\section{Questions}\label{sec:questions}

\begin{question}\label{q:both}
Let $\Lambda$ be the set of Theorem~\ref{thm:D}. Is $\dim\M_\Lambda=\aleph_0$? More
generally, is there a locally finite set $\Lambda$ for which both $\M_\Lambda$ and
$\M^{\mathrm{str}}_\Lambda$ have dimension $\aleph_0$?
\end{question}

For the sets of Theorems~\ref{thm:A} and~\ref{thm:B} we have
$\M^{\mathrm{str}}_\Lambda=\{0\}$ (Theorem~\ref{thm:nonstrong}), and for the set of
Theorem~\ref{thm:D} the space $\M^{\mathrm{str}}_\Lambda$ is determined but $\M_\Lambda$ is
not. If $A$ and $B$ are separated at a polynomial rate, every element of
$\Ncal(A,B)$ is a Fourier quasicrystal by Favorov's theorem
\cite[Theorem~1]{Fav24a}, so that $\Ncal(A,B)=\Ncal^{\mathrm{str}}(A,B)$. If, moreover,
the growth exponents $\inf\{N:\int(1+|x|)^{-N}d|\mu|<\infty\}$ of the elements of
$\Ncal(A,B)$ and of their Fourier transforms were bounded, then $\Ncal(A,B)$ would be
contained in a single strong layer $X_N(A,B)$, and
Proposition~\ref{prop:strongtrichotomy} would exclude the countable case.
Theorem~\ref{thm:D} shows that these exponents can be unbounded on one set, but its
set is not separated at a polynomial rate (Remark~\ref{rem:fqcount}). This suggests
the following question.

\begin{question}\label{q:separated}
Let $A,B\subset\R$ be locally finite and separated at a polynomial rate. Is
$\dim\Ncal(A,B)$ always finite or equal to $\cc$?
\end{question}

Under strong separation assumptions every element of $\Ncal(A,B)$ is a
generalized Dirac comb: when $A$ and $B$ are uniformly discrete \cite{LO15}, and
when the support is uniformly discrete and the spectrum is relatively separated
\cite[Theorem~2.2]{LO17}; see \cite[Theorem~6.1]{LR21b} for distributions. A
natural test case for Question~\ref{q:separated} and for Question~\ref{q:geometry}
below is Meyer's set
$\Lambda=\bigcup_{j\ge0}\{x\in2^{-j}\Z:\ |x|>2^{j-2}\}$, which carries
Kolountzakis's crystalline measure and for which $(\Lambda,\Lambda)$ is a uniqueness
pair for $\Sc$ \cite[Section~8]{Mey22}; it is separated at a polynomial rate.

\begin{question}
Can Theorem~\ref{thm:A} be achieved with a set $\Lambda$ given by an explicit
formula, without the probabilistic choice of the scales $s_i$ of
Section~\ref{sec:exhaustion} and the adaptive choice of the radii $R_i$? Can the layer dimensions be prescribed exactly: for
which nondecreasing unbounded $F$ is there a locally finite $\Lambda$ with
$r_p(\Lambda)=F(p)$ for all $p\ge1$?
\end{question}

Theorem~\ref{thm:B} realizes every growth rate up to the changes of order
$p\mapsto\lo p$ and $p\mapsto6p$, which come from the window $[P,18P+4]$ for the
Hermite order of a block and from the loss in \eqref{eq:selector}; in particular
the minimal Hermite orders of the basis elements can be made to grow arbitrarily
slowly. The question is whether these losses can be removed, for instance by blocks
whose Hermite order is known exactly. In the construction of Theorem~\ref{thm:D} no
random choices are made, but the radii are again chosen adaptively, by a
compactness argument.

\begin{question}\label{q:geometry}
Find geometric conditions on $\Lambda$ that characterize each of the three
alternatives of Propositions~\ref{prop:trichotomy} and~\ref{prop:strongtrichotomy}.
Lemma~\ref{lem:window} shows that countable dimension requires new elements to
appear at larger and larger distances, and Favorov's theorem
\cite[Theorem~1]{Fav24a} shows that measures that are not Fourier quasicrystals,
such as those of Theorem~\ref{thm:nonstrong}, require a super-polynomial
coalescence of atoms. The set of Theorem~\ref{thm:D} also contains points at
super-polynomially small distances, in the cone $C_\beta/c$, although the atoms of
each of its blocks are separated at a polynomial rate; we do not know whether such
points are necessary for $\dim\M^{\mathrm{str}}_\Lambda=\aleph_0$.
\end{question}

Meyer \cite{Mey23} describes the theory of crystalline measures in several
variables as being in its infancy; see also \cite{Mey23b} and, for Fourier
interpolation in higher dimensions, \cite{Sto21,RSt22}. Products with lattices give
analogues of Theorems~\ref{thm:A}, \ref{thm:B} and~\ref{thm:D} and of
Corollary~\ref{thm:C} in $\R^d$ (Proposition~\ref{prop:product}), but every element
of the resulting spaces is periodic in $d-1$ directions.

\begin{question}\label{q:higher}
Is there, for some $d\ge2$, a locally finite set $\Lambda\subset\R^d$ with
$\dim\M_\Lambda=\aleph_0$, or with $\dim\M^{\mathrm{str}}_\Lambda=\aleph_0$, such that no
nonzero element of $\M_\Lambda$ is invariant under a nonzero translation?
\end{question}