\section{Introduction}\label{sec:intro}

\subsection{Fourier quasicrystals and Meyer's question}
The Poisson summation formula $\sum_{n\in\Z}\varphi(n)=\sum_{n\in\Z}\widehat\varphi(n)$
says that the Dirac comb $\delta_\Z=\sum_{n\in\Z}\delta_n$ is its own Fourier
transform. Following Meyer \cite{Mey16,Mey17a}, a \emph{crystalline measure} is a
locally finite complex Radon measure $\mu=\sum_\lambda c_\lambda\delta_\lambda$ on
$\R$ with locally finite support, which is a tempered distribution and whose
distributional Fourier transform $\widehat\mu$ is again a measure with locally
finite support. Each crystalline measure is a generalized Poisson summation
formula,
\begin{equation}\label{eq:summation}
 \mu(\widehat\varphi)=\sum_{s}\widehat\mu(\{s\})\,\varphi(s)\qquad(\varphi\in C_c^\infty(\R));
\end{equation}
the right-hand side is a finite sum, and the left-hand side is the distributional
pairing, equal to $\lim_{R\to\infty}\sum_\lambda\chi(\lambda/R)c_\lambda\widehat\varphi(\lambda)$
for any $\chi\in C_c^\infty$ equal to $1$ near $0$ (Lemma~\ref{lem:atomic}). Such
formulas go back to Kahane--Mandelbrojt \cite{KM58} and Guinand \cite{Gui59},
and were revived by the theory of mathematical quasicrystals
\cite{Lag00,BG13,Dys09} and by Meyer's work \cite{Mey16,Mey17a,Mey17b,Mey21,Mey22}.
Only $\mu$ itself is required to be tempered. If $|\mu|$ and $|\widehat\mu|$ are
also tempered, that is, if the total variations of $\mu$ and $\widehat\mu$ grow at
most polynomially, $\mu$ is a \emph{Fourier quasicrystal}
\cite{LO17,KS20,OU20b}; this is the class in which much of the theory takes place
\cite{LO17,KS20,OU20b,ACV24,Fav24a}, and every crystalline measure whose support
and spectrum are separated at a polynomial rate belongs to it \cite[Theorem~1]{Fav24a}.

For a locally finite set $\Lambda\subset\R$, let $\M_\Lambda$ be the space of
crystalline measures with support and spectrum ($=\supp\widehat\mu$) in
$\Lambda$, and $\M^{\mathrm{str}}_\Lambda\subseteq\M_\Lambda$ the subspace of Fourier
quasicrystals. Meyer asked: ``what is the dimension of $\M_\Lambda$?''
\cite[Section~6]{Mey17a}. Kulikov, Nazarov and Sodin \cite[Section~2.3]{KNS25}
consider the two-set version
\begin{equation}\label{eq:N}
 \Ncal(A,B)=\{\mu:\ \mu\text{ crystalline},\ \supp\mu\subseteq B,\ \supp\widehat\mu\subseteq A\},
\end{equation}
so that $\M_\Lambda=\Ncal(\Lambda,\Lambda)$; as they do, we put the spectrum
first. The known answers are small or very large. The integers carry one
crystalline measure up to scalars, $\M_\Z=\C\delta_\Z$; so does
$\{0\}\cup\{\pm\sqrt n:n\ge1\}$, by the interpolation theorem of Radchenko and
Viazovska \cite{RV19} (see \cite[Section~1]{LR21a}); for the shifted square roots
$\{\pm\sqrt{n+\theta}:n\ge0\}$, $0<\theta<1$, about which Meyer also asks
\cite[Section~6]{Mey17a}, the dimension is $2$, $1$ or $0$, according as
$\theta\le\frac18$, $\frac18<\theta\le\frac38$ or $\theta>\frac38$ \cite{PalSSR};
uniformly discrete supports and spectra allow only generalized Dirac combs
\cite{Cor89,LO13,LO15};
and Kulikov, Nazarov and Sodin proved that $\Ncal(\Lambda,M)$ is
finite-dimensional for their subcritical pairs and infinite-dimensional for
their supercritical pairs \cite[Section~2.3, Remark~7.5]{KNS25}. For the
supercritical pairs that are separated at a polynomial rate, these
infinite-dimensional spaces consist of Fourier quasicrystals by Favorov's theorem
(Section~\ref{sec:literature}).

\subsection{Three possible sizes}
``Infinite-dimensional'' hides two different possibilities. For Fourier
quasicrystals the natural grading is by growth. For an integer $N\ge0$ let the
\emph{strong layer} $X_N(\Lambda)$ consist of the $\mu\in\M_\Lambda$ with
\[
 \sum_{x\in\Lambda}\frac{|\mu(\{x\})|}{(1+|x|)^N}
 +\sum_{\xi\in\Lambda}\frac{|\widehat\mu(\{\xi\})|}{(1+|\xi|)^N}<\infty,
\]
so that $\M^{\mathrm{str}}_\Lambda=\bigcup_NX_N(\Lambda)$. The strong layers are Banach
spaces, and a classical argument of Mackey \cite{Mac45} shows
(Proposition~\ref{prop:strongtrichotomy}) that the Hamel dimension of
$\M^{\mathrm{str}}_\Lambda$ is
\begin{enumerate}[label=(\roman*)]
\item finite, if the dimensions of the $X_N(\Lambda)$ are bounded;
\item $\aleph_0$, if they are all finite but unbounded;
\item $\cc=2^{\aleph_0}$, if some $X_N(\Lambda)$ is infinite-dimensional.
\end{enumerate}
(Countable dimension does not mean countably many measures: the coefficients are
complex, so even a one-dimensional space has uncountably many elements. The
dimension counts linearly independent directions.) In case~(iii) one layer
contains infinitely many independent measures, and their convergent infinite
combinations produce continuum many. In case~(ii) every Fourier quasicrystal on
$\Lambda$ is a finite combination of a fixed sequence, each new member of which
grows faster than the previous ones. Constructing infinitely many Fourier
quasicrystals on one set cannot distinguish (ii) from (iii): one has to determine
\emph{all} of them. The question is whether case~(ii) occurs at all.

For the space $\M_\Lambda$ of all crystalline measures the same trichotomy holds with
the strong layers replaced by the layers of fixed Hermite order. Let
\[
 H_s=\Bigl\{\sum_{n\ge0}u_nh_n:\ \sum_{n\ge0}(n+1)^{2s}|u_n|^2<\infty\Bigr\}
 \qquad(s\in\R)
\]
be the Hermite--Sobolev scale built on the Hermite functions $h_n$, so that
$\Sc(\R)=\bigcap_pH_p$ and $\Sc'(\R)=\bigcup_pH_{-p}$ \cite{Sim71}. We say that a
tempered distribution has \emph{Hermite order at most $p$} if it lies in $H_{-p}$;
large Hermite order means large growth or low regularity. The layers
\[
 V_p(\Lambda)=\M_\Lambda\cap H_{-p},\qquad r_p(\Lambda)=\dim V_p(\Lambda),
\]
are closed subspaces of the Hilbert spaces $H_{-p}$, they exhaust $\M_\Lambda$, and
$\dim\M_\Lambda$ is finite, $\aleph_0$ or $\cc$ according as the $r_p$ are bounded,
finite and unbounded, or not all finite (Proposition~\ref{prop:trichotomy}). In
case~(ii) no series in the fixed sequence with infinitely many nonzero coefficients
converges in $\Sc'(\R)$ (Corollary~\ref{cor:nosums}).

\subsection{The main result}
\begin{introthm}\label{thm:D}
There exist a locally finite set $\Lambda\subset\R$, Fourier quasicrystals
$\nu_1,\nu_2,\dots$ with support and spectrum in $\Lambda$, and integers
$2\le s_1<s_2<\cdots$ such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item every Fourier quasicrystal with support and spectrum in $\Lambda$ is a finite
      linear combination of the $\nu_k$, and the $\nu_k$ are linearly independent;
      in particular $\dim_\C\M^{\mathrm{str}}_\Lambda=\aleph_0$;
\item for every integer $N\ge0$,
      \[
       2\,\#\{i:\ s_i+3\le N\}\le\dim X_N(\Lambda)\le2\,\#\{i:\ s_i\le N\};
      \]
\item for every $i\ge1$, the measures $\nu_{2i-1}$ and $\nu_{2i}$ belong to
      $X_{s_i+3}(\Lambda)$ but not to $X_{s_i-1}(\Lambda)$.
\end{enumerate}
\end{introthm}

So case~(ii) occurs for Fourier quasicrystals: $\M^{\mathrm{str}}_\Lambda$ is a complete
space of summation formulas with polynomially growing masses that is
finite-dimensional at every fixed growth exponent and infinite-dimensional overall.
The existence assertion of part~(i) has been formalised in Lean~4
(Section~\ref{subsec:introlean}). Theorem~\ref{thm:D} determines
$\M^{\mathrm{str}}_\Lambda$, but not the space $\M_\Lambda$ of all crystalline measures on
$\Lambda$, which may be larger; we do not know whether $\M_\Lambda$ is also countably
infinite-dimensional (Question~\ref{q:both}).

\subsection{How measures imitate derivatives}\label{subsec:mechanism}
If derivatives of Dirac masses were allowed, case~(ii) would be easy. The
tempered distributions $T$ with $\supp T\subseteq\Z$ and $\supp\widehat T\subseteq\Z$
are the finite combinations of the $x^j\delta_\Z^{(k)}$, $j,k\ge0$
(Example~\ref{ex:Z}, a special case of a theorem of Lev and Reti
\cite[Theorem~1.1]{LR21b}); this space has countable dimension, and the
$k$-th derivatives enter at higher and higher order. A measure cannot
contain derivatives, and indeed $\M_\Z=\C\delta_\Z$.

The starting point of this paper is that clusters of atoms can imitate
derivatives. For $a\in\R$,
\begin{equation}\label{eq:toy}
 \frac{\delta_{a+\eps}-\delta_{a-\eps}}{2\eps}\ \longrightarrow\ -\delta_a'
 \qquad(\eps\to0)\quad\text{in }\Sc'(\R),
\end{equation}
since pairing with $\varphi$ gives the difference quotient
$(\varphi(a+\eps)-\varphi(a-\eps))/(2\eps)\to\varphi'(a)$. In the same way, $m+1$
atoms within distance $\eps$ of $a$, with masses of size $\eps^{-m}$ given by
divided differences, imitate $\delta_a^{(m)}$. Two features of \eqref{eq:toy}
matter below. The more atoms a cluster has, the higher the derivative it can
imitate, hence the higher the order. And the masses are large while the
distribution stays bounded: if the clusters tighten at a polynomial rate, the
masses grow polynomially and the measure can still be a Fourier quasicrystal; if
they tighten faster, the measure stays tempered but its total variation does not.
Meyer already observed, in a variant of Guinand's construction, that ``a
derivative of the Dirac mass at $\pm1/2$ peeps out from the Fourier transform''
\cite{Mey16}; Olevskii and Ulanovskii note that clustering zeros of exponential
polynomials produce derivatives and unbounded coefficients \cite{OU20a}.

Theorem~\ref{thm:D} is proved in Section~\ref{sec:fqcount}, with proofs of the
technical lemmas in Appendix~\ref{app:fq}. Its building blocks are Poisson formulas
on the zero sets of exponential polynomials, in the spirit of Kurasov and Sarnak
\cite{KS20} and Olevskii and Ulanovskii \cite{OU20a}. The atoms of a block lie on
$s$ sheets, the real zeros of
$\prod_{j\le s}p_{a_j}\bigl(e(x/\lambda),e(\beta x/\lambda-\frac14)\bigr)$, where
$p_a(z,w)=1-aw-z(w-a)$ is a Lee--Yang polynomial, $e(t)=e^{2\pi\ii t}$, the $a_j$ are
distinct numbers in $(0,1)$, and $\beta$ is a Liouville number. The sheets never
meet, but along a sequence of points they come within a distance of order
$|x|^{-1}$ of one another, and there the masses are of size $|x|^{s-1}$. These
clusters imitate derivatives at a polynomial rate, so the block measures are
Fourier quasicrystals whose growth exponent is determined by $s$ up to a bounded
error (Figure~\ref{fig:sheets}); the arithmetic of $\beta$ controls the
near-crossings from both sides. The blocks are placed at distinct prime scales and
overlaid with growing inner radii, a device of Kolountzakis \cite[Section~2]{Kol16},
so that the union stays locally finite. Infinitely many independent Fourier
quasicrystals give only a lower bound, and a union of sets may well support new
ones; the heart of the proof is to show that the union carries \emph{nothing else}.
Every finite union of blocks is determined completely, by a uniqueness theorem for
measures on cones at a scale whose square is a quadratic irrational and an algebraic
splitting of rational functions on $\mathbb P^1\times\mathbb P^1$, and the infinite
union is then handled by a compactness argument.

\input{fig_sheets}

\subsection{All crystalline measures}
None of the measures of Theorem~\ref{thm:D} needs masses that grow faster than a
polynomial. If arbitrary crystalline measures are allowed, the clusters can tighten
at any rate, and the countable case also occurs for the whole space $\M_\Lambda$, on
a set where every crystalline measure is determined.

\begin{introthm}\label{thm:A}
There exist a symmetric locally finite set $\Lambda\subset\R$ and nonzero
crystalline measures $\mu_1,\mu_2,\dots$ with support and spectrum in $\Lambda$
such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item every $\mu\in\M_\Lambda$ is a finite linear combination of the $\mu_k$, and
      the $\mu_k$ are linearly independent; in particular $\dim_\C\M_\Lambda=\aleph_0$;
\item every layer $V_p(\Lambda)$ is finite-dimensional, and the dimensions grow
      linearly: for every integer $p\ge1$,
      \begin{equation}\label{eq:rankA}
       2\lo p\le r_p(\Lambda)\le 12p-2,
      \end{equation}
      where $\pfl x=\max\{0,\lfloor x\rfloor\}$;
\item the Hermite orders of the $\mu_k$ grow linearly: for every $i\ge1$, the
      measures $\mu_{2i-1}$ and $\mu_{2i}$ belong to $H_{-(18i+4)}$ but not to
      $H_{-i}$.
\end{enumerate}
\end{introthm}

No nonzero element of this space is a Fourier quasicrystal; in fact neither $\mu$ nor
$\widehat\mu$ is strongly tempered for any nonzero $\mu\in\M_\Lambda$
(Theorem~\ref{thm:nonstrong}). Single crystalline measures with this property were
constructed by Maz\'a\v{c}, Richard and Strungaru \cite[Theorem~4.11,
Remark~5.2]{MRS26}; here it holds for every element of an infinite-dimensional
space, as a consequence of the mechanism above: a cluster that imitates a derivative
of high Hermite order needs masses that are large compared with the distances
within the cluster. Theorems~\ref{thm:D} and~\ref{thm:A} thus show that the
countable case occurs both among Fourier quasicrystals and among crystalline
measures, on different sets, and in Theorem~\ref{thm:A} the whole space $\M_\Lambda$
is determined. The existence assertion of part~(i), without the symmetry of
$\Lambda$, has been formalised in Lean~4 (Section~\ref{subsec:introlean}). Adding the
point $0$ to $\Lambda$, as in Meyer's normalization, changes $\M_\Lambda$ by at most
two dimensions (Proposition~\ref{prop:covariance}).

The set $\Lambda$ of Theorem~\ref{thm:A} is a union of rescaled copies of
one-dimensional ``blocks'' (Figure~\ref{fig:overlay}), and $\mu_{2i-1},\mu_{2i}$ are
the two copies of the measure of the $i$-th block. In a block, the atoms
$n\pm\gamma_j$ have phases $\gamma_j\to\frac12$ at a doubly exponential rate, so that
near every half-integer the atoms of two adjacent cells form a cluster, and cells
farther out carry more atoms (Figure~\ref{fig:block}). We show that each block
carries exactly one crystalline measure up to scalars, that the speed of clustering
controls its Hermite order, and that the block can avoid an arbitrarily large
central interval (Section~\ref{sec:blocks}, with the proof of the main analytic
input in Appendix~\ref{app:rapid}). The blocks are again overlaid with growing inner
radii, and the exhaustion (Section~\ref{sec:exhaustion}) rests on a short
implication. If $T\in\M_\Lambda$ has Hermite order at most $p$, then its component on
each block has Hermite order at most a fixed multiple of $p$ (we obtain $6p$); since
the measure of the $i$-th block has Hermite order larger than $P_i$, this component
vanishes as soon as $P_i\ge6p$. As $P_i\to\infty$, only finitely many blocks
contribute at each order, while every block whose measure has Hermite order at most
$p$ does (this holds when $18P_i+4\le p$). The difficulty lies in the phrase ``the
component of $T$ on a block'': one has to show that restricting the spectrum of $T$
to one block gives a tempered distribution whose inverse Fourier transform lives on
the reciprocal block, with no additional distribution at the half-integer seams
where the atoms accumulate. Random scalings of the blocks, periodic annihilators and
the large gaps between the blocks are used for this. One feature of the
construction is forced: in any countably infinite-dimensional $\Ncal(A,B)$, the
elements of Hermite order at most $p$ are determined by their masses in a bounded
window, whose size tends to infinity with $p$ (Lemma~\ref{lem:window}), so new
elements must appear farther and farther out. What we do not supply, for
either theorem, is a general geometric characterization of the sets with
countably infinite-dimensional spaces (Question~\ref{q:geometry}).

\input{fig_block}
\input{fig_overlay}

The Hermite orders of the blocks can be prescribed, which gives any rate of growth
of the layer dimensions, and the construction has a natural consequence for pairs
of sets.

\begin{introthm}\label{thm:B}
For every nondecreasing unbounded $F:\NN\cup\{0\}\to\NN\cup\{0\}$ with $F(0)=0$
there is a symmetric locally finite $\Lambda$ with $\dim_\C\M_\Lambda=\aleph_0$ and
$F\bigl(\lo p\bigr)-1\le r_p(\Lambda)\le F(6p)$ for all $p\ge1$.
\end{introthm}

The same construction yields $\dim\M_\Lambda=\cc$ when the block orders stay
bounded (Corollary~\ref{cor:exhaust}). Together with $\M_{N^{-1}\Z}$, of
dimension $N^2$ (Example~\ref{ex:Z}), and the subsets considered below, the
dimensions $0$, $N^2$, every even integer, $\aleph_0$ and $\cc$ all occur for spaces
$\M_\Lambda$.

\begin{introcor}\label{thm:C}
There exist disjoint nonempty locally finite sets $A,B\subset\R$ with
$\dim_\C\Ncal(A,B)=\aleph_0$ and $\M_A=\M_B=\{0\}$. Moreover, for the set $\Lambda$
of Theorem~\ref{thm:A} and all $A',B'\subseteq\Lambda$, the space $\Ncal(A',B')$ is
spanned by the $\mu_k$ it contains, so every finite dimension and $\aleph_0$ occur
as $\dim\Ncal(A',B')$.
\end{introcor}

Taking products with the lattice $\Z^{d-1}$ gives the analogous sets in $\R^d$ for
every $d\ge2$, for Theorems~\ref{thm:D} and~\ref{thm:A} alike
(Proposition~\ref{prop:product}).

\subsection{Formal verification}\label{subsec:introlean}
The existence statements of Theorems~\ref{thm:D} and~\ref{thm:A} have been
formalised in Lean~4 with Mathlib and checked by Palomar, a public registry of machine-checked Lean results; each record
fixes a statement of record, a repository at a pinned commit, the toolchain and the
admitted axioms (\texttt{propext}, \texttt{Classical.choice} and \texttt{Quot.sound}).
\begin{itemize}[leftmargin=2em]
\item \emph{Fourier quasicrystals} \cite{Mar26,PalStr}: there is a locally finite
$\Lambda$ with $\dim_\C\M^{\mathrm{str}}_\Lambda=\aleph_0$, with $\M^{\mathrm{str}}_\Lambda$
defined through the strong layers above and Mathlib's Fourier transform. This is the
existence assertion of Theorem~\ref{thm:D}(i). The formal witness is built from
blocks of the kind used in Section~\ref{sec:fqcount}, but with a different choice of
the primes, sheet numbers and parameters, and with radii selected by classical
choice; it is therefore not the set of Theorem~\ref{thm:fqcount}.
\item \emph{All crystalline measures} \cite{MS26,Pal26}: there is a locally finite
$\Lambda$ with $\dim_\C\M_\Lambda=\aleph_0$, every element of $\M_\Lambda$ being a unique
finite combination of a fixed sequence. This is the existence and finite-synthesis
assertion of Theorem~\ref{thm:A}(i), without the symmetry of $\Lambda$.
\end{itemize}
The explicit bounds in parts~(ii) and~(iii) of both theorems, the identification of
the bases, Theorems~\ref{thm:B} and~\ref{thm:nonstrong}, Corollary~\ref{thm:C}, and
Section~\ref{sec:pairs} are not part of either statement of record.
Appendix~\ref{app:lean} describes the two statements of record and indicates where
the main steps of the proofs are formalised.

\subsection{Organization}
The paper has two parts, which share the framework of Section~\ref{sec:prelim}:
the layers, and the trichotomies for Hermite layers and for strong layers.

\emph{Part~I, Fourier quasicrystals (Section~\ref{sec:fqcount} and
Appendix~\ref{app:fq}).} The logical chain is
\[
 \text{Lems.~\ref{lem:beta}--\ref{lem:fqparams}}\ \longrightarrow\
 \text{Prop.~\ref{prop:finiteunion}}\ \longrightarrow\
 \text{Cor.~\ref{cor:finitestrong}}\ \longrightarrow\
 \text{Thm.~\ref{thm:fqcount}}\ \longrightarrow\ \text{Thm.~\ref{thm:D}},
\]
where Proposition~\ref{prop:finiteunion} also uses the uniqueness theorem
Proposition~\ref{prop:noreturn} and the splitting Lemma~\ref{lem:splitting}; a more
detailed guide is given at the beginning of Section~\ref{sec:fqcount}.

\emph{Part~II, all crystalline measures (Sections~\ref{sec:blocks}--\ref{sec:fq}).}
Section~\ref{sec:blocks} constructs the blocks; its main analytic input,
Theorem~\ref{thm:rapid}, is proved in Appendix~\ref{app:rapid}.
Section~\ref{sec:exhaustion} proves Theorems~\ref{thm:A} and~\ref{thm:B}, with
technical lemmas in Appendices~\ref{app:lift} and~\ref{app:sec5}. The logical chain is
\[
 \text{Thm.~\ref{thm:rapid}}\ \longrightarrow\ \text{Prop.~\ref{prop:block}}
 \ \longrightarrow\ \text{Thm.~\ref{thm:exhaust}}\ \longrightarrow\
 \text{Cor.~\ref{cor:exhaust}}\ \longrightarrow\ \text{Thms.~\ref{thm:A}
 and~\ref{thm:B}}.
\]
Section~\ref{sec:pairs} derives Corollary~\ref{thm:C}, a reformulation in terms of
Schwartz space and the analogues in higher dimensions, and Section~\ref{sec:fq}
shows that no nonzero measure of Part~II is a Fourier quasicrystal.

Section~\ref{sec:literature} relates our results to the literature,
Section~\ref{sec:questions} collects open questions, and Appendix~\ref{app:lean}
describes the two formal developments.