# A Fractal Uncertainty Principle for the Short-Time Fourier Transform and
Gabor Multipliers

Helge Knutsen<EMAIL_ADDRESS>Department of Mathematical Sciences,
Norwegian University of Science and Technology,
7034 Trondheim, Norway

(Date: April 2022)

###### Abstract.

We study the fractal uncertainty principle in the joint time-frequency
representation, and we prove a version for the Short-Time Fourier transform
with Gaussian window on the modulation spaces. This can equivalently be
formulated in terms of projection operators on the Bargmann-Fock spaces of
entire functions. Specifically for signals in $L^{2}(\mathbb{R}^{d})$, we
obtain norm estimates of Daubechies’ time-frequency localization operator
localizing on porous sets. The proof is based on the maximal Nyquist density
of such sets, and for multidimensional Cantor iterates we derive explicit
upper bound asymptotes. Finally, we translate the fractal uncertainty
principle to discrete Gaussian Gabor multipliers.

###### Key words and phrases:

Fractal Uncertainty Principle, Short-Time Fourier Transform, Daubechies’
localization operator, Gabor frames, Gabor multipliers, Cantor set, Nyquist
density

###### 2010 Mathematics Subject Classification:

47A30, 30H20

##  1\. Introduction

The fractal uncertainty principle (FUP) was first introduced and developed for
the separate time-frequency representation in [10], [5], [9], see also [18]
for explicit estimates. It states that no signal in $L^{2}(\mathbb{R})$ can be
concentrated near fractal sets in both time and frequency. We reference
Dyatlov’s detailed introduction to the topic [8], where fractals sets are
defined broadly in terms of either $\delta$-regular sets or almost
equivalently in terms of $\nu$-porous sets within the scale bounds $0<h$ to
$1$ (see Definition 2.1). In both definitions, Dyatlov considers families of
subsets $T(h),\Omega(h)\subseteq[0,1]$ and formulates the FUP for said
families as the lower bound scale $h\to 0$. The FUP is presented as a norm
estimate for the localization operator
$\chi_{\Omega(h)}\pazocal{F}_{h}\chi_{T(h)}$, where $\chi_{E}$ denotes the
characteristic function of a subset $E$, and $\pazocal{F}_{h}$ denotes the
dilated Fourier transform
$\pazocal{F}_{h}f(\omega):=\sqrt{h}^{-1}\pazocal{F}f(\omega h^{-1})$. In terms
of $\nu$-porosity (see Theorem 2.19 in [8]), for signals in $f\in
L^{2}(\mathbb{R})$ and for families $T(h),\Omega(h)\subseteq[0,1]$ of
$\nu$-porous sets on scales $h$ to $1$, there exist constants $C,\beta>0$ only
dependent on $\nu>0$ such that

$\displaystyle\|\chi_{\Omega(h)}\pazocal{F}_{h}\chi_{T(h)}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\leq Ch^{\beta}\ \ \forall\ \ 0<h\leq 1.$

Alternatively, if we disentangle $h$ from the Fourier transform and write
$\sqrt{h}$ as $h$, we obtain a statement with regard to families
$T(h),\Omega(h)\subseteq[0,h^{-1}]$ of $\nu$-porous sets on scales $h$ to
$h^{-1}$, to which there exist constants $C,\beta>0$ only dependent on $\nu>0$
so that

(1.1)
$\displaystyle\|\chi_{\Omega(h)}\pazocal{F}\chi_{T(h)}\|_{L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R})}\leq Ch^{\beta}\ \ \forall\ \ 0<h\leq 1.$

On this form, the FUP more clearly reads as an uncertainty principle as,
depending on our choice of $\nu$, the measures of our time and frequency set,
$|T(h)|$ and $|\Omega(h)|$, respectively, might tend to infinity as $h\to 0$.

Inspired by the FUP in the separate representation and motivated by the
understanding that uncertainty principles should be present regardless of
time-frequency representation, we search for analogous results in the joint
representation. In particular, we consider and have considered Daubechies’
localization operator $P_{\Omega}$ based on the Short-Time Fourier Transform
(STFT) with the Gaussian window that projects onto some subset $\Omega$ of the
time-frequency plane. In previous installments [19], [20], we have restricted
our attention to radially symmetric subsets in $\mathbb{R}^{2}$, as this
yields a known eigenbasis, the Hermite functions, and explicit formulas for
the associated eigenvalues. With such insights, we have been able to derive
estimates for the operator norm when localizing on radial Cantor iterates that
mirror estimate (1.1) but with explicit estimates for the exponent, sometimes
even precise estimates. The radial assumption has also proved effective for
Bergman spaces and by extension for analytic wavelets in [2], where direct
knowledge of the eigenvalues of the localization operator have produced
similar estimates when localizing on the mid-third radial Cantor set.

In the present paper, however, we abandon the radial assumption and instead
consider the more general problem of optimal localization on $\nu$-porous sets
in phase space $\mathbb{R}^{2d}$ for arbitrary $d\geq 1$. Although we no
longer have knowledge of the eigenvalues of such an operator, the Gaussian
window in the STFT in and of itself provides additional structure. Namely,
Daubechies’ operator $P_{\Omega}:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$ can equivalently be viewed as a Toeplitz operator on
Bargmann-Fock spaces or simply Fock spaces, $\pazocal{F}^{2}(\mathbb{C}^{d})$,
of square integrable entire functions with respect to a Gaussian measure. With
this perspective, we utilize the subaveraging properties of entire functions
to derive estimates of the operator norm $\|P_{\Omega}\|_{\text{op}}$ in terms
of the maximal Nyquist density of $\Omega$. These estimates bear resemblance
to the estimates in Abreu and Speckbacher’ paper [3], which in large part
served as inspiration for our approach. By an inductive scheme, we find that
for a family $\Omega(h)\subseteq\mathbb{R}^{2d}$ of $\nu$-porous sets on
scales $h$ to $1$, there exists constants $C,\beta>0$ only dependent on
$\nu>0$ (and $d$) such that

(1.2) $\displaystyle\|P_{\Omega(h)}\|_{\text{op}}\leq Ch^{\beta}\ \ \forall\ \
0<h\leq 1,$

which represents a direct analogue of (1.1), now in the joint representation.
In fact, these estimates extend to norm estimates on the Fock space
$\pazocal{F}^{p}(\mathbb{C}^{d})$ for generic $p\geq 1$, which in turn yield
an FUP not only for $L^{2}(\mathbb{R}^{d})$ but also for the modulation
spaces, $M^{p}(\mathbb{R}^{d})$. For more explicit estimates of the exponent
in (1.2), we specifically consider multidimensional Cantor set constructions,
and here the upper bound asymptotes relies on our ability to directly compute
the Nyquist density of such sets.

In addition, we present an FUP for Gabor multipliers, which represents a
discrete alternative to Daubechies’ localization operator based on Gabor
frames (see [12] for an introduction to Gabor multipliers). Approximation
properties of such operators been studied in [15], [6], and spectral
properties have been studied [13]. For our purpose, we consider the closest
comparison to the Daubechies’ operator with a Gaussian window. Namely, we
consider the case when the generating function of the Gabor multiplier also
equals Gaussian.

The paper is organized as follows: Section 2 contains necessary background
theory. This includes, a formal introduction to the Fock space and the
connection to the STFT (section 2.1), an introduction to Gabor frames and
Gabor multipliers (section 2.2) and a precise description of what we mean by
”fractal” with the Cantor set constructions as concrete examples (section
2.3). The results are divided into three sections 3, 4 and 5. The general FUP
for Fock spaces and modulation spaces are formulated in section 3. The next
section 4 is focused on the multidimensional Cantor set constructions, with an
FUP formulated specifically for these sets. In the last section 5 we show how
the FUP can be translated to Gaussian Gabor multipliers.

##  2\. Preliminaries

###  2.1. From the Short-Time Fourier Transform to the Fock space

Consider some fixed window function $\phi:\mathbb{R}^{d}\to\mathbb{C}$, and
introduce the basic operations $T_{x}\phi(t):=\phi(t-x)$ and
$M_{\omega}\phi(t):=e^{2\pi i\omega\cdot t}\phi(t)$, i.e., time-translation
and frequency-modulation, respectively. The Short-Time Fourier Transform
(STFT) of some signal $f\in L^{2}(\mathbb{R}^{d})$, with respect to window
$\phi$, evaluated at point $(x,\omega)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$,
is then given by the inner product

$\displaystyle V_{\phi}f(x,\omega):=\langle f,M_{\omega}T_{x}\phi\rangle.$

Observe that if $\phi\equiv 1$, the STFT coincides with the regular Fourier
transform. For non-constant windows, however, we obtain a joint time-frequency
description of our signal. Furthermore, for $\|\phi\|_{2}=1$, the STFT becomes
an isometry onto some subspace of $L^{2}(\mathbb{R}^{2d})$, i.e., $\langle
V_{\phi}f,V_{\phi}g\rangle_{L^{2}(\mathbb{R}^{2d})}=\langle f,g\rangle$. In
this case, we have an inversion formula, namely

(2.1) $\displaystyle
f=\int_{\mathbb{R}^{2d}}V_{\phi}f(x,\omega)M_{\omega}T_{x}\phi\
\mathrm{d}x\mathrm{d}\omega,$

where the integral is interpreted in the weak-sense. Daubechies’ time-
frequency localization operator, $P_{S}^{\phi}:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$, with some bounded symbol $S$, is then obtained by
modifying the above integrand by the multiplicative weight function
$S(x,\omega)$, i.e.,

$\displaystyle P_{S}^{\phi}f:=\int_{\mathbb{R}^{2d}}S(x,\omega)\cdot
V_{\phi}f(x,\omega)M_{\omega}T_{x}\phi\ \mathrm{d}x\mathrm{d}\omega$ (2.2)
$\displaystyle\iff\langle$ $\displaystyle P_{S}^{\phi}f,g\rangle=\langle
S\cdot V_{\phi}f,V_{\phi}g\rangle_{L^{2}(\mathbb{R}^{2d})}\ \ \forall\ \ g\in
L^{2}(\mathbb{R}^{d}).$

This could equivalently be viewed as modifying the resulting STFT by
multiplication by $S$ before inversion. Oftentimes, we consider
$S=\chi_{\Omega}$, i.e., the characteristic function of a subset $\Omega$ of
the phase space $\mathbb{R}^{2d}$, so that the operator
$P_{\Omega}^{\phi}:=P_{\chi_{\Omega}}^{\phi}$ is interpreted as projecting
signals onto said time-frequency domain. The associated operator norm
$\|P_{\Omega}^{\phi}\|_{\text{op}}$ then measures the optimal localization on
$\Omega$.

A popular choice for window function is the Gaussian function, which on
$\mathbb{R}^{d}$ reads

$\displaystyle\phi_{0}(x):=2^{d/4}e^{-\pi x^{2}},\ \ \text{where
}x^{2}=x_{1}^{2}+\dots+x_{d}^{2}\ \ \text{for }\ x=(x_{1},\dots,x_{d}).$

With this window choice, we can, in fact, replace Daubechies’ operator by a
Toeplitz operator on the Bargmann-Fock space or simply Fock space,
$\pazocal{F}^{2}(\mathbb{C}^{d})$.

We reference Zhu’s book [29] for a detailed introduction to Fock spaces in
$\mathbb{C}$. For a complex vector
$z=x+i\omega=(z_{1},\dots,z_{d})\in\mathbb{C}^{d}$, we distinguish between
$z^{2}=z\cdot z=z_{1}^{2}+\dots+z_{d}^{2}$ and
$|z|^{2}=z\cdot\overline{z}=|z_{1}|^{2}+\dots+|z_{d}|^{2}$. Now, for arbitrary
$p\geq 1$, let
$\mathrm{d}\mu_{p}(z):=e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z)$ denote the
Gaussian measure on $\mathbb{C}^{d}$, where $\mathrm{d}A(z)$ is the volume
measure $\mathrm{d}x\mathrm{d}\omega$. The associated $L^{p}$-space,
$L^{p}(\mathbb{C}^{d},\mathrm{d}\mu_{p})$, is simply denoted by
$\pazocal{L}^{p}(\mathbb{C}^{d})$. The Fock space
$\pazocal{F}^{p}(\mathbb{C}^{d})$ is then defined as the Banach space of
entire functions $F\in\pazocal{L}^{p}(\mathbb{C}^{d})$, with norm111For
consistent and simple notation, we denote the norm in the Fock space by
$\|\cdot\|_{\pazocal{L}^{p}}$ rather than $\|\cdot\|_{\pazocal{F}^{p}}$. In
particular, this is to avoid switching notation for functions in
$\pazocal{L}^{p}(\mathbb{C}^{d})\setminus\pazocal{F}^{p}(\mathbb{C}^{d})$,
e.g., when we consider $F\cdot\chi_{\Omega}$ for
$F\in\pazocal{F}^{p}(\mathbb{C}^{d})\setminus\\{0\\}$ and
$\Omega\subsetneq\mathbb{C}^{d}$.

$\displaystyle\|F\|_{\pazocal{L}^{p}}=\left(\int_{\mathbb{C}^{d}}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z)\right)^{1/p}<\infty.$
For $p=\infty$, we let $\pazocal{L}^{\infty}(\mathbb{C}^{d})$ denote the space
of measurable functions on $\mathbb{C}^{d}$ such that
$\displaystyle\|F\|_{\pazocal{L}^{\infty}}=\operatorname*{ess\,sup}\left\\{|F(z)|e^{-\frac{\pi}{2}|z|^{2}}\
\big{|}\ z\in\mathbb{C}^{d}\right\\}<\infty.$

Again, the Fock space $\pazocal{F}^{\infty}(\mathbb{C}^{d})$ is the Banach
space of entire functions in $\pazocal{L}^{\infty}(\mathbb{C}^{d})$. For
$p=2$, we find that the Fock space forms a reproducing kernel Hilbert space,
with reproducing kernel $K_{\xi}(z)=e^{\pi z\cdot\overline{\xi}}$ so that
$\langle F,K_{\xi}\rangle_{\pazocal{L}^{2}}=F(\xi)$. Utilizing this kernel, we
obtain an orthogonal projection
$P:\pazocal{L}^{2}(\mathbb{C}^{d})\to\pazocal{F}^{2}(\mathbb{C}^{d})$, defined
by

$\displaystyle
PF(z):=\int_{\mathbb{C}^{d}}K_{\xi}(z)F(\xi)e^{-\pi|\xi|^{2}}\mathrm{d}A(\xi).$

For a bounded measurable function
$S:\mathbb{C}^{d}(\cong\mathbb{R}^{2d})\to\mathbb{C}$, we define the Toeplitz
operator
$\pazocal{T}_{S}:\pazocal{F}^{2}(\mathbb{C}^{d})\to\pazocal{F}^{2}(\mathbb{C}^{d})$,
with symbol $S$, by

$\displaystyle\pazocal{T}_{S}F(z):=P(SF)(z)=\int_{\mathbb{C}^{d}}S(\xi)\cdot
K_{\xi}(z)F(\xi)e^{-\pi|\xi|^{2}}\mathrm{d}A(\xi).$

If we initially consider test functions
$G\in\pazocal{L}^{1}(\mathbb{C}^{d})\cap\pazocal{F}^{2}(\mathbb{C}^{d})$,
then, by Fubini’s theorem and the reproducing property of the kernel
$K_{\xi}(z)$, the inner product attains the simple form

(2.3) $\displaystyle\langle\pazocal{T}_{S}F,G\rangle_{\pazocal{L}^{2}}=\langle
S\cdot F,G\rangle_{\pazocal{L}^{2}}$

By a density argument, it follows that (2.3) holds for all
$G\in\pazocal{F}^{2}(\mathbb{C}^{d})$, more akin to the inner product (2.2).

The precise connection to Daubechies’ time-frequency operator with a Gaussian
window is established through the Bargmann transform. The Bargmann transform,
first introduced in [4], is an isometric isomorphism
$\pazocal{B}:L^{2}(\mathbb{R}^{d})\to\pazocal{F}^{2}(\mathbb{C}^{d})$, given
by

(2.4) $\displaystyle\pazocal{B}f(z):=2^{d/4}\int_{\mathbb{R}^{d}}f(t)e^{2\pi
t\cdot z-\pi t^{2}-\frac{\pi}{2}z^{2}}\mathrm{d}t.$

The transform is related to the STFT with Gaussian window via the following
identity (see Proposition 3.4.1 in [14])

(2.5) $\displaystyle V_{\phi_{0}}f(x,-\omega)=e^{\pi
ix\cdot\omega}\pazocal{B}f(z)e^{-\frac{\pi}{2}|z|^{2}}\ \ \text{for
}z=x+i\omega.$

Define $S^{c}(z):=S(\overline{z})$, so that $\langle
P_{S}^{\phi_{0}}f,g\rangle$
$=\langle\pazocal{T}_{S^{c}}\pazocal{B}f,\pazocal{B}g\rangle_{\pazocal{L}^{2}}$,
from which the one-to-one correspondence between Daubechies’ operator
$P_{S}^{\phi_{0}}$ and the Toeplitz operator $\pazocal{T}_{S^{c}}$ is evident.
In particular, we have that their norms coincide.

In the subsequent discussion, we shall consider Toeplitz operators projecting
onto $\Omega\subseteq\mathbb{C}^{d}$, which we will simply denote by
$\pazocal{T}_{\Omega}=\pazocal{T}_{\chi_{\Omega}}$. By (2.3), the operator
norm is given by

(2.6)
$\displaystyle\|\pazocal{T}_{\Omega}\|_{\text{op}}=\sup_{\|F\|_{\pazocal{F}^{2}}=1}\int_{\Omega}|F(z)|^{2}e^{-\pi|z|^{2}}\mathrm{d}A(z).$

For general $p\geq 1$, we consider upper bounds for the quantity

$\displaystyle\frac{\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{p}}^{p}}{\|F\|_{\pazocal{L}^{p}}^{p}}=\frac{\int_{\Omega}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z)}{\|F\|_{\pazocal{L}^{p}}^{p}}\
\ \text{with }\ F\in\pazocal{F}^{p}(\mathbb{C}^{d})\setminus\\{0\\}.$ By the
use of complex interpolation (see Appendix A for details), the above quotients
are actually bounded by the estimate for $p=1$, i.e., (2.7)
$\displaystyle\sup_{F\in\pazocal{F}^{p}(\mathbb{C}^{d})\setminus\\{0\\}}\frac{\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{p}}^{p}}{\|F\|_{\pazocal{L}^{p}}^{p}}\leq\sup_{F\in\pazocal{F}^{1}(\mathbb{C}^{d})\setminus\\{0\\}}\frac{\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{1}}}{\|F\|_{\pazocal{L}^{1}}},$

which allows for estimates without any $p$-dependence. Unsurprisingly, the
added structure provided by the Fock space turns out to be beneficial when
estimating the norm. Namely, we shall exploit subaveraging properties of
subharmonic functions.

###  2.2. Gabor frames and Gabor multipliers

In general, a family of vectors $\\{\phi_{\lambda}\\}_{\lambda\in\Lambda}$ in
the Hilbert space $\pazocal{H}$ is called a frame if there exist constants
$0<A\leq B<\infty$, i.e., frame bounds, such that

$\displaystyle A\|f\|^{2}\leq\sum_{\lambda\in\Lambda}|\langle
f,\phi_{\lambda}\rangle|^{2}\leq B\|f\|^{2}\ \ \forall\ \ f\in\pazocal{H}.$

The associated frame operator $\pazocal{S}:\pazocal{H}\to\pazocal{H}$ is given
by $\pazocal{S}f:=\sum_{\lambda}\langle f,\phi_{\lambda}\rangle\phi_{\lambda}$
with norm $\|\pazocal{S}\|_{\pazocal{H}}\in[A,B]$. If $A=B$, the frame is
called a tight frame. By renormalizing the vectors
$\phi_{\lambda}\mapsto\phi_{\lambda}/\sqrt{A}$, any tight frame can be turned
into a Parseval frame, i.e., $A=B=1$, where we also have the representation
$f=\sum_{\lambda}\langle f,\phi_{\lambda}\rangle\phi_{\lambda}$, i.e.,
$\pazocal{S}=\text{id}$.

The Gabor frame for $L^{2}(\mathbb{R}^{d})$ is based on the idea of
discretizing the STFT inversion formula (2.1). For this purpose consider a
lattice $\Lambda\subseteq\mathbb{R}^{2d}$ of sampled points. Oftentimes, we
consider rectangular lattices of the form $\Lambda=a\mathbb{Z}^{d}\times
b\mathbb{Z}^{d}$ with parameters $a,b>0$. Further, fix a window function
$\phi\in L^{2}(\mathbb{R}^{d})\setminus\\{0\\}$ and define the time-frequency
shifts $\pi(x,\omega)\phi:=M_{\omega}T_{x}\phi$. If the family of time-
frequency shifts $\\{\pi(\lambda)\phi\\}_{\lambda\in\Lambda}$ forms a frame,
we call this system a Gabor frame with generating function $\phi$ over lattice
$\Lambda$. We may also include a normalization factor based on the density of
the lattice. More precisely, to each lattice we can associate a connected
neighbourhood of the origin $A_{\Lambda}$ called the fundamental region such
that $\cup_{\lambda\in\Lambda}(\lambda+A_{\Lambda})=\mathbb{R}^{2d}$ and
$|(\lambda+A_{\Lambda})\cap(\xi+A_{\Lambda})|=0$ whenever
$\lambda\neq\xi\in\Lambda$. With the normalization
$\sqrt{|A_{\Lambda}|}\\{\pi(\lambda)\phi\\}_{\lambda\in\Lambda}$, the
associated frame operator
$\pazocal{S}_{\Lambda}^{\phi}:L^{2}(\mathbb{R}^{d})\to L^{2}(\mathbb{R}^{d})$
reads

(2.8)
$\displaystyle\pazocal{S}_{\Lambda}^{\phi}f=|A_{\Lambda}|\sum_{\lambda\in\Lambda}\langle
f,\pi(\lambda)\phi\rangle\pi(\lambda)\phi.$

We immediately recognize $\pazocal{S}_{\Lambda}^{\phi}f$ as a Riemann sum of
the integral (2.1). Thus, for a sequence of Gabor frames
$\sqrt{|A_{\Lambda_{n}}}|\\{\pi(\lambda)\phi\\}_{\lambda\in\Lambda_{n}}$ where
$A_{\Lambda_{n}}\to 0$ as $n\to\infty$, we expect the discretization (2.8) to
converge weakly to the integral (2.1) and the frame bounds to tighten. Notably
for the sequence of square lattices
$\Lambda_{n}=\frac{1}{n}\left(\mathbb{Z}\times\mathbb{Z}\right)$ and $f,\phi$
in Feichtinger’s algebra $M^{1}(\mathbb{R}^{d})$, Weisz shows in [27] that
$S^{\phi}_{\Lambda_{n}}f$ converges to $f$ in the $M^{1}$-norm as
$n\to\infty$. While $\phi$ remains in $M^{1}(\mathbb{R}^{d})$, this result is
extended to $f$ in the modulation space $M^{p,q}(\mathbb{R}^{d})$ with
convergence in the $M^{p,q}$-norm. In particular, we have convergence in the
$L^{2}$-norm when $f\in L^{2}(\mathbb{R}^{d})$.

The Gabor multiplier, $\mathscr{G}_{\Lambda,b}^{\phi}:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$, represents a discretization of Daubechies’ operator in
(2.2), where the sum (2.8) is weighted by a bounded symbol $b$ defined on the
lattice $\Lambda$, i.e.,

$\displaystyle\mathscr{G}_{\Lambda,b}^{\phi}f:=|A_{\Lambda}|\sum_{\lambda\in\Lambda}b(\lambda)\langle
f,\pi(\lambda)\phi\rangle\pi(\lambda)\phi.$

For localization on a specific subset $\Omega\subseteq\mathbb{R}^{2d}$, we
shall consider Gabor symbols $b$ that mimic the behaviour of the
characteristic function $\chi_{\Omega}$. One natural option is to consider the
portion of a lattice point region $\lambda+A_{\Lambda}$ containing the subset
$\Omega$, i.e.,

$\displaystyle
b_{\Omega}(\lambda):=\frac{|\Omega\cap(\lambda+A_{\Lambda})|}{|A_{\Lambda}|}\in[0,1].$

Alternatively, we only distinguish between whether the lattice point region
$\lambda+A_{\Lambda}$ contains a non-zero part of $\Omega$. That is, we apply
the ceiling function $\lceil\cdot\rceil$, rounding up to the nearest integer,
so that

$\displaystyle\lceil b_{\Omega}(\lambda)\rceil\in\\{0,1\\}.$

Evidently $0\leq b_{\Omega}(\lambda)\leq\lceil b_{\Omega}(\lambda)\rceil$,
from which it is easily verified that the operator norms also satisfy
$\|\mathscr{G}_{\Lambda,b_{\Omega}}^{\phi}\|_{\text{op}}\leq\|\mathscr{G}_{\Lambda,\lceil
b_{\Omega}\rceil}^{\phi}\|_{\text{op}}$. Thus, when estimating upper bounds
for the operator norm in section 5, we only consider the second symbol
suggestion, $\lceil b_{\Omega}\rceil$. Notice that utilizing symbol $\lceil
b_{\Omega}\rceil$ is the same as restricting the summation (2.8) to a subset
of the lattice $\Lambda$, namely

(2.9) $\displaystyle\Lambda_{\Omega}:=\big{\\{}\lambda\in\Lambda\ \big{|}\
|\Omega\cap(\lambda+A_{\Lambda})|>0\big{\\}}.$

For simplicity, we denote the Gabor multiplier with symbol $\lceil
b_{\Omega}\rceil$ by $\mathscr{G}^{\phi}_{\Lambda,\Omega}$, which, by the
above observation, is given by

(2.10)
$\displaystyle\mathscr{G}_{\Lambda,\Omega}^{\phi}f=|A_{\Lambda}|\sum_{\lambda\in\Lambda_{\Omega}}\langle
f,\pi(\lambda)\phi\rangle\pi(\lambda)\phi.$

###  2.3. Porous sets and Cantor sets

We shall define ”fractal sets” in terms of the general notion of
$\nu$-porosity. It is based on Definition 2.7 in [8], adjusted to higher
dimensions. Informally, in order for a set to be classified as porous, we
require the set to contain gaps or pores within certain scale bounds.

######  Def. 2.1.

($\nu$-porosity) Suppose $0<\nu<1$,
$0\leq\alpha_{\min}\leq\alpha_{\max}\leq\infty$ and
$\Omega\subseteq\mathbb{R}^{d}$ is closed. We say that $\Omega$ is
$\nu$-porous on scales $\alpha_{\min}$ to $\alpha_{\max}$ if for every ball
$B_{r}(x)$ of radius $r\in[\alpha_{\min},\alpha_{\max}]$ there exists a ball
$B_{\nu r}(y)\subseteq B_{r}(x)$ of radius $\nu r$ such that $|\Omega\cap
B_{\nu r}(y)|=0$.

Notice that in one dimension, the $\nu$-porous set resembles a Cantor type
set, where we are able to remove a $\nu$-portion of any interval inductively
down to the lower bound scale. The Cantor sets represent a popular and easy to
understand family of fractal sets, which we construct as follows:

Let $M>1$ be a fixed integer, and let $\mathscr{A}$ be a non-empty proper
subset of $\\{0,1,\dots,M-1\\}$. The $n$-iterate ($n$-order) discrete Cantor
set with base $M$ and alphabet $\mathscr{A}$ is then defined as

$\displaystyle\pazocal{C}_{n}^{(d)}(M,\mathscr{A}):=\left\\{\sum_{j=0}^{n-1}a_{j}M^{j}\
\Big{|}\ a_{j}\in\mathscr{A},\
j=0,1,\dots,n-1\right\\}\subseteq\\{0,1,\dots,M^{n}-1\\}.$

The ”continuous” $n$-iterate Cantor set based in the interval $[0,L]$ is given
by

$\displaystyle\pazocal{C}_{n}(L,M,\mathscr{A}):=LM^{-n}\cdot\pazocal{C}_{n}^{(d)}(M,\mathscr{A})+[0,LM^{-n}]\
\ \text{for }\ n=0,1,2,\dots$

The iterates are nested, i.e., $\pazocal{C}_{n+1}\subseteq\pazocal{C}_{n}$,
and the (limit) Cantor set is then given by the intersection of all the
$n$-iterates. While the Cantor set itself has measure zero, each $n$-iterate
does not. If we let $|\mathscr{A}|$ denote the cardinality of the alphabet
$\mathscr{A}$, the measure of the $n$-iterate Cantor set is given by

$\displaystyle|\pazocal{C}_{n}(L,M,\mathscr{A})|=\left(\frac{|\mathscr{A}|}{M}\right)^{n}L.$

Note that for $M=3$ and $\mathscr{A}=\\{0,2\\}$, we obtain the standard mid-
third $n$-iterate Cantor set, with measure $(2/3)^{n}L$.

Unsurprisingly, the Cantor sets are indeed $\nu$-porous. Below we present a
simple estimate for the porosity constant and the scales (see Appendix B for
details):

######  Lemma 2.1.

The $n$-iterate Cantor set $\pazocal{C}_{n}(L,M,\mathscr{A})$ with base $M>1$
and alphabet size $|\mathscr{A}|<M$, based in the interval $[0,L]$, is
$\nu$-porous on scales $M^{-n+1}$ to $\infty$, with any $\nu\leq M^{-2}$.

In multiple dimensions $\mathbb{R}^{2d}$, we consider two possible Cantor set
constructions:

  1. (1)

For a ball of radius $R>0$ centered at the origin, we consider the radially
symmetric $n$-iterate Cantor set as a subset of the form

$\displaystyle\mathscr{C}_{n}^{2d}(R,M,\mathscr{A}):=\left\\{(x_{1},\dots,x_{2d})\in\mathbb{R}^{2d}\
\big{|}\
\left(x_{1}^{2}+\dots+x_{2d}^{2}\right)^{d}\in\pazocal{C}_{n}(R^{2d},M,\mathscr{A})\right\\}.$

In particular, localizing on $\mathscr{C}_{n}^{1}(R,M,\mathscr{A})$ has been
discussed extensively in [20]. In general, these radially symmetric Cantor
iterates are constructed such that all annuli that make up the set have the
same measure. The total measure is given by

$\displaystyle|\mathscr{C}_{n}^{2d}(R,M,\mathscr{A})|=\left|\pazocal{C}_{n}\left((\pi
R^{2})^{d}/d!,M,\mathscr{A}\right)\right|=\left(\frac{|\mathscr{A}|}{M}\right)^{n}\frac{(\pi
R^{2})^{d}}{d!},$

where we recognize $\frac{(\pi R^{2})^{d}}{d!}$ as the volume of the
$2d$-dimensional ball of radius $R$.

  2. (2)

Alternatively, we can consider the Cartesian product of 1-dimensional
$n_{j}$-iterate Cantor sets,
$\\{\pazocal{C}_{n_{j}}(L_{j},M_{j},\mathscr{A}_{j})\\}_{j}$, that is,

$\displaystyle\pazocal{C}_{n_{1}}(L_{1},M_{1},\mathscr{A}_{1})\times\pazocal{C}_{n_{2}}(L_{2},M_{2},\mathscr{A}_{2})\times\dots\times\pazocal{C}_{n_{2d}}(L_{2d},M_{2d},\mathscr{A}_{2d}).$

If all iterates coincide, the above Cartesian product reduces to
$\pazocal{C}_{n}(L,M,\mathscr{A})^{2d}$ based in the hypercube $[0,L]^{2d}$.

######  Remark.

In [8] fractal sets are originally defined in terms of $\delta$-regularity for
some $0<\delta<1$ (see Definition 2.2 in [8]). While only formulated in
$1$-dimension, the notion of $\delta$-regularity can also be extended to
higher dimensions. Compared to $\nu$-porosity, this concept offers a different
perspective on fractal sets: For instance with regard to the $n$-iterate
Cantor set $\pazocal{C}_{n}(L,M,\mathscr{A})$, we find that the $\delta$
corresponds to fractal dimension (or Hausdorff dimension) of the iterate,
namely $\frac{\ln|\mathscr{A}|}{\ln M}$. However, the notion of
$\delta$-regularity might appear more abstract than $\nu$-porosity as it does
not immediately read as a set containing gaps, and less so the size of those
gaps. Nonetheless, as shown in [8] Proposition 2.10, any $\delta$-regular set
is $\nu$-porous, and the scales associated to $\delta$-regularity coincide
with the $\nu$-porous scales up to multiplicative constants. Thus, formulating
the FUP on $\nu$-porous sets directly translates to an FUP on $\delta$-regular
sets.

##  3\. Fractal Uncertainty Principle in Joint Representation

In this section we present the FUP for the joint time-frequency
representation. Initially, in section 3.1 we derive the FUP for the Fock
spaces. In the subsequent section 3.2 we introduce the modulation spaces, and
translate the FUP for the Fock spaces to an uncertainty principle for the STFT
on modulation spaces.

###  3.1. Fractal Uncertainty Principle for Fock spaces

######  Theorem 3.1.

(FUP for Fock Spaces $\pazocal{F}^{p}(\mathbb{C}^{d})$) Let $0<h\leq 1$, and
suppose $\Omega(h)\subseteq\mathbb{C}^{d}$ is an $h$-dependent family of sets
which is $\nu$-porous on scales $h$ to $1$. Then for all $p\geq 1$ and all
$F\in\pazocal{F}^{p}(\mathbb{C}^{d})\setminus\\{0\\}$ there exist constants
$C,\beta>0$ only dependent on $\nu$ (and $d$) such that

(3.1)
$\displaystyle\frac{\|F\cdot\chi_{\Omega(h)}\|_{\pazocal{L}^{p}}^{p}}{\|F\|_{\pazocal{L}^{p}}^{p}}\leq
Ch^{\beta}\ \ \forall\ \ 0<h\leq 1.$ In particular, for $p=2$, the Toeplitz
operator $\pazocal{T}_{\Omega(h)}$ satisfies (3.2)
$\displaystyle\|\pazocal{T}_{\Omega(h)}\|_{\textnormal{op}}\leq Ch^{\beta}\ \
\forall\ \ 0<h\leq 1.$

The essential property of the Fock space that we utilize is the subaveraging
property, where point evaluations can be bounded by an average over the ball.
The statement is found in [29] for $\pazocal{F}^{p}(\mathbb{C})$, which
generalized to $\pazocal{F}^{p}(\mathbb{C}^{d})$ (see Appendix C for details)
reads:

######  Lemma 3.2.

For any $F\in\pazocal{F}^{p}(\mathbb{C}^{d})$ and any point
$z\in\mathbb{C}^{d}$, we have for all $R>0$ that

(3.3)
$\displaystyle|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\leq\left(\frac{p}{2}\right)^{d}\left(1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi
R^{2}\right)^{j}}{j!}\right)^{-1}\int_{B_{R}(z)}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi).$

Proceeding, we require the following two concepts: For a set
$\Omega\subseteq\mathbb{C}^{d}$ and $R>0$, we define the maximal Nyquist
density, $\rho(\Omega,R)$, by

(3.4) $\displaystyle\rho(\Omega,R)$
$\displaystyle:=\sup_{z\in\mathbb{C}^{d}}|\Omega\cap
B_{R}(z)|\leq\max\\{|\Omega|,|B_{R}(0)|\\}.$ Further, we define the
$R$-thickened set, $\Omega_{R}$, by $\displaystyle\Omega_{R}$
$\displaystyle:=\Omega+B_{R}(0)=\cup_{z\in\Omega}B_{R}(z).$

With these notions and Lemma 3.2, we present an upper bound estimate for the
integral over $\Omega$.

######  Lemma 3.3.

Suppose $\Omega\subseteq\mathbb{C}^{d}$ measurable. Then for any
$F\in\pazocal{F}^{p}(\mathbb{C}^{d})$ and any $R>0$, we have that

(3.5)
$\displaystyle\int_{\Omega}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z)\leq\left(\frac{p}{2}\right)^{d}\frac{\rho(\Omega,R)}{1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi
R^{2}\right)^{j}}{j!}}\int_{\Omega_{R}}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z).$

###### Proof.

By Lemma 3.2,

$\displaystyle\int_{\Omega}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z)\leq\left(\frac{p}{2}\right)^{d}$
$\displaystyle\left(1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi
R^{2}\right)^{j}}{j!}\right)^{-1}$
$\displaystyle\cdot\int_{\Omega}\int_{B_{R}(z)}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi)\mathrm{d}A(z).$

Since $z\in\Omega$, we have that

$\displaystyle\int_{B_{R}(z)}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi)=\int_{\Omega_{R}}\chi_{B_{R}(z)}(\xi)\cdot|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi).\
\ \ \ \ \ $

By Fubini’s theorem, we obtain

$\displaystyle\int_{\Omega}\int_{B_{R}(z)}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi)\mathrm{d}A(z)=\int_{\Omega_{R}}\left(\int_{\Omega}\chi_{B_{R}(\xi)}(z)\mathrm{d}A(z)\right)|F(\xi)|^{p}e^{-\pi\frac{p}{2}|\xi|^{2}}\mathrm{d}A(\xi)$
$\displaystyle=\int_{\Omega_{R}}|\Omega\cap
B_{R}(\xi)|\cdot|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi)\leq\rho(\Omega,R)\int_{\Omega_{R}}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi).$

∎

In the above lemma notice that the integral over $\Omega_{R}$ is always
bounded by the integral over $\mathbb{C}^{d}$, that is,

$\displaystyle\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{p}}^{p}\leq\left(\frac{p}{2}\right)^{d}$
$\displaystyle\frac{\rho(\Omega,R)}{1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi R^{2}\right)^{j}}{j!}}\
\|F\|_{\pazocal{L}^{p}}^{p}\ \ \forall\ \ R>0.$

Further, by the interpolation result (2.7), we can optimize the right-hand
side with the estimate for $p=1$ so that

(3.6) $\displaystyle\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{p}}^{p}\leq 2^{-d}$
$\displaystyle\frac{\rho(\Omega,R)}{1-e^{-\frac{1}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{1}{2}\pi R^{2}\right)^{j}}{j!}}\
\|F\|_{\pazocal{L}^{p}}^{p}\ \ \forall\ \ R>0.$

Regardless, this shows that estimates of the quotient
$\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{p}}^{p}/\|F\|_{\pazocal{L}^{p}}^{p}$
can be solely based on estimates of the maximal Nyquist density. E.g., as
shown in the subsequent section 4, under certain growth conditions, we are
able to obtain good estimates for Nyquist density for the standard Cartesian
product of Cantor sets and radial Cantor sets. For the general case of porous
sets, however, we also take into account the integral over the thickened set.

Besides the trivial upper bounds of (3.4), if we suppose
$\Omega\subseteq\mathbb{C}^{d}$ is $\nu$-porous on scales $\alpha_{\min}$ to
$\alpha_{\max}$ and consider a radius $R$ within the scale bounds, we obtain,
by Def. 2.1, the simple estimate

$\displaystyle\rho(\Omega,R)\leq\sup_{|z-\xi|\leq(1-\nu)R}\big{|}B_{R}(z)\setminus
B_{\nu
R}(\xi)\big{|}=\left(1-\nu^{2d}\right)|B_{R}(0)|=\left(1-\nu^{2d}\right)\frac{\left(\pi
R^{2}\right)^{d}}{d!}.$

We apply this upper bound to Lemma 3.3, which yields the corollary:

######  Corollary 3.1.

Suppose $\Omega\subseteq\mathbb{C}^{d}$ is $\nu$-porous on scales
$\alpha_{\min}$ to $\alpha_{\max}$. Then for any radius
$R\in[\alpha_{\min},\alpha_{\max}]$ and any function
$F\in\pazocal{F}^{p}(\mathbb{C}^{d})$, we have that

(3.7)
$\displaystyle\int_{\Omega}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z)\leq\kappa_{d}\left(\frac{p}{2}\pi
R^{2}\right)\cdot\left(1-\nu^{2d}\right)\int_{\Omega_{R}}|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\mathrm{d}A(z),$

where the function $\kappa_{d}$ is given by

(3.8)
$\displaystyle\kappa_{d}(x):=\frac{x^{d}}{d!}\left(1-e^{-x}\sum_{j=0}^{d-1}\frac{x^{j}}{j!}\right)^{-1}.$

The crucial observation, moving forward, is that for some choices of $R>0$,
the thickened set $\Omega_{R}$ is itself a porous set provided the original
set $\Omega$ is porous. A special case of this observation is presented in
Proposition 2.11 [8].

######  Lemma 3.4.

(Thickening of porous set) Suppose $\Omega\subseteq\mathbb{C}^{d}$ is
$\nu$-porous on scales $\alpha_{\min}$ to $\alpha_{\max}$. For any $0\leq
r<\nu\cdot\alpha_{\max}$, consider the $r$-thickened set
$\Omega_{r}:=\Omega+B_{r}(0)=\cup_{z\in\Omega}B_{r}(z)$. Then for any
$R\in[\alpha_{\min},\alpha_{\max}]$ with $\frac{r}{\nu}<R$, the set
$\Omega_{r}$ is $\left(\nu-\frac{r}{R}\right)$-porous on scales $R$ to
$\alpha_{\max}$.

###### Proof.

Consider $\alpha_{\min}\leq R\leq L\leq\alpha_{\max}$. By $\nu$-porosity of
$\Omega$, for any $B_{L}(z)$ there exists $B_{\nu L}(\xi)\subseteq B_{L}(z)$
such that $|B_{\nu L}(\xi)\cap\Omega|=0$. After $r$-thickening, we maintain a
zero intersection with $\Omega_{r}$ if the radius of $B_{\nu L}(\xi)$ is
reduced to $\nu L-r$. Since $L$ is arbitrary and

$\displaystyle\left(\nu-\frac{r}{R}\right)L\leq\nu
L-r\implies|B_{\left(\nu-\frac{r}{R}\right)L}(\xi)\cap\Omega_{r}|\leq|B_{\nu
L-r}(\xi)\cap\Omega_{r}|=0,$

the set $\Omega_{r}$ satisfies the claimed porosity properties. The conditions
on $r$ and $R$ ensure that the new porosity is positive and that the scale
bounds are valid. ∎

We are now ready to prove Theorem 3.1.

###### Proof.

(Theorem 3.1) By the interpolation result (2.7), it suffices to prove the
statement for the case $p=1$. Nonetheless, for notational convenience, we will
rather consider the case $p=2$ as the only difference in the proof between
$p=2$ and general $p\geq 1$ is that $\kappa_{d}(x)$ is replaced by
$\kappa_{d}(\frac{p}{2}x)$.

Recall that for some $0<h\leq 1$ we presume $\Omega=\Omega(h)$ is $\nu$-porous
on scales $h$ to $1$. For some finite sequence of radii
$\\{R_{j}\\}_{j=1}^{n}$, we consider an associated sequence of thickened sets
$\\{\Omega^{(j)}\\}_{j=0}^{n-1}$, given by

$\displaystyle\Omega^{(0)}:=\Omega\ \ \text{and }\
\Omega^{(j)}:=\Omega_{R_{1}+R_{2}+\dots+R_{j}}\ \ \text{for }j=1,2,\dots,n-1.$

We chose the radii such that the set $\Omega^{(j)}$ maintains the porous
property, in the sense that

$\displaystyle\Omega^{(j)}\ \text{is }\nu_{j}\text{-porous on scales }R_{j+1}\
\text{to }1.$

By Lemma 3.4, the new porosity constants $\\{\nu_{j}\\}_{j}$ are given by

(3.9) $\displaystyle\nu_{j}=\nu-\frac{R_{1}+\dots+R_{j}}{R_{j+1}},\ \
\text{where }\ R_{j+1}>\frac{R_{1}+\dots+R_{j}}{\nu}\ \ \text{for
}j=0,1,\dots,n-1.$

Since the radii must also satisfy $h\leq R_{j}\leq 1$, the length of the
sequences $n$ cannot be infinite and must depend on $h$. From the above
conditions, it is clear that radii must at least grow geometrically, i.e., (in
the equality case) $R_{j}=c\cdot a^{j}$ for $j=1,2,\dots,n$ where $a>1$ and
$c>0$. In particular, the conditions are satisfied for the sequence of radii
$R_{j}=c\cdot\left(\frac{3}{\nu}\right)^{j}$ for $j=1,2,\dots,n$ such that
$c\cdot\left(\frac{3}{\nu}\right)^{n}\leq
1<c\cdot\left(\frac{3}{\nu}\right)^{n+1}$. By choosing the initial radius to
be as small as possible, namely $R_{1}=h$, it follows that $0<n+\ln
h\cdot\left(\ln\frac{3}{\nu}\right)^{-1}\leq 1$, which means $n\sim|\ln h|$.

Now we make repeated use of Corollary 3.1 to the sequence of thickened sets
$\\{\Omega^{(j)}\\}_{j}$, which reveals that the integral over $\Omega$ is
bounded by

$\displaystyle\int_{\Omega}|F(z)|^{2}e^{-\pi|z|^{2}}\mathrm{d}A(z)$
$\displaystyle\leq\left[\prod_{k=1}^{j}\kappa_{d}\left(\pi
R_{k}^{2}\right)\cdot\left(1-\nu_{k-1}^{2d}\right)\right]\cdot\int_{\Omega^{(j)}}|F(z)|^{2}e^{-\pi|z|^{2}}\mathrm{d}A(z)$
$\displaystyle\leq\left[\prod_{k=1}^{j}\kappa_{d}\left(\pi
R_{k}^{2}\right)\cdot\left(1-\nu_{k-1}^{2d}\right)\right]\cdot\|F\|_{\pazocal{L}^{2}}^{2}\
\ \text{for }j=1,2,\dots,n.$

It is straightforward to verify that $\lim_{x\to 0}\kappa_{d}(x)=1$ and the
derivative $\kappa_{d}^{\prime}(x)>0$ for all $x>0$. Combined with the fact
that the sequence of porosity constants $\\{\nu_{j}\\}_{j}$ is decreasing, we
conclude that the sequence $\\{\kappa_{d}\left(\pi
R_{j}^{2}\right)\cdot\left(1-\nu_{j-1}^{2d}\right)\\}_{j}$ is monotonically
increasing. Thus, we are interested in the index reduction
$n_{0}\in\mathbb{N}$ such that

$\displaystyle\kappa_{d}\left(\pi
R_{j}^{2}\right)\cdot\left(1-\nu_{j-1}^{2d}\right)<1\ \ \text{for
}j=1,2,\dots,n-n_{0}.$

Since $\lim_{h\to 0}\kappa_{d}\left(\pi
R_{1}^{2}\right)\cdot\left(1-\nu_{0}^{2d}\right)=1-\nu^{2d}<1$, for
sufficiently small $h<h_{0}$, we have that there exists an $n_{0}$ such that
the difference $n-n_{0}>0$. Next, we claim that there exists an index
reduction $n_{0}$ which is independent of $h<h_{0}$, implying $n-n_{0}\sim|\ln
h|$. Note that the procedure below does not produce an optimal $n_{0}$, merely
shows that such an $n_{0}$ exists.

From (3.9) with $R_{j}=c\left(\frac{3}{\nu}\right)^{j}$, we find the following
lower bound for the porosities

$\displaystyle\nu_{j}=\nu-\left(\frac{\nu}{3}\right)^{j+1}\sum_{k=1}^{j}\left(\frac{3}{\nu}\right)^{k}\geq\frac{\nu}{2}\
\ \ \forall\ \ \ j\in\mathbb{N}\ \ \text{and }\ 0<\nu<1.$

Now, let $r=r(\nu)>0$ denote the (unique) solution to $\kappa_{d}(\pi
r^{2})\cdot\left(1-\left(\frac{\nu}{2}\right)^{2d}\right)=1$, from which it is
evident that $\kappa_{d}\left(\pi
R_{j}^{2}\right)\cdot\left(1-\nu_{j-1}^{2d}\right)<1$ for all $R_{j}<r$. Now,
consider any fixed $r_{0}<r$, and observe that since $r_{0}$ is independent of
$h$, and the radius $R_{n}(\approx 1)$ is bounded, there must exist a finite
number $n_{0}$ also independent of $h$ such that
$R_{n-n_{0}}=R_{n}\cdot(\frac{\nu}{3})^{n_{0}}\leq r_{0}$.

In summary, by our choice of index reduction $n_{0}$, we obtain

$\displaystyle\kappa_{d}\left(\pi
R_{j}^{2}\right)\cdot\left(1-\nu_{j-1}^{2d}\right)\leq\kappa_{d}(\pi
r_{0}^{2})\cdot\left(1-\left(\frac{\nu}{2}\right)^{2d}\right)=:(1-\epsilon)<1\
\ \text{for }\ j=1,2,\dots,n-n_{0}.$

Since $n-n_{0}\sim|\ln h|$, we conclude

$\displaystyle\int_{\Omega}|F(z)|^{2}e^{-\pi|z|^{2}}\mathrm{d}A(z)\leq(1-\epsilon)^{n-n_{0}}\|F\|_{\pazocal{L}^{2}}^{2}\sim
h^{\beta}\|F\|_{\pazocal{L}^{2}}^{2}\ \ \text{for some }\beta>0.$

∎

###  3.2. Fractal Uncertainty Principle for Modulation Spaces

Recall that for signals $f$ and windows $\phi$ in $L^{2}(\mathbb{R}^{d})$ the
STFT $V_{\phi}f\in L^{2}(\mathbb{R}^{2d})$. For general $p\geq 1$, we instead
consider modulation spaces, introduced in [11]. Let
$\pazocal{S}^{\prime}(\mathbb{R}^{d})$ denote the space of tempered
distributions, and suppose $\phi:\mathbb{R}^{d}\to\mathbb{C}$ is a window
function such that $0<\|V_{\phi}\phi\|_{L^{1}(\mathbb{R}^{2d})}<\infty$. Then
the modulation space $M^{p}(\mathbb{R}^{d})$ is defined as the subspace

$\displaystyle
M^{p}(\mathbb{R}^{d}):=\left\\{f\in\pazocal{S}^{\prime}(\mathbb{R}^{d})\
\big{|}\ \|V_{\phi}f\|_{L^{p}(\mathbb{R}^{2d})}<\infty\right\\}.$

Depending on our particular choice of window, we induce equivalent norms on
$M^{p}(\mathbb{R}^{d})$, namely
$\|f\|_{M^{p}}:=\|V_{\phi}f\|_{L^{p}(\mathbb{R}^{2d})}$. Hence, we can
consider the Gaussian window $\phi_{0}$ and establish a connection between the
modulation spaces and the Fock spaces. In fact, the Bargmann transform,
defined in (2.4), extends to an isometric isomorphism
$\pazocal{B}:M^{p}(\mathbb{R}^{d})\to\pazocal{F}^{p}(\mathbb{C}^{d})$ (see
[1], [25]). Utilizing identity (2.5), we find that

$\displaystyle\|V_{\phi_{0}}f\cdot\chi_{\Omega}\|_{L^{p}(\mathbb{R}^{2d})}=\|\pazocal{B}f\cdot\chi_{\Omega^{*}}\|_{\pazocal{L}^{p}},$

where $\Omega^{*}:=\\{z\in\mathbb{C}^{d}\ |\ \overline{z}\in\Omega\\}$ denotes
the complex conjugate subset. Thus, Theorem 3.1 can be rephrased in terms of
the STFT on modulation spaces.

######  Theorem 3.5.

(FUP for modulation spaces $M^{p}(\mathbb{R}^{d})$) Let $0<h\leq 1$, and
suppose $\Omega(h)\subseteq\mathbb{C}^{d}$ is an $h$-dependent family of sets
which is $\nu$-porous on scales $h$ to $1$. Then for all $p\geq 1$ and all
$f\in M^{p}(\mathbb{R}^{d})\setminus\\{0\\}$ there exist constants $C,\beta>0$
only dependent on $\nu$ (and $d$) such that

(3.10)
$\displaystyle\frac{\|V_{\phi_{0}}f\cdot\chi_{\Omega(h)}\|_{L^{p}(\mathbb{R}^{2d})}^{p}}{\|V_{\phi_{0}}f\|_{L^{p}(\mathbb{R}^{2d})}^{p}}\leq
Ch^{\beta}\ \ \forall\ \ 0<h\leq 1.$ In particular, for $p=2$, Daubechies’
time-frequency localization operator $P^{\phi_{0}}_{\Omega(h)}$ satisfies
(3.11) $\displaystyle\|P^{\phi_{0}}_{\Omega(h)}\|_{\textnormal{op}}\leq
Ch^{\beta}\ \ \forall\ \ 0<h\leq 1.$

Another noteworthy rephrasement is result (3.6), which in terms of the STFT
reads

(3.12)
$\displaystyle\|V_{\phi_{0}}f\cdot\chi_{\Omega}\|_{L^{p}(\mathbb{R}^{2d})}^{p}\leq
2^{-d}$ $\displaystyle\frac{\rho(\Omega,R)}{1-e^{-\frac{1}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{1}{2}\pi R^{2}\right)^{j}}{j!}}\
\|V_{\phi_{0}}f\|_{L^{p}(\mathbb{R}^{2d})}^{p}\ \ \forall\ \ R>0.$

######  Remark.

For $d=1$, in [3] Theorem 3, there is a similar but more general result, valid
for all Hermite windows $\\{h_{j}\\}_{j}$, namely

$\displaystyle\|V_{h_{j}}f\cdot\chi_{\Omega}\|_{L^{p}(\mathbb{R}^{2})}^{p}\leq\frac{\rho(\Omega,R)}{1-e^{-\pi
R^{2}}P_{j}(\pi R^{2})}\|V_{h_{j}}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\ \ \forall\
\ p\geq 1\ \text{and}\ R>0,$

where $P_{j}$ is a specified polynomial of degree $2j$. In particular, for
$j=0$, we have that $h_{0}=\phi_{0}$ and $P_{0}\equiv 1$ so that the above
inequality reduces to

(3.13)
$\displaystyle\|V_{\phi_{0}}f\cdot\chi_{\Omega}\|_{L^{p}(\mathbb{R}^{2})}^{p}\leq\frac{\rho(\Omega,R)}{1-e^{-\pi
R^{2}}}\|V_{\phi_{0}}f\|_{L^{p}(\mathbb{R}^{2})}^{p}\ \ \forall\ \ p\geq 1\
\text{and}\ R>0.$

By close inspection, for $d=1$ result (3.12) turns out to be an improvement of
(3.13).222In [19] (section 3.2 page 10) there is a similar comparison for a
specific set $\Omega$, where $\rho(\Omega,R)$ is known, with the claim that
the upper bound obtained in [19] is also an improvement of (3.13). By closer
examination, this claim is incorrect, and the upper bound for the localization
operator is in fact the same as result (3.13), i.e., the special case
$h_{0}=\phi_{0}$ of Theorem 3 in [3].

##  4\. Density of Cantor sets

In this section we consider the Cantor sets specifically and show, under
certain conditions, how the FUP in Theorem 3.5 (or equivalently Theorem 3.1)
can be refined for this family of fractal sets. While the general FUP is
formulated in terms of a continuous parameter $h\to 0^{+}$, it is more
convenient to consider discrete iterations $n\to\infty$ when working with the
Cantor sets. The relation between $h$ and $n$ is made clear by Lemma 2.1,
where for the Cantor iterate $\pazocal{C}_{n}(M,\mathscr{A},L)$, we have that

$\displaystyle h\sim LM^{-n}.$

From the above relation, we observe that $h$ is of the same order of magnitude
as the intervals that make up the Cantor iterate. Furthermore, similarly to
the condition imposed in [8], we consider $L\sim h^{-1}$. Thus, we have that
$L$ is dependent on the iterates $n$ so that $L(n)\sim M^{\frac{n}{2}}$. More
precisely, we specify the multiplicative constants of the asymptotes so that
the interval condition now reads:

######  Def. 4.1.

Let the interval length $L$ be a function $\mathbb{N}\to\mathbb{R}_{+}$. The
interval length $L$ satisfies condition $(I_{M})$ with constants $0<c_{1}\leq
c_{2}<\infty$ if

(4.1) $\displaystyle c_{1}M^{\frac{n}{2}}\leq L(n)\leq c_{2}M^{\frac{n}{2}}\ \
\text{for }\ n=0,1,2,\dots$

For the $2d$-dimensional radially symmetric Cantor iterate
$\mathscr{C}_{n}^{d}(R,M,\mathscr{A})$, we adjust the above condition to the
radius $R>0$, similar to condition (1.3) in [19] and (1.4) in [20].

######  Def. 4.2.

Let the radius $R$ be a function $\mathbb{N}\to\mathbb{R}_{+}$. The radius $R$
satisfies condition $(D_{M})$ with constants $0<c_{1}\leq c_{2}<\infty$ if

(4.2) $\displaystyle c_{1}M^{\frac{n}{2}}\leq R^{2d}(n)\leq
c_{2}M^{\frac{n}{2}},\ \ \text{for }\ n=0,1,2,\dots$

With these conditions, we present an FUP for the two multidimensional Cantor
set constructions from section 2.3.

######  Theorem 4.1.

(FUP for Cantor sets) We consider the two constructions separately:

  1. (i)

Let $\mathscr{C}_{n}^{2d}(R,M,\mathscr{A})$ denote the $2d$-dimensional
radially symmetric Cantor iterate, and suppose the radius $R=R(n)>0$ satisfies
condition $(D_{M})$ with constants $c_{1}\leq c_{2}$. Then there exists a
positive constant $\gamma$ only dependent on $M$, $|\mathscr{A}|$ and
$c_{1},c_{2}$ such that for all $f\in M^{p}(\mathbb{R}^{d})$ with $p\geq 1$,

(4.3)
$\displaystyle\|V_{\phi_{0}}f\cdot\chi_{\mathscr{C}^{2d}_{n}(R(n),M,\mathscr{A})}\|_{L^{p}(\mathbb{R}^{2d})}^{p}\leq\gamma\cdot\left(\frac{|\mathscr{A}|}{M}\right)^{\frac{n}{2}}\|V_{\phi_{0}}f\|_{L^{p}(\mathbb{R}^{2d})}^{p}\
\ \text{for }\ n=0,1,2,\dots$

  2. (ii)

Let
$\Omega:=\pazocal{C}_{n_{1}}(L_{1},M_{1},\mathscr{A}_{1})\times\dots\times\pazocal{C}_{n_{2d}}(L_{2d},M_{2d},\mathscr{A}_{2d})$
denote the $2d$-dimensional Cartesian product of Cantor iterates, and suppose
the interval lengths $L_{j}=L_{j}(n_{j})$ satisfy condition $(I_{M_{j}})$ with
the same constants $c_{1}\leq c_{2}$. Then there exists a positive constant
$\gamma$ only dependent on $\\{(M_{j},|\mathscr{A}_{j}|)\\}_{j}$ and
$c_{1},c_{2}$ such that for all $f\in M^{p}(\mathbb{R}^{d})$ with $p\geq 1$
and all iterations $\\{n_{j}\\}_{j}\in\mathbb{N}_{0}^{2d}$,

$\displaystyle\|V_{\phi_{0}}f\cdot\chi_{\Omega}\|_{L^{p}(\mathbb{R}^{2d})}^{p}\leq\gamma\cdot\left[\prod_{j=1}^{2d}\left(\frac{|\mathscr{A}_{j}|}{M_{j}}\right)^{n_{j}/2}\right]\|V_{\phi_{0}}f\|_{L^{p}(\mathbb{R}^{2d})}^{p}.$

In particular, when all iterates are equal to say
$\pazocal{C}_{n}(L(n),M,\mathscr{A})$, we find that

(4.4)
$\displaystyle\|V_{\phi_{0}}f\cdot\chi_{\pazocal{C}_{n}(L,M,\mathscr{A})^{2d}}\|_{L^{p}(\mathbb{R}^{2d})}^{p}\leq\gamma\cdot\left(\frac{|\mathscr{A}|}{M}\right)^{n\cdot
d}\|V_{\phi_{0}}f\|_{L^{p}(\mathbb{R}^{2d})}^{p}\ \ \text{for }\ n=0,1,2\dots$

######  Remark.

For localization on the radially symmetric Cantor set with $d=1$ and $p=2$, we
retrieve the same upper bound asymptote as in [20] Theorem 3.3. In fact,
Theorem 3.3 reveals that the general asymptote is optimal in the sense that
there are alphabets where the asymptote is precise. However, the same theorem
also state that there are alphabets where the asymptote is not precise.

The proof of Theorem 4.1 is based on inequality (3.12), combined with
estimates of the maximal Nyquist density of the multidimensional Cantor
iterates. To begin with, we consider the 1-dimensional $n$-iterate
$\pazocal{C}_{n}(M,\mathscr{A}):=\pazocal{C}_{n}(1,M,\mathscr{A})$ based in
$[0,1]$, to which we associate the so-called Cantor function
$\pazocal{G}_{n,M,\mathscr{A}}:\mathbb{R}\to[0,1]$, given by

(4.5)
$\displaystyle\pazocal{G}_{n,M,\mathscr{A}}(x):=|\pazocal{C}_{n}(M,\mathscr{A})|^{-1}\begin{cases}0,\
\ x\leq 0,\\\ |\pazocal{C}_{n}(M,\mathscr{A})\cap[0,x]|,\ \ x>0.\end{cases}$

For the iterates based in $[0,L]$, we consider the dilated Cantor function
$\pazocal{G}_{n,M,\mathscr{A}}(\cdot\ L^{-1})$. The Cantor function is a
useful concept as the difference
$\pazocal{G}_{n,M,\mathscr{A}}(y)-\pazocal{G}_{n,M,\mathscr{A}}(x)$ measures
the portion of the $n$-iterate $\pazocal{C}_{n}(M,\mathscr{A})$ contained in
the interval $[x,y]$. By definition, the Cantor function is said to be
subadditive if the difference is bounded by
$\pazocal{G}_{n,M,\mathscr{A}}(y-x)$. While this is the case for the mid-third
Cantor set (see [7]), we cannot guarantee subadditivity with an arbitrary
alphabet. Nonetheless, for our purpose, we only require a weaker version
utilizing the canonical alphabet
$\overline{\mathscr{A}}:=\\{0,1,\dots,|\mathscr{A}|\\}$ (see [20] Appendix A).

######  Lemma 4.2.

Let $\pazocal{G}_{n,M,\mathscr{A}}$ denote the Cantor-function, defined in
(4.5). Then for any $x\leq y$,

(4.6)
$\displaystyle\pazocal{G}_{n,M,\mathscr{A}}(y)-\pazocal{G}_{n,M,\mathscr{A}}(x)\leq\pazocal{G}_{n,M,\overline{\mathscr{A}}}(y-x).$

Re-scaling to Cantor iterates based $[0,L]$, we find, by the above lemma, that
the maximal Nyquist density is bounded by

$\displaystyle\rho\big{(}\pazocal{C}_{n}(L,M,\mathscr{A}),x\big{)}=\max_{a\in\mathbb{R}}|\pazocal{C}_{n}(L,M,\mathscr{A})\cap[a,a+x]|\leq|\pazocal{C}_{n}(L,M,\overline{\mathscr{A}})\cap[0,x]|.$

In the next lemma, we enforce condition (4.1), which yields a more explicit
estimate for the upper bound.

######  Lemma 4.3.

Suppose that the alphabet $\mathscr{A}$ is a proper subset of
$\\{0,1,\dots,M-1\\}$, and suppose that the length $L(n)>0$ satisfies
condition $(I_{M})$ with constants $0<c_{1}\leq c_{2}<\infty$. Then for any
fixed $x>0$, there exists a finite constant $\gamma>0$ dependent only on
$x,c_{1},c_{2}$ such that

(4.7)
$\displaystyle|\pazocal{C}_{n}(L,M,\mathscr{A})\cap[0,x]|=\pazocal{G}_{n,M,\mathscr{A}}\big{(}xL^{-1}\big{)}\left(\frac{|\mathscr{A}|}{M}\right)^{n}L\leq\gamma\cdot\left(\frac{|\mathscr{A}|}{M}\right)^{\frac{n}{2}}\to
0\ \text{as }n\to\infty.$

###### Proof.

Fix some positive integer $n_{0}<n$, and observe that for coefficients
$a_{j}\in\mathscr{A}$, the sum $\sum_{j=0}^{n-1}a_{j}M^{j-n}<M^{n_{0}-n}$ only
if $a_{j}=0$ for $j\geq n_{0}$. Thus, the Cantor function is bounded by

$\displaystyle\pazocal{G}_{n,M,\mathscr{A}}(x)\leq|\mathscr{A}|^{n_{0}-n}\ \
\forall\ \ x<M^{n_{0}-n}.$

Evidently, we also have that $xM^{-m}<M^{n_{0}+m-(n-2m)}$, which, by the above
estimate, yields

(4.8)
$\displaystyle\pazocal{G}_{n,M,\mathscr{A}}(xM^{-m})\leq|\mathscr{A}|^{n_{0}+m-(n+2m)}=|\mathscr{A}|^{-m}|\mathscr{A}|^{n_{0}-n}\
\ \text{for }\ m=0,1,2,\dots$

Now, suppose $x$ is any fixed number, to which there exists a threshold
iterate $N\geq 0$ so that $xM^{-\frac{N}{2}}<M^{-1}$, and we are in a position
to apply (4.8). From here, we deduce that there exists a constant only
dependent on $x$, say $\kappa=\kappa(x)$ such that

$\displaystyle\pazocal{G}_{n,M,\mathscr{A}}(xM^{-\frac{n}{2}})\leq\kappa(x)|\mathscr{A}|^{-\frac{n}{2}}\
\ \text{for }\ n=0,1,2,\dots$

Enforce the lower bound condition $L(n)\geq c_{1}M^{\frac{n}{2}}$. By
monotonicity of the Cantor function,

$\displaystyle\pazocal{G}_{n,M,\mathscr{A}}(xL^{-1})\leq\pazocal{G}_{n,M,\mathscr{A}}(xc_{1}^{-1}M^{-\frac{n}{2}})\leq\kappa(xc_{1}^{-1})|\pazocal{A}|^{-\frac{n}{2}}.$

Finally, enforce the upper bound condition $L(n)\leq c_{2}M^{\frac{n}{2}}$,
and inequality (4.7) follows with constant $\gamma=\kappa(xc_{1}^{-1})c_{2}$.
Since $0<c_{1}\leq c_{2}<\infty$, the constant $\gamma$ is indeed finite. ∎

We are now ready to estimate upper bounds for the maximal Nyquist densities
for the multidimensional Cantor iterates, which, combined with (3.12), proves
Theorem 4.1.

######  Lemma 4.4.

(Nyquist density of Cantor sets) Let $r>0$ be fixed.

  1. (i)

Let $\mathscr{C}_{n}^{2d}(R,M,\mathscr{A})$ denote the radial Cantor iterate,
and suppose the radius $R(n)>0$ satisfies condition $(D_{M})$ with constants
$c_{1}\leq c_{2}$. Then there exist a constant $\gamma_{r}>0$ only dependent
on $M,|\mathscr{A}|$ and $c_{1},c_{2}$ such that the maximal Nyquist density

$\displaystyle\rho\left(\mathscr{C}_{n}^{2d}(R,M,\mathscr{A}),r\right)\leq\gamma_{r}\cdot\left(\frac{|\mathscr{A}}{M}\right)^{\frac{n}{2}}\
\ \text{for }\ n=0,1,2,\dots$

  2. (ii)

Let
$\Omega:=\pazocal{C}_{n_{1}}(L_{1},M_{1},\mathscr{A}_{1})\times\dots\times\pazocal{C}_{n_{2d}}(L_{2d},M_{2d},\mathscr{A}_{2d})$
denote the Cartesian product, and suppose the interval lengths $L_{j}(n_{j})$
satisfy condition $(I_{M_{j}})$ with the same constants $c_{1}\leq c_{2}$.
Then there exist a constant $\gamma_{r}>0$ only dependent on
$\\{(M_{j},|\mathscr{A}_{j}|)\\}_{j}$ and $c_{1},c_{2}$ such that the maximal
Nyquist density is bounded by

$\displaystyle\rho\left(\Omega,r\right)\leq\gamma_{r}\cdot\prod_{j=1}^{2d}\left(\frac{|\mathscr{A}_{j}|}{M_{j}}\right)^{n_{j}/2}\
\ \text{for }\ n=0,1,2,\dots\ \ \ \ \ \ \ \ $

###### Proof.

Without loss of generality, we can assume that $r\geq 1$. For part (i), fix a
cutoff value $N\gg 1$ and distinguish between two cases for
$B_{r}(a)\in\mathbb{R}^{2d}$, (1) $|a|\leq Nr$ and (2) $|a|>Nr$:

For case (1), $B_{r}(a)\subseteq B_{(N+1)r}(0)$ so that
$|\mathscr{C}^{2d}_{n}(\dots)\cap
B_{r}(a)|\leq|\mathscr{C}^{2d}_{n}(\dots)\cap B_{(N+1)r}(0)|$. By definition
of the radial Cantor iterate, it follows that

$\displaystyle\left|\mathscr{C}^{2d}_{n}(R,M,\mathscr{A})\cap B_{r}(a)\right|$
$\displaystyle\leq\frac{\pi^{d}}{d!}\left|\pazocal{C}_{n}(R^{2d},M,\mathscr{A})\cap\left[0,(N+1)^{2d}r^{2d}\right]\right|$
$\displaystyle\leq\gamma_{1}\cdot\left(\frac{|\mathscr{A}|}{M}\right)^{\frac{n}{2}}\
\ \text{(by Lemma \ref{lemma_cantor_function_decay})}$

for some constant $\gamma_{1}>0$.

For case (2), $B_{r}(a)\subseteq A_{r}(|a|):=\\{z\in\mathbb{R}^{2d}\ |\
0<|a|-r\leq|z|\leq|a|+r\\}$. For the annulus $A_{r}(|a|)$ with $r<|a|$, we
have again, by definition of the radial Cantor iterate, that

$\displaystyle\big{|}\mathscr{C}^{2d}_{n}(R,M,\mathscr{A})\cap
A_{r}(|a|)\big{|}=\frac{\pi^{d}}{d!}\big{|}\pazocal{C}_{n}(R^{2d},M,\mathscr{A})\cap\left[(|a|-r)^{2d},(|a|+r)^{2d}\right]\big{|}$
(4.9)
$\displaystyle\leq\frac{\pi^{d}}{d!}\big{|}\pazocal{C}_{n}(R^{2d},M,\overline{\mathscr{A}})\cap\left[0,(|a|+r)^{2d}-(|a|-r)^{2d}\right]\big{|}\
\ \text{(by Lemma \ref{lemma_Cantor_function_weak_subadditivity_property})}.$

Since $|a|>1$ and since the leading term $|a|^{2d}$ cancels, it follows that

$\displaystyle(|a|+r)^{2d}-(|a|-r)^{2d}\leq|a|^{2d-1}\left((1+r)^{2d}-(1-r)^{2d}\right).$
Furthermore, by subadditivity, the Cantor function
$\pazocal{G}_{n,M,\overline{\mathscr{A}}}(m\cdot x)\leq
m\cdot\pazocal{G}_{n,M,\overline{\mathscr{A}}}(x)$ for any $m\in\mathbb{N}$.
In general, this means
$\displaystyle\pazocal{G}_{n,M,\overline{\mathscr{A}}}(m\cdot
x)\leq(m+1)\cdot\pazocal{G}_{n,M,\overline{\mathscr{A}}}(x)\ \ \forall\ \
m>0.$

With these observations inequality (4.9) simplifies to

$\displaystyle\big{|}\mathscr{C}^{2d}_{n}(R,M,\mathscr{A})\cap
A_{r}(|a|)\big{|}$
$\displaystyle\leq\frac{\pi^{d}}{d!}\left(|a|^{2d-1}+1\right)\big{|}\pazocal{C}_{n}(R^{2d},M,\overline{\mathscr{A}})\cap\left[0,(1+r)^{2d}-(1-r)^{2d}\right]\big{|}$
(4.10)
$\displaystyle\leq\gamma_{2}\cdot\left(|a|^{2d-1}+1\right)\left(\frac{|\mathscr{A}|}{M}\right)^{\frac{n}{2}}\
\ \text{(by Lemma \ref{lemma_cantor_function_decay})}$

for some constant $\gamma_{2}>0$ independent of $|a|$. Evidently, the right-
hand side of the above inequality is unbounded in terms of $|a|$. The ball
$B_{r}(a)$, however, only represents a fraction of the annulus $A_{r}(|a|)$,
which warrants a closer comparison between $|\mathscr{C}^{2d}_{n}(\dots)\cap
A_{r}(|a|)|$ and $|\mathscr{C}^{2d}_{n}(\dots)\cap B_{r}(a)|$. Let $\partial
B_{\eta}:=\\{z\in\mathbb{R}^{2d}\ |\ |z|=\eta\\}$ denote the
$(2d-1)$-dimensional sphere of radius $\eta>0$ with associated surface area
$|\partial B_{\eta}|$. We consider the optimal surface quotient

$\displaystyle\max\left\\{\frac{|\partial B_{\eta}\cap B_{r}(a)|}{|\partial
B_{\eta}|}\ \Big{|}\
\eta\in\big{[}|a|-r,|a|+r\big{]}\right\\}\leq\alpha\cdot\left(\frac{r}{|a|}\right)^{2d-1}$

for some constant $\alpha>0$ only dependent on $N\gg 1$ and $d$. Hence,

$\displaystyle|\mathscr{C}^{2d}_{n}(R,M,\mathscr{A})\cap
B_{r}(a)|\leq\alpha\cdot\left(\frac{r}{|a|}\right)^{2d-1}|\mathscr{C}^{2d}_{n}(R,M,\mathscr{A})\cap
A_{r}(|a|)|,$

which, combined with (4.10), yields the desired conclusion of part (i).

Part (ii) turns out to be much simpler. Since any ball $B_{r}(a)$ is contained
in some shifted hypercube $a+[-r,r]^{2d}$, we consider the maximal Nyquist
density $\rho\big{(}\pazocal{C}_{n_{j}}(L_{j},M_{j},\mathscr{A}_{j}),r)$ in
each direction $j=1,2,\dots,2d$. In total, we obtain

$\displaystyle\rho(\Omega,r)\leq\prod_{j=1}^{2d}\rho\big{(}\pazocal{C}_{n_{j}}(L_{j},M_{j},\mathscr{A}_{j}),r),$

where the desired conclusion follows once we apply the 1-dimensional result
Lemma 4.3. ∎

##  5\. Fractal Uncertainty Principle for Gabor Multipliers

In this section we present one simple translation of Theorem 3.1 to Gabor
multipliers. Specifically, since the previous theorems are based on the
Gaussian window $\phi_{0}$, we proceed with Gabor multipliers on the form
(2.10) with $\phi_{0}$ as generating function, i.e.,

(5.1)
$\displaystyle\mathscr{G}_{\Lambda,\Omega}^{\phi_{0}}f=|A_{\Lambda}|\sum_{\lambda\in\Lambda_{\Omega}}\langle
f,\pi(\lambda)\phi_{0}\rangle\pi(\lambda)\phi_{0}$

for some subset $\Omega\subseteq\mathbb{R}^{2d}$ and lattice $\Lambda$.
Furthermore, since we consider an $h$-dependent family $\Omega(h)$ of sets for
the FUP, we also let the lattice $\Lambda$ depend on the parameter $0<h\leq
1$, meaning, we let $\Lambda(h)$ become sufficiently dense so to capture the
fractal details of $\Omega(h)$. In particular, we consider the following
condition:

######  Def. 5.1.

We say that an $h$-dependent family of lattices
$\Lambda(h)\subseteq\mathbb{R}^{2d}$ for $0<h\leq 1$ satisfies condition $(H)$
with constant $L>0$ if the fundamental region, $A_{\Lambda(h)}$, satisfies the
inclusion

$\displaystyle A_{\Lambda(h)}\subseteq B_{hL}(0)\ \ \forall\ \ 0<h\leq 1.$

Utilizing condition ($H$), we formulate the FUP for Gaussian Gabor
multipliers:

######  Theorem 5.1.

(FUP Gaussian Gabor multipliers) Let $0<h\leq 1$, and suppose
$\Omega(h)\subseteq\mathbb{R}^{2d}$ is an $h$-dependent family of sets which
is $\nu$-porous on scales $h$ to $1$. Let
$\pazocal{G}_{\Lambda(h),\Omega(h)}^{\phi_{0}}:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$ denote the Gaussian Gabor multiplier defined in (5.1),
whose lattice $\Lambda(h)$ satisfies condition $(H)$ with constant $L>0$. Then
there exists constants $C,\beta>0$ only dependent on $\nu,L$ (and $d$) such
that the operator norm is bounded by

$\displaystyle\|\mathscr{G}_{\Lambda(h),\Omega(h)}^{\phi_{0}}\|_{\text{op}}\leq
Ch^{\beta}\ \ \forall\ \ 0<h\leq 1.$

First, we reformulate the problem to an estimate in the Fock space.

######  Lemma 5.2.

Let $\Omega^{*}:=\\{z\in\mathbb{C}^{d}\ |\ \overline{z}\in\Omega\\}$ denote
the complex conjugate set of $\Omega\subseteq\in\mathbb{R}^{2d}$. The operator
norm of the Gaussian Gabor multiplier
$\mathscr{G}_{\Lambda,\Omega}^{\phi_{0}}:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$ is given by

$\displaystyle\|\mathscr{G}_{\Lambda,\Omega}^{\phi_{0}}\|_{\text{op}}=\sup_{F\in\pazocal{F}^{2}(\mathbb{C}^{d}),\
\|F\|_{\pazocal{L}^{2}}=1}|A_{\Lambda}|\sum_{\lambda\in\Lambda_{\Omega^{*}}}|F(\lambda)|^{2}e^{-\pi|\lambda|^{2}}.$

###### Proof.

By Cauchy-Schwarz’ inequality on $\ell^{2}$-sequences, the operator norm is
given by

$\displaystyle\|\mathscr{G}_{\Lambda,\Omega}^{\phi_{0}}\|_{\text{op}}=\sup_{\|f\|_{2}=1}\langle\mathscr{G}_{\Lambda,\Omega}^{\phi_{0}}f,f\rangle=\sup_{\|f\|_{2}=1}|A_{\Lambda}|\sum_{\lambda\in\Lambda_{\Omega}}|\langle
f,\pi(\lambda)\phi_{0}\rangle|^{2}.$

By identity (2.5), each term can be expressed in terms of the Bargmann
transform, namely $|\langle
f,\pi(\overline{\lambda})\phi_{0}\rangle|^{2}=|\pazocal{B}f(\lambda)|^{2}e^{-\pi|\lambda|^{2}}$,
and since the Bargmann transform
$\pazocal{B}:L^{2}(\mathbb{R}^{d})\to\pazocal{F}^{2}(\mathbb{C}^{d})$ is an
isometry onto the Fock space, the result follows. ∎

###### Proof.

(Theorem 5.1) By Lemma 5.2, we consider the sum
$|A_{\Lambda}|\sum_{\lambda}|F(\lambda)|^{2}e^{-\pi|\lambda|^{2}}$ for any
normalized $F\in\pazocal{F}^{2}(\mathbb{C}^{d})$. By the subaveraging property
in Lemma 3.2, for every $R>0$

$\displaystyle\sum_{\lambda}|F(\lambda)|^{2}e^{-\pi|\lambda|^{2}}$
$\displaystyle\leq\left(1-e^{-\pi R^{2}}\sum_{j=0}^{d-1}\frac{(\pi
R^{2})^{j}}{j!}\right)^{-1}\sum_{\lambda}\int_{B_{R}(\lambda)}|F(\xi)|^{2}e^{-\pi|\xi|^{2}}\mathrm{d}A(\xi)$
$\displaystyle\leq\left(1-e^{-\pi R^{2}}\sum_{j=0}^{d-1}\frac{(\pi
R^{2})^{j}}{j!}\right)^{-1}\gamma(R,\Lambda)\cdot\int_{\bigcup_{\lambda}B_{R}(\lambda)}|F(\xi)|^{2}e^{-\pi|\xi|^{2}}\mathrm{d}A(\xi)$

where

$\displaystyle\gamma(R,\Lambda):=\sup_{z\in
A_{\Lambda}}\mathbf{card}\big{\\{}\lambda\in\Lambda\ \big{|}\
|B_{R}(z)\cap(A_{\Lambda}+\lambda)|>0\big{\\}}$

takes into account the possible overlap between the balls $B_{R}(\lambda)$. In
particular, for $R=h$ and by the condition $A_{\Lambda(h)}\subseteq
B_{Lh}(0)$, there clearly exists some finite constant $C_{d}>0$ dependent on
$d,L$ and otherwise independent of the lattice structure so that

$\displaystyle\gamma(h,\Lambda(h))\leq
C_{d}\frac{|B_{h}(0)|}{|A_{\Lambda(h)}|}\ \ \forall\ \ 0<h\leq 1.$

In total, we obtain

$\displaystyle|A_{\Lambda(h)}|\sum_{\lambda\in\Lambda_{\Omega^{*}}(h)}|F(\lambda)|^{2}e^{-\pi|\lambda|^{2}}\leq
C_{d}\cdot\kappa_{d}(\pi
h^{2})\int_{\bigcup_{\lambda\in\Lambda_{\Omega^{*}(h)}}B_{h}(\lambda)}|F(\xi)|^{2}e^{-\pi|\xi|^{2}}\mathrm{d}A(\xi),$

where $\kappa_{d}(x)$ was defined in (3.8). Since $\kappa_{d}(x)$ is
continuous for $x>0$ and $\lim_{x\to 0}\kappa_{d}(x)=1$, this factor is simply
absorbed by the multiplicative constant $C>0$ of the theorem.

Now, by definition (2.9) for the subset
$\Lambda_{\Omega^{*}}(h)\subseteq\Lambda(h)$ and since
$A_{\Lambda(h)}\subseteq B_{hL}(0)$, the union
$\bigcup_{\lambda\in\Lambda_{\Omega^{*}}(h)}B_{h}(\lambda)$ must be contained
in the $r$-thickened set $\Omega_{r}(h)$ for $r:=(1+L)h$. By Lemma 3.4, the
thickened set $\Omega_{r}(h)$ is itself $\frac{\nu}{2}$-porous on scales
$h_{1}:=\frac{4r}{\nu}h$ to $1$. Thus, we simply apply the FUP for the Fock
space in Theorem 3.1 on the family of sets $\Omega_{hr}(h)$ as $h_{1}\to 0$,
from which the statement follows. ∎

######  Remark.

The question immediately arises whether the Gabor systems
$\\{\pi(\lambda)\phi_{0}\\}_{\lambda\in\Lambda(h)}$ with $\Lambda(h)$
satisfying condition ($H$) actually are Gabor frames for all $0<h\leq 1$ or
for sufficiently small $h$. In $d=1$, by the results obtained independently by
Lyubarskii [22] and Seip and Wallsten [23], [24], we have a simple density
criterion. Namely, that the system
$\\{\pi(\lambda)\phi_{0}\\}_{\lambda\in\Lambda}$ is a Gabor frame if and only
if the fundamental region satisfies $|A_{\Lambda}|<1$. In higher dimensions
$d>1$, characterizations of lattices $\Lambda$ that yield Gaussian Gabor
frames becomes much more intricate, which has been studied in [16], [17],
[21]. In particular, the density criterion does not translate, e.g., as shown
in [21] Theorem 1.5, the lattice
$\Lambda=(\mathbb{Z}\times\frac{1}{2}\mathbb{Z})^{2}$ does not generate a
Gaussian Gabor frame even though $|A_{\Lambda}|=\frac{1}{4}<1$. Further, a
sufficient condition is formulated in [21] Theorem 1.2 combining a density
criterion with the notion of transcendental lattices, which, as remarked by
the authors, represents a large family of lattices in $\mathbb{C}^{d}$.
Nonetheless, our estimates of the Gabor multiplier are not reliant on the
operator being associated to a Gabor frame.

We conclude this section with a simple example of Gabor multipliers based on
Cantor sets, which also illustrates an alternative approach to choosing the
lattice restriction $\Lambda_{\Omega}$.

######  Example 5.1.

(Cantor set) For simplicity, we let $d=1$ and consider the symmetric Cartesian
product of Cantor iterates
$\Omega:=\pazocal{C}_{n}(L,M,\mathscr{A})\times\pazocal{C}_{n}(L,M,\mathscr{A})$.
For this case one obvious choice of lattices are square lattices
$\Lambda=a\left(\mathbb{Z}\times\mathbb{Z}\right)$ with density $a\sim
LM^{-n}$. Unsurprisingly, the restriction $\Lambda_{\Omega}$ closely resembles
the Cartesian product of scaled discrete Cantor iterates
$\Omega^{(d)}:=[L\cdot\pazocal{C}_{n}^{(d)}(M,\mathscr{A})]^{2}$. Thus, for
Gabor multipliers localizing on such Cartesian products, it seems more natural
to consider sampled points directly from the already available discrete set,
i.e., we consider the operator

$\displaystyle\mathscr{G}_{\phi_{0}}(\Omega^{(d)})f:=(LM^{-n})^{2}\sum_{\lambda\in\left[L\cdot\pazocal{C}^{(d)}(M,\mathscr{A})\right]^{2}}\langle
f,\pi(\lambda)\phi_{0}\rangle\pi(\lambda)\phi_{0}.$ In order to estimate the
operator norm, we follow the same procedure as in the proof of Theorem 5.1.
After one $\frac{1}{2}LM^{-n}$-thickening and then utilizing inequality
(3.12), we obtain for all $f\in L^{2}(\mathbb{R})$ and $r>0$ that
$\displaystyle\|\mathscr{G}_{\phi_{0}}(\Omega^{(d)})f\|_{2}\leq\frac{\kappa_{1}\big{(}\frac{\pi}{4}(LM^{-n})^{2}\big{)}}{1-e^{\frac{1}{2}\pi
r^{2}}}\rho\left(\Omega^{(d)}+B_{\frac{1}{2}LM^{-n}}(0),r\right)\|f\|_{2}^{2}.$

By definition of the discrete and ”continuous” Cantor iterate, the Nyquist
density must satisfy
$\rho(\Omega^{(d)}+B_{\frac{1}{2}LM^{-n}}(0),r)\leq\rho(\Omega,r)$, so the
operator norm is in turn bounded by

$\displaystyle\|\mathscr{G}_{\phi_{0}}(\Omega^{(d)})\|_{\text{op}}\leq\frac{\kappa_{1}\big{(}\frac{\pi}{4}(LM^{-n})^{2}\big{)}}{1-e^{\frac{1}{2}\pi
r^{2}}}\rho(\Omega,r).$

If we now suppose the length $L$ depends on the iterates $n$ according to
condition ($I_{M}$), we simply apply the estimate for the Nyquist density in
Lemma 4.4 to retrieve the same asymptotic estimate for the operator norm as in
Theorem 4.1.

##  Appendix A Complex interpolation in Fock space

We prove the following inequality between the Fock spaces
$\pazocal{F}^{1}(\mathbb{C}^{d})$ and $\pazocal{F}^{p}(\mathbb{C}^{d})$ for
$p\geq 1$.

######  Lemma A.1.

For any measurable subset $\Omega\subseteq\mathbb{C}^{d}$ and any $p\geq 1$,
we have the inequality

$\displaystyle\sup_{F\in\pazocal{F}^{p}(\mathbb{C}^{d})\setminus\\{0\\}}\frac{\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{p}}^{p}}{\|F\|_{\pazocal{L}^{p}}^{p}}\leq\sup_{F\in\pazocal{F}^{1}(\mathbb{C}^{d})\setminus\\{0\\}}\frac{\|F\cdot\chi_{\Omega}\|_{\pazocal{L}^{1}}}{\|F\|_{\pazocal{L}^{1}}}.$

The proof combines central results from complex interpolation. An introduction
to complex interpolation, based on Hadamard’s three line theorem, can be found
in [28] Chapter 2. We follow the notation in [28] and let
$X_{\theta}=[X_{0},X_{1}]_{\theta}$ for $\theta\in[0,1]$ denote the complex
interpolation space between two (compatible) Banach spaces $X_{0}$ and
$X_{1}$. To begin with, we present a general result regarding linear operators
between interpolation spaces and the associated operator norm (from [28]
Theorem 2.4 (c) combined with subsequent remark).

######  Theorem A.2.

Suppose $X_{0},X_{1}$ and $Y_{0},Y_{1}$ are compatible pairs of Banach spaces,
and suppose the mapping

$\displaystyle T:X_{0}+X_{1}\to Y_{0}+Y_{1}$

is bounded, linear such that $T:X_{k}\to Y_{k}$ is bounded with norm
$\|T\|_{k}\leq M_{k}$ for $k=0,1$. Then the mapping $T$ satisfies
$T:[X_{0},X_{1}]_{\theta}\to[Y_{0},Y_{1}]_{\theta}$ with norm estimate
$\|T\|_{\theta}\leq M_{0}^{1-\theta}M_{1}^{\theta}$ for all $\theta\in[0,1]$.

In order to relate the above theorem to our context, we need to consider
interpolation between weighted $L^{p}$-spaces and between Fock spaces. The
next statements are all found in [29] Chapter 2.4, formulated for spaces over
$\mathbb{C}$ but easily generalized to $\mathbb{C}^{d}$. First, we consider
the Stein-Weiss interpolation theorem, first published in [26], for weighted
$L^{p}$-spaces.

######  Theorem A.3.

Suppose $w,w_{0}$ and $w_{1}$ are positive weight functions on
$\mathbb{C}^{d}$. Then for any $1\leq p_{0}\leq p_{1}\leq\infty$ and
$\theta\in[0,1]$, we have that

$\displaystyle\left[L^{p_{0}}(\mathbb{C}^{d},w_{0}\mathrm{d}A),L^{p_{1}}(\mathbb{C}^{d},w_{1}\mathrm{d}A)\right]_{\theta}=L^{p}(\mathbb{C}^{d},w\mathrm{d}A),$

where

$\displaystyle\frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}\ \
\text{and }\
w^{\frac{1}{p}}=w_{0}^{\frac{1-\theta}{p_{0}}}w_{1}^{\frac{\theta}{p_{1}}}.$

In particular, for the $L^{p}$-spaces with Gaussian measures,
$\pazocal{L}^{p}(\mathbb{C}^{d})$, we obtain:

######  Corollary A.1.

For any $1\leq p_{0}\leq p_{1}\leq\infty$ and $\theta\in[0,1]$, we have that

$\displaystyle\left[\pazocal{L}^{p_{0}}(\mathbb{C}^{d}),\pazocal{L}^{p_{1}}(\mathbb{C}^{d})\right]_{\theta}=\pazocal{L}^{p}(\mathbb{C}^{d}),\
\ \text{where}\ \ \frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}.$

For Fock spaces we have the following interpolation:

######  Theorem A.4.

For any $1\leq p_{0}\leq p_{1}\leq\infty$ and $\theta\in[0,1]$, we have that

$\displaystyle\left[\pazocal{F}^{p_{0}}(\mathbb{C}^{d}),\pazocal{F}^{p_{1}}(\mathbb{C}^{d})\right]_{\theta}=\pazocal{F}^{p}(\mathbb{C}^{d}),\
\ \text{where}\ \ \frac{1}{p}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}}.$

With these interpolation results, we are ready to prove Lemma A.1.

###### Proof.

(Lemma A.1) Note that the statement is equivalent to the linear mapping

$\displaystyle
T_{\Omega}:\pazocal{F}^{p}(\mathbb{C}^{d})\to\pazocal{L}^{p}(\mathbb{C}^{d}),\
\ F\mapsto F\cdot\chi_{\Omega}$ satisfying the operator norm inequality
$\|T_{\Omega}\|_{\pazocal{L}^{p}}^{p}\leq\|T_{\Omega}\|_{\pazocal{L}^{1}}$ for
$p\geq 1$. Therefore, we consider the mapping $T_{\Omega}$ between the spaces
$\displaystyle
T_{\Omega}:\pazocal{F}^{1}(\mathbb{C}^{d})+\pazocal{F}^{\infty}(\mathbb{C}^{d})\to\pazocal{L}^{1}(\mathbb{C}^{d})+\pazocal{L}^{\infty}(\mathbb{C}^{d}),$
which is clearly bounded. By Theorem A.2, $\displaystyle
T_{\Omega}:\left[\pazocal{F}^{1}(\mathbb{C}^{d}),\pazocal{F}^{\infty}(\mathbb{C}^{d})\right]_{\theta}\to\left[\pazocal{L}^{1}(\mathbb{C}^{d}),\pazocal{L}^{\infty}(\mathbb{C}^{d})\right]_{\theta},$

and the associated operator norm is bounded by
$\|T_{\Omega}\|_{\pazocal{L}^{1}}^{1-\theta}\|T_{\Omega}\|_{\pazocal{L}^{\infty}}^{\theta}$.
By Corollary A.1 and Theorem A.4, these interpolation spaces correspond to
$\pazocal{F}^{p}(\mathbb{C}^{d})$ and $\pazocal{L}^{p}(\mathbb{C}^{d})$,
respectively, with $p^{-1}=1-\theta$. In addition,
$\|T_{\Omega}\|_{\pazocal{L}^{\infty}}\leq 1$, from which the norm estimate
readily follows. ∎

##  Appendix B Omitted proof: Simple porosity estimate of Cantor sets

We shall prove the following simple porosity estimate for the $n$-iterate
Cantor set in $1$-dimension.

######  Lemma B.1.

The $n$-iterate Cantor set $\pazocal{C}_{n}(L,M,\mathscr{A})$ with base $M>1$
and alphabet size $|\mathscr{A}|<M$, based in the interval $[0,L]$, is
$\nu$-porous on scales $LM^{-n+1}$ to $\infty$, with any $\nu\leq M^{-2}$.

###### Proof.

Without loss of generality, we assume $L=1$. Consider an interval
$I\subseteq\mathbb{R}$ of size $M^{-m+1}\leq|I|\leq M^{-m+2}$ for some integer
$1\leq m\leq n$. Suppose first that the intersection
$I\cap\pazocal{C}_{m}(\dots)$ does not form an interval. Then, by the Cantor
set construction, there exists an interval $J\subseteq I$ of size
$|J|=M^{-m}\geq M^{-2}|I|$ such that $|J\cap\pazocal{C}_{m}(\dots)|=0$.
Conversely, suppose the intersection forms an interval, effectively dividing
the remainder $I\setminus\pazocal{C}_{m}(\dots)$ into two intervals
$J_{1},J_{2}$. Again, by the Cantor set construction, we have the upper bound
$|I\cap\pazocal{C}_{m}(\dots)|\leq|\mathscr{A}|M^{-m}\leq(M-1)M^{-m}$, so that
$\max\\{|J_{1}|,|J_{2}|\\}\geq\frac{1}{2}(|I|-(M-1)M^{-m})$. Hence, we
conclude that there exists an interval $J\subseteq I$ with
$|I\cap\pazocal{C}_{n}(\dots)|=0$ of size

$\displaystyle|J|=\left(\frac{|I|-(M-1)M^{-m}}{2|I|}\right)|I|\geq M^{-2}|I|.$

Since $m$ is arbitrary and
$\pazocal{C}_{n}(\dots)\subseteq\pazocal{C}_{m}(\dots)$, the statement holds
for all intervals $I$ with $M^{-n+1}\leq|I|\leq M$. For $|I|\geq M$ the
statement becomes trivial as $\pazocal{C}_{n}(\dots)\subset[0,1]$. ∎

##  Appendix C Subaveraging in Fock space

We shall generalize the subaveraging statement made in Lemma 2.32 in [29] for
the Fock space over $\mathbb{C}$ to the space over $\mathbb{C}^{d}$, namely:

######  Lemma C.1.

For any $F\in\pazocal{F}^{p}(\mathbb{C}^{d})$ and any point
$z\in\mathbb{C}^{d}$, we have for all $R>0$ that

(C.1)
$\displaystyle|F(z)|^{p}e^{-\frac{p}{2}\pi|z|^{2}}\leq\left(\frac{p}{2}\right)^{d}\left(1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi
R^{2}\right)^{j}}{j!}\right)^{-1}\int_{B_{R}(z)}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi).$

In one dimension, the proof is based on the subaveraging property of
subharmonic functions in $\mathbb{C}$. More precisely, for
$u:\mathbb{C}\to\mathbb{R}\cup\\{-\infty\\}$ subharmonic, we have that

$\displaystyle
u(z)\leq\frac{1}{2\pi}\int_{0}^{2\pi}u(re^{i\theta}+z)\mathrm{d}\theta\ \
\forall\ r>0.$

In $d$-dimensions, we instead consider plurisubharmonic functions.

######  Def. C.1.

(Plurisubharmonic) Let $X$ be a domain in $\subseteq\mathbb{C}^{d}$. We say
that a function $u:X\to\mathbb{R}\cup\\{-\infty\\}$ is plurisubharmonic if

  1. (a)

$u$ is upper semi-continuous, and

  2. (b)

for every $z,\xi\in\mathbb{C}^{d}$ the function $\tau\mapsto u(\xi\tau+z)$ is
subharmonic in the open subset of $\mathbb{C}$ where it is defined.

Many well-known examples of plurisubharmonic functions are based on the
holomorphic functions. In particular, if $F$ is entire, then $|F|^{p}$ is a
plurisubharmonic function in $\mathbb{C}^{d}$ for every $p>0$. Hence, by point
(b) in Def. C.1, every $F\in\pazocal{F}^{p}(\mathbb{C}^{d})$ satisfies

(C.2) $\displaystyle|F(z)|^{p}\leq\frac{1}{2\pi}\int_{0}^{2\pi}|F|^{p}(\xi
re^{i\theta}+z)\mathrm{d}\theta\ \ \forall\ \xi\in\mathbb{C}^{d}\ \
\text{and}\ \ r>0.$

Notice that since left-hand side of (C.2) is independent of $\xi$ (and $r$),
we may multiply both sides by a positive function of $\xi$, integrate with
respect to $\xi$ and keep the inequality intact.

Introduce the notation

$\displaystyle\partial B_{1}:=\big{\\{}z\in\mathbb{C}^{d}\ \big{|}\
|z|=1\big{\\}}$

for the $(2d-1)$-dimensional unit sphere, and let $|\partial B_{1}|$ denote
the associated surface area. Further, let $\mathrm{d}S$ denote the surface
measure on $\partial B_{1}$. To begin with, we show that the point evaluation
$|F(z)|^{2}$ is bounded by an average over the sphere.

######  Lemma C.2.

For every entire function $F$, we have that

(C.3) $\displaystyle|F(z)|^{p}\leq\frac{1}{|\partial
B_{1}|}\int_{|\xi|=1}|F|^{p}(\xi r+z)\mathrm{d}S(\xi)\ \ \forall\ r>0.$

###### Proof.

Consider the equivalence relation $\sim$ in $\mathbb{C}^{d}$, where

$\displaystyle\xi_{1}\sim\xi_{2}\iff\exists\ \ \theta\in[0,2\pi]\ \text{such
that}\ \xi_{1}=e^{i\theta}\xi_{2}.$

This induces the quotient space $\partial B_{1}/\sim$ of ”remaining angles”
over the unit sphere, with measure $\mathrm{d}\Phi$ such that
$\mathrm{d}S=\mathrm{d}\theta\mathrm{d}\Phi$ and

$\displaystyle|\partial B_{1}|=\int_{0}^{2\pi}\mathrm{d}\theta\int_{\partial
B_{1}/\sim}\mathrm{d}\Phi=2\pi\int_{\partial B_{1}/\sim}\mathrm{d}\Phi.$

With the above decomposition, we may integrate both sides of (C.2) over the
space $\partial B_{1}/\sim$, which leaves the desired result. ∎

Proceeding, we relate subaverage over the sphere to an subaverage over the
ball $B_{R}(0)$.

######  Lemma C.3.

For every entire function $F$, we have for all $R>0$ that

(C.4)
$\displaystyle|F(z)|^{p}\leq\left(\frac{p}{2}\right)^{d}\left(1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi
R^{2}\right)^{j}}{j!}\right)^{-1}\int_{B_{R}(0)}|F|^{p}(z+\xi)e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi).$

###### Proof.

For any $\omega\in\mathbb{C}^{d}$, we have the basic decomposition
$\omega=r\xi$ for $|\xi|=1$ and $r=|\omega|$, so that the volume measure can
be expressed $\mathrm{d}A(\omega)=r^{2d-1}\mathrm{d}r\mathrm{d}S(\xi)$. Since
we consider the Gaussian measure of the Fock space
$\pazocal{F}^{p}(\mathbb{C}^{d})$, we integrate both sides of inequality (C.3)
against $e^{-\frac{p}{2}\pi r^{2}}r^{2d-1}\mathrm{d}r$. For the right-hand
side, we find that

$\displaystyle\frac{1}{|\partial B_{1}|}\int_{0}^{R}\int_{|\xi|=1}|F|^{p}(\xi
r+z)e^{-\frac{p}{2}\pi
r^{2}}r^{2d-1}\mathrm{d}S(\xi)\mathrm{d}r=\frac{1}{|\partial
B_{1}|}\int_{B_{R}(0)}|F|^{p}(\omega+z)e^{-\frac{p}{2}\pi|\omega|^{2}}\mathrm{d}A(\omega).$

While for the left-hand side, using the formula

$\displaystyle\int_{0}^{L}r^{k}e^{-r}\mathrm{d}r=k!\left(1-e^{-L}\sum_{j=0}^{k}\frac{L}{j!}\right)\
\ \text{for }k=0,1,2,\dots,$

we obtain

$\displaystyle|F(z)|^{p}\int_{0}^{R}e^{-\frac{p}{2}\pi
r^{2}}r^{2d-1}\mathrm{d}r=|F(z)|^{p}\frac{(d-1)!}{2}\left(\frac{2}{p\pi}\right)^{d}\left(1-e^{-\frac{p}{2}\pi
R^{2}}\sum_{j=0}^{d-1}\frac{\left(\frac{p}{2}\pi
R^{2}\right)^{j}}{j!}\right).$

Since the surface area of the $(2d-1)$-dimensional unit sphere is given by
$|\partial B_{1}|=2\pi^{d}/(d-1)!$, inequality (C.4) follows. ∎

We are now ready to prove Lemma C.1.

###### Proof.

(Lemma C.1) Consider the integral over the ball centered at
$z\in\mathbb{C}^{d}$,

(C.5)
$\displaystyle\int_{B_{R}(z)}|F(\xi)|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi)=$
$\displaystyle\int_{B_{R}(0)}|F|^{p}(z+\xi)e^{-\frac{p}{2}\pi|z+\xi|^{2}}\mathrm{d}A(\xi)$
$\displaystyle=e^{-\frac{p}{2}\pi|z|^{2}}$
$\displaystyle\int_{B_{R}(0)}|F(z+\xi)e^{-\pi\xi\cdot\overline{z}}|^{p}e^{-\frac{p}{2}\pi|\xi|^{2}}\mathrm{d}A(\xi).$

Define $G_{z}(\xi):=F(z+\xi)e^{-\pi\xi\cdot\overline{z}}$, which is an entire
function with respect to $\xi$. Since $G_{z}(0)=F(z)$, the inequality follows
once we apply Lemma C.3 to $|G_{z}(0)|^{p}$, combined with (C.5). ∎

## Acknowledgements

The research of the author was supported by Grant 275113 of the Research
Council of Norway. The author would like to extend thanks to prof. Eugenia
Malinnikova for many insightful discussions that helped shape the ideas in the
manuscript. In addition, the author would like to thank prof. Franz Luef for
the support and who suggested that fractal uncertainty principles for the
Daubechies’ operator could be translated to the context of Gabor multipliers.

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