\section{One-dimensional blocks of prescribed Hermite order}\label{sec:blocks}

We now turn to the second part of the paper, on all crystalline measures.
Sections~\ref{sec:blocks} and~\ref{sec:exhaustion} prove Theorems~\ref{thm:A}
and~\ref{thm:B}; the notation is independent of that of Section~\ref{sec:fqcount}.
In this section we construct, for all positive integers $P$ and $R$, a symmetric
locally finite set $C(P,R)$ which avoids $[-R+\frac12,R-\frac12]$ and carries,
up to scalars, exactly one crystalline measure, whose Hermite order lies between
$P$ and $18P+4$. The point is that the Hermite order and the inner radius can be
chosen independently. Figure~\ref{fig:block} shows such a block. Heuristically,
the clusters of atoms near the half-integers imitate derivatives of Dirac masses
there, as in \eqref{eq:toy}, and the speed at which the clusters tighten
determines the Hermite order; Theorem~\ref{thm:rapid} below makes this precise.
As in Kolountzakis's construction \cite[Section~2]{Kol16}, the gap around the
origin is produced by a finite-dimensional argument (Lemma~\ref{lem:holes}).

\subsection{Half-endpoint carriers}
For a sequence $\gamma=(\gamma_j)_{j\ge1}$ of distinct numbers in $(0,\frac12)$
put
\[
 L_\gamma=\{n\pm\gamma_j:\ j\ge1,\ n\in\Z,\ |n|\ge j\}.
\]
This set is locally finite because of the condition $|n|\ge j$: the cell
$[n,n+1)$ contains at most $2(|n|+2)$ of its points. If $\gamma_j\to\frac12$, it is
not uniformly discrete: as the cells move outward, they contain increasingly
tight finite clusters around the half-integers, formed by the points $n+\gamma_j$
and $n+1-\gamma_j$; there is no accumulation in any bounded interval.

For positive integers $P$ and $k$ we fix the parameters
\begin{equation}\label{eq:blockparams}
 b=1+\frac1{6P},\quad Q=2^{64}k,\quad d=6P\log Q,\quad
 \eps_j=\tfrac1{16}e^{-db^j},\quad \gamma_j=\tfrac12-\eps_j\quad(j\ge1),
\end{equation}
and the \emph{head phases} $\beta_j=(2j-1)/(4k)$, $1\le j\le k$. Note that
$b/(b-1)=6P+1$, that $\eps_1=\frac1{16}Q^{-(6P+1)}\le2^{-452}$, that
$\eps_{j+1}/\eps_j=Q^{-b^j}\le2^{-64}$, and that $\eps_1<\frac1{4k}$, so that all
the $\beta_j$ and $\gamma_j$ are distinct. Let
$\gamma'=(\beta_1,\dots,\beta_k,\gamma_1,\gamma_2,\dots)$; thus in $L_{\gamma'}$ the
phase $\beta_j$ occurs in the cells $|n|\ge j$ and the phase $\gamma_j$ in the
cells $|n|\ge k+j$.

The analytic input of the construction is the following theorem, which is
proved in Appendix~\ref{app:rapid}.

\begin{theorem}\label{thm:rapid}
Let $P,k\ge1$ and let $\gamma,\gamma'$ be as above.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $\M_{L_\gamma}$ is one-dimensional; it is spanned by an odd measure $\mu$
      with $\F\mu=-\ii\mu$.
\item $\M_{L_\gamma}\cap H_{-p}=\{0\}$ for $1\le p\le P$.
\item $\M_{L_\gamma}\subset H_{-(18P+2)}$.
\item \textup{(Head removal.)} There is a linear bijection
      $\mathcal R:\M_{L_{\gamma'}}\to\M_{L_\gamma}$ such that $\mathcal R$ maps
      $V_p(L_{\gamma'})$ into $V_p(L_\gamma)$ and $\mathcal R^{-1}$ maps
      $V_p(L_\gamma)$ into $V_{p+2}(L_{\gamma'})$, for every integer $p\ge0$.
\item \textup{(Finite heads carry nothing.)} If $w$ belongs to the space $W_k$
      of \eqref{eq:Wk} below and neither $w$ nor $\F w$ has mass at the points
      $n\pm\beta_j$ with $j\le k$ and $|n|<j$, then $w=0$.
\end{enumerate}
\end{theorem}

Here $\mathcal R$ is built from finitely many integer translations and
multiplications by a trigonometric polynomial (Section~\ref{subsec:rapidproof}), so
it preserves every $H_{-p}$ and preserves strong temperedness on both sides; the
content of~(d) is that its inverse loses at most two Hermite orders.

\subsection{Outline of the proof of Theorem~\ref{thm:rapid}}\label{subsec:rapidproof}
The proof is given in Appendix~\ref{app:rapid}; we describe its structure here.
Part~(e) concerns finitely many lattice combs and is proved by a triangularity
argument: in suitable coordinates the conditions on $w$ and on $\F w$ are
complementary triangular systems, and a determinant computation shows that they
admit only $w=0$ (Lemma~\ref{lem:finitecap}). Parts~(a)--(c) are proved in Zak
coordinates. After a shift by $\frac12$, the Zak transform of $T\in\M_{L_\gamma}$ is a
compactly supported distribution $V$ on a small square, and the atoms of $T$ and
of $\F T$ become point masses on the grid $\{0,\pm\eps_j\}^2$. We test $V$ against
products of the Newton-type functions $\cos(\pi t)D_i(t)$ and $\sin(\pi t)D_i(t)$,
where $D_i=\prod_{l\le i}(1-\cos2\pi t-a_l)$ and $a_l=1-\cos2\pi\eps_l$ vanishes at
the first $i$ pairs of points. In the resulting moment arrays the support condition
on $T$ says that the array $\mathbf A$ of $V$ is upper triangular, the support
condition on $\F T$ says that the array $\mathbf B$ of $e^{-2\pi\ii xy}V$ is lower
triangular, and $\mathbf B=E\mathbf A$ for an explicit bounded operator $E$ (the
\emph{gauge}). The proof has four stages.
\begin{enumerate}[label=\textup{\arabic*.},leftmargin=2em]
\item \emph{Uniqueness} (Appendix~\ref{subsec:A-stage1}). With geometric weights
the two triangularity conditions leave a single free parameter: after
elimination they reduce to $\mathcal Ku=0$ for an operator
$\mathcal K=c_0S+\mathcal E$ on $\ell^2(\NN_0)$, where $S$ is the backward shift and
$\norm{\mathcal E}<|c_0|/10$. Hence $\dim\M_{L_\gamma}\le1$.
\item \emph{Exclusion at low order} (Appendix~\ref{subsec:A-stage2}). If
$T\in H_{-p}$, then $V$ has distribution order at most $2p$, and when
$6p<b/(b-1)=6P+1$ this allows moments with weights $\prod_{l\le i}\eps_l$, which
shrink as fast as the clusters. In these weights the relevant part of $E-I$
carries a factor $\eps_{i+1}^2$, which is small compared with the coupling
$\tan(\pi\eps_{i+1})$ created by the support conditions, and a contraction argument
gives $T=0$. This is~(b).
\item \emph{Construction} (Appendix~\ref{subsec:A-stage3}). Keeping only the first
$N$ phases and adding the half-integers outside $[-N+\frac12,N-\frac12]$ gives a
finite system, whose space of crystalline measures is a line spanned by a
combination $T_N$ of lattice combs. Its moments decay like $\prod_{l\le m}\eps_l$
uniformly in $N$, and an interpolation argument converts this into
$\sup_N\norm{T_N}_{-(2L+2)}<\infty$ whenever $b^L>4$. A weak limit is a nonzero
element of $\M_{L_\gamma}$; with $L=9P$ this gives~(a) and~(c).
\item \emph{Head transfer} (Appendix~\ref{subsec:A-stage4}). The map $\mathcal R$ in~(d)
is the composition of an operator $\sum_{|l|\le k}c_l\tau_l$, whose symbol has simple
zeros exactly at $e^{\pm2\pi\ii\beta_j}$, $j\le k$, with multiplication by the
trigonometric polynomial with the same coefficients. Each factor has a right
inverse built from bounded Green sequences that loses one Hermite order, and a
finite correction in $W_k$, controlled by~(e), makes the inverse exact.
\end{enumerate}
The parameters \eqref{eq:blockparams} enter as follows: in~(b) through the
condition $6p<b/(b-1)$, and in~(c) through $b^L>4$ with $L=9P$, since
$b^{9P}=(1+\frac1{6P})^{9P}\ge(\frac76)^9=40353607/10077696>4$ by the monotonicity
of $(1+1/x)^x$. The arguments are also carried out in the formal development
\cite{MS26} (Appendix~\ref{app:lean}). For odd $k\le11$ we have, in addition,
checked~(e) by computing the rank of the corresponding $4k^2\times4k^2$ matrix
exactly, modulo a prime $p\equiv1\pmod{16k^2}$; this is not needed for the proof.

\subsection{Exchanging finitely many holes}
Let $k\ge1$. The points $n\pm\beta_j$, $n\in\Z$, $j\le k$, form the lattice coset
$G_k=\{x_m=(m+\frac12)/(2k):m\in\Z\}$. Let
\begin{equation}\label{eq:Wk}
 W_k=\Span\Bigl\{\sum_{n\in\Z}e^{2\pi\ii nb}\delta_{n+a}:\ a,b\in\{\pm\beta_j:j\le k\}\Bigr\}.
\end{equation}
These modulated translated combs are crystalline with support and spectrum in
$G_k$, since the Fourier transform of $\sum_ne^{2\pi\ii nb}\delta_{n+a}$ has mass
$e^{-2\pi\ii a(\ell+b)}$ at $\ell+b$. One checks that $\dim W_k=4k^2$, that
$W_k\subset H_{-1}$, and that $W_k$ consists of the measures
$\sum_mc_m\delta_{x_m}$ with $c_{m+4k^2}=-c_m$.

\begin{lemma}\label{lem:holes}
Let $Y\subset\R\setminus G_k$ be locally finite, let $H^0,H^1\subset G_k$ be
finite sets with $\#H^0=\#H^1=2k^2$, and put $\Lambda^j=(G_k\setminus H^j)\cup Y$.
Assume that, for $j=0,1$, the map $\mathcal L_j:W_k\to\C^{H^j}\times\C^{H^j}$
recording the masses of $w$ and of $\F w$ at the points of $H^j$ is injective.
Then there is a linear bijection $\M_{\Lambda^0}\to\M_{\Lambda^1}$ which maps
$V_p(\Lambda^0)$ onto $V_p(\Lambda^1)$ for every $p\ge1$. In particular
$r_p(\Lambda^0)=r_p(\Lambda^1)$ for all $p\ge1$.
\end{lemma}

\begin{proof}
Since $\dim W_k=4k^2$, the maps $\mathcal L_j$ are bijective. For
$T\in\M_{\Lambda^0}$ let $\mathcal E_1T\in\C^{H^1}\times\C^{H^1}$ record the masses
of $T$ and $\F T$ at $H^1$, and put $S_1T=T-\mathcal L_1^{-1}\mathcal E_1T$. Then
$S_1T$ and $\F S_1T$ have no mass at $H^1$, and they are supported in
$G_k\cup Y$; hence $S_1T\in\M_{\Lambda^1}$. Define $S_0:\M_{\Lambda^1}\to\M_{\Lambda^0}$
symmetrically. The difference $S_0S_1T-T$ lies in $W_k$ and has no mass, on
either side, at $H^0$, so it vanishes by the injectivity of $\mathcal L_0$;
similarly $S_1S_0=I$. Finally, each mass is a continuous functional on $H_{-p}$
(pair with a fixed bump function), and $W_k\subset H_{-1}$, so $S_0$ and $S_1$
preserve $V_p$ for $p\ge1$. (The norms of these maps may grow rapidly with $k$;
only the qualitative preservation of each $V_p$ is used.)
\end{proof}

We apply the lemma with the triangular holes
$H^0=\{n\pm\beta_j:j\le k,\ |n|<j\}$ and the central holes
$H^1=\{n\pm\beta_j:j\le k,\ |n|\le R-1\}$, where $k=2R-1$. Injectivity of
$\mathcal L_0$ is Theorem~\ref{thm:rapid}(e). For $\mathcal L_1$, note that
$H^1=\{x_m:m\in I_0\}$ with $I_0=\{-k^2,\dots,k^2-1\}$, and that the elements of
$W_k$ are determined by $(c_m)_{-k^2\le m<3k^2}$. If $w=\sum c_m\delta_{x_m}\in W_k$
has no mass on $H^1$, then $c_m=0$ for $m\in I_0$, and by Poisson summation the
mass of $\F w$ at $x_s$ is
$\frac1{2k}\sum_{m\in I_1}e^{-2\pi\ii(m+1/2)(s+1/2)/(4k^2)}c_m$ with
$I_1=\{k^2,\dots,3k^2-1\}$. Requiring these to vanish for $s\in I_0$ gives a
square system which, after removing nonzero row and column factors, has
Vandermonde matrix $(z_m^{\,s+k^2})_{s\in I_0,m\in I_1}$ with distinct nodes
$z_m=e^{-2\pi\ii m/(4k^2)}$, $m\in I_1$. Hence $c=0$ and $\mathcal L_1$ is
injective. (This matrix is extremely ill-conditioned, as its nodes fill only
half of the unit circle; its invertibility is exact, not numerical.)

\subsection{The blocks}
For positive integers $P,R$ put $k=2R-1$, take the parameters
\eqref{eq:blockparams}, and let
\begin{equation}\label{eq:block}
 C(P,R)=\{n\pm\beta_j:\ 1\le j\le k,\ |n|\ge R\}\cup
 \{n\pm\gamma_j:\ j\ge1,\ |n|\ge k+j\}.
\end{equation}

\begin{proposition}\label{prop:block}
For all positive integers $P,R$ the following hold.
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\M_{C(P,R)}=\C\mu_{P,R}$ for a nonzero $\mu_{P,R}$ with
      $\F\mu_{P,R}=\zeta\mu_{P,R}$, $\zeta^4=1$.
\item $C(P,R)\cap[-R+\frac12,R-\frac12]=\varnothing$.
\item $\M_{C(P,R)}\cap H_{-p}=\{0\}$ for $1\le p\le P$, and
      $\mu_{P,R}\in H_{-(18P+4)}$.
\item $\#\bigl(C(P,R)\cap[-r,r]\bigr)\le32(1+r)^2$ for $r\ge0$.
\item The set of phases $\Theta\subset(-\frac12,\frac12)$ of $C(P,R)$ has
      closure $K=\Theta\cup\{\frac12\}$ in $\R/\Z$; $K$ is countable, its only
      accumulation point is $\frac12$, and $K\cap(-\frac1{4k},\frac1{4k})=\varnothing$.
\end{enumerate}
\end{proposition}

\begin{proof}
By Theorem~\ref{thm:rapid}(a)--(c), $\M_{L_\gamma}$ is a line contained in
$H_{-(18P+2)}$ which meets $H_{-p}$ only in $0$ for $p\le P$. By
Theorem~\ref{thm:rapid}(d), $\M_{L_{\gamma'}}$ is a line contained in
$H_{-(18P+4)}$, again meeting $H_{-p}$ only in $0$ for $p\le P$, since
$\mathcal R$ is injective and preserves Hermite orders. Now
$L_{\gamma'}=(G_k\setminus H^0)\cup Y$ and $C(P,R)=(G_k\setminus H^1)\cup Y$ with
$Y=\{n\pm\gamma_j:|n|\ge k+j\}$ and $H^0,H^1$ as above, and
$\#H^0=\#H^1=2k^2$. Lemma~\ref{lem:holes} transfers (i) and (iii) to $C(P,R)$;
the line is invariant under $\F$ because $C(P,R)$ is symmetric
(Proposition~\ref{prop:covariance}(b)), and $\F^4=I$. Points of the first set in
\eqref{eq:block} satisfy $|x|\ge R-\beta_k>R-\frac12$, and points of the second
satisfy $|x|>k+\frac12\ge R$, which gives~(ii). For~(iv), points of the first
set in $[-r,r]$ exist only if $k<2r$, and there are at most $2k(2r+2)$ of them;
points of the second set satisfy $j\le r+1$, and there are at most $4(r+1)^2$
of them. Finally (v) holds because $\gamma_j\to\frac12$, $\pm\frac12$ coincide
on $\R/\Z$, and $\beta_1=\frac1{4k}$ is the smallest phase.
\end{proof}