\section{Relation to the literature}\label{sec:literature}

We discuss the literature under three headings: what was known about the
dimension of spaces of crystalline measures, why the restriction to measures is
essential, and what is new in the constructions. Work on strong temperedness and on
the classification of summation formulas is discussed in Section~\ref{sec:fq},
together with Theorem~\ref{thm:nonstrong}.

\subsection{What was known about dimensions}
Poisson-type summation formulas not coming from lattices go back to
Kahane--Mandelbrojt \cite{KM58} and Guinand \cite{Gui59}, and interest in them was
renewed by the theory of mathematical quasicrystals; see Lagarias \cite{Lag00},
Baake and Grimm \cite{BG13}, and Dyson's proposal to classify one-dimensional
quasicrystals \cite{Dys09}. Meyer \cite{Mey16,Mey17a} introduced crystalline
measures in the form used here, gave a new proof of Guinand's formula, and
constructed crystalline measures that are not generalized Dirac combs; see also
\cite{Mey21,Mey22,Mey23,Mey23b} and, for formulas involving $r_k(n)$, Lev and Reti
\cite{LR21a}. Further constructions use stable and Lee--Yang polynomials: Kurasov
and Sarnak \cite{KS20} found positive crystalline measures and Fourier
quasicrystals that are not generalized Dirac combs; Olevskii and Ulanovskii
characterized Fourier quasicrystals with unit masses \cite{OU20b} and showed that
pairs of exponential polynomials generate Poisson-type formulas \cite{OU20a}; and
Alon, Cohen and Vinzant \cite{ACV24} completed the description of Fourier
quasicrystals with positive integer masses (see also \cite{AV24,AKKV25}). Modular
forms \cite{AKM25,BSS23} and inner functions \cite{SZ26} give further examples, and
Tselishchev \cite{Tse26} found a new crystalline-type measure with unit masses,
with an application to tilings of the line by translates of a function.
These works construct individual measures or explicit families, often under
restrictions on the masses, such as positivity or integrality, which do not define
vector spaces. Meyer's question \cite[Section~6]{Mey17a} concerns instead the whole
space $\M_\Lambda$ attached to a prescribed set. (Crystalline measures should not be
confused with measures with pure point diffraction; see Richard and Strungaru
\cite{RiS17}.)

Three kinds of answers were known. The first is \emph{rigidity}. If the support
and the spectrum are uniformly discrete, the structure is rigid: Córdoba
\cite{Cor89} proved that the support is a finite union of translated lattices if
the masses take finitely many values, and Lev and Olevskii \cite{LO13,LO15} proved
that in dimension one every such measure is a generalized Dirac comb, that is, a
finite combination of translated and modulated lattice combs; this answered a
problem of Lagarias \cite[Problem~4.1]{Lag00}. In dimension one,
uniform discreteness of the spectrum can be relaxed to bounded density
\cite[Theorem~2.2]{LO17}. Lev and Reti \cite{LR21b} extended the structure theorem
to tempered distributions, and Favorov \cite{Fav16,Fav21,Fav23} proved further
rigidity and uniqueness theorems.

The second is \emph{interpolation}. By duality, $\Ncal(A,B)$ is the space of linear
relations between the restrictions $\varphi|_A$ and $\widehat\varphi|_B$ of Schwartz
functions (Proposition~\ref{prop:annihilator}), so Fourier interpolation theorems
compute dimensions. Radchenko and Viazovska \cite{RV19} proved that an even
Schwartz function is determined by its values and those of its Fourier transform
at $\pm\sqrt n$, and that the data map is an isomorphism onto a closed subspace of
codimension one, cut out by Poisson's formula \cite[Theorem~2]{RV19}. This was
extended to powers of integers and perturbed nodes by Ramos and Sousa
\cite{RS22,RS23}, whose Fredholm-index argument yields a dichotomy between
interpolation and a one-dimensional defect \cite[Theorem~5.3]{RS23}; to zeros of
zeta and $L$-functions by Bondarenko, Radchenko and Seip \cite{BRS23}; and to
spheres in $\R^d$ by Stoller \cite{Sto21} and Ramos and Stoller \cite{RSt22}.
For Meyer's shifted square roots $\{\pm\sqrt{n+\theta}:n\ge0\}$, $0<\theta<1$, which
lie in the critical case of \cite{KNS25} and are not covered by the perturbation
results of \cite{RS23}, the space has been determined in a formalised
classification \cite{PalSSR}: it is spanned by an even measure with Fourier
eigenvalue $1$ when $\theta\le\frac18$ and an odd measure with eigenvalue $-\ii$ when
$\theta\le\frac38$, both built from generalized eta products.
Density thresholds were obtained by Kulikov \cite{Kul21} and Adve \cite{Adv23},
critical and asymmetric pairs were treated by Lysen \cite{Lys25}, the
interpolation basis was studied in \cite{BBRSS25}, and Bondarenko and Seip
\cite{BS26} showed, with nonzero Fourier-invariant functions vanishing on
$\{\sqrt n/(\log(e+n))^{\beta}\}$, $0\le\beta\le\frac12$, that uniqueness depends on
the function class. See Cohn \cite{Coh24} and Viazovska \cite{Via18,Via23} for
surveys, including the connection with sharp sphere packings,
and Meyer \cite[Section~8]{Mey22} for the relation with the problem of recovering
$f$ from $f|_\Lambda$ and $\widehat f|_F$.

The third is \emph{infinite dimension without a determination of the space}.
Kulikov, Nazarov and Sodin \cite{KNS25} proved that supercritical discrete pairs
are uniqueness pairs and subcritical pairs are not, and deduced that
$\Ncal(\Lambda,M)$ is finite-dimensional for subcritical and infinite-dimensional
for supercritical pairs \cite[Section~2.3, Remark~7.5]{KNS25}. Their model pairs,
such as $\Lambda=M=\{\operatorname{sign}(j)\sqrt{a|j|}:j\in\Z\}$ with $0<a<1$, are
separated at a polynomial rate, and every supercritical pair contains a
supercritical subpair with this property \cite[Claim~1]{KNS25}; for such pairs
every element of
$\Ncal(\Lambda,M)$ is a Fourier quasicrystal by Favorov's theorem
\cite[Theorem~1]{Fav24a} (Section~\ref{sec:fq}). Thus infinite-dimensional spaces of
Fourier quasicrystals were known, again without a determination of their
dimension. The closest
precedent for an infinite-dimensional space of relations built from
finite-dimensional pieces is due to Stoller \cite[Proposition~7.1]{Sto21}: for
$d\ge5$, the relations between the restrictions of a Schwartz function on $\R^d$
and of its Fourier transform to the spheres $\sqrt n\,S^{d-1}$ form an
infinite-dimensional space containing all finite sums of suitable modular forms of
weights $d/2+m$, $m\ge0$, each weight contributing finitely many dimensions. There
the grading is by modular weight and the whole relation space is not determined.
By Proposition~\ref{prop:trichotomy}, deciding between dimensions $\aleph_0$ and
$\cc$ requires control of every layer, and we are not aware of an earlier example
in which the whole space was determined and found to be countably
infinite-dimensional, either for crystalline measures or for Fourier
quasicrystals.

\subsection{Why the restriction to measures is essential}
For tempered distributions the countable case occurs already for the lattice:
the distributions with support and spectrum in $\Z$ form a space of countable
dimension (Example~\ref{ex:Z}), a special case of the structure theorem of Lev and
Reti \cite[Theorem~1.1]{LR21b}, by which tempered distributions with uniformly
discrete support and spectrum are obtained from Poisson's formula by finitely many
shifts, modulations, differentiations, multiplications by polynomials and linear
combinations. Derivatives raise the Hermite order, and countable dimension comes
from adding more and more of them. For measures nothing of this kind is available,
and $\M_\Z=\C\delta_\Z$. Derivatives do appear at the edge of the theory of
measures. Guinand's formula with nodes $\pm\sqrt n$ and weights $r_3(n)$ contains a
derivative of the Dirac mass at the origin \cite{Gui59,Mey16}; Meyer observed that
a natural variant of his construction is not crystalline, because ``a derivative of
the Dirac mass at $\pm1/2$ peeps out from the Fourier transform'' \cite{Mey16}; and
Olevskii and Ulanovskii
\cite{OU20a} noted that multiple zeros of exponential polynomials, together with
the failure of uniform discreteness, lead to derivatives in the formula and to
unbounded coefficients. Our construction turns this phenomenon into a mechanism:
clusters of atoms imitate derivatives, but remain measures
(Section~\ref{subsec:mechanism}).

\subsection{What is new in the constructions}
The blocks of our main construction (Section~\ref{sec:fqcount}) belong to a
well-studied family. Kurasov and
Sarnak \cite{KS20} obtained Fourier quasicrystals with positive integer masses from
Lee--Yang polynomials restricted to lines in a torus, and Alon, Cohen and Vinzant
\cite{ACV24} completed the description of Fourier quasicrystals with positive
integer masses in these terms. Olevskii and Ulanovskii \cite{OU20a} proved that
every pair $(\psi,\varphi)$ of exponential polynomials, where the zeros of $\varphi$
are simple and form a uniformly discrete set, gives a Poisson-type formula with
masses $\psi(\lambda)/\varphi'(\lambda)$; they remark that without these assumptions
the formula may contain derivatives and the masses may be unbounded, and set this
case aside. Our block measures have masses
of exactly this form, with zero sets that are not uniformly discrete and masses that
grow polynomially; related constructions with Blaschke factors are Meyer's curved
model sets \cite{Mey23c} and the measures of Szehr and Zarouf \cite{SZ26}. Every
element of the spaces of Theorem~\ref{thm:D} satisfies the growth hypotheses of
Gon\c{c}alves and Vedana \cite{GV25}, so each individual summation formula is
covered by their classification; the classification concerns single formulas,
not the space attached to a prescribed set. What is new in
Section~\ref{sec:fqcount} is the control of the growth exponent of every element of
a block, from above and from below, through the near-crossings of its sheets, and
the complete determination of the Fourier quasicrystals on unions of blocks, which
uses a uniqueness theorem for measures on cones (Proposition~\ref{prop:noreturn}) and
an algebraic splitting lemma on $\mathbb P^1\times\mathbb P^1$
(Lemma~\ref{lem:splitting}).

Overlaying crystalline measures whose supports and spectra avoid larger and larger
intervals is a device of Kolountzakis \cite[Section~2]{Kol16}, who used it to give
a simpler construction of crystalline measures, found by Lev and Olevskii
\cite{LO16}, whose support meets every arithmetic progression in finitely many
points; the device is also used in \cite{Fav24b,MRS26}. We use it in both
constructions; the growing gaps keep the union locally finite and produce
infinitely many independent measures. Without an exhaustion the overlay gives only a
lower bound for the dimension, which cannot distinguish $\aleph_0$ from $\cc$. For
all crystalline measures, what is new is, first, the blocks: whole spaces of crystalline measures that are
one-dimensional and of prescribed Hermite order, produced by clustering atoms
(Section~\ref{sec:blocks}); and second, the exhaustion: a proof that the union of the
blocks carries no crystalline measures other than finite combinations of the block
measures (Section~\ref{sec:exhaustion}). The sets of Theorems~\ref{thm:A} and~\ref{thm:B} are not uniformly discrete, and the coalescence of their atoms is
essential.