[a,b]Samuel Kováčik

# Microscopic black holes as probes for quantum gravity

###### Abstract

One of the main goals of contemporary theoretical physics is to find the
quantum theory of gravity. There are various working hypotheses, mostly
operating in the regime of high-energy physics well above the reach of
particle accelerators. So far, strong experimental or observational evidence
to guide the theory is missing. A possible consequence of quantum gravity and
quantum spacetime that is often discussed is the vacuum dispersion effect. In
this paper, we consider a different line of quantum space phenomenology, the
behaviour of microscopic black holes. Even though their exact nature is
unknown, some of their features are very model-independent, allowing us to
draw conclusions about their role in the current cosmological models.

##  1 Introduction

Fundamental constants hint at important physical scales. We understand the
importance of the speed of light, $c$, the gravitational constant, $G$, or the
Planck constant, $\hbar$. The importance of their combinations, such as the
Planck length

$l_{P}=\sqrt{\frac{G\hbar}{c^{3}}}\approx 1.616255(18)\times 10^{-35}\mbox{m}$
(1)

is still being discussed. Many different perspectives view it as the minimal
length of space(time) interval [1]. This length scale, or dual to it the
Planck energy scale, is inaccessible for current particle accelerators.
Phenomenology is therefore more focused on observational data of astrophysical
origin. For example, gamma-ray bursts (GRB) travel through space for billions
of years and minuscule effects of Planck-length physics can accumulate and
manifest themselves as the vacuum dispersion [2, 3, 4, 5, 6].

Objects that can be sensitive to the existence of minimal length scale are
black holes, as in the current general-relativity description their mass
resides in a point-like singularity. The existence of a minimal length scale
should lead to the absence of point-like objects, considerable research has
been done on some aspects of regular black holes [7, 8, 9]. In this line of
research, the singular matter distribution is replaced with a blurred one—for
example described by the Gaussian distribution.

The solution to Einstein field equations with such matter distribution is then
found and analysed. Mostly, the effects are negligible for macroscopic black
holes of astrophysical origin. They are, however, important for microscopic
black holes whose size—that is the Schwarzschild radius—approaches the Planck
length. In many cases, the Hawking temperature of a regular black hole has an
upper boundary and has a minimal Schwarzschild radius [10, 11]. When this
minimum size is reached, the Hawking temperature vanishes. The resulting
object has the cross-section of the order of the Planck area and has a
considerable mass—of the order of the Planck mass. Therefore, it was suggested
that such objects are viable dark matter candidates [12].

What would be the origin of such small black holes, since astrophysical ones
could not have evaporated this much during the current existence of the
Universe? A possible source of small black holes are the overdensities in
primordial fluctuations. Their mass at the time of production has no lower
boundary and their mass spectrum can be derived from various models of
cosmological inflation [13, 14].

Given our current knowledge, the existence of small black holes is neither
guaranteed nor disproved. In this paper, we discuss their general properties,
their role in the cosmological context, and outline future lines of research.

##  2 Regular black holes

Ordinary black holes are vacuum solutions of the Einstein field equations with
curvature singularity at the origin. Regular black holes lack this singularity
and have positive mass density, $\rho$, in some nonzero space volume. In
principle, this mass distribution could be derived from the fundamental theory
of quantum gravity, which is, however, currently unknown.

Regular black holes appear naturally in theories with quantum spaces that have
a minimal length scale, denoted $\lambda$, which is expected to be of the
Planck-length order. The construction goes as follows. First, one needs to
find the minimal-length analogue of the point-particle Delta function; perhaps
using coherent states as in the following references. For example, in the case
of the 2D Moyal plane [15, 16, 17, 18], this yields $\rho(r)\sim
e^{-\left(r/\lambda\right)^{2}}$, see [10]. In the case of 3D noncommutative
rotationally invariant space [19, 20], it gives $\rho(r)\sim
e^{-\left(r/\lambda\right)}$, see [21]. This matter density is then completed
into the stress-energy tensor in a way so the solution to field equations can
be of the Schwarzschild-like form,
$g=\mbox{diag}\left(f(r),-f^{-1}(r),r^{2},r^{2}\sin\theta\right)$. This ansatz
is then plugged into the field equations which can be expressed in the form of
the following differential equation for the redshift factor $f(r)$

$\frac{1+f(r)+rf^{\prime}(r)}{r^{2}}=8\pi\rho(r).$ (2)

The solution of this equation can be found in the form

$f(r)=-1+\frac{2m}{r}\int\limits_{0}^{r}\tilde{\rho}(r^{\prime})4\pi r^{\prime
2}dr^{\prime},$ (3)

where $\tilde{\rho}(r)$ is a density normalised to unity,
$\rho(r)=m\tilde{\rho}(r)$. Regularity of this solution is obvious from the
construction.

How does it behave? Close to the origin is the function $f(r)$ growing from
the value of $-1$. Far away from the origin, $r\gg\lambda$, it asymptotes the
Schwarzschild solution, $-1+\frac{2m}{r}$. It is obvious that $f(r)$ has a
maximum for a finite and positive value of $r$ and its value at this point
grows linearly with $m$. Let us, for simplicity, consider only monotonic
matter densities $\rho(r)$. In that case, there is only one maximum and the
value of $m$ determines whether is the maximum value positive, negative, or
zero [11].

In case it is positive, since at limiting cases $r=0$ and $r\rightarrow\infty$
is the function $f(r)$ negative, there have to be two horizons such that
$f(r_{\pm})=0$. This follows from the continuity of $f(r)$. When the mass is
very large, $m\gg\lambda$ (the Planck mass in our units), one of the horizons
is close to the Schwarzschild value and one is close to the origin, see Figure
1. As the mass is being decreased, they move toward each other and merge at
some specific value $m=m_{0}$. For $m<m_{0}$, there are no horizons—we have a
blurred naked singularity. For $m=m_{0}$, the two horizons merge $r_{+}=r_{-}$
and $f(r)$ reaches maximum at this point. From this follows that the
derivative $f^{\prime}(r)$ vanishes there and so does the Hawking temperature.

Also, since the derivative of $f(r)$ is finite everywhere, as is obvious from
the regularity of $\rho(r)$ and the form of the solution (3), we see that the
infinite Hawking temperature is avoided. Actually, in cases studied in [11],
the maximal temperature is an order or two below the Planck temperature,
$T\sim\lambda^{-1}$, see Figure 2.

The existence of black hole remnants have been derived from very few
assumptions: having monotonically decreasing regular matter density $\rho(r)$
and Schwarzschild-like form of the solution. The microscopic black holes,
before freezing into remnants, radiate with near-Planck temperatures;
steepness of their temperature profile for the case of $\tilde{\rho}(r)\sim
e^{-\left(r/\lambda\right)}$ has been derived in [21]. The temperature reached
after adding a small mass $\delta m$ to the minimal mass $m_{0}$ is

$T(m_{0}+\delta m)\stackrel{{\scriptstyle.}}{{=}}\frac{\sqrt{\delta
m/\lambda}}{41.01}\lambda^{-1}.$ (4)

Parameters of these remnants, such as the mass or the cross-section, are
naturally close to the Plank units. Typical temperature profiles are shown in
Figure 2.

This leads to two possible observational consequences. The first is that such
black hole remnants are a good dark matter candidate [12]. If the minimal
scale is the Planck length, then their cross-section is of the order of
$10^{-70}m^{2}$—this makes them nearly impossible to interact with. However,
with the mass of the order of the Planck mass, only a relatively small number
of them would be needed to explain the observed amount of the dark matter
density, approximately of the order of $n_{\mbox{\scriptsize{mbh}}}\sim
10^{-20}m^{-3}$ [21]. If this is indeed the case, it could be difficult to
verify as there is no straightforward way of detecting them. We will return to
this question in the next section.

The second possible observational consequence is the production of GRB during
the final stage of the evaporation process. Contrary to the ordinary black
holes, in the case of regular black holes is the infinite temperature avoided.
There are very few quanta radiated during the peak emission. To put this into
perspective, the Large hadron collider currently operates at energies
$E\approx 10^{-15}E_{p}$, where $E_{p}$ is the Planck energy. The GRB with
highest ever detected energy have reached and even surpassed this energy scale
[22, 23] and the cosmic ray with the largest energy had $E\approx
10^{-9}E_{P}$, [24, 25]. Also, protons with $E>10^{-9}E_{P}$ are prevented
from traveling galactic distances by Greisen-Zatsepin-Kuz’min limit [26, 27].
In the studied models of regular black holes, the energy of Hawking-radiated
particles peaked around $10^{-2}E_{p}$.

Lack of observational evidence of the Hawking radiation from miscroscopic
black holes does not disprove their existence but only puts some constraints
on their mass spectrum [28]. However, only a small number, on the order of
$10-100$, are radiated during the peak emission. Most of the energy is
radiated at energy scales that overlap with more conventional astrophysical
mechanisms. Also, the microscopic black holes could have finished the
evaporation process during the earliest moments of the universe when the
radiated energy would have been quickly absorbed. The search for a signal from
evaporating primordial black holes is still ongoing [29, 30, 31].

Lacking observational signals, the other option is to try to connect the
observed amount of dark matter with the theoretical predictions of primordial
black hole production. However, their production could have taken place during
the period of inflation, physics of which we only begin to understand [13].
The total number of black holes produced depends on the details of the
inflation field [32, 33, 34, 35]—another aspect that is still not perfectly
understood. However, if at some point in the future the details of the
inflation process will be known, we can use them to calculate the expected
number of black hole remnants and compare them to the observed dark matter
density; keeping in mind that the Planckian remnants could contribute only
partially to the overall dark matter density.

Figure 1:  Comparison of the redshift factor of a regular black hole (black
solid line) and the Schwarzschild solution (red dashed line) with the same
mass. In this case, the mass of the regular black hole is above the minimal
mass, $m>m_{0}$, since there are two horizons. This solution is of the form
$f(r)=-1+\frac{2m}{r}\int\limits_{0}^{r}\tilde{\rho}(r^{\prime})4\pi r^{\prime
2}dr^{\prime}$, so it is obvious that decreasing the mass would lower the peak
of this function and for a sufficiently small mass, there would no horizon at
all. Figure 2: Temperature profiles for the densities (3) with $q\leq 15$
where $g_{1}$ are the red lines, $g_{2}$ are the blue lines and $g_{3}$ are
the green lines—they mostly overlap. The plots have been rescaled so the
maximal temperature is reached at the same point. The unscaled value of the
maximal temperature in the considered cases is between
$0.02\lambda^{-1}-0.1\lambda^{-1}$, where $\lambda^{-1}$ is of the order of
the Planck temperature.

##  3 Radiation recoil effect

Recently, an important piece of the puzzle was added to this picture. In [11],
it was observed that the last fraction of the mass of microscopic black holes
is radiated in a relatively small amount of quanta. From the point of having
maximal temperature to the point of zero temperature, the black hole radiates
approximately $10-100$ particles, this estimate is only slightly model-
dependent.

The radiated particles have large momenta and random directions. Radiating
each of them gives the black hole an opposite momentum kick. As a result,
during the final moments of the evaporation process the microscopic black hole
makes a random walk in the momentum space. Therefore, black hole remnants are
formed with considerable velocity, typically close to $0.1$ of the speed
light.

This effect was tested in [11] on three (infinite) classes of matter
distributions

$\displaystyle\tilde{\rho}_{1}(r;q)$ $\displaystyle=$ $\displaystyle
n_{1}e^{-\left(r/\lambda\right)^{q}},$ $\displaystyle\tilde{\rho}_{2}(r;q)$
$\displaystyle=$ $\displaystyle
n_{2}\left(1+\left(r/\lambda\right)\right)^{-q},$ (5)
$\displaystyle\tilde{\rho}_{3}(r;q)$ $\displaystyle=$ $\displaystyle
n_{3}\left(1+\left(r/\lambda\right)^{q}\right)^{-1}.$

Solutions to the corresponding Einstein field equations are, using the
parametrisation $f_{i}(r;m,q)=-1+\frac{2m}{r}g_{i}(r;q)$, following

$\displaystyle g_{1}(r;q)$ $\displaystyle=$ $\displaystyle\frac{3\
\left(\Gamma\left(3\ q^{-1}\right)-\Gamma\left(3\
q^{-1},r^{q}\right)\right)}{q\ \Gamma\left(1+3\ q^{-1}\right)},$
$\displaystyle g_{2}(r;q)$ $\displaystyle=$ $\displaystyle
1-\frac{\left(2+\left(q-1\right)r\left(2+\left(q-2\right)r\right)\right)}{2\left(1+r\right)^{q-1}},$
(6) $\displaystyle g_{3}(r;q)$ $\displaystyle=$ $\displaystyle\frac{q\ r^{3}\
{}_{2}F_{1}\left(1,3\ q^{-1},1+3\ q^{-1},-r^{q}\right)\sin\left(3\pi\
q^{-1}\right)}{3\pi},$

where $\Gamma(a,b)$ is the upper incomplete Gamma function and ${}_{2}F_{1}$
is the hypergeometric function.

Note that the cases of $\tilde{\rho}_{1}(r;q)$ with $q=1,2,3$ have been
studied before, $q=1,2$ in the context of quantum spaces and $q=3$ in the
context of vacuum polarisation and some others were studied in the context of
dark matter halos [36]. The recoil velocity for each of those cases are shown
in Figure 3, its value has been derived in [11]

$v_{\mbox{\small{rec}}}\approx\frac{\Delta m}{m_{0}\sqrt{N_{q}}},$ (7)

where $\Delta m=m_{\mbox{\scriptsize{Tmax}}}-m_{0}$ is the mass difference
between black holes with maximal and zero temperature and $N_{q}$ is the
number of quanta radiated during the transition between them.

Figure 3: Recoil velocities for the densities (3) with $q\leq 12$ expressed in
units where $c=1$. As in the previous plot: the result for $g_{1}$ is shown by
the red line, $g_{2}$ by the blue line, and $g_{3}$ by the green line. The
values were computer for integer values of $q$, the lines are only joining the
points to help to see the overall trend. As discussed in [11], the results for
$g_{3}$ can be trusted only for $q\approx 10$ as above this point the
description breaks down, the mass $\Delta m$ is radiated away in a single
quantum and the description needs to be refined.

Such velocities would make microscopic black holes incompatible with the
current picture of cold dark matter. How to avoid this issue? The study [37]
suggested that if the evaporation finished a long time ago–––as also
corroborated from the theory of big bang nucleosynthesis that is sensitive to
possible energy injections—the black hole remnants would have been slowed down
by the expansion of space. In [11], it was calculated that for this the
happen, the primordial black holes would need to have formed before the
universe was $10^{-28}\mbox{s}$ old, which is shortly after the end of the
inflation era.

Details of the Hawking radiation and therefore also the recoil effect are not
well-understood at the moment and have been investigated in a different
context in [38, 39].

##  4 Conclusion

Despite the fact that we expect the effects of quantum gravity to be dominant
at the Planck energy scale, various mechanisms make this theory relevant well
below this scale. The exact formulation of quantum gravity is not understood
at the moment, however, some of its properties are rather general and
therefore we can hope to make some phenomenological predictions without
knowing the exact details.

We have discussed the effect of the quantum structure of space on microscopic
black holes that have been proposed as possible dark matter candidates. Even
though there is some effort to observe GRB signal from the final moments of
evaporation, there are two arguments why the evaporation process has to have
ended at very early times. The first is the agreement between the theory and
observation of the Big bang nucleosynthesis process which is sensitive to
energy injection [40, 41]. The second is the recoil effect which would make
the late-evaporation black holes incompatible with the current cold dark
matter hypothesis.

This estimate is in line with works investigating the formation of black holes
during the inflation epoch or during the reheating phase when the inflation
ceased [42, 43]. However, there is yet another option—could the Planck
remnants have been formed during the Planck epoch? If this is the case, there
is no obvious way how to compute their properties, such as their mass
distribution, given our current understanding.

One thing needs to be stressed, there are in principle two types of black
primordial black holes that have been conjectured to be dark matter
constituents. One of them are the Planck remnants, which, as we have discussed
in this paper, finished their evaporation process at a very early time, [11,
37, 40, 41]. The other class consists of heavy black holes for which the
evaporation is not so important and perhaps outpaced by accretion [44]. Our
discussion only concerns the first type, the Planck-mass black holes.

The black hole models were derived from the mass density functions $\rho(r)$
which were either defined by hand or derived from simple models of quantum
spaces. Currently, the best candidate for the quantum theory of gravity is the
M-theory. It would be therefore interesting to use this theory to derive
$\rho(r)$ and to study semi-classical consequences of the M-theory. Another
option is to compare the presented results with the quantum loop
gravity—another prominent quantum gravity hypothesis—description of black
holes [45]. If not for anything else then just to understand better the
generality of the results presented here.

## Acknowledgement

The author wish to thank the Corfu Summer Institute for its hospitality. This
research was supported by VEGA 1/0703/20 and the MUNI Award for Science and
Humanities funded by the Grant Agency of Masaryk University. Valuable comments
from V. Balek, J. Tekel and N. Werner were greatly appreciated.

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for some $\Upsilon_{\triangleleft}$ of bounded perimeter. Then there is a
sequence $\left\\{t_{n}\right\\}$ of positive real numbers such that
$t_{n}\to\infty$ and a Lebesgue conull subset $A\subset(0,\infty)$ such that
for all $r\in A$ and all functions $f:(0,\infty)\to(0,\infty)$ diverging at
infinity

$\varlimsup_{n\to\infty}\,\frac{\mathrm{NV}_{t_{n}}(\Gamma,\Omega_{\triangleleft})}{t_{n}^{2d_{\triangledown}}\psi(t_{n})^{-2}f(t_{n})^{-1}}=\infty.$

###  A.2 The diffraction formula for the number variance

It will be convenient for us to use the language of point processes; see e.g.
[Kal17] for background. Given $n\in\mathbb{N}$, we denote by
$M(\mathbb{R}^{n})$ the space of Radon measures on $\mathbb{R}^{n}$ and by
$\mathcal{LF}(\mathbb{R}^{n})$ the space of locally finite (i.e. closed and
discrete) subsets of $\mathbb{R}^{n}$. We consider
$\mathcal{LF}(\mathbb{R}^{n})$ as a subset of $M(\mathbb{R}^{n})$ by
identifying each set $\Lambda$ with its Dirac comb
$\delta_{\Lambda}:=\sum_{x\in\Lambda}\delta_{x}$. Every bounded Borel function
$f:\mathbb{R}^{n}\to\mathbb{C}$ with bounded support defines a _linear
statistic_

$\mathcal{P}f:M(G)\to\mathbb{C},\quad p\mapsto p(f),$

and we equip $\mathcal{LF}(\mathbb{R}^{n})\subset M(\mathbb{R}^{n})$ with the
smallest $\sigma$-algebra $\mathcal{B}$ for which all of these linear
statistics are measurable. Now assume that $(\Omega,\mathcal{F},\mathbb{P})$
is some auxiliary probability space on which $\mathbb{R}^{n}$ acts measurably,
preserving $\mathbb{P}$; then an equivariant measurable map

$\Lambda:\Omega\to\mathcal{LF}(\mathbb{R}^{n}),\quad\omega\mapsto\Lambda_{\omega}.$

is called a _stationary simple point process_ with _distribution_
$\mu=\Lambda_{*}\mathbb{P}$. We will only consider processes which are
_locally square-integrable_ in the sense that the real-value random variable

$\Lambda\cap B:\Omega\to\mathbb{R},\quad z\mapsto\\#(\Lambda_{z}\cap B)$

is square-integrable for every bounded Borel set $B\subset\mathbb{R}^{n}$;
note that this is automatically the case if $\Lambda_{\omega}$ is almost
surely $r$-uniformly discrete for some fixed $r>0$.

######  Definition A.7.

If $\Lambda:\Omega\to\mathcal{LF}(\mathbb{R}^{n})$ is locally square-
integrable, then the random variable

$\mathrm{disc}_{B}(\Lambda):\Omega\to\mathbb{R},\quad\mathrm{disc}_{B}(\Lambda)(z):=\\#(\Lambda_{z}\cap
B)-E[\Lambda\cap B].$

is called the _discrepancy_ of $\Lambda$ on $B$, and the variance

$N_{B}(\Lambda):=\mathrm{Var(\Lambda\cap
B)}=\int_{\Omega}|\mathrm{disc}_{B}(\Lambda)(\omega)|^{2}\,\mathrm{d}\mathbb{P}(\omega)$

If $\Lambda:\Omega\to\mathcal{LF}(\mathbb{R}^{n})$ is locally square-
integrable, then the random variable

$\mathrm{disc}_{B}(\Lambda):\Omega\to\mathbb{R},\quad\mathrm{disc}_{B}(\Lambda)(z):=\\#(\Lambda_{z}\cap
B)-E[\Lambda\cap B].$

is called the _discrepancy_ of $\Lambda$ on $B$, and the variance

$N_{B}(\Lambda):=\mathrm{Var(\Lambda\cap
B)}=\int_{\Omega}|\mathrm{disc}_{B}(\Lambda)(\omega)|^{2}\,\mathrm{d}\mathbb{P}(\omega)$

is called the _number variance_ of $\Lambda$ at $B$.

Note that the number variance is finite, since $\Lambda\cap B$ is square-
integrable.

######  Remark A.8 (Diffraction formula).

According to [BH23, Prop. 2.3] there exists a positive-definite signed Radon
measure $\eta_{\Lambda}$, the _(reduced) autocorrelation measure_ of
$\Lambda$, such that

$N_{B}(\Lambda)=\eta_{\Lambda}(\chi_{B}\ast\chi_{B}^{*}),$ (A.10)

where $\chi_{B}$ denotes the characteristic function of $B$. The Fourier
transform $\mathcal{F}\eta_{\Lambda}$ is a positive Radon measure on
$\mathbb{R}^{n}$, called the _(reduced) diffraction_ of $\Lambda$, and it
follows from (A.10) that for sufficiently regular $B$ we have

$N_{B}(\Lambda)=[\mathcal{F}\eta_{\Lambda}](|\mathcal{F}\chi_{B}|^{2}).$
(A.11)

This is called the _diffraction formula_ for the number variance.

######  Example A.9.

We return to the general setting of Subsection A.1. If we fix the lattice
$\Gamma\subset\mathbb{E}$ and the window $\Omega_{\triangleleft}$ then the map

$\Lambda(-;\Omega_{\triangleleft}):\Gamma\backslash\mathbb{E}\to\mathcal{LF}(\mathbb{E}_{\triangledown}),\quad\Gamma+\mathbf{z}\mapsto\Lambda(\Gamma+\mathbf{z};\Omega_{\triangleleft})$

is a stationary simple point process with respect to the unique
$\mathbb{E}$-invariant probability measure on $\Gamma\backslash\mathbb{E}$,
called the _cut and project process_ associated with $\Gamma$ and
$\Omega_{\triangleleft}$. For a bounded Borel set
$\Omega_{\triangledown}\subset\mathbb{E}_{\triangledown}$ we have

$E[\Lambda(-;\Omega_{\triangleleft})\cap
B]=\frac{\operatorname{vol}_{\triangledown}(\Omega_{\triangledown})\operatorname{vol}_{\triangleleft}(\Omega_{\triangleleft})}{\operatorname{covol}(\Gamma)},$

and hence the discrepancy of $\Lambda(-;\Omega_{\triangleleft})$ is given by

$\mathrm{disc}_{\Omega_{\triangledown}}(\Lambda(-;\Omega_{\triangleleft}))(\Gamma+\mathbf{z})=\Delta_{\Omega_{\triangleleft}}(\Gamma+\mathbf{z};\Omega_{\triangledown}).$

On the other hand, it follows from classical results of Meyer [Mey70] that the
reduced diffraction of $\Lambda(-;\Omega_{\triangleleft})$ is given by [BH23,
Thm. 2.9]

$\mathcal{F}\eta=\frac{1}{\operatorname{covol}(\Gamma)^{2}}\,\sum_{\xi=(\xi_{\triangledown},\xi_{\triangleleft})\in\Gamma^{\dagger}\setminus\\{(0,0)\\}}|[\mathcal{F}\chi_{\Omega_{\triangleleft}}](\xi_{\triangleleft})|^{2}\cdot\delta_{\xi_{\triangledown}}.$

The diffraction formula (A.11) for the number variance of
$\Lambda(-;\Omega_{\triangleleft})$ may thus be stated as follows; here we use
that since
$\chi_{t\Omega_{\triangledown}}(x_{\triangledown})=\chi_{\Omega_{\triangledown}}(x_{\triangledown}/t)$
we have
$[\mathcal{F}\chi_{t\Omega_{\triangledown}}](\xi_{\triangledown})=t^{d_{\triangledown}}[\mathcal{F}\chi_{\Omega_{\triangledown}}](t\xi_{\triangledown})$.

######  Proposition A.10.

If $\Omega_{\triangleleft}$ has sufficient Fourier decay, then for all $t>0$
we have

$\mathrm{NV}_{t}(\Gamma,\Omega_{\triangleleft})=\frac{t^{2d_{\triangledown}}}{\operatorname{covol}(\Gamma)^{2}}\,\sum_{\xi=(\xi_{\triangledown},\xi_{\triangleleft})\in\Gamma^{\dagger}\setminus\\{(0,0)\\}}|[\mathcal{F}\chi_{\Omega_{\triangledown}}](t\xi_{\triangledown})|^{2}|[\mathcal{F}\chi_{\Omega_{\triangleleft}}](\xi_{\triangleleft})|^{2}.\qed$

###  A.3 Proof of Theorem A.5

We now return to the setting of Theorem A.5; thus $\Gamma^{\dagger}$ is
assumed to be $\psi$-Liouvillean with parameters
$\psi,\xi_{\triangledown}^{(n)},\xi_{\triangleleft}^{(n)}$ as in (A.5) and
$\Omega_{\triangleleft}=\mathbb{B}_{\triangleleft}(0,r)$ for some $r>0$. Let
$f:(0,\infty)\to(0,\infty)$ be a function such that
$\lim_{t\to\infty}f(t)=\infty$. If we abbreviate
$\mathbb{B}_{\triangleleft}:=\mathbb{B}_{\triangleleft}(0,1)$, then by
Proposition A.10 we have

$\Psi_{f}(t,r):=\frac{\mathrm{NV}_{t}(\Gamma,\mathbb{B}_{\triangleleft}(0,r))}{t^{2d_{\triangledown}}\psi(t)^{-d_{\triangleleft}-1}f(t)^{-1}}=\frac{\psi(t)^{d_{\triangleleft}+1}f(t)r^{2d_{\triangleleft}}}{\operatorname{covol}(\Gamma)^{2}}\,\sum_{\xi=\in\Gamma^{\dagger}\setminus\\{(0,0)\\}}|[\mathcal{F}\chi_{\Omega_{\triangledown}}](t\xi_{\triangledown})|^{2}|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi_{\triangleleft})|^{2}.$
(A.12)

We have to prove that there exists a sequence $t_{n}$ such that for all $r$ in
a Lebesgue conull subset $E\subset(0,\infty)$ we have

$\varlimsup_{n\to\infty}\Psi_{f}(t_{n},r)=\infty.$ (A.13)

Following (8.7),

$\inf\left\\{|\widehat{\chi}_{\mathbb{B}_{\triangledown}}(\xi_{\triangledown})|:|\xi_{\triangledown}|\leq(2\operatorname{diam}(\Omega_{\triangledown}))^{-1}\right\\}\geq\frac{1}{2}\operatorname{vol}_{\triangledown}(\mathbb{B}_{\triangledown}).$
(A.14)

If we set

$\mathcal{C}_{t}:=\\{\xi=(\xi_{\triangledown},\xi_{\triangleleft})\in\Gamma^{\dagger}\setminus\\{(0,0)\\}:|\xi_{\triangledown}|\leq
1/t\\},$

we thus obtain for any
$t\geqslant(2\operatorname{diam}(\Omega_{\triangledown}))^{-1}$

$\displaystyle\Psi_{f}(t,r)$ $\displaystyle\geq$
$\displaystyle\frac{\psi(t)^{d_{\triangleleft}+1}f(t)r^{2d_{\triangleleft}}}{\operatorname{covol}(\Gamma)^{2}}\,\sum_{\xi\in\mathcal{C}_{t}}|[\mathcal{F}{\chi}_{\Omega_{\triangledown}}](t\xi_{\triangledown})|^{2}|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi_{\triangleleft})|^{2}$
$\displaystyle\geq$
$\displaystyle\frac{\psi(t)^{d_{\triangleleft}+1}f(t)r^{2d_{\triangleleft}}\operatorname{vol}_{\triangledown}(\Omega_{\triangledown})^{2}}{4\operatorname{covol}(\Gamma)^{2}}\,\sum_{\xi\in\mathcal{C}_{t}}|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi_{\triangleleft})|^{2}$
$\displaystyle\geq$
$\displaystyle\frac{\psi(t)^{d_{\triangleleft}+1}f(t)r^{2d_{\triangleleft}}\operatorname{vol}_{\triangledown}(\Omega_{\triangledown})^{2}}{4\operatorname{covol}(\Gamma)^{2}}\,\sup_{\xi\in\mathcal{C}_{t}}|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi_{\triangleleft})|^{2}.$

######  Lemma A.11.

If we define $t_{n}:=\|\xi_{\triangledown}^{(n)}\|^{-1}$, then there exists a
constant $C>0$ depending only on $d_{\triangledown}$ and the covolume of
$\Gamma$ such that

$\Psi_{f}(t_{n},r)\geq
C\cdot|\xi_{\triangleleft}^{(n)}|f(\psi^{-1}(|\xi_{\triangleleft}^{(n)}|))\cdot|J_{d_{\triangleleft}/2}(2\pi
r|\xi^{(n)}_{\triangleleft}|)|^{2}.$ (A.15)

###### Proof.

Since $\xi_{\triangledown}^{(n)}\to 0$ we have $t_{n}\to\infty$, and for all
$n\in\mathbb{N}$ sufficiently large we have $|\xi_{\triangledown}^{(n)}|\leq
1/t_{n}$, and hence $\xi_{n}\in\mathcal{C}_{t_{n}}$. We deduce that, for all
$n\in\mathbb{N}$ large enough we have

$\sup_{\xi\in\mathcal{C}_{t_{n}}}|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi_{\triangleleft})|^{2}\geq|[\mathcal{F}\chi_{\mathbb{B}_{\triangleleft}}](r\xi^{(n)}_{\triangleleft})|^{2}.$
(A.16)

Moreover, by (A.5) we have $t_{n}\geq\psi^{-1}(|\xi_{\triangleleft}^{(n)}|)$.
Plugging both into the inequality above, we deduce that

$\displaystyle\Psi_{f}(t_{n},r)$
$\displaystyle\geq\frac{\psi(t_{n})^{d_{\triangleleft}+1}f(t_{n})r^{2d_{\triangleleft}}\operatorname{vol}_{\triangledown}(\Omega_{\triangledown})^{2}}{4\operatorname{covol}(\Gamma)^{2}}\,|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi^{(n)}_{\triangleleft})|^{2}$
(A.17)
$\displaystyle\geq\frac{\|\xi_{\triangleleft}^{(n)}\|^{d_{\triangleleft}+1}f(\psi^{-1}(|\xi_{\triangleleft}^{(n)}|))r^{2d_{\triangleleft}}\operatorname{vol}_{\triangledown}(\mathbb{B}_{\triangledown})^{2}}{4\operatorname{covol}(\Gamma)^{2}}\,|[\mathcal{F}{\chi}_{\mathbb{B}_{\triangleleft}}](r\xi^{(n)}_{\triangleleft})|^{2}.$

We recall, following [GR07, Formula 8.145.1], that the Fourier transform of
the indicator of a unit ball can be expressed in terms of Bessel functions as

$[\mathcal{F}{\chi}_{B_{\triangleleft}}](\xi_{\triangleleft})=\frac{J_{d_{\triangleleft}/2}(2\pi|\xi_{\triangleleft}|)}{|\xi_{\triangleleft}|^{d_{\triangleleft}/2}},$

hence we obtain the desired inequality with
$C=\frac{\operatorname{vol}_{\triangledown}(\Omega_{\triangledown})^{2}}{4\operatorname{covol}(\Gamma)^{2}}$.
∎

Let us put $\tau_{n}:=2\pi|\xi^{(n)}_{\triangleleft}|$ and
$\widetilde{f}=f\circ\psi^{-1}$. By (A.15) we thus have

$\Psi_{f}(t_{n},r)\gtrsim[\tau_{n}^{1/2}\widetilde{J}_{d_{\triangleleft}/2}(r\tau_{n})]^{2}\widetilde{f}(\tau_{n}).$

To deduce (A.13) and thereby prove Theorem A.5 it thus suffices to establish
the following lemma.

######  Lemma A.12.

Let $(\tau_{n})$ be a sequence of positive real numbers such that
$\tau_{n}\to\infty$. Then there is a Lebesgue conull subset
$E\subset(0,\infty)$ such that for all $r\in E$, and
$d_{\triangleleft}\in\mathbb{N}$

$\varlimsup_{n\to\infty}\tau_{n}^{1/2}\cdot|J_{d_{\triangleleft}/2}(r\tau_{n})|\cdot\widetilde{f}(\tau_{n})=\infty.$

###### Proof.

Since $\tau_{n}\to\infty$ we may without loss of generality assume that
$\tau_{n}>1$ for all $n$. By [Wat95, Subsection 7.21], there exist two real
constants $\varphi$ and $\kappa$ depending on $d_{\triangleleft}$ such that

$\left|J_{d_{\triangleleft}/2}(\tau)-\left(\frac{2}{\pi\tau}\right)^{1/2}\cos(\tau-\varphi)\right|\leq\frac{\kappa}{\tau^{3/2}},\quad\textrm{for
all $\tau\geq 1$.}$

In particular, for all $r>0$ and $n$,

$\displaystyle|J_{d_{\triangleleft}/2}(r\tau_{n})|$
$\displaystyle=\left|J_{d_{\triangleleft}/2}(r\tau_{n})-\left(\frac{2}{\pi
r\tau_{n}}\right)^{1/2}\cos(r\tau_{n}-\varphi)+\left(\frac{2}{\pi
r\tau_{n}}\right)^{1/2}\cos(r\tau_{n}-\varphi)\right|$
$\displaystyle\geq\left(\frac{2}{\pi
r\tau_{n}}\right)^{1/2}|\cos(r\tau_{n}-\varphi)|-\frac{\kappa}{(r\tau_{n})^{3/2}},$

and thus,

$\tau_{n}^{\frac{1}{2}}\widetilde{f}(\tau_{n})\cdot|J_{d_{\triangleleft}/2}(r\tau_{n})|\geq\widetilde{f}(\tau_{n})\cdot\left(\frac{2}{\pi
r}\right)^{1/2}|\cos(r\tau_{n}-\varphi)|-\widetilde{f}(\tau_{n})\tau_{n}^{-1}\frac{\kappa}{r^{3/2}}.$

Since $\tau_{n}\to\infty$, [KN74, Chapter 1, Theorem 4.1] tells us that we can
find a Lebesgue conull subset $A\subset(0,\infty)$ such that the sequence
$(r\tau_{n})$ is equidistributed modulo $2\pi$ for all $r\in A$. Hence, for
all $r\in A$, we can find a further sub-sequence $(\tau_{n_{k}})$ such that
$r\tau_{n_{k}}-\varphi\to 0$ (modulo $2\pi$) as $k\to\infty$. In particular,

$|\cos(r\tau_{n_{k}}-\varphi)|\geq\frac{1}{2}\quad\textrm{for all large enough
$k$},$

and thus

$\tau_{n_{k}}^{\frac{1}{2}}\widetilde{f}(\tau_{n_{k}})\cdot|J_{d_{\triangleleft}/2}(r\tau_{n_{k}})|\geq\frac{1}{2}\cdot\widetilde{f}(\tau_{n_{k}})\cdot\left(\frac{2}{\pi
r}\right)^{1/2}-\widetilde{f}(\tau_{n_{k}})\tau_{n_{k}}^{-1}\frac{\kappa}{r^{3/2}}.$

The right-hand side clearly tends to infinity as $k\to\infty$, which finishes
the proof. ∎

###  A.4 Proof of Theorem A.6

We now turn our eye to Theorem A.6; thus $\Gamma^{\dagger}$ is the lattice
from Example A.4. In particular there is a $\psi$-Liouville number $0<a<2$ and
unit length vectors $e_{\triangleleft}\in\mathbb{E}_{\triangleleft}$ and
$e_{\triangledown}\in\mathbb{E}_{\triangledown}$ such that
$\operatorname{span}(ae_{\triangledown}+e_{\triangleleft},e_{\triangledown}+e_{\triangleleft})\subset\Gamma^{\dagger}$,
and the rest of the lattice is orthogonal to this span. We fix once and for
all the sequence
$\xi^{(n)}=q_{n}(ae_{\triangledown}+e_{\triangleleft})-m_{n}(e_{\triangledown}+e_{\triangleleft})$
where $q_{n}\in\mathbb{N}$ satisfies $\psi(\|aq_{n}\|^{-1})\geqslant q_{n}$
and $m_{n}$ is the nearest integer to $aq_{n}$. Note that
$|\xi^{(n)}_{\triangleleft}|\sim|a-1|q_{n}\to\infty$. Our goal is to use the
specific knowledge of the sequence realising the $\psi$-Liouvilleanness of
$\Gamma^{\dagger}$ in order to choose a window making the discrepancy
particularly large; the proof is almost identical to the proof of Theorem A.5
but in fact more direct because we can put our hands on the explicit sequence.

We put $\Omega_{\triangledown}$ again to be any set of bounded perimeter, and
$\Omega_{\triangleleft}=[-r,r]\times\Upsilon_{\triangleleft}$, where the
interval is taken to be in the direction of $e_{\triangleleft}$. This time, we
have that for every $n\in\mathbb{N}$,

$[\mathcal{F}\chi_{\Omega_{\triangleleft}}](\xi^{(n)})=2r\operatorname{vol}(\Upsilon_{\triangleleft})\frac{\sin(2\pi
r(q_{n}-m_{n}))}{2\pi r(q_{n}-m_{n})}$ (A.18)

Let $f:(0,\infty)\to(0,\infty)$ be any function such that
$\lim_{t\to\infty}f(t)=\infty$. Set
$t_{n}=|q_{n}a_{n}-m_{n}|^{-1}=|\xi^{(n)}_{\triangledown}|^{-1}$, and assume
$n$ is large enough so that the bound in (A.14) holds. By Proposition A.10 we
have just as in (A.12) that

$\displaystyle\Psi_{f}(t_{n},r):=\frac{\mathrm{NV}_{t_{n}}(\Gamma,r\Omega_{\triangleleft})}{t_{n}^{2d_{\triangledown}}\psi(t_{n})^{-2}f(t_{n})^{-1}}$
$\displaystyle=\frac{\psi(t_{n})^{2}f(t_{n})}{\operatorname{covol}(\Gamma)^{2}}\,\sum_{\xi\in\Gamma^{\dagger}\setminus\\{(0,0)\\}}|[\mathcal{F}\chi_{\Omega_{\triangledown}}](t_{n}\xi_{\triangledown})|^{2}|[\mathcal{F}\chi_{\Omega_{\triangleleft}}](\xi_{\triangleleft})|^{2}$
(A.19)
$\displaystyle\geqslant\frac{\psi(t_{n})^{2}f(t_{n})r^{2}\operatorname{vol}(\Omega_{\triangledown})^{2}\operatorname{vol}(\Upsilon_{\triangleleft})^{2}}{\operatorname{covol}(\Gamma)^{2}}\frac{\sin^{2}(2\pi
r(q_{n}-m_{n}))}{(2\pi r(q_{n}-m_{n}))^{2}}$
$\displaystyle\geqslant\frac{f(t_{n})\operatorname{vol}(\Omega_{\triangledown})^{2}\operatorname{vol}(\Upsilon_{\triangleleft})^{2}}{4\pi^{2}\operatorname{covol}(\Gamma)^{2}}\sin^{2}(2\pi
r(q_{n}-m_{n}))$

where we have used that $|q_{n}-m_{n}|\leqslant\psi(t_{n})$. But again, [KN74,
Chapter 1, Theorem 4.1] tells us that we can find a Lebesgue conull set
$A\subset(0,\infty)$ for which the sequence $\left\\{r(q_{n}-m_{n})\right\\}$
is equidistributed modulo $1$ for all $r\in A$. Hence, for all $r\in A$ we can
find a further subsequence such that $r(q_{n_{k}}-m_{n_{k}})-1/4\to 0$ (modulo
$1$) as $k\to\infty$. In particular, the last line in (A.19) diverges to
infinity as $k\to\infty$ since $f$ is divergent. ∎

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