\section{The functional-analytic framework}\label{sec:prelim}

\subsection{Conventions}
We write $\widehat f=\F f$, $\F f(\xi)=\int_\R f(x)e^{-2\pi\ii x\xi}\,dx$, on
$\Sc=\Sc(\R)$ and extend $\F$ to $\Sc'=\Sc'(\R)$ by $(\F T)(\varphi)=T(\F\varphi)$,
using the complex-bilinear pairing. Thus $\F^2T$ is the reflection
$\check T(\varphi)=T(\varphi(-\,\cdot))$. For $h\in\R$ let $\tau_hf(x)=f(x+h)$,
extended to $\Sc'$ by transposition, so that
$\F(e^{2\pi\ii\omega x}T)=\tau_{-\omega}\F T$. For $t>0$ let
\[
 (D_tT)(\varphi)=\sqrt t\,T\bigl(\varphi(t\,\cdot)\bigr),
\]
so that $D_tf(x)=t^{-1/2}f(x/t)$ on functions, $D_t\delta_x=\sqrt t\,\delta_{tx}$,
and $\F D_t=D_{1/t}\F$.

Let $(h_n)_{n\ge0}$ be the Hermite functions, normalized in $L^2(\R)$, so that
$\F h_n=(-\ii)^nh_n$, and let $H_s$ be the Hermite--Sobolev spaces defined in
the introduction, with norm $\norm u_s^2=\sum_n(n+1)^{2s}|u_n|^2$. Then
$\Sc=\bigcap_{p\in\NN}H_p$ with its Fr\'echet topology, $\Sc'=\bigcup_{p}H_{-p}$,
and the pairing of $H_{-p}$ with $H_p$ extends the distributional pairing,
$|T(\varphi)|\le\norm T_{-p}\norm\varphi_p$ \cite{Sim71}, \cite[Section~V.3 and
its appendix]{RS80}. For integers $p\ge0$ the norm $\norm\cdot_p$ is equivalent
to $\bigl(\sum_{a+b\le2p}\norm{x^a\partial^b\varphi}_{L^2}^2\bigr)^{1/2}$. The
Fourier transform is unitary on every $H_s$. Consequently translations,
modulations $e^{2\pi\ii ax}$, and dilations $D_t$ with $t$ in a compact subset of
$(0,\infty)$ are bounded on $H_{\pm p}$ for every integer $p$, with constants
that are locally uniform in the parameters; translation by $h$ has norm at most
$C_p(1+|h|)^{2p}$ on $H_{\pm p}$. A tempered distribution has \emph{Hermite order
at most $p$} if it belongs to $H_{-p}$. The word \emph{order} without
qualification refers to the classical order of a distribution (the number of
derivatives in a local representation).

A set $C\subset\R$ is \emph{locally finite} if it meets every bounded interval in
a finite set; it is then closed and at most countable. A tempered distribution
$T$ is \emph{locally atomic on $C$} if there are coefficients $a_c\in\C$ such
that $T(f)=\sum_{c\in C}a_cf(c)$ for all $f\in C_c^\infty(\R)$ (a finite sum
for each $f$); the $a_c$ are the \emph{masses} of $T$. For locally finite $A,B$
we let $\Ncal(A,B)$ be the space of $T\in\Sc'$ that are locally atomic on $B$
and whose Fourier transform is locally atomic on $A$; this is the space
\eqref{eq:N} of crystalline measures with support in $B$ and spectrum in $A$.
We put $\M_\Lambda=\Ncal(\Lambda,\Lambda)$,
\[
 V_p(A,B)=\Ncal(A,B)\cap H_{-p},\qquad r_p(A,B)=\dim V_p(A,B),
\]
where $r_p$ is a Hilbert dimension (which coincides with the Hamel dimension when
it is finite), and $V_p(\Lambda)=V_p(\Lambda,\Lambda)$, $r_p(\Lambda)=r_p(\Lambda,\Lambda)$.
Unless stated otherwise, $\dim$ of $\Ncal(A,B)$ denotes the complex Hamel
dimension. A measure $\nu$ is \emph{strongly tempered} if
$\int(1+|x|)^{-N}\,d|\nu|(x)<\infty$ for some $N\ge0$; equivalently,
$|\nu|([-R,R])$ grows at most polynomially in $R$.

\subsection{The annihilator description}
For a locally finite set $C$ let
$I_C=\{\varphi\in\Sc:\varphi(c)=0\text{ for all }c\in C\}$.
It is essential that $I_C$ is defined by the vanishing of values: distributions
supported in $C$ may contain derivatives of Dirac masses, and these are not
annihilated by $I_C$.

\begin{lemma}\label{lem:atomic}
Let $C$ be locally finite. A tempered distribution $T$ is locally atomic on $C$
if and only if $T(\varphi)=0$ for every $\varphi\in I_C$. In that case
$T(\varphi)=\lim_{R\to\infty}\sum_{c\in C}a_c\chi(c/R)\varphi(c)$ for every
$\varphi\in\Sc$ and every $\chi\in C_c^\infty$ equal to $1$ near $0$.
\end{lemma}

\begin{proof}
Suppose $T$ is locally atomic on $C$. For $\varphi\in\Sc$ let
$\chi_R(x)=\chi(x/R)$; then $T(\chi_R\varphi)=\sum_ca_c\chi_R(c)\varphi(c)$, and
$\chi_R\varphi\to\varphi$ in $\Sc$, which proves the last formula; in particular
$T(\varphi)=0$ if $\varphi\in I_C$. Conversely, suppose $T(I_C)=0$. For $c\in C$
choose $\eta_c\in C_c^\infty$ with $\eta_c(c)=1$ whose support contains no other
point of $C$. For $f\in C_c^\infty$, the sum $\sum_cf(c)\eta_c$ is finite and
$f-\sum_cf(c)\eta_c\in I_C$. Hence $T(f)=\sum_cT(\eta_c)f(c)$, and $T(\eta_c)$
does not depend on the choice of $\eta_c$.
\end{proof}

\begin{proposition}\label{prop:annihilator}
For locally finite $A,B\subset\R$,
\[
 \Ncal(A,B)=(I_B+\F I_A)^\perp=\{T\in\Sc':T(\varphi)=0\text{ for all }
 \varphi\in I_B+\F I_A\}.
\]
In particular $\Ncal(A,B)\ne\{0\}$ if and only if the closure of $I_B+\F I_A$ in
$\Sc$ is a proper subspace.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{lem:atomic} to $T$ and $C=B$, and to $\F T$ and $C=A$; the
second condition reads $T(\F\psi)=0$ for $\psi\in I_A$. The last assertion
follows from the Hahn--Banach theorem.
\end{proof}

Thus an element of $\Ncal(A,B)$ is the same as a continuous linear functional on
$\Sc$ that depends only on $\varphi|_B$ and also only on $(\F^{-1}\varphi)|_A$;
it encodes the summation formula \eqref{eq:summation}.

\subsection{Layers and the trichotomy}
Lemma~\ref{lem:layers}, Proposition~\ref{prop:trichotomy}, Lemma~\ref{lem:BS} and
Corollary~\ref{cor:nosums} use only that $\Ncal(A,B)$ is a weak$^*$-closed subspace
of $\Sc'$, and they hold, with the same proofs, for every such subspace (for the
upper bound in Proposition~\ref{prop:trichotomy} one uses that $\Sc'$ has
cardinality $\cc$). Lemma~\ref{lem:window} is specific to $\Ncal(A,B)$.

\begin{lemma}\label{lem:layers}
Each $V_p(A,B)$ is a closed subspace of $H_{-p}$. The spaces $V_p(A,B)$
increase with $p$ and their union is $\Ncal(A,B)$. Moreover $r_p(A,B)$ equals
the codimension in $H_p$ of the $H_p$-closure of $I_B+\F I_A$.
\end{lemma}

\begin{proof}
For fixed $\varphi\in\Sc$ the map $T\mapsto T(\varphi)$ is continuous on
$H_{-p}$, so by Proposition~\ref{prop:annihilator} $V_p(A,B)$ is an intersection
of closed hyperplanes. The inclusions $H_{-p}\subset H_{-p-1}$ preserve the
distribution represented, and $\Sc'=\bigcup_pH_{-p}$. Finally $V_p(A,B)$ is the
annihilator in $H_{-p}=(H_p)'$ of $I_B+\F I_A$, hence is isomorphic to the dual
of the quotient of $H_p$ by the closure of $I_B+\F I_A$.
\end{proof}

\begin{proposition}\label{prop:trichotomy}
Let $A,B\subset\R$ be locally finite. Exactly one of the following holds.
\begin{enumerate}[label=\textup{(\roman*)}]
\item All $r_p(A,B)$ are finite and bounded. Then $V_p(A,B)=\Ncal(A,B)$ for all
large $p$, and $\dim\Ncal(A,B)=\sup_pr_p(A,B)<\infty$.
\item All $r_p(A,B)$ are finite and $\sup_pr_p(A,B)=\infty$. Then
$\dim\Ncal(A,B)=\aleph_0$.
\item Some $r_p(A,B)$ is infinite. Then $\dim\Ncal(A,B)=\cc$.
\end{enumerate}
\end{proposition}

\begin{proof}
If all layers are finite-dimensional, the union of bases of the layers is a
countable spanning set of $\Ncal(A,B)$. If the ranks are bounded, the increasing
sequence of layers stabilizes, which gives~(i); otherwise $\Ncal(A,B)$ is
infinite-dimensional and countably spanned, which gives~(ii). In case~(iii) let
$(e_n)$ be an orthonormal sequence in an infinite-dimensional layer $V_p$. For
$0<t<1$ the vector $v_t=\sum_nt^ne_n$ lies in $V_p$, and pairing a relation
between $v_{t_1},\dots,v_{t_k}$ ($t_j$ distinct) with $e_0,\dots,e_{k-1}$ gives
a Vandermonde system; hence the $v_t$ are linearly independent. This is
Mackey's argument \cite[Theorem~I-1]{Mac45}; see also \cite{Lac73}. Conversely,
an element of $\Ncal(A,B)$ is determined by its masses on the countable set $B$
(Lemma~\ref{lem:atomic}), so $|\Ncal(A,B)|\le|\C^B|\le\cc$.
\end{proof}

\subsection{The countable case}
\begin{lemma}\label{lem:BS}
Let $T_n\in\Ncal(A,B)$ and suppose that $T_n(\varphi)$ converges for every
$\varphi\in\Sc$. Then there is $p$ such that all $T_n$ and the limit
$T=\lim_nT_n$ belong to $V_p(A,B)$.
\end{lemma}

\begin{proof}
Since $\Sc$ is a Fr\'echet space, the Banach--Steinhaus theorem shows that
$\{T_n\}$ is equicontinuous: there are $C$ and $p$ with
$|T_n(\varphi)|\le C\norm\varphi_p$ for all $n$ and $\varphi$. Hence all $T_n$,
and $T$, lie in the ball of radius $C$ of $H_{-p}$, and $T\in\Ncal(A,B)$ because
$\Ncal(A,B)$ is an annihilator.
\end{proof}

\begin{corollary}\label{cor:nosums}
Suppose $\dim\Ncal(A,B)=\aleph_0$, and let $(\nu_k)$ be a basis of $\Ncal(A,B)$
such that each $H_{-p}$ contains only finitely many $\nu_k$. If the partial sums
of $\sum_kc_k\nu_k$ converge in $\Sc'$, then $c_k=0$ for all but finitely many
$k$.
\end{corollary}

\begin{proof}
By Lemma~\ref{lem:BS} all partial sums lie in some $V_p$, hence so does
$c_k\nu_k$ for every $k$, and $c_k=0$ whenever $\nu_k\notin H_{-p}$.
\end{proof}

\begin{lemma}\label{lem:window}
Let $V_p(A,B)$ be finite-dimensional. Then there is $R>0$ such that every
$T\in V_p(A,B)$ is determined by its masses at the points of $B\cap(-R,R)$; in
particular $r_p(A,B)\le\#(B\cap(-R,R))$. If $\dim\Ncal(A,B)=\aleph_0$ and $R_p$
denotes the infimum of such $R$, then $R_p\to\infty$.
\end{lemma}

\begin{proof}
For $R\in\NN$ let $K_R$ be the subspace of $T\in V_p(A,B)$ with no mass in
$B\cap(-R,R)$. These finite-dimensional spaces decrease in $R$, so they are
eventually constant, and their intersection is $\{0\}$ because an element of
$\Ncal(A,B)$ is determined by its masses. In case~(ii) the ranks $r_p$ are
unbounded, so $R_p\to\infty$.
\end{proof}

In words: in a countably infinite-dimensional space the elements of bounded
Hermite order are detected in a bounded window, and new independent elements
must appear farther and farther from the origin. In the construction of
Section~\ref{sec:exhaustion} the
elements of Hermite order at most $p$ are combinations of the generators of
finitely many blocks, whose inner radii tend to infinity.

\subsection{Strong layers}\label{subsec:strong}
For $T\in\Ncal(A,B)$ we write $T(\{x\})$ for the mass of $T$ at $x$. For an integer
$N\ge0$ put
\[
 \norm T_{(N)}=\sum_{x\in B}\frac{|T(\{x\})|}{(1+|x|)^N}
 +\sum_{\xi\in A}\frac{|\F T(\{\xi\})|}{(1+|\xi|)^N}\in[0,\infty],
\]
and let $X_N(A,B)=\{T\in\Ncal(A,B):\norm T_{(N)}<\infty\}$ and
$\Ncal^{\mathrm{str}}(A,B)=\bigcup_NX_N(A,B)$. Thus $\Ncal^{\mathrm{str}}(A,B)$ is the
space of Fourier quasicrystals with support in $B$ and spectrum in $A$. We write
$X_N(\Lambda)=X_N(\Lambda,\Lambda)$ and $\M^{\mathrm{str}}_\Lambda=\Ncal^{\mathrm{str}}(\Lambda,\Lambda)$. If
$\norm T_{(N)}<\infty$, then $T$ is the measure $\sum_xT(\{x\})\delta_x$ and
$T(\varphi)=\sum_xT(\{x\})\varphi(x)$ for $\varphi\in\Sc$, with absolute convergence, by
Lemma~\ref{lem:atomic} and dominated convergence; the same holds for $\F T$.

\begin{proposition}\label{prop:strongtrichotomy}
Let $A,B\subset\R$ be locally finite. Each $X_N(A,B)$ is a Banach space with the
norm $\norm\cdot_{(N)}$, and $X_N(A,B)\subseteq X_{N+1}(A,B)$. Exactly one of the
following holds.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The dimensions of the $X_N(A,B)$ are finite and bounded. Then
$\dim\Ncal^{\mathrm{str}}(A,B)=\sup_N\dim X_N(A,B)<\infty$.
\item The dimensions of the $X_N(A,B)$ are finite and unbounded. Then
$\dim\Ncal^{\mathrm{str}}(A,B)=\aleph_0$.
\item Some $X_N(A,B)$ is infinite-dimensional. Then
$\dim\Ncal^{\mathrm{str}}(A,B)=\cc$.
\end{enumerate}
\end{proposition}

\begin{proof}
Since $(1+|x|)^{-N-1}\le(1+|x|)^{-N}$, the layers increase with $N$, and
$\norm\cdot_{(N)}$ is a norm because an element of $\Ncal(A,B)$ is determined by its
masses. Let $(T_n)$ be a Cauchy sequence in $X_N(A,B)$. The records
$(T_n(\{x\}))_{x\in B}$ and $(\F T_n(\{\xi\}))_{\xi\in A}$ converge in the weighted
$\ell^1$ norms to limits $(c_x)$ and $(d_\xi)$, and we put $\mu=\sum_xc_x\delta_x$ and
$\nu=\sum_\xi d_\xi\delta_\xi$. For $\varphi\in\Sc$ the function
$(1+|x|)^N\F\varphi$ is bounded, so
$T_n(\F\varphi)=\sum_xT_n(\{x\})\F\varphi(x)\to\mu(\F\varphi)$, and likewise
$\F T_n(\varphi)\to\nu(\varphi)$. As $T_n(\F\varphi)=\F T_n(\varphi)$, we get $\F\mu=\nu$.
Hence $\mu\in X_N(A,B)$ and $T_n\to\mu$, so $X_N(A,B)$ is complete. It is
separable, being isometric to a subspace of a weighted $\ell^1$ space over the
countable set $B\sqcup A$.

If all layers are finite-dimensional, cases~(i) and~(ii) follow as in the proof of
Proposition~\ref{prop:trichotomy}. In case~(iii), an infinite-dimensional
separable Banach space has Hamel dimension $\cc$ \cite{Lac73}, so
$\dim\Ncal^{\mathrm{str}}(A,B)\ge\cc$, and the upper bound follows from
$|\Ncal(A,B)|\le\cc$.
\end{proof}

The Hermite layers $V_p$ and the strong layers $X_N$ measure different things,
growth in the sense of distributions and growth of the total variation.
Theorem~\ref{thm:D} gives a set for which Proposition~\ref{prop:strongtrichotomy} is
in case~(ii). By Theorem~\ref{thm:nonstrong}, the set of Theorem~\ref{thm:A} lies in
case~(ii) of Proposition~\ref{prop:trichotomy} and in case~(i), with
$\M^{\mathrm{str}}_\Lambda=\{0\}$, of Proposition~\ref{prop:strongtrichotomy}.

\subsection{Covariance and finite modifications}

\begin{proposition}\label{prop:covariance}
Let $A,B\subset\R$ be locally finite.
\begin{enumerate}[label=\textup{(\alph*)}]
\item For $a,b\in\R$ and $t>0$, the map induced by
$f(x)\mapsto e^{2\pi\ii ax}t^{-1/2}f((x-b)/t)$ is a linear isomorphism
$\Ncal(A,B)\to\Ncal(a+t^{-1}A,\,b+tB)$ which maps $V_p$ onto $V_p$ for every
integer $p$.
\item $\F$ is a linear isomorphism $\Ncal(A,B)\to\Ncal(-B,A)$ mapping $V_p$ onto
$V_p$. In particular $\F\M_\Lambda=\M_\Lambda$ if $\Lambda=-\Lambda$.
\item Let $A_1,B_1\subset\R$ be locally finite with
$k=\#(A_1\triangle A)+\#(B_1\triangle B)<\infty$. Then
$\Ncal(A\cap A_1,B\cap B_1)$ has codimension at most $k$ in both
$\Ncal(A,B)$ and $\Ncal(A_1,B_1)$, and the same holds for every layer $V_p$. In
particular $\Ncal(A,B)$ and $\Ncal(A_1,B_1)$ are both finite-dimensional, both
of dimension $\aleph_0$, or both of dimension $\cc$.
\end{enumerate}
\end{proposition}

\begin{proof}
(a) The map sends an atom of $T$ at $x$ to $b+tx$, and by the computation
$\F[e^{2\pi\ii ax}t^{-1/2}f((\cdot-b)/t)](\xi)=t^{1/2}e^{-2\pi\ii b(\xi-a)}\widehat
f(t(\xi-a))$ an atom of $\F T$ at $\xi$ to $a+\xi/t$; its inverse is of the same
form, and it preserves $H_{-p}$ by the boundedness of translations, modulations
and dilations. (b) $\F$ preserves $H_{-p}$ and $\F^2$ is the reflection. (c)
$\Ncal(A\cap A_1,B\cap B_1)$ is the kernel of the map on $\Ncal(A,B)$ recording
the masses of $T$ on the finite set $B\setminus B_1$ and of $\F T$ on
$A\setminus A_1$; the same argument applies to $\Ncal(A_1,B_1)$ and to each
layer.
\end{proof}

\subsection{Crystalline distributions on $\Z$}
If derivatives of Dirac masses are admitted, countable dimension occurs already
for the lattice. The following is a special case of
\cite[Theorem~1.1]{LR21b}; we include the short argument.

\begin{example}\label{ex:Z}
Let $\mathcal D_\Z$ be the space of $T\in\Sc'$ with $\supp T\subseteq\Z$ and
$\supp\widehat T\subseteq\Z$. Then $\mathcal D_\Z$ is the linear span of the
linearly independent distributions $x^j\delta_\Z^{(k)}$, $j,k\ge0$; hence
$\dim\mathcal D_\Z=\aleph_0$, and every layer $\mathcal D_\Z\cap H_{-p}$ is
finite-dimensional. On the other hand $\M_\Z=\C\delta_\Z$.

Indeed, $\F(x^jT)=(\ii/2\pi)^j\partial^j\F T$ and
$\F(\delta_\Z^{(k)})=(2\pi\ii\xi)^k\delta_\Z$, so $\Span\{x^j\delta_\Z^{(k)}\}$ is
invariant under $\F$. Conversely, if $T\in\mathcal D_\Z$, then $\widehat T$ has
finite order $K$, so it is annihilated by $(e^{2\pi\ii\xi}-1)^{K+1}$, i.e.\
$(\tau_1-I)^{K+1}T=0$; an induction on the exponent, using that
$(\tau_1-I)^{L-1}T$ is a $1$-periodic distribution supported in $\Z$, shows that
$T$ is a polynomial combination of the $\delta_\Z^{(k)}$. Linear independence
follows by comparing the coefficients of $\delta_n^{(k)}$, which are polynomials
in $n$. Each layer is a closed subspace of $H_{-p}$ of countable Hamel dimension
and is therefore finite-dimensional (Proposition~\ref{prop:trichotomy}).
Finally, if $T\in\mathcal D_\Z$ is a measure, the difference equation shows that
its masses at $n\in\Z$ are given by a polynomial $q(n)$, so $T=q(x)\delta_\Z$, and
$\F T=q(\ii\partial/2\pi)\delta_\Z$ is a measure only if $q$ is constant.

More generally, for $N\ge1$ the space $\M_{N^{-1}\Z}$ consists of the
$N$-periodic measures supported in $N^{-1}\Z$, and has dimension $N^2$. Indeed,
if $\mu\in\M_{N^{-1}\Z}$, the continuous function $e^{2\pi\ii N\xi}-1$ vanishes on
$N^{-1}\Z$, so it annihilates the measure $\widehat\mu$, i.e.\ $\tau_N\mu=\mu$.
Conversely, the Fourier transform of $\sum_{t\in\Z}\delta_{a+Nt}$ is
$N^{-1}e^{-2\pi\ii a\xi}\sum_{k\in\Z}\delta_{k/N}$, so the $N^2$ linearly
independent measures $\sum_{t\in\Z}\delta_{r/N+Nt}$, $0\le r<N^2$, belong to
$\M_{N^{-1}\Z}$ and span it.
\end{example}