, because the map
$g^{T}\to\\{[g,x]\,|x\in T\\}$ defined by $g^{x}\mapsto g^{-1}g^{x}$ is a
bijection.

Proposition 1.3. Let $A,B$ be subsets and $T$ a subgroup of a finite group
$G$. Set $\gamma=|A\cap B|/|A|$, $n=[G:T]$ and $m=|[T,\langle B\rangle]|$.
Then $\mathrm{Pr}(A,G)\geq\frac{\gamma}{nm}$.

###### Proof.

Note that

$|A\cap\langle B\rangle|\geq|A\cap B|\geq\gamma|A|,$

so $|A\cap\langle B\rangle|/|A|\geq\gamma$ and without loss of generality we
can assume that $B$ is a subgroup. Moreover, if $g\in A\cap B$ , then

$\displaystyle[G:C_{G}(g)]\leq[G:T]\,[T:C_{T}(G)]\leq nm,$

since $|[T,\langle B\rangle]|=m$. Therefore

$\displaystyle\mathrm{Pr}(A,G)$ $\displaystyle=$
$\displaystyle\frac{1}{|A|}\sum_{a\in
A}\frac{|C_{G}(a)|}{|G|}\geq\frac{1}{|A|}\sum_{g\in A\cap
B}\frac{|C_{G}(g)|}{|G|}$ $\displaystyle\geq$ $\displaystyle\frac{|A\cap
B|}{nm|A|}\geq\frac{\gamma}{nm}.$

∎

Proposition 1.4. Let $A$ be a $K$-approximate subgroup of a finite group $G$,
and let $C\leq G$ be a subgroup. Set $\gamma=|C\cap A^{2}|/|A|$ and
$s=|C^{\prime}|$. Then

$\mathrm{Pr}(A^{2},A^{2})\geq\frac{\gamma^{2}}{K^{4}s}.$

###### Proof.

Since $|C\cap A^{2}|=\gamma|A|$, it follows from [10, Lemma 7.1.5] that $A$ is
contained in the union of $n\leq\gamma^{-1}K^{2}$ left cosets of $C$, say

$A\subseteq\bigcup_{1\leq i\leq n}f_{i}C.$

As $A$ is symmetric, for every $a\in A$ we have $a^{-1}\in\bigcup_{i}f_{i}C$,
whence $a=(f_{i}c)^{-1}=c^{-1}f_{i}^{-1}$ for some $i\leq n$ and $c\in C$. It
follows that

$A\subseteq\bigcup_{1\leq i\leq n}Cf_{i}^{-1}.$

Therefore, for every $g\in G$ we have

$|g^{A}|\leq|g^{\bigcup_{1\leq i\leq
n}Cf_{i}^{-1}}|\leq\sum_{i-1}^{n}|(g^{C})^{f_{i}^{-1}}|=n|g^{C}|.$

When $g\in C\cap A^{2}$ we also have $|g^{C}|\leq s$, since $|C^{\prime}|=s$,
hence

(6) $\frac{1}{|g^{A}|}\geq\frac{1}{ns}.$

Moreover, by Lemma 2.4 (b),
${|g^{A}|}\geq|A|/{|C_{A^{2}}(g)|}\geq|A^{2}|/(K{|C_{A^{2}}(g)|})$, which
gives

(7) $\frac{|C_{A^{2}}(g)|}{|A^{2}|}\geq\frac{1}{K|g^{A}|}.$

Now, by (7) and (6)

$\displaystyle\mathrm{Pr}(A^{2},A^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{|A^{2}|}\sum_{a\in
A^{2}}\frac{|C_{A^{2}}(a)|}{|A^{2}|}\geq\frac{1}{|A^{2}|}\sum_{g\in C\cap
A^{2}}\frac{|C_{A^{2}}(g)|}{|A^{2}|}$ $\displaystyle\geq$
$\displaystyle\frac{1}{|A^{2}|}\sum_{g\in C\cap A^{2}}\frac{1}{K|g^{A}|}$
$\displaystyle\geq$ $\displaystyle\frac{|C\cap A^{2}|}{Kns|A^{2}|}.$

Moreover, as $|A^{2}|\leq K|A|$,

$|C\cap A^{2}|=\gamma|A|\geq\gamma|A^{2}|/K.$

Therefore

$\mathrm{Pr}(A^{2},A^{2})\geq\frac{|C\cap
A^{2}|}{Kns|A^{2}|}\geq\frac{\gamma}{K^{2}ns}.$

Since $n\leq\gamma^{-1}K^{2}$, we conclude that
$\mathrm{Pr}(A^{2},A^{2})\geq{\gamma^{2}}/({K^{4}s}),$ as claimed. ∎

Acknowledgements. The first and second authors are members of GNSAGA (INDAM),
and the third author was supported by FAPDF and CNPq. The authors are grateful
to the anonymous referee for many helpful suggestions.

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