\section{Consequences: pairs, Schwartz space and higher dimensions}\label{sec:pairs}

\subsection{Disjoint support and spectrum, and all subpairs}
Let $\Lambda$, $(P_i)$ and $(e_\ell)$ be as in Corollary~\ref{cor:exhaust}, with
$P_i\to\infty$. Recall that $\supp e_\ell\subseteq\Gamma_\ell$ and
$\F e_\ell=\zeta_ie_{\ell^*}$, where $\Gamma_{\ell^*}$ is the reciprocal sector.

\begin{proposition}\label{prop:subpairs}
For all subsets $A',B'\subseteq\Lambda$,
\begin{equation}\label{eq:subpairs}
 \Ncal(A',B')=\bigoplus_{\ell:\ \supp e_\ell\subseteq B',\ \supp\F e_\ell\subseteq A'}\C e_\ell .
\end{equation}
In particular, for unions of sectors $B_I=\bigcup_{\ell\in I}\Gamma_\ell$ and
$A_J=\bigcup_{\ell\in J}\Gamma_\ell$,
\[
 \Ncal(A_J,B_I)=\bigoplus_{\ell\in I,\ \ell^*\in J}\C e_\ell ,
\]
so every finite dimension and $\aleph_0$ occur among the spaces $\Ncal(A',B')$,
$A',B'\subseteq\Lambda$, and every even dimension and $\aleph_0$ occur among the
spaces $\M_{B_I}$ with $I=I^*$.
\end{proposition}

\begin{proof}
Since $A',B'\subseteq\Lambda$, every $T\in\Ncal(A',B')$ lies in $\M_\Lambda$ and is
therefore a finite combination $\sum c_\ell e_\ell$ by \eqref{eq:exhaust}. The
measures $e_\ell$ have pairwise disjoint supports, and so do the $\F e_\ell$. If
$c_\ell\ne0$, then every atom of $e_\ell$ is an atom of $T$, so
$\supp e_\ell\subseteq B'$, and likewise $\supp\F e_\ell\subseteq A'$. Conversely
every $e_\ell$ with these two properties lies in $\Ncal(A',B')$. For unions of
sectors, $\supp e_\ell\subseteq\Gamma_\ell$ is nonempty, so $\supp e_\ell\subseteq B_I$
if and only if $\ell\in I$, and similarly $\supp\F e_\ell\subseteq A_J$ if and only
if $\ell^*\in J$. Taking $J=I^*$ with $I$ finite of any cardinality, or infinite,
gives all finite dimensions and $\aleph_0$; taking $J=I=I^*$ gives the diagonal
statement.
\end{proof}

\begin{theorem}\label{thm:pairs}
For a choice of orientations $\eta=(\eta_i)\in\{+,-\}^{\NN}$ put
\[
 B_\eta=\bigcup_{i\ge1}\Gamma_{(i,\eta_i)},\qquad
 A_\eta=\bigcup_{i\ge1}\Gamma_{(i,-\eta_i)},\qquad u_i=e_{(i,\eta_i)}.
\]
Then $A_\eta$ and $B_\eta$ are disjoint, nonempty, symmetric and locally finite,
$A_\eta\cup B_\eta=\Lambda$, and
\begin{gather*}
 \Ncal(A_\eta,B_\eta)=\bigoplus_{i\ge1}\C u_i,\qquad
 \dim\Ncal(A_\eta,B_\eta)=\aleph_0,\\
 \M_\Lambda=\Ncal(A_\eta,B_\eta)\oplus\Ncal(B_\eta,A_\eta),\qquad
 \M_{A_\eta}=\M_{B_\eta}=\{0\},\\
 \#\{i:18P_i+4\le p\}\le r_p(A_\eta,B_\eta)\le\#\{i:P_i<6p\}\qquad(p\ge1).
\end{gather*}
\end{theorem}

\begin{proof}
Everything except the rank bounds follows from
Proposition~\ref{prop:subpairs}: $\supp u_i\subseteq\Gamma_{(i,\eta_i)}\subseteq B_\eta$
and $\supp\F u_i\subseteq\Gamma_{(i,-\eta_i)}\subseteq A_\eta$, whereas
$e_{(i,-\eta_i)}$ is supported in $A_\eta$, which is disjoint from $B_\eta$; and
an element of $\M_{B_\eta}$ would have to be a combination of $u_i$'s whose
Fourier transforms lie in $A_\eta$, hence zero. The lower rank bound counts the
$u_i$ with $18P_i+4\le p$. For the upper bound, an element of
$V_p(A_\eta,B_\eta)\subseteq V_p(\Lambda)$ is a combination of the $e_\ell$ with
$P_i<6p$ (Corollary~\ref{cor:exhaust}), and only the $u_i$ can occur, one for
each such $i$.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{thm:C}]
Take $P_i=i$ and any $\eta$ in Theorem~\ref{thm:pairs}, and apply
Proposition~\ref{prop:subpairs} with $\mu_{2i-1}=e_{(i,+)}$ and
$\mu_{2i}=e_{(i,-)}$.
\end{proof}

The upper bound $6p-1$ is not obtained by halving the bound $12p-2$ of
Theorem~\ref{thm:A}: projections onto the summands in the decomposition of
$\M_\Lambda$ need not preserve the Hermite order, and the argument uses instead
that an element of Hermite order at most $p$ involves only blocks with $P_i<6p$.
By Proposition~\ref{prop:covariance}(a), translating $A_\eta$ and $B_\eta$
independently, and dilating them by reciprocal factors, gives non-symmetric
examples with the same layer dimensions.

\subsection{Schwartz space modulo the relations}
The countable case has a reformulation in terms of Schwartz space. Let
$\mathcal J(A,B)$ be the closure of $I_B+\F I_A$ in $\Sc$. By the Hahn--Banach
theorem and Proposition~\ref{prop:annihilator}, the dual of the Fr\'echet space
$\Sc/\mathcal J(A,B)$ is $\Ncal(A,B)$. Let $\omega=\C^{\NN}$ be the Fr\'echet space of
all sequences, whose dual is the space of finitely supported sequences.

\begin{proposition}\label{prop:omega}
Let $A,B\subset\R$ be locally finite. Then $\dim\Ncal(A,B)=\aleph_0$ if and only
if $\Sc/\mathcal J(A,B)$ is isomorphic to $\omega$. More precisely, if
$(T_k)_{k\ge1}$ is a Hamel basis of $\Ncal(A,B)$ in case~\textup{(ii)} of
Proposition~\ref{prop:trichotomy}, then $\varphi\mapsto(T_k(\varphi))_{k\ge1}$ is a
continuous linear surjection of $\Sc$ onto $\omega$ with kernel $\mathcal J(A,B)$.
In particular this holds for the sets of Theorems~\ref{thm:A} and~\ref{thm:B} and
Corollary~\ref{thm:C}.
\end{proposition}

\begin{proof}
If $\Sc/\mathcal J\cong\omega$, then $\Ncal(A,B)\cong\omega'$ has dimension
$\aleph_0$. Conversely let $(T_k)$ be a Hamel basis in case~(ii). The map
$\Phi:\Sc/\mathcal J\to\omega$, $\Phi(\varphi+\mathcal J)=(T_k(\varphi))_k$, is
continuous, and it is injective because $\mathcal J=\Ncal(A,B)^\perp$ by the
Hahn--Banach theorem. Every continuous seminorm on $\Sc/\mathcal J$ is dominated by
a multiple of the quotient seminorm $q_p$ of some $\norm\cdot_p$, and the
functionals in $(\Sc/\mathcal J)'$ that are bounded with respect to $q_p$ form the
space $V_p(A,B)$, which is finite-dimensional in case~(ii). By Eidelheit's theorem
\cite{Eid36}, see \cite[Satz~26.27]{MV92}, $\Phi$ is surjective, and by the open
mapping theorem it is an isomorphism.
\end{proof}

\begin{remark}\label{rem:omega}
More generally, a Fr\'echet space $E$ whose dual has countably infinite Hamel
dimension is isomorphic to $\omega$: for each zero-neighbourhood $U$, the
functionals bounded on $U$ form a Banach space of countable Hamel dimension, which
is therefore finite-dimensional, and Eidelheit's theorem applies as above. Thus in
case~(ii) the countably many independent summation formulas $T_k$ are independent
in the strongest sense: their values on Schwartz functions can be prescribed
arbitrarily, and they are the only relations. This is a dual analogue of the
codimension-one statement of Radchenko and Viazovska \cite[Theorem~2]{RV19}: it
does not describe the image of the data map
$\varphi\mapsto(\varphi|_\Lambda,\widehat\varphi|_\Lambda)$, but it shows that, modulo the
closure of $I_\Lambda+\F I_\Lambda$, Schwartz space is isomorphic to $\omega$.
\end{remark}

\subsection{Higher dimensions}\label{subsec:higher}
For $d\ge1$ we use $\F f(\xi)=\int_{\R^d}f(x)e^{-2\pi\ii x\cdot\xi}\,dx$ on $\Sc(\R^d)$
and $\Sc'(\R^d)$. For locally finite $A,B\subset\R^d$ (meeting every bounded set in a
finite set) we let $\Ncal(A,B)$ be the space of tempered distributions on $\R^d$
that are locally atomic on $B$ and whose Fourier transforms are locally atomic on
$A$, and we put $\M_\Lambda=\Ncal(\Lambda,\Lambda)$; Fourier quasicrystals and
$\M^{\mathrm{str}}_\Lambda$ are defined as in Section~\ref{subsec:strong}, with the
weights $(1+|x|)^{-N}$ on $\R^d$. Let $\delta_{\Z^{d-1}}=\sum_{n\in\Z^{d-1}}\delta_n$, so
that $\F\delta_{\Z^{d-1}}=\delta_{\Z^{d-1}}$ by Poisson's formula.

\begin{proposition}\label{prop:product}
Let $A,B\subset\R$ be locally finite and $d\ge2$. Then
$\mu\mapsto\mu\otimes\delta_{\Z^{d-1}}$ is a linear isomorphism of $\Ncal(A,B)$ onto
$\Ncal(A\times\Z^{d-1},B\times\Z^{d-1})$, and $\mu\otimes\delta_{\Z^{d-1}}$ is a Fourier
quasicrystal if and only if $\mu$ is one.
\end{proposition}

\begin{proof}
Since $\F(\mu\otimes\delta_{\Z^{d-1}})=\widehat\mu\otimes\delta_{\Z^{d-1}}$, the map is well
defined; it is linear and injective. Let
$T\in\Ncal(A\times\Z^{d-1},B\times\Z^{d-1})$ and $2\le j\le d$. The function
$e^{2\pi\ii\xi_j}-1$ vanishes on $A\times\Z^{d-1}$, where $\F T$ is locally atomic, so
$(e^{2\pi\ii\xi_j}-1)\F T=0$; that is, $T$ is invariant under the translation by the
$j$-th unit vector. Hence the mass of $T$ at $(b,n)$ does not depend on
$n\in\Z^{d-1}$; call it $c_b$, and put $\mu=\sum_{b\in B}c_b\delta_b$. Let
$\psi\in C_c^\infty(\R^{d-1})$ be supported in the ball of radius $\frac12$ about $0$,
with $\psi(0)=1$. Then $\mu(\varphi)=T(\varphi\otimes\psi)$ for $\varphi\in C_c^\infty(\R)$,
and the right-hand side is continuous on $\Sc(\R)$, so $\mu$ is tempered; as
$\mu\otimes\delta_{\Z^{d-1}}$ and $T$ agree on $C_c^\infty(\R^d)$, they are equal. Hence
$\F T=\widehat\mu\otimes\delta_{\Z^{d-1}}$ and $\widehat\mu(\varphi)=\F T(\varphi\otimes\psi)$, and
since $\F T$ is locally atomic on $A\times\Z^{d-1}$, $\widehat\mu$ is locally atomic on $A$.
Thus $\mu\in\Ncal(A,B)$. Finally
$|\mu\otimes\delta_{\Z^{d-1}}|=|\mu|\otimes\delta_{\Z^{d-1}}$, and
$\sum_{n\in\Z^{d-1}}(1+|(x,n)|)^{-N}$ is comparable to $(1+|x|)^{d-1-N}$ for $N>d$, so
$\mu\otimes\delta_{\Z^{d-1}}$ is strongly tempered if and only if $\mu$ is; the same
holds for the Fourier transforms.
\end{proof}

\begin{corollary}\label{cor:higher}
For every $d\ge1$ there are a symmetric locally finite set $\Lambda\subset\R^d$ with
$\dim\M_\Lambda=\aleph_0$, disjoint locally finite sets $A,B\subset\R^d$ with
$\dim\Ncal(A,B)=\aleph_0$ and $\M_A=\M_B=\{0\}$, and a locally finite set
$\Lambda'\subset\R^d$ with $\dim\M^{\mathrm{str}}_{\Lambda'}=\aleph_0$.
\end{corollary}

\begin{proof}
Apply Proposition~\ref{prop:product} to the sets of Theorems~\ref{thm:A}
and~\ref{thm:D} and Corollary~\ref{thm:C}; the product of a symmetric set with
$\Z^{d-1}$ is symmetric.
\end{proof}

\begin{remark}\label{rem:higher}
Define $H_s(\R^d)$ through the products $h_\alpha$ of Hermite functions and the
weights $(|\alpha|+1)^{2s}$, $|\alpha|=\alpha_1+\dots+\alpha_d$. Since
$\max\{\alpha_1+1,|\alpha'|+1\}\le|\alpha|+1\le(\alpha_1+1)(|\alpha'|+1)$ for
$\alpha=(\alpha_1,\alpha')$, the map of Proposition~\ref{prop:product} sends
$\Ncal(A,B)\cap H_{-p}$ into $H_{-(p+q)}$ when $\delta_{\Z^{d-1}}\in H_{-q}(\R^{d-1})$, and its
inverse sends elements of $H_{-p}$ to $H_{-p}$. Hence the layer dimensions in $\R^d$
and in $\R$ agree up to the fixed shift $q$ of the order, and the layer bounds of
Theorems~\ref{thm:A} and~\ref{thm:B} transfer with this shift. The proof of
Proposition~\ref{prop:trichotomy} applies verbatim in $\R^d$.
\end{remark}