# Spectral sets and weak tiling

Mihail N. Kolountzakis  M.K.: Department of Mathematics and Applied
Mathematics, University of Crete, Voutes Campus, GR-700 13, Heraklion, Crete,
Greece<EMAIL_ADDRESS>,  Nir Lev  N.L.: Department of Mathematics, Bar-Ilan
University, Ramat-Gan 5290002, Israel<EMAIL_ADDRESS>and  Máté
Matolcsi  M.M.: Department of Analysis, Institute of Mathematics, Budapest
University of Technology and Economics (BME), Müegyetem rkp. 3., H-1111
Budapest, Hungary (also at Alfred Renyi Institute of Mathematics, ELKH,
H-1053, Realtanoda u 13-15, Budapest, Hungary<EMAIL_ADDRESS>
(Date: September 9, 2022)

###### Abstract.

A set $\Omega\subset\mathbb{R}^{d}$ is said to be spectral if the space
$L^{2}(\Omega)$ admits an orthogonal basis of exponential functions. Fuglede
(1974) conjectured that $\Omega$ is spectral if and only if it can tile the
space by translations. While this conjecture was disproved for general sets,
it was recently proved that the Fuglede conjecture does hold for the class of
convex bodies in $\mathbb{R}^{d}$. The proof was based on a new geometric
necessary condition for spectrality, called “weak tiling”. In this paper we
study further properties of the weak tiling notion, and present applications
to convex bodies, non-convex polytopes, product domains and Cantor sets of
positive measure.

###### Key words and phrases:

Fuglede’s conjecture, spectral set, tiling, polytopes, equidecomposability

###### 2020 Mathematics Subject Classification:

42B10, 52B11, 52B45, 52C07, 52C22

M.K. was supported by the Hellenic Foundation for Research and Innovation,
Project HFRI-FM17-1733 and by University of Crete Grant 4725.

N.L. was supported by ISF Grant No. 1044/21 and ERC Starting Grant No. 713927.

M.M. was supported by NKFIH grants K129335 and K132097.

##  1\. Introduction

###  1.1. Spectral sets

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded, measurable set of positive
measure. We say that $\Omega$ is _spectral_ if there exists a countable set
$\Lambda\subset\mathbb{R}^{d}$ such that the system of exponential functions
$\\{\exp 2\pi i\langle\lambda,x\rangle\\}$, $\lambda\in\Lambda$, forms an
orthogonal basis in $L^{2}(\Omega)$, that is, the system is orthogonal and
complete in the space. In this case the set $\Lambda$ is called a _spectrum_
for $\Omega$.

The study of spectral sets has a long history, which goes back to Fuglede
[Fug74] who conjectured that the spectral sets could be characterized
geometrically as the sets which can tile the space by translations. We say
that the set $\Omega$ _tiles the space by translations_ if there exists a
countable set $\Lambda\subset\mathbb{R}^{d}$ such that the translated copies
$\\{\Omega+\lambda\\}$, $\lambda\in\Lambda$, constitute a partition of
$\mathbb{R}^{d}$ up to measure zero.

Fuglede’s conjecture inspired extensive research over the years, and a number
of interesting results establishing connections between spectrality and tiling
had since been obtained. On the other hand, also counterexamples were found to
both directions of the conjecture in dimensions $d\geqslant 3$, see [Tao04],
[KM10, Section 4].

It was recently proved in [LM22] that the Fuglede conjecture holds for the
class of _convex bodies_ in $\mathbb{R}^{d}$ (compact, convex sets with
nonempty interior). That is, a convex body $\Omega\subset\mathbb{R}^{d}$ is
spectral if and only if $\Omega$ can tile the space by translations.

###  1.2. Weak tiling

To prove the Fuglede conjecture for convex domains, the authors in [LM22]
established a link between the analytic notion of spectrality and a geometric
notion which was termed “weak tiling”. It is defined as follows:

######  Definition 1.1.

We say that the set $\Omega$ can _weakly tile_ another measurable (possibly
unbounded) set $\Sigma\subset\mathbb{R}^{d}$ by translations, if there exists
a positive, locally finite Borel measure $\nu$ on $\mathbb{R}^{d}$ such that
$\mathds{1}_{\Omega}\ast\nu=\mathds{1}_{\Sigma}$ a.e.

Note that in the special case where the measure $\nu$ is the sum of (finitely
or countably many) unit masses, the weak tiling becomes a _proper tiling_ of
$\Sigma$ by translates of $\Omega$.

The following result was proved in [LM22].

######  Theorem 1.2 (see [LM22, Theorem 1.5]).

Let $\Omega$ be a bounded, measurable set in $\mathbb{R}^{d}$. If $\Omega$ is
spectral, then it can weakly tile its complement
$\Omega^{\complement}=\mathbb{R}^{d}\setminus\Omega$ by translations.

This result thus gives a geometric condition necessary for spectrality. It
establishes a weak form of the “spectral implies tiling” part of Fuglede’s
conjecture. We observe that the weak tiling conclusion cannot in general be
strengthened to proper tiling since there exist examples of spectral sets
which cannot tile by translations.

Several applications of Theorem 1.2 were given in [LM22]. The main one is the
proof that the Fuglede conjecture holds for convex bodies in $\mathbb{R}^{d}$.
As another application, it was proved in [LM22, Theorem 3.6] that if a
bounded, open set $\Omega\subset\mathbb{R}^{d}$ is spectral then its boundary
$\partial\Omega$ must be a set of Lebesgue measure zero.

In the present paper we study further properties and applications of weak
tiling.

###  1.3. Equidecomposability

Let $A,B$ be two (not necessarily convex) polytopes in $\mathbb{R}^{d}$. We
say that $A$ and $B$ are _equidecomposable_ if $A$ can be partitioned, up to
measure zero, into a finite number of smaller polytopes which can be
rearranged using rigid motions to form, again up to measure zero, a partition
of $B$. If the pieces of the partition can be rearranged using translations
only, then $A$ and $B$ are said to be _equidecomposable by translations_.

It has long been known that if a polytope $A\subset\mathbb{R}^{d}$ can
(properly) tile the space by translations, then $A$ must be equidecomposable
by translations to a cube of the same volume. This result was first
established by Mürner in [Mür75], and was later rediscovered in [LM95a].
Recently, it was proved in [LL21] that the same conclusion holds also for the
class of spectral polytopes, that is, any spectral polytope
$A\subset\mathbb{R}^{d}$ must be equidecomposable by translations to a cube of
the same volume.

We will prove the following simultaneous strengthening of the latter two
results:

######  Theorem 1.3.

Let $A$ be a (not necessariy convex) polytope in $\mathbb{R}^{d}$. Assume that
$A$ can weakly tile its complement by translations. Then $A$ is
equidecomposable by translations to a cube of the same volume.

###  1.4. Convex domains

It is a classical result due to Mürner, see [Mür77, Section 3.3], that a
_convex_ polytope $A\subset\mathbb{R}^{d}$ is equidecomposable by translations
to a cube if and only if $A$ is centrally symmetric and all the facets of $A$
(that is, the $(d-1)$-dimensional faces of $A$) are also centrally symmetric.
Together with Theorem 1.3 this implies that a convex polytope which can weakly
tile its complement by translations must be centrally symmetric and have
centrally symmetric facets.

Actually, this conclusion can be significantly strengthened as follows. It was
proved in [LM22, Theorem 4.1] that if $A$ is a general convex body in
$\mathbb{R}^{d}$ and if $A$ can weakly tile its complement by translations,
then $A$ must in fact be a convex polytope. It was also proved [LM22, Theorem
6.1] that if, in addition, $A$ is centrally symmetric and has centrally
symmetric facets, then each belt of $A$ must have either $4$ or $6$ facets. So
in the latter case, it follows by the Venkov-McMullen theorem [Ven54], [McM80]
(see also [Gru07, Section 32.2]) that $A$ can tile its complement not only
weakly, but even properly, by translations. We thus arrive at the following
result:

######  Theorem 1.4.

Let $A$ be a convex body in $\mathbb{R}^{d}$, and assume that $A$ can weakly
tile its complement by translations. Then $A$ must be a convex polytope which
can also tile the space properly by translations.

We observe that for general sets (i.e. not assumed to be convex) the result
does not hold, since there exist examples of spectral sets which cannot tile
by translations.

###  1.5. Product domains

Let $A\subset\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{m}$ be two bounded,
measurable sets. It is known that if $A$ and $B$ are both spectral sets in
$\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ respectively, then their cartesian
product $\Omega=A\times B$ is spectral in
$\mathbb{R}^{n}\times\mathbb{R}^{m}$. Indeed, if $U\subset\mathbb{R}^{n}$ is a
spectrum for $A$, and $V\subset\mathbb{R}^{m}$ is a spectrum for $B$, then the
product set $\Lambda=U\times V$ serves as a spectrum for $\Omega$ (see e.g.
[JP99, Theorem 3]).

In [Kol16] the question was posed as to whether the converse statement is also
true.

######  Conjecture 1.5.

Let $A\subset\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{m}$ be two bounded,
measurable sets. Then their product $\Omega=A\times B$ is spectral if and only
if $A$ and $B$ are both spectral sets.

The “only if” part of this conjecture is the non-trivial one. The difficulty
lies in that we assume the product set $\Omega$ to be spectral, but we do not
make any a priori assumption that the spectrum $\Lambda$ also has a product
structure, so it is not obvious which sets $U$ and $V$ may serve as spectra
for the factors $A$ and $B$, respectively. (One can show, see [JP99, Lemma 2],
that if $\Omega$ happens to admit a spectrum $\Lambda$ with a product
structure, $\Lambda=U\times V$, then $U$ is a spectrum for $A$ and $V$ is a
spectrum for $B$.)

It was proved in [GL16] that Conjecture 1.5 holds in the case where one of the
factors, say $A$, is an interval in $\mathbb{R}$. In [Kol16] it was
established, using a different approach, that the conjecture is true also if
the set $A$ is the union of two intervals in $\mathbb{R}$. In [GL20] the
conjecture was proved in the case where the factor $A$ is a convex polygon in
$\mathbb{R}^{2}$.

As an application of the weak tiling method, we will prove the following
result:

######  Theorem 1.6.

Let $\Omega=A\times B$ where $A$ is a convex body in $\mathbb{R}^{n}$, while
$B$ is any bounded, measurable set in $\mathbb{R}^{m}$. If $\Omega$ is a
spectral set then $A$ must be spectral, or equivalently, $A$ must be a convex
polytope which can (properly) tile the space by translations.

This significantly strengthens [GL20, Theorems 2.1 and 2.2] where it was
proved that $A$ cannot have a smooth boundary, and that if $A$ is assumed a
priori to be a convex polytope, then $A$ must be centrally symmetric and have
centrally symmetric facets.

It is known, see [Kol16, Section 1.2], that the product set $\Omega=A\times B$
can tile the space $\mathbb{R}^{n}\times\mathbb{R}^{m}$ by translations if and
only if both $A$ tiles $\mathbb{R}^{n}$ and $B$ tiles $\mathbb{R}^{m}$. In
order to prove Theorem 1.6 we shall use an analogous result for weak tiling:

######  Theorem 1.7.

Let $A\subset\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{m}$ be two bounded,
measurable sets. Then the product set $\Omega=A\times B$ can weakly tile its
complement in $\mathbb{R}^{n}\times\mathbb{R}^{m}$ by translations if and only
if both $A$ and $B$ weakly tile their complements in $\mathbb{R}^{n}$ and
$\mathbb{R}^{m}$ respectively.

Theorem 1.6 is thus obtained by a combination of Theorems 1.2, 1.7 and 1.4.

Similarly, by a combination of Theorems 1.2, 1.7 and 1.3 we obtain that if $A$
is a (not necessariy convex) polytope in $\mathbb{R}^{n}$, and if $B$ is any
bounded, measurable set in $\mathbb{R}^{m}$, then the spectrality of the
product $\Omega=A\times B$ implies that $A$ is equidecomposable by
translations to a cube. Alternatively, it is possible to obtain this result by
a combination of [GL20, Lemma 5.1] and [LL21, Theorem 7.1].

###  1.6. Nowhere dense sets

Assume now that $\Omega\subset\mathbb{R}^{d}$ is a _bounded, nowhere dense
set_ of positive measure. It is not hard to show that such a set $\Omega$
cannot tile the space by translations. To see this, suppose to the contrary
that $\Omega+\Lambda$ is a tiling, then $\Lambda$ must be a locally finite
set. Since $\Omega$ is bounded, any ball can thus intersect only finitely many
translated copies $\Omega+\lambda$, $\lambda\in\Lambda$. Since $\Omega$ is a
nowhere dense set, then also any finite union of translates of $\Omega$ is a
nowhere dense set. Hence the union of all the translated copies
$\Omega+\lambda$ intersecting a given ball is a nowhere dense set, so this
union does not cover a set of full measure in the ball, a contradiction.

In [Mat05], the following question was posed: can a bounded, nowhere dense set
be spectral? The answer is expected to be negative, but so far this has been
proved only in dimension one. In fact, it was proved in [IK13, Corollary 1.6]
that if a bounded, measurable set $\Omega\subset\mathbb{R}$ is spectral, then
it can _multi-tile_ the real line by translations. This excludes the
possibility that $\Omega$ is a nowhere dense set, by the same argument as
above.

In this paper we use the weak tiling method as a different approach to the
spectrality problem for bounded, nowhere dense sets. We will prove the
following result:

######  Theorem 1.8.

Let $E\subset\mathbb{R}$ be a symmetric Cantor set of positive measure. Then
$E$ cannot weakly tile its complement by translations. As a consequence, $E$
is not spectral.

The definition of a symmetric Cantor set will be given in Section 5.

If we combine Theorem 1.8 with Theorem 1.7 then we obtain the following
conclusion for multi-dimensional nowhere dense sets with a cartesian product
structure:

######  Theorem 1.9.

Let $\Omega=E\times B$ be the product of a symmetric Cantor set
$E\subset\mathbb{R}$ of positive measure, and an arbitrary bounded, measurable
set $B\subset\mathbb{R}^{m}$. Then $\Omega$ is not a spectral set in
$\mathbb{R}\times\mathbb{R}^{m}$.

We give two proofs of Theorem 1.8. In Section 5 we prove it by closely
examining the essential difference set of a symmetric Cantor set in
$\mathbb{R}$ and proving that the Cantor set admits a packing region which is
strictly larger than the set itself (see Section 4). In Section 6 we prove
directly that the Cantor set cannot weakly tile its complement by
translations, by establishing in a quantitative way the same property for the
$n$-th generation set that approximates the Cantor set.

##  2\. Weak tiling and equidecomposability

In this section we prove Theorem 1.3, that is, we show that if a (not
necessarily convex) polytope $A\subset\mathbb{R}^{d}$ can weakly tile its
complement by translations, then $A$ is equidecomposable by translations to a
cube. As a consequence, any _convex_ polytope that can weakly tile its
complement by translations must be centrally symmetric and have centrally
symmetric facets.

### 2.1.

The Fourier transform of a function $f\in L^{1}(\mathbb{R}^{d})$ is defined by

$\widehat{f}(\xi)=\int_{\mathbb{R}^{d}}f(x)\,e^{-2\pi
i\langle\xi,x\rangle}dx,\quad\xi\in\mathbb{R}^{d}.$

We will need a result from [LL21] concerning the zeros of the Fourier
transform $\widehat{\mathds{1}}_{A}$ of the indicator function
$\mathds{1}_{A}$ of a (not necessarily convex) polytope
$A\subset\mathbb{R}^{d}$.

Given $\delta>0$ and real numbers $\tau_{1},\dots,\tau_{k}$ we consider the
set

$T=T(\delta;\tau_{1},\dots,\tau_{k})=\\{n\in\mathbb{Z}:|e^{2\pi
in\tau_{j}}-1|<\delta,\;1\leqslant j\leqslant k\\}.$ (2.1)

If we are also given $R>0$ then we let

$J=J(R)=\\{n\in\mathbb{Z}:0<|n|<R\\}.$ (2.2)

Finally, given also $\varepsilon>0$ and a vector $v\in\mathbb{R}^{d}$ we
define

$S=S(T,J,v,\varepsilon)=\\{nv+w:n\in T\setminus
J,\;w\in\mathbb{R}^{d},\;|w|<\varepsilon\\}.$ (2.3)

The following result was proved in [LL21] although it was not explicitly
stated there in this form.

######  Theorem 2.1 ([LL21]).

Let $A$ be a polytope in $\mathbb{R}^{d}$. If $A$ is not equidecomposable by
translations to a cube, then there exist $\delta>0$, real numbers
$\tau_{1},\dots,\tau_{k}$, a nonzero vector $v\in\mathbb{R}^{d}$,
$\varepsilon>0$ and $R>0$ such that the Fourier transform
$\widehat{\mathds{1}}_{A}$ has no zeros in the set $S$, where $T$, $J$ and $S$
are the three sets defined by (2.1), (2.2) and (2.3).

We briefly indicate how this can be inferred from [LL21]. If $A$ is not
equidecomposable by translations to any cube, then there exists a Hadwiger
functional $H_{\Phi}$ such that $H_{\Phi}(A)\neq 0$. Let $p(u)$ be the
trigonometric polynomial constructed in [LL21, Section 6.1], and expand it in
the form $p(u)=\sum_{j=1}^{k}c_{j}e^{2\pi iu\tau_{j}}$. Let
$\delta=\delta(p,\eta)>0$ be small enough such that $|p(n)-p(0)|<\eta$ for
every $n$ in the set (2.1). We then continue the proof as in [LL21, Section 6]
to conclude that there exist a nonzero vector $v\in\mathbb{R}^{d}$,
$\varepsilon>0$ and $R>0$ such that $\widehat{\mathds{1}}_{A}$ has no zeros in
the set (2.3).

The results in [LL21] are based on an intricate analysis of the asymptotic
behavior of the Fourier transform $\widehat{\mathds{1}}_{A}$, see [LL21,
Theorem 4.1]. The analysis becomes considerably simpler if the polytope $A$ is
assumed to be convex, see [GL17, Sections 3 and 4].

### 2.2.

A measure $\mu$ on $\mathbb{R}^{d}$ is said to be _translation-bounded_ if for
every (or equivalently, for some) open ball $B$ we have

$\sup_{x\in\mathbb{R}^{d}}|\mu|(B+x)<+\infty.$

If $\mu$ is a translation-bounded measure on $\mathbb{R}^{d}$, then $\mu$ is a
tempered distribution.

If $f$ is a function in $L^{1}(\mathbb{R}^{d})$ and $\mu$ is a translation-
bounded measure on $\mathbb{R}^{d}$, then the convolution $f\ast\mu$ is a
locally integrable function on $\mathbb{R}^{d}$ defined uniquely (up to
equality a.e.) by the condition that
$(f\ast\mu)\ast\varphi=f\ast(\mu\ast\varphi)$ for every continuous, compactly
supported function $\varphi$ on $\mathbb{R}^{d}$.

######  Theorem 2.2.

Let $f\in L^{1}(\mathbb{R}^{d})$, $\int f\neq 0$, and let $\mu$ be a
translation-bounded measure on $\mathbb{R}^{d}$. If we have $f\ast\mu=1$ a.e.,
then $\widehat{\mu}=(\int f)^{-1}\cdot\delta_{0}$ in the open set
${Z(\widehat{f}\,)}^{\complement}$.

Here we let
${Z(\widehat{f}\,)}:=\\{\xi\in\mathbb{R}^{d}:\widehat{f}(\xi)=0\\}$ denote the
(closed) set of zeros of $\widehat{f}$.

Theorem 2.2 can be proved in a similar way to [KL16, Theorem 4.1].

### 2.3.

We now turn to the proof of the main result of this section.

###### Proof of Theorem 1.3.

Let $A$ be a polytope in $\mathbb{R}^{d}$, and assume that $A$ can tile its
complement weakly by translations, that is, there exists a positive, locally
finite measure $\nu$ on $\mathbb{R}^{d}$ such that
$\mathds{1}_{A}\ast\nu=\mathds{1}_{A^{\complement}}$ a.e. By [LM22, Lemma 2.4]
the measure $\nu$ is not only locally finite, but in fact $\nu$ must be
translation-bounded. It follows that the measure $\mu:=\delta_{0}+\nu$ is
translation-bounded and satisfies $\mathds{1}_{A}\ast\mu=1$ a.e. In turn,
Theorem 2.2 implies that $\widehat{\mu}=m(A)^{-1}\cdot\delta_{0}$ in the open
set ${{Z(\widehat{\mathds{1}}_{A})}}^{\complement}$.

We must prove that $A$ is equidecomposable by translations to a cube of the
same volume. Suppose to the contrary that this is not the case. Then by
Theorem 2.1 there exist $\delta>0$, real numbers $\tau_{1},\dots,\tau_{k}$, a
nonzero vector $v\in\mathbb{R}^{d}$, $\varepsilon>0$ and $R>0$ such that
$\widehat{\mathds{1}}_{A}$ has no zeros in the set $S$, where $T$, $J$ and $S$
are the three sets defined by (2.1), (2.2) and (2.3). It follows that we have
$\widehat{\mu}=m(A)^{-1}\cdot\delta_{0}$ in the open set $S$.

Now suppose that we are given a real-valued Schwartz function $g$ on
$\mathbb{R}^{d}$ satisfying

$\operatorname{supp}(g)\subset S,\quad\widehat{g}\geqslant 0.$ (2.4)

Then we have

$\int_{\mathbb{R}^{d}}g(x)dx=\widehat{g}(0)\leqslant\widehat{g}(0)+\int\widehat{g}(\xi)d\nu(\xi)=\int\widehat{g}(\xi)d\mu(\xi),$
(2.5)

where the inequality in (2.5) is due to $\widehat{g}$ being a nonnegative
function and $\nu$ being a positive measure. On the other hand, we have

$\int\widehat{g}(\xi)d\mu(\xi)=\mu(\widehat{g})=\widehat{\mu}(g)=m(A)^{-1}g(0),$
(2.6)

where the last equality holds since we have $\operatorname{supp}(g)\subset S$
and $\widehat{\mu}=m(A)^{-1}\cdot\delta_{0}$ in the open set $S$. We conclude
that

$\int_{\mathbb{R}^{d}}g(x)dx\leqslant m(A)^{-1}g(0)$ (2.7)

for every real-valued Schwartz function $g$ satisfying (2.4). We will show
that this leads to a contradiction, by constructing an example of a real-
valued Schwartz function $g$ satisfying (2.4), but such that (2.7) does not
hold.

We choose a nonnegative Schwartz function $\varphi$ such that $\int\varphi=1$,
$\varphi$ is supported in the open ball of radius $\varepsilon$ centered at
the origin, and $\widehat{\varphi}\geqslant 0$. Next, let $\psi$ be a
nonnegative, smooth function on the $k$-dimensional torus
$\mathbb{T}^{k}=(\mathbb{R}/\mathbb{Z})^{k}$, such that $\int\psi=1$, $\psi$
is supported in the set

$\\{(t_{1},\dots,t_{k}):|e^{2\pi it_{j}}-1|<\delta,\;1\leqslant j\leqslant
k\\},$ (2.8)

and the Fourier coefficients $\widehat{\psi}(m)$, $m\in\mathbb{Z}^{k}$, are
nonnegative. We may assume that $R$ is an integer (by enlarging it if
necessary) and define also the trigonometric polynomial
$p_{N}(t):=K_{N}(R\,t)$, where $K_{N}$ is the classical Fejér kernel,

$K_{N}(t)=\sum_{|n|<N}\Big{(}1-\frac{|n|}{N}\Big{)}e^{2\pi int},\quad
t\in\mathbb{T}=\mathbb{R}/\mathbb{Z}.$ (2.9)

Then $p_{N}$ is nonnegative, $p_{N}(0)=N$, the Fourier coefficients
$\widehat{p}_{N}(n)$, $n\in\mathbb{Z}$, are also nonnegative,
$\widehat{p}_{N}(0)=1$, and we have $\widehat{p}_{N}(n)=0$ for every $n\in J$.

Finally, we define the function

$g_{N}(x):=\sum_{n\in\mathbb{Z}}\psi(n\tau)\,\widehat{p}_{N}(n)\,\varphi(x-nv),\quad
x\in\mathbb{R}^{d},$ (2.10)

where we denote $\tau=(\tau_{1},\dots,\tau_{k})$. Notice that there are only
finitely many nonzero terms in the sum (2.10), and that the nonzero terms
correspond to integers $n$ belonging to the set $T\setminus J$. Hence $g_{N}$
is a real-valued (in fact, nonnegative) Schwartz function such that
$\operatorname{supp}(g_{N})$ is contained in the set $S$. The Fourier
transform of $g$ is given by

$\widehat{g}_{N}(\xi)=\widehat{\varphi}(\xi)\sum_{n\in\mathbb{Z}}\psi(n\tau)\,\widehat{p}_{N}(n)\,e^{-2\pi
in\langle v,\xi\rangle}.$ (2.11)

Using the (absolutely convergent) Fourier expansion of the function $\psi$, we
obtain

$\psi(n\tau)=\sum_{m\in\mathbb{Z}^{k}}\widehat{\psi}(m)e^{2\pi in\langle
m,\tau\rangle}.$ (2.12)

If we plug (2.12) into (2.11) and exchange the order of summation (which is
justified as $n$ goes through a finite set of values) we obtain

$\widehat{g}_{N}(\xi)=\widehat{\varphi}(\xi)\sum_{m\in\mathbb{Z}^{k}}\widehat{\psi}(m)\,p_{N}(\langle
m,\tau\rangle-\langle v,\xi\rangle).$ (2.13)

Since all the terms on the right hand side of (2.13) are nonnegative, we
obtain that $\widehat{g}_{N}$ is a nonnegative function. We conclude that
$g_{N}$ satisfies the conditions (2.4).

To complete the proof we will show that if $N$ is sufficiently large, then
$g_{N}$ does not satisfy (2.7). Indeed, we may assume that
$\varepsilon<\frac{1}{2}|v|$ which ensures that the terms in the sum (2.10)
have pairwise disjoint supports. This implies that

$g_{N}(0)=\varphi(0)\psi(0),$ (2.14)

so that the value $g_{N}(0)$ does not depend on $N$. On the other hand, using
(2.13) we have

$\int_{\mathbb{R}^{d}}g_{N}(x)dx=\widehat{g}_{N}(0)\geqslant\widehat{\varphi}(0)\widehat{\psi}(0)p_{N}(0)=N,$
(2.15)

which can be arbitrarily large, contradicting (2.7). We have thus arrived at
the desired contradiction and so Theorem 1.3 is proved. ∎

##  3\. Weak tiling and product domains

In this section we prove Theorem 1.7. That is, we show that if
$A\subset\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{m}$ are two bounded,
measurable sets, then the product set $\Omega=A\times B$ weakly tiles its
complement in $\mathbb{R}^{n}\times\mathbb{R}^{m}$ by translations if and only
if both $A$ and $B$ can weakly tile their complements in $\mathbb{R}^{n}$ and
$\mathbb{R}^{m}$ respectively.

###### Proof of Theorem 1.7.

Suppose first that both $A$ and $B$ weakly tile their complements. Then there
exist two positive, locally finite measures $\nu_{1}$ on $\mathbb{R}^{n}$ and
$\nu_{2}$ on $\mathbb{R}^{m}$, such that

$\text{$\mathds{1}_{A}\ast(\delta_{0}+\nu_{1})=1$ a.e.\ in
$\mathbb{R}^{n}$},\quad\text{$\mathds{1}_{B}\ast(\delta_{0}+\nu_{2})=1$ a.e.\
in $\mathbb{R}^{m}$}.$ (3.1)

It follows that the (also locally finite) product measure

$\mu:=(\delta_{0}+\nu_{1})\times(\delta_{0}+\nu_{2})$ (3.2)

satisfies $\mathds{1}_{\Omega}\ast\mu=1$ a.e. in
$\mathbb{R}^{n}\times\mathbb{R}^{m}$. Let $\nu$ be the positive measure

$\nu:=\nu_{1}\times\delta_{0}+\delta_{0}\times\nu_{2}+\nu_{1}\times\nu_{2},$
(3.3)

and observe that we have $\mu=\delta_{0}+\nu$. Hence
$\mathds{1}_{\Omega}\ast\nu=\mathds{1}_{\Omega^{\complement}}$ a.e. and
$\Omega$ weakly tiles its complement in $\mathbb{R}^{n}\times\mathbb{R}^{m}$
by translations.

Conversely, suppose that $\nu$ is a positive, locally finite measure in
$\mathbb{R}^{n}\times\mathbb{R}^{m}$ satisfying

$\mathds{1}_{\Omega}\ast\nu=\mathds{1}_{\Omega^{\complement}}\quad\text{a.e.}$
(3.4)

For each $y\in\mathbb{R}^{m}$ we define a measure $\nu_{y}$ on
$\mathbb{R}^{n}$ by the condition

$\int_{\mathbb{R}^{n}}\varphi(u)\,d\nu_{y}(u)=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{m}}\varphi(u)\mathds{1}_{B}(y-v)\,d\nu(u,v)$
(3.5)

for any bounded, compactly supported Borel function $\varphi$ on
$\mathbb{R}^{n}$. It follows that

$\displaystyle(\mathds{1}_{A}\ast\nu_{y})(x)=\int_{\mathbb{R}^{n}}\mathds{1}_{A}(x-u)\,d\nu_{y}(u)$
(3.6)
$\displaystyle\qquad=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{m}}\mathds{1}_{A}(x-u)\mathds{1}_{B}(y-v)\,d\nu(u,v)=(\mathds{1}_{\Omega}\ast\nu)(x,y).$
(3.7)

We use this together with (3.4) to conclude by an application of Fubini’s
theorem that there is a set $Y\subset\mathbb{R}^{m}$ of full measure, such
that for any $y\in Y$ we have

$(\mathds{1}_{A}\ast\nu_{y})(x)=\mathds{1}_{\Omega^{\complement}}(x,y),\quad
x\in X(y),$ (3.8)

where $X(y)$ is a set of full measure in $\mathbb{R}^{n}$. In particular, for
$y\in Y\cap B$ this yields
$\mathds{1}_{A}\ast\nu_{y}=\mathds{1}_{A^{\complement}}$ a.e. in
$\mathbb{R}^{n}$, which shows that $A$ weakly tiles its complement in
$\mathbb{R}^{n}$. In the same way one can show that also $B$ weakly tiles its
complement in $\mathbb{R}^{m}$. ∎

##  4\. Packing regions

In this section we introduce the notion of a _packing region_ , which we use
to establish a geometric condition necessary for the spectrality of a set
$A\subset\mathbb{R}^{d}$. This geometric condition is different from the weak
tiling condition of Theorem 1.2.

### 4.1.

In order to define the notion of a packing region, we first need to introduce
the following definition:

######  Definition 4.1.

If $A\subset\mathbb{R}^{d}$ is a bounded, measurable set, then the set

$\Delta(A):=\\{t\in\mathbb{R}^{d}:\operatorname{mes}(A\cap(A+t))>0\\}$ (4.1)

will be called the _essential difference set_ of $A$.

The set $\Delta(A)$ is a bounded open set, symmetric with respect to the
origin.

We use the name “essential difference set” because $\Delta(A)$ serves as the
measure-theoretic analog of the set of differences $A-A$. In particular, if
$A$ is an open set then $\Delta(A)=A-A$. In general we have $\Delta(A)\subset
A-A$, but this inclusion can be strict.

######  Definition 4.2.

We say that a bounded, measurable set $D\subset\mathbb{R}^{d}$ is a _packing
region for $A$_ if we have $\Delta(D)\subset\Delta(A)$.

Let us explain the reason for the name “packing region”. If $\Lambda$ is a
finite or countable set in $\mathbb{R}^{d}$, then we say that $A+\Lambda$ is a
_packing_ if the translated copies $\\{A+\lambda\\}$, $\lambda\in\Lambda$, are
disjoint up to measure zero. Then a bounded, measurable set $D$ is a packing
region for $A$ if and only if whenever we have that $A+\Lambda$ is a packing,
then also $D+\Lambda$ is a packing.

### 4.2.

The following theorem is the main result of this section.

######  Theorem 4.3.

Let $A$ be a bounded, measurable set in $\mathbb{R}^{d}$. Suppose that $A$
admits a packing region $D$ such that $m(D)>m(A)$. Then $A$ cannot be
spectral.

###### Proof.

Suppose to the contrary that $A$ is spectral, and let $\gamma$ be the measure
given by Theorem 3.1 in [LM22]. Define $f:=|\widehat{\mathds{1}}_{D}|^{2}$,
then the Fourier transform of $f$ is the function
$\widehat{f}=\mathds{1}_{D}\ast\mathds{1}_{-D}$, that is, we have
$\widehat{f}(x)=m(D\cap(D+x))$ for every $x\in\mathbb{R}^{d}$. In particular,
$\widehat{f}(0)=m(D)$, and $\widehat{f}$ vanishes on the set
$\Delta(D)^{\complement}$. On the other hand, we have
$\widehat{\gamma}=m(A)\,\delta_{0}$ in the open set $\Delta(A)$, so due to the
assumption that $\Delta(D)\subset\Delta(A)$ this implies that
$\widehat{\gamma}\cdot\widehat{f}=m(A)m(D)\,\delta_{0}$. The measure
$\widehat{\gamma}\cdot\widehat{f}$ is the Fourier transform of the function
$\gamma\ast f$ (see [KL21, Section 2.3]), and it follows that $\gamma\ast
f=m(A)m(D)$ a.e.

Recall now that $\gamma$ is a positive measure, and that $\gamma=\delta_{0}$
in the open ball $B_{\varepsilon}$ for some $\varepsilon>0$. Since the
function $f$ is positive as well, this implies that

$f=\delta_{0}\ast f\leqslant\gamma\ast f=m(A)m(D)\quad\text{a.e.}$

But since $f$ is a continuous function, it follows that the inequality
$f(t)\leqslant m(A)m(D)$ must in fact hold for every $t\in\mathbb{R}^{d}$.
However $f(0)=m(D)^{2}$, so this is possible only if $m(D)\leqslant m(A)$.
Since we assumed that $m(D)>m(A)$, we arrive at a contradiction. ∎

######  Remark 4.4.

Let us comment that the proofs of Theorems 1.3 and 4.3 share a common idea: in
order to arrive at a contradiction we construct a positive definite function
with prescribed support, whose integral is “too large” compared to its value
at the origin.

### 4.3.

We give two examples demonstrating how to use Theorem 4.3.

######  Example 4.5.

Let $A$ and $D$ be the two planar domains illustrated respectively on the left
and right hand sides of Figure 4.1. It is straightforward to verify that
$\Delta(D)$ is the open rectangle $(-2,2)\times(-3,3)$, and that $\Delta(A)$
contains this rectangle. Hence $D$ serves as a packing region for $A$. But
notice that $m(A)=5$, $m(D)=6$, so that $m(D)>m(A)$. It thus follows from
Theorem 4.3 that $A$ is not spectral.

Figure 4.1. The planar domain $A$ shown on the left is not spectral, since its
has a packing region $D$ (shown on the right) satisfying $m(D)>m(A)$.

######  Example 4.6.

Let $A$ be a convex body in $\mathbb{R}^{d}$, and assume that $A$ is not
centrally symmetric. Then the convex body $D=\frac{1}{2}(A-A)$ is a packing
region for $A$, and we have $m(D)>m(A)$ by the Brunn-Minkowski inequality.
Using Theorem 4.3 this yields that a spectral convex body must be centrally
symmetric, a result first proved in [Kol00] based on the same consideration.

### 4.4.

The assumptions in Theorem 4.3 imply not only that $A$ cannot be spectral, but
also that $A$ cannot tile the space by translations. Indeed, if $A+\Lambda$ is
a tiling then $D+\Lambda$ must be a packing, and it follows e.g. from [GL20,
Lemma 3.2] that $m(D)\leqslant m(A)$, a contradiction.

The following result connects the notions of a packing region and weak tiling.

######  Theorem 4.7.

Let $A$ be a bounded, measurable set in $\mathbb{R}^{d}$. Suppose that $A$
admits a packing region $D$ such that not only $m(D)>m(A)$, but moreover
$D\supset A$. Then $A$ cannot weakly tile its complement by translations.

###### Proof.

We use the same argument as in [LM22, Theorem 3.5]. Suppose to the contrary
that $\mathds{1}_{A}\ast\nu=\mathds{1}_{A^{\complement}}$ a.e., where $\nu$ is
a positive, locally finite measure on $\mathbb{R}^{d}$. Then

$m(D\cap
A^{\complement})=\int_{D}\mathds{1}_{A^{\complement}}\,dm=\int_{D}(\mathds{1}_{A}\ast\nu)\,dm=\int\varphi\,d\nu,$
(4.2)

where $\varphi(t):=m((A+t)\cap D)$. The assumptions that $D$ is a packing
region for $A$ and $D\supset A$ imply that $\varphi$ must vanish on the set
$\Delta(A)^{\complement}$. On the other hand, the measure $\nu$ is supported
on the set $\Delta(A)^{\complement}$ due to [LM22, Corollary 2.6]. Hence the
integral on the right hand side of (4.2) vanishes. We conclude that $m(D\cap
A^{\complement})=0$, or equivalently, $m(D)=m(D\cap A)$. But this contradicts
our assumption that $m(D)>m(A)$. ∎

##  5\. Cantor sets of positive measure: via the essential difference set

In this section we prove Theorem 1.8, which asserts that a symmetric Cantor
set of positive measure in $\mathbb{R}$ cannot weakly tile its complement by
translations. The approach is based on the construction of a packing region
$D$ for the set $E$ with $m(D)>m(E)$ and moreover $D\supset E$. The conclusion
then follows from Theorem 4.7.

### 5.1.

We start by recalling the definition of a symmetric Cantor set in
$\mathbb{R}$. Let $\\{\xi_{k}\\}$, $k=1,2,\dots$, be a sequence of real
numbers such that $0<\xi_{k}<\frac{1}{2}$ for all $k$. The _symmetric Cantor
set associated to the ratio sequence $\\{\xi_{k}\\}$_ is the closed set
$E(\xi_{1},\xi_{2},\dots)\subset\mathbb{R}$ obtained from the interval $[0,1]$
by a Cantor type construction, where at the $k$’th step we remove from each
one of the intervals of the previous step a central open interval of relative
length $1-2\xi_{k}$, so that at the $k$’th step we obtain $2^{k}$ closed
intervals of common length $\xi_{1}\cdots\xi_{k}$.

The Lebesgue measure of the set $E(\xi_{1},\xi_{2},\dots)$ is

$m(E(\xi_{1},\xi_{2},\dots))=\lim_{k\to\infty}2^{k}\xi_{1}\cdots\xi_{k}.$
(5.1)

We are interested in the case where $E(\xi_{1},\xi_{2},\dots)$ is a set of
positive measure.

### 5.2.

Our proof of Theorem 1.8 is based on the following lemma:

######  Lemma 5.1.

Let $E$ be a symmetric Cantor set of positive measure in $\mathbb{R}$. Then
$E$ admits a packing region $D$ satisfying the two conditions $D\supset E$ and
$m(D)>m(E)$.

We note that there exist results in the literature which concern the set of
differences $E-E$ of a symmetric Cantor set $E$, see e.g. [NF20] and the
references therein. For example, it is well known that if $E$ is the classical
ternary Cantor set (associated to the constant ratio sequence
$\xi_{k}=\frac{1}{3}$) then $E-E=[-1,1]$. However, such results do not suffice
for our present purpose, as in order to prove Lemma 5.1 we must establish that
the _essential difference set_ $\Delta(E)$ (and not only $E-E$) is quite
large.

We now turn to the proof of Lemma 5.1. This will be done in several steps.

### 5.3.

We will need the following proposition.

######  Proposition 5.2.

Let $A$ be a bounded, measurable set in $\mathbb{R}^{d}$, and let
$B=\bigcup_{j=1}^{n}(A+t_{j})$ where $t_{1},\dots,t_{n}\in\mathbb{R}^{d}$.
Then

$\Delta(B)=\bigcup_{i,j}(\Delta(A)+t_{i}-t_{j}).$ (5.2)

The proof is straightforward and we omit the details.

### 5.4.

Let $E(\xi_{1},\xi_{2},\dots)$ be the symmetric Cantor set associated to a
given ratio sequence $\\{\xi_{k}\\}$ (that is, a sequence of real numbers such
that $0<\xi_{k}<\frac{1}{2}$ for all $k$). Then the set
$E(\xi_{1},\xi_{2},\dots)$ consists of all points $x$ of the form

$x=\sum_{k=1}^{\infty}r_{k}\varepsilon_{k},$ (5.3)

where

$r_{k}:=\xi_{1}\cdots\xi_{k-1}(1-\xi_{k})$ (5.4)

and each $\varepsilon_{k}$ is either $0$ or $1$. If we let $T_{k}$ denote the
finite set of points $x$ of the form

$x=\sum_{j=1}^{k}r_{j}\varepsilon_{j}$ (5.5)

where $\varepsilon_{j}=0$ or $1$, then we have

$E(\xi_{1},\xi_{2},\dots)=T_{k}+\xi_{1}\cdots\xi_{k}E(\xi_{k+1},\xi_{k+2},\dots),$
(5.6)

so that $E(\xi_{1},\xi_{2},\dots)$ is the union of $2^{k}$ disjoint homothetic
copies of $E(\xi_{k+1},\xi_{k+2},\dots)$.

It follows from Proposition 5.2 that

$\Delta(E(\xi_{1},\xi_{2},\dots))=T_{k}-T_{k}+\xi_{1}\cdots\xi_{k}\Delta(E(\xi_{k+1},\xi_{k+2},\dots)).$
(5.7)

### 5.5.

The following result establishes that Lemma 5.1 holds in the special case
where the set $E(\xi_{1},\xi_{2},\dots)$ has measure close to $1$.

######  Lemma 5.3.

Suppose that $E(\xi_{1},\xi_{2},\dots)$ has measure at least $\frac{9}{10}$.
Then

$\Delta(E(\xi_{1},\xi_{2},\dots))=\Delta(I),$ (5.8)

where $I:=[0,1]$.

Note that $\Delta(I)$ is simply the interval $(-1,1)$.

###### Proof.

Write $m(E(\xi_{1},\xi_{2},\dots))=1-\varepsilon$. From (5.1) it follows that

$2\xi_{1}\geqslant 1-\varepsilon,$ (5.9)

and

$\lim_{n\to\infty}2^{n}\xi_{k+1}\cdots\xi_{k+n}\geqslant 1-\varepsilon$ (5.10)

for every $k$. Note that $0<\varepsilon\leqslant\tfrac{1}{10}$ by assumption.

The set $\Delta(E(\xi_{1},\xi_{2},\dots))$ is symmetric with respect to the
origin and it is a subset of $(-1,1)$, so to establish (5.8) it would be
enough to show that $\Delta(E(\xi_{1},\xi_{2},\dots))$ contains the interval
$[0,1)$. This will be done in two steps.

_Step 1_ : We first show that $\Delta(E(\xi_{1},\xi_{2},\dots))$ contains the
interval $[0,1-\xi_{1})$.

Indeed, let $t\in[0,1-\xi_{1})$. For any two bounded, measurable sets $E$ and
$F$ we have

$m(E\cap(E+t))\geqslant m(F\cap(F+t))-2m(E\triangle F),$ (5.11)

where $E\triangle F$ is the symmetric difference of $E$ and $F$. We apply this
estimate to the two sets $E=E(\xi_{1},\xi_{2},\dots)$ and
$F=[0,\xi_{1}]\cup[1-\xi_{1},1]$. On one hand, we have

$F\cap(F+t)\supset[t,t+\xi_{1}]\setminus(\xi_{1},1-\xi_{1}),$ (5.12)

and hence

$m(F\cap(F+t))\geqslant\xi_{1}-(1-2\xi_{1})=3\xi_{1}-1>\tfrac{1}{2}-2\varepsilon,$
(5.13)

where the last inequality follows from (5.9). On the other hand, notice that

$m(F\triangle
E)=2\xi_{1}(1-\lim_{n\to\infty}2^{n}\xi_{2}\cdots\xi_{n+1})<\varepsilon,$
(5.14)

where this time the inequality follows from (5.10). So (5.11), (5.13) and
(5.14) imply

$m(E\cap(E+t))\geqslant\tfrac{1}{2}-4\varepsilon>0,$ (5.15)

hence $E\cap(E+t)$ is a set of positive measure and $t\in\Delta(E)$.

_Step 2_ : Next we show that $\Delta(E(\xi_{1},\xi_{2},\dots))$ contains also
the interval $[1-\xi_{1},1)$.

Let therefore $t\in[1-\xi_{1},1)$, then we have $r_{1}\leqslant t<1$. Since
$\sum_{j=1}^{\infty}r_{j}=1$, there is $k$ such that

$r_{1}+\dots+r_{k}\leqslant t<r_{1}+\dots+r_{k}+r_{k+1}.$ (5.16)

Then we can write

$t=r_{1}+\dots+r_{k}+\xi_{1}\cdots\xi_{k}s,\quad 0\leqslant s<1-\xi_{k+1}.$
(5.17)

The set $E(\xi_{k+1},\xi_{k+2},\dots)$ also has measure at least
$\tfrac{9}{10}$, due to (5.10), so by what we have already shown in Step 1 we
have $s\in\Delta(E(\xi_{k+1},\xi_{k+2},\dots))$. Moreover, the number
$r_{1}+\dots+r_{k}$ belongs to the set $T_{k}-T_{k}$. Hence using (5.7) we
conclude that

$t\in
T_{k}-T_{k}+\xi_{1}\cdots\xi_{k}\Delta(E(\xi_{k+1},\xi_{k+2},\dots))=\Delta(E(\xi_{1},\xi_{2},\dots)),$
(5.18)

which completes the proof. ∎

### 5.6.

Let $E_{k}(\xi_{1},\dots,\xi_{k})$ be the set consisting of the $2^{k}$
disjoint closed intervals of common length $\xi_{1}\dots\xi_{k}$ obtained
after the $k$’th step of the Cantor type construction. Then

$E_{k}(\xi_{1},\dots,\xi_{k})=T_{k}+\xi_{1}\cdots\xi_{k}I$ (5.19)

where (as before) $I=[0,1]$, and we have

$E(\xi_{1},\xi_{2},\dots)=\bigcap_{k=1}^{\infty}E_{k}(\xi_{1},\dots,\xi_{k}).$
(5.20)

It follows from Proposition 5.2 that

$\Delta(E_{k}(\xi_{1},\dots,\xi_{k}))=T_{k}-T_{k}+\xi_{1}\cdots\xi_{k}\Delta(I).$
(5.21)

The next result concludes the proof of Theorem 1.8 by establishing that Lemma
5.1 holds in the general case.

######  Lemma 5.4.

Let $E(\xi_{1},\xi_{2},\dots)$ have positive measure. Then there is $k$ such
that

$\Delta(E(\xi_{1},\xi_{2},\dots))=\Delta(E_{k}(\xi_{1},\dots,\xi_{k})).$
(5.22)

###### Proof.

We choose $k$ sufficiently large such that

$\lim_{n\to\infty}2^{n}\xi_{k+1}\xi_{k+2}\cdots\xi_{k+n}\geqslant\tfrac{9}{10}.$
(5.23)

Then by Lemma 5.3 we have

$\Delta(E(\xi_{k+1},\xi_{k+2},\dots))=\Delta(I).$ (5.24)

Together with (5.7) and (5.21) this implies that

$\Delta(E(\xi_{1},\xi_{2},\dots))=T_{k}-T_{k}+\xi_{1}\cdots\xi_{k}\Delta(I)=\Delta(E_{k}(\xi_{1},\dots,\xi_{k})),$
(5.25)

and thus (5.22) is proved. ∎

##  6\. Cantor sets of positive measure: via weak tiling

In this section we give another proof of Theorem 1.8, which states that a
symmetric Cantor set of positive measure cannot weakly tile its complement by
translations. In this proof the approach is direct and does not go through the
essential difference set.

### 6.1.

Suppose that we have a Cantor set $E\subset{\mathbb{R}}$ which is the
intersection of the compact sets $E_{n}$, each of which is a finite collection
of closed intervals of length $\ell_{n}$. The set $E_{n}$ is constructed from
$E_{n-1}$ by throwing away an open middle interval of length $d_{n}$ from each
of the intervals of $E_{n-1}$. For $E$ to have positive measure it is
necessary that $d_{n}/\ell_{n}\to 0$.

The idea of this proof is to use the fact that the sets $E_{n}$ obviously
cannot weakly tile their complement (the holes are too small compared to the
intervals making up $E_{n}$). Making this observation quantitative is the key.

Suppose $\mu=\delta_{0}+\nu$, with $\nu$ being a nonnegative Borel measure and

${\mathds{1}}_{E}\ast\mu=1\text{ on ${\mathbb{R}}$ so also
}{\mathds{1}}_{E}\ast\nu=0\text{ on $E$}.$ (6.1)

Write $A_{n}$ for the union of intervals (each of length $d_{n}$) that we
threw away from the intervals of $E_{n-1}$ in order to obtain the intervals of
$E_{n}$ (each of length $\ell_{n}$).

Observe that

$m(A_{n})=\frac{d_{n}}{2\ell_{n}}m(E_{n}).$ (6.2)

Since $E_{n+1}\subset E_{n}$ we have by the monotone convergence theorem that

$\int_{E_{n}}{\mathds{1}}_{E_{n}}\ast\nu\to\int_{E}{\mathds{1}}_{E}\ast\nu=0,\text{
as $n\to\infty$.}$ (6.3)

The last equality is due to (6.1).

We also have the crucial inequality (for all $n$ for which $d_{n}/\ell_{n}<1$)

$\frac{\ell_{n}-d_{n}}{d_{n}}\int_{A_{n}}{\mathds{1}}_{E_{n}}\ast\nu\leqslant\int_{E_{n}}{\mathds{1}}_{E_{n}}\ast\nu.$
(6.4)

To see this note that every time we “load” an interval of $A_{n}$ with a
fractional copy (via the measure $\nu$) of some interval of $E_{n}$ we also
“load” one or both the intervals of $E_{n}$ on each side of $A_{n}$ by at
least a multiple $(\ell_{n}-d_{n})/d_{n}$ of the load that goes onto the
$A_{n}$-interval.

More formally, we first observe that for each interval $J$ of $E_{n-1}$ and
for all $t\in{\mathbb{R}}$ we have

$\frac{\ell_{n}-d_{n}}{d_{n}}m((E_{n}+t)\cap A_{n}\cap J)\leqslant
m((E_{n}+t)\cap E_{n}\cap J).$ (6.5)

Summing over all intervals $J$ comprising $E_{n-1}$ we obtain

$\frac{\ell_{n}-d_{n}}{d_{n}}m((E_{n}+t)\cap A_{n})\leqslant m((E_{n}+t)\cap
E_{n})$ (6.6)

or, equivalently,

$\frac{\ell_{n}-d_{n}}{d_{n}}\int_{A_{n}}{\mathds{1}}_{E_{n}}(x-t)\,dx\leqslant\int_{E_{n}}{\mathds{1}}_{E_{n}}(x-t)\,dx,$
(6.7)

and integration $d\nu(t)$ followed by Fubini’s theorem gives (6.4).

We continue from (6.4) using (6.2):

$\displaystyle\int_{E_{n}}{\mathds{1}}_{E_{n}}\ast\nu$
$\displaystyle\geqslant\frac{\ell_{n}-d_{n}}{d_{n}}\int_{A_{n}}{\mathds{1}}_{E_{n}}\ast\nu$
$\displaystyle\geqslant\frac{\ell_{n}-d_{n}}{d_{n}}\int_{A_{n}}{\mathds{1}}_{E}\ast\nu\text{\
\ (since $E\subset E_{n}$)}$
$\displaystyle=\frac{\ell_{n}-d_{n}}{d_{n}}m(A_{n})\text{\ \ (due to
\eqref{wtiling} since $A_{n}\subset E^{\complement}$)}$
$\displaystyle=\frac{\ell_{n}-d_{n}}{d_{n}}\frac{d_{n}}{2\ell_{n}}m(E_{n})\text{\
\ (from \eqref{ratio})}$
$\displaystyle=\left(\frac{1}{2}-\frac{d_{n}}{2\ell_{n}}\right)m(E_{n})$
$\displaystyle\to\frac{1}{2}m(E)\text{ as $n\to\infty$.}$

This positive lower bound contradicts the limit (6.3) and finishes the proof.

### 6.2.

It should be apparent that the proof above is quite flexible and does not
impose much rigidity on the Cantor sets of positive measure to which it
applies. Instead of trying to state the most general result possible let us
indicate this flexibility by giving an example in two dimensions, to which the
method applies. This Cantor set is not a cartesian product.

Define a Cantor set $E\subset[0,1]^{2}$ of positive measure as follows (refer
to Figure 6.1).

The set $E$ will be the intersection of the decreasing sequence of compact
sets $E_{n}$, with $E_{0}=[0,1]^{2}$. The $n$-th stage set $E_{n}$ will be a
union of non-overlapping cubes of the same side-length $s_{n}$, all of them
aligned at multiples of $s_{n}$. To obtain the set $E_{n}$ from $E_{n-1}$ we
visit each of the cubes of side-length $s_{n-1}$ comprising $E_{n-1}$, we
subdivide such a cube $Q$ into cubes of side-length $s_{n}=s_{n-1}/M_{n}$
(here $M_{n}>0$ is a fast increasing integer sequence) and we throw away any
one of these cubes situated in the middle third of $Q$.

If the integer sequence $M_{n}$ grows sufficiently fast then the resulting
Cantor set has positive measure. The proof of the previous section then
applies with no essential changes.

$s_{n-1}$remove$s_{n}$

Figure 6.1. To go from $E_{n-1}$ to $E_{n}$ we remove one of the small cubes
near the center of the big cube.

### 6.3.

Finally let us mention that there do exist spectral _unbounded_ nowhere dense
sets of positive and finite measure. One way to obtain such a set is to
construct a set that tiles by $\mathbb{Z}^{d}$ translates, and which therefore
admits $\mathbb{Z}^{d}$ also as a spectrum. We describe the construction in
dimension one, but a similar idea works also in several dimensions.

Suppose we can partition, up to measure zero, the interval $[0,1]$ into a
countable union of nowhere dense sets $E_{n}$, $n\in{\mathbb{Z}}$. In other
words

${\mathds{1}}_{[0,1]}(x)=\sum_{n\in{\mathbb{Z}}}{\mathds{1}}_{E_{n}}(x),$

for a.e. $x\in{\mathbb{R}}$. It follows then that the union

$E=\bigcup_{n\in{\mathbb{Z}}}(E_{n}+n)$

is a nowhere dense set which tiles with ${\mathbb{Z}}$, and is therefore
spectral.

To construct this partition $E_{n}$ we first pick a fat Cantor set in $[0,1]$
(a Cantor set of positive measure – we can construct such sets of arbitrarily
large measure in any given interval) as our first set. At any stage in the
construction the complement of the (finitely many) closed sets we have
selected so far is an open subset of $(0,1)$, which is therefore a disjoint
countable union of intervals. In each of these intervals we select another fat
Cantor set, taking care for the total measure of the complement to go to zero.
This process exhausts the measure and it is clear the the resulting set
coincides a.e. with a fundamental domain of the lattice ${\mathbb{Z}}$ in
${\mathbb{R}}$. In other words $E+{\mathbb{Z}}$ is a tiling.

##  7\. Open problems

We conclude the paper by posing some open problems.

### 7.1.

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded, nowhere dense set of positive
measure. Can $\Omega$ be a spectral set? As we have mentioned in Section 1.6,
the answer is known to be negative in dimension one, but in dimensions two and
higher the problem is open.

### 7.2.

Let $\Omega=A\times B$ where $A$ is a convex body in $\mathbb{R}^{n}$, and $B$
is a bounded, measurable set in $\mathbb{R}^{m}$. If $\Omega$ is a spectral
set, then $A$ must be spectral according to Theorem 1.6. Is it true that also
$B$ must be spectral?

At present this is proved only for dimensions $n=1$ and $2$ [GL16], [GL20].

### 7.3.

Recall from Example 4.6 that if $A$ is a convex body in $\mathbb{R}^{d}$ then
the convex body $D=\frac{1}{2}(A-A)$ is a packing region for $A$, and if $A$
is not centrally symmetric then we have $m(D)>m(A)$ by the Brunn-Minkowski
inequality.

Assume now that $A\subset\mathbb{R}^{d}$ is a convex polytope, which is
centrally symmetric but has a non-centrally symmetric facet (so we know that
$A$ can neither tile nor be spectral). Does $A$ admit a packing region $D$
with $m(D)>m(A)$?

### 7.4.

Let $K$ be a convex body in $\mathbb{R}^{d}$, and assume that $K$ can weakly
tile its complement by translations. Let $W(K)$ be the (nonempty, convex) set
of all positive, locally finite measures $\nu$ such that
$\mathds{1}_{K}\ast\nu=\mathds{1}_{K^{\complement}}$ a.e. If we endow $W(K)$
with the topology of vague convergence, then $W(K)$ is also compact, so by the
Krein-Milman theorem $W(K)$ is the closed convex hull of its extremal points.
In particular, $W(K)$ has extremal points.

It is not difficult to verify that any proper tiling (that is, any measure
$\nu\in W(K)$ which is the sum of unit masses) is an extremal point of $W(K)$.
Is the converse true?

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# Late-time cosmic evolution of dust: solving the puzzle

Andrea Ferrara Scuola Normale Superiore, Piazza dei Cavalieri 7, 50126 Pisa,
Italy  Celine Peroux European Southern Observatory, Karl-Schwarzschild-Straße
2, 85748 Garching bei München, Germany and
Aix Marseille Univ, CNRS, LAM, (Laboratoire d’Astrophysique de Marseille), UMR
7326, 13388, Marseille, France

###### Abstract

Dust is an essential ingredient of galaxies, determining the physical and
chemical conditions in the interstellar medium. Several complementary
observational evidences indicate that the cosmic dust mass density
significantly drops from redshift $z=1$ to $z=0$. Clearly, and for the first
time during cosmic evolution, dust must be destroyed more rapidly than it is
formed. By considering the dust production/destruction processes acting in
this cosmic time lapse, we find that the drop can be explained if dust is
mainly destroyed by astration (49% contribution in the fiducial case) and
supernova shocks within galaxies (42%). Our results further imply that on
average each supernova destroys only $M_{d,sn}=0.45\,M_{\odot}$ of dust, i.e.
$5-10$ times less than usually assumed, with a hard upper limit of
$M_{d,sn}<{3.0}M_{\odot}$ set by the available metal budget and maximal grain
growth. The lower efficiency might be explained by effective shielding of dust
against shock processing in pre-supernova wind shells.

cosmology: observations – intergalactic medium – ISM: evolution – dust,
extinction

††journal: MNRAS

##  1 Introduction

Dust plays a crucial role in the thermal balance, dynamics and visibility of
galaxies throughout cosmic times. Importantly, dust has a strong influence on
the physical processes of the insterstellar medium (ISM) of galaxies in
several ways.

Grain surfaces and the Polycyclic Aromatic Hydrocarbons (PAH) participate in a
large number of chemical reaction networks in different phases of ISM, and act
as catalyst for important chemical processes such as the formation of H2
(Tielens, 2010), which in turn drives molecular chemistry.

Dust governs the ISM thermal balance (Draine, 2003; Galliano et al., 2018) by
providing photoelectric heating, and cooling which can alter the shape of the
Initial Mass Function (IMF) by favouring cloud fragmentation, thus inhibiting
the formation of massive stars and fostering the formation of low-mass stars
(Schneider et al., 2003; Omukai et al., 2005).

Finally, grains absorb the stellar ultraviolet light and re-radiate it in the
infrared, shielding the dense gas, and by these means triggering the formation
of molecular clouds where new stars are born.

In spite of the almost 80-years history of dust studies, relatively little is
known about the origin and build-up history of the solid component of the ISM.
The naïve expectation is that cosmic dust abundance should be tied to the
metal abundance. However, recent data (presented in Sec. 2) suggest that this
is not the case. Indeed in the last $7-8$ Gyr the dust abundance has
significantly decreased (see Fig. 12 of Péroux & Howk, 2020) despite of the
increasing availability of heavy elements, the primary components of dust
grains.

Theoretically, a few studies have addressed the dust evolution issue. Aoyama
et al. (2018, see also ()), by performing cosmological simulations including
dust evolution, found that the cosmic dust density111The cosmic evolution of
the cosmic dust mass density, $\Omega_{d}$, is defined as the comoving density
of dust in the Universe normalised by the critical density at redshift zero,
$\Omega_{d}\equiv\rho_{d}/\rho_{\rm cr}(z=0)$., $\Omega_{d}(z)$, peaks at
$z\approx 1$. They suggest that the slight decline afterwards is due to
astration. A similar type of simulation has been presented by Li et al.
(2019), who found that the total $\Omega_{d}$ (i.e. dust in galaxies and
outside them) always increases with time; however, the comoving dust mass
density excluding dust ejected out of galaxies via galactic winds peaks at
$z=2$ and then declines. They interpret this trend as a result of the reduced
availability of gas-phase metals to be accreted on grains due to the
decreasing star formation rate at $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2$. In another study based on EAGLE
(Schaye et al., 2015) simulations which assesses the reliability of SED
fitting to recover the input dust mass, Baes et al. (2020) derived an
evolution of $\Omega_{d}(z)$ which fits well the flat trend measured by Driver
et al. (2018) data, but less so the steeper decline originally found by Dunne
et al. (2011).

In any case, the physical nature of the decline cannot be addressed by current
hydrodynamical cosmological simulations which lack the detailed treatment of
the dust formation/destruction processes, but instead simply scale the dust
content with the metal abundance. While other numerical studies (e.g. Bekki,
2015; McKinnon et al., 2018, 2019; Aoyama et al., 2020; Osman et al., 2020)
have included some of these processes, these simulations concentrate on
single, isolated galaxies, thus hampering the ability to use such important
results in a cosmological framework.

The reported decrease is not predicted by some semi-analytical models (e.g.
Popping et al., 2017), but resembles that obtained by Gioannini et al. (2017),
although their model does not account for dust destruction in the hot
intracluster (ICM) and intragroup (IGrM) medium (see also Vijayan et al.,
2019; Triani et al., 2020).

Notwithstanding these many modelling efforts, the current lack of convergence
in the predictions indicates that the important issue of dust cosmic evolution
is still, at best, poorly understood.

Our work is motivated by a single important question: why does the dust
abundance – for the first time during cosmic evolution – decrease from $z=1$
to $z=0$ in spite of the increased availability of metals? We answer this
question by combining new/recent observational data with simple but solid
physical arguments. As a byproduct, we set novel constraints on dust
destruction efficiency. The strength of the method is based on its simplicity.
It complements more general models which need to make a larger number of
assumptions and/or do not fully include dust physics.

##  2 Cosmic Dust Density: Observations

The last decade has brought a wealth of new measurements of $\Omega_{d}^{\rm
obs}$ based on different techniques, which together draw a coherent picture of
the global evolution of the dust mass with cosmic times.

Figure 1: Observations of dust-to-metal ratio in neutral gas traced by Damped
Lyman-$\alpha$ Absorbers at $z<$1.2 (Péroux & Howk, 2020). The individual
measurements are shown by stars color-coded with metallicity, $\log$[M/H]. The
purple error bars indicate the Fe-weighted mean dust-to-metal ratio, $f_{\rm
d}$, for two redshift bins: $0<z<0.6$ and $0.6<z<1.2$.

Initially, measurements came from infrared (IR) Spectral Energy Distribution
(SED) fits of the extinction of individual galaxies. We first describe
techniques measuring the amount of dust in galaxies. Making educated
assumptions on the slope of the opacity power-law and the dust temperature,
the IR emission of galaxies has been widely used to estimate their dust mass.
The modelling of the IR SED has been especially improved in the past few
decades with the arrival of far-IR (Spitzer, Herschel), submillimetre (SCUBA,
BLAST), and ALMA ground instrumentation, adding much better constraints on the
cold dust regime.

In a work of reference, Dunne et al. (2011) performed a measurement of the
evolution of the dust mass density from a large sample of galaxies detected
both at 250 $\mu$m in Herschel-ATLAS and in the Sloan survey. They reproduced
the SED from temperature-based models fitted on the photometric data points.
This work has been later complemented with large spectroscopic samples
(including GAMA) and advanced SED fitting processes (see also Clemens et al.,
2013; Clark et al., 2015; Beeston et al., 2018; Driver et al., 2018; Bellstedt
et al., 2020). Here, we use the value of $\Omega_{d}^{\rm
obs}$=$1.0^{+0.8}_{-0.5}\times 10^{-6}$ at $z=0$ derived from Driver et al.
(2018), who despite a poorly constrained dust temperature, have a large sample
leading to high statistical significance. We note that our results are largely
unsensitive to this choice though this is a conservative approach, as the
value of $\Omega_{d}^{\rm obs}$ at $z=0$ derived by Dunne et al. (2011) is
even lower.

Recently, Pozzi et al. (2019) derived the evolution of the dust mass density
from a far-IR (160$\mu$m) Herschel selected catalogue in the COSMOS field,
pushing estimates of $\Omega_{d}^{\rm obs}$ to $z=2.5$. They also find a broad
peak at $z=1$, with a $\Omega_{d}$ decrease by a factor of $\approx 3$ from
$z=1\to 0$. Alternatively, ALMA deep fields allow to stack the contribution
from, e.g., H-band selected galaxies and use the continuum detection at 1.2mm
to derive the averaged dust mass in redshift bins over large lookback times
(Magnelli et al., 2020).

Table 1: Observed and expected cosmic dust and metal budget at late cosmic times. For the observed quantities, the redshift intervals are driven by the availability of data. The observed gas and dust values refer to neutral gas. They are recomputed refering to Péroux & Howk (2020) figures as follows: log [M/H] (their Fig. 7), $\Omega_{\rm gas}^{\rm obs}$ (their Fig. 3), $f_{\rm d}^{\rm obs}$ (their Fig. 10 & 11; see eqn 1) and finally $\Omega_{d}^{\rm obs}$ (their Fig. 12). The metallicity values for stars ($\Omega_{Z,*}^{\rm obs}$) and hot halos ($\Omega_{Z,h}^{\rm obs}$) are directly taken from Péroux & Howk (2020) (their Fig. 8). The expected quantities are derived from the following equations in the present work: $\Omega_{*}$ (eqn 4), $\Omega_{Z}$ (eqn 5) and $\Omega_{d}$ (eqn 6). The last column refers to the difference between $z\sim$0 and $z\sim$1 values defined as: $\Delta\Omega_{j}=\Omega_{j}(z\sim 0)-\Omega_{j}(z\sim 1)$. A positive $\Delta\Omega_{j}$ indicates an increase of the quantity with cosmic time.  |  $z\sim$0 |  $z\sim$1 | $\Delta\Omega_{j}$
---|---|---|---
Observed |  |  |
$z_{\rm mean}$ | $0.17^{+0.43}_{-0.17}$ | $0.84^{+0.36}_{-0.24}$ |
$\log[M/H]$ | $-0.26^{+0.14}_{-0.11}$ | $-0.20^{+0.09}_{-0.09}$ |
$\Omega_{\rm gas}^{\rm obs}$ | $5.0^{+0.2}_{-0.3}\times 10^{-4}$ | $6.4^{+0.5}_{-0.4}\times 10^{-4}$ |
$f_{\rm d}^{\rm obs}$ | $0.30^{+0.05}_{-0.05}$ | $0.32^{+0.02}_{-0.02}$ |
$\Omega_{d}^{\rm obs}$ | $1.0^{+0.8}_{-0.5}\times 10^{-6}$ | $1.6^{+0.3}_{-0.2}\times 10^{-6}$ | $-0.6^{+0.8}_{-0.5}\times 10^{-6}$
$z_{\rm mean}$ | $0.10^{+0.12}_{-0.09}$ | $0.70^{+0.05}_{-0.05}$ |
$\Omega_{Z,*}^{\rm obs}$ | $3.39^{+1.18}_{-0.82}\times 10^{-5}$ | $0.91^{+0.95}_{-0.28}\times 10^{-5}$ | $+2.48^{+1.51}_{-0.87}\times 10^{-5}$
$\Omega_{Z,h}^{\rm obs}$ | $10.0^{+3.5}_{-2.6}\times 10^{-6}$ | $5.2^{+2.2}_{-1.5}\times 10^{-6}$ | $+4.8^{+4.1}_{-3.0}\times 10^{-6}$
Expected |  |  |
$z_{\rm mean}$ | 0.0 | 1.0 |
$\Omega_{*}$ | $3.76\times 10^{-3}$ | $2.33\times 10^{-3}$ | $+1.43\times 10^{-3}$
$\Omega_{Z}$ | $1.24\times 10^{-4}$ | $0.77\times 10^{-4}$ | $+0.47\times 10^{-4}$
$\Omega_{d}$ |  |  | $+(0.43,4.72)\times 10^{-5}$

An alternative technique aims at estimating the total dust mass in the
Universe. To this end, a number of Herschel surveys has been utilized to
measure the far-infrared background anisotropy which captures the full
population of grains responsible for thermal dust emission in galaxies. These
measurements then provide an estimate of the global quantity of dust in the
Universe (De Bernardis & Cooray, 2012; Thacker et al., 2013). These
observations recover remarkably well the global evolution of $\Omega_{d}^{\rm
obs}$ with cosmic times traced by individual galaxies.

Lastly, a powerful approach to studying the dust content of the Universe is
provided by cold gas traced by quasar absorbers. Indeed, the dust content of
intervening gas has been assessed from the analysis of unrelated background
objects. Ménard et al. (2010) derived an estimate for $\Omega_{d}^{\rm obs}$
using the reddening of SDSS quasars due to foreground Mg II absorbers,
extending such measurements to higher redshift (see also Ménard & Fukugita,
2012). This estimate includes strong Mg II absorbers (with equivalent widths
EW $>$0.8 Å), which also trace the circum-galactic medium (CGM) of galaxies.
Peek et al. (2015) provided further estimates of the dust content of the CGM
by integrating the stellar mass over the galaxy stellar mass function of
Wright et al. (2018). These measurements provide an assessment of the dust
content of galaxies’ halos and therefore is complementary to SED-fitting
techniques described above.

In all of these cases, the dust mass is derived assuming an extinction curve
and scaling the results based on the dust-to-gas ratio within the Small
Magellanic Cloud. Adopting Milky Way dust properties increases the masses by a
factor of $\approx 1.8$. We note that Zafar et al. (2011, 2013), De Cia et al.
(2013) and Wiseman et al. (2017) show that depletion-based $A_{V}$ estimates
differ from reddening-based (i.e. SED fit) and postulate that extinction and
reddening do not trace the same type of dust. Wiseman et al. (2017) hint that
reddening-based measurements might differ, since reddening is measuring the
cumulative effect along the line-of-sight (Ménard & Fukugita, 2012).

Until recently, models describing the chemical evolution in galaxies rely on
dust studies in local metal-poor dwarfs (e.g. Rémy-Ruyer et al., 2014; De Vis
et al., 2019), which are usually adopted as a benchmark case for the
interstellar medium in galaxies at high-redshift (see also Shapley et al.,
2020). An alternative approach was proposed by Jenkins (2009); De Cia et al.
(2016); Jenkins & Wallerstein (2017); De Cia et al. (2018); De Cia (2018);
Roman-Duval et al. (2019) who utilised multi-element methods to correct
elemental depletion to estimate the amount of the metals locked into dust
grains in neutral gas. These techniques enable to derive the dust-to-metal
ratio ($f_{d}$) in quasar absorbers, extending such measurements to lower
metallicities than are currently available in the local Universe and to higher
redshifts than possible before. By combining these estimates with measurements
of $\Omega_{\rm gas}^{\rm obs}$, Péroux & Howk (2020) uniquely derive the
global dust density of the neutral gas. The individual dust-to-metal ratio
measurements are displayed in Fig. 1 for $z<1.2$. We compute the Fe-weighted
mean dust-to-metal ratios as follows:

$\langle f^{obs}_{d}\rangle=\frac{\Sigma(f_{d}\ \times N({\rm Fe}))}{\Sigma
N({\rm Fe})},$ (1)

where the errors are estimated from the standard deviation, $\sigma^{\prime}$:

$\sigma^{\prime 2}=\left(\sum(f_{d}-\langle f_{d}^{\rm
obs}\rangle)^{2}\right)/(n-1).$ (2)

Therefore, the uncertainties do not take into account errors on the slopes of
depletions versus [Zn/Fe] (De Cia et al., 2016), and on the contribution of
carbon, an important contributor to dust mass. An additional uncertainty not
considered here, is related to differential carbon depletion with respect to
other elements (Jenkins, 2009).

Eq. 1 represents the metal-weighted mean of the points binned by redshift
interval. The results indicate that $f^{obs}_{d}$ in the cold phase remains
constant over that redshift range. These values are shown in Fig. 1 and
tabulated in Table 1. Taken together, this large set of observations depict a
consistent picture (see Fig. 12 of Péroux & Howk, 2020). Notwithstanding a
continuous rise in the metal content of the Universe, the data indicate a
surprising global decrease of the dust mass density from $z=1\to 0$, of the
order $\Delta\Omega_{d}^{\rm obs}=-0.6\times 10^{-6}$. This trend is readily
apparent in all the results described above despite the many observational
methods utilised. To investigate this issue, we next turn into making
quantitative predictions of the expected amount of dust at late cosmic times.

##  3 Expected Dust Density

The basic ingredient of the calculation is represented by the cosmic star
formation history (CSFH), $\psi(z)$. Adopt the analytical fit to the available
data provided by Madau & Dickinson (2014):

$\psi(z)=\psi_{0}\frac{(1+z)^{2.7}}{1+[(1+z)/2.9]^{5.6}}\quad M_{\odot}{\rm
yr}^{-1}{\rm Mpc}^{-3},$ (3)

with $\psi_{0}=0.015$. We can then compute the stellar mass density at any
given redshift, $z$, by integrating:

$\rho_{*}(z)=(1-R)\int_{z}^{\infty}\psi(z^{\prime})\frac{dz^{\prime}}{H(z^{\prime})(1+z^{\prime})}\quad
M_{\odot}{\rm Mpc}^{-3}.$ (4)

In the previous expression
$H(z)=H_{0}[\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$ is the Hubble
parameter; for consistency with data, we will adopt the following values of
the cosmological parameters $(\Omega_{m},\Omega_{\Lambda},h)=(0.3,0.7,0.7)$.
The return fraction of gas from stars appropriate for a Chabrier initial mass
function and Instantaneous Recycling Approximation is $R=0.41$. The previous
expression can be cast in a more handy form,

$\rho_{*}(z)=(1-R)\frac{\psi_{0}}{H_{0}}{\cal I}(z)=1.26\times 10^{8}{\cal
I}(z)\quad M_{\odot}{\rm Mpc}^{-3},$ (5)

where the ${\cal I}(z)$ is the nondimensional integral in eq. 4; further
define ${\cal I}^{x}\equiv{\cal I}(z=x)$, and note that $({\cal I}^{1},{\cal
I}^{0})=(2.33,3.76)$, and $\Delta{\cal I}\equiv{\cal I}^{0}-{\cal
I}^{1}=1.43$. For the adopted cosmology, the stellar density parameter is
$\Omega_{*}=\rho_{*}/(3H_{0}^{2}/8\pi G)=10^{-3}{\cal I}(z)$. The density of
stars formed from $z=1$ to the present is $\Delta\Omega_{*}=1.43\times
10^{-3}$.

The metal density associated with $\Omega_{*}$ is

$\Omega_{Z}=y\,\Omega_{*}=3.3\times 10^{-5}{\cal I}(z);$ (6)

we have assumed a metal yield $y=0.033\pm 0.01$, with errors accounting for
uncertainties in the nucleosynthetic yields (Peeples et al., 2014; Vincenzo et
al., 2016).

Finally, the dust mass can be computed by knowing the dust-to-metal ratio,
i.e. the fraction of metals locked into dust, $f_{d}={\cal D}/Z$, where ${\cal
D}$ and $Z$ are the dust-to-gas ratio and gas metallicity, respectively. In
the Milky Way, ${\cal D}=1/162$ (Zubko et al., 2004); assuming solar
metallicity, $Z=0.0142$ (Asplund et al., 2009), then $f_{d}=0.43$ (Draine,
2011). At higher redshifts $f_{d}$ in measured from the gas depletion patterns
in, e.g. Damped Lyman-$\alpha$ (DLA) systems (Péroux & Howk, 2020). In the
redshift range $0<z<1$, $f_{d}^{\rm obs}\approx 0.31$ (see Figure 1 and Table
1) in the cold, neutral medium out of which stars form.

We note that $f_{d}^{\rm obs}$ combines the effects of dust production, growth
and destruction, while we aim here at isolating the first process. Hence, we
write the dust density associated with the metal density as

$\Omega_{d}=f_{d}\Omega_{Z},$ (7)

where, in general222We stress that $f_{d}^{\rm obs}$ is refers to neutral gas
only, while $f_{d}$ and $\Omega_{d}$ denote the total dust abundance in all
phases., $f_{d}\neq f_{d}^{\rm obs}$. To determine $f_{d}$ we proceed as
follows.

The minimum value, $f_{d}^{\rm min}$, is obtained by neglecting growth after
grains are injected in the ISM by sources (SNe and AGB stars). SNe with
progenitor mass $12-40\,M_{\odot}$ form on average $0.3M_{\odot}$ of dust, 80%
of which is typically destroyed by the reverse shock on site (Todini &
Ferrara, 2001; Bianchi & Schneider, 2007; Leśniewska & Michałowski, 2019). The
AGB stars ($>2M_{\odot}$) contribution per unit stellar mass formed for
$Z>0.2-0.3Z_{\odot}$ is $\approx 4$ times higher than the SN one (Zhukovska et
al., 2008; Dell’Agli et al., 2017; Valiante et al., 2017). Hence, the combined
effective dust yield per supernova is $y_{d}=0.3M_{\odot}$, which gives
$f_{d}^{\rm min}=\nu y_{d}/y=0.09$, where $\nu^{-1}=102\,M_{\odot}$ is the
number of SNe produced per stellar mass formed, according to the adopted
Chabrier (2003, eq. 17) IMF.

Larger values of $f_{d}$ might arise as a result of grain growth, which
depends on ambient conditions (Asano et al., 2013; Ferrara et al., 2016), and
it is very difficult to estimate reliably. The maximum $f_{d}^{\rm max}=1$
value is obtained when all the metals are depleted (maximal growth
efficiency). Note that typical values observed in galaxies fall conveniently
in the range $f_{d}^{\rm min}<f_{d}^{\rm obs}<f_{d}^{\rm max}$. To account for
dust growth uncertainties we write the dust density as

$\Omega_{d}=(f_{d}^{\rm min},f_{d}^{\rm max})\Omega_{Z}=(0.3,3.3)\times
10^{-5}{\cal I}(z).$ (8)

Fig. 2 shows the detailed balance of dust production and destruction
(discussed in the next Sec.) mechanisms, and their relative importance. As
${\cal I}(z)$ is a decreasing function of redshift, the expected cosmic dust
content should steadily increase with time, reaching
$\Omega_{d}=(1.13,12.41)\times 10^{-5}$ at $z=0$, with a variation
$\Delta\Omega_{d}^{+}=+(0.43,4.72)\times 10^{-5}$ from $z=1$ in the case of
(zero, maximal) dust growth efficiency.

This conclusion is in striking contrast with observations, which indicate a
decrease of $\Omega_{d}$ from $z=1\to 0$ ($\Delta\Omega_{d}^{\rm
obs}=-0.6\times 10^{-6}$, see Table 1). Clearly, and for the first time during
cosmic evolution, dust must be destroyed more rapidly than it is formed during
this time span of 7.8 Gyr (Gjergo et al., 2020).

##  4 Dust destruction

We now consider the various dust destruction processes at play. Before we
proceed, we justify the assumption we will make that $f_{d}=0$ in hot
($T\lower 2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}10^{5}$ K) gas, such as
the ICM/IGrM, or in SN-driven galactic outflows. In these environments dust is
destroyed by thermal sputtering with ions and electrons of the plasma. The
rate at which the grain radius decreases is described by a simple fit to the
numerical results by Draine & Salpeter (1979a); Tsai & Mathews (1995); Dwek et
al. (1996):

$\frac{da}{dt}=-AnT_{6}^{-1/4}e^{-BT_{6}^{-1/2}},$ (9)

where $a$ is the grain size, $n$ and $T_{6}=T/10^{6}\,{\rm K}$ are the gas
density and temperature; we have adopted material-averaged values for the
constants $(A,B)=(1.2\times 10^{-5}\mu$m yr${}^{-1},3.85)$. From eq. 9 the
survival time of a typical grain ($a=0.1\,\mu$m) in a $T=10^{6}$ K gas is
$\tau_{s}=(0.4/n)$ Myr. Provided $n>5\times 10^{-5}\,\rm cm^{-3}$, which
applies to ICM/IGrM, grains produced at $z=1$ are destroyed by thermal
sputtering well before $z=0$. While the details of the process depend on the
exact gas temperature, grain size distribution, and residence time, assuming
$f_{d}=0$ in the hot cosmic gas appears warranted.

Let us go back to the three main dust destruction processes and quantify their
impact. These are: (a) astration, (b) thermal sputtering, (c) supernova (SN)
shocks; they are discussed separately in the following. Their combined action
must lead to a decrease the dust mass density at $z=0$ to the observed value
$\Omega_{d}^{\rm obs}=10^{-6}$.

Figure 2: Sketch of dust production and destruction mechanisms and
corresponding yields described in Sec. 4. The diagonally-hatched blue region
in the production histogram denotes the uncertainty on the $f_{d}$ value of
dust sources; errors on each individual $\Delta\Omega_{d}^{(i)}$ contribution
are given in Tab. 1 and in Sec. 4.. Relative contributions to the destruction
budget refer to the fiducial, low SN destruction efficiency case
($M_{d,sn}=0.45M_{\odot}$).

###  4.1 Destruction by astration

Astration involves the incorporation of gas and dust into a stellar interior
during star formation. As stars forms in cold, neutral gas, we assume that the
stellar build-up material has $f_{d}=f_{d}^{\rm obs}=0.31$. The data in Table
1 show that the increase of metals in stars is $\Delta\Omega_{Z,*}^{\rm
obs}=2.48\times 10^{-5}$. Then, the (negative) variation of dust density due
to astration is

$\Delta\Omega_{d}^{(a)}=-f_{d}^{\rm obs}\Delta\Omega_{Z,*}^{\rm
obs}=-7.7{{}^{+4.9}_{-3.0}}\times 10^{-6}.$ (10)

Astration contributes ($11-49$)% (for a high or low SN dust destruction
efficiency, respectively; see Sec. 4.4) to the total amount of dust
destruction; the rest must be removed by the other two mechanisms.

###  4.2 Destruction by hot gas

As already mentioned, we assume that as dust gets embedded in the hot phase,
it is – for our purposes – instantaneously and completely eroded by
sputtering. This implies that dust associated with metals contained in the hot
cosmic gas at $z=0$ must be removed from the total budget. The metal content
of hot gas has increased from $z=1\to 0$ by $\Delta\Omega_{Z,h}^{\rm
obs}=4.8\times 10^{-6}$ (see Table 1). This term contributes a negative
variation equal to

$\Delta\Omega_{d}^{(b)}=-f_{d}^{\rm obs}\Delta\Omega_{Z,h}^{\rm
obs}=-1.4{{}^{+1.3}_{-1.0}}\times 10^{-6},$ (11)

corresponding to a mere ($2-9$)% of the total dust destruction budget.

###  4.3 Destruction by SN shocks in galaxies

Finally, we consider dust destruction by SN shocks in the ISM of galaxies,
which we identify here with the cold, neutral gas. Recent detailed numerical
simulations of dust production and destruction in SN explosions (Martínez-
González et al., 2019) find that, once the presence of a pre-supernova wind-
driven cavity is properly included, dust destruction is strongly suppressed.
The physical reason for this is that the dust is collected in a dense shell by
the wind; the shell represents an almost insurmountable barrier that prevents
the SN blast wave from processing the majority of the ambient dust protected
by the shell. As a result, under typical ambient conditions (gas
density333Cases with $n=10^{3}\,\rm cm^{-3}$ have also been explored, showing
a $\sim 25$% decrease in the amount of dust destroyed. $n\approx 1\,\rm
cm^{-3}$), the amount of dust destroyed444Hu et al. (2019) performed similar
simulations finding $M_{d,sn}\approx 5M_{\odot}$ for $n=1\,\rm cm^{-3}$. They
do not treat the pre-supernova wind-driven cavity self-consistently, but when
they allow SNe to occur in hot ($T>10^{4}$ K) bubbles carved by previous SN
explosions, the destruction rate is decreased by a factor $\approx 2.5$. We
point out that reduced destruction due to the pre-SN wind affects each SN, not
only those exploding in pre-existing hot bubbles. per SN event is
$M_{d,sn}=0.45M_{\odot}$.

An alternative calculation, which however does not include the effects of the
wind-driven shell discussed above, might be performed as follows. First, dust
sputtering requires projectiles (electrons, ions) with kinetic energies
$E_{t}>100$ eV (Draine & Salpeter, 1979b), which can be produced by shocks
with velocities $v_{s}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}200\,{\rm km\,s}^{-1}\,$. As the
transition from the energy-conserving, Sedov-Taylor phase to the radiative one
occurs at $v_{s}=200(n^{2}E_{51})^{1/14}\,{\rm km\,s}^{-1}\,$ (McKee, 1989),
we conclude that dust destruction essentially terminates with the first phase,
unless the density is very high (typically, though, the diffuse gas component
in galaxies has the largest filling factor; hence, expansion primarily occurs
in low density gas). The mass swept-up by the shock, $M_{e}$, as a function of
$v_{s}$ is

$M_{e}(v_{s})=\frac{E}{\sigma v_{s}^{2}}=6800\frac{E_{51}}{v_{s7}^{2}}\quad
M_{\odot};$ (12)

where $\sigma=0.736$, $E_{51}=E/10^{51}\rm erg\,{=1}$ is the explosion energy,
and $v_{s7}=v_{s}/100\,{\rm km\,s}^{-1}\,$. Hence,
$M_{e}(v_{s})=1700\,M_{\odot}$ for $v_{s}=200\,{\rm km\,s}^{-1}\,$. The dust
destruction efficiency, $\gamma_{d}(v_{s})$, by a shock depends on its
velocity. For $n=0.25\,\rm cm^{-3}$, Slavin et al. (2015) give the following
fit to their numerical results, valid for $1.85<v_{s7}<5$:

$\gamma_{d}=-1.9+2.02v_{s7}-0.641v_{s7}^{2}+0.092v_{s7}^{3}-0.05v_{s7}^{4}$
(13)

By mass-averaging $\gamma_{d}$ using expression eq. 12, we find
$\bar{\gamma}_{d}=0.41$. Hence, the dust mass destroyed per SN in this case,
assuming solar metallicity gas, would be $M_{d,sn}=\bar{\gamma}_{d}\,{\cal
D}\,M_{e}(v_{t})=4.3\,M_{\odot}$, i.e. a factor about $10$ times higher than
obtained by Martínez-González et al. (2019). Given these uncertainties we will
use these values to bracket our results.

The number density of SN exploded in $0<z<1$ is $\Delta{\cal
N}=\nu\Delta\rho_{*}=1.26\times 10^{8}\nu\Delta{\cal I}=1.8\times 10^{6}\,{\rm
Mpc}^{-3}$, or

$\Delta\Omega_{d}^{(c)}=-M_{d,sn}\Delta{\cal
N}\rho_{cr}^{-1}=-(6.5-62.3)\times 10^{-6},$ (14)

depending on the adopted value of $M_{d,sn}$.

###  4.4 Total dust destruction

Figure 2 displays the results in graphic form. In summary, processes (a)–(c)
account for a total dust destruction corresponding to

$\Delta\Omega_{d}^{-}\equiv\Sigma_{i}\Delta\Omega_{d}^{(i)}=-(0.91{{}^{+0.5}_{-0.3}}+1.45\,M_{d,sn})\times
10^{-5},$ (15)

where $M_{d,sn}$ is in solar masses, and $i$=(a,b,c). This value must balance
the dust mass produced in $0<z<1$, $\Delta\Omega_{d}^{+}$, augmented by the
observed dust decrease during the same epoch, $\Delta\Omega_{d}^{\rm
obs}=-0.6{{}^{+0.8}_{-0.5}}\times 10^{-6}$:

$\Delta\Omega_{d}^{+}-\Delta\Omega_{d}^{\rm obs}+\Delta\Omega_{d}^{-}=0$ (16)

From eqs. 15 and 16 we can then conclude that: (a) if dust growth does not
occur ($f_{d}=f_{d}^{\rm min}$), then $\Delta\Omega_{d}^{-}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}\Delta\Omega_{d}^{\rm
obs}-\Delta\Omega_{d}^{+}$ within errors, implying that observed $\Omega_{d}$
decrease can be explained by astration and sputtering in hot gas only, without
the need for dust destruction by SNe (i.e. $M_{d,sn}=0$); (b) the low-
efficiency SN destruction ($M_{d,sn}=0.45M_{\odot}$) would yield
$\Delta\Omega_{d}^{-}=-1.56{{}^{+0.5}_{-0.3}}\times 10^{-5}$, which falls
exactly in the middle of the allowed production/growth range. Then, by
combining eqs. 16 and 8, we get a nominal value $f_{d}=0.34$, a value
tantalizingly close to the observed one (0.31); (c) assuming that all the
newly produced metals in $0<z<1$ are incorporated into dust ($f_{d}=f_{d}^{\rm
max}=1$), we can get a hard upper limit on $M_{d,sn}<3.0M_{\odot}$. Larger
values, such as those ($4.3M_{\odot}$) predicted by the high SN destruction
efficiency case, would produce a steeper $\Omega_{d}$ decrease at late times,
and are therefore inconsistent with the data.

##  5 Implications for dust physics

Most likely, the usually adopted destruction rates in SN shocks have been
significantly overestimated as they result in an apparent inconsistency with
the observed cosmic dust evolution. From our calculation, combined with the
available data, we conclude that each SN can destroy at most
${3.0}\,M_{\odot}$ of dust. This upper limit assumes that all the metals are
locked into dust; most likely, the actual value is a factor 4-5$\times$ lower,
in agreement with recent theoretical findings (Martínez-González et al.,
2019).

At the same time, Ferrara et al. (2016) noted that dust growth, particularly
at high-$z$, is problematic and invoked solutions in which a lower growth rate
is balanced by a reduced destruction rate as we suggest here. Finally, we
notice that our empirical argument, based on dust cosmic evolution, resonates
with theoretical down-revaluations of the dust destruction rates by SN
presented by Jones & Nuth (2011).

##  6 Summary

We have investigated the evolution of the cosmic dust density in the last
$\approx 8$ Gyr. During this time stretch (corresponding to $z=1\to 0$),
observations show that $\Omega_{d}$ has decreased by about $37.5\%$ in spite
of the fact that the cosmic metal abundance has increased by about a factor
1.6. Thus, dust must have been efficiently destroyed during this period.

By evaluating different dust destruction mechanisms, we conclude that
astration and SN shocks in the ISM of galaxies are the dominant factors, with
sputtering in hot gas playing a sub-dominant role. All these processes were
obviously at work also at $z>1$, but the decrease of $\Omega_{d}$ at later
times is driven by the declining cosmic star formation rate and associated
metal production.

An implication of our study is that the dust destruction efficiency required
to explain the data is $\approx 10$ times lower than usually adopted (i.e.
0.45 $M_{\odot}$ vs. 4.3 $M_{\odot}$ of destroyed dust/SN) as suggested by
recent hydrodynamical simulations (Martínez-González et al., 2019) leading to
a reduced efficiency caused by the shielding effects of pre-SN wind-driven
shells. By assuming a maximally efficient grain growth in the ISM, we find
that the available metal budget sets a hard upper limit
$M_{d,sn}<{3.0}M_{\odot}$ on the dust mass destroyed per SN.

We thank V. D’Odorico, G. Popping, L. Sommovigo for insightful comments. AF
acknowledges support from the ERC Advanced Grant INTERSTELLAR H2020/740120.
Any dissemination of results must indicate that it reflects only the author’s
view and that the Commission is not responsible for any use that may be made
of the information it contains. Generous support from the Carl Friedrich von
Siemens-Forschungspreis der Alexander von Humboldt-Stiftung Research Award is
kindly acknowledged (AF). AF thanks the European Southern Observatory (ESO)
and Max-Planck for Astrophysics (MPA) in Garching for a warm hospitality
during part of this research. CP is grateful to the Alexander von Humboldt-
Stiftung for the granting of a Bessel Research Award. CP is indebted to the
Max-Planck for Astrophysics (MPA) in Garching for a fruitful visit. All plots
in this paper were built with the matplotlib (Hunter, 2007) package for
PYTHON.
Data Availability
Data available on request.

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