\section{A countably infinite-dimensional space of Fourier quasicrystals}\label{sec:fqcount}

In this section we prove Theorem~\ref{thm:D}, the main result. The architecture,
which the construction of Theorem~\ref{thm:A} in Sections~\ref{sec:blocks}
and~\ref{sec:exhaustion} shares, has three parts: blocks whose complete spaces are
of small dimension and of prescribed order, an assembly with growing gaps, and an
exhaustion argument showing that the union carries nothing else. The order of an
element is its position in the strong layers $X_N$ of Section~\ref{subsec:strong}, and the blocks come from Poisson formulas on the zero
sets of exponential polynomials, in the spirit of Kurasov and Sarnak \cite{KS20}
and Olevskii and Ulanovskii \cite{OU20a}. Table~\ref{tab:analogy} compares the two
constructions. The notation of this section is independent of that of
Sections~\ref{sec:blocks} and~\ref{sec:exhaustion}.

\begin{table}[!ht]
\centering\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.2\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}}
\hline
 & Theorem~\ref{thm:D} (this section) & Theorem~\ref{thm:A} (Sections~\ref{sec:blocks}--\ref{sec:exhaustion})\\
\hline
order of an element & growth exponent of $|\mu|$ and $|\widehat\mu|$ & Hermite order\\
block & $s$ sheets of zeros of a product of Lee--Yang polynomials & clusters of atoms at the half-integers\\
order of a block & between $s$ and $s+3$ & between $P$ and $18P+4$\\
separation of blocks & distinct prime scales and generic parameters & random scales\\
exhaustion & complete description of finite unions, then compactness & the component of an element on each block\\
\hline
\end{tabular}
\caption{The two constructions.}\label{tab:analogy}
\end{table}

\emph{The mechanism.} We write $e(t)=e^{2\pi\ii t}$. For $0<a<1$ put
\[
 p_a(z,w)=1-aw-z(w-a)=(1-aw)\bigl(1-zB_a(w)\bigr),\qquad B_a(w)=\frac{w-a}{1-aw}.
\]
Along a line $(z,w)=(e(t),e(\beta t-\frac14))$ in the torus, $p_a$ has only real
simple zeros. A product of $s$ such factors with distinct parameters has $s$ sheets
of zeros. The sheets never meet, but they come close where the line passes near the
point $(1,1)$, since $p_b-p_a=(b-a)(z-w)$. There the derivative of the product, which
is the denominator of the residues in the Poisson formula, is of size
$|z-w|^{s-1}$, so the masses become large. For a suitable
slope $\beta$ they grow like a power $|t|^{s-1}$, with the exponent controlled from
above and from below (Lemma~\ref{lem:strongorder}). The number of sheets thus plays
the role that, for the blocks of Section~\ref{sec:blocks} (Theorem~\ref{thm:rapid}),
the clustering parameter $P$ plays for the Hermite order; see
Figure~\ref{fig:sheets}.

\emph{Plan.} The proof uses three tools, each for one purpose.
Continued fractions (Section~\ref{subsec:beta}) choose the slope $\beta$ so that
the sheets approach one another at a controlled polynomial rate, from above and from
below; this fixes the growth exponent of a block (Lemma~\ref{lem:strongorder}).
Poisson formulas on Lee--Yang curves (Section~\ref{subsec:poisson}) turn every
crystalline measure on a block, or on a finite union of blocks, into a rational
function $R/Q$ whose numerator ranges over a finite-dimensional space; on one block,
finite linear algebra then cuts the space down to a line (Lemma~\ref{lem:fqline}).
For a finite union of blocks (Section~\ref{subsec:finiteunions}) two further inputs
are needed. A uniqueness theorem (Proposition~\ref{prop:noreturn}), proved by
harmonic analysis at a scale whose square is a quadratic irrational, separates the part of an element that
lives on the cones from the part that lives on the roots. An algebraic splitting
lemma (Lemma~\ref{lem:splitting}), proved on $\mathbb P^1\times\mathbb P^1$ with a
group action and the vanishing of $H^1(\mathbb P^1\times\mathbb P^1,\mathcal O)$,
splits the rational function of the root part into the contributions of the
individual blocks; this is needed because the zero sets of the denominators of
different blocks intersect, so that partial fractions are not available. Finally,
a compactness argument (Section~\ref{subsec:infiniteunion}) passes from finite to
infinite unions. In terms of dependencies: Lemma~\ref{lem:beta} is used in
Lemmas~\ref{lem:sheets} and~\ref{lem:fqparams} and in
Proposition~\ref{prop:noreturn}; Lemmas~\ref{lem:poisson} and~\ref{lem:sheets} give
Lemma~\ref{lem:strongorder}, and Lemma~\ref{lem:poisson} gives
Lemma~\ref{lem:fqline}; Lemmas~\ref{lem:fqline} and~\ref{lem:fqparams},
Proposition~\ref{prop:noreturn} and Lemma~\ref{lem:splitting}, together with
Lemmas~\ref{lem:poisson} and~\ref{lem:sheets}, give
Proposition~\ref{prop:finiteunion}, which with Lemma~\ref{lem:strongorder} gives
Corollary~\ref{cor:finitestrong}; and Corollary~\ref{cor:finitestrong} with
Lemma~\ref{lem:compact} gives Theorem~\ref{thm:fqcount}, hence
Theorem~\ref{thm:D}. The proofs of Lemmas~\ref{lem:beta}, \ref{lem:sheets}
and~\ref{lem:fqparams}, Proposition~\ref{prop:noreturn} and
Lemma~\ref{lem:splitting} are given in Appendix~\ref{app:fq}.

Throughout this section $c=2^{1/4}$, $\Gamma_{\Q}=(\Q+\beta\Q)/c$ with $\beta$ as in
Lemma~\ref{lem:beta}, and
\[
 C_\beta=\{k+\beta n:\ k,n\in\Z,\ k,n\ge0\}\cup\{k+\beta n:\ k,n\in\Z,\ k,n\le0\}.
\]
Since $\beta$ is irrational, the labels $(k,n)$ of the points of $C_\beta$ are unique.

\subsection{An arithmetic choice}\label{subsec:beta}
Let $\beta=[3;1,a_2,a_3,\dots]$ be the continued fraction with partial quotients
$a_j=2q_{j-1}^{\,j}$ for $j\ge2$, where $p_j/q_j$ are the convergents of $\beta$.

\begin{lemma}\label{lem:beta}
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every $q_j$ is odd, and $|2k\beta-u|\ge1/(4|k|)$ for all integers $k\ne0$ and
all odd integers $u$.
\item There are integers $n$ of arbitrarily large modulus, and integers $k$, with
$|\beta n-k-\frac14|\le1/|n|$.
\item For every $j\ge1$, $\eps_j=\beta q_j-p_j$ satisfies $0<|\eps_j|<q_j^{-j-2}$.
\item $\beta$ is transcendental.
\end{enumerate}
\end{lemma}

Part~(i) bounds the near-crossings of the sheets from below, so that the masses are
at most polynomial; part~(ii) produces near-crossings, so that the masses really
grow; part~(iii) drives the no-return theorem (Proposition~\ref{prop:noreturn});
and part~(iv) is used to choose the parameters of the blocks. The proof is given in
Appendix~\ref{app:fq}.

\subsection{Poisson formulas on Lee--Yang curves}\label{subsec:poisson}
We call a polynomial $Q(z,w)$ \emph{admissible} if it is a product of factors
$p_a(\rho z^n,\sigma w^n)$ with $a\in(0,1)$, $n\ge1$ and $|\rho|=|\sigma|=1$. It has
bidegree $(S,S)$, where $S$ is the sum of the exponents $n$, and nonzero
coefficients at $1$ and at $z^Sw^S$. We write
\[
 V_S=\Span\{z^iw^j:\ 0\le i,j\le S,\ (i,j)\ne(S,S)\}.
\]
For $\alpha>0$ and $\varphi_0\in\R$ put $\gamma(x)=(e(\alpha x),e(\alpha\beta x+\varphi_0))$.
The following lemma is a version of the Poisson formulas of \cite{KS20,OU20a} that
does not assume that the zero set is uniformly discrete.

\begin{lemma}\label{lem:poisson}
Let $Q$ be admissible of bidegree $(S,S)$, and suppose that the real zeros of
$f=Q\circ\gamma$ are simple; let $A$ be the set of these zeros. For $R\in V_S$ the
function $G_R=(R/Q)\circ\gamma$ is meromorphic in $\C$, with poles only at points
of $A$, all simple. Put $\mu_R=\sum_{x\in A}\operatorname{Res}_x(G_R)\,\delta_x$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $\mu_R$ is a crystalline measure with support in $A$ and spectrum in
$\alpha C_\beta$. If $\sum c_{kn}z^kw^n$ and $\sum c'_{kn}z^{-k}w^{-n}$ ($k,n\ge0$)
are the expansions of $R/Q$ at $(0,0)$ and at $(\infty,\infty)$, then $c'_{00}=0$,
\[
 \F\mu_R(\{\alpha(k+\beta n)\})=-\frac{e(n\varphi_0)\,c_{kn}}{2\pi\ii},\qquad
 \F\mu_R(\{-\alpha(k+\beta n)\})=\frac{e(-n\varphi_0)\,c'_{kn}}{2\pi\ii}
\]
for $k,n\ge0$ (with $(k,n)\ne(0,0)$ in the second formula), and
$|c_{kn}|+|c'_{kn}|\le C_R(1+k+n)^S$.
\item Let $u:\Z^2\to\C$ vanish outside $\Z_{\ge0}^2\cup\Z_{\le0}^2$ and satisfy
$\sum_{i,j}Q_{ij}\,u(k-i,n-j)=0$ for all $(k,n)\in\Z^2$. Then there is a unique
$R\in V_S$ with $e(-n\varphi_0)\,\F\mu_R(\{\alpha(k+\beta n)\})=u(k,n)$ for all
$(k,n)\in\Z_{\ge0}^2\cup\Z_{\le0}^2$.
\item $R\mapsto\mu_R$ is a linear bijection of $V_S$ onto $\Ncal(\alpha C_\beta,A)$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) For $|u|,|v|<1$ we have $|B_a(v)|<1$, hence
$|p_a(u,v)|=|1-av|\,|1-uB_a(v)|\ge(1-a)(1-\max\{|u|,|v|\})$. Consequently
$|Q|\ge\kappa(1-r)^S$ on the polydisc $\{|z|,|w|\le r\}$, $r<1$, for some $\kappa>0$.
Moreover $p_a(u,v)=-uv\,p_a(1/u,1/v)$, so that
$Q(z,w)=\zeta_Qz^Sw^SQ^\sharp(1/z,1/w)$ with $|\zeta_Q|=1$ and $Q^\sharp$ admissible
(replace $\rho,\sigma$ by their conjugates). Thus $Q$ has no zeros in $\{|z|,|w|<1\}$
or in $\{|z|,|w|>1\}$. For $\operatorname{Im}x>0$ the point $\gamma(x)$ lies in the
first of these sets, so $G_R$ is holomorphic there and
$G_R(x)=\sum_{k,n\ge0}c_{kn}e(n\varphi_0)e(\alpha(k+\beta n)x)$, with absolute
convergence. Writing $R(z,w)=z^Sw^SR^\sharp(1/z,1/w)$, where $R^\sharp(0,0)=R_{SS}=0$,
we get $G_R(x)=\sum_{k,n\ge0}c'_{kn}e(-n\varphi_0)e(-\alpha(k+\beta n)x)$ for
$\operatorname{Im}x<0$, with $c'$ the Taylor coefficients of
$\zeta_Q^{-1}R^\sharp/Q^\sharp$; in particular $c'_{00}=0$. Cauchy's estimate on the
torus of radius $r=1-1/(k+n+2)$ gives the coefficient bound. Since the cone has
$O((1+T)^2)$ points in $[-T,T]$, the two series converge in $\Sc'(\R)$ to tempered
distributions $G_\pm$, and $G_R(\cdot\pm\ii y)\to G_\pm$ in $\Sc'$ as $y\downarrow0$
(pair with a Schwartz function and use dominated convergence). On the other hand,
near the real axis $G_R$ is meromorphic with simple poles at the points of $A$, so on
every bounded interval the Sokhotski--Plemelj formula gives
$G_+-G_-=-2\pi\ii\sum_{x\in A}\operatorname{Res}_x(G_R)\delta_x$. Hence
$\mu_R=(G_--G_+)/(2\pi\ii)$ is tempered, and the formulas for $\F\mu_R$ follow from
$\F(e(\omega x))=\delta_\omega$.

(b) Write $u=u^++u^-$, where $u^+$ is the restriction of $u$ to $\Z_{\ge0}^2$ and $u^-$
that to $\Z_{\le0}^2\setminus\{0\}$. The convolution $\widetilde R=Q*u^+=-Q*u^-$ is
supported in $\Z_{\ge0}^2$ by the first expression and in $\{(k,n):k,n\le S\}$ by
the second, and its coefficient at $(S,S)$ is $-Q_{SS}u^-(0,0)=0$. Thus
$\widetilde R\in V_S$, $\sum u^+(k,n)z^kw^n$ is the Taylor series of
$\widetilde R/Q$ at the origin (as $Q_{00}\ne0$), and
$\sum u^-(-k,-n)z^{-k}w^{-n}$ is minus its expansion at infinity (as $Q_{SS}\ne0$).
By (a), $R=-2\pi\ii\widetilde R$ has the required property. Uniqueness follows because
$u$ determines $\widetilde R=Q*u^+$.

(c) If $\mu_R=0$, then $u=0$ in (b), so $R=0$. Conversely let $T\in\Ncal(\alpha C_\beta,A)$.
Since $f$ vanishes on $A$ and $T$ is locally atomic there, $fT=0$. As
$f=\sum Q_{ij}e(j\varphi_0)e(\alpha(i+\beta j)x)$, the equation $\F(fT)=0$ says that
$u(k,n)=e(-n\varphi_0)\F T(\{\alpha(k+\beta n)\})$ satisfies the recurrence of (b). If
$R$ is given by (b), then $\F(T-\mu_R)=0$, so $T=\mu_R$.
\end{proof}

\subsection{Blocks}\label{subsec:fqblocks}
A \emph{block} is given by an odd prime $m$, an integer $s\ge2$ and distinct
parameters $a_1,\dots,a_s\in(0,1)$. We put $\lambda=cm$ and
\[
 Z(x)=e(x/\lambda),\quad W(x)=e(\beta x/\lambda-\tfrac14),\quad
 P=\prod_{j\le s}p_{a_j},\quad A=\{x\in\R:\ P(Z(x),W(x))=0\},
\]
and we let $\mathcal C=C_\beta/\lambda$ and $\mathcal C_0=C_\beta/c$. Since $m$ is an
integer, $\mathcal C_0\subset\mathcal C$; precisely, $(k+\beta n)/\lambda\in\mathcal C_0$
if and only if $m$ divides both $k$ and $n$. For $0<a<1$ let $\psi_a:\R\to\R$ be the
continuous function with $\psi_a(0)=0$ and $B_a(e(y))=e(\psi_a(y))$; it is increasing,
$\psi_a(y+1)=\psi_a(y)+1$, and $\psi_a'(y)=(1-a^2)/|1-ae(y)|^2$.

\begin{lemma}\label{lem:sheets}
Let $D_\beta=z\partial_z+\beta w\partial_w$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item $P$ is admissible of bidegree $(s,s)$. The zeros of $P(Z(x),W(x))$ in $\C$ are
real and simple, and $A$ is the disjoint union of the sets
$A_j=\{x:\ x/\lambda+\psi_{a_j}(\beta x/\lambda-\frac14)\in\Z\}$. Moreover
$\#(A\cap[-r,r])\le s(2(1+\beta)r/\lambda+2)$.
\item $A\cap(-\delta\lambda,\delta\lambda)=\varnothing$, where $\delta=1/(8(1+\beta))$.
\item There is $\kappa>0$ with $|D_\beta P(Z(x),W(x))|\ge\kappa(1+|x|)^{-(s-1)}$ for
all $x\in A$.
\item There are $x\in A$ of arbitrarily large modulus with
$|D_\beta P(Z(x),W(x))|\le C|x|^{-(s-1)}$ and $|Z(x)-1|+|W(x)-1|\le C/|x|$.
\item There is $\kappa'>0$ with $|x-x'|\ge\kappa'(1+|x|)^{-1}$ for all distinct
$x,x'\in A$.
\end{enumerate}
\end{lemma}

The proof is given in Appendix~\ref{app:fq}. We apply Lemma~\ref{lem:poisson} to $P$
with $\alpha=1/\lambda$ and $\varphi_0=-\frac14$, so that $\gamma=(Z,W)$ and $\mu_R$ is
defined for $R\in V_s$. The strong layers $X_N$ were defined in
Section~\ref{subsec:strong}.

\begin{lemma}\label{lem:strongorder}
For $R\in V_s$ we have $\mu_R\in X_{s+3}(\mathcal C,A)$. If $R(1,1)\ne0$, then
$\sum_{x\in A}|\mu_R(\{x\})|(1+|x|)^{-N}=\infty$ for every $N\le s-1$.
\end{lemma}

\begin{proof}
Along the real line the derivative of $P(Z,W)$ is $2\pi\ii\lambda^{-1}D_\beta P(Z,W)$, so
the mass of $\mu_R$ at $x\in A$ is $\lambda R(Z,W)/(2\pi\ii\,D_\beta P(Z,W))$, evaluated
at $(Z(x),W(x))$. By
Lemma~\ref{lem:sheets}(iii) these masses are $O((1+|x|)^{s-1})$, and by
Lemma~\ref{lem:sheets}(i) the physical sum converges for $N\ge s+1$. By
Lemma~\ref{lem:poisson}(a) the Fourier masses at $\pm(k+\beta n)/\lambda$ are
$O((1+k+n)^s)=O((1+|\xi|)^s)$, and $\mathcal C$ has $O((1+r)^2)$ points in $[-r,r]$, so
the Fourier sum converges for $N\ge s+3$. If $R(1,1)\ne0$, then at the points of
Lemma~\ref{lem:sheets}(iv) we have $R(Z,W)\to R(1,1)$, and the masses are at least
$\kappa''|x|^{s-1}$ for some $\kappa''>0$; for $N\le s-1$ the corresponding summands do not tend to zero.
\end{proof}

\begin{lemma}\label{lem:fqline}
Let $b=\lfloor s/2\rfloor/2$ and
$B=\mathcal C\setminus\bigl([-b/\lambda,b/\lambda]\setminus\mathcal C_0\bigr)$. There are a
finite set $E\subset A$ and $R_0\in V_s$ with $R_0(1,1)\ne0$ such that
\[
 \Ncal(B,A\setminus E)=\C\mu_{R_0}.
\]
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:poisson}(c), for every finite $E\subset A$ the space
$\Ncal(B,A\setminus E)$ consists of the $\mu_R$, $R\in V_s$, that have no mass on $E$
and whose Fourier transforms vanish on $\mathcal C\setminus B$. The points of
$\mathcal C\setminus B$ are the $\pm(k+\beta n)/\lambda$ with
$(k,n)\in\Z_{\ge0}^2\setminus m\Z^2$ and $k+\beta n\le b$; for them $k,n<s$, since
$b<s/2$ and $\beta>1$. By Lemma~\ref{lem:poisson}(a), the mass of $\F\mu_R$ at
$(k+\beta n)/\lambda$ is a nonzero multiple of $c_{kn}$, and division by $Q=P$ shows
that $c_{kn}$ equals $R_{kn}/P_{00}$ plus a combination of the $R_{k'n'}$ with $k'\le k$,
$n'\le n$, $(k',n')\ne(k,n)$. Similarly the mass at $-(k+\beta n)/\lambda$ is a nonzero
multiple of $R_{s-k,s-n}$ plus a combination of the $R_{s-k',s-n'}$ with $k'\le k$,
$n'\le n$, $(k',n')\ne(k,n)$. The first family of conditions involves only the
coefficients $R_{ij}$ with $i+\beta j\le b$, the second only those with
$i+\beta j\ge s(1+\beta)-b>b$, and each family is triangular with nonzero diagonal.
Hence these conditions are linearly independent; let $H\subset V_s$ be the subspace
they define. The monomial $z^hw^h$, $h=\lfloor s/2\rfloor$, lies in $V_s$ and in $H$:
the expansion of $z^hw^h/P$ at the origin involves only monomials $z^kw^n$ with
$k,n\ge h$, its expansion at infinity only monomials $z^{-k}w^{-n}$ with
$k,n\ge s-h\ge h$, and $h(1+\beta)>b$. As $z^hw^h$ takes the value $1$ at $(1,1)$, the
subspace $H_0=\{R\in H:R(1,1)=0\}$ has codimension one in $H$.

The functionals $R\mapsto R(Z(x),W(x))$, $x\in A$, separate the points of $V_s$: if
they all vanish, then $\mu_R=0$, and $R=0$ by Lemma~\ref{lem:poisson}(c). Choose a set
$E$ of $\dim H_0$ points of $A$ whose functionals restrict to a basis of $H_0^*$.
Their common kernel in $H$ is a line $\ell$ with $\ell\cap H_0=\{0\}$. Since $\mu_R$ has
no mass at $x$ if and only if $R(Z(x),W(x))=0$, we obtain
$\Ncal(B,A\setminus E)=\{\mu_R:R\in\ell\}=\C\mu_{R_0}$ for any nonzero $R_0\in\ell$, and
$R_0(1,1)\ne0$.
\end{proof}

We call $\mu=\mu_{R_0}$ the \emph{block measure}. The set $B$ is symmetric, so
$\F^{-1}\mu$ has support in $B$ and spectrum in $A\setminus E$. By
Lemma~\ref{lem:strongorder}, $\mu$ and $\F^{-1}\mu$ belong to $X_{s+3}$, while the
physical record of $\mu$, which is the spectral record of $\F^{-1}\mu$, has infinite
$N$-weighted variation for $N\le s-1$. The spectral gap of a block is only
partial: $B$ contains the fixed coarse cone $\mathcal C_0$, and only its other points
avoid $[-b/\lambda,b/\lambda]$. The coarse cone is shared by all blocks; including it
from the start is what allows different blocks to be combined.

\subsection{Finite unions}\label{subsec:finiteunions}
We now consider blocks $i=1,\dots,r$, with data $m_i,s_i,a_{i,j}$ and $\lambda_i$, $P_i$,
$A_i$, $\mathcal C_i$, $B_i$, $E_i$ and $\mu_i$ as above, and with pairwise distinct
primes $m_i$. We put $L_i=(A_i\setminus E_i)\cup B_i$ and
$L_{[r]}=\bigcup_{i\le r}L_i$. Three further ingredients are needed; their proofs are
given in Appendix~\ref{app:fq}. The first is a choice of the parameters that
separates the roots of different blocks arithmetically.

We call blocks $1,\dots,r$ \emph{independent} if their root sets $A_1,\dots,A_r$ are
pairwise disjoint and disjoint from $\Gamma_{\Q}$, and no difference of two distinct
points of $A_1\cup\dots\cup A_r$ lies in $\Gamma_{\Q}$.

\begin{lemma}\label{lem:fqparams}
Let blocks $1,\dots,r-1$ be independent, let $m_r$ be an odd prime and let
$s_r\ge2$. Then the parameters $a_{r,1},\dots,a_{r,s_r}$ can be chosen,
successively and each outside a countable subset of $(0,1)$, so that blocks
$1,\dots,r$ are independent.
\end{lemma}

The second is a uniqueness theorem: at a scale whose square is a quadratic
irrational, no nonzero crystalline measure has both its support and its spectrum
in finitely many translates of the cone. It is used to separate the part of an
element that lives on the cones from the part that lives on the roots. The
hypothesis on the scale cannot be weakened to irrationality: since $\Z\subset C_\beta$,
the measure $\delta_{\Z/\sqrt2}$ has support in $C_\beta/\sqrt2$, and its Fourier
transform $\sqrt2\,\delta_{\sqrt2\Z}$ is supported in $\Z/\sqrt2\subset C_\beta/\sqrt2$. This
is why the blocks are placed at the scales $cm$ with $c=2^{1/4}$.

\begin{proposition}\label{prop:noreturn}
Let $\lambda>0$ be such that $\lambda^2$ is a quadratic irrational, and let
$F_1,F_2\subset\Z+\beta\Z$ be finite. If $T$ is a crystalline measure with support in
$(C_\beta+F_1)/\lambda$ and spectrum in $(C_\beta+F_2)/\lambda$, then $T=0$.
\end{proposition}

The third is an algebraic splitting of rational functions with sparse Taylor
series. The zero sets of the denominators of different blocks intersect in
$\C^2$, so the splitting does not reduce to partial fractions; it is proved on
$\mathbb P^1\times\mathbb P^1$ with a group action and the vanishing of
$H^1(\mathbb P^1\times\mathbb P^1,\mathcal O)$.

\begin{lemma}\label{lem:splitting}
Let $m_1,\dots,m_r$ be distinct odd primes, $M=m_1\cdots m_r$ and $n_i=M/m_i$. For
each $i$ let $P_i$ be a product of $s_i$ factors $p_a$ with distinct $a\in(0,1)$,
let $Q_i(z,w)=P_i(z^{n_i},-\ii w^{n_i})$, $Q=\prod_iQ_i$ and $S=\sum_in_is_i$. If
$R\in V_S$ and the Taylor coefficients of $R/Q$ at the origin vanish outside
$\bigcup_in_i\Z_{\ge0}^2$, then
\[
 \frac RQ=\sum_{i\le r}\frac{R_i(z^{n_i},-\ii w^{n_i})}{Q_i(z,w)}
 \qquad\text{with }R_i\in V_{s_i}.
\]
\end{lemma}

\begin{proposition}\label{prop:finiteunion}
Let blocks $1,\dots,r$ be independent, with pairwise distinct primes $m_i$. Then
\[
 \M_{L_{[r]}}=\Span\{\mu_i,\ \F^{-1}\mu_i:\ i\le r\},
\]
and these $2r$ measures are linearly independent.
\end{proposition}

\begin{proof}
Put $M=\prod_im_i$, $n_i=M/m_i$, $\lambda_*=cM$, $\Gamma_*=(\Z+\beta\Z)/\lambda_*$ and
$\mathcal C_*=C_\beta/\lambda_*$. Then $\lambda_i=\lambda_*/n_i$, so
$\mathcal C_i=n_iC_\beta/\lambda_*\subset\mathcal C_*\subset\Gamma_*\subset\Gamma_{\Q}$. In the
variables $z=e(x/\lambda_*)$, $w=e(\beta x/\lambda_*)$ we have $Z_i=z^{n_i}$ and
$W_i=-\ii w^{n_i}$, so $P_i(Z_i,W_i)=Q_i(z,w)$ with $Q_i$ as in
Lemma~\ref{lem:splitting}. The polynomial $Q=\prod_iQ_i$ is admissible, and by
Lemma~\ref{lem:sheets} and the independence of the blocks, the real zeros of
$f=Q(z(x),w(x))$ are simple and form the set $A_*=\bigcup_iA_i$. Thus
Lemma~\ref{lem:poisson} applies to $Q$ with $\alpha=1/\lambda_*$, $\varphi_0=0$. Note
that $L_{[r]}\subset A_*\cup\mathcal C_*$, that $A_*\cap\Gamma_*=\varnothing$, and that
$\mathcal C_*\subset\Gamma_*$.

Let $T\in\M_{L_{[r]}}$, and write $f=\sum_{\omega\in\Omega}f_\omega e(\omega x)$ with a finite
$\Omega\subset\Gamma_*$.

\emph{Step 1: clearing the roots.} The measure $U=fT$ is supported in $\mathcal C_*$,
since $f$ vanishes on $A_*$. Its Fourier transform
$\F U=\sum_\omega f_\omega\tau_{-\omega}\F T$ is supported in
$(A_*\cup\mathcal C_*)+\Omega$. The trigonometric polynomial
$g(x)=\prod_{\omega\in\Omega}f(x-\omega)$ vanishes on $A_*+\Omega$, so $Y=g\,\F U$ is
supported in finitely many $\Gamma_*$-translates of $\mathcal C_*$. As the frequencies of
$g$ lie in $\Gamma_*$, the measure $\F^{-1}Y$ is a finite combination of
$\Gamma_*$-translates of $U$, so it is supported in finitely many $\Gamma_*$-translates of
$\mathcal C_*$ as well. Since $\lambda_*^2=M^2\sqrt2$ is a quadratic irrational,
Proposition~\ref{prop:noreturn} gives $Y=0$.

\emph{Step 2: the cone part.} For $y\in\Gamma_*$ we have $y-\omega\in\Gamma_*$ for
$\omega\in\Omega$, so $g(y)\ne0$, and therefore $\F U(\{y\})=0$, i.e.
$\sum_\omega f_\omega\F T(\{y-\omega\})=0$. The points of $\Gamma_*$ that carry mass of
$\F T$ lie in $\mathcal C_*$. Hence $u(k,n)=\F T(\{(k+\beta n)/\lambda_*\})$ vanishes
outside $\Z_{\ge0}^2\cup\Z_{\le0}^2$ and satisfies the recurrence of
Lemma~\ref{lem:poisson}(b) for $Q$. By that lemma there is a measure $\rho=\mu_R$,
$R\in V_S$, supported in $A_*$, such that $\F\rho$ is the restriction of $\F T$ to
$\Gamma_*$.

\emph{Step 3: the literal decomposition.} The measure $\sigma=\F(T-\rho)=\F T-\F\rho$ is
the restriction of $\F T$ to $\R\setminus\Gamma_*$, so it is supported in $A_*$ and
$f\sigma=0$. Its Fourier transform $\F\sigma=(T-\rho)^\vee$ is supported in
$(-A_*)\cup\mathcal C_*$ and, by $\F(f\sigma)=0$, satisfies
$\sum_\omega f_\omega\F\sigma(\{y-\omega\})=0$ for all $y\in\R$. A coset $y+\Gamma_*$ with
$y\notin\Gamma_*$ contains no point of $\mathcal C_*$ and, by the independence of the
blocks, at most one point of $-A_*$. A nonzero measure carried by one point cannot satisfy this
recurrence, because the term with the largest frequency in $\Omega$ is not cancelled.
Hence $\F\sigma$ is supported in $\Gamma_*\cap((-A_*)\cup\mathcal C_*)=\mathcal C_*$, and $T-\rho$
is supported in $\mathcal C_*$. Since $\rho$ is supported in $A_*$, we conclude that
\[
 \rho=T|_{A_*},\qquad T-\rho=T|_{\mathcal C_*},\qquad
 \F\rho=(\F T)|_{\mathcal C_*},\qquad\sigma=(\F T)|_{A_*}.
\]
In particular $\rho$ and $\sigma$ are measures with support in
$\bigcup_i(A_i\setminus E_i)$ and spectrum in $\bigcup_iB_i$; for $\sigma$ this uses
that the sets $B_i$ are symmetric.

\emph{Step 4: splitting.} Let $\varrho\in\{\rho,\sigma\}$. By
Lemma~\ref{lem:poisson}(c), $\varrho=\mu_R$ for some $R\in V_S$, and the Taylor
coefficients of $R/Q$ at the origin are multiples of the masses of $\F\varrho$ at the
points $(k+\beta n)/\lambda_*$, $k,n\ge0$. These masses vanish unless the point lies
in some $\mathcal C_i$, that is, unless $(k,n)\in n_i\Z_{\ge0}^2$ for some $i$. By
Lemma~\ref{lem:splitting}, $R/Q=\sum_iR_i(z^{n_i},-\ii w^{n_i})/Q_i(z,w)$ with
$R_i\in V_{s_i}$. Along the real line the $i$-th summand is the function
$(R_i/P_i)(Z_i,W_i)$, so $\varrho=\sum_i\varrho_i$, where $\varrho_i$ is the measure of
Lemma~\ref{lem:poisson} for $P_i$ and $R_i$; it has support in $A_i$ and spectrum in
$\mathcal C_i$. As the $A_i$ are disjoint, $\varrho_i=\varrho|_{A_i}$ has no mass on
$E_i$. Let $\xi\in\mathcal C_i\setminus B_i$. Then $\xi\notin\mathcal C_0$, since
$\mathcal C_0\subseteq B_i$, and as $\mathcal C_i\cap\mathcal C_j=\mathcal C_0$ for $j\ne i$
(the primes are distinct), $\xi\notin\mathcal C_j$. Thus $\xi$ lies in no $B_j$, so
$\F\varrho(\{\xi\})=0$, and $\F\varrho_j(\{\xi\})=0$ for $j\ne i$; hence
$\F\varrho_i(\{\xi\})=0$. Therefore
$\varrho_i\in\Ncal(B_i,A_i\setminus E_i)=\C\mu_i$ by Lemma~\ref{lem:fqline}.

Applying this to $\rho$ and to $\sigma$ we get $T|_{A_*}=\sum_i\theta_i\mu_i$ and
$\F(T|_{\mathcal C_*})=\sum_i\theta'_i\mu_i$, that is,
$T=\sum_i(\theta_i\mu_i+\theta'_i\F^{-1}\mu_i)$. Conversely every $\mu_i$ and every
$\F^{-1}\mu_i$ lies in $\M_{L_{[r]}}$. The coefficients $\theta_i$ and $\theta'_i$ are
determined by the physical and the Fourier records of $T$ on the disjoint sets $A_i$,
which gives linear independence.
\end{proof}

\begin{corollary}\label{cor:finitestrong}
In the situation of Proposition~\ref{prop:finiteunion}, for every $N\ge0$
\[
 \Span\{\mu_i,\F^{-1}\mu_i:\ s_i+3\le N\}\subseteq X_N(L_{[r]})
 \subseteq\Span\{\mu_i,\F^{-1}\mu_i:\ s_i\le N\}.
\]
\end{corollary}

\begin{proof}
The first inclusion is Lemma~\ref{lem:strongorder}. Let
$T=\sum_i(\theta_i\mu_i+\theta'_i\F^{-1}\mu_i)\in X_N(L_{[r]})$. The physical record of
$T$ on $A_i$ is $\theta_i$ times that of $\mu_i$, and the Fourier record of $T$ on $A_i$ is
$\theta'_i$ times the physical record of $\mu_i$. By Lemma~\ref{lem:strongorder},
$\theta_i=\theta'_i=0$ whenever $N\le s_i-1$.
\end{proof}

There is no cancellation in this argument: the masses that diverge are those of the
private root set $A_i$ of a block, which no other block can reach.

\subsection{The infinite union}\label{subsec:infiniteunion}
It remains to exclude elements of $X_N$ that involve infinitely many blocks. Unlike the
construction of Theorem~\ref{thm:A}, where this is the main difficulty
(Section~\ref{sec:exhaustion}), here a soft compactness argument suffices, because Proposition~\ref{prop:finiteunion} describes
every finite union completely. For an integer $N\ge0$ write $w_N(x)=(1+|x|)^{-N}$, and
let $\mathcal Q_N$ be the set of pairs $(\mu,\nu)$ of complex Radon measures on $\R$
with $\nu=\F\mu$ and $\norm{w_N\mu}+\norm{w_N\nu}\le1$, where $\norm\cdot$ is the total
variation. We give $\mathcal Q_N$ the topology induced by the weak$^*$ topology of
$(w_N\mu,w_N\nu)$ in $C_0(\R)^*\times C_0(\R)^*$. For a closed set $L\subset\R$ and
$R>0$ let $\mathcal A_R(L)$ be the set of pairs in $\mathcal Q_N$ such that $\mu$ and $\nu$
are supported in $L\cup\{|x|\ge R\}$.

\begin{lemma}\label{lem:compact}
$\mathcal Q_N$ is compact and metrizable, and every $\mathcal A_R(L)$ is closed. If $L$
is locally finite, then $\bigcap_R\mathcal A_R(L)$ consists of the pairs $(T,\F T)$ with
$T\in X_N(L)$ and $\norm T_{(N)}\le1$. Consequently, if $\Psi$ is a continuous function on
$\mathcal Q_N$ that vanishes on these pairs, then $\sup_{\mathcal A_R(L)}|\Psi|\to0$ as
$R\to\infty$.
\end{lemma}

\begin{proof}
The unit ball of $C_0(\R)^*\times C_0(\R)^*$ is weak$^*$ compact and metrizable, since
$C_0(\R)$ is separable. For $\varphi\in\Sc$ we have $\varphi/w_N\in C_0(\R)$, so
$\mu\mapsto\mu(\varphi)$ is continuous; hence the condition $\nu=\F\mu$ is closed, and so
are the support conditions. A Radon measure carried by a locally finite closed set is
locally atomic on it, which identifies the intersection. If the last assertion
failed, there would be $\epsilon>0$ and pairs in $\mathcal A_R(L)$, $R\to\infty$, with
$|\Psi|\ge\epsilon$; a convergent subsequence would have a limit in the intersection of
the decreasing closed sets $\mathcal A_R(L)$, where $\Psi=0$, a contradiction.
\end{proof}

\begin{theorem}\label{thm:fqcount}
There are blocks $i=1,2,\dots$, with pairwise distinct primes $m_i$ and strictly
increasing sheet numbers $s_i\ge i$, such that blocks $1,\dots,r$ are independent for
every $r$, the set $\Lambda=\bigcup_iL_i$ is locally finite, and
\begin{equation}\label{eq:fqlayers}
 \Span\{\mu_i,\F^{-1}\mu_i:\ s_i+3\le N\}\subseteq X_N(\Lambda)
 \subseteq\Span\{\mu_i,\F^{-1}\mu_i:\ s_i\le N\}\qquad(N\ge0).
\end{equation}
Consequently
$\M_\Lambda^{\mathrm{str}}=\bigoplus_i\Span\{\mu_i,\F^{-1}\mu_i\}$, and this space has
Hamel dimension $\aleph_0$.
\end{theorem}

\begin{proof}
We construct the blocks recursively, together with points $r_i\in A_i$, functions
$\phi_i\in C_c^\infty(\R)$ and radii $R_i$. Let $R_0=1$, and let $(h_l)_{l\ge0}$ be the
Hermite functions. At stage $i\ge1$:
\begin{enumerate}[label=(\alph*),leftmargin=2em]
\item choose an odd prime $m_i$ different from $m_1,\dots,m_{i-1}$ with
$\delta cm_i>R_{i-1}$, where $\delta$ is as in Lemma~\ref{lem:sheets}(ii);
\item choose an integer $s_i\ge\max\{i,2,s_{i-1}+1\}$ with
$\lfloor s_i/2\rfloor>2cm_iR_{i-1}$;
\item choose the parameters $a_{i,j}$ by Lemma~\ref{lem:fqparams}, and then $E_i$ and
$\mu_i$ by Lemma~\ref{lem:fqline};
\item choose $r_i\in A_i\setminus E_i$ with $\mu_i(\{r_i\})\ne0$, normalize
$\mu_i(\{r_i\})=1$, and choose $\phi_i$ with $\phi_i(r_i)=1$, supported in
$(r_i-\frac12,r_i+\frac12)$ and containing no other point of $L_{[i]}$;
\item choose $R_i\ge\max\{R_{i-1}+1,|r_i|+1\}$ such that
$\sup_{\mathcal A_{R_i}(L_{[i]})}|\Psi_{N,l}|<2^{-i}$ for all $N,l\le i$, where
\[
 \Psi_{N,l}(\mu,\nu)=\mu(h_l)-\sum_{k\le N}\Bigl(\mu(\phi_k)\,\mu_k(h_l)
 +\nu(\phi_k)\,(\F^{-1}\mu_k)(h_l)\Bigr).
\]
\end{enumerate}
By (a) and Lemma~\ref{lem:sheets}(ii), the roots of block $k$ lie outside
$[-R_{k-1},R_{k-1}]$, and by (b) the points of $B_k$ in $[-R_{k-1},R_{k-1}]$ lie in
$\mathcal C_0\subseteq L_1$. As the $R_k$ increase, it follows that
$L_{[i']}\cap(-R_k,R_k)=L_{[k]}\cap(-R_k,R_k)$ for $k\le i'$, and
$\Lambda\cap(-R_k,R_k)=L_{[k]}\cap(-R_k,R_k)$ for every $k$. In particular $\Lambda$ is
locally finite, $\Lambda\subseteq L_{[k]}\cup\{|x|\ge R_k\}$, and, since the support of
$\phi_k$ lies in $(-R_k,R_k)$, the function $\phi_k$ isolates $r_k$ in every $L_{[i']}$,
$i'\ge k$, and in $\Lambda$.

Step (e) is possible by Lemma~\ref{lem:compact}. Indeed, $\Psi_{N,l}$ is continuous on
$\mathcal Q_N$. Let $N\le i$ and $T\in X_N(L_{[i]})$. By
Corollary~\ref{cor:finitestrong} and $s_k\ge k$, $T$ lies in
$W_N=\Span\{\mu_k,\F^{-1}\mu_k:k\le N\}$. The point $r_k$ lies in no $A_{k'}$ with
$k'\ne k$ and in no $B_{k'}$, because $A_k$ is disjoint from the other root sets and
from $\Gamma_{\Q}\supseteq B_{k'}$. Since moreover $\phi_k$ isolates $r_k$, the generators
other than $\mu_k$ have no mass at $r_k$, and those other than $\F^{-1}\mu_k$ have
Fourier transforms without mass at $r_k$, so the numbers
$T(\phi_k)$ and $\F T(\phi_k)$ are the coefficients of $\mu_k$ and $\F^{-1}\mu_k$ in $T$.
Hence $\Psi_{N,l}(T,\F T)=0$.

Let $T\in X_N(\Lambda)$ with $\norm T_{(N)}\le1$. Then $(T,\F T)\in\mathcal A_{R_i}(L_{[i]})$
for every $i$, so $|\Psi_{N,l}(T,\F T)|<2^{-i}$ for all $i\ge\max\{N,l\}$, and therefore
$\Psi_{N,l}(T,\F T)=0$ for every $l$. As the $h_l$ span a dense subspace of $\Sc$, the
distribution $T$ coincides with $\sum_{k\le N}(T(\phi_k)\mu_k+\F T(\phi_k)\F^{-1}\mu_k)$,
so $T\in W_N$. The argument of Corollary~\ref{cor:finitestrong}, applied to the
records of $T$ on the sets $A_k$, now gives the right-hand inclusion in
\eqref{eq:fqlayers}, and the left-hand inclusion is Lemma~\ref{lem:strongorder}. The
$\mu_i$ and $\F^{-1}\mu_i$ are linearly independent by their records on the disjoint
sets $A_i$, and every element of $\M_\Lambda^{\mathrm{str}}$ lies in some $X_N(\Lambda)$.
Hence the $\mu_i$ and $\F^{-1}\mu_i$ form a basis of $\M_\Lambda^{\mathrm{str}}$, which
has Hamel dimension $\aleph_0$; in accordance with
Proposition~\ref{prop:strongtrichotomy}, the layers $X_N(\Lambda)$ are
finite-dimensional and their dimensions are unbounded.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:D}]
Take $\Lambda$ and the $s_i$ as in Theorem~\ref{thm:fqcount}, $\nu_{2i-1}=\mu_i$ and
$\nu_{2i}=\F^{-1}\mu_i$. These are Fourier quasicrystals with support and spectrum in
$\Lambda$, and part~(i) is the last assertion of Theorem~\ref{thm:fqcount}. Part~(ii)
follows from \eqref{eq:fqlayers} and the linear independence of the $\nu_k$. For
part~(iii), the norms $\norm{\nu_{2i-1}}_{(N)}$ and $\norm{\nu_{2i}}_{(N)}$ on $\Lambda$ are
equal to $\norm{\mu_i}_{(N)}$ on $(\mathcal C_i,A_i)$, which is finite for $N=s_i+3$ and
infinite for $N=s_i-1$ by Lemma~\ref{lem:strongorder}, since $\mu_i=\mu_{R_0}$ with
$R_0(1,1)\ne0$.
\end{proof}

\begin{remark}\label{rem:fqcount}
(a) No random choices are made: the primes, the sheet numbers, the parameters
(outside countable sets) and the radii are chosen by deterministic conditions,
although the existence of suitable radii is obtained by compactness. The
formal proof of the existence statement in Theorem~\ref{thm:D}(i)
\cite{Mar26,PalStr} uses a set of the same kind, with a different schedule of primes,
sheet numbers and parameters; there too the radii come from a compactness statement
and are selected by classical choice, so the formal witness is not given by an
algorithm.
(b) The set $\Lambda$ is not separated at a polynomial rate: it contains the cone
$\mathcal C_0=C_\beta/c$, and since $\beta$ is a Liouville number, distinct points
$k+\beta n$ of $C_\beta$ can be closer than any power of their size. This is consistent
with Question~\ref{q:separated}. The atoms of a single block, on the other hand, are
separated at a polynomial rate by Lemma~\ref{lem:sheets}(v), and the clusters of
Lemma~\ref{lem:sheets}(iv) consist of atoms at mutual distances of order
$|x|^{-1}$, bounded below by Lemma~\ref{lem:sheets}(v).
(c) Theorem~\ref{thm:fqcount} determines the space of Fourier quasicrystals on
$\Lambda$, not the space $\M_\Lambda$ of all crystalline measures, which might be larger;
see Question~\ref{q:both}.
\end{remark}