# Vortex structure deformation of rotating Lifshitz Holographic
Superconductors

Jhony A. Herrera-Mendoza<EMAIL_ADDRESS>Instituto de Física,
Benemérita Universidad Autónoma de Puebla, Edificio IF-1, Ciudad
Universitaria, CP 72570, Puebla, México.  Daniel F. Higuita-Borja
<EMAIL_ADDRESS>Instituto de Física, Benemérita Universidad Autónoma de
Puebla, Edificio IF-1, Ciudad Universitaria, CP 72570, Puebla, México.  Julio
A. Méndez-Zavaleta<EMAIL_ADDRESS>Facultad de Física, Universidad
Veracruzana, Paseo No. 112, Desarrollo Habitacional Nuevo Xalapa, C.P. 91097,
Xalapa-Enriquez, México.  Alfredo Herrera-Aguilar<EMAIL_ADDRESS>Instituto de Física, Benemérita Universidad Autónoma de Puebla, Edificio IF-1,
Ciudad Universitaria, CP 72570, Puebla, México.  Felipe Pérez-Rodríguez
<EMAIL_ADDRESS>Instituto de Física, Benemérita Universidad Autónoma de
Puebla, Edificio IF-1, Ciudad Universitaria, CP 72570, Puebla, México.

###### Abstract

We construct a type-II holographic superconductor from an anisotropic rotating
background metric. We study the effects of the magnetic field on the vortex
structure, including continuous deformations both from triangular to square
lattices or vice versa. Our holographic model reproduces known experimental
vortex lattice deformations and the increment of the vortex population by
increasing the external magnetic field.

_Introduction._ — Nowadays, the Gauge/Gravity duality has become an important
tool to describe different physical phenomena. Since the original proposal,
_AdS/CFT correspondence_ [1], the framework has been extended to explore
different phenomena such as quark-gluon plasma [2], superfluidity [3] and
superconductivity [4, 5, 6]. The seminal works in superconductivity
qualitatively reproduce the condensation operator for different boundary
conditions. Ever since, major progress has been made to extend the holographic
description of the phenomena. The vortex structure was explored for the first
time in [7], while a clearer analytical procedure was implemented in [8],
reproducing the Abrikosov vortex structure [9]. A degree of anisotropy has
been also considered via Lifshitz black holes in arbitrary dimension [10]. For
both the condensate and the magnetic fields, a good agreement between the
semi-analytical matching method and the numerical methods has been reported.
The holographic superconductor for rotating AdS-type spacetimes was
constructed in [11, 12] where the introduction of a magnetic potential
component $A_{\varphi}$ results mandatory to support the configuration and
make compatible the boundary conditions. A noteworthy re-visit to the problem
is presented in [13], where analytical methods were pursued instead of a
numerical approach. Several details of the Lifshitz holography were
collectively worked out in [14, 15], including both the condensate operator
and the vortex structure. To the best of the authors’ knowledge, this is the
first report on vortex lattice deformations in type II superconductors,
realized from a rotating holographic description. In previous works, the
consideration of anisotropy and angular momentum achieve solely a modification
of the lattice scale.

_The setup._ — Our starting point is the 5-dimensional anisotropic rotating
metric

$\displaystyle ds^{2}=$
$\displaystyle-\left(\dfrac{r}{\ell}\right)^{2z}f(r)\left(\Xi dt-
ad\phi\right)^{2}+\dfrac{r^{2}}{\ell^{4}}\left(adt-\Xi\ell^{2}d\phi\right)^{2}$
$\displaystyle+\dfrac{dr^{2}}{\left(\dfrac{r}{\ell}\right)^{2}f(r)}+\left(\dfrac{r}{\ell}\right)^{2}d\vec{y}{}^{2},$
(1)

where $\Xi=\sqrt{1+a^{2}/\ell^{2}}$, $d\vec{y}{}^{2}=dx^{2}+dy^{2}$,
$f(r)=1-\left(\dfrac{r_{h}}{r}\right)^{z+3}$, $z$ is the critical exponent,
$a$ is the rotation parameter, $\ell$ is the Lifshitz radius and $r_{h}$ is
the horizon radius. This metric describes a 5-dimensional black hole, and it
falls into a particular case of the solutions reported in [16]. The black hole
temperature, a fundamental quantity for the upcoming analysis, is given by

$T=\dfrac{1}{4\pi}\dfrac{(z+3)r_{h}^{z}}{\ell^{z+1}\Xi}.$ (2)

The holographic model is well described by the action

$S=\int{d^{5}x\sqrt{-g}\left[\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}+\dfrac{1}{\ell^{2}}\left(|D_{\mu}\Psi|^{2}+m^{2}|\Psi|^{2}\right)\right]},$
(3)

with equations of motion

$\displaystyle\nabla_{\mu}F^{\mu\nu}-\dfrac{1}{\ell^{2}}\left(2A^{\nu}|\Psi|^{2}-i\Psi\nabla^{\nu}\overline{\Psi}+i\overline{\Psi}\nabla^{\nu}\Psi\right)$
$\displaystyle=0,$
$\displaystyle\nabla^{\mu}\nabla_{\mu}\Psi-2iA^{\mu}\nabla_{\mu}\Psi-i\Psi\nabla_{\mu}A^{\mu}-\left(m^{2}+A_{\mu}A^{\mu}\right)\Psi$
$\displaystyle=0,$ (4)

where $D_{\mu}\equiv\nabla_{\mu}-iA_{\mu}$ is the gauge covariant derivative.

_The effect of an external magnetic field._ — In this section, we investigate
the effects of placing the type II holographic s-wave superconductor under the
influence of an external magnetic field. It is well known that type II
superconductors are characterized by manifesting two critical magnetic fields,
a lower one $B_{c1}$ and an upper one $B_{c2}$. The vorticity phenomenon
appears when the external magnetic field surpasses the lower critical value
$B>B_{c1}$. However, when the external magnetic field approaches from below to
the upper critical value $B_{c2}$, a vortex lattice is prone to be formed. On
the other hand, by decreasing the external magnetic field in the direction
perpendicular to the boundary ($u=0$) while keeping the temperature and
chemical potential fixed, it can be seen that the scalar field $\Psi$ begins
to condensate bellow the upper critical value $B_{c2}$.

To construct the vortex lattice, we will use a generalized version of the
procedure introduced in [8]. Firstly, we need to solve system (Vortex
structure deformation of rotating Lifshitz Holographic Superconductors) near
the upper critical magnetic field. With this aim, we perform a series
expansion of the scalar and gauge fields on the parameter
$\epsilon=(B_{c2}-B)/B$

$\displaystyle\Psi(u,\vec{y})=\epsilon^{1/2}\Psi_{1}(u,\vec{y})+\epsilon^{3/2}\Psi_{2}(u,\vec{y})+\ldots,$
(5a) $\displaystyle A_{\mu}(u,\vec{y})=A_{\mu}^{(0)}(u,\vec{y})+\epsilon
A_{\mu}^{(1)}(u,\vec{y})+\ldots,$ (5b)

in which $\mu=(t,\phi,x,y)$. We assume the leading order contribution of the
scalar field in the form of $\Psi_{1}(u,\vec{y})=\Phi(u,y)\text{e}^{ipx}$,
with $p$ constant. Moreover, to have a constant magnetic field normal to the
boundary, we shall consider $A_{x}^{(0)}(u,\vec{y})=B\,y$. In addition, a
constant chemical potential at the boundary imposes
$A_{t}^{(0)}(u,\vec{y})=A_{t}^{(0)}(u)$ and
$A_{\phi}^{(0)}(u,\vec{y})=A_{\phi}^{(0)}(u)$. Thus, the nontrivial lowest-
order Maxwell equations, coming from (Vortex structure deformation of rotating
Lifshitz Holographic Superconductors), read

$\displaystyle{A_{t}^{(0)}}^{\prime\prime}$
$\displaystyle-\Biggl{[}\quantity(\Xi^{2}-1)\dfrac{f^{\prime}}{f}-\dfrac{\Big{(}2(z-1)\Xi^{2}-z\Big{)}}{u}\Biggr{]}{A_{t}^{(0)}}^{\prime}$
$\displaystyle-\dfrac{a\Xi}{\ell^{2}}\Biggl{[}\dfrac{f^{\prime}}{f}-\dfrac{2(z-1)}{u}\Biggr{]}{A_{\phi}^{(0)}}^{\prime}=0,$
(6a) $\displaystyle{A_{\phi}^{(0)}}^{\prime\prime}$
$\displaystyle+\Biggl{[}\Xi^{2}\dfrac{f^{\prime}}{f}-\dfrac{\Big{(}2(z-1)\Xi^{2}+2-z\Big{)}}{u}\Biggr{]}{A_{\phi}^{(0)}}^{\prime}$
$\displaystyle+a\Xi\Biggl{[}\dfrac{f^{\prime}}{f}-\dfrac{2(z-1)}{u}\Biggr{]}{A_{t}^{(0)}}^{\prime}=0,$
(6b)

while the remaining ones imply $A_{y}^{(0)}(u,\vec{y})=0$. At this order, the
scalar and the Maxwell fields are not coupled, as expected from expansion (5).
These two equations can be decoupled and analytically integrated. The
following regular solutions arise

$A_{t}^{(0)}(u)=\mu-\rho\quantity(\dfrac{u}{r_{h}})^{3-z},\quad
A_{\phi}^{(0)}(u)=\nu-\zeta\quantity(\dfrac{u}{r_{h}})^{3-z},$ (7)

with $\zeta=-\dfrac{a}{\Xi}\,\rho$. Clearly, functions satisfy the required
boundary conditions provided $1\leq z<3$. The regularity of the gauge field at
the horizon translates to the vanishing of these components at $u=1$,
requiring the fixing $\rho=r_{h}^{3-z}\mu$ and $\zeta=r_{h}^{3-z}\nu$.

On the other hand, the lowest-order scalar equation has separable solutions of
the form $\Phi(u,y)=R_{n}(u)\gamma_{n}(y;p)$, with the functions $R_{n}(u)$
and $\gamma_{n}(y;p)$ defined by the following equations

$\displaystyle\Bigl{(}-\partial^{2}_{YY}+\dfrac{Y^{2}}{4}\Bigr{)}\gamma_{n}(y;p)=\dfrac{\lambda_{n}}{2}\gamma_{n}(y;p),$
(8a) $\displaystyle R_{n}^{\prime\prime}(u)$
$\displaystyle+\Bigl{(}\dfrac{f^{\prime}}{f}-\dfrac{z+2}{u}\Bigr{)}R_{n}^{\prime}(u)=\Biggl{[}\quantity(\dfrac{\ell^{2}}{r_{h}})^{2}\dfrac{B\lambda_{n}}{f}$
$\displaystyle+\dfrac{\ell^{2}}{f}\Big{(}\dfrac{m^{2}}{u^{2}}+\dfrac{(aA_{t}^{(0)}+\Xi
A_{\phi}^{(0)})^{2}}{r_{h}^{2}}\Big{)}$ $\displaystyle-\quantity(\dfrac{\ell
u}{r_{h}})^{2(z-1)}\dfrac{(\ell^{2}\Xi
A_{t}^{(0)}+aA_{\phi}^{(0)})^{2}}{r_{h}^{2}f^{2}}\Biggr{]}R_{n}(u),$ (8b)

where $Y:=\sqrt{2B}\quantity(y-p/B)$. Equation (8a) is identified as a one-
dimensional Schrödinger-like equation and is interpreted as the distribution
of the order parameter on the $(x,y)$ plane. Moreover, the radial equation
(8b) defines the superconducting phase transition.

Let us now consider the regular and normalizable solutions of (8a). These are
spanned by the Hermite polynomials as

$\gamma_{n}(y;p)=\text{e}^{-Y^{2}/4}H_{n}(Y),$ (9)

with eigenvalues defined by $\lambda_{n}=2n+1$, for a non-negative integer
$n$.

Now, we derive a general expression for the upper critical magnetic field
$B_{c2}$ as a function of the background parameters. Near the boundary, the
radial solution is given according to

$R(u)=J_{-}u^{\Delta_{-}}+J_{+}u^{\Delta_{+}},$ (10)

with $\Delta_{\pm}=\dfrac{z+3\pm\sqrt{(z+3)^{2}+4\ell^{2}m^{2}}}{2}$. To keep
arguments as simple as possible, we also impose the boundary condition
$J_{-}=0$, and set $J_{+}=J$ and $\Delta_{+}=\Delta$.

The solution of (8b) near the horizon is tackled through an expansion of
$R(u)$ around $u=1$

$R(u)=R(1)+R^{\prime}(1)(u-1)+\dfrac{1}{2}R^{\prime\prime}(1)(u-1)^{2}+\ldots\
$ (11)

so that boundary conditions are met

$\displaystyle
R^{\prime}(1)=-\dfrac{\ell^{2}\quantity(m^{2}r_{h}^{2}+\ell^{2}B)}{r_{h}^{2}(z+3)}R(1),$
(12a) $\displaystyle
R^{\prime\prime}(1)=\dfrac{1}{2(z+3)^{2}}\Biggl{\\{}-\quantity(\dfrac{\ell}{\Xi})^{2}\quantity(\dfrac{\ell}{r_{h}})^{2z}[{A_{t}^{(0)}}^{\prime}(1)]^{2}$
(12b)
$\displaystyle\,+\ell^{2}m^{2}\Big{[}\ell^{2}m^{2}+2(z+3)\Big{]}+\dfrac{\ell^{4}B}{r_{h}^{2}}\Big{[}2\ell^{2}m^{2}+\dfrac{\ell^{4}B}{r_{h}^{2}}\Big{]}\Biggr{\\}}R(1).$

In what follows, we will use the semi-analytical matching method [10] which
has shown good agreement with the numerical computations. This method is based
on the mild assumption that there is an intermediate point $u=u_{m}$ where the
horizon and boundary solutions, (10) and (11), as well as their first
derivatives match. The resulting algebraic system can be solved consistently
to give

$B=\dfrac{r_{h}^{2}}{\ell^{4}\eta}\Biggl{\\{}-\delta+\Biggl{[}\chi+\eta^{2}\quantity(\dfrac{\ell}{\Xi})^{2}\quantity(\dfrac{\ell}{r_{h}})^{2z}[{A_{t}^{(0)}}^{\prime}(1)]^{2}\Biggr{]}^{1/2}\Biggr{\\}},$
(13)

with

$\displaystyle\qquad\eta$
$\displaystyle=(1-u_{m})\Bigl{[}(2-\Delta)u_{m}+\Delta\Bigr{]},$
$\displaystyle\quad\chi$
$\displaystyle=2(z+3)\Big{(}2(z+3)u_{m}^{2}-\ell^{2}m^{2}\eta^{2}\Big{)},$
$\displaystyle\delta$
$\displaystyle=2u_{m}\Bigl{[}(z+3)+\ell^{2}m^{2}(1-u_{m})\Bigr{]}$
$\displaystyle{}+\Delta(1-u_{m})\Bigl{[}2(z+3)+\ell^{2}m^{2}(1-u_{m})\Bigr{]}.$
(14)

Near to the upper critical magnetic field value, $B\approx B_{c2}$ is a good
approximation. Hence, letting (7) in (13), we disclose the expression

$\displaystyle B_{c2}=$
$\displaystyle\quantity(\dfrac{\ell^{z+1}\Xi}{z+3})^{2/z}\dfrac{T^{2/z}}{\ell^{4}\eta}\Biggl{\\{}-\delta$
$\displaystyle\qquad\qquad+\Biggl{[}\chi+\quantity(\delta^{2}-\chi)\quantity(\dfrac{T_{c}}{T})^{6/z}\Biggr{]}^{1/2}\Biggr{\\}},$
(15)

wherein the critical temperature is found to be

$T_{c}=\dfrac{z+3}{\ell^{z+1}\Xi}\dfrac{1}{\quantity(\delta^{2}-\chi)^{z/6}}\quantity[\eta^{2}\ell^{2z}\quantity(z-3)^{2}\quantity(\dfrac{\ell}{\Xi})^{2}\rho^{2}]^{z/6}.$
(16)

Moreover, it can be shown that when $T\approx T_{c}$ , the upper critical
magnetic field has a linear dependence with the temperature

$B_{c2}\propto\quantity(1-\dfrac{T}{T_{c}}),$ (17)

in agreement with the well-known results in the Ginzburg-Landau theory of
superconductivity.

_Vortex lattice deformation._ —

(a)

(b)

(c)

Figure 1: The transformation of the vortex lattice from the triangular (a),
passing through the rhombic (b), to the square one (c), as the external
magnetic field is increased. The vortex cores are located at the center of the
blue regions, wherein the function $\absolutevalue{\gamma_{L}}^{2}$ vanishes.

We are now in position to address the construction of the vortex lattice. We
will follow a slightly similar procedure to the one implemented on [8, 14], by
generalizing their symmetry arguments. We should note that it is sufficient to
consider the lowest mode $n=0$ in the solution for $\gamma_{n}$ since it is
expected to correspond with the most stable case [8]. Namely, we take

$\gamma_{0}(y;p_{k})=\exp[-\dfrac{1}{2r_{0}^{2}}(y-p_{k}r_{0}^{2})^{2}],$ (18)

with $r_{0}=1/\sqrt{B}$. In this sense the lattice is constructed by a
superposition of the lowest order solutions

$\Psi_{1}(u,\vec{y})=R_{0}(u)\sum_{k=-\infty}^{\infty}c_{k}e^{ip_{k}x}\gamma_{0}(y;p_{k}),$
(19)

where $R_{0}(u)$ is the lowest order solution of the radial equation (8b). At
this point one finds useful to define the spatial factor in (19) as

$\gamma_{L}(\vec{y})=\sum_{k=-\infty}^{\infty}c_{k}e^{ip_{k}x}\exp[-\dfrac{1}{2r_{0}^{2}}(y-p_{k}r_{0}^{2})^{2}].$
(20)

This expression is very similar to the one emerging in Ginzburg-Landau theory
for the order parameter of a type II superconductor under the influence of a
constant magnetic field. The coherence length is identified as $r_{0}=\xi$,
and close to the critical temperature it approaches

$\xi\propto\quantity(1-\dfrac{T}{T_{c}})^{-1/2},$ (21)

which also agrees with the known result in the Ginzburg-Landau theory.

Note that the function (20) is periodic along the $x$ direction but not
necessarily along the $y$ direction. However, for this function to represent a
vortex lattice, we need periodicity (at least up to a phase factor) along the
$y$ direction as well. This is possible if we choose the parameters $c_{k}$,
$q_{k}$, and $p_{k}$ to be

$c_{k}\equiv e^{iq_{k}},\qquad q_{k}\equiv\alpha k^{2},\qquad p_{k}\equiv\beta
k,$ (22)

with $\alpha$ and $\beta$ denoting arbitrary real constants. The above
definition allows us to write $\gamma_{L}$ in terms of the elliptic theta
function $\vartheta_{3}$ as

$\gamma_{L}(\vec{y})=e^{-y^{2}/2r_{0}^{2}}\vartheta_{3}(v,\tau),$ (23)

in which

$v=\dfrac{\beta}{2\pi}\quantity(x-iy),\qquad\tau=\dfrac{1}{2\pi}\quantity(2\alpha+i\beta^{2}r_{0}^{2}).$
(24)

In this form, the symmetries of $\gamma_{L}$ are dictated by the well-known
symmetries of $\vartheta_{3}$. Using this fact, we can easily show that
$\gamma_{L}$ has a pseudo-periodicity in the following directions:

$\vec{b}_{1}=\dfrac{2\pi}{\beta}\partial_{x},\qquad\vec{b}_{2}=\dfrac{2\alpha}{\beta}\partial_{x}-\beta
r_{0}^{2}\partial_{y}.$ (25)

Here lies the keystone of the uttermost result. In contrast to [8, 14], we
determine the most general form of (25), which accommodates a lattice
deformation beyond a rescaling by $r_{0}$. In consequence, the pseudo-
periodicity of $\gamma_{L}$ reads

$\displaystyle\gamma_{L}(x+\dfrac{2\pi}{\beta},y)$
$\displaystyle=\gamma_{L}(x,y),$ (26a)
$\displaystyle\gamma_{L}\Bigl{(}x+\dfrac{2\alpha}{\beta},y-\beta
r_{0}^{2}\Bigr{)}$ $\displaystyle=\exp[-i\quantity(\alpha+\beta
x)]\gamma_{L}(x,y).$ (26b)

Altogether, the properties of $\gamma_{L}$ guarantee a lattice structure on
$\absolutevalue{\gamma_{L}}^{2}$ with fundamental regions spanned by the
vectors (25). Therein, the magnetic flux penetrating a unit cell is quantized
and given by $B\times 2\pi r_{0}^{2}=2\pi$. This coincides with the well-known
results in the Ginzburg-Landau theory. From (25), we find the following
relations between two adjacent vectors in the fundamental cell

$\cos{\theta}=\sqrt{1-\dfrac{\beta^{4}}{4\pi^{2}B^{2}}},\qquad\alpha=\pi\cos{\theta},$
(27)

wherein $\theta$ denotes the angle between these two vectors and we have
chosen their magnitudes to be the same. Concerning the above relations, it is
crucial to note that they provide a mechanism to continuously deform a given
vortex lattice driven by the external magnetic field. Note that $\beta$
remains arbitrary in Eq. (27), hence it can be chosen appropriately as a
function of the external magnetic field.

_Experimental realization._ — Beyond the theoretical scopes of the model here
conceived, we present a direct application to the local vortex lattice
deformations of a FeSe-based superconductor induced by an external magnetic
field, as presented in [17]. The experimental results showed a transition from
a nearly triangular lattice at $B=1$ T to a rhombic lattice at $B=3$ T, and an
almost square one at $B=4$ T, at fixed temperature. Herein T standing for the
Tesla unit which is linked to the length scale in the bulk gravitational
theory through T$=[\ell]^{-1}$. In our model, it suffices to determine an
appropriate form of the $\beta$ parameter in terms of the external magnetic
field. Our case of interest, the FeSe lattice of [17], is adequately comprised
through the quadratic regression

$\beta=a_{1}+a_{2}\,B+a_{3}\,B^{2},$ (28)

with the coefficients given by $a_{1}=2.37693\ \text{T}^{1/2}$,
$a_{2}=-0.27870\ \text{T}^{-1/2}$ and $a_{3}=0.23444\ \text{T}^{-3/2}$.
Fig. 1 illustrates the evolution of a vortex structure when the external
magnetic field increases. The observed configurations take place when the
magnetic field adopts the values: $B_{tr}=1$ T for the triangular, $B_{rh}=3$
T for the rhombic, and $B_{sq}=4$ T for the square one. Besides that, the
vortex density increases when increasing the magnetic field, which also is in
agreement with the experimental observations of [17].

On the other hand, if we consider $\beta$ as a non-zero arbitrary constant,
the vortex deformation will take place from a triangular to a square lattice
but now by decreasing the external magnetic field. In this case, the
triangular lattice corresponds to $B_{tr}=\beta^{2}/{\sqrt{3}\pi}$ while the
square one to $B_{sq}=\beta^{2}/{2\pi}$, so that the transition is complete
when the magnetic field has decreased about $13.4\ \%$ with respect to
$B_{tr}$. However, to the best of our knowledge, this behavior has not been
observed in experiments using superconducting materials.

Finally, we advert a less interesting case in which $\beta\sim\sqrt{B}$. In
this particular setting, the external magnetic field has no other effect but
scaling the vortex structure, similar to already known models.

_Conclusions and discussion._ — In this work, we have constructed a
holographic superconductor stemmed from a rotating Lifshitz background that
mimics a type II s-wave superconductor.

In constructing the Abrikosov vortex lattice, a generalized approach to that
in [8, 14] proved fruitful. We managed to describe continuous transformations
of the vortex lattice by varying the magnetic field as a prediction of the
holographic model. In particular, with a proper election of the parameter
$\beta$ as a function of the external magnetic field, we faithfully reproduced
the experimentally observed vortex transformation of the FeSe-based
superconductor investigated in [17]. Likewise, the transition of the lattice
morphology reported in [18] for a LiFeAs superconductor can also be modeled by
our framework. Yet, the freedom in choosing $\beta$ means a further modeling
power, worth to investigate in detail with application to other
phenomenological results.

An interesting point of comparison with our findings is provided by [19].
There, the authors propose a numerical model capable of predicting the
triangular and rectangular cases, at different temperatures. However, no other
lattice configurations are reported nor a mechanism for continuous
deformations.

Finally, we emphasize that though our model was shaped from a near $B_{c2}$
approximation, the theory accurately describes transitions in the vortex
lattice beyond the critical point. Thus, this work also paves the way to
reproduce experimental magnetization curves by adjusting the free parameter
$\beta(B)$.

Besides, this holographic approach can be extended to other studies, namely,
to describe holographic ferromagnetic or ferroelectric materials, and to
analyze the magnetic response of materials simultaneously exhibiting both
superconductivity and ferromagnetism.

###### Acknowledgements.

All the authors are grateful to Manuel de la Cruz for enriching discussions.
The authors acknowledge financial support from a CONACYT Grant No. A1-S-38041
and from the VIEP-BUAP Grant No. 122. JAHM also acknowledges support from
CONACYT through a PhD Grant No. 750974. DFHB and JAMZ are also grateful to
CONACYT for a _Estancias posdoctorales por México_ Grant No. 372516 and
project 898686, respectively.

## References

  * Maldacena [1998] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200 .
  * Casalderrey-Solana  _et al._ [2014] J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, and U. A. Wiedemann,  _Gauge/String Duality, Hot QCD and Heavy Ion Collisions_ (Cambridge University Press, 2014) arXiv:1101.0618 [hep-th] .
  * Herzog  _et al._ [2009] C. P. Herzog, P. K. Kovtun, and D. T. Son, Holographic model of superfluidity, Phys. Rev. D 79, 066002 (2009), arXiv:0809.4870 [hep-th] .
  * Hartnoll  _et al._ [2008a] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101, 031601 (2008a), arXiv:0803.3295 [hep-th] .
  * Hartnoll  _et al._ [2008b] S. A. Hartnoll, C. P. Herzog, and G. T. Horowitz, Holographic Superconductors, JHEP 12, 015, arXiv:0810.1563 [hep-th] .
  * Hartnoll [2009] S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26, 224002 (2009), arXiv:0903.3246 [hep-th] .
  * Montull  _et al._ [2009] M. Montull, A. Pomarol, and P. J. Silva, The Holographic Superconductor Vortex, Phys. Rev. Lett. 103, 091601 (2009), arXiv:0906.2396 [hep-th] .
  * Maeda  _et al._ [2010] K. Maeda, M. Natsuume, and T. Okamura, Vortex lattice for a holographic superconductor, Phys. Rev. D 81, 026002 (2010), arXiv:0910.4475 [hep-th] .
  * Abrikosov [1957] A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solids 2, 199 (1957).
  * Zhao  _et al._ [2014] Z. Zhao, Q. Pan, and J. Jing, Notes on analytical study of holographic superconductors with Lifshitz scaling in external magnetic field, Phys. Lett. B 735, 438 (2014).
  * Sonner [2009] J. Sonner, A Rotating Holographic Superconductor, Phys. Rev. D 80, 084031 (2009), arXiv:0903.0627 [hep-th] .
  * Lin and Abdalla [2014] K. Lin and E. Abdalla, Holographic Superconductors in a Rotating Spacetime, Eur. Phys. J. C 74, 3144 (2014), arXiv:1403.7407 [hep-th] .
  * Srivastav and Gangopadhyay [2019] A. Srivastav and S. Gangopadhyay, Analytic investigation of rotating holographic superconductors, Eur. Phys. J. C 79, 340 (2019), arXiv:1902.01628 [hep-th] .
  * Guo  _et al._ [2016] H. Guo, F.-W. Shu, J.-H. Chen, H. Li, and Z. Yu, A holographic model of d-wave superconductor vortices with Lifshitz scaling, Int. J. Mod. Phys. D 25, 1650021 (2016), arXiv:1410.7020 [hep-th] .
  * Natsuume and Okamura [2018] M. Natsuume and T. Okamura, Holographic Lifshitz superconductors: Analytic solution, Phys. Rev. D 97, 066016 (2018), arXiv:1801.03154 [hep-th] .
  * Herrera-Aguilar  _et al._ [2021] A. Herrera-Aguilar, J. A. Herrera-Mendoza, and D. F. Higuita-Borja, Rotating Spacetimes generalizing Lifshitz Black Holes, Eur. Phys. J. C 81, 874 (2021), arXiv:2104.14514 [hep-th] .
  * Putilov  _et al._ [2019] A. V. Putilov, C. Di Giorgio, V. L. Vadimov, D. J. Trainer, E. M. Lechner, J. L. Curtis, M. Abdel-Hafiez, O. S. Volkova, A. N. Vasiliev, D. A. Chareev, G. Karapetrov, A. E. Koshelev, A. Y. Aladyshkin, A. S. Mel’nikov, and M. Iavarone, Vortex-core properties and vortex-lattice transformation in FeSe, Phys. Rev. B 99, 144514 (2019).
  * Zhang  _et al._ [2019] S. S. Zhang, J.-X. Yin, G. Dai, H. Zheng, G. Chang, I. Belopolski, X. Wang, H. Lin, Z. Wang, C. Jin, and M. Z. Hasan, Vector field controlled vortex lattice symmetry in LiFeAs using scanning tunneling microscopy, Phys. Rev. B 99, 161103 (2019).
  * Xia  _et al._ [2022] C.-Y. Xia, H.-B. Zeng, Y. Tian, C.-M. Chen, and J. Zaanen, Holographic Abrikosov lattice: Vortex matter from black hole, Phys. Rev. D 105, L021901 (2022), arXiv:2111.07718 [hep-th] .


# On an application of higher energies to Sidon sets ††This work is supported
by the Russian Science Foundation under grant 19–11–00001.

Shkredov I.D

Annotation.

We show that for any finite set $A$ and an arbitrary $\varepsilon>0$ there is
$k=k(\varepsilon)$ such that the higher energy $\mathsf{E}_{k}(A)$ is at most
$|A|^{k+\varepsilon}$ unless $A$ has a very specific structure. As an
application we obtain that any finite subset $A$ of the real numbers or the
prime field either contains an additive Sidon–type subset of size
$|A|^{1/2+c}$ or a multiplicative Sidon–type subset of size $|A|^{1/2+c}$.

##  1 Introduction

Sidon sets is a classical object of Combinatorial Number theory, which was
introduced by S. Sidon in [24]. A subset $S$ of an abelian group
${\mathbf{G}}$ is a Sidon set iff all its non–zero differences are distinct.
Being ”more random than random” this interesting class of sets were
extensively studied by various authors, see [1]–[9], [13]–[19] and many others
papers. Detailed information about Sidon sets can be found in survey [17], for
example.

Let $\mathsf{Sid}(A)$ be size of the maximal (by cardinality) Sidon subset of
a set $A\subseteq{\mathbf{G}}$. If we want to underline the group operation we
write $\mathsf{Sid}^{+}(A)$ or $\mathsf{Sid}^{\times}(A)$. In [11] (also, see
paper [21]) it was proved that for any $A\subseteq{\mathbb{R}}$ one has

$\mathsf{Sid}(A)\geqslant c\sqrt{|A|}\,,$ (1)

where $c>0$ is an absolute constant. Of course, this result is tight (just
take $A$ equals a segment of integers to see that
$\mathsf{Sid}^{+}(A)\ll\sqrt{|A|}$). Oleksiy Klurman and Cosmin Pohoata (see
[10]) asked the following sum–product-type question (on the sum–product
phenomenon see, e.g., [25]): is it true that bound (1) can be improved either
for $\mathsf{Sid}^{+}(A)$ or for $\mathsf{Sid}^{\times}(A)$, where $A$ is any
finite subset of the real numbers?

We write $\mathsf{Sid}_{k}(A)$ for size of maximal subset of $A$ having at
most $k$ representations of any non–zero element as a difference. Thus
$\mathsf{Sid}_{1}(A)=\mathsf{Sid}(A)$ and bound (1) cannot be improved for the
quantity $\mathsf{Sid}_{k}(A)$ (again take $A$ equals a segment of integers).
Our main result is

###### Theorem 1

Let $A\subseteq\mathbb{F}$ be a set, where $\mathbb{F}={\mathbb{R}}$ or
$\mathbb{F}=\mathbb{F}_{p}$ (in the prime field case suppose, in addition,
that $|A|<\sqrt{p}$, say). Then there are some absolute constants $c>0$,
$K\geqslant 1$ such that

$\max\\{\mathsf{Sid}^{+}_{K}(A),\mathsf{Sid}_{K}^{\times}(A)\\}\gg|A|^{1/2+c}\,.$
(2)

On the other hand, for any integer $k\geqslant 1$ there is
$A\subseteq\mathbb{F}$ with

$\max\\{\mathsf{Sid}^{+}_{k}(A),\mathsf{Sid}^{\times}_{k}(A)\\}\ll
k^{1/2}|A|^{2/3}\,.$ (3)

Oliver Roche-Newton and Audie Warren obtained a bound similar to (3) and
another estimate of the same form was obtained by Green–Peluse, see [10]
(also, see [6, page 57]). Our construction is different from these
counterexamples and we give our own proof of (3) at the end of section 3 for
completeness.

Actually, Theorem 3 is a consequence of a more general fact about so–called
higher energies [20] (all required definitions can be found in section 2). We
think that Theorem 2 below is interesting in its own right and it can find
further applications in Additive Combinatorics.

###### Theorem 2

Let $A\subseteq{\mathbf{G}}$ be a set, $\delta,\varepsilon\in(0,1]$ be
parameters. Then there is $k=k(\delta,\varepsilon)$ such that either
$\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$ or there is
$H\subseteq{\mathbf{G}}$, $|H|\gg|A|^{\delta(1-\varepsilon)}$,
$|H+H|\ll|A|^{\varepsilon}|H|$ and there exists $Z\subseteq{\mathbf{G}}$,
$|Z||H|\ll|A|^{1+\varepsilon}$ with $|(H\dotplus Z)\cap
A|\gg|A|^{1-\varepsilon}$.

In other words, we can always take large $k$ to make $\mathsf{E}_{k}(A)$ as
small as possible unless the set $A$ has a very rigid structure. It is easy to
see (or consult [22, Theorem 22]) that Theorem 2 is, actually, a criterion.

Finally, at our last section we study $k$–Sidon sets (and its
generalizations), that is, sets $A$ having at most $k$ representations of any
non–zero element as a difference. This class of sets were introduced by Erdős
in [6] and it is strongly connected with the higher energies. We show that
such sets are even more natural than usual $B_{2}[g]$–sets, see the definition
from [17]. In particular, size of such sets can be estimated relatively easily
(unlike to $B_{2}[g]$–sets), they have heritability properties, they have a
natural reinterpretation in terms of its Cayley graph and so on.

We thank Oliver Roche–Newton who communicated the question of Oleksiy Klurman
and Cosmin Pohoata to the author. Also, we thank him for very useful comments,
discussions and remarks.

##  2 Definitions

By ${\mathbf{G}}$ we denote an abelian group. Sometimes we underline the group
operation writing $+$ or $\times$ in the considered quantities (as the energy,
the representation function and so on, see below). Let $\mathbb{F}$ be the
field ${\mathbb{R}}$ or $\mathbb{F}=\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ for
a prime $p$.

We use the same capital letter to denote set $A\subseteq\mathbb{F}$ and its
characteristic function $A:\mathbb{F}\to\\{0,1\\}$. Given two sets
$A,B\subset{\mathbf{G}}$, define the sumset of $A$ and $B$ as

$A+B:=\\{a+b~{}:~{}a\in{A},\,b\in{B}\\}\,.$

In a similar way we define the difference sets and higher sumsets, e.g.,
$2A-A$ is $A+A-A$. We write $\dotplus$ for a direct sum, i.e., $|A\dotplus
B|=|A||B|$. For an abelian group ${\mathbf{G}}$ the Plünnecke–Ruzsa inequality
(see, e.g., [25]) holds stating

$|nA-mA|\leqslant\left(\frac{|A+A|}{|A|}\right)^{n+m}\cdot|A|\,,$ (4)

where $n,m$ are any positive integers. We use representation function
notations like $r_{A+B}(x)$ or $r_{A-B}(x)$ and so on, which counts the number
of ways $x\in{\mathbf{G}}$ can be expressed as a sum $a+b$ or $a-b$ with $a\in
A$, $b\in B$, respectively. For example, $|A|=r_{A-A}(0)$.

For any two sets $A,B\subseteq{\mathbf{G}}$ the additive energy of $A$ and $B$
is defined by

$\mathsf{E}(A,B)=\mathsf{E}^{+}(A,B)=|\\{(a_{1},a_{2},b_{1},b_{2})\in A\times
A\times B\times B~{}:~{}a_{1}+b_{1}=a_{2}+b_{2}\\}|\,.$

If $A=B$, then we simply write $\mathsf{E}(A)$ for $\mathsf{E}(A,A)$. For
$k\geqslant 2$ put

$\mathsf{E}_{k}(A)=\sum_{x}r^{k}_{A-A}(x)=\sum_{\alpha_{1},\dots,\alpha_{k-1}}|A\cap(A+\alpha_{1})\cap\dots\cap(A+\alpha_{k-1})|^{2}\,.$
(5)

Clearly, $|A|^{k}\leqslant\mathsf{E}_{k}(A)\leqslant|A|^{k+1}$. Also, we write
$\hat{\mathsf{E}}_{k}(A)=\sum_{x}r^{k}_{A+A}(x)$.

The signs $\ll$ and $\gg$ are the usual Vinogradov symbols. When the constants
in the signs depend on a parameter $M$, we write $\ll_{M}$ and $\gg_{M}$. All
logarithms are to base $2$. If we have a set $A$, then we will write
$a\lesssim b$ or $b\gtrsim a$ if $a=O(b\cdot\log^{c}|A|)$, $c>0$. Let us
denote by $[n]$ the set $\\{1,2,\dots,n\\}$. By $K_{s,t}$ denote the complete
subgraph with two parts of sizes $s$ and $t$.

##  3 Proof of the main result

We start with a probability lemma, which was known before in the case $k=2$,
see [1], say. Roughly speaking, ignoring the presence of the parameter $k$ in
$\mathsf{Sid}_{O(k)}$ from formula (6) below, Lemma 6 works better than
inequality (1) if $\mathsf{E}_{k}(A)\leqslant|A|^{k+\omega}$ for a certain
$\omega<1/2$.

###### Lemma 3

Let $A\subseteq{\mathbf{G}}$ be a set. Then for any $k\geqslant 2$ one has

$\mathsf{Sid}_{3k-3}(A)\gg\left(\frac{|A|^{2k}}{\mathsf{E}_{k}(A)}\right)^{1/(2k-1)}\,,\quad\quad\mbox{
and
}\quad\quad\mathsf{Sid}_{2k-2}(A)\gg\left(\frac{|A|^{2k}}{\hat{\mathsf{E}}_{k}(A)}\right)^{1/(2k-1)}\,.$
(6)

P r o o f. Form a random set $A_{*}\subseteq A$ piking elements of $A_{*}$
from $A$ independently at random with probability $q$. Then the expectation of
the solutions to the equation

$\mathsf{E}^{\prime}_{k}(A):=|\\{x_{1}-x^{\prime}_{1}=\dots=x_{k}-x^{\prime}_{k}~{}:~{}x_{j},x^{\prime}_{j}\in
A,\,~{}~{}x_{j},x^{\prime}_{j}\mbox{ are different}\\}|$ (7)

is $q^{2k}\mathsf{E}^{\prime}_{k}(A)\leqslant q^{2k}\mathsf{E}_{k}(A)$. If
$q^{2k}\mathsf{E}_{k}(A)\leqslant q|A|/2$, then we can delete roughly a half
of elements of $A_{*}$ to find a subset $B$ of $A_{*}$, $|B|\gg|A_{*}|$,
having no solutions to equation (7). Let us take any $z\neq 0$ and prove that
$r_{B-B}(z)<3k-2:=g$. Suppose that for some $x_{j},x^{\prime}_{j}\in B$ the
following holds

$0\neq z=x_{1}-x^{\prime}_{1}=\dots=x_{g}-x^{\prime}_{g}\,.$ (8)

Clearly, any $x_{j}$ or $x^{\prime}_{j}$ presents in (8) in at most two
equations and hence a pair $(x_{j},x^{\prime}_{j})$ presents in at most three
equations. Also, by the definition of the set $B$ any $k$ equations from (8)
must have some equal variables. It follows that $r_{B-B}(z)\leqslant
3\cdot(k-1)$ as required. Finally, taking
$q=(|A|\mathsf{E}^{-1}_{k}(A)/2)^{1/(2k-1)}$, we obtain our result. The second
bound in (6) can be obtained similarly. In this case the correspondent
analogue of (8) is

$x_{1}+x^{\prime}_{1}=\dots=x_{g}+x^{\prime}_{g}$ (9)

and we see that a pair $(x_{j},x^{\prime}_{j})$ presents in at most two
equations from (9). Again by the definition of the set $B$ any $k$ equations
from (9) must have some equal variables. It follows that $r_{B+B}(z)\leqslant
2k-2$ as required. $\hfill\Box$

We now obtain our driving result (Theorem 2 from the introduction is a
particular case of Theorem 4). The proof is in the spirit of [22, Theorem 21].

###### Theorem 4

Let $A\subseteq{\mathbf{G}}$ be a set, $\delta,\varepsilon\in(0,1]$ be
parameters, $\varepsilon\leqslant\delta$.
$1)~{}$ Then there is
$k=k(\delta,\varepsilon)=\exp(O(\varepsilon^{-1}\log(1/\delta)))$ such that
either $\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$ or there is
$H\subseteq{\mathbf{G}}$, $|H|\gtrsim|A|^{\delta(1-\varepsilon)}$,
$|H+H|\ll|A|^{\varepsilon}|H|$ and there exists $Z\subseteq{\mathbf{G}}$,
$|Z||H|\ll|A|^{1+\varepsilon}$ with

$|(H\dotplus Z)\cap A|\gg|A|^{1-\varepsilon}\,.$

$2)~{}$ Similarly, either there is a set $A^{\prime}\subseteq A$,
$|A^{\prime}|\gg|A|^{1-\varepsilon}$ and $P\subseteq{\mathbf{G}}$,
$|P|\gtrsim|A|^{\delta}$ such that for all $x\in A^{\prime}$ one has
$r_{A-P}(x)\gg|P||A|^{-\varepsilon}$ or
$\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$ with $k\ll 1/\varepsilon$.

P r o o f. Let $\mathsf{E}_{l}=\mathsf{E}_{l}(A):=|A|^{l+\kappa_{l}}$, where
$\kappa_{l}\in[0,1]$ and $l\geqslant 2$ be any integer. We assume that
$\kappa_{l}\geqslant\delta$. By the pigeon–hole principle there is a number
$\Delta>0$ and a set $P\subseteq{\mathbf{G}}$ such that
$P=\\{x~{}:~{}\Delta<r_{A-A}(x)\leqslant 2\Delta\\}$ and
$\mathsf{E}_{l+1}\lesssim\Delta^{l+1}|P|$. In particular,

$|P|\gtrsim\mathsf{E}_{l+1}|A|^{-(l+1)}=|A|^{\kappa_{l}}\geqslant|A|^{\delta}\,.$
(10)

Let $M$ be a parameter, which we will choose later in an appropriate way in
each case $1)$ and $2)$. Suppose that

$\mathsf{E}_{l+1}\geqslant\frac{|A|\mathsf{E}_{l}}{M}\,.$ (11)

Then

$\frac{|A|\Delta^{l}|P|}{M}\leqslant\frac{|A|\mathsf{E}_{l}}{M}\leqslant\mathsf{E}_{l+1}\lesssim\Delta^{l+1}|P|\,.$
(12)

Hence

$\frac{|A||P|}{M}\lesssim\sum_{x\in P}r_{A-A}(x)=\sum_{a\in A}|A\cap(P+a)|\,,$
(13)

and the inequality $\Delta\gtrsim|A|/M$ follows from (12). Take
$A^{\prime}=\\{a\in A~{}:~{}|A\cap(P+a)|\gtrsim|P|/CM\\}$ with a sufficiently
large constant $C$. Then from (13), we get $|A^{\prime}|\gtrsim|A|/M$ and by
the definition of the set $A^{\prime}$, we have $r_{A-P}(x)\gtrsim|P|/M$ for
all $x\in A^{\prime}$. To obtain $2)$ we put $M=|A|^{\varepsilon/2}$ and we
are done under assumption (11). Now suppose that
$\mathsf{E}_{l+1}\leqslant|A|\mathsf{E}_{l}/M$. Then apply the previous
argument to $\mathsf{E}_{l}$ instead of $\mathsf{E}_{l+1}$. Again, if
$\mathsf{E}_{l}\geqslant|A|\mathsf{E}_{l-1}/M$, then we are done otherwise
$\mathsf{E}_{l}\leqslant|A|\mathsf{E}_{l-1}/M$ and we repeat the argument.
Clearly, after at most $k\ll 1/\varepsilon$ steps our algorithm finishes and
we obtain $\mathsf{E}_{k}(A)\leqslant|A|^{k+\delta}$.

It remains to obtain $1)$. Returning to (13) and using the Hölder inequality
several times, we get

$\frac{|A||P|^{2}}{M^{2}}\lesssim\mathsf{E}(A,P)=\sum_{x}r_{A-A}(x)r_{P-P}(x)\,,$

and hence

$\left(\frac{|A||P|^{2}}{M^{2}}\right)^{l+1}\lesssim\mathsf{E}_{l+1}\left(\sum_{x}r^{1+1/l}_{P-P}(x)\right)^{l}\lesssim|P|\Delta^{l+1}\cdot\mathsf{E}(P)|P|^{2l-2}\,.$

In other words,

$\mathsf{E}(P)\gtrsim\frac{|P|^{3}}{M^{2l+2}}:=\frac{|P|^{3}}{M_{*}}\,.$

By the Balog–Szemerédi–Gowers Theorem (e.g., see, [25]), we find $H\subseteq
P$, $|H|\gg|P|/M^{C}_{*}$, $|H+H|\ll M^{C}_{*}|H|$, where $C>0$ is an absolute
constant which may change from line to line. As above

$|A||H|/M\lesssim|H|\Delta\ll\sum_{x\in H}r_{A-A}(x)=\sum_{a\in
A}|A\cap(H+a)|\,.$

Again, we define $W\subseteq A$ similarly to the set $A^{\prime}$ and obtain
in particular, that $|W|\gtrsim|A|/M$. Also, let $Z\subseteq W$ be a set such
that the sets $\\{H+z\\}_{z\in Z}$ form the maximal system of disjoint sets.
By maximality, $W\subseteq Z+H-H$ and hence by the Plünnecke inequality (4),
we get

$|A|/M\lesssim|W|\leqslant|Z||H-H|\ll M^{C}_{*}|Z||H|$

and thus

$|(H\dotplus Z)\cap A|=\sum_{z\in
Z}|A\cap(H+z)|\gtrsim|Z||H|/M\gtrsim|A|/M^{C}_{*}\,.$

In particular,

$M|A|\gtrsim|Z||H|\,.$

We now put $M=|A|^{\frac{\varepsilon\kappa_{l}}{C_{*}l}}$, where $C_{*}>0$ is
a sufficiently large constant. Hence we get $|H+H|\ll|A|^{\varepsilon}|H|$ and
$|(H\dotplus Z)\cap A|\gg|A|^{1-\varepsilon}$. Also, in view of (10) one has

$|H|\gg|P|/M^{C}_{*}\gtrsim|A|^{\delta}/M^{C_{*}}\gtrsim|A|^{\delta(1-\varepsilon)}$

as required. Now suppose that inequality (11) fails, i.e.,
$\mathsf{E}_{l+1}\leqslant|A|\mathsf{E}_{l}/M$. It gives us
$\kappa_{l+1}\leqslant\kappa_{l}(1-\frac{\varepsilon}{C_{*}l})$ and thus
iterating, we see that our algorithm stops after at most
$\exp(O(\varepsilon^{-1}\log(1/\delta)))$ steps. This completes the proof.
$\hfill\Box$

###### Remark 5

Notice that Lemma 6, as well as the first part of Theorem 4 work for sums as
for differences and thus the first part of Theorem 3 can be obtained for the
quantity $\mathsf{Sid}^{*}_{k}$ defined via pluses but not differences.

A consequence of Theorem 4 is Corollary 6 below, which is a direct result
having no the sum–product flavour. Roughly speaking, it says that basic
estimate (1) can be easily improved for a wide class of sets, if we ignore the
presence of the parameter $K$ in formula (14).

###### Corollary 6

Let $A\subseteq{\mathbf{G}}$ be a set, $\delta\in(0,1/2)$ be a parameter. Then
either for some $K=K(\delta)$, $c=c(\delta)$ the following holds

$\mathsf{Sid}_{K}(A)\gg|A|^{1/2+c}$ (14)

or there is a set of shifts $T$, $|T|\gg|A|^{\delta/8}$ such that for any
$t\in T$ one has $|A\cap(A+t)|\gg|A|^{1-\delta}$.

P r o o f. We apply the first part of Theorem 4 with $\delta=\delta/2$ and
$\varepsilon=\delta/16$. If the first alternative holds, then we are done in
view of Lemma 6 because $\mathsf{E}^{+}_{k}(A)\leqslant|A|^{k+\delta}$ implies
(14). Otherwise there is $H\subseteq{\mathbf{G}}$,
$|H|\gtrsim|A|^{\delta(1-\varepsilon)/2}$, $|H+H|\ll|A|^{\varepsilon}|H|$ and
there exists $Z\subseteq{\mathbf{G}}$, $|Z||H|\ll|A|^{1+\varepsilon}$ with
$|(H\dotplus Z)\cap A|\gg|A|^{1-\varepsilon}\,.$ For any $z\in Z$ put
$A_{z}=A\cap(H+z)\subseteq A$. Applying the Cauchy–Schwarz inequality several
times, we have

$\sum_{x}r_{A-A}(x)r_{H-H}(x)=\mathsf{E}(A,H)\geqslant\sum_{z\in
Z}\mathsf{E}(A_{z},H)\gg\sum_{z\in
Z}\frac{|A_{z}|^{2}|H|^{2}}{|A|^{\varepsilon}|H|}\gg$
$\gg(|H||Z|)^{-1}|A|^{2-3\varepsilon}|H|^{2}\gg|A|^{1-4\varepsilon}|H|^{2}\,.$
(15)

Taking $T=\\{t\in{\mathbf{G}}~{}:~{}|A\cap(A+t)|\gg|A|^{1-4\varepsilon}\\}$,
we obtain from (15) that

$|T|\gg|H||A|^{-4\varepsilon}\gg|A|^{\delta/8}\,.$

This completes the proof. $\hfill\Box$

Now we are ready to obtain bound (2) of Theorem 3. We give even two proofs
using both statements of our driving Theorem 4.

P r o o f. Take any $\delta<1/2$, e.g., $\delta=1/4$ and let
$\varepsilon\leqslant\delta/4$ be a parameter, which we will choose later. In
view of Lemma 6 we see that $\mathsf{E}^{+}_{k}(A)\leqslant|A|^{k+\delta}$
implies

$\mathsf{Sid}^{+}_{3k-3}(A)\gg|A|^{\frac{1}{2}+\frac{1-2\delta}{2(2k-1)}}=|A|^{\frac{1}{2}+\frac{1}{4(2k-1)}}$
(16)

and we are done. Here $k=k(\varepsilon)$. Otherwise there is
$H\subseteq\mathbb{F}$,
$|H|\gtrsim|A|^{\delta(1-\varepsilon)}\geqslant|A|^{\delta/2}$,
$|H+H|\ll|A|^{\varepsilon}|H|$ and there exists $Z\subseteq\mathbb{F}$,
$|Z||H|\ll|A|^{1+\varepsilon}$ with $|(H\dotplus Z)\cap
A|\gg|A|^{1-\varepsilon}\,.$ Put $A_{*}=(H\dotplus Z)\cap A$,
$|A_{*}|\gg|A|^{1-\varepsilon}$ and we want to estimate
$\mathsf{E}^{\times}_{l+1}(A_{*})$ or $\hat{\mathsf{E}}_{l+1}^{\times}(A_{*})$
for large $l$. After that having a good upper bound for
$\mathsf{E}^{\times}_{l+1}(A_{*})$ or
$\hat{\mathsf{E}}_{l+1}^{\times}(A_{*})$, we apply Lemma 6 again to find large
multiplicative Sidon subset of $A_{*}$.

First of all, notice that
$|A_{*}+H|\leqslant|H+H||Z|\ll|A|^{\varepsilon}|H||Z|\ll|A|^{1+2\varepsilon}$.
In other words, the set $A_{*}$ almost does not grow after the addition with
$H$. Let $Q=A_{*}+H$, $|Q|\ll|A|^{1+2\varepsilon}$. Secondly, fix any
$\lambda\neq 0$. The number of the solutions to the equation
$a_{1}a_{2}=\lambda$, where $a_{1},a_{2}\in A_{*}$ does not exceed

$\sigma_{\lambda}:=|H|^{-2}|\\{h_{1},h_{2}\in H,\,q_{1},q_{2}\in
Q~{}:~{}(h_{1}-q_{1})(h_{2}-q_{2})=\lambda\\}|\,.$

The last equation can be interpreted as a question about the number of
incidences between points and modular hyperbolas (see [23]) and for each
non–zero $\lambda$ the quantity $\sigma_{\lambda}$ can be estimated as

$\sigma_{\lambda}\ll|H|^{-2}\cdot|Q||H|^{2-\kappa}\ll|A|^{1+2\varepsilon}|H|^{-\kappa}\,,$

see [14] in the case $\mathbb{F}={\mathbb{R}}$ and [23, Theorem 22] in the
case of the prime field. Here $\kappa=\kappa(\delta)>0$. Recalling that
$|H|\gg|A|^{\delta/2}$, $|A_{*}|\gg|A|^{1-\varepsilon}$ and taking any
$\varepsilon\leqslant\delta\kappa/100$, we obtain after some calculations that
$\sigma_{\lambda}\ll|A_{*}|^{1-\delta\kappa/4}$. Hence taking sufficiently
large $l\gg(\delta\kappa)^{-1}$, we derive

$\hat{\mathsf{E}}_{l+1}^{\times}(A_{*})=\sum_{\lambda}r^{l+1}_{A_{*}A_{*}}(\lambda)\ll|A_{*}|^{l+1}+(|A_{*}|^{1-\delta\kappa/2})^{l}|A_{*}|^{2}\ll|A_{*}|^{l+1}+|A|^{l+2-\delta\kappa
l/2}\ll|A_{*}|^{l+1}\,.$

Applying Lemma 6 and choosing $\varepsilon\ll l^{-1}$, we see that

$\mathsf{Sid}^{\times}_{2l}(A)\geqslant\mathsf{Sid}^{\times}_{2l}(A_{*})\gg|A_{*}|^{\frac{l+1}{2l+1}}\gg|A|^{\frac{(1-\varepsilon)(l+1)}{2l+1}}=|A|^{\frac{1}{2}+\frac{1-2\varepsilon(l+1)}{2(2l+1)}}\gg|A|^{\frac{1}{2}+c}\,,$

where $c=c(\delta)>0$ is an absolute constant.

Now let us give the second proof applying the last part of Theorem 4. The
argument is almost the same but we estimate
$\hat{\mathsf{E}}^{\times}_{l+1}(A^{\prime})$. Again, the number of the
solutions to the equation $a_{1}a_{2}=\lambda$, where $a_{1},a_{2}\in
A^{\prime}$ does not exceed

$\sigma_{\lambda}:=(|P||A|^{-\varepsilon})^{-2}\cdot|\\{a_{1},a_{2}\in
A,\,p_{1},p_{2}\in P~{}:~{}(a_{1}-p_{1})(a_{2}-p_{2})=\lambda\\}|\,.$

By results from [14], [23] one has

$\sigma_{\lambda}\ll(|P||A|^{-\varepsilon})^{-2}\cdot|A||P|^{2-\kappa}\ll|A|^{1+2\varepsilon}|P|^{-\kappa}\,,$

where $\kappa=\kappa(\delta)>0$. Again $|P|\gtrsim|A|^{\delta}$ and we can use
the arguments as above. This concludes the proof. $\hfill\Box$

To complete the proof of Theorem 3 we need a simple lemma on upper estimates
of sizes of Sidon sets in sumsets.

###### Lemma 7

Let $A\subseteq{\mathbf{G}}$ be a set, $A=B+C$, and $k\geqslant 1$ be an
integer. Then

$\mathsf{Sid}_{k}(A)\leqslant\min\\{|C|\sqrt{k|B|}+|B|,|B|\sqrt{k|C|}+|C|\\}\,.$

More generally, if $r_{B+C}(a)\geqslant\sigma$ for any $a\in A$, then

$\mathsf{Sid}_{k}(A)\leqslant\sigma^{-1}\min\\{|C|\sqrt{k|B|}+|B|,|B|\sqrt{k|C|}+|C|\\}\,.$

P r o o f. We give two proofs. The first proof uses theory of graphs and it
demonstrates transparently how $\mathsf{Sid}_{k}$–sets are naturally connected
with Cayley graphs.

Let $\Lambda$ be the maximal subset of $A$ such that
$r_{\Lambda-\Lambda}(x)\leqslant k$ for any $x\neq 0$. Consider the graph
$G=G(V,E)$, where $V$ is the disjoint union of $B$ and $C$, the edge $(b,c)\in
E$ iff $b+c\in\Lambda$. Moreover, we assume that $|E|=|\Lambda|$ ignoring
elements of $\Lambda$, which have several representations as $b+c$. Using the
Cauchy–Schwartz inequality, we obtain

$|\Lambda|^{2}\leqslant|B|\sum_{b\in B}\sum_{c,c^{\prime}\in
C}E(b,c)E(b,c^{\prime})=$ $=|B|\sum_{c,c^{\prime}\in C}\,\sum_{b\in
B}E(b,c)E(b,c^{\prime})=|B||\Lambda|+|B|\sum_{c,c^{\prime}\in C,\,c\neq
c^{\prime}}\,\sum_{b\in B}E(b,c)E(b,c^{\prime})\,.$

If the last sum over $b$ is at least $k+1$, then we find a complete bipartite
subgraph $K_{2,k+1}$ in $G$ and hence there are different elements
$\lambda_{1},\lambda^{\prime}_{1},\dots,\lambda_{k+1},\lambda^{\prime}_{k+1}\in\Lambda$
such that
$\lambda_{1}-\lambda^{\prime}_{1}=\dots=\lambda_{k+1}-\lambda^{\prime}_{k+1}$.
The last fact contradicts the assumption that $r_{\Lambda-\Lambda}(x)\leqslant
k$ for any $x\neq 0$. Hence

$|\Lambda|^{2}\leqslant|B|\leqslant|B||\Lambda|+k|B||C|^{2}$

as required.

To obtain the second part of our lemma we use a little bit different method.
Again, let $\Lambda$ be the set as before. Then by the Cauchy–Schwarz
inequality, we get

$(\sigma|\Lambda|)^{2}\leqslant\mathcal{S}^{2}:=\left(\sum_{x\in\Lambda}r_{B+C}(x)\right)^{2}=\left(\sum_{b\in
B}r_{\Lambda-C}(b)\right)^{2}\leqslant|B|\sum_{x,y\in\Lambda}|B\cap(x-C)\cap(y-C)|\leqslant$
$\leqslant|B|\mathcal{S}+|B|\sum_{x\neq
y\in\Lambda}|(x-C)\cap(y-C)|\leqslant|B|\mathcal{S}+k|B||C|^{2}\,.$

This completes the proof. $\hfill\Box$

Now we can easily obtain a non–trivial upper bound for size of maximal Sidon
set in any difference set or sumset.

###### Corollary 8

Let $A\subseteq{\mathbf{G}}$ be a set and $D=A-A$, $S=A+A$. Then for any
positive integer $k$ one has
$\mathsf{Sid}_{k}(D)\ll\sqrt{k}\min\\{|A|^{3/2},\sqrt{|D|^{3}}|A|^{-1}\\}$ and

$\mathsf{Sid}_{k}(S)\ll\sqrt{k}\min\\{|A|^{3/2},|A|^{-1}\min\\{|D|\sqrt{|S|}+|S|,|S|\sqrt{|D|}+|D|\\}\\}\,.$

P r o o f. The bound
$\mathsf{Sid}_{k}(D),\mathsf{Sid}_{k}(S)\ll\sqrt{k}|A|^{3/2}$ follows
immediately from the first part of Lemma 7. Further it is easy to see (or
consult [20]) that for any $d\in D$ one has $r_{D-D}(d)\geqslant|A|$. Applying
the second part of Lemma 7 with $A=B=C=D$ and $\sigma=|A|$, we obtain
$\mathsf{Sid}_{k}(D)\ll\sqrt{k|D|^{3}}|A|^{-1}$. To prove the second part of
our lemma notice that for any $s\in S$ the following holds
$r_{D+S}(s)\geqslant|A|$. This concludes the proof. $\hfill\Box$

To complete the proof of Theorem 3 we need to obtain upper bound (3). In the
case $\mathbb{F}={\mathbb{R}}$ we put $B=\Gamma$, $C=H\Gamma$, where
$\Gamma=\\{1,g,\dots,g^{n}\\}$, $g\geqslant 2$ is an integer,
$\bar{\Gamma}=\\{g^{-n},\dots,g^{-1},1,g,\dots,g^{n}\\}$,
$H=\\{g^{n+1},g^{2(n+1)},\dots,g^{n(n+1)}\\}$. Then $A=B+C=\Gamma+H\Gamma$ is
contained in $\Gamma(1+H\bar{\Gamma})$ and in view of Lemma 7 any
additive/multiplicative $k$–Sidon subset of $A$ has size at most
$O_{k}(|\Gamma|^{2})=O_{k}(|A|^{2/3})$ because as one can easily see
$|A|=|\Gamma|^{3}$. Similarly, in the case $\mathbb{F}=\mathbb{F}_{p}$ we
apply Lemma 7 with $B=\Gamma$, $C=H\Gamma$, where
$\Gamma\leqslant\mathbb{F}_{p}^{*}$, $H\subseteq\mathbb{F}_{p}^{*}/\Gamma$ and
$|H|=|\Gamma|$ is sufficiently small relatively to $p$. Then
$A:=B+C=\Gamma(1+\Gamma H)$ and hence by Lemma 7 any additive/multiplicative
$k$–Sidon subset of $A$ has size at most $2\sqrt{k}|\Gamma|^{2}$. To obtain
the required bound $|A|\gg|\Gamma+H\Gamma|$ for an appropriate $H$ one can use
the random choice (we leave the details to the interested reader).

##  4 On $B^{\circ}_{2}[g]$–sets

In the previous section we have obtained some results about the family of sets
$S$ with

$r_{S-S}(x)\leqslant g\,,\quad\quad\forall x\in{\mathbf{G}}\,,\quad x\neq
0\,.$ (17)

This class of sets were introduced by Erdős in [6, page 57] (also, see [7])
and we denote this family as $B^{\circ}_{2}[g]$ (Erdős used the symbol
$B^{{}^{\prime}(k)}_{2}$). It was said in [6] that ”V.T. Sós and I considered
$B^{{}^{\prime}(k)}_{2}$ sequences… We could not decide whether there is a
$B^{{}^{\prime}(k)}_{2}$ sequence which is not the union of a finite number of
Sidon sequences.” According to the author’s knowledge this paper of Erdős and
Sós was not published. The question from [6] is a nice problem of Erdős, which
is open and if it has a negative answer, then the original question of
Klurman–Pohoata would be closed thanks to our Theorem 3. Let us underline it
one more time that it is possible to construct sets $S$ with bounded
$r_{S+S}(x)$ which are not the union of a finite number of Sidon sequences
(see [6] and [1]). It seems like condition (17) has another nature and that is
why we devote this section studying some further properties of
$B^{\circ}_{2}[g]$–sets. To see that this is a special family, notice that,
for example, the condition $r_{S-S}(x)\ll 1$, $x\neq 0$ has an interpretation
in terms of the Cayley graph of $S$ but $r_{S+S}(x)\ll 1$ cannot be expressed
in terms of any Cayley graph.

First of all, notice that if $S$ is a random subset of $[N]$, which was
obtained piking elements from $[N]$ independently at random with probability
$q\sim N^{-1/2}$, then with probability $1-o(1)$ one has $r_{S-S}(x)\ll\log N$
see, e.g., [5, Lemma 4.3] and a similar lower bound for the function
$r_{S-S}(x)$ takes place. Thus for any fixed $g$ sets belonging to
$B^{\circ}_{2}[g]$ are far from random sets.

Secondly, as it was noted in the proof of Lemma 7 a set $S$ belongs to the
family $B^{\circ}_{2}[g]$ iff its Cayley graph $\mathsf{Cay}(S,{\mathbf{G}})$
has no complete bipartite subgraphs $K_{2,g+1}$. Recall that the vertex set of
$\mathsf{Cay}(A,{\mathbf{G}})$ is ${\mathbf{G}}$ and $(x,y)$ is an edge of
$\mathsf{Cay}(A,{\mathbf{G}})$ iff $x-y\in A$. Another equivalent
interpretation of (17) is

$|S\cap(S+x_{1})\cap\dots\cap(S+x_{g})|\leqslant 1\quad\quad\mbox{for all
distincts and non--zero }\quad x_{1},\dots,x_{g}\in{\mathbf{G}}\,.$ (18)

This reinterpretation of $B^{\circ}_{2}[g]$–sets says that the considered
family is naturally connected with the higher energies [20] (also, see the
second formula in definition (5)). Also, if we put for an arbitrary
$A\subseteq{\mathbf{G}}$

$\Delta_{g}(A):=\\{(a,\dots,a)\in A^{g}~{}:~{}a\in
A\\}\subseteq{\mathbf{G}}^{g}\,,$

then (18) means that the sum $S^{g}$ and $\Delta_{g}(S)$ is direct, in other
words, $S^{g}$ and $\Delta_{g}(S)$ form a co–Sidon pair, see [5]. Further
formula (18) suggests us the following definition for $B^{\circ}_{k}[g]$ sets

$|S\cap(S+x_{1})\cap\dots\cap(S+x_{g})|<k\quad\quad\mbox{for all distincts and
non--zero }\quad x_{1},\dots,x_{g}\in{\mathbf{G}}\,.$ (19)

Similarly, $S\in B^{\circ}_{k}[g]$ iff $\mathsf{Cay}(S,{\mathbf{G}})$ does not
contain $K_{k,g+1}$ or, in other words, $S$ does not contain any sumsets
$X+Y$, where $|X|=g+1$, $Y=k$. Since the number of edges in
$\mathsf{Cay}(S,{\mathbf{G}})$ equals $|S||{\mathbf{G}}|$ for any finite group
${\mathbf{G}}$, it follows that to estimate size of $S$ it is enough to bound
the number of edges in $K_{k,g+1}$–free graphs. Such results are discussed in
detail in survey [8], for example. Of course there is a direct approach to
estimate cardinality of $S\in B^{\circ}_{2}[g]$. Namely, if $S\in
B^{\circ}_{2}[g]$ belongs to a group ${\mathbf{G}}$ of size $N$, then,
obviously,

$|S|^{2}=\sum_{x}r_{S-S}(x)\leqslant|S|+g(N-1)$

and hence

$|S|<\sqrt{gN}+1\,.$ (20)

Similarly, if $S\in B^{\circ}_{k}[g]$ and $S$ belongs to a group
${\mathbf{G}}$ of size $N$, then

$|S|^{g+1}=\sum_{x_{1},\dots,x_{g}}|S\cap(S+x_{1})\cap\dots\cap(S+x_{g})|<kN^{g}+\binom{g+1}{2}|S|^{g}$

and thus $|S|<k^{\frac{1}{g+1}}N^{\frac{g}{g+1}}+\binom{g+1}{2}$. Of course in
the case when $S\subseteq[N]$ there are no such good upper bounds for size of
$S$ even if $S$ is a classical Sidon set. Nevertheless, we easily obtain a
generalization of Linstrom’s result [13] for $B^{\circ}_{2}[g]$–sets in the
segment (as well as for $B^{\circ}_{k}[g]$–sets but it is not the main topic
of our paper, for better bounds see [16]).

###### Proposition 9

Let $S\subseteq[N]$ belongs to the family $B^{\circ}_{2}[g]$. Then

$|S|<\sqrt{gN}+(gN)^{1/4}+1\,.$ (21)

Generally, if $S\in B^{\circ}_{k}[g]$, then

$|S|<k^{\frac{1}{g+1}}N^{\frac{g}{g+1}}+\binom{g+1}{2}^{\frac{1}{g+1}}k^{\frac{g}{(g+1)^{2}}}N^{\frac{g^{2}}{(g+1)^{2}}}+1\,.$
(22)

P r o o f. We use the method from [9]. Let $u=[N^{3/4}g^{-1/4}]$ be a
parameter and $I=[u]$. Embed $S$ into $\mathbb{Z}/(N+u)\mathbb{Z}$. It is easy
to check that for any $x\in[-u,u]\setminus\\{0\\}$ one has
$r_{S-S}(x)\leqslant g$, where now $x$ runs over $\mathbb{Z}/(N+u)\mathbb{Z}$.
Hence using the Cauchy–Schwarz inequality to estimate the common energy
$\mathsf{E}(S,I)$, we get

$\frac{|S|^{2}u^{2}}{N+u}\leqslant\mathsf{E}(S,I)=\sum_{x}r_{S-S}(x)r_{I-I}(x)<|S|u+gu^{2}$

or, in other words,

$gu^{2}+u(gN+|S|-|S|^{2})+|S|N>0$

and substituting $u=[N^{3/4}g^{-1/4}]<N^{3/4}g^{-1/4}$ and
$|S|=\sqrt{gN}+(gN)^{1/4}+C$, we find after some calculations that the
condition $C\leqslant 1$ is enough. To obtain (22) we use a similar argument
(with another parameter $u$, of course) to estimate an analogue of the higher
common energy of $S$ and $I$, namely,

$\frac{(|S|u)^{g+1}}{(N+u)^{g}}\leqslant\sum_{x}r_{S-I}^{g+1}(x)=\sum_{x_{1},\dots,x_{g}}|S\cap(S+x_{1})\cap\dots\cap(S+x_{g})||I\cap(I+x_{1})\cap\dots\cap(I+x_{g})|<$
(23) $<ku^{g+1}+\binom{g+1}{2}\sum_{x}r_{S-I}^{g}(x)\leqslant
ku^{g+1}+\binom{g+1}{2}|S|^{g}u:=ku^{g+1}+C_{g}|S|^{g}u\,.$

We have used in (23) that all variables $x_{j}$ belong to $[-u,u]$. Finally,
in the case of $B^{\circ}_{k}[g]$–sets an appropriate choice of the parameter
$u$ is $u=[C^{1/(g+1)}_{g}k^{-1/(g+1)^{2}}N^{1-g/(g+1)^{2}}]$ and after some
calculations we obtain (22) (we roughly estimate the third term in this
formula by one and actually, we do not optimize the constant in the middle
term of (22), taking it for the simplest way to check). This completes the
proof. $\hfill\Box$

As for lower bounds on size of maximal subsets of $B^{\circ}_{k}[g]$, again
one can consult [8] to find correspondent lower bound for the number of edges
in graphs having no $K_{s,t}$. Our graphs must be Cayley graphs and such
constructions are known for $K_{2,2}$, $K_{3,3}$ (Brown’s construction, see
[2]) and for $K_{s,t}$, $t\geqslant s!+1$ (so–called, norm–graphs), see [8].
As for $K_{2,t}$, $t>2$ one can obtain a result similar to [4, Theorem 1.6].
Namely, define for any $g$ the quantity

$\alpha_{g}=\limsup_{N\to\infty}\frac{\max\\{|S|~{}:~{}S\subseteq\mathbb{Z}/N\mathbb{Z}\,,S\in
B^{\circ}_{2}[g]\\}}{\sqrt{N}}\,.$

###### Theorem 10

We have

$\alpha_{g}=\sqrt{g}+O(g^{3/10})\,.$

P r o o f. Actually, our argument almost coincides with the approach of the
proof [4, Theorem 1.6], so we omit the details.

By the method of [4, section 4] it is enough to construct a set
$A\subseteq(\mathbb{Z}/p\mathbb{Z})^{2}$ (here $p$ is a prime number) such
that $A\in B^{\circ}_{2}[g]$ with $g=k^{2}+O(k^{3/2})$ and $|A|=kp+O(k)$. We
put $A=\bigsqcup_{u\in U}A_{u}$, where $U$ is an appropriate arithmetic
progression, $U=t+[k]$ and for any non–zero $u\in\mathbb{Z}/p\mathbb{Z}$ we
define

$A_{u}=\\{(x,x^{2}/u)~{}:~{}x\in\mathbb{Z}/p\mathbb{Z}\\}\subset(\mathbb{Z}/p\mathbb{Z})^{2}\,.$

Clearly, $|A|=kp-k+1$. Let $r_{u,v}(x)=r_{A_{u}-A_{v}}(x)$. Our task is to
estimate for any $x$ the sum

$r_{A-A}(x)\leqslant\sum_{u,v\in U}r_{u,v}(x)\,.$

By [4, Lemma 3.2] one has $r_{u,v}(x)+r_{u^{\prime},v^{\prime}}(x)=2$,
provided $u+v=u^{\prime}+v^{\prime}$ and
$\left(\frac{uvu^{\prime}v^{\prime}}{p}\right)=-1$. Using this lemma and
acting exactly as on pages 2794–2795 of [4], we find

$r_{A-A}(x)\leqslant
k^{2}+\sum_{|l|<k}\left|\sum_{i+j=k+1+l}\left(\frac{(t+i)(t+j)}{p}\right)\right|\,.$

After that taking the summation over $t$ (to find an appropriate $U$) and
applying the Cauchy–Schwarz inequality and Weil’s bound on sum of Legendre
symbols, we obtain the required estimate (see the rest of the argument from
[4]). This completes the proof. $\hfill\Box$

Finally, let us mention that the construction of Linström [12] works for
$B^{\circ}_{2}[g]$, as well as for usual Sidon sets. Namely, having a Sidon
set $A$ in $[(N-g)/g]$ one can see that the set $S:=g\cdot
A+\\{0,1,\dots,g-1\\}\subseteq[N]$ belongs to the family $B^{\circ}_{2}[g]$
and hence it gives us another construction of sets $S\in B^{\circ}_{2}[g]$,
$S\subseteq[N]$ of size $\sqrt{gN}(1+o(1))$.

We continue this section considering heritability properties of
$B^{\circ}_{2}[g]$–sets. For different $x_{1}\dots,x_{s}$ and any set
$A\subseteq{\mathbf{G}}$ put $A_{X}:=(A+x_{1})\cap\dots\cap(A+x_{s})$, where
$X=\\{x_{1},\dots,x_{s}\\}$. Taking the same set $X_{1}=\dots=X_{l}=X$ in
inequality (24) below, one can see that any $B^{\circ}_{k}[g]$–set generates
$B^{\circ}_{k}[g_{1}]$–sets with a smaller $g_{1}$. In particular, in Brown’s
construction [2] of the set $S\in B^{\circ}_{2}[2]$, this set $S$ is a (almost
disjoint) union of classical Sidon sets $S_{w}=S\cap(S+w)$,
$w\in(S-S)\setminus\\{0\\}$.

###### Proposition 11

Let $S\in B^{\circ}_{k}[g]$, $l\geqslant 2$ and take any sets
$X_{1},\dots,X_{l}$ with $|X_{1}|+\dots+|X_{l}|\geqslant g+\binom{l}{2}+1$ and
such that any $(X_{i},X_{j})$ forms a co–Sidon pair. Then

$|S_{X_{1}}\cap(S_{X_{2}}+z_{1})\cap\dots\cap(S_{X_{l}}+z_{l-1})|<k\quad\quad\mbox{
for any different }\quad\quad z_{1},\dots,z_{l-1}\neq 0\,.$ (24)

In particular, if $S\in B^{\circ}_{2}[2]$ and ${\mathbf{G}}$ has no elements
of order two, then for any non–zero $w$ the set $S_{w}=S\cap(S+w)$ is a Sidon
set.

P r o o f. Let $X_{j}=\\{x^{(j)}_{1},\dots,x^{(j)}_{s_{j}}\\}$, $j\in[l]$. For
an arbitrary $z\in{\mathbf{G}}$, we have

$\mathcal{S}:=S_{X_{1}}\cap(S_{X_{2}}+z_{1})\cap\dots\cap(S_{X_{l}}+z_{l-1})=$
$(S+x^{(1)}_{1})\cap\dots\cap(S+x^{(1)}_{s_{1}})\cap(S+x^{(2)}_{1}+z_{1})\cap\dots\cap(S+x^{(2)}_{1}+z_{1})\cap\dots\cap(S+x^{(l)}_{1}+z_{l-1})\cap\dots\cap(S+x^{(l)}_{1}+z_{l-1})\,.$

Since $\sum_{j=1}^{l}s_{j}\geqslant g+\binom{l}{2}+1$, then either
$|\mathcal{S}|<k$ by (18), (19) or we have for some indices
$x^{(i^{\prime})}_{i}+z_{i}=x^{(j^{\prime})}_{j}+z_{j}$ and
$x^{(i^{\prime\prime})}_{i}+z_{i}=x^{(j^{\prime\prime})}_{j}+z_{j}$. The last
alternative implies that
$x^{(i^{\prime})}_{i}-x^{(j^{\prime})}_{j}=x^{(i^{\prime\prime})}_{i}-x^{(j^{\prime\prime})}_{j}$
but $X_{i}$ and $X_{j}$ form a co–Sidon pair by the assumption.

In the case $S_{w}=S\cap(S+w)$ our ground set $X$ is $\\{0,w\\}$ and it is
easy to see that this is a Sidon set. This completes the proof. $\hfill\Box$

In Proposition 11 we assume that each $(X_{i},X_{j})$ forms a co–Sidon pair.
Again (and this is in the spirit of this section) one can make more general
assumptions to the intersections of some shifts of the sets $X_{j}$ as in
(18), (19) to obtain higher order Sidon sets.

We finish this section discussing the tightness of Lemma 6. In the next
proposition we show that any set $A$ contains rather large (in terms of its
energy) subset with controllable size of the maximal $B^{\circ}_{k}[g]$
subset.

###### Proposition 12

Let $A\subseteq{\mathbf{G}}$ be a set. Then there is $A_{*}\subseteq A$ such
that $\mathsf{E}_{g+1}(A_{*})\gg_{g}\mathsf{E}_{g+1}(A)$ and the maximal
$B^{\circ}_{k}[g]$ subset of $A_{*}$ has size
$O_{g}(k^{1/(g+1)}|A|^{2}\mathsf{E}^{-1/(g+1)}_{g+1}(A))$.
In particular, for any $n$ one has $\mathsf{Sid}_{n}(A_{*})\ll
n^{1/2}|A|^{2}\mathsf{E}^{-1/2}(A)$.

P r o o f. We have

$\mathsf{E}_{g+1}(A)=\sum_{y}r^{g+1}_{A-A}(y)=\sum_{a\in A}\sum_{x\in
A}r^{g}_{A-A}(x-a)\,.$

Put $A_{*}=\\{a\in A~{}:~{}\sum_{x\in
A}r^{g}_{A-A}(x-a)\geqslant\mathsf{E}_{g+1}(A)/(2|A|)\\}$. Then

$\mathsf{E}_{g+1}(A)\leqslant 2\sum_{a\in A_{*}}\sum_{x\in
A}r^{g}_{A-A}(x-a)=2\sum_{y}r^{g}_{A-A}(y)r_{A-A_{*}}(y)$

and using the Hölder inequality several times, we obtain

$\mathsf{E}_{g+1}(A)\leqslant 2^{g+1}\sum_{y}r^{g+1}_{A-A_{*}}(y)\leqslant
2^{g+1}(\mathsf{E}_{g+1}(A_{*})\mathsf{E}^{2g+1}_{g+1}(A))^{1/(2g+2)}$ (25)

or, in other words, $\mathsf{E}_{g+1}(A_{*})\gg_{g}\mathsf{E}_{g+1}(A)$. Now
let $\Lambda$ be the maximal $B^{\circ}_{k}[g]$ subset of $A_{*}$. Then by the
definition of the set $A_{*}$, one has

$2^{-1}|\Lambda|\mathsf{E}_{g+1}(A)|A|^{-1}\leqslant\sum_{y}r^{g}_{A-A}(y)r_{A-\Lambda}(y)\,.$
(26)

Hence as in (25)

$2^{-(g+1)}|\Lambda|^{g+1}\mathsf{E}_{g+1}(A)|A|^{-(g+1)}\leqslant\sum_{y}r^{g+1}_{A-\Lambda}(y)=$
$=\sum_{x_{1},\dots,x_{g}}|A\cap(A+x_{1})\cap\dots\cap(A+x_{g})||\Lambda\cap(\Lambda+x_{1})\cap\dots\cap(\Lambda+x_{g})|\,.$

Applying the fact that $\Lambda\in B^{\circ}_{k}[g]$, we get

$2^{-(g+1)}|\Lambda|^{g+1}\mathsf{E}_{g+1}(A)|A|^{-(g+1)}\leqslant
k|A|^{g+1}+\binom{g+1}{2}|A||\Lambda|^{g}\,.$

Noting that the second term in the last formula is negligible, we obtain the
required result. $\hfill\Box$

###### Remark 13

From (26) it follows that the conclusion of Proposition 12 remains true for a
wider family of sets $\Lambda$, namely, for $\Lambda$ with
$\mathsf{E}_{g+1}(\Lambda)\ll|\Lambda|^{g+1}$. This class of sets and its
connection with Sidon sets was discussed in [18, Section 3.2].

## References

  * [1] N. Alon, P. Erdős,  An application of graph theory to additive number theory,  European Journal of Combinatorics 6.3 (1985): 201–203.
  * [2] W.G. Brown,  On graphs that do not contain a Thomsen graph,  Canadian Mathematical Bulletin 9.3 (1966): 281–285.
  * [3] J. Cilleruelo,  Gaps in dense Sidon sets,  Integers (2000): Paper-A11.
  * [4] J. Cilleruelo, I.Z. Ruzsa, C. Vinuesa,  Generalized Sidon sets,  Advances in Mathematics 225.5 (2010): 2786–2807.
  * [5] A. Dubickas, T. Schoen, M. Silva, P. Sarka,  Finding large co-Sidon subsets in sets with a given additive energy,  European Journal of Combinatorics 34.7 (2013): 1144–1157.
  * [6] P. Erdős,  Extremal problems in number theory, combinatorics and geometry,  Proc. Inter. Congress in Warsaw. 1983.
  * [7] P. Erdős, E. Harzheim,  Congruent subsets of infinite sets of natural numbers,  J. Reine Angew. Math., 367 (1986) 207–214.
  * [8] Z. Füredi, M. Simonovits,  The history of degenerate (bipartite) extremal graph problems,  Erdős Centennial. Springer, Berlin, Heidelberg, 2013. 169–264.
  * [9] B. Green,  The number of squares and $B_{h}[g]$ sets,  Acta Arithmetica 100.4 (2001): 365–390.
  * [10] https://pohoatza.wordpress.com/2021/01/23/sidon-sets-and-sum-product-phenomena/
  * [11] J. Komlós, M. Sulyok, E. Szemerédi,  Linear problems in combinatorial number theory,  Acta Mathematica Academiae Scientiarum Hungarica 26.1–2 (1975): 113–121.
  * [12] B. Linström,  $B_{h}[g]$–sequences from $B_{h}$–sequences,  Proc. Amer. Math. Soc. 128(2000), 657–659.
  * [13] B. Linström,  An inequality for $B_{2}$–sequences,  Journal of Combinatorial Theory 6.2 (1969): 211–212.
  * [14] B. Murphy,  Upper and lower bounds for rich lines in grids,  arXiv:1709.10438v1 [math.CO] 29 Sep 2017.
  * [15] M.B. Nathanson,  Every function is the representation function of an additive basis for the integers,  Portugaliae Mathematica. Nova Série 62.1 (2005): 55–72.
  * [16] X. Peng, R. Tesoro, C. Timmons,  Bounds for generalized Sidon sets,  Discrete Mathematics 338.3 (2015): 183–190.
  * [17] K. O’Bryant,  A complete annotated bibliography of work related to Sidon sequences,  arXiv preprint math/0407117 (2004).
  * [18] O. Roche–Newton, A. Warren,  Additive and multiplicative Sidon sets,  arxiv.org/abs/2103.13066
  * [19] I.Z. Ruzsa,  An infinite Sidon sequence,  Journal of Number Theory 68.1 (1998): 63–71.
  * [20] T. Schoen, I. D. Shkredov,  Higher moments of convolutions,  J. Number Theory 133:5 (2013), 1693–1737.
  * [21] A. S. Semchankau,  Maximal Subsets Free of Arithmetic Progressions in Arbitrary Sets,  Math. Notes, 102:3 (2017), 396–402.
  * [22] I. D. Shkredov,  Energies and structure of additive sets,  Electronic Journal of Combinatorics, 21:3 (2014), #P3.44, 1–53.
  * [23] I. D. Shkredov,  Modular hyperbolas and bilinear forms of Kloosterman sums,  J. Number Theory, 220 (2021), 182–211.
  * [24] S. Sidon,  Ein Satz über trigonomietrische Polynome und seine Anwendungen in der Theorie der Fourier–Reihen,  Math. Annalen 106 (1932), 536–539.
  * [25] T. Tao, V. Vu, Additive combinatorics,  Cambridge University Press 2006.

I.D. Shkredov
Steklov Mathematical Institute,
ul. Gubkina, 8, Moscow, Russia, 119991
and
IITP RAS,
Bolshoy Karetny per. 19, Moscow, Russia, 127994
and
MIPT,
Institutskii per. 9, Dolgoprudnii, Russia, 141701
<EMAIL_ADDRESS>