# Hawking radiation and stability of the canonical acoustic black holes

M. A. Anacleto<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  F. A. Brito<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina
Grande, Paraíba, Brazil  Departamento de Física, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil  E. Passos
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Campina
Grande Caixa Postal 10071, 58429-900 Campina Grande, Paraíba, Brazil

###### Abstract

In this paper we determine the Hawking temperature and entropy of a modified
canonical acoustic black hole. In our results we obtain an entropy-area
relation that has a logarithmic correction term in leading order. We also
analyze the stability condition of the noncommutative canonical acoustic black
hole and also with quantum corrections implemented by the generalized
Heisenberg uncertainty principle.

##  I Introduction

Black holes are fascinating objects predicted by General Relativity and have
attracted enormous interest in high-energy physics, astrophysics and
astronomy. In addition, in the physics of black holes some phenomena have been
extensively analyzed such as Hawking radiation [1, 2], superradiance and
quasinormal modes. The thermodynamics of black holes with quantum corrections
implemented by the generalized uncertainty principle was analyzed in [3, 4, 5,
6] and Hawking radiation of black holes in arbitrary dimensions with or
without Gauss-Bonnet term have been investigated in [7, 8, 9]. It is well
known that when a black hole interacts with the surrounding matter the
disturbances cause the black hole to suffer oscillation, leading to the
emission of gravitational waves [10, 11, 12, 13]. Such gravitational waves
that are ripples in the spacetime fabric were detected on September 14, 2015
by the collaboration LIGO-VIRGO [14, 15]. Also, in 2019 the image of a
supermassive black hole at the center of galaxy M87 was captured by the Event
Horizon Telescope [16, 17]. Therefore, these observations usher in a new era
in the development of black hole physics.

In order to examine Hawking radiation and other phenomena to understand
quantum gravity, acoustic black holes were proposed for this purpose by Unruh
in 1981 [18, 19]. Since then, great advances have been achieved in the
investigation of Hawking radiation in the theoretical field as well as in the
experimental field. As we know, an acoustic black hole can be built when the
fluid motion becomes greater than the local speed of sound. Therefore,
acoustic event horizons are formed by showing similar characteristics to the
gravitational black hole event horizon. Moreover, these acoustic black holes,
as well as analogous models, are quite useful in various branches of physics
with applications, for example, in high energy physics, condensed matter and
quantum physics [20, 21]. Experimentally, the Hawking temperature of acoustic
black holes has been successfully measured [22, 23]. This has also been done
in other physical systems [24, 25, 26, 27, 28, 29]. The first experimental
study of Hawking radiation from acoustic black holes was carried out in the
Bose-Einstein condensate system [30].

In recent years, by considering Gross-Pitaevskii relativistic and Yang-Mills
theories, effective acoustic black hole metrics for analogous gravity models
have been constructed [31]. However, based on a holographic approach, the
acoustic black hole of a D3-black brane was obtained [32]. Furthermore,
relativistic metrics of acoustic black holes have been obtained from the
Abelian-Higgs model [33, 34, 35, 36] and also from other models [37, 38, 39,
40]. Besides, by applying these relativistic effective metrics, many works
have been carried out to explore, for example, analogous Hawking radiation
[41], superradiance [42, 43, 44, 45, 46], entropy [47, 48, 49, 50, 51],
quasinormal modes [52, 53, 54, 55, 56, 57], and as well as, in other models
[58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. In addition, analogous Hawking
radiation and the analytical solution of a massless scalar field have been
addressed in [68]. Although, due to equivalent mathematical descriptions
between acoustic black holes and gravitational black holes, it has been
conjectured that if a physical phenomenon occurs in gravitational black holes,
it can also happen in acoustic black holes. In this sense, in [69], it has
been shown that there is a thermodynamic-like description for acoustic black
holes in two dimensions; entropy and specific heat capacity have also been
calculated. Hence, unlike Hawking radiation, an analogous form for the
Bekenstein-Hawking entropy has been little investigated so far. However, it
was explored in [70], that such an analogy arises in a Bose-Einstein
condensate system. Thus, the Bekenstein-Hawking entropy can be understood as
an entanglement entropy. This entanglement entropy is related to the phonons
generated in the Hawking mechanism. Furthermore, the dependence of entropy on
the area of the event horizon of the acoustic black hole was examined in [71].
Moreover, the entanglement entropy of an acoustic black hole was also explored
in 1D degenerate ideal fermi fluids [72]. So, with the examples mentioned
above in ultracold quantum gases, we can look at other systems that exhibit
entanglement entropy.

Therefore, guided by all these works, in this paper, we are interested in
examining whether such a conjecture occurs in models of canonical acoustic
black holes with modified metrics. In this sense, we will address the effect
of noncommutativity on the computation of Hawking radiation, entropy, and
stability of the canonical acoustic black hole, as well as the effect of
minimum length on such quantities. As a result, we show that for a given
minimum radius (which depends on the noncommutative parameter) the
noncommutative canonical acoustic black hole becomes stable. In the absence of
noncommutativity the acoustic black hole presents instability. In addition, we
verify that for a certain minimum radius that depends on the deformation
parameter, the canonical acoustic black hole with minimum length effect also
becomes stable.

The paper is organized as follows. In Sec. II we briefly review the steps to
find the noncommutative acoustic black hole metrics and we calculate the
Hawking temperature, entropy and specific heat capacity of the noncommutative
canonical acoustic black hole. In Sec. III we compute the Hawking temperature,
entropy and specific heat capacity of the canonical acoustic black hole with
quantum correction. Finally in Sec. IV we present our final considerations.

##  II noncommutative canonical acoustic black hole

The metric of a noncommutative canonical acoustic black hole has been found by
us in [35]. Here we will briefly show the steps to obtain the noncommutative
acoustic black hole metric in (3+1) dimensions from quantum field theory. We
start from the the Lagrangian of the noncommutative Abelian Higgs model in
flat space given by [73]

$\displaystyle\hat{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\left(1+\frac{1}{2}\theta^{\alpha\beta}F_{\alpha\beta}\right)+\left(1-\frac{1}{4}\theta^{\alpha\beta}F_{\alpha\beta}\right)\left(|D_{\mu}\phi|^{2}+m^{2}|\phi|^{2}-b|\phi|^{4}\right)$
(1) $\displaystyle+$
$\displaystyle\frac{1}{2}\theta^{\alpha\beta}F_{\alpha\mu}\left[(D_{\beta}\phi)^{\dagger}D^{\mu}\phi+(D^{\mu}\phi)^{\dagger}D_{\beta}\phi\right],$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$,
$D_{\mu}\phi=\partial_{\mu}\phi-ieA_{\mu}\phi$ and the parameter
$\theta^{\alpha\beta}$ is a constant, real-valued antisymmetric $D\times D$\-
matrix in $D$-dimensional spacetime with dimensions of length squared.

Now, we applying $\phi=\sqrt{\rho(x,t)}\exp{(iS(x,t))}$ into the original
Lagrangian, we have

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\left(1-2\vec{\theta}\cdot\vec{B}\right)+\rho(\tilde{\theta}g^{\mu\nu}+\Theta^{\mu\nu}){\cal
D}_{\mu}S{\cal
D}_{\nu}S+\tilde{\theta}m^{2}\rho-\tilde{\theta}b\rho^{2}+\frac{\rho}{\sqrt{\rho}}(\tilde{\theta}g^{\mu\nu}+\Theta^{\mu\nu})\partial_{\mu}\partial_{\nu}\sqrt{\rho},$
(2)

where ${\cal D}_{\mu}=\partial_{\mu}-eA_{\mu}/S$,
$\tilde{\theta}=(1+\vec{\theta}\cdot\vec{B})$, $\vec{B}=\nabla\times\vec{A}$
and $\Theta^{\mu\nu}=\theta^{\alpha\mu}{F_{\alpha}}^{\nu}$. In our analysis we
consider the case where there is no noncommutativity between space and time,
that is $\theta^{0i}=0$ and use $\theta^{ij}=\varepsilon^{ijk}\theta^{k}$,
$F^{i0}=E^{i}$ and $F^{ij}=\varepsilon^{ijk}B^{k}$.

Hence, by linearizing the equations of motion around the background
$(\rho_{0},S_{0})$, with $\rho=\rho_{0}+\rho_{1}$ and $S=S_{0}+\psi$ we find
the equation of motion for a linear acoustic disturbance $\psi$ given by

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})\psi=0,$
(3)

where $g_{\mu\nu}=\frac{b\rho_{0}}{2c_{s}\sqrt{f}}\tilde{g}_{\mu\nu}$ is the
relativistic acoustic metric with noncommutative corrections in (3+1)
dimensions and with $\tilde{g}_{\mu\nu}$ given in the form [35]

$\displaystyle\tilde{g}_{tt}$ $\displaystyle=$
$\displaystyle-[(1-3\vec{\theta}\cdot\vec{B})c^{2}_{s}-(1+3\vec{\theta}\cdot\vec{B})v^{2}+2(\vec{\theta}\cdot\vec{v})(\vec{B}\cdot\vec{v})-(\vec{\theta}\times\vec{E})\cdot\vec{v}],$
$\displaystyle\tilde{g}_{tj}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\vec{\theta}\times\vec{E})^{j}(c^{2}_{s}+1)-\left[2(1+2\vec{\theta}\cdot\vec{B})-(\vec{\theta}\times\vec{E})\cdot\vec{v}\right]\frac{v^{j}}{2}+\frac{B^{j}}{2}(\vec{\theta}\cdot\vec{v})+\frac{\theta^{j}}{2}(\vec{B}\cdot\vec{v}),$
$\displaystyle\tilde{g}_{it}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\vec{\theta}\times\vec{E})^{i}(c^{2}_{s}+1)-\left[2(1+2\vec{\theta}\cdot\vec{B})-(\vec{\theta}\times\vec{E})\cdot\vec{v}\right]\frac{v^{i}}{2}+\frac{B^{i}}{2}(\vec{\theta}\cdot\vec{v})+\frac{\theta^{i}}{2}(\vec{B}\cdot\vec{v}),$
$\displaystyle\tilde{g}_{ij}$ $\displaystyle=$
$\displaystyle[(1+\vec{\theta}\cdot\vec{B})(1+c^{2}_{s})-(1+\vec{\theta}\cdot\vec{B})v^{2}-(\vec{\theta}\times\vec{E})\cdot\vec{v}]\delta^{ij}+(1+\vec{\theta}\cdot\vec{B})v^{i}v^{j}.$
$\displaystyle f$ $\displaystyle=$
$\displaystyle[(1-2\vec{\theta}\cdot\vec{B})(1+c^{2}_{s})-(1+4\vec{\theta}\cdot\vec{B})v^{2}]-3(\vec{\theta}\times\vec{E})\cdot\vec{v}+2(\vec{B}\cdot\vec{v})(\vec{\theta}\cdot\vec{v}).$
(4)

Here, in deducing the metric, we assume linear perturbations in the scalar
sector and keep the vector field $A_{\mu}$ unchanged.

In the following we shall focus on the noncommutative canonical acoustic black
hole metrics in (3+1) dimensions [35] to address the issues of Hawking
temperature and entropy. Now we consider an incompressible fluid with
spherical symmetry. In this case the density $\rho$ is a position independent
quantity and the continuity equation implies that $v\sim\frac{1}{r^{2}}$. The
sound speed is also a constant.

The noncommutative acoustic metric can be written as a Schwarzschild metric
type, up to an irrelevant position-independent factor, in the nonrelativistic
limit as follows,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\tilde{{\cal
F}}(v_{r})d\tau^{2}+\frac{[v_{r}^{2}\Gamma+\Sigma+\tilde{{\cal
F}}(v_{r})\Lambda]}{\tilde{\cal
F}(v_{r})}dr^{2}+\frac{r^{2}(d{\vartheta}^{2}+\sin^{2}\vartheta
d\phi^{2})}{\sqrt{f}},$ (5)

where

$\displaystyle\tilde{{\cal F}}(v_{r})$ $\displaystyle=$
$\displaystyle\frac{{\cal
F}(v_{r})}{\sqrt{f(v_{r})}}=\frac{1}{\sqrt{f(v_{r})}}\left[(1-3\vec{\theta}\cdot\vec{B})c^{2}_{s}-(1+3\vec{\theta}\cdot\vec{B})v^{2}_{r}-\theta{\cal
E}_{r}{v}_{r}+2(\theta_{r}B_{r}v^{2}_{r})\right],$ (6) $\displaystyle
f(v_{r})$ $\displaystyle=$ $\displaystyle
1-2\vec{\theta}\cdot\vec{B}-3\theta{\cal E}_{r}v_{r},$ (7)
$\displaystyle\Lambda(v_{r})$ $\displaystyle=$ $\displaystyle
1+\vec{\theta}\cdot\vec{B}-\theta{\cal E}_{r}v_{r},$ (8)
$\displaystyle\Gamma(v_{r})$ $\displaystyle=$ $\displaystyle
1+4\vec{\theta}\cdot\vec{B}-2\theta{\cal E}_{r}{v}_{r},$ (9)
$\displaystyle\Sigma(v_{r})$ $\displaystyle=$ $\displaystyle\left[\theta{\cal
E}_{r}-({B}_{r}{v}_{r})\theta_{r}-({\theta}_{r}{v}_{r})B_{r}\right]v_{r},$
(10)

being $\theta{\cal E}_{r}=\theta(\vec{n}\times\vec{E})_{r}$. Now, by applying
the relation $v_{r}=c_{s}\frac{r^{2}_{h}}{r^{2}}$, where $r_{h}$ is the radius
of the event horizon of the canonical acoustic black hole and making $c_{s}=1$
and so, the metric function of the noncommutative canonical acoustic black
hole becomes

$\displaystyle\tilde{\cal
F}(r)=\left[1-3\vec{\theta}\cdot\vec{B}-(1+3\vec{\theta}\cdot\vec{B}-2\theta_{r}B_{r})\frac{r^{4}_{h}}{r^{4}}-\theta{\cal
E}_{r}\frac{r^{2}_{h}}{r^{2}}\right]\left[1-2\vec{\theta}\cdot\vec{B}-3\theta{\cal
E}_{r}\frac{r^{2}_{h}}{r^{2}}\right]^{-1/2}.$ (11)

Next, we will do our analysis considering the pure magnetic sector first and
then we will investigate the pure electric sector.

###  II.1 Magnetic Sector

Hence, for $\theta_{r}=0$, $\vec{\theta}\cdot\vec{B}=\theta_{3}B_{3}\neq 0$,
$\theta{\cal E}_{r}=0$ (or $E=0$) with small $\theta_{3}B_{3}$,

$T_{H}=\frac{\tilde{\cal
F}^{\prime}(r_{h})}{4\pi}=\frac{(1+3\theta_{3}B_{3})}{\sqrt{1-2\theta_{3}B_{3}}}\frac{1}{(\pi
r_{h})}=\frac{\left(1+4\theta_{3}B_{3}\right)}{(\pi r_{h})}.$ (12)

For $\theta=0$ the usual result is obtained. Here the temperature has its
value increased when we vary the parameter $\theta$.

However, for the temperature in (12) we can find the entropy (entanglement
entropy [51]) of the modified canonical acoustic black hole by applying the
first law of thermodynamics such that

$\displaystyle S=\int\frac{dE}{T}=\int\frac{dA}{4\pi
r_{h}T_{H}}=\frac{(1-4\theta_{3}B_{3})}{4}A,$ (13)

where $A=4\pi r^{2}_{h}$ is the horizon area of the canonical acoustic black
hole.

Now for $\theta_{r}=0$, $\theta{\cal E}_{r}=0$ (or $E=0$) and
$\vec{\theta}\cdot\vec{B}=\theta_{3}B_{3}={\theta}r^{2}_{0}/r^{2}$ ($r_{0}$ is
a parameter with length dimension and $\theta$ is now dimensionless), in (11),
we have

$\displaystyle\tilde{\cal
F}(r)=\left[1-3\theta\frac{r^{2}_{0}}{r^{2}}-\left(1+3\theta\frac{r^{2}_{0}}{r^{2}}\right)\frac{r^{4}_{h}}{r^{4}}\right]\left[1-2\theta\frac{r^{2}_{0}}{r^{2}}\right]^{-1/2}.$
(14)

At this point we call attention to the correction term $3\theta
r^{2}_{0}/r^{2}$. This is a Reissner-Nordström type term with an effective
charge $Q^{2}=3\theta r^{2}_{0}$ generated due to the noncommutativity.

For the Hawking temperature, we obtain

$\displaystyle
T_{H}=\frac{1}{\pi\left(r_{h}+3\sqrt{\theta}r_{0}\right)\displaystyle\left(1-\frac{3\sqrt{\theta}r_{0}}{2r_{h}}+\frac{2\theta
r^{2}_{0}}{r^{2}_{h}}\right)^{2}},$ (15)

which can be rewritten as

$\displaystyle T_{H}=\frac{1}{\pi r_{h}}\left(1+\frac{7\theta
r^{2}_{0}}{r^{2}_{h}}\right).$ (16)

Again, we see that noncommutativity has the effect of increasing the
temperature when we vary the parameter $\theta$.

In this case the entropy is now given by

$\displaystyle S=\frac{1}{4}\int\left(1-\frac{28\pi\theta
r^{2}_{0}}{A}\right)dA=\frac{1}{4}\left(A-28\pi\theta r^{2}_{0}\ln A\right).$
(17)

Thus, we have found a logarithmic correction for entropy.

Next to analyze the stability of the canonical acoustic black hole we will
compute the specific heat capacity using the relation [69]

$\displaystyle C=\frac{\partial E}{\partial T_{H}}=\left(\frac{\partial
E}{\partial r_{h}}\right)\left(\frac{\partial T_{H}}{\partial
r_{h}}\right)^{-1}=T_{H}\left(\frac{\partial S}{\partial T_{H}}\right)=-2\pi
r^{2}_{h}\left(1-\frac{21\theta r^{2}_{0}}{r^{2}_{h}}\right).$ (18)

Hence, for

$\displaystyle r_{h}=\sqrt{21\theta}\,r_{0}$ (19)

the specific heat capacity is zero and the noncommutative canonical acoustic
black hole becomes stable with a temperature given by

$\displaystyle T_{Hmax}=\frac{4}{3\pi r_{h}}.$ (20)

In the situation where $\theta=0$ the specific heat capacity is $C=-2\pi
r^{2}_{h}$ and so the canonical acoustic black hole is unstable by presenting
a good analogy with the gravitational black hole. Note that due to the effect
of noncommutativity via application of an external magnetic field a stability
condition for the noncommutative canonical acoustic black hole is generated.

In the absence of noncommutativity the temperature of the canonical acoustic
black hole is given by

$\displaystyle T_{h}=\frac{1}{\pi r_{h}},$ (21)

for $r_{h}=0$ the temperature diverges. On the other hand, the temperature of
the canonical acoustic black hole (15) is now regularized by the
noncommutativity effect. Thus, by taking $r_{h}\rightarrow 0$ in (15) the
temperature goes to zero, before it passes through a maximum point, as we can
see in Fig. 1.

Figure 1: The Hawking temperature $\pi r_{0}T_{H}$ in function of
$r_{h}/r_{0}$. For $r_{h}=0$ the temperature $Th1$ (21) diverges. The
temperature $Th2$ (15) reaches a maximum value $T_{max}$, and then decreases
to zero as $r_{h}/r_{0}\rightarrow 0$.

###  II.2 Electric Sector

At this point, we will consider the situation where $B=0$ and $\theta{\cal
E}_{r}\neq 0$. So, from (11), we have

$\displaystyle\tilde{\cal F}(r)=\left[1-\theta{\cal
E}_{r}\frac{r^{2}_{h}}{r^{2}}-\frac{r^{4}_{h}}{r^{4}}\right]\left[1-3\theta{\cal
E}_{r}\frac{r^{2}_{h}}{r^{2}}\right]^{-1/2}.$ (22)

Here, we also have the Reissner-Nordström type correction term $\theta{\cal
E}_{r}/r^{2}$ with an effective charge $Q^{2}=\theta{\cal E}_{r}$ generated
due to the noncommutativity.

We also noticed that the temperature is increased when we vary the $\theta$
parameter

$T_{H}=\frac{\left[1+\theta{\cal E}_{r}/2\right]}{\sqrt{1-3\theta{\cal
E}_{r}}}\frac{1}{\pi r_{h}}=\frac{\left(1+2\theta{\cal E}_{r}\right)}{\pi
r_{h}}.$ (23)

For entropy we have

$\displaystyle S=\left(1-2\theta{\cal E}_{r}\right)\frac{A}{4}.$ (24)

Now considering $\theta{\cal E}_{r}=\theta\lambda^{2}/r^{2}$, being $\lambda$
a parameter with length dimension and $\theta$ a dimensionless parameter, so
we have

$\displaystyle\tilde{\cal
F}(r)=\left[1-\theta\frac{\lambda^{2}r^{2}_{h}}{r^{4}}-\frac{r^{4}_{h}}{r^{4}}\right]\left[1-3\theta\frac{\lambda^{2}r^{2}_{h}}{r^{4}}\right]^{-1/2}.$
(25)

In this case for the Hawking temperature we find

$\displaystyle
T_{H}=\frac{1}{\pi\displaystyle\left(r_{h}+2\sqrt{\theta}\lambda+\frac{\theta\lambda^{2}}{2r_{h}}\right)\left(1-\frac{\sqrt{\theta}\lambda}{r_{h}}+\frac{\theta\lambda^{2}}{2r^{2}_{h}}\right)^{2}},$
(26)

hence we have

$\displaystyle T_{H}=\frac{1}{\pi
r_{h}}\left(1+\frac{5\theta\lambda^{2}}{2r^{2}_{h}}\right).$ (27)

Here again, we note the effect of noncommutativity on the Hawking temperature,
that is, by increasing the parameter $\theta$, the intensity of the Hawking
temperature increases.

Therefore, for entropy we obtain

$\displaystyle
S=\frac{1}{4}\int\left(1-\frac{10\pi\theta\lambda^{2}}{A}\right)dA=\frac{1}{4}\left(A-10\pi\theta\lambda^{2}\ln
A\right).$ (28)

Now to investigate the stability of the noncommutative canonical acoustic
black hole, we determine the specific heat capacity which is given by

$\displaystyle C=-2\pi
r^{2}_{h}\left(1-\frac{15\theta\lambda^{2}}{2r^{2}_{h}}\right).$ (29)

The specific heat capacity is zero for

$\displaystyle r_{h}=\sqrt{\frac{15\theta}{2}}\,\lambda.$ (30)

Consequently, the noncommutative canonical acoustic black hole reaches its
stability with a temperature given by

$\displaystyle T_{Hmax}=\frac{4}{3\pi r_{h}}.$ (31)

Again, we verify the stability condition of the canonical acoustic black hole
implemented by the effect of noncommutativity via application of an external
electric field.

In Fig. 2 we show the behavior of temperature (26) as a function of
$r_{h}/\lambda$. In the limit of $r_{h}/\lambda\rightarrow 0$ the temperature
passes through a maximum value and then goes to zero.

Figure 2: The Hawking temperature $\lambda T_{H}$ in function of
$r_{h}/\lambda$. Note that the temperature $Th3$ (26) reaches a maximum value
$T_{max}$, and then decreases to zero as $r_{h}/\lambda\rightarrow 0$.

##  III Quantum-Corrected Canonical Acoustic Black Hole

In [74] we have found an effective metric for acoustic black holes with
quantum corrections implemented by the generalized Heisenberg uncertainty
principle (GUP) [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93] in the Abelian Higgs model. However, the line element written
in the Schwarzschild-type form is given by

$\displaystyle
ds^{2}=-\mathcal{F}(v_{r})d\tau^{2}+\frac{c^{2}_{s}\left(1+6\alpha-2\alpha
e\Phi\right)}{f(v_{r})\mathcal{F}(v_{r})}dr^{2}+\frac{\left[1-2\alpha(e\Phi-2)\right]}{\sqrt{f(v_{r})}}r^{2}d\Omega^{2},$
(32)

where

$\displaystyle\mathcal{F}(v_{r})=\frac{F(v_{r})}{\sqrt{f(v_{r})}}\approx
c^{2}_{s}[1-\alpha(1-e\Phi+ev_{r}\Lambda_{r})]-v^{2}_{r}[1-\alpha(ev_{r}\Lambda_{r}-e\Phi-1)]+2e\alpha
v_{r}\Lambda_{r}.$ (33)

The metric of the canonical acoustic black hole with quantum corrections
obtained in [74] is given by

$\displaystyle ds^{2}=-\mathcal{F}(r)d\tau^{2}+\frac{\left(1+6\alpha-2\alpha
e\Phi\right)}{f(r)\mathcal{F}(r)}dr^{2}+\frac{\left[1-2\alpha(e\Phi-2)\right]}{\sqrt{f(r)}}r^{2}d\Omega^{2},$
(34)

where

$\displaystyle\mathcal{F}(r)$ $\displaystyle=$ $\displaystyle
1-\alpha\left(1-e\Phi(r)+\frac{er^{2}_{h}\Lambda_{r}(r)}{r^{2}}\right)-\frac{r^{4}_{h}}{r^{4}}\left[1-\alpha\left(\frac{er^{2}_{h}\Lambda_{r}(r)}{r^{2}}-e\Phi(r)-1\right)\right]+\frac{2e\alpha
r^{2}_{h}\Lambda_{r}(r)}{r^{2}},$ (35) $\displaystyle f(r)$ $\displaystyle=$
$\displaystyle
1+6\alpha-2e\alpha\Phi+2e\alpha\Lambda_{r}\frac{r^{2}_{h}}{r^{2}},$ (36)

being $r_{h}$ the radius of the event horizon, $\alpha$ the dimensionless GUP
parameter, $\Phi$ and $\Lambda_{r}$ the scalar and vector potentials
respectively. Now choosing $\Lambda_{r}=0$ and $\Phi=r_{0}/r$ (being $r_{0}$ a
parameter with length dimension), the Hawking temperature and entropy are
given respectively by

$\displaystyle
T_{H}=\frac{1+\alpha}{\pi\displaystyle\left(r_{h}+2r_{h}\sqrt{\frac{\alpha
er_{0}}{r_{h}}}+\frac{\alpha er_{0}}{2}\right)\left(1-\sqrt{\frac{\alpha
er_{0}}{2r_{h}}}+\frac{\alpha er_{0}}{4r_{h}}\right)^{2}},$ (37)

or

$\displaystyle{T}_{H}=\frac{1}{\pi r_{h}}\left(1+\alpha+\frac{\alpha
er_{0}}{r_{h}}\right)\qquad\Rightarrow\qquad\tilde{T}_{H}\approx\frac{1}{\pi
r_{h}}\left(1+\frac{\alpha er_{0}}{r_{h}}\right),$ (38) $\displaystyle
S=\int\frac{dA}{4\pi r_{h}\tilde{T}_{H}}=\frac{A}{4}-\frac{4\sqrt{\pi}\alpha
er_{0}\sqrt{A}}{4}+\frac{4\pi\alpha^{2}e^{2}r^{2}_{0}}{4}\ln\left(\frac{A}{4r^{2}_{0}}\right).$
(39)

At this point we will analyze the stability condition of the modified
canonical acoustic black hole due to quantum corrections computing the
specific heat capacity. Thus, we obtain

$\displaystyle C=-2\pi r^{2}_{h}\left(1-\frac{2e\alpha r_{0}}{r_{h}}\right).$
(40)

Note that by setting

$\displaystyle r_{h}=2e\alpha r_{0},$ (41)

the specific heat capacity vanishes and the modified canonical acoustic black
hole becomes stable with a temperature given by

$\displaystyle T_{Hmax}=\frac{3}{2\pi r_{h}}.$ (42)

The behavior of temperature (37) as a function of $r_{h}/r_{0}$ can be seen in
Fig. 3. In the limit of $r_{h}/r_{0}\rightarrow 0$ the temperature reaches a
maximum value and then tends to zero.

Figure 3: The Hawking temperature $r_{0}T_{H}$ in function of $r_{h}/r_{0}$.
Note that the temperature $Th4$ (37) reaches a maximum value $T_{max}$, and
then decreases to zero as $r_{h}/r_{0}\rightarrow 0$.

Next, by choosing $\Phi=0$ and $\Lambda_{r}=\lambda/r$ (being $\lambda$
parameter with length dimension), we have [74]

$\displaystyle T_{H}=\frac{1+\alpha}{\pi\displaystyle\left(r_{h}+\frac{3\alpha
e\lambda}{2}\right)\left(1+\frac{\alpha e\lambda}{r_{h}}\right)},$ (43)

or

$\displaystyle{T}_{H}=T_{H}\left(1+\alpha-\frac{5\alpha
e\lambda}{2r_{h}}\right)\qquad\Rightarrow\qquad\tilde{T}_{H}=\frac{1}{\pi
r_{h}}\left(1-\frac{5\alpha e\lambda}{2r_{h}}\right),$ (44)

and

$\displaystyle S=\frac{A}{4}+\frac{10\sqrt{\pi}\alpha
e\lambda\sqrt{A}}{4}+\frac{25\pi\alpha^{2}e^{2}\lambda^{2}}{4}\ln\left(\frac{A}{4\lambda^{2}}\right).$
(45)

For the specific heat capacity we find

$\displaystyle C=-2\pi
r^{2}_{h}\left(1-\sqrt{\frac{5e\alpha\lambda}{r_{h}}}\right)\left(1+\sqrt{\frac{5e\alpha\lambda}{r_{h}}}\right).$
(46)

In this situation, for

$\displaystyle r_{h}=5e\alpha\lambda,$ (47)

the specific heat capacity vanishes. Thus, indicating that the modified
canonical acoustic black hole become stable. Then, replacing (47) in (44), we
find the maximum temperature given by

$\displaystyle T_{Hmax}=\frac{1}{2\pi r_{h}}.$ (48)

Therefore, due to the GUP effect, we obtain the stability conditions of the
canonical acoustic black hole via applications of the scalar and vector
potentials. Note that the temperature (43) is regularized by the effect of the
GUP as we can see in Fig. 4. For $r_{h}/\lambda\rightarrow 0$ the temperature
reaches a maximum value before going to zero.

It is worth mentioning that the stability condition analyzed here for the
noncommutative and GUP cases shows similarity with the result that arises in
the gravitational case. Black hole remnant formation emerges from the minimum
radius (or minimum mass) condition that nullifies the specific heat capacity,
and thus, the black hole stops to evaporate completely [90]. Therefore, in our
results we would have, for a certain minimum radius, an analogous condition
for the formation of an acoustic black hole remnant due to the effect of
noncommutativity and GUP. In addition, we have found logarithmic corrections
for entropy in the noncommutative and GUP cases. We highlight that the
logarithmic correction term for entropy is fundamental to exploring the
process of evaporation and formation of gravitational black hole remnant and,
analogously, we have verified this possibility in the physics of a canonical
acoustic black hole in the noncommutative background and with GUP.

Also, as shown in [74], we point out that by making $\Phi=r_{0}/r$ and
$\Lambda_{r}=\lambda_{0}$ (a dimensionless constant) in (35), we find

$\displaystyle\mathcal{F}(r)=1-\alpha\left(1-\frac{er_{0}}{r}-\frac{e\lambda_{0}r^{2}_{h}}{r^{2}}\right)-\frac{r^{4}_{h}}{r^{4}}\left[1+\alpha\left(1+\frac{er_{0}}{r}-\frac{e\lambda_{0}r^{2}_{h}}{r^{2}}\right)\right],$
(49)

with contributions from a Reissner-Nordström type metric being generated for
the canonical acoustic black hole metric function. Hence we identify
$Q^{2}=\alpha e\lambda_{0}r^{2}_{h}$ as the effective charge.

Figure 4: The Hawking temperature $\pi\lambda T_{H}$ in function of
$r_{h}/\lambda$. Note that the temperature $Th5$ (43) reaches a maximum value
$T_{max}$, and then decreases to zero as $r_{h}/\lambda\rightarrow 0$.

##  IV conclusions

In summary, by considering the noncommutative canonical acoustic black hole
metric we found the Hawking temperature, entropy and we have verified the
stability condition by calculating the specific heat capacity. We have been
analyzing the stability in the electric and magnetic sectors. In both sectors
we show that stability occurs for a certain value of the minimum horizon
radius. In the case of the canonical acoustic black hole with quantum
corrections we have determined the Hawking temperature and entropy. In the
situation where the metric depends only on the correction due to the scalar
potential, we have shown that for a given value of the horizon radius the
specific heat capacity goes to zero, and the canonical acoustic black hole
becomes stable. Besides, when the metric depends only on the correction due to
the vector potential, we have also found that the specific heat capacity
vanishes, and the canonical acoustic black hole becomes stable. In addition,
for the cases where stability was analyzed, we have shown that temperatures
reach maximum values before going to zero when the horizon radius is zero.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant nos.
165/2018 and 015/2019), for partial financial support. MAA, FAB and EP
acknowledge support from CNPq (Grant nos. 306398/2021-4, 309092/2022-1,
304290/2020-3).

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# On higher dimensional point sets in general position

Andrew Suk Department of Mathematics, University of California at San Diego,
La Jolla, CA, 92093 USA. Supported by NSF CAREER award DMS-1800746 and NSF
award DMS-1952786. Email<EMAIL_ADDRESS>Ji Zeng Department of Mathematics,
University of California at San Diego, La Jolla, CA, 92093 USA. Supported by
NSF grant DMS-1800746<EMAIL_ADDRESS>
###### Abstract

A finite point set in $\mathbb{R}^{d}$ is in general position if no $d+1$
points lie on a common hyperplane. Let $\alpha_{d}(N)$ be the largest integer
such that any set of $N$ points in $\mathbb{R}^{d}$ with no $d+2$ members on a
common hyperplane, contains a subset of size $\alpha_{d}(N)$ in general
position. Using the method of hypergraph containers, Balogh and Solymosi
showed that $\alpha_{2}(N)<N^{5/6+o(1)}$. In this paper, we also use the
container method to obtain new upper bounds for $\alpha_{d}(N)$ when $d\geq
3$. More precisely, we show that if $d$ is odd, then
$\alpha_{d}(N)<N^{\frac{1}{2}+\frac{1}{2d}+o(1)}$, and if $d$ is even, we have
$\alpha_{d}(N)<N^{\frac{1}{2}+\frac{1}{d-1}+o(1)}$.

We also study the classical problem of determining the maximum number
$a(d,k,n)$ of points selected from the grid $[n]^{d}$ such that no $k+2$
members lie on a $k$-flat. For fixed $d$ and $k$, we show that

$a(d,k,n)\leq
O\left(n^{\frac{d}{2\lfloor(k+2)/4\rfloor}(1-\frac{1}{2\lfloor(k+2)/4\rfloor
d+1})}\right),$

which improves the previously best known bound of
$O\left(n^{\frac{d}{\lfloor(k+2)/2\rfloor}}\right)$ due to Lefmann when $k+2$
is congruent to 0 or 1 mod 4.

##  1 Introduction

A finite point set in $\mathbb{R}^{d}$ is said to be in _general position_ if
no $d+1$ members lie on a common hyperplane. Let $\alpha_{d}(N)$ be the
largest integer such that any set of $N$ points in $\mathbb{R}^{d}$ with no
$d+2$ members on a hyperplane, contains $\alpha_{d}(N)$ points in general
position.

In 1986, Erdős [8] proposed the problem of determining $\alpha_{2}(N)$ and
observed that a simple greedy algorithm shows
$\alpha_{2}(N)\geq\Omega(\sqrt{N})$. A few years later, Füredi [10] showed
that

$\Omega(\sqrt{N\log N})<\alpha_{2}(N)<o(N),$

where the lower bound uses a result of Phelps and Rödl [20] on partial Steiner
systems, and the upper bound relies on the density Hales-Jewett theorem [11,
12]. In 2018, a breakthrough was made by Balogh and Solymosi [3], who showed
that $\alpha_{2}(N)<N^{5/6+o(1)}$. Their proof was based on the method of
hypergraph containers, a powerful technique introduced independently by
Balogh, Morris, and Samotij [1] and by Saxton and Thomason [24], that reveals
an underlying structure of the independent sets in a hypergraph. We refer
interested readers to [2] for a survey of results based on this method.

In higher dimensions, the best lower bound for $\alpha_{d}(N)$ is due to
Cardinal, Tóth, and Wood [5], who showed that $\alpha_{d}(N)\geq\Omega((N\log
N)^{1/d})$, for every fixed $d\geq 2$. For upper bounds, Milićević [18] used
the density Hales-Jewett theorem to show that $\alpha_{d}(N)=o(N)$ for every
fixed $d\geq 2$. However, these upper bounds in [18], just like that in [10],
are still almost linear in $N$. Our main result is the following.

######  Theorem 1.1.

Let $d\geq 3$ be a fixed integer. If $d$ is odd, then
$\alpha_{d}(N)<N^{\frac{1}{2}+\frac{1}{2d}+o(1)}$. If $d$ is even, then
$\alpha_{d}(N)<N^{\frac{1}{2}+\frac{1}{d-1}+o(1)}.$

Our proof of Theorem 1.1 is also based on the hypergraph container method. A
key ingredient in the proof is a new supersaturation lemma for $(k+2)$-tuples
of the grid $[n]^{d}$ that lie on a $k$-flat, which we shall discuss in the
next section. Here, by a _$k$ -flat_ we mean a $k$-dimensional affine subspace
of $\mathbb{R}^{d}$.

One can consider a generalization of the quantity $\alpha_{d}(N)$. We let
$\alpha_{d,s}(N)$ be the largest integer such that any set of $N$ points in
$\mathbb{R}^{d}$ with no $d+s$ members on a hyperplane, contains
$\alpha_{d,s}(N)$ points in general position. Hence,
$\alpha_{d}(N)=\alpha_{d,2}(N)$. A simple argument of Erdős [8] shows that
$\alpha_{d,s}(N)\geq\Omega(N^{1/d})$ for fixed $d$ and $s$ (see Section 6, or
[5] for large $s$). In the other direction, following the arguments in our
proof of Theorem 1.1 with a slight modification, we show the following.

######  Theorem 1.2.

Let $d,s\geq 3$ be fixed integers. If $d$ is odd and
$\frac{2d+s-2}{2d+2s-2}<\frac{d-1}{d}$, then $\alpha_{d,s}(N)\leq
N^{\frac{1}{2}+o(1)}$. If $d$ is even and
$\frac{2d+s-2}{2d+2s-2}<\frac{d-2}{d-1}$, then $\alpha_{d,s}(N)\leq
N^{\frac{1}{2}+o(1)}$.

For example, when we fix $d=3$ and $s\geq 5$, we have $\alpha_{d,s}(N)\leq
N^{\frac{1}{2}+o(1)}$.

We also study the classical problem of determining the maximum number of
points selected from the grid $[n]^{d}$ such that no $k+2$ members lie on a
$k$-flat. The key ingredient of Theorem 1.1 mentioned above can be seen as a
supersaturation version of this Turán-type problem. When $k=1$, this is the
famous _no-three-in-line problem_ raised by Dudeney [7] in 1917: Is it true
that one can select $2n$ points in $[n]^{2}$ such that no three are collinear?
Clearly, $2n$ is an upper bound as any vertical line must contain at most 2
points. For small values of $n$, many authors have published solutions to this
problem obtaining the bound of $2n$ (e.g. see [9]), but for large $n$, the
best known general construction is due to Hall et al. [13] with slightly fewer
than $3n/2$ points.

More generally, we let $a(d,k,r,n)$ denote the maximum number of points from
$[n]^{d}$ such that no $r$ points lie on a $k$-flat. Since $[n]^{d}$ can be
covered by $n^{d-k}$ many $k$-flats, we have the trivial upper bound
$a(d,k,r,n)\leq(r-1)n^{d-k}$. For certain values $d$, $k$, and $r$ fixed and
$n$ tends to infinity, this bound is known to be asymptotically best possible:
Many authors [22, 4, 16] noticed that $a(d,d-1,d+1,n)=\Theta(n)$ by looking at
the modular moment curve over a finite field $\mathbb{Z}_{p}$; In [21], Pór
and Wood proved that $a(3,1,3,n)=\Theta(n^{2})$; Very recently, Sudakov and
Tomon [25] showed that $a(d,k,r,n)=\Theta(n^{d-k})$ when $r>d^{k}$.

We shall focus on the case when $r=k+2$ and write $a(d,k,n):=a(d,k,k+2,n)$.
Surprisingly, Lefmann [16] (see also [15]) showed that $a(d,k,n)$ behaves much
differently than $\Theta(n^{d-k})$. In particular, he showed that

$a(d,k,n)\leq O\left(n^{\frac{d}{\lfloor(k+2)/2\rfloor}}\right).$

Our next result improves this upper bound when $k+2$ is congruent to 0 or 1
mod 4.

######  Theorem 1.3.

For fixed $d$ and $k$, as $n\to\infty$, we have

$a(d,k,n)\leq
O\left(n^{\frac{d}{2\lfloor(k+2)/4\rfloor}(1-\frac{1}{2\lfloor(k+2)/4\rfloor
d+1})}\right).$

For example, we have $a(4,2,n)\leq O(n^{\frac{16}{9}})$ while Lefmann’s bound
in [16] gives us $a(4,2,n)\leq O(n^{2})$, which coincides with the trivial
upper bound. In particular, Theorem 1.3 tells us that, if $4$ divides $k+2$,
then $a(d,k,n)$ only behaves like $\Theta(n^{d-k})$ when $d=k+1$. This is
quite interesting compared to the fact that $a(3,1,n)=\Theta(n^{2})$ proved in
[21]. Lastly, let us note that the current best lower bound for $a(d,k,n)$ is
also due to Lefmann [16], who showed that
$a(d,k,n)\geq\Omega\left(n^{\frac{d}{k+1}-k-\frac{k}{k+1}}\right)$.

For integer $n>0$, we let $[n]=\\{1,\dots,n\\}$, and
$\mathbb{Z}_{n}=\\{0,1,\dots,n-1\\}$. We systemically omit floors and ceilings
whenever they are not crucial for the sake of clarity in our presentation. All
logarithms are in base two.

##  2 $(k+2)$-tuples of $[n]^{d}$ on a $k$-flat

In this section, we establish two lemmas that will be used in the proof of
Theorem 1.1.

Given a set $T$ of $k+2$ points in $\mathbb{R}^{d}$ that lie on a $k$-flat, we
say that $T$ is _degenerate_ if there is a subset $S\subset T$ of size $j$,
where $3\leq j\leq k+1$, such that $S$ lies on a $(j-2)$-flat. Otherwise, we
say that $T$ is _non-degenerate_. We establish a supersaturation lemma for
non-degenerate $(k+2)$-tuples of $[n]^{d}$.

######  Lemma 2.1.

For real number $\gamma>0$ and fixed positive integers $d,k$, such that $k$ is
even and $d-2\gamma>(k-1)(k+2)$, any subset $V\subset[n]^{d}$ of size
$n^{d-\gamma}$ spans at least $\Omega(n^{(k+1)d-(k+2)\gamma})$ non-degenerate
$(k+2)$-tuples that lie on a $k$-flat.

###### Proof.

Let $V\subset[n]^{d}$ such that $|V|=n^{d-\gamma}$. Set $r=\frac{k}{2}+1$ and
$E_{r}=\binom{V}{r}$ to be the collection of $r$-tuples of $V$. Notice that
the sum of a $r$-tuple from $V$ belongs to $[rn]^{d}$. For each
$v\in[rn]^{d}$, we define

$E_{r}(v)=\\{\\{v_{1},\dots,v_{r}\\}\in E_{r}:v_{1}+\dots+v_{r}=v\\}.$

Then for $T_{1},T_{2}\in E_{r}(v)$, where $T_{1}=\\{v_{1},\dots,v_{r}\\}$ and
$T_{2}=\\{u_{1},\dots,u_{r}\\}$, we have

$v_{1}+\dots+v_{r}=v=u_{1}+\dots+u_{r},$

which implies that $T_{1}\cup T_{2}$ lies on a common $k$-flat. Let

$E_{2r}=\bigcup_{v\in[rn]^{d}}\ \bigcup_{T_{1},T_{2}\in
E_{r}(v)}\\{T_{1},T_{2}\\}.$

Hence, for each $\\{T_{1},T_{2}\\}\in E_{2r}$, $T_{1}\cup T_{2}$ lies on a
$k$-flat. Moreover, by Jensen’s inequality, we have

$|E_{2r}|=\sum_{v\in[rn]^{d}}\binom{|E_{r}(v)|}{2}\geq(rn)^{d}\binom{\frac{\sum_{v}|E_{r}(v)|}{(rn)^{d}}}{2}=(rn)^{d}\binom{|E_{r}|/(rn)^{d}}{2}\geq\frac{|E_{r}|^{2}}{4(rn)^{d}}.$

Since $k$ and $d$ are fixed and $r=\frac{k}{2}+1$ and $|V|=n^{d-\gamma}$,

$|E_{r}|^{2}=\binom{|V|}{r}^{2}=\binom{|V|}{(k/2)+1}^{2}\geq\Omega(n^{(k+2)(d-\gamma)}).$

Combining the two inequalities above gives

$|E_{2r}|\geq\Omega(n^{(k+1)d-(k+2)\gamma}).$

We say that $\\{T_{1},T_{2}\\}\in E_{2r}$ is _good_ if $T_{1}\cap
T_{2}=\emptyset$, and the $(k+2)$-tuple $(T_{1}\cup T_{2})$ is non-degenerate.
Otherwise, we say that $\\{T_{1},T_{2}\\}$ is _bad_. In what follows, we will
show that at least half of the pairs (i.e. elements) in $E_{2r}$ are good. To
this end, we will need the following claim.

######  Claim 2.2.

If $\\{T_{1},T_{2}\\}\in E_{2r}$ is bad, then $T_{1}\cup T_{2}$ lies on a
$(k-1)$-flat.

###### Proof.

Write $T_{1}=\\{v_{1},\dots,v_{r}\\}$ and $T_{2}=\\{u_{1},\dots,u_{r}\\}$. Let
us consider the following cases.

_Case 1._ Suppose $T_{1}\cap T_{2}\neq\emptyset$. Then, without loss of
generality, there is an integer $j<r$ such that

$v_{1}+\dots+v_{j}=u_{1}+\dots+u_{j},$

where $v_{1},\dots,v_{j},u_{1},\dots,u_{j}$ are all distinct elements, and
$v_{t}=u_{t}$ for $t>j$. Thus $|T_{1}\cup T_{2}|=2j+(r-j)$. The $2j$ elements
above lie on a $(2j-2)$-flat. Adding the remaining $r-j$ points implies that
$T_{1}\cup T_{2}$ lies on a $(j-2+r)$-flat. Since $r=\frac{k}{2}+1$ and
$j\leq\frac{k}{2},$ $T_{1}\cup T_{2}$ lies on a $(k-1)$-flat.

_Case 2._ Suppose $T_{1}\cap T_{2}=\emptyset$. Then $T_{1}\cup T_{2}$ must be
degenerate, which means there is a subset $S\subset T_{1}\cup T_{2}$ of $j$
elements such that $S$ lies on a $(j-2)$-flat, for some $3\leq j\leq k+1$.
Without loss of generality, we can assume that $v_{1}\not\in S$. Hence,
$(T_{1}\cup T_{2})\setminus\\{v_{1}\\}$ lies on a $(k-1)$-flat. On the other
hand, we have

$v_{1}=u_{1}+\dots+u_{r}-v_{2}-\dots-v_{r}.$

Hence, $v_{1}$ is in the affine hull of $(T_{1}\cup
T_{2})\setminus\\{v_{1}\\}$ which implies that $T_{1}\cup T_{2}$ lies on a
$(k-1)$-flat. ∎

We are now ready to prove the following claim.

######  Claim 2.3.

At least half of the pairs in $E_{2r}$ are good.

###### Proof.

For the sake of contradiction, suppose at least half of the pairs in $E_{2r}$
are bad. Let $H$ be the collection of all the $j$-flats spanned by subsets of
$V$ for all $j\leq k-1$. Notice that if $S\subset V$ spans a $j$-flat $h$,
then $h$ is also spanned by only $j+1$ elements from $S$. So we have

$|H|\leq\sum_{j=0}^{k-1}|V|^{j+1}\leq kn^{k(d-\gamma)}.$

For each bad pair $\\{T_{1},T_{2}\\}\in E_{2r}$, $T_{1}\cup T_{2}$ lies on a
$j$-flat from $H$ by Claim 2.2. By the pigeonhole principle, there is a
$j$-flat $h$ with $j\leq k-1$ such that at least

$\frac{|E_{2r}|/2}{|H|}\geq\frac{\Omega(n^{(k+1)d-(k+2)\gamma})}{2kn^{k(d-\gamma)}}=\Omega(n^{d-2\gamma})$

bad pairs from $E_{2r}$ have the property that their union lies in $h$. On the
other hand, since $h$ contains at most $n^{k-1}$ points from $[n]^{d}$, $h$
can correspond to at most $O(n^{(k-1)(k+2)})$ bad pairs from $E_{2r}$. Since
we assumed $d-2\gamma>(k-1)(k+2)$, we have a contradiction for $n$
sufficiently large. ∎

Each good pair $\\{T_{1},T_{2}\\}\in E_{2r}$ gives rise to a non-degenerate
$(k+2)$-tuple $T_{1}\cup T_{2}$ that lies on a $k$-flat. On the other hand,
any such $(k+2)$-tuple in $V$ will correspond to at most $\binom{k+2}{r}$ good
pairs in $E_{2r}$. Hence, by Claim 2.3, there are at least

$\left.\frac{|E_{2r}|}{2}\middle/\binom{k+2}{r}\right.=\Omega(n^{(k+1)d-(k+2)\gamma})$

non-degenerate $(k+2)$-tuples that lie on a $k$-flat, concluding the proof. ∎

In the other direction, we will use the following upper bound.

######  Lemma 2.4.

For real number $\gamma>0$ and fixed positive integers $d,k,\ell$, such that
$\ell<k+2$, suppose $U,V\subset[n]^{d}$ satisfy $|U|=\ell$ and
$|V|=n^{d-\gamma}$, then $V$ contains at most $n^{(k+1-\ell)(d-\gamma)+k}$
non-degenerate $(k+2)$-tuples that lie on a $k$-flat and contain $U$.

###### Proof.

If $U$ spans a $j$-flat for some $j<\ell-1$, then by definition no non-
degenerate $(k+2)$-tuple contains $U$. Hence we can assume $U$ spans a
$(\ell-1)$-flat. Observe that a non-degenerate $(k+2)$-tuple $T$, which lies
on a $k$-flat and contains $U$, must contain a $(k+1)$-tuple
$T^{\prime}\subset T$ such that $T^{\prime}$ spans a $k$-flat and $U\subset
T^{\prime}$. Then there are at most $n^{(k+1-\ell)(d-\gamma)}$ ways to add
$k+1-\ell$ points to $U$ from $V$ to obtain such $T^{\prime}$. After
$T^{\prime}$ is determined, there are at most $n^{k}$ ways to add a final
point from the affine hull of $T^{\prime}$ to obtain $T$. So we conclude the
proof by multiplication. ∎

##  3 The container method: Proof of Theorem 1.1

In this section, we use the hypergraph container method to prove Theorem 1.1.
We follow the method outlined in [3]. Let
$\mathcal{H}=(V(\mathcal{H}),E(\mathcal{H}))$ denote a $(k+2)$-uniform
hypergraph. For any $U\subset V(\mathcal{H})$, its degree $\delta(U)$ is the
number of edges containing $U$. For each $\ell\in[k+2]$, we use
$\Delta_{\ell}(\mathcal{H})$ to denote the maximum $\delta(U)$ among all $U$
of size $\ell$. For parameter $\tau>0$, we define the following quantity

$\Delta(\mathcal{H},\tau)=\frac{2^{\binom{k+2}{2}-1}|V(\mathcal{H})|}{(k+2)|E(\mathcal{H})|}\sum_{\ell=2}^{k+2}\frac{\Delta_{\ell}(\mathcal{H})}{\tau^{\ell-1}2^{\binom{\ell-1}{2}}}.$

Then we have the following hypergraph container lemma from [3], which is a
restatement of Corollary 3.6 in [24].

######  Lemma 3.1.

Let $\mathcal{H}$ be a $(k+2)$-uniform hypergraph and $0<\epsilon,\tau<1/2$.
Suppose that $\tau<1/(200\cdot(k+2)\cdot(k+2)!)$ and
$\Delta(\mathcal{H},\tau)\leq\epsilon/(12\cdot(k+2)!)$. Then there exists a
collection $\mathcal{C}$ of subsets (containers) of $V(\mathcal{H})$ such that

  1. 1.

Every independent set in $\mathcal{H}$ is a subset of some $C\in\mathcal{C}$;

  2. 2.

$\log|\mathcal{C}|\leq
1000\cdot(k+2)\cdot((k+2)!)^{3}\cdot|V(\mathcal{H})|\cdot\tau\cdot\log(1/\epsilon)\cdot\log(1/\tau)$;

  3. 3.

For every $C\in\mathcal{C}$, the induced subgraph $\mathcal{H}[C]$ has at most
$\epsilon|E(\mathcal{H})|$ many edges.

The main result in this section is the following theorem.

######  Theorem 3.2.

Let $k,r$ be fixed integers such that $r\geq k\geq 2$ and $k$ is even. Then
for any $0<\alpha<1$, there are constants $c=c(\alpha,k,r)$ and
$d=d(\alpha,k,r)$ such that the following holds. For infinitely many values of
$N$, there is a set $V$ of $N$ points in $\mathbb{R}^{d}$ such that no $r+3$
members of $V$ lie on an $r$-flat, and every subset of $V$ of size
$cN^{\frac{r+2}{2(k+1)}+\alpha}$ contains $k+2$ members on a $k$-flat.

Before we prove Theorem 3.2, let us show that it implies Theorem 1.1. In
dimensions $d_{0}\geq 3$ where $d_{0}$ is odd, we apply Theorem 3.2 with
$k=r=d_{0}-1$ to obtain a point set $V$ in $\mathbb{R}^{d}$ with the property
that no $d_{0}+2$ members lie on a $(d_{0}-1)$-flat, and every subset of size
$cN^{\frac{1}{2}+\frac{1}{2d_{0}}+\alpha}$ contains $d_{0}+1$ members on a
$(d_{0}-1)$-flat. By projecting $V$ to a generic $d_{0}$-dimensional subspace
of $\mathbb{R}^{d}$, we obtain $N$ points in $\mathbb{R}^{d_{0}}$ with no
$d_{0}+2$ members on a common hyperplane, and no
$cN^{\frac{1}{2}+\frac{1}{2d_{0}}+\alpha}$ members in general position.

In dimensions $d_{0}\geq 4$ where $d_{0}$ is even, we apply Theorem 3.2 with
$k=d_{0}-2$ and $r=d_{0}-1$ to obtain a point set $V$ in $\mathbb{R}^{d}$ with
the property that no $d_{0}+2$ members on a $(d_{0}-1)$-flat, and every subset
of size $cN^{\frac{1}{2}+\frac{1}{d_{0}-1}+\alpha}$ contains $d_{0}$ members
on a $(d_{0}-2)$-flat. By adding another point from this subset, we obtain
$d_{0}+1$ members on a $(d_{0}-1)$-flat. Hence, by projecting to $V$ a generic
$d_{0}$-dimensional subspace of $\mathbb{R}^{d}$, we obtain $N$ points in
$\mathbb{R}^{d_{0}}$ with no $d_{0}+2$ members on a common hyperplane, and no
$cN^{\frac{1}{2}+\frac{1}{d_{0}-1}+\alpha}$ members in general position. This
completes the proof of Theorem 1.1.

###### Proof of Theorem 3.2.

We set $d=d(\alpha,k,r)$ to be a sufficiently large integer depending on
$\alpha$, $k$, and $r$. Let $\mathcal{H}$ be the hypergraph with
$V(\mathcal{H})=[n]^{d}$ and $E(\mathcal{H})$ consists of non-degenerate
$(k+2)$-tuples $T$ such that $T$ lies on a $k$-flat. Let $C^{0}=[n]^{d}$,
$\mathcal{C}^{0}=\\{C^{0}\\}$, and $\mathcal{H}^{0}=\mathcal{H}$. In what
follows, we will apply the hypergraph container lemma to $\mathcal{H}^{0}$ to
obtain a family of containers $\mathcal{C}^{1}$. For each
$C_{j}^{1}\in\mathcal{C}^{1}$, we consider the induced hypergraph
$\mathcal{H}_{j}^{1}=\mathcal{H}[C^{1}_{j}]$, and we apply the hypergraph
container lemma to it. The collection of containers obtained from all
$\mathcal{H}_{j}^{1}$ will form another collection of containers
$\mathcal{C}^{2}$. We iterate this process until each container in
$\mathcal{C}^{i}$ is sufficiently small, and moreover, we will only produce a
small number of containers. As a final step, we apply the probabilistic method
to show the existence of the desired point set. We now flesh out the details
of this process.

We start by setting $C^{0}=[n]^{d},\mathcal{C}^{0}=\\{C^{0}\\}$, and set
$\mathcal{H}^{0}=\mathcal{H}[C^{0}]=\mathcal{H}$. Having obtained a collection
of containers $\mathcal{C}^{i}$, for each container
$C_{j}^{i}\in\mathcal{C}^{i}$ with $|C_{j}^{i}|\geq n^{\frac{k}{k+1}d+k}$, we
set $\mathcal{H}^{i}_{j}=\mathcal{H}[C_{j}^{i}]$. Let $\gamma=\gamma(i,j)$ be
defined by $|V(\mathcal{H}^{i}_{j})|=n^{d-\gamma}$. So,
$\gamma\leq\frac{d}{k+1}-k$. We set
$\tau=\tau(i,j)=n^{-\frac{k}{k+1}d+\gamma+\alpha}$ and
$\epsilon=\epsilon(i,j)=c_{1}n^{-\alpha}$, where $c_{1}=c_{1}(d,k)$ is a
sufficiently large constant depending on $d$ and $k$. Then we can verify the
following condition.

######  Claim 3.3.

$\Delta(\mathcal{H}^{i}_{j},\tau)\leq\epsilon/(12\cdot(k+2)!)$.

###### Proof.

Since $|V(\mathcal{H}^{i}_{j})|=n^{d-\gamma}$, $\gamma\leq\frac{d}{k+1}-k$,
and $d$ is sufficiently large, Lemma 2.1 implies that
$|E(\mathcal{H}^{i}_{j})|\geq c_{2}n^{(k+1)d-(k+2)\gamma}$ for some constant
$c_{2}=c_{2}(d,k)$. Hence, we have

$\frac{|V(\mathcal{H}^{i}_{j})|}{|E(\mathcal{H}^{i}_{j})|}\leq\frac{n^{d-\gamma}}{c_{2}n^{(k+1)d-(k+2)\gamma}}=\frac{1}{c_{2}n^{kd-(k+1)\gamma}}.$

On the other hand, by Lemma 2.4, we have

$\Delta_{\ell}(\mathcal{H}_{j}^{i})\leq n^{(d-\gamma)(k+1-\ell)+k}\text{\quad
for $\ell<k+2$},$

and obviously $\Delta_{k+2}(\mathcal{H}_{j}^{i})\leq 1$.

Applying these inequalities together with the definition of $\Delta$, we
obtain

$\displaystyle\Delta(\mathcal{H}^{i}_{j},\tau)$
$\displaystyle=\frac{2^{\binom{k+2}{2}-1}|V(\mathcal{H}^{i}_{j})|}{(k+2)|E(\mathcal{H}_{j}^{i})|}\sum_{\ell=2}^{k+2}\frac{\Delta_{\ell}(\mathcal{H}_{j}^{i})}{\tau^{\ell-1}2^{\binom{\ell-1}{2}}}$
$\displaystyle\leq\frac{c_{3}}{n^{kd-(k+1)\gamma}}\left(\sum_{\ell=2}^{k+1}\frac{n^{(k+1-\ell)(d-\gamma)+k}}{\tau^{\ell-1}}+\frac{1}{\tau^{k+1}}\right)$
$\displaystyle=\sum_{\ell=2}^{k+1}\frac{c_{3}}{\tau^{\ell-1}n^{(\ell-1)d-k-\ell\gamma}}+\frac{c_{3}}{\tau^{k+1}n^{kd-(k+1)\gamma}},$

for some constant $c_{3}=c_{3}(d,k)$. Let us remark that the summation above
is where we determined our $\tau$ and $\gamma$. In order to make the last term
small, we choose $\tau=n^{-\frac{k}{k+1}d+\gamma+\alpha}$. Having determined
$\tau$, in order for the first term in the summation to be small, we choose
$\gamma\leq\frac{d}{k+1}-k$.

By setting $\epsilon=c_{1}n^{-\alpha}$ with $c_{1}=c_{1}(d,k)$ sufficiently
large, we have

$\displaystyle\Delta(\mathcal{H}^{i}_{j},\tau)$ $\displaystyle\leq
c_{3}\left(\sum_{\ell=2}^{k+1}n^{-\frac{\ell-1}{k+1}d+\gamma+k-(\ell-1)\alpha}+n^{-(k+1)\alpha}\right)$
$\displaystyle\leq c_{3}kn^{-\alpha}+c_{3}n^{-(k+1)\alpha}$
$\displaystyle<\frac{\epsilon}{12(k+2)!}.$

This verifies the claimed condition. ∎

Given the condition above, we can apply Lemma 3.1 to $\mathcal{H}_{j}^{i}$
with chosen parameters $\tau$ and $\epsilon$. Hence we obtain a family of
containers $\mathcal{C}_{j}^{i+1}$ such that

$\displaystyle|\mathcal{C}_{j}^{i+1}|$ $\displaystyle\leq
2^{10^{3}(k+2)((k+2)!)^{3}|V(\mathcal{H}_{i}^{j})|\tau\log(1/\epsilon)\log(1/\tau)}$
$\displaystyle\leq 2^{c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n},$

for some constant $c_{4}=c_{4}(d,k)$. In the other case where
$|C_{j}^{i}|<n^{\frac{k}{k+1}d+k}$, we just define
$\mathcal{C}_{j}^{i+1}=\\{C_{j}^{i}\\}$. Then, for each container
$C\in\mathcal{C}_{j}^{i+1}$, we have either $|C|<n^{\frac{k}{k+1}d+k}$ or
$|E(\mathcal{H}[C])|\leq\epsilon|E(\mathcal{H}_{j}^{i})|\leq\epsilon^{i}|E(\mathcal{H})|$.
After applying this procedure for each container in $\mathcal{C}^{i}$, we
obtain a new family of containers
$\mathcal{C}^{i+1}=\bigcup\mathcal{C}^{i}_{j}$ such that

$|\mathcal{C}^{i+1}|\leq|\mathcal{C}^{i}|2^{c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n}\leq
2^{(i+1)c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n}.$

Notice that the number of edges in $\mathcal{H}_{j}^{i}$ shrinks by a factor
of $c_{1}n^{-\alpha}$ whenever $i$ increases by one, while on the other hand,
Lemma 2.1 tells us that every large subset $C\subset[n]^{d}$ induces many
edges in $\mathcal{H}$. Hence, after at most $t\leq c_{5}/\alpha$ iterations,
for some constant $c_{5}=c_{5}(d,k)$, we obtain a collection of containers
$\mathcal{C}=\mathcal{C}^{t}$ such that: each container $C\in\mathcal{C}$
satisfies $|C|<n^{\frac{k}{k+1}d+k}$; every independent set of $\mathcal{H}$
is a subset of some $C\in\mathcal{C}$; and

$|\mathcal{C}|\leq 2^{(c_{5}/\alpha)c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n}.$

Before we construct the desired point set, we make the following crude
estimate.

######  Claim 3.4.

The grid $[n]^{d}$ contains at most $O(n^{(r+1)d+2r})$ many $(r+3)$-tuples
that lie on an $r$-flat.

###### Proof.

Let $T$ be an arbitrary $(r+3)$-tuple that spans a $j$-flat. There are at most
$n^{(j+1)d}$ ways to choose a subset $T^{\prime}\subset T$ of size $j+1$ that
spans the affine hull of $T$. After this $T^{\prime}$ is determined, there are
at most $n^{(r+2-j)j}$ ways to add the remaining $r+2-j$ points from the
$j$-flat spanned by $T^{\prime}$. Then the total number of $(r+3)$-tuples that
lie on a $r$-flat is at most

$\sum_{j=1}^{r}n^{(j+1)d+(r+2-j)j}\leq\sum_{j=1}^{r}n^{(j+1)d+(r+2-j)r}\leq
rn^{(r+1)d+2r},$

since we can assume $d>r$. ∎

Now, we randomly select a subset of $[n]^{d}$ by keeping each point
independently with probability $p$. Let $S$ be the set of selected elements.
Then for each $(r+3)$-tuple $T$ in $S$ that lies on an $r$-flat, we delete one
point from $T$. We denote the resulting set of points by $S^{\prime}$. By the
claim above, the number of $(r+3)$-tuples in $[n]^{d}$ that lie on a $r$-flat
is at most $c_{6}n^{(r+1)d+2r}$ for some constant $c_{6}=c_{6}(r)$. Therefore,

$\mathbb{E}[|S^{\prime}|]\geq pn^{d}-c_{6}p^{r+3}n^{(r+1)d+2r}.$

By setting $p=(2c_{6})^{-\frac{1}{r+2}}n^{-\frac{r}{r+2}(d+2)}$, we have

$\mathbb{E}[|S^{\prime}|]\geq\frac{pn^{d}}{2}=\Omega(n^{\frac{2(d-r)}{r+2}}).$

Finally, we set $m=(c_{7}/\alpha)n^{\frac{d}{k+1}+2\alpha}$ for some
sufficiently large constant $c_{7}=c_{7}(d,k,r)$. Let $X$ denote the number of
independent sets of size $m$ in $S^{\prime}$. Using the family of containers
$\mathcal{C}$, we have

$\displaystyle\mathbb{E}[X]$
$\displaystyle\leq|\mathcal{C}|\cdot\binom{n^{\frac{k}{k+1}d+k}}{m}p^{m}$
$\displaystyle\leq\left(2^{(c_{5}/\alpha)c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n}\right)\left(\frac{en^{\frac{k}{k+1}d+k}p}{m}\right)^{m}$
$\displaystyle\leq\left(2^{(c_{5}/\alpha)c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n}\right)\left(c_{8}\alpha\frac{n^{\frac{k}{k+1}d+k}\cdot
n^{-\frac{r}{r+2}(d+2)}}{n^{\frac{d}{k+1}+2\alpha}}\right)^{m}$
$\displaystyle\leq\left(2^{(c_{5}/\alpha)c_{4}n^{\frac{d}{k+1}+\alpha}\log^{2}n}\right)\left(c_{8}\alpha
n^{\frac{2(k-r-1)d}{(k+1)(r+2)}+k-\frac{2r}{r+2}-2\alpha}\right)^{(c_{7}/\alpha)n^{\frac{d}{k+1}+2\alpha}},$

for some constant $c_{8}=c_{8}(d,k,r)$. Since $r\geq k$, $0<\alpha<1$, and $d$
is large, for $n$ sufficiently large, we have

$c_{8}\alpha n^{\frac{2(k-r-1)d}{(k+1)(r+2)}+k-\frac{2r}{r+2}-2\alpha}<1/2.$

Hence, we have $\mathbb{E}[X]\leq o(1)$ as $n$ tends to infinity. Notice that
$|S^{\prime}|$ is exponentially concentrated around its mean by Chernoff’s
inequality. Therefore, some realization of $S^{\prime}$ satisfies:
$|S^{\prime}|=N=\Omega(n^{2(d-r)/(r+2)})$; $S^{\prime}$ contains no
$(r+3)$-tuples on a $r$-flat; and $\mathcal{H}[S^{\prime}]$ does not contain
an independent set of size

$m=(c_{7}/\alpha)n^{\frac{d}{k+1}+2\alpha}\leq
cN^{\frac{r+2}{2(k+1)}+\frac{(r+2)r}{2(k+1)(d-r)}+\frac{r+2}{d}2\alpha}\leq
cN^{\frac{r+2}{2(k+1)}+\alpha},$

for some constant $c=c(\alpha,d,k,r)$. Here we assume $d$ is sufficiently
large so that

$\frac{(r+2)r}{2(k+1)(d-r)}+\frac{r+2}{d}2\alpha\leq\alpha.$

This completes the proof. ∎

##  4 No $d+s$ points on a hyperplane: Proof of Theorem 1.2

In this section, we prove Theorem 1.2. The proof is essentially the same as in
the previous section with a different choice of parameters. For the reader’s
convenience, we include the details here. We start by proving the following
theorem.

######  Theorem 4.1.

Let $k,r,s$ be fixed integers such that $r\geq k\geq 2$, $s\geq 2$, $k$ is
even, and $\frac{r+\frac{s-1}{2}}{r+s-1}<\frac{k}{k+1}$. Then for any
$0<\alpha<1$, there are constants $c=c(\alpha,k,r,s)$ and $d=d(\alpha,k,r,s)$
such that the following holds. For infinitely many values of $N$, there is a
set $V$ of $N$ points in $\mathbb{R}^{d}$ such that no $r+s$ members of $V$
lie on an $r$-flat, and every subset of $V$ of size $cN^{\frac{1}{2}+\alpha}$
contains $k+2$ members on a $k$-flat.

###### Proof.

Just as before, let $\mathcal{H}$ be the hypergraph with
$V(\mathcal{H})=[n]^{d}$ and $E(\mathcal{H})$ consists of non-degenerate
$(k+2)$-tuples $T$ such that $T$ lies on a $k$-flat. Let $x=x(k,r,s)$ be a
constant that will be determined later, and set
$C^{0}=[n]^{d},\mathcal{C}^{0}=\\{C^{0}\\}$, and
$\mathcal{H}^{0}=\mathcal{H}[C^{0}]=\mathcal{H}$. Having obtained a collection
of containers $\mathcal{C}^{i}$, for each container
$C_{j}^{i}\in\mathcal{C}^{i}$ with $|C_{j}^{i}|\geq n^{xd+k}$, we set
$\mathcal{H}^{i}_{j}=\mathcal{H}[C_{j}^{i}]$. Let $\gamma=\gamma(i,j)$ be
defined by $|V(\mathcal{H}^{i}_{j})|=n^{d-\gamma}$. So, $\gamma\leq d-xd-k$.
We set $\tau=\tau(i,j)=n^{-xd+\gamma+\alpha}$ and
$\epsilon=\epsilon(i,j)=c_{1}n^{-\alpha}$, where $c_{1}=c_{1}(d,k,x)$ is a
sufficiently large constant. We now make the following claim.

######  Claim 4.2.

If $x\leq\frac{k}{k+1}$, then
$\Delta(\mathcal{H}^{i}_{j},\tau)\leq\epsilon/(12\cdot(k+2)!)$.

###### Proof.

Since $|V(\mathcal{H}^{i}_{j})|=n^{d-\gamma}$, $\gamma\leq d-xd-k$, and $d$ is
sufficiently large, Lemma 2.1 implies that $|E(\mathcal{H}^{i}_{j})|\geq
c_{2}n^{(k+1)d-(k+2)\gamma}$ for some constant $c_{2}=c_{2}(d,k)$. Hence, we
have

$\frac{|V(\mathcal{H}^{i}_{j})|}{|E(\mathcal{H}^{i}_{j})|}\leq\frac{n^{d-\gamma}}{c_{2}n^{(k+1)d-(k+2)\gamma}}=\frac{1}{c_{2}n^{kd-(k+1)\gamma}}.$

On the other hand, by Lemma 2.4, we have

$\Delta_{\ell}(\mathcal{H}_{j}^{i})\leq n^{(d-\gamma)(k+1-\ell)+k}\text{\quad
for $\ell<k+2$},$

and obviously $\Delta_{k+2}(\mathcal{H}_{j}^{i})\leq 1$. Applying these
inequalities together with the definition of $\Delta$, we obtain

$\displaystyle\Delta(\mathcal{H}^{i}_{j},\tau)$
$\displaystyle=\frac{2^{\binom{k+2}{2}-1}|V(\mathcal{H}^{i}_{j})|}{(k+2)|E(\mathcal{H}_{j}^{i})|}\sum_{\ell=2}^{k+2}\frac{\Delta_{\ell}(\mathcal{H}_{j}^{i})}{\tau^{\ell-1}2^{\binom{\ell-1}{2}}}$
$\displaystyle\leq\frac{c_{3}}{n^{kd-(k+1)\gamma}}\left(\sum_{\ell=2}^{k+1}\frac{n^{(k+1-\ell)(d-\gamma)+k}}{\tau^{\ell-1}}+\frac{1}{\tau^{k+1}}\right)$
$\displaystyle=\sum_{\ell=2}^{k+1}\frac{c_{3}}{\tau^{\ell-1}n^{(\ell-1)d-k-\ell\gamma}}+\frac{c_{3}}{\tau^{k+1}n^{kd-(k+1)\gamma}},$

for some constant $c_{3}=c_{3}(d,k,x)$. Let us remark that the summation above
is where we determined our $\tau$ and $\gamma$. In order to make the last term
small, we set $\tau=n^{-xd+\gamma+\alpha}$, and recall that we assumed
$x\leq\frac{k}{k+1}$. Having determined $\tau$, in order for the first term in
the summation to be small, we choose $\gamma\leq d-xd-k$.

By setting $\epsilon=c_{1}n^{-\alpha}$ with $c_{1}=c_{1}(d,k,x)$ sufficiently
large, we have

$\displaystyle\Delta(\mathcal{H}^{i}_{j},\tau)$ $\displaystyle\leq
c_{3}\left(\sum_{\ell=2}^{k+1}n^{-(\ell-1)(1-x)d+k+\gamma-(\ell-1)\alpha}+n^{-(k-x(k+1))d-(k+1)\alpha}\right)$
$\displaystyle\leq c_{3}kn^{-\alpha}+c_{3}n^{-(k+1)\alpha}$
$\displaystyle<\frac{\epsilon}{12(k+2)!}.$

This verifies the claimed condition. ∎

Given the claim above, we can apply Lemma 3.1 to $\mathcal{H}_{j}^{i}$ with
the chosen parameters $\tau$ and $\epsilon$. Hence, we obtain a family of
containers $\mathcal{C}_{j}^{i+1}$ such that

$\displaystyle|\mathcal{C}_{j}^{i+1}|$ $\displaystyle\leq
2^{10^{3}(k+2)((k+2)!)^{3}|V(\mathcal{H}_{i}^{j})|\tau\log(1/\epsilon)\log(1/\tau)}$
$\displaystyle\leq 2^{c_{4}n^{d-xd+\alpha}\log^{2}n},$

for some constant $c_{4}=c_{4}(d,k)$. In the other case, where
$|C_{j}^{i}|<n^{xd+k}$, we just define
$\mathcal{C}_{j}^{i+1}=\\{C_{j}^{i}\\}$. Then, for each container
$C\in\mathcal{C}_{j}^{i+1}$, we have either $|C|<n^{xd+k}$ or
$|E(\mathcal{H}[C])|\leq\epsilon|E(\mathcal{H}_{j}^{i})|\leq\epsilon^{i}|E(\mathcal{H})|$.
After applying this procedure for each container in $\mathcal{C}^{i}$, we
obtain a new family of containers
$\mathcal{C}^{i+1}=\bigcup\mathcal{C}^{i}_{j}$ such that

$|\mathcal{C}^{i+1}|\leq|\mathcal{C}^{i}|2^{c_{4}n^{d-xd+\alpha}\log^{2}n}\leq
2^{(i+1)c_{4}n^{d-xd+\alpha}\log^{2}n}.$

Notice that the number of edges in $\mathcal{H}_{j}^{i}$ shrinks by a factor
of $c_{1}n^{-\alpha}$ whenever $i$ increases by one, while on the other hand,
Lemma 2.1 tells us that every large subset $C\subset[n]^{d}$ induces many
edges in $\mathcal{H}$. Hence, after at most $t\leq c_{5}/\alpha$ iterations,
for some constant $c_{5}=c_{5}(d,k,x)$, we obtain a collection of containers
$\mathcal{C}=\mathcal{C}^{t}$ such that: each container $C\in\mathcal{C}$
satisfies $|C|<n^{xd+k}$; every independent set of $\mathcal{H}$ is a subset
of some $C\in\mathcal{C}$; and

$|\mathcal{C}|\leq 2^{(c_{5}/\alpha)c_{4}n^{d-xd+\alpha}\log^{2}n}.$

Now, we randomly select a subset of $[n]^{d}$ by keeping each point
independently with probability $p$. Let $S$ be the set of selected elements.
Then for each $(r+s)$-tuple $T$ in $S$ that lies on an $r$-flat, we delete one
point from $T$. We denote the resulting set of points by $S^{\prime}$. By the
proof of Claim 3.4, the number of $(r+s)$-tuples in $[n]^{d}$ that lie on a
$r$-flat is at most $c_{6}n^{(r+1)d+(s-1)r}$ for some constant
$c_{6}=c_{6}(r)$. Therefore,

$\mathbb{E}[|S^{\prime}|]\geq pn^{d}-c_{6}p^{r+s}n^{(r+1)d+(s-1)r}.$

By setting $p=(2c_{6})^{-\frac{1}{r+s-1}}n^{-\frac{r}{r+s-1}(d+s-1)}$, we have

$\mathbb{E}[|S^{\prime}|]\geq\frac{pn^{d}}{2}=\Omega(n^{\frac{(s-1)(d-r)}{r+s-1}}).$

Finally, we set $m=(c_{7}/\alpha)n^{d-xd+2\alpha}$ for some sufficiently large
constant $c_{7}=c_{7}(d,k,r,x)$. Let $X$ denote the number of independent sets
of size $m$ in $S^{\prime}$. Using the family of containers $\mathcal{C}$, we
have

$\displaystyle\mathbb{E}[X]$
$\displaystyle\leq|\mathcal{C}|\cdot\binom{n^{xd+k}}{m}p^{m}$
$\displaystyle\leq\left(2^{(c_{5}/\alpha)c_{4}n^{d-xd+\alpha}\log^{2}n}\right)\left(\frac{en^{xd+k}p}{m}\right)^{m}$
$\displaystyle\leq\left(2^{(c_{5}/\alpha)c_{4}n^{d-xd+\alpha}\log^{2}n}\right)\left(c_{8}\alpha\frac{n^{xd+k}\cdot
n^{-\frac{r}{r+s-1}(d+s-1)}}{n^{d-xd+2\alpha}}\right)^{m}$
$\displaystyle\leq\left(2^{(c_{5}/\alpha)c_{4}n^{d-xd+\alpha}\log^{2}n}\right)\left(c_{8}\alpha
n^{2xd+k-2\alpha-\frac{r(s-1)}{r+s-1}-\frac{2r+s-1}{r+s-1}d}\right)^{(c_{7}/\alpha)n^{d-xd+2\alpha}},$

for some constant $c_{8}=c_{8}(d,k,r,x)$. Since $r\geq k$, $0<\alpha<1$, and
$d$ is large, for $n$ sufficiently large, we have

$c_{8}\alpha
n^{2xd+k-2\alpha-\frac{r(s-1)}{r+s-1}-\frac{2r+s-1}{r+s-1}d}<1/2,$

as long as

$x\leq\frac{r+\frac{s-1}{2}}{r+s-1}+\frac{1}{2d}\frac{r(s-1)}{r+s-1}-\frac{k}{2d}.$

We now set $x(k,r,s)$ equal to the right-hand side of the inequality above.
Moreover, using our assumption that
$\frac{r+\frac{s-1}{2}}{r+s-1}<\frac{k}{k+1}$, we have $x\leq\frac{k}{k+1}$
for $d=d(\alpha,k,r,s)$ sufficiently large. Thus, satisfying the condition for
Claim 4.2.

Hence, we have $\mathbb{E}[X]=o(1)$ as $n$ tends to infinity. Notice that
$|S^{\prime}|$ is exponentially concentrated around its mean by Chernoff’s
inequality. Therefore, some realization of $S^{\prime}$ satisfies:
$|S^{\prime}|=N=\Omega(n^{(s-1)(d-r)/(r+s-1)})$, $S^{\prime}$ contains no
$(r+s)$-tuples on an $r$-flat, and $\mathcal{H}[S^{\prime}]$ does not contain
an independent set of size

$m=(c_{7}/\alpha)n^{d-xd+2\alpha}\leq
cN^{\frac{1}{2}+\frac{k/2}{\frac{(s-1)(d-r)}{r+s-1}}+\frac{2\alpha}{\frac{(s-1)(d-r)}{r+s-1}}}\leq
N^{\frac{1}{2}+\alpha},$

for some constant $c=c(\alpha,d,k,r,x)$. Here we assume $d$ is sufficiently
large so that

$\frac{k/2}{\frac{(s-1)(d-r)}{r+s-1}}+\frac{2\alpha}{\frac{(s-1)(d-r)}{r+s-1}}\leq\alpha.$

This completes the proof. ∎

###### Proof of Theorem 1.2.

In dimensions $d_{0}\geq 3$, where $d_{0}$ is odd, we obtain an upper bound
for $\alpha_{d_{0},s_{0}}(N)$, where

$\frac{2d_{0}+s_{0}-2}{2d_{0}+2s_{0}-2}<\frac{d_{0}-1}{d_{0}},$

by applying Theorem 4.1 with $k=r=d_{0}-1$ and $s=s_{0}+1$. Hence, we have

$\frac{r+\frac{s-1}{2}}{r+s-1}<\frac{k}{k+1},$

and therefore, we obtain a point set $V$ in $\mathbb{R}^{d}$ with the property
that no $d_{0}+s_{0}$ members lie on a $(d_{0}-1)$-flat, and every subset of
size $cN^{\frac{1}{2}+\alpha}$ contains $d_{0}+1$ members on a
$(d_{0}-1)$-flat. By projecting $V$ to a generic $d_{0}$-dimensional subspace
of $\mathbb{R}^{d}$, we obtain $N$ points in $\mathbb{R}^{d_{0}}$ with no
$d_{0}+s_{0}$ members on a common hyperplane, and no $cN^{\frac{1}{2}+\alpha}$
members in general position.

In dimensions $d_{0}\geq 4$ where $d_{0}$ is even, we obtain an upper bound
for $\alpha_{d_{0},s_{0}}(N)$, where

$\frac{2d_{0}+s_{0}-2}{2d_{0}+2s_{0}-2}<\frac{d_{0}-2}{d_{0}-1},$

by applying Theorem 4.1 with $k=d_{0}-2$, $r=d_{0}-1$, and $s=s_{0}+1$. Hence,
we have

$\frac{r+\frac{s-1}{2}}{r+s-1}<\frac{k}{k+1},$

and therefore, we obtain a point set $V$ in $\mathbb{R}^{d}$ with the property
that no $d_{0}+s_{0}$ members on a $(d_{0}-1)$-flat, and every subset of size
$cN^{\frac{1}{2}+\alpha}$ contains $d_{0}$ members on a $(d_{0}-2)$-flat. By
adding another point from this subset, we obtain $d_{0}+1$ members on a
$(d_{0}-1)$-flat. Hence, by projecting to $V$ a generic $d_{0}$-dimensional
subspace of $\mathbb{R}^{d}$, we obtain $N$ points in $\mathbb{R}^{d_{0}}$
with no $d_{0}+s_{0}$ members on a common hyperplane, and no
$cN^{\frac{1}{2}+\alpha}$ members in general position. This completes the
proof of Theorem 1.2. ∎

##  5 Avoiding non-trivial solutions: Proof of Theorem 1.3

In this section, we will give a proof of Theorem 1.3. Let $V\subset[n]^{d}$
such that there are no $k+2$ points that lie on a $k$-flat. In [16], Lefmann
showed that $|V|\leq O\left(n^{\frac{d}{\lfloor(k+2)/2\rfloor}}\right)$. To
see this, assume that $k$ is even and consider all elements of the form
$v_{1}+\dots+v_{\frac{k}{2}+1}$, where $v_{i}\neq v_{j}$ and $v_{i}\in V$. All
of these elements are distinct, since otherwise we would have $k+2$ points on
a $k$-flat. In other words, the equation

$\left(\textbf{x}_{1}+\dots+\textbf{x}_{\frac{k}{2}+1}\right)-\left(\textbf{x}_{\frac{k}{2}+2}+\dots+\textbf{x}_{k+2}\right)=\textbf{0},$

does not have a solution with
$\\{\textbf{x}_{1},\dots,\textbf{x}_{\frac{k}{2}+1}\\}$ and
$\\{\textbf{x}_{\frac{k}{2}+2},\dots,\textbf{x}_{k+2}\\}$ being two different
$(\frac{k}{2}+1)$-tuples of $V$. Therefore, we have
$\binom{|V|}{\frac{k}{2}+1}\leq(kn)^{d}$, and this implies Lefmann’s bound.

More generally, let us consider the equation

$c_{1}\textbf{x}_{1}+c_{2}\textbf{x}_{2}+\dots+c_{r}\textbf{x}_{r}=\textbf{0},$
(5.1)

with constant coefficients $c_{i}\in\mathbb{Z}$ and $\sum_{i}c_{i}=0$. Here,
the variables $\textbf{x}_{i}$ takes value in $\mathbb{Z}^{j}$. A solution
$(\textbf{x}_{1},\dots,\textbf{x}_{r})$ to equation (5.1) is called _trivial_
if there is a partition
$\mathcal{P}:[r]=\mathcal{I}_{1}\cup\dots\cup\mathcal{I}_{t}$, such that
$\textbf{x}_{j}=\textbf{x}_{\ell}$ if and only if $j,\ell\in\mathcal{I}_{i}$,
and $\sum_{j\in\mathcal{I}_{i}}c_{j}=0$ for all $i\in[t]$. In other words,
being trivial means that, after combining like terms, the coefficient of each
$\textbf{x}_{i}$ becomes zero. Otherwise, we say that the solution
$(\textbf{x}_{1},\dots,\textbf{x}_{r})$ is _non-trivial_. A natural extremal
problem is to determine the maximum size of a set $A\subset[n]^{d}$ with only
trivial solutions to (5.1). When $d=1$, this is a classical problem in
additive number theory, and we refer the interested reader to [23, 19, 17, 6].

By combining the arguments of Cilleruelo and Timmons [6] and Jia [14], we
establish the following theorem.

######  Theorem 5.1.

Let $d,r$ be fixed positive integers. Suppose $V\subset[n]^{d}$ has only
trivial solutions to each equation of the form

$c_{1}\left((\textbf{x}_{1}+\dots+\textbf{x}_{r})-(\textbf{x}_{r+1}+\dots+\textbf{x}_{2r})\right)=c_{2}\left((\textbf{x}_{2r+1}+\dots+\textbf{x}_{3r})-(\textbf{x}_{3r+1}+\dots+\textbf{x}_{4r})\right),$
(5.2)

for integers $c_{1},c_{2}$ such that $1\leq c_{1},c_{2}\leq
n^{\frac{d}{2rd+1}}$. Then we have

$|V|\leq O\left(n^{\frac{d}{2r}\left(1-\frac{1}{2rd+1}\right)}\right).$

Notice that Theorem 1.3 follows from Theorem 5.1. Indeed, when $k+2$ is
divisible by $4$, we set $r=(k+2)/4$. If $V\subset[n]^{d}$ contains $k+2$
points $\\{v_{1},\dots,v_{k+2}\\}$ that is a non-trivial solution to (5.2)
with $\textbf{x}_{i}=v_{i}$, then $\\{v_{1},\dots,v_{k+2}\\}$ must lie on a
$k$-flat. Hence, when $k+2$ is divisible by $4$, we have

$a(d,k,n)\leq
O\left(n^{\frac{d}{(k+2)/2}\left(1-\frac{1}{(k+2)d/2+1}\right)}\right).$

Since we have $a(d,k,n)<a(d,k-1,n)$, this implies that for all $k\geq 2$, we
have

$a(d,k,n)\leq
O\left(n^{\frac{d}{2\lfloor(k+2)/4\rfloor}\left(1-\frac{1}{2\lfloor(k+2)/4\rfloor
d+1}\right)}\right).$

In the proof of Theorem 5.1, we need the following well-known lemma (see e.g.
[6]Lemma 2.1 and [23]Theorem 4.1). For $U,T\subset\mathbb{Z}^{d}$ and
$x\in\mathbb{Z}^{d}$, we define

$\Phi_{U-T}(x)=\\{(u,t):u-t=x,u\in U,t\in T\\}.$

######  Lemma 5.2.

For finite sets $U,T\subset\mathbb{Z}^{d}$, we have

$\frac{(|U||T|)^{2}}{|U+T|}\leq\sum_{x\in\mathbb{Z}^{d}}|\Phi_{U-U}(x)|\cdot|\Phi_{T-T}(x)|.$

###### Proof of Theorem 5.1.

Let $d$, $r$, and $V$ be as given in the hypothesis. Let $m\geq 1$ be an
integer that will be determined later. We define

$S_{r}=\\{v_{1}+\dots+v_{r}:v_{i}\in V,v_{i}\neq v_{j}\\},$

and a function

$\sigma:\binom{V}{r}\rightarrow S_{r},\ \\{v_{1},\dots,v_{r}\\}\mapsto
v_{1}+\dots+v_{r}.$

Notice that $\sigma$ is a bijection. Indeed, suppose on the contrary that

$v_{1}+\dots+v_{r}=v^{\prime}_{1}+\dots+v^{\prime}_{r}$

for two different $r$-tuples in $V$. Then by setting
$(\textbf{x}_{1},\dots,\textbf{x}_{r})=(v_{1},\dots,v_{r})$,
$(\textbf{x}_{r+1},\dots,\textbf{x}_{2r})=(v^{\prime}_{1},\dots,v^{\prime}_{r})$,
$(\textbf{x}_{2r+1},\dots,\textbf{x}_{3r})=(\textbf{x}_{3r+1},\dots,\textbf{x}_{4r})$
arbitrarily, and $c_{1}=c_{2}=1$, we obtain a non-trivial solution to (5.2),
which is a contradiction. In particular, we have $|S_{r}|=\binom{|V|}{r}$.

For $j\in[m]$ and $w\in\mathbb{Z}_{j}^{d}$, we let

$U_{j,w}=\\{u\in\mathbb{Z}^{d}:ju+w\in S_{r}\\}.$

Notice that for fixed $j\in[m]$, we have

$\sum_{w\in\mathbb{Z}_{j}^{d}}|U_{j,w}|=\sum_{w\in\mathbb{Z}_{j}^{d}}|\\{v\in
S_{r}:v\equiv w\text{ mod $j$}\\}|=|S_{r}|.$

Applying Jensen’s inequality to above, we have

$\sum_{w\in\mathbb{Z}_{j}^{d}}|U_{j,w}|^{2}\geq|S_{r}|^{2}/j^{d}.$ (5.3)

For $i\geq 0$, we define

$\Phi^{i}_{U_{j,w}-U_{j,w}}(x)=\\{(u_{1},u_{2})\in\Phi_{U_{j,w}-U_{j,w}}(x):|\sigma^{-1}(ju_{1}+w)\cap\sigma^{-1}(ju_{2}+w)|=i\\}.$

It’s obvious that these sets form a partition of $\Phi_{U_{j,w}-U_{j,w}}(x)$.
We also make the following claims.

######  Claim 5.3.

For a fixed $x\in\mathbb{Z}^{d}$, we have

$\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}|\Phi^{0}_{U_{j,w}-U_{j,w}}(x)|\leq
1,$

###### Proof.

For the sake of contradiction, suppose the summation above is at least two,
then we have $(u_{1},u_{2})\in\Phi^{0}_{U_{j,w}-U_{j,w}}(x)$ and
$(u_{3},u_{4})\in\Phi^{0}_{U_{j^{\prime},w^{\prime}}-U_{j^{\prime},w^{\prime}}}(x)$
such that either $(u_{1},u_{2})\neq(u_{3},u_{4})$ or
$(j,w)\neq(j^{\prime},w^{\prime})$.

Let $s_{1},s_{2},s_{3},s_{4}\in S_{r}$ such that $s_{1}=ju_{1}+w$,
$s_{2}=ju_{2}+w$, $s_{3}=j^{\prime}u_{3}+w^{\prime}$,
$s_{4}=j^{\prime}u_{4}+w^{\prime}$ and write
$\sigma^{-1}(s_{i})=\\{v_{i,1},\dots,v_{i,r}\\}$. Notice that
$u_{1}-u_{2}=x=u_{3}-u_{4}$. Putting these equations together gives us

$j^{\prime}((v_{1,1}+\dots+v_{1,r})-(v_{2,1}+\dots+v_{2,r}))=j((v_{3,1}+\dots+v_{3,r})-(v_{4,1}+\dots+v_{4,r})).$
(5.4)

It suffices to show that (5.4) can be seem as a non-trivial solution to (5.2).
The proof now falls into the following cases.

_Case 1._ Suppose $j\neq j^{\prime}$. Without loss of generality we can assume
$j^{\prime}>j$. Notice that $(u_{1},u_{2})\in\Phi^{0}_{U_{j,w}-U_{j,w}}(x)$
implies

$\\{v_{1,1},\dots,v_{1,r}\\}\cap\\{v_{2,1},\dots,v_{2,r}\\}=\emptyset.$

Then after combining like terms in (5.4), the coefficient of $v_{1}^{1}$ is at
least $j^{\prime}-j$, which means this is indeed a non-trivial solution to
(5.2).

_Case 2._ Suppose $j=j^{\prime}$, then we must have $s_{1}\neq s_{3}$. Indeed,
if $s_{1}=s_{3}$, we must have $w=w^{\prime}$ (as $s_{1}$ modulo $j$ equals
$s_{3}$ modulo $j^{\prime}$) and $s_{2}=s_{4}$ (as
$j^{\prime}(s_{1}-s_{2})=j(s_{3}-s_{4})$). This is a contradiction to either
$(u_{1},u_{2})\neq(u_{3},u_{4})$ or $(j,w)\neq(j^{\prime},w^{\prime})$.

Given $s_{1}\neq s_{3}$, we can assume, without loss of generality,
$v_{1,1}\not\in\\{v_{3,1},\dots,v_{3,r}\\}$. Again, we have
$\\{v_{1,1},\dots,v_{1,r}\\}\cap\\{v_{2,1},\dots,v_{2,r}\\}=\emptyset$. Hence,
after combining like terms in (5.4), the coefficient of $v_{1}^{1}$ is
positive and we have a non-trivial solution to (5.2).∎

######  Claim 5.4.

For a finite set $T\subset\mathbb{Z}^{d}$, and fixed integers $i,j\geq 1$, we
have

$\sum_{w\in\mathbb{Z}_{j}^{d}}\sum_{x\in\mathbb{Z}^{d}}|\Phi^{i}_{U_{j,w}-U_{j,w}}(x)|\cdot|\Phi_{T-T}(x)|\leq|V|^{2r-i}|T|.$

###### Proof.

The summation on the left-hand side counts all (ordered) quadruples
$(u_{1},u_{2},t_{1},t_{2})$ such that
$(u_{1},u_{2})\in\Phi^{i}_{U_{j,w}-U_{j,w}}(t_{1}-t_{2})$. For each such a
quadruple, let $s_{1},s_{2}\in S_{r}$ such that

$s_{1}=ju_{1}+w\text{\quad and\quad}s_{2}=ju_{2}+w.$

There are at most $|V|^{2r-i}$ ways to choose a pair $(s_{1},s_{2})$
satisfying $|\sigma^{-1}(s_{1})\cap\sigma^{-1}(s_{2})|=i$. Such a pair
$(s_{1},s_{2})$ determines $(u_{1},u_{2})$ uniquely. Moreover, $(s_{1},s_{2})$
also determines the quantity

$t_{1}-t_{2}=u_{1}-u_{2}=\frac{s_{1}-w}{j}-\frac{s_{2}-w}{j}=\frac{1}{j}(s_{1}-s_{2}).$

After such a pair $(s_{1},s_{2})$ is chosen, there are at most $|T|$ ways to
choose $t_{1}$ and this will also determine $t_{2}$. So we conclude the claim
by multiplication.∎

Now, we set $T=\mathbb{Z}_{\ell}^{d}$ for some integer $\ell$ to be determined
later. Notice that $U_{j,w}+T\subset\\{0,1,\dots,\lfloor
rn/j\rfloor+\ell-1\\}^{d}$, which implies

$|U_{j,w}+T|\leq(rn/j+\ell)^{d}.$ (5.5)

By Lemma 5.2, we have

$\frac{|U_{j,w}|^{2}||T|^{2}}{|U_{j,w}+T|}\leq\sum_{x\in\mathbb{Z}^{d}}|\Phi_{U_{j,w}-U_{j,w}}(x)|\cdot|\Phi_{T-T}(x)|.$

Summing over all $j\in[m]$ and $w\in\mathbb{Z}_{j}^{d}$, and using Claims 5.3
and 5.4, we can compute

$\displaystyle\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}\frac{|U_{j,w}|^{2}||T|^{2}}{|U_{j,w}+T|}$
$\displaystyle\leq\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}\sum_{x\in\mathbb{Z}^{d}}|\Phi_{U_{j,w}-U_{j,w}}(x)|\cdot|\Phi_{T-T}(x)|$
$\displaystyle=\sum_{x\in\mathbb{Z}^{d}}\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}\left(|\Phi^{0}_{U_{j,w}-U_{j,w}}(x)|+\sum_{i=1}^{r}|\Phi^{i}_{U_{j,w}-U_{j,w}}(x)|\right)|\Phi_{T-T}(x)|$
$\displaystyle\leq\sum_{x\in\mathbb{Z}^{d}}|\Phi_{T-T}(x)|\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}|\Phi^{0}_{U_{j,w}-U_{j,w}}(x)|+\sum_{j\in[m]}\sum_{i=1}^{r}|V|^{2r-i}\ell^{d}$
$\displaystyle\leq\sum_{x\in\mathbb{Z}^{d}}\Phi_{T-T}(x)+\sum_{j\in[m]}\sum_{i=1}^{r-1}|V|^{2r-i}\ell^{d}$
$\displaystyle\leq\ell^{2d}+rm|V|^{2r-1}\ell^{d},$

On the other hand, using (5.3) and (5.5), we can compute

$\displaystyle\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}\frac{|U_{j,w}|^{2}||T|^{2}}{|U_{j,w}+T|}$
$\displaystyle\geq\sum_{j\in[m]}\sum_{w\in\mathbb{Z}_{j}^{d}}\frac{|U_{j,w}|^{2}\ell^{2d}}{(rn/j+\ell)^{d}}$
$\displaystyle\geq\sum_{j\in[m]}\frac{|S_{r}|^{2}\ell^{2d}}{j^{d}(rn/j+\ell)^{d}}$
$\displaystyle=\sum_{j\in[m]}\frac{|S_{r}|^{2}\ell^{2d}}{(rn+j\ell)^{d}}$
$\displaystyle\geq\frac{m|S_{r}|^{2}\ell^{2d}}{(rn+m\ell)^{d}},$

Combining the two inequalities above gives us

$\displaystyle\frac{m|S_{r}|^{2}\ell^{2d}}{(rn+m\ell)^{d}}\leq\ell^{2d}+rm|V|^{2r-1}\ell^{d}$
$\displaystyle\implies$
$\displaystyle|S_{r}|^{2}\leq\frac{(rn+m\ell)^{d}}{m}+r|V|^{2r-1}\frac{(rn+m\ell)^{d}}{\ell^{d}}.$

By setting $m=n^{\frac{d}{2rd+1}}$ and $\ell=n^{1-\frac{d}{2rd+1}}$, we get

$\binom{|V|}{r}^{2}=|S_{r}|^{2}\leq
cn^{d-\frac{d}{2rd+1}}+c|V|^{2r-1}n^{\frac{d^{2}}{2rd+1}},$

for some constant $c$ depending only on $d$ and $r$. We can solve from this
inequality that

$|V|=O\left(n^{\frac{d}{2r}\left(1-\frac{1}{2rd+1}\right)}\right),$

completing the proof.∎

##  6 Concluding remarks

1\. It is easy to see that $\alpha_{d,s}(N)\geq\Omega(N^{1/d})$ for any fixed
$d,s\geq 2$. Let $S$ be a set consisting of $N$ points in $\mathbb{R}^{d}$
with no $d+s$ members on a hyperplane. Suppose $V$ is a maximal subset of $S$
in general position, then $V$ generates at most $\binom{|V|}{d}$ hyperplanes
and each of them covers at most $s$ points from $S\setminus V$. Hence we have
the inequality

$s\binom{|V|}{d}+|V|\geq|S|=N,$

which justifies the claimed lower bound of $\alpha_{d,s}(N)$.

######  Problem 6.1.

Are there fixed integers $d,s\geq 3$ such that $\alpha_{d,s}(N)\leq
o(N^{\frac{1}{2}})$?

2\. We call a subset $V\subset[n]^{d}$ a _$m$ -fold $B_{g}$-set_ if $V$ only
contains trivial solutions to the equations

$c_{1}\textbf{x}_{1}+c_{2}\textbf{x}_{2}+\dots+c_{g}\textbf{x}_{g}=c_{1}\textbf{x}^{\prime}_{1}+c_{2}\textbf{x}^{\prime}_{2}+\dots+c_{g}\textbf{x}^{\prime}_{g},$

with constant coefficients $c_{i}\in[m]$. We call $1$-fold $B_{g}$-sets simply
_$B_{g}$ -sets_. By counting distinct sums, we have an upper bound $|V|\leq
O(n^{\frac{d}{g}})$ for any $B_{g}$-set $V\subset[n]^{d}$.

Our Theorem 5.1 can be interpreted as the following phenomenon: by letting $m$
grow as some proper polynomial in $n$, we have an upper bound for $m$-fold
$B_{g}$-sets, where $g$ is even, which gives a polynomial-saving improvement
from the trivial $O(n^{\frac{d}{g}})$ bound. We believe this phenomenon should
also hold without the parity condition on $g$.

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